# Reto B. Keller

# Design for Electromagnetic Compatibility - In a Nutshell

# Theory and Practice

Design for Electromagnetic Compatibility–In a Nutshell

Reto B. Keller

# Design for Electromagnetic Compatibility–In a Nutshell

Theory and Practice

Reto B. Keller Trachslau, Switzerland

ISBN 978-3-031-14185-0 ISBN 978-3-031-14186-7 (eBook) https://doi.org/10.1007/978-3-031-14186-7

This work was supported by Reto Keller

© The Editor(s) (if applicable) and The Author(s) 2023. This book is an open access publication. **Open Access** This book is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this book are included in the book's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the book's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

*To my parents Martha and Hans Keller-Camenzind, my wife Nicole Keller-Holdener, and our daughter Valeria Svenja Keller. Thank you for your love and support.*

# **Preface**

Electromagnetic compatibility is a topic that is becoming more and more important in our modern and high-tech society. EMC no longer just helps to ensure that the television screen does not flicker when the vacuum cleaner is running. EMC also helps critical applications—such as space missions or medical devices—function reliably. This means that EMC-compliant design receives increased attention, both from engineers and from management.

*Design for Electromagnetic Compatibility*— *In a Nutshell* is a book dedicated to development engineers, who design EMC-compliant electrical equipment and electronic devices. This book is to be regarded both as an introduction to EMC for beginners and as a quick reference guide for experts.

The book's content is compressed and focuses on the basics of EMC. Where appropriate, reference is made to further and more advanced literature in the respective chapters. This book contains theoretical and practical knowledge:

**Theory**. Introduction to EMC, the essential know-how, and the basic concepts of EMC-compliant design

**Practice**. Discussion of the biggest hurdles to get a product through the EMC test right away and proposal of some universal EMC design guidelines

For years, I have been passionate about EMC. Yes, it sounds odd. I mean: EMC is not a fancy topic, and you cannot talk with your friends about it (at least not with non-engineers), as they get bored quickly. Nevertheless, what is it that makes EMC a fascinating topic for many engineers? For me, EMC means sophisticated theoretical knowledge and practical experience brought together and a responsibility for us—the product development engineers—towards the users and society to release products to the market which are safe and reliable. Enjoy the read!

Einsiedeln, SZ, Switzerland Reto B. Keller April 2022

# **Acknowledgments**

*Relationships are based on four principles: respect, understanding, acceptance and appreciation.*

*—*Mahatma Gandhi

I firmly believe that education is the foundation of prosperity and a free society. Therefore, freely available education is vital in our world. This is why I decided to publish this book open-access. The publication of this book would not have been possible without help. I would like to take this opportunity to express my sincere thanks to all of those who have supported me:


**Mr. Ignacio de Mendizábal** for his feedback about the book's content.

**Mr. Christian Grätzer** for supporting this book.

**Dr. Heik Heinrich Hellmich** for supporting this book: "I am very excited about the activities of the *Academy of EMC* and I am convinced that this book will be another major step to close the gap of lacking EMC awareness which is required for the upcoming energy and digital transformation in the near future."

**Prof. Dr. Hans-Dieter Lang** for supporting this book.

**Mr. Claudio Pfister** for supporting this book.

**Mr. Simon Durrer** for supporting this book.

**Mr. Simon Saado** for supporting this book.

**Ms. Veronika Schärer** for supporting this book.

**Mr. Benyamin Farzaneh Aghajarie** for supporting this book.


Einsiedeln, SZ, Switzerland Reto B. Keller April 2022













# **Acronyms**

*The key test for an acronym is to ask whether it helps or hurts communication.*

*—*Elon Musk

List of the most common *abbreviations* and *acronyms* in the field of electromagnetic compatibility used throughout this book:




# **Symbols and Units**

*In science there is only physics; all the rest is stamp collecting. —*Lord Kelvin


**Table 1** Symbols, dimensions, and units. Part 1/2




# **Chapter 1 Introduction**

*I have not failed. I've just found 10'000 ways that won't work. —*Thomas A. Edison

# **1.1 What Is EMC?**

*Electromagnetic compatibility* (*EMC*) is an established discipline within the field of electrical and electronics engineering. EMC is the ability of equipment or a system to function satisfactorily in its electromagnetic environment without introducing intolerable electromagnetic disturbances to anything in that environment [1]. In other words, each device must have a certain immunity against electromagnetic disturbances, and on the other hand, each device must keep its own electromagnetic emissions low enough to not disturb other devices in its environment (Fig. 1.1).

# **1.2 EMC vs. EMI**

*EMI* stands for *electromagnetic interference* and is often mixed up with EMC. EMI means that one electronic device A is causing disturbance to another electronic device B, which is in the surrounding of device A.

What is the difference between EMC and EMI? Now, an EMC-compliant product has to be tested on EMI during its development. For an EMC-compliant product, EMI should not happen anymore. This is because EMC-compliant products proved their electromagnetic immunity to be high enough and their electromagnetic emission to be low enough to work seamlessly in its predefined environment.

**Fig. 1.1** How an EMC design engineer sees a product

#### **1.3 Why Is EMC Important?**

EMC-compliant products reduce the risk of undesired interference and disturbances. However, why is that so important? EMC is important because interference and disturbances of electrical equipment can seriously harm people, infrastructure, and the environment. For example, in 1992, a woman died because the heart machine of the ambulance shut down every time the technicians turned on their radio transmitter to ask for advice [12]. Another example is the explosion of the Texaco refinery in Milford Haven UK, on the 24th of July 1994, which was caused by an electrical storm giving rise to power surges which tripped out a number of pump motors while leaving others running. The explosion led to 26 people being sustainably injured and damage of £48 million [12]. These are just two of numerous examples and show us that taking care of EMC-compliant design is not just a necessity for selling products; it means a safer world with reliable products and satisfied customers.

#### **1.4 EMC Terms and Definitions**

The discipline of EMC can primarily be divided into the subjects of *emission* and *immunity* (or *susceptibility*). Figure 1.2 shows the most important terms and definitions in EMC. Other EMC abbreviations and acronyms can be found in chapter "Acronyms" in the front matter of this book on page xxiii.

**Fig. 1.2** EMC terms and definitions

#### *1.4.1 Emission*

It is distinguished between *conducted emissions* (CE) and *radiated emissions* (RE).

	- **RF conducted emissions.** Goal: prevent connected cables from radiating and avoid the interference of connected equipment. The frequency range of the RF emission measurements for commercial and industrial products goes from 150 kHz to 30 MHz (CISPR 32 [2], FCC 47 CFR Part 15 [3]). However, the frequency range depends on the industry and may be as wide as from 30 Hz to 40 GHz for defense and military products (MIL-STD-461G [13]).
	- **Harmonics.** Goal: limitation of harmonic currents injected into the public mains supply system. The mains supply input current of many devices and machines is often not a harmonic sine wave.
	- **Flicker.** Goal: limitation of voltage changes, voltage fluctuations, and flicker in public low-voltage supply systems. *Flicker* is the impression of unsteadiness of visual sensation induced by a light stimulus whose luminance or spectral distribution fluctuates with time [1].

# *1.4.2 Immunity*

It is distinguished between conducted immunity/susceptibility (CI, CS), radiated immunity/susceptibility (RI, RS), and immunity against phenomena like electrostatic discharge (ESD) or electromagnetic pulses (EMP).

	- **RF conducted immunity.** Goal: functional immunity to conducted disturbances induced by RF fields. The frequency range of the conducted RF immunity tests for commercial and industrial products goes from 150 kHz to 80 MHz (IEC 61000-4-6 [9]).
	- **EFT.** Goal: functional immunity to repetitive electrical fast transients (EFT, bursts), such as those originating from switching transients (interruption of inductive loads, relay contact bounce, etc.). Test voltages vary from 0*.*5 kV to 4 kV (IEC 61000-4-4 [8]).
	- **Surge.** Goal: functional immunity to unidirectional surges caused by overvoltages from switching and lightning transients. Test voltages vary from 0*.*5 kV to 4 kV (IEC 61000-4-5 [11]).
	- **Dips.** Goal: functional immunity to voltage dips, short interruptions, and voltage variations at the power supply ports.

### **1.5 Design for EMC**

Design for EMC means considering EMC early during product development, not just at the end of the project. If you test your product for EMC just shortly before the product is to be launched, project delays and budget overruns result. Figure 1.3 shows that the later a defect in the product life cycle is fixed, the more expensive it is. Here are some essential points regarding EMC and product development:

	- **Grounding.** Define a grounding concept for the product (system grounding), the subsystems of the product (intrasystem grounding), and PCBAs (boardlevel grounding).
	- **Shielding.** Define if and how to shield sensitive circuits and cables (intrasystem and external cables). Define the bonding of the shields.
	- **Filtering.** Define if and how cables and wires (intrasystem and external cables) have to be filtered. Especially consider ESD, EFT, and surge for cables that leave your product. RF-filtering should be considered for every cable.

**Fig. 1.3** Life cycle costs [4] and the costs of fixing defects at different life cycle stages

### **1.6 Summary**


### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 2 Regulations and Standards**

*If you make 10'000 regulations you destroy all respect for the law.*

*—* Winston Churchill

# **2.1 Big Picture**

Products which are sold on the market should have the following properties:


In order to have only products on the market which fulfill the bullet points above, governments and their legislative bodies issue laws and directives, and products must be regulatory compliant according to this legislation.

A company that is the legal manufacturer of a product must prove compliance with the regulations. This is done with *conformity assessments*. In case a product contains electronics, an EMC conformity assessment is necessary. Such an EMC conformity assessment usually comprises the proof of:


In nearly every country, electronic devices or machines on the market must be EMC compliant, meaning they must fulfill the EMC regulations and standards for the intended use of the products. What EMC regulations and standards are applicable for which product is defined by the country where the product is sold to the end customer.

#### **2.2 EMC Compliance**

*EMC compliance* means that an electronic or electromechanical product is compliant with the laws, directives, and regulations of the country where it is sold to the end customer.

#### **2.3 EMC Regulations**

Every government defines its own *EMC regulations* (laws, directives) for its country. However, these national regulations often adopt multinational regulations (e.g., countries in the European Union refer to the EMC directive 2014/30/EU).

The government usually sets up or appoints an organization, commission, or committee responsible for defining the applicable EMC standards (see Fig. 2.1). Such organizations or committees define the applicable EMC standards so that products, which pass the tests defined in the applicable EMC standards, are then compliant with the EMC regulations (laws, directives).

Some authorities (administrations, organizations, commissions, committees) develop the applicable EMC standards for their countries or customs unions themselves. However, in most cases, the authorities adopt the content from the respective international EMC standards into their EMC standards or refer to the respective international EMC standards. Further information about specific countries and their responsible authorities for EMC regulations and standards can be found in Sect. 2.5.

### **2.4 EMC Standards**

The following chapters focus primarily on international *EMC standards* (IEC, CISPR) and not on local or country-specific standards.

#### *2.4.1 What Are EMC Standards?*

EMC standards define terms, rules, and test methods for EMC. Furthermore, they specify limits and minimum test levels for electric and electromagnetic emissions and immunity. Figure 2.2 shows a product and some commonly applied EMC standards for consumer electronic products.

#### *2.4.2 Why Do We Need EMC Standards?*

EMC standards help make measurements comparable and repeatable by defining the test methods, test equipment, and test environment. Most importantly, EMC standards aim to bring harmonization to EMC testing, in the best case: global harmonization. Global harmonization of standards reduces trade barriers, and as a most important consequence for society, harmonized EMC standards help increase global prosperity and wealth.

#### *2.4.3 Who Writes EMC Standards?*

Standards in EMC are either developed by international, national, or regional organizations and committees on behalf of administrative bodies, or the administrative and/or regulatory bodies word the EMC standards and regulations themselves (e.g., FCC MP-5). Usually, the appointed administrative bodies (e.g., CENELEC for the EU, BSI for the UK, or ACMA for Australia) adopt the international EMC standards written by IEC/CISPR or ISO, word for word. The *International Electrotechnical Commission* (*IEC*) is the most important organization when it comes to EMC standards development. Within the IEC, the following committees have the lead:

	- Immunity and related items, over the whole frequency range: Basic EMC Publications and Generic EMC Standards.
	- Emission in the low-frequency range (*f <*= 9 kHz, e.g., harmonics and voltage fluctuations): Basic, Generic, and Product (Family) EMC Standards.


Here is a (not exhaustive) list of international and national organizations and committees which develop and/or define the applicable EMC standards:

	- IEC Committees for Basic and Generic EMC Standards:
		- . Technical Committee 77, Electromagnetic Compatibility (TC 77)
		- . International Special Committee on Radio Interference (CISPR)
	- IEC Committees for Product and Product Family EMC Standards.
	- IEC liaison partners:
		- . International Organization for Standardization (ISO)
		- . ITU Telecommunication Standardization Sector (ITU-T)
		- . International Council on Large Electric Systems (CIGRE)
		- . The Union of the Electricity Industry (Eurelectric)
		- . International Organization of Legal Metrology (OIML)
	- Comité Européen de Normalisation Electrotechniques (CENELEC)
	- European Telecommunications Standards Institute (ETSI)
	- International Council on Large Electric Systems (CIGRE)
	- European Committee for Standardization (CEN)
	- Japanese Industrial Standards Committee (JISC)
	- Japanese Standards Association (JAS)
	- Association for the Advancement of Medical Instrumentation (AAMI)
	- American National Standards Institute (ANSI)
	- Department of Defense (DoD)
	- Federal Communications Commission (FCC)
	- Radio Technical Commission for Aeronautics (RTCA)
	- Society of Automotive Engineers (SAE)

# *2.4.4 EMC Emission Standards*

*EMC emission standards* define test methods and emission limits for conducted and radiated electromagnetic emissions. Examples of EMC emission standards for residential, commercial, and industrial products are:


# *2.4.5 EMC Immunity Standards*

*EMC immunity standards* define test methods and immunity test levels for conducted and radiated electromagnetic disturbances. Examples of EMC immunity standards for residential, commercial, and industrial products are:


*Performance criteria* are used to evaluate the immunity characteristics of the equipment under test (EUT) in the course of an EMC immunity test. In other words, performance criteria describe the loss of function or degradation of performance of the EUT. The pass/fail criteria of an immunity test case are specified in the EMC test plan before testing the EUT. These criteria are typically set to A, B, C, or D. The criteria A, B, C, and D are described in detail in the respective EMC immunity standard [3]:


# *2.4.6 Types of EMC Standards*

The following are classes or types of EMC standards (see Fig. 2.3):


**Fig. 2.3** Types of EMC standards and their dependencies

#### **2.4.6.1 Basic EMC Publications**

*Basic EMC Publications* specify terms and conditions for EMC testing, define the rules necessary for achieving electromagnetic compatibility, specify test methods (testing techniques, test setup, test equipment, and environment), and so on. Basic EMC Publications are the EMC standards to which other EMC standards (EMC Product Standards, Generic EMC Standards, etc.) refer.

The IEC Basic EMC Publications are structured into the following categories [1]:


• **Installation/Mitigation.** Installation and mitigation guidelines regarding earthing and cabling, mitigation of external electromagnetic influences, HEMP protection concepts, and so on. Examples are IEC TR 61000-5-1, IEC TR 61000- 5-2, IEC 61000-5-xx, etc.

#### **2.4.6.2 EMC Product Standards**

*EMC Product Standards* apply to particular products. EMC Product Standards refer to the relevant Basic EMC Publications (for that particular product) and specify the limits of emission and immunity (the minimum test levels).

Examples of EMC Product Standards are IEC 62104 (DAB receivers), IEC 61851-21 (electric road vehicles charging system), IEC 62599-2 (alarm end electronic security systems), etc.

#### **2.4.6.3 EMC Product Family Standards**

*EMC Product Family Standards* apply to a group of products with common general characteristics and may operate in the same environment and have neighboring fields of application. EMC Product Family Standards refer to the relevant Basic EMC Publications (for that particular product) and specify the limits of emission and immunity (the minimum test levels).

Examples of EMC Product Family Standards are IEC 61967 (integrated circuits), IEC 61326 (electrical equipment for measurement, control, and laboratory use), and IEC 60947 (switchgear and controlgear).

#### **2.4.6.4 Generic EMC Standards**

The *Generic EMC Standards* are for products operating in a particular EMC environment (residential/industrial), where a specific EMC Product (Family) Standard does not exist. They are general and somewhat simplified EMC Product Standards while referring to Basic EMC Publications for detailed measurement and test methods. Generic EMC Standards specify a limited number of emission and immunity tests and minimum test levels.

Examples of Generic EMC Standards are IEC 61000-6-1 (immunity standard for residential, commercial, and light-industrial environments), IEC 61000-6-2 (immunity standard for industrial environments), IEC 61000-6-3 (emission standard for equipment in residential environments), IEC 61000-6-4 (emission standard for industrial environments), and IEC 61000-6-8 (emission standard for professional equipment in commercial and light-industrial locations).

# *2.4.7 EMC Standards in Different Industries*

Some industries require their own industry-specific EMC standards. Sometimes they are also called EMC Product (Family) Standards. Table 2.1 lists some popular international and industry-specific EMC standards, e.g., automotive, aviation, lightning, or medical.



### *2.4.8 Which EMC Standards to Apply?*

As mentioned above in Sect. 2.3, the applicable EMC standards are defined by the responsible governmental administrations, organizations, commissions, or committees. Therefore, the process of finding the applicable standards often differs from country to country. As an example, Fig. 2.4 shows a flow chart based on the EMCD Guide published by the European Commission [4].

Other useful tips:


### **2.5 Compliance Marks**

Which compliance marks to go for? To decide if and which *compliance marks* you need for your product depends on which countries or customs unions (e.g., EU) you want to sell your product to.

Let's take a step back to the big picture of conformity marks: mandatory marks vs. nonmandatory marks.

	- **Australia and New Zealand.** RCM mark (Regulatory Compliance Mark).
	- **Brazil.** ANATEL label (Agência Nacional de Telecomunicações).
	- **Canada.** ISED label (Innovation, Science, and Economic Development).
	- **China.** CCC mark (China Compulsory Certificate).
	- **EEU (Russia).** EAC mark (Eurasian Conformity).
	- **EU.** CE mark (Conformité Européenne).
	- **Japan.** PSE mark (Product Safety Electrical Appliance and Material).
	- **Republic of Korea (South Korea).** KC mark (Korean Certification).
	- **Singapore.** IMDA label (Infocomm Media Development Authority).
	- **Switzerland.** CH mark (Swiss Conformity Mark).
	- **Taiwan.** BSMI mark (Bureau of Standards, Metrology, and Inspection).
	- **UK.** UKCA (United Kingdom Conformity Assessed).
	- **USA.** FCC mark (Federal Communications Commission).
	- **Canada.** CSA (Canadian Standards Association).

**Fig. 2.4** Flow chart based on the EMCD Guide published by the European Commission [4]


More country-specific EMC regulations and requirements can be found in the Appendix in Chap.J.

#### **2.6 Global Market Access**

Section 2.5 and Appendix J show that the situation regarding EMC regulations and standards can be confusing. However, there is a great way to get access to the global market with a single EMC test report: test your product according to the International Electrotechnical Commission for Electrical Equipment (IECEE) Certified Body (CB) Scheme. The IECEE CB Scheme is a system for mutual recognition of safety and EMC test certificates of conformity for over 50 countries. It is also a tool for accessing global markets directly when national authorities and regulators, retailers, buyers, and vendors accept CB test certificates and the associate test reports. The same concept of the IECEE CB Scheme does also apply for safety regulations and standards. Figure 2.5 shows the organizational hierarchy of the Member Body (there is one MB per country), the National Certification Body (NCB issues the CB test certificate), and the Certified Body Test Laboratories (CBTLs do perform the actual EMC testing).

This is how the EMC testing for global market access works:

1. **EMC testing.** Find a Certified Body EMC Test Lab (CBTL) and test your product according to the CB Scheme at this CBTL and obtain a CB EMC Test Report for your product. CBTLs are listed on the IECEE website [2].


#### **2.7 Summary**


#### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 3 Decibel**

*If you cannot measure it, you cannot improve it.*

*—*Lord Kelvin

#### **3.1 Gain and Loss [dB]**

Let's have a look at the amplifier or attenuation network in Fig. 3.1. The power, voltage, and current gain of this network can be expressed in [dB] as [2]:

$$\text{Power gain in [dB]} = 10 \cdot \log\_{10} \left( \frac{P\_{out}}{P\_{in}} \right) = 10 \cdot \log\_{10} \left[ \left( \frac{V\_{out}}{V\_{in}} \right)^2 \cdot \left( \frac{R\_{in}}{R\_{load}} \right) \right] \tag{3.1}$$

$$\text{Voltage gain in [dB]} = 20 \cdot \log\_{10} \left( \frac{V\_{out}}{V\_{in}} \right) + 10 \cdot \log\_{10} \left( \frac{R\_{in}}{R\_{load}} \right) \tag{3.2}$$

$$\text{Current gain in [dB]} = 20 \cdot \log\_{10} \left( \frac{I\_{out}}{I\_{in}} \right) + 10 \cdot \log\_{10} \left( \frac{R\_{load}}{R\_{in}} \right) \tag{3.3}$$

In case *Rin* and *Rload* are equal (typically 50 ), then the following term is equal to zero:

$$10 \cdot \log\_{10} \left( \frac{R\_{in}}{R\_{load}} \right) = 10 \cdot \log\_{10} \left( \frac{R\_{load}}{R\_{in}} \right) = 10 \cdot \log\_{10}(1) = 0$$

Now we can write the following for power/voltage/current gain:

$$\text{Power gain in [dB]} = 10 \cdot \log\_{10} \left( \frac{P\_{out}}{P\_{in}} \right) \tag{3.4}$$

23

**Fig. 3.1** An arbitrary amplifier or attenuation network

$$\text{Voltage gain in [dB]} = 20 \cdot \log\_{10} \left( \frac{V\_{out}}{V\_{in}} \right) \tag{3.5}$$

$$\text{Current gain in [dB]} = 20 \cdot \log\_{10} \left( \frac{I\_{out}}{I\_{in}} \right) \tag{3.6}$$

Points to remember when it comes to gain and loss calculations in *decibel*:


$$\text{Power gain in [dB] at } f\_c = 10 \cdot \log\_{10} \left(\frac{1}{2}\right) = -\text{3dB}$$

$$\text{Voltage gain in [dB] at } f\_c = 20 \cdot \log\_{10} \left( \frac{1}{\sqrt{2}} \right) = -3 \text{dB}$$

$$\text{Current gain in [dB] at } f\_c = 20 \cdot \log\_{10} \left( \frac{1}{\sqrt{2}} \right) = -3 \text{dB}$$

• **Ratio to [dB].** Table 3.2 presents some common ratio to [dB] value conversions. For example, if power increases by factor 2, the power/voltage/current gain increases by 3 dB.


**Fig. 3.2** Commonly used absolute decibel levels in EMC

#### **3.2 Absolute Levels [dBm, dBµV, dBµA]**

The most common absolute power, voltage, and current levels in EMC are [dBm], [dBµV], and [dBµA] (Fig. 3.2). They are calculated like this [2]:

$$\text{Absolute power level in [dBm]} = 10 \cdot \log\_{10} \left( \frac{P}{1 \text{ mW}} \right) \tag{3.7}$$

$$\text{Absolute voltage level in } [\mathbf{dB} \mu \mathbf{V}] = 20 \cdot \log\_{10} \left( \frac{V}{1 \,\mu \text{V}} \right) \tag{3.8}$$

$$\text{Absolute current level in [dB}\mu\text{A]} = 20 \cdot \log\_{10} \left(\frac{I}{1\,\mu\text{A}}\right) \tag{3.9}$$

More information about physical quantities and their units (also in decibel) can be found in the Appendix D. The same concept of absolute levels can also be applied to electrical fields *E* [V/m], magnetic fields *H* [A/m], or radiated power density *S* [W/m2]:

$$\text{Electric field strength in } [\text{dB} \mu \text{V/m}] = 20 \cdot \log\_{10} \left( \frac{E}{1 \,\mu \text{V/m}} \right) \tag{3.10}$$

$$\text{Magnetic field strength in } \text{[dB\mu A/m]} = 20 \cdot \log\_{10} \left( \frac{H}{1 \,\mu\text{A/m}} \right) \tag{3.11}$$

$$\text{radiated power density in } [\text{dBmW/m}^2] = 10 \cdot \log\_{10} \left( \frac{S}{1 \,\text{mW/m}^2} \right) \qquad (3.12)$$

Points to remember when it comes to calculations with absolute power levels in decibel:

	- 0 dBm def = 1 mW
	- 0 dBµV def = 1 µV
	- 0 dBµA def = 1 µA
	- 0 dBµV/m def = 1 µV/m
	- 0 dBµA/m def = 1 µA/m

$$P\_{out} = P\_{ln} \cdot G$$

In decibel, the output power *Pout,dB* is the sum of the input power *Pin,dB* and the gain *GdB*:

$$P\_{out,dB} = 10 \cdot \log\_{10}(P\_{out}) = 10 \cdot \log\_{10}(P\_{ln}) + 10 \cdot \log\_{10}(G) = P\_{ln,dB} + G\_{dB}$$

Let's assume a signal with *Pin,dBm* = 0 dBm at the input of an amplifier with gain *GdB* = 20 dB. The output power is:

$$P\_{out,dBm} = P\_{in,dBm} + G\_{dB} = 0 \,\text{dBm} + 20 \,\text{dB} = 20 \,\text{dBm}$$

• **Never sum up absolute levels.** Do never sum up absolute decibel levels, because adding decibels means multiplying the linear values and therefore:

$$10\,\text{dBm} + 0\,\text{dBm} \stackrel{\text{def}}{=} 1\,\text{mW} \cdot 1\,\text{mW} = 1\,\text{mW}^2 = ?$$

What does power squared mean? Thus, never add up absolute decibel levels.

• **Sum of absolute levels and decibel.** It is allowed and useful to sum up [dBm]-, [dBµV]-, or [dBµA]-values with gain *G* values in [dB]:

$$P\_{out,lin} = P\_{in,lin} \cdot G\_{lin} \stackrel{\text{def}}{=} P\_{out,d\,Bm} = P\_{in,d\,Bm} + G\_{dB} = -3\,\text{dBm} + 20\,\text{dB} = 17\,\text{dBm}$$

$$V\_{out,lin} = V\_{in,lin} \cdot V\_{lin} \stackrel{\text{def}}{=} V\_{out,d\,BV} = V\_{in,d\,BV} + G\_{dB} = -3\,\text{dBV} + 20\,\text{dB} = 17\,\text{dBV}$$

• **Subtraction of two absolute levels.** Subtracting two absolute [dBm]-, [dBµV]-, or [dBµA]-values is equivalent to computing the ratio of their linear values:

$$\frac{P\_{out,lin}}{P\_{in,lin}} = \frac{100\,\text{mW}}{1\,\text{mW}} = 100 \stackrel{\text{def}}{=} P\_{out,dBm} - P\_{in,dBm} = 20\,\text{dBm} - 0\,\text{dBm} = 20\,\text{dB}$$

$$\frac{V\_{out,lin}}{V\_{in,lin}} = \frac{10\,\text{V}}{1\,\text{V}} = 10 \stackrel{\text{def}}{=} V\_{out,dBV} - V\_{in,dBV} = 20\,\text{dBV} - 0\,\text{dBV} = 20\,\text{dB}$$

#### **3.3 Summary**

The Tables 3.1, 3.2, 3.3 and 3.4 present conversions between different absolute decibel levels, linear rations to relative decibel values and vice versa.

**Table 3.1** Conversion between [dBµV], [dBµA], and [dBm] for systems with system impedance *Z*<sup>0</sup> = 50 [1]



**Table 3.2** Conversion from (linear) ratios to [dB] values and vice versa


**Table 3.3** Conversion between voltages in [V] and [dBµV], between currents in [A] and [dBµA], and between power in [dBm] and [mW] for different system impedances *Z*<sup>0</sup> [1]


**Table 3.4** Conversion between field strengths [V/m], [A/m], [dBµV/m], [dBµA/m], [dBpT], and [µT] for free-space (far-field) where *Z*<sup>0</sup> = 377 [1]

#### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 4 Frequency and Wavelength**

*If you want to find the secrets of the universe, think in terms of energy, frequency and vibration.*

*—*Nikola Tesla

# **4.1 EMC and Frequencies**

In general, *conducted EMC emission and immunity tests* take place at lower frequencies than *radiated EMC emission and immunity tests*.



**Table 4.1** ITU frequency bands and their corresponding wavelengths in free-space [11]

• **Common-Mode Low-Frequency Disturbance**. In some areas, there are also conducted EMC immunity tests specified from *f* = 0 Hz to *f* = 150 kHz [2]. This is intended to demonstrate the immunity of electrical and electronic equipment when subjected to conducted common-mode disturbances such as those originating from power line currents, frequency converters, and return leakage currents in the earthing/grounding system.

The radio spectrum managed by the *International Telecommunication Union (ITU)* goes up to 3000 GHz<sup>1</sup> and is divided into 12 *ITU frequency bands*. Table 4.1) shows the ITU frequency bands with their corresponding wavelength and potential applications.

#### **4.2 Wavelength vs. Frequency**

#### *4.2.1 Wavelength in Any Media*

The *frequency f* [Hz] of a sinusoidal electromagnetic wave (Fig. 4.1) and its *wavelength λ* [m] have the following relationship [6]:

$$\lambda = \frac{v}{f} = \frac{2\pi}{\beta} = \frac{1}{f \sqrt{\frac{\left(\epsilon' \mu' - \epsilon'' \mu''\right)}{2} \cdot \left(\sqrt{1 + \left(\frac{\epsilon' \mu'' + \epsilon'' \mu'}{\epsilon' \mu' - \epsilon'' \mu''}\right)^2} + 1\right)}}\tag{4.1}$$

<sup>1</sup> Radio waves are electromagnetic waves of frequencies arbitrarily lower than 3000 GHz (3 THz), propagated in space without artificial guide. 3 THz is already in the infrared frequency band (300 GHz–430 THz); visible light starts at 430 THz.

**Fig. 4.1** Wavelength *λ* of a sine wave signal

where:

*v* = propagation velocity of the signal in [m/sec]


#### *4.2.2 Wavelength in Insulating Media*

In case of an insulator (*μ*- *<sup>r</sup>* = 1) and negligible dielectric and magnetic losses (-- = 0, *μ*-- = 0), the wavelength of a sinusoidal electromagnetic wave with frequency *f* [Hz] can be written as:

$$
\lambda = \frac{v}{f} = \frac{1}{f\sqrt{\mu\_0 \epsilon\_0 \epsilon\_r'}} = \frac{c}{f\sqrt{\epsilon\_r'}} \tag{4.2}
$$

where:

*v* = propagation velocity of the signal in [m/sec] *f* = frequency of the sinusoidal signal in [Hz] *<sup>c</sup>* <sup>=</sup> <sup>1</sup>*/(*√*μ*00*)* <sup>=</sup> <sup>2</sup>*.*<sup>998</sup> · 108 m/sec <sup>=</sup> speed of light *<sup>μ</sup>*<sup>0</sup> <sup>=</sup> 12.57 ·10−<sup>7</sup> H/m <sup>=</sup> permeability of vacuum, absolute permeability <sup>0</sup> <sup>=</sup> <sup>8</sup>*.*<sup>854</sup> · <sup>10</sup>−<sup>12</sup> F/m <sup>=</sup> permittivity of vacuum, absolute permittivity - *<sup>r</sup>* = relative permittivity, dielectric constant of the insulator

#### *4.2.3 Wavelength in Vacuum*

For vacuum (and approximately air), the calculation of the wavelength of a sinusoidal electromagnetic wave reduces to [10]:

$$
\lambda = \frac{c}{f} = \frac{1}{f\sqrt{\mu\_0 \epsilon\_0}}\tag{4.3}
$$

where:

*<sup>c</sup>* <sup>=</sup> <sup>1</sup>*/(*√*μ*00*)* <sup>=</sup> <sup>2</sup>*.*<sup>998</sup> · 108 m/sec <sup>=</sup> speed of light *f* = frequency of the sinusoidal signal in [Hz]

#### *4.2.4 Wavelength in Good Conducting Media*

In case the electromagnetic sinusoidal wave travels through a good conductor (through and not along(!), e.g., through a shield) with negligible magnetic losses (*μ*--= 0), the wavelength can be calculated as [10]:

$$
\lambda = \sqrt{\frac{4\pi}{f\mu'\sigma}}\tag{4.4}
$$

where:

*f* = frequency of the sinusoidal signal in [Hz] *μ*- = *μ*- *rμ*<sup>0</sup> = real part of the complex permeability (*μ* = *μ*- − *jμ*--) in [H/m] *σ* = specific conductance of the medium where the wave is propagating through in [S/m]

# **4.3 Wavelength of Signals Along Wires, Cables, and PCB Traces**

It is important to understand that the signal propagation velocity *v* [m/sec] depends on the transport medium through which the electromagnetic field is traveling. Therefore, the same signal with the same frequency *f* [Hz] has a different wavelength *λ* [m] in a blank wire (surrounded by air) than in a cable or PCB trace (surrounded by one or multiple insulation materials). The wavelength *λ* of signals traveling along wires, cables, and PCB traces—where the dielectric and magnetic losses can be neglected (-- = 0, *μ*-- = 0) and the materials around the conductors are assumed to be nonmagnetic (*μ*- *<sup>r</sup>* = 1)—is given as [9]:

#### 4.3 Wavelength of Signals Along Wires, Cables, and PCB Traces 37

$$
\lambda = \frac{v}{f} = \frac{c}{f\sqrt{\epsilon\_{ref}}} = \frac{c}{f} \cdot \text{VF} \tag{4.5}
$$

where:

*v* = propagation velocity of the signal in [m/sec] *f* = frequency of the sinusoidal signal in [Hz] *<sup>c</sup>* <sup>=</sup> <sup>2</sup>*.*<sup>998</sup> · 108 m/sec <sup>=</sup> speed of light

*reff* = effective relative permittivity (dielectric constant) of the material(s) through which the electromagnetic field is propagating

VF = velocity factor

### *4.3.1 Wavelength of Signals Along Blank Wires*

The *wavelength λ* [m] of a signal with frequency *f* [Hz] which travels along a blank wire (or antenna surrounded by air) depends only on the speed of light *c* [m/sec] and the signal frequency *f* [Hz] (*v* = *c*, because - *<sup>r</sup>* = 1 and *μ*- *<sup>r</sup>* = 1 and therefore VF = 1) [9]:

$$
\lambda\_{blankwire} = \frac{c}{f} \tag{4.6}
$$

where:

*<sup>c</sup>* <sup>=</sup> <sup>1</sup>*/(*√*μ*00*)* <sup>=</sup> <sup>2</sup>*.*<sup>998</sup> · 108 m/sec <sup>=</sup> speed of light *f* = frequency of the sinusoidal signal in [Hz]

#### *4.3.2 Wavelength of Signals Along Cables and PCB Traces*

The wavelength *λ* of a signal with frequency *f* which travels along a wire, *cable*, or a *printed circuit board* (PCB) trace is [9]:

$$
\lambda\_{cable/PCBtrace} = \frac{c}{f \cdot \sqrt{\epsilon\_{ref}}} \tag{4.7}
$$

where:

*<sup>c</sup>* <sup>=</sup> <sup>1</sup>*/(*√*μ*00*)* <sup>=</sup> <sup>2</sup>*.*<sup>998</sup> · 108 m/sec <sup>=</sup> speed of light *f* = frequency of the sinusoidal signal in [Hz] *reff* = the effective dielectric constant (relative permittivity) through which the electromagnetic wave is propagating

**Fig. 4.2** Transmission line examples. (**a**) PCB trace: microstrip line. (**b**) Twisted pair cable. (**c**) PCB trace: stripline. (**d**) PCB trace: coplanar waveguide with reference plane

The effective dielectric constant *reff* is defined as the uniform equivalent dielectric constant for a transmission line, even in the presence of different dielectrics (e.g., FR-4 and air for a microstrip line; see Fig. 4.2a). The relative permeability *μ*- *<sup>r</sup>* is assumed to be equal 1.0 for cables and PCBs, because the insulation materials are nonmagnetic. Thus, the *velocity factor* (VF) depends primarily on the effective relative permittivity *reff* through which the electromagnetic wave is propagating.

The calculation of the effective dielectric constant *reff* depends on the insulation material and the geometry of the transmission line (e.g., ribbon cable, microstrip, coplanar waveguide, etc.), because the amount of electric field lines in the different media depends on the geometry of the transmission line. Figure 4.2 shows some common transmission lines and Table 4.2 the corresponding *reff*

The velocity factor (VF) of a transmission medium is the ratio of the velocity *v* [m/sec] at which a wavefront of an electromagnetic signal passes through the medium, compared to the speed of light in vacuum *<sup>c</sup>* <sup>=</sup>2.998 ·108 m/sec:

$$\text{VF} = \frac{v}{c} \tag{4.8}$$

Thus, the smaller the velocity factor (VF), the smaller the wavelength *λ* [m].

During EMC emission measurement and troubleshooting, it is often necessary to determine the wavelength *λ* [m] of a certain unintended disturbance with frequency *f* [Hz] because once you know the wavelength of the disturbance, you can look for potential antennas of the disturbance (e.g., looking for cables with length *l* = *λ/*4 or *l* = *λ/*2 of the disturbance). Table 4.3 presents the velocity factors for different *reff* and the resulting wavelength *λ* [m] for a given frequency *f* [Hz].

**Table 4.2** Approximate velocity factor (VF) for different transmission lines and insulation materials. Calculation of *reff* according to: [7, 8, 12]


**Table 4.3** Wavelength *λ* [m] for given frequencies *f* [Hz] and dielectric constants - *r*


#### *4.3.3 Summary*


$$
\lambda = v/f = c/(f\sqrt{\epsilon\_{ref}})\tag{4.9}
$$

where:

*v* = propagation velocity of the signal in [m/sec]

*f* = frequency of the sinusoidal signal in [Hz]

*reff* = the effective dielectric constant (relative permittivity) through which the electromagnetic wave is propagating

#### • **Wavelength of electromagnetic waves in free-space.**

$$
\lambda = c/f \tag{4.10}
$$

where:

*<sup>c</sup>* <sup>=</sup> <sup>1</sup>*/(*√*μ*00*)* <sup>=</sup> <sup>2</sup>*.*<sup>998</sup> · 108 m/sec <sup>=</sup> speed of light *f* = frequency of the sinusoidal electromagnetic wave in [Hz]

#### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 5 Time-Domain and Frequency-Domain**

*Fourier's theorem has all the simplicity and yet more power than other familiar explanations in science. Stated simply, any complex pattern, whether in time or space, can be described as a series of overlapping sine waves of multiple frequencies and various amplitudes.*

*—*Bruce Hood

#### **5.1 Fourier Analysis**

Figure 5.1 presents measurement instruments for measuring electrical signals in the time-domain (oscilloscope) and the frequency-domain (spectrum analyzer). *Fourier analysis* refers to the mathematical principle that every signal can be represented by the sum of simple trigonometric functions (sine, cosine, etc.). The Fourier analysis enables a transformation of a signal in the time-domain *x(t)* to a signal in the frequency-domain *X(ω)*, where *ω* = 2*πf* . In other words, a Fourier analysis is a mathematical operation for calculating the frequency-domain representation (frequency spectrum) of a signal in the time-domain. Two common Fourier analysis notations are:

$$\times (\iota) \circ \xrightarrow{\smile \circ} X(a) \circ$$

$$\mathcal{F}\{\mathbf{x}(t)\} = X(a)\mathbf{y}$$

where:

*x(t)* = a signal in the time-domain *X(ω)* = the Fourier transform of *x(t)* (frequency-domain)

Figure 5.2 shows a representation of a square wave signal (1 V amplitude) with the sum of only four harmonic sine waves (first, third, fifth, and seventh) and a direct current (DC) component of 0*.*5 V. For representing an ideal square wave, an

**Fig. 5.1** Time-domain and frequency-domain measurement equipment. However, some oscilloscopes do also have built-in Fourier analysis functionality (FFT). (**a**)Time domain: high-end oscilloscope R&S®RTO64 by Rohde & Schwarz. (**b**) Frequency domain: signal and spectrum analyzer R&S®FSVA3030 by Rohde & Schwarz

**Fig. 5.2** Fourier analysis of a square wave; up to the seventh harmonic

indefinite number of sine waves would be necessary (because the rise- and fall-time of an ideal square wave are 0 s).

There are different types of integral transforms. These transforms have in common that they define the necessary math for converting a signal from timeto the frequency-domain and vice versa. Every integral transform has its field of application:


**Fig. 5.3** Time-domain to frequency-domain transforms


Figure 5.3 shows some common integral transforms used in the field of EMC. Further details about the different integral transforms can be found in the Appendix F.

#### **5.2 Frequency Spectra of Digital Signals**

Digital signals like clocks and data interfaces are the primary cause for EMC emissions of electronic circuits and electrical equipment and systems. Digital signals in the time-domain can be represented by trapezoid-shaped pulses with a period time *T* [sec], a pulse width *tpw* [sec], a rise-time *tr* [sec], and a fall-time *tf* [sec]. Figure 5.4 shows an extract of a digital waveform and the corresponding frequency spectrum (Fourier analysis). It can be seen that the pulse width *tpw* [sec] defines the frequency at which the amplitude spectrum starts to drop with −20 dB/decade, whereas the minimum fall- and/or rise-time defines the frequency at which the spectrum starts to drop with −40 dB/decade. Figure 5.5a compares the frequency spectrum of clock signals with different duty-cycles [%] (pulse

**Fig. 5.4** Digital signal in the time- and frequency-domain. (**a**) Digital waveform with period time *T* [sec], pulse width *tpw* [sec], rise-time *tr* [sec] and fall-time *tf* [sec]. (**b**) Amplitude frequency spectrum of a digital signal and its envelope curve [3]

widths). It is remarkable that a clock signal with a 90 % duty-cycle has a lower amplitude of the first harmonic than a clock signal with a 50 % duty-cycle (of the same frequency [Hz] and with the same rise-/fall-time [sec]). However, as a 90 % clock signal has more power, this power adds to the DC component (0 Hz). For EMC design engineers, Fig. 5.5b is even more important than Fig. 5.5a because it shows how high-frequency harmonics of a clock signal can be reduced effectively. In this example, an increase of the rise- and fall-time [sec] by factor 10 reduces the amplitude of the high-frequency harmonics (*f >* 32 MHz) also by factor 10 (20 dB).

**Fig. 5.5** Clock signal parameter analysis in the frequency-domain [1]. (**a**) Frequency spectrum of 100 kHz clock signals with rise-/fall-times of 10 nsec and duty-cycles of 50% vs. 90%. (**b**) Frequency spectrum of 100 kHz clock signals with 50% duty-cycle and rise-/fall-times of 10 nsec vs. 100 nsec

#### **5.3 Bandwidth of Digital Signals**

*Bandwidth* [Hz] of a digital signal means what is the highest significant sine wave frequency component in the digital signal? Significant in this case means that the power in [W] in the frequency component is bigger than 50 % of the power in an ideal square wave's signal with the same amplitude *A* in [V] and duty-cycle *D* = *(tpw/T )* in [%]. A drop in 50 % of the power [W] is the same as a drop of 70 % in amplitude [V] or a drop of 3 dB.

The rule of thumb for calculating the bandwidth [Hz]—or the highest significant sine wave frequency—of a trapezoid digital signal is [2]:

$$B = \frac{0.35}{t\_{10-90\%}}\tag{5.1}$$

**Fig. 5.6** The 3dB-bandwidth of a clock signal with a pulse width of *tpw* [sec], rise- and fall-times of *tr* = *tf* [sec]

where:


Figure 5.6 shows the frequency spectrum envelop curves of an ideal square wave *tr* = *tf* = 0 sec and a trapezoid with *tr >* 0 sec and *tf >* 0 sec. The 3 dB-bandwidth of the trapezoid waveform can be found at *f*3*dB* = 0*.*35*/t*10−90%.

Note: it is assumed that there is no ringing in the waveform and *tr* = *tf* . In case of ringing, the frequency spectrum envelope for *f >* 1*/(πtr)* would not drop off with −40 dB/decade. Instead, an increase in the frequency spectrum is assumed at the ringing frequency (see Fig. 5.7).

#### **5.4 Ringing and Frequency Spectrum**

Impedance mismatch along a transmission line could cause *ringing*. One could argue that ringing has nothing to do with EMC and that ringing is a signal integrity topic. However, this is not entirely correct. Due to ringing, higher frequency components (harmonics) are introduced, and it may be possible that the conductor where the signal is propagating along—acts as an efficient antenna at these higher frequencies.

**Fig. 5.7** Simulation data of a signal with ringing (blue) vs. without ringing (gray). Both signals have an amplitude of 3*.*3 V, a fundamental frequency of 100 kHz, and a rise-/fall-time of 10 nsec [1]. (**a**) Time-domain. (**b**) Frequency-domain

In the example of Fig. 5.7, the harmonics around 50 MHz show as much as 100 times higher voltage amplitudes (40 dB) in the signal with ringing than compared to the signal without ringing. This example illustrates that impedance matching does also matter for EMC and that EMC design engineers must know which signal connections must be considered as transmission lines and which not.

#### **5.5 Summary**


$$B \approx 0.35/t\_{10-90\%} \tag{5.2}$$

where:

*B* = highest significant sine wave frequency (harmonic) in a digital signal [Hz] *t*10−90% = rise- and/or fall-time (whichever is smaller) from 10 % to 90 % of the digital signal in [sec]

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 6 RF Parameters**

*The term impedance matching can be a very useful metaphor for connoting important aspects of social interactions. For example, the smooth and efficient functioning of social networks, whether in a society, a company, a group activity, and especially in relationships such as marriages and friendships, requires good communication in which information is faithfully transmitted between groups and individuals. When information is dissipated or "reflected," such as when one side is not listening, it cannot be faithfully or efficiently processed, inevitably leading to misinterpretation, a process analogous.*

*—*Geoffrey West

#### **6.1 Reflection Coefficient** *Γ*

We speak of matched impedances in case the load impedance *Zload* is the complex conjugate of the source impedance *Zsource* = *Z*<sup>∗</sup> *load* (Fig. 6.1). In radiated emission and immunity EMC testing, it is important to understand the term matching and how to quantify it. All EMC RF measurement test setup receiver and transmitter antennas must be matched to their receiver and transmitter equipment impedance (typically *Z*<sup>0</sup> = 50 ).

The *reflection coefficient Γ* is defined as [6]:

$$
\underline{\varGamma} = \frac{\underline{V}\_{reflected}}{\underline{V}\_{forward}} \tag{6.1}
$$

$$\underline{\mathbf{L}} = \frac{\underline{\mathbf{Z}\_{load}} - \underline{\mathbf{Z}\_{source}}}{\underline{\mathbf{Z}\_{load}} + \underline{\mathbf{Z}\_{source}}} \tag{6.2}$$

where:

*V f orward* = complex forward voltage wave to the load in [V] *V ref lected* = complex reflected voltage wave by the load in [V]

49

*Zsource* = complex source impedance in [] *Zload* = complex load impedance in []

The reflection coefficient *Γ* is often given in [dB]:

$$
\Gamma\text{ [dB]} = 10 \cdot \log\_{10} \left(\frac{P\_{reflected}}{P\_{forward}}\right) \tag{6.3}
$$

where:

*Pref lect ion* = reflected power by the load in [W] *Pf orward* = power sent to the load (at the load terminals) in [W]

#### **6.2 Voltage Standing Wave Ratio (VSWR)**

VSWR is dimensionless and means voltage standing wave ratio. The VSWR expresses the ratio of maximum and minimum voltage of a standing wave pattern along a transmission line. It is also a parameter for measuring the degree of impedance matching. Standing waves occur in case of impedance mismatch. A VSWR value of 1 means perfectly matched. A VSWR value of ∞ means complete mismatch (100% of the forward wave is reflected). Figure 6.2 shows an example of a standing wave pattern.

The VSWR can be calculated by using the reflection coefficient from above [6]:

$$\text{VSWR} = \frac{|V|\_{\text{max}}}{|V|\_{\text{min}}} = \frac{1 + |\underline{\varGamma}|}{1 - |\underline{\varGamma}|} \tag{6.4}$$

where:


#### **6.3 Return Loss (RL)**

The *return loss* RL [dB] is a measure for the transferred power to a load by a source (Fig. 6.3). A low RL value indicates that not much power is transferred to the load and is reflected instead. Return loss [dB] is always a positive value and it is equal the reflection coefficient *Γ* [dB] multiplied by −1 [5, 9].

$$\text{RL [dB]} = 10 \cdot \log\_{10} \left( \frac{P\_{forward}}{P\_{reflected}} \right) = -20 \cdot \log\_{10} \left( |\underline{\Gamma}| \right) \tag{6.5}$$

$$\text{RL [dB]} = -\Gamma \text{[dB]} \tag{6.6}$$

where:

*Pf orward* = power sent to the load (at the load terminals) in [W] *Pref lect ion* = reflected power by the load in [W] |*Γ* | = is the magnitude of the reflection coefficient

#### **6.4 Insertion Loss (IL)**

The term *insertion loss* IL [dB] is generally used for describing the amount of power loss due to the insertion of one or several of the following components (passive twoport networks):


The insertion loss is the ratio in [dB] of the power *P*<sup>1</sup> and *P*<sup>2</sup> in Fig. 6.4. *P*<sup>1</sup> represents the power, which would be transferred to the load in case the source is directly connected to the load. The power *P*<sup>2</sup> represents the power, which is transferred to the load in case the passive two-port network is inserted between the source and the load [2, 6].

$$\text{IL [dB]} = 10 \cdot \log\_{10} \left( \frac{P\_1}{P\_2} \right) = -10 \cdot \log\_{10} \left( \frac{P\_2}{P\_1} \right) \tag{6.7}$$

where:

*P*<sup>1</sup> = power, which would be transferred to the load in case the source is directly connected to the load (without passive two-port network) in [W]

*P*<sup>2</sup> = power, which is transferred to the load in case the passive two-port network is inserted between the source and the load in [W]

**Fig. 6.4** Insertion loss

#### **6.5 Scattering Parameters**

*Scattering parameters* —also called S-parameters—are commonly used in highfrequency or microwave engineering to characterize a *two-port circuit* (see Figs. 6.5 and 6.6). The scattering parameters describe the relation of the power wave parts *a*1, *b*1, *a*2, *b*<sup>2</sup> that are transferred and reflected from a two-port input and output. The physical dimension for the incident *a* and reflected *b* power waves is not Watt, but it is <sup>√</sup>Watt. The generic definitions for the forward power wave *ai* and the reflected power wave *bi* of an arbitrary port *i* are given as [1]:

$$\underline{a}\_{l} = \frac{\underline{V}\_{l} + \underline{Z}\_{l}\underline{I}\_{l}}{2\sqrt{|\mathrm{Re}(\underline{Z}\_{l})|}}\tag{6.8}$$

$$\underline{b}\_{l} = \frac{\underline{V}\_{l} - \underline{Z}\_{l}^{\*}\underline{I}\_{l}}{2\sqrt{|\operatorname{Re}(\underline{Z}\_{l})|}}\tag{6.9}$$

where:

*V <sup>i</sup>* = complex voltage at the input of the *i*th port of a junction in [V] *I <sup>i</sup>* = current flowing into the *i*th port of a junction in [A] *Zi* = complex impedance of the *i*th port in [] *Z*∗ *<sup>i</sup>* = complex conjugate of *Zi* The positive real value is chosen for the square root in the denominators

The forward traveling power toward port 1 *P*1*f wd* and the reflected power of port 1 *P*1*ref* can be written as:

$$P\_{1fwd} = \frac{1}{2} \cdot |\underline{a}\_1|^2 = \frac{|\hat{\mathcal{V}}\_{1fwd}|^2}{2\operatorname{Re}(\underline{Z}\_1)}\tag{6.10}$$

$$P\_{lref} = \frac{1}{2} \cdot |\underline{b}\_{\rm l}|^2 = \frac{|\hat{V}\_{lref}|^2}{2\operatorname{Re}(\underline{Z}\_{\rm l})} \tag{6.11}$$

**Fig. 6.6** S-parameters are measured with *vector network analyzers*. The picture shows the R&S®ZNL6 by Rohde & Schwarz

where:

*V*ˆ <sup>1</sup>*f wd* = peak value of the forward voltage to port 1 in [V] *V*ˆ <sup>1</sup>*ref* = peak value of the reflected voltage from port 1 in [V] Re*(Z*1*)* = real part of the complex impedance *Z*<sup>1</sup> of port 1 in []

Generally speaking, the S-parameter *sij* is determined by driving port *j* with an incident wave of voltage *V* + *<sup>j</sup>* and measuring the outgoing voltage wave *V* <sup>−</sup> *<sup>i</sup>* at port *i*. Considering Fig. 6.5, the four scattering parameters can be computed as follows:

$$\underline{a}\_{11} = \frac{\underline{b}\_1}{\underline{a}\_1}\bigg|\_{\underline{a}\_2 = 0} = \text{input reflection coefficient} \tag{6.12}$$

$$\underline{s}\_{21} = \frac{\underline{b}\_2}{\underline{a}\_1}\bigg|\_{\underline{a}\_2 = 0} = \text{forward gain, forward transmission}\tag{6.13}$$

$$\underline{a}\_{12} = \frac{\underline{b}\_1}{\underline{a}\_2}\bigg|\_{\underline{a}\_1=0} = \text{backward gain, reverse transmission}\tag{6.14}$$

$$
\underline{a}\_{22} = \frac{\underline{b}\_2}{\underline{a}\_2}\bigg|\_{\underline{a}\_1=0} = \text{output reflection coefficient}\tag{6.15}
$$

where:

*<sup>a</sup>*<sup>1</sup> <sup>=</sup> incoming power wave at the input port of a two-port in [√*W*] *<sup>b</sup>*<sup>1</sup> <sup>=</sup> outgoing power wave at the input port of a two-port in [√*W*] *<sup>a</sup>*<sup>2</sup> <sup>=</sup> incoming power wave at the output port of a two-port in [√*W*] *<sup>b</sup>*<sup>2</sup> <sup>=</sup> outgoing power wave at the output port of a two-port in [√*W*]

The formulas 6.12, 6.14, 6.13, 6.15 showed that the S-parameters can be determined by loading the ports with a reference impedance, e.g., *Z*<sup>0</sup> = 50 . This means that *s*<sup>11</sup> = *Γ* when the output port 2 is matched with its load. For a one-port *s*<sup>11</sup> is always equal the reflection coefficient *Γ* . The input reflection coefficient *Γ* <sup>1</sup> of a two-port with a certain load (given by the reflection factor *Γ <sup>L</sup>*) is given by:

$$
\underline{\Gamma}\_{1} = \underline{\underline{s}}\_{11} + \frac{\underline{s}\_{12}\underline{s}\_{21}\underline{\Gamma}\_{L}}{1 - \underline{s}\_{22}\underline{\Gamma}\_{L}}\tag{6.16}
$$

#### **6.6 Signal-to-Noise Ratio**

The *signal-to-noise-ratio* SNR [dB] is a key figure in analog or digital signal processing applications. The bigger the SNR value, the better. The SNR is defined as the ratio of the desired signal power to the power of the undesired signal (noise, interference) [3]:

$$\text{SNR} = \frac{P\_{signal}}{P\_{noise}}\tag{6.17}$$

$$\text{SNR [dB]} = 10 \log\_{10} \left( \frac{P\_{signal}}{P\_{noise}} \right) \tag{6.18}$$

where:

*Psignal* = power of the desired signal in [W] *Pnoise* = power of the undesired signal (noise, interference) in [W]

#### **6.7 Noise Factor and Noise Figure**

*Noise factor F* and *noise figure* NF [dB] are measures of degradation of the signalto-noise ratio (SNR), caused by components in a signal chain. The noise factor of a device (e.g., amplifier) is defined as the ratio of the SNR at the input of the device to the SNR at the output of the device [3]:

$$F = \frac{\text{SNR}\_{\ell}}{\text{SNR}\_{o}} \tag{6.19}$$

where:

SNR*<sup>i</sup>* = SNR of the signal at the input of the device (linear, not in [dB]) SNR*<sup>o</sup>* = SNR of the signal at the output of the device (linear, not in [dB])

The noise figure NF [dB] is defined as the noise factor *F* in [dB]:

$$\text{NF} = 10 \log\_{10}(F) = 10 \log\_{10} \left( \frac{\text{SNR}\_l}{\text{SNR}\_o} \right) \tag{6.20}$$

where:

*F* = noise factor of the device (linear, not in [dB]) SNR*<sup>i</sup>* = SNR of the signal at the input of the device (linear, not in [dB]) SNR*<sup>o</sup>* = SNR of the signal at the output of the device (linear, not in [dB])

**Fig. 6.7** Cascade of devices with gain *Gn* and noise factor *Fn*

If several devices are cascaded (see Fig. 6.7), the total noise factor *F* can be calculated with Friis formula [3]:

$$F = F\_1 + \frac{F\_2 - 1}{G\_1} + \frac{F\_3 - 1}{G\_1 G\_2} + \frac{F\_4 - 1}{G\_1 G\_2 G\_3} + \dots + \frac{F\_n - 1}{G\_1 G\_2 G\_3 \dotsm G\_{n-1}}\tag{6.21}$$

where:

*Fn* = noise figure of the *n*-th device (linear, not in [dB]) *Gn* = gain of the *n*-th device (linear, not in [dB])

The first amplifier in a chain usually has the most significant effect on the total noise figure NF. Equation 6.21 shows that the first device should have the lowest noise factor *F*<sup>1</sup> and a high gain *G*1, since the noise figure of any subsequent stage will be divided by the gains of the preceding stages. As a consequence, the noise figure requirements of subsequent stages are more relaxed.

The noise figure NF [dB] of a passive component (cable, attenuator, filter) is always its loss *L*, and one can set *F* = *L* and *G* = 1*/L* (linear, not in [dB]) for a passive component [3], where *G* is the gain.

#### **6.8 1 dB Compression Point**

*The 1 dB compression point* P1dB [dB] is a key specification for amplifiers. Besides many factors, the gain *G* [dB] of an amplifier is a function of the amplifier's input power *Pin* [dBm]. P1dB is defined as the output power *Pout* [dBm] (or sometimes as the input power *Pin*) of an amplifier at which the output is 1 dB lower than it is supposed to be, if it were ideal. In other words, P1dB is the output power *Pout* [dBm] when the amplifier is at the 1 dB compression point (Fig. 6.8).

Once an amplifier reaches its P1dB, it goes into compression and becomes a nonlinear device, producing distortion, harmonics, and intermodulation products. Therefore, amplifiers should always be operated below the P1dB.

#### **6.9 Spectrum Analyzer Terms**

*Spectrum analyzers* measure the power of a signal's frequency components. Today, spectrum analyzers apply these measurement principles:


Basically, there are three types of spectrum analyzers:


**Fig. 6.9** Block diagram of spectrum analyzer principles . (**a**) Superheterodyne principle. (**b**) FFT principle

• **Hybrid superheterodyne-FFT analyzers** combine the principle of superheterodyne measurement and FFT calculation. They have shorter scan times than purely swept-tuned analyzers and a wider bandwidth then FFT analyzers.

Sections 6.9.1–6.9.5 explain the basic terms of spectrum analyzers. Note: *EMI receivers* are spectrum analyzers that fulfill the requirements according to CISPR 16-1-1 [8] (dynamic range, peak-detection, etc.).

#### *6.9.1 Frequency Range*

A spectrum analyzer measures the input signal from the *start frequency* to the *stop frequency*. The frequency between the start and the stop frequency is called *center frequency*, and the difference is the frequency *span*.

$$\text{center frequency} = \text{(start frequency} + \text{stop frequency)} / 2 \tag{6.22}$$

$$\text{span} = \text{stop frequency} - \text{start frequency} \tag{6.23}$$

#### *6.9.2 Resolution Bandwidth*

For spectrum analyzers that employ the superheterodyne principle, the *resolution bandwidth* (RBW) is the bandwidth of the band-pass IF filter in Fig. 6.9a. The RBW determines how close two signals can be resolved into two separate peaks. The smaller the RBW, the better the resolution in the frequency range and the lower the noise floor (see Fig. 6.10).

#### *6.9.3 Video Bandwidth*

For spectrum analyzers that employ the superheterodyne principle, the *video bandwidth* (VBW) is the bandwidth of the low-pass filter after the envelope detector in Fig. 6.9a. The VBW is used to discriminate between a low-power harmonic signal and the noise floor. The smaller the VBW, the smoother the envelope signal and the lower the noise floor (see Fig. 6.11).

#### *6.9.4 Sweep Time*

For spectrum analyzers that employ the superheterodyne principle, the *sweep time tsweep* [sec] is the time required to record the entire frequency range (span) [4]:

**Fig. 6.10** Measurement of the same signal with different resolution bandwidths (RBWs). Measurements performed with R&S®FSV3013 by Rohde & Schwarz. (**a**) RBW <sup>=</sup> 120 kHz. (**b**) RBW = 9 kHz

**Fig. 6.11** Measurement of the same signal with different video bandwidths (VBWs). Measurements performed with R&S®FSV3013 by Rohde & Schwarz. (**a**) VBW <sup>=</sup> 200 kHz. (**b**) VBW <sup>=</sup> 1 Hz

where:

*k* = Factor, depending on the type of filter and on the required level of accuracy span = frequency range to be displayed in [Hz] RBW = resolution bandwidth in [Hz] VBW = video bandwidth in [Hz]

# *6.9.5 Detectors*

Different detector types are used in spectrum analyzers (Fig. 6.12):


#### **6.10 Summary**

Table 6.1 summarize the impedance matching parameters and Table 6.2 compares the impedance matching parameters for matched vs. unmatched.

**Table 6.1** Impedance matching parameters. *Z*<sup>0</sup> [] denotes the system impedance. Typically, the system impedance is *Z*<sup>0</sup> = 50 (RF), *Z*<sup>0</sup> = 60 (CAN-bus), *Z*<sup>0</sup> = 75 (television) or *Z*<sup>0</sup> = 100 (Ethernet)


**Table 6.2** Qualitative analysis of impedance matching


#### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

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# **Chapter 7 Transmission Lines**

*In the mid-1980s or so, connecting conductors were of no consequence; i.e., the voltage and current at the input to the line are almost identical to the voltage and current at the output of the line. Today this is no longer true. As clock and data speeds continue to increase, seemingly without bound, these interconnect conductors will have a significant effect on the signal transmission and cannot be ignored.*

*—* Dr. Clayton R. Paul

### **7.1 What Is a Transmission Line**

A *transmission line* is a series of conductors, often but not necessarily two, used to guide electromagnetic energy from one place to the other [3]. It's that simple. The more complicated part is the math behind it (Maxwell's Equations; see Appendix E) because we no longer consider a transmission line a lumped-element *RLC* network (see Fig. 7.9). Rather more, we consider the signal conductor as a transmission line through which an electromagnetic field is moved from one point to another. Transmission lines are characterized or described by their characteristic impedance *Z*<sup>0</sup> [] and distributed parameter model (see Fig. 7.2).

Figure 7.1 shows some common transmission line geometries:


**Fig. 7.1** Transmission line examples


# **7.2 When to Consider Transmission Lines**

Every signal interconnection is a transmission line. However, it is not necessary to treat every signal path as a transmission line. Two different rules of thumb—if a conductor should be treated as transmission line or not—are explained in Sect. 7.2.1 (frequency-domain) and Sect. 7.2.2 (time-domain).

# *7.2.1 Rule of Thumb for* **lcritical** *in the Frequency Domain*

A common rule of thumb when working in the frequency domain is [3]:


When determining the shortest wavelength *λmin* [m] in a digital signal (e.g., clock), it is necessary to know the maximum frequency *fmax* [Hz] of the signal by considering the rising and falling time (rather than the fundamental frequency). The rule of thumb for calculating the bandwidth—or the highest significant sine wave frequency *fmax* [Hz]—of a rectangular digital signal and the corresponding minimum wavelength *λmin* [m] [1]:

$$f\_{max} = \text{bandwidth} = \frac{0.35}{t\_{10-90\%}}\tag{7.1}$$

$$
\lambda\_{\min} = \frac{v}{f\_{\max}}\tag{7.2}
$$

where,

*fmax* = highest significant sine wave frequency harmonic in a digital signal (what significant means is described in Sect. 5.3) in [Hz]

*t*10−90% = rise and/or fall time (whichever is smaller) from 10 % to 90 % of the slope of a digital signal in [sec]

*λmin* = wavelength of the highest significant harmonic frequency in [m] *v* = propagation velocity of the signal along the transmission line in [m/sec]

The frequency domain approximation for the critical length *l*critical*,*fd [m] can be calculated with respect to the digital signal rise /fall time *t*10−90%:

$$I\_{\rm critical,fd} = \frac{\lambda\_{\rm min}}{10} = \frac{v}{10 \cdot f\_{\rm max}} = \frac{v \cdot t\_{10-90\%}}{10 \cdot 0.35} = \frac{c \cdot t\_{10-90\%}}{3.5 \cdot \sqrt{\epsilon'\_{r,eff}}} \tag{7.3}$$

where,

*<sup>c</sup>* <sup>=</sup> <sup>1</sup>*/(*√*μ*00*)* <sup>=</sup> <sup>2</sup>*.*<sup>998</sup> · 108 m/sec <sup>=</sup> speed of light *v* = propagation velocity of the signal along the signal line in [m/sec] *fmax* = highest significant sine wave frequency harmonic in a digital signal (what significant means is described in Sect. 5.3) in [Hz]


# *7.2.2 Rule of Thumb for* **lcritical** *in the Time Domain*

There is also a rule of thumb for the time domain [3]:


If we multiply both sides of the equation of the time domain rule of thumb with the signal propagation velocity *v* [m/sec], we get:

$$t\_{10-90\%} \le 2 \cdot t\_{pd}$$

$$t\_{10-90\%} \cdot v \le 2 \cdot t\_{pd} \cdot v$$

$$l\_{10-90\%} \le 2 \cdot l\_{\text{critical,td}}$$

where,

*t*10−90% = rise and/or fall time (whichever is smaller) from 10 % to 90 % of the digital signal in [sec]

*tpd* = propagation delay of the signal along the signal line in [sec]

*l*10−90% = distance which the signal travels along the signal line during the rise and/or fall time (whichever is smaller) *t*10−90% in [m]

*l*critical*,*td = critical length of the time domain rule of thumb in [m]

The time domain approximation for the critical length *l*critical*,*td [m] can be calculated with respect to the digital signal rise/fall time *t*10−90%:

$$I\_{\rm critical,td} = \frac{I\_{10-90\%}}{2} = \frac{v \cdot t\_{10-90\%}}{2} = \frac{c \cdot t\_{10-90\%}}{2 \cdot \sqrt{\epsilon'\_{r,eff}}} \tag{7.4}$$

where,

*l*10−90% = distance which the signal travels along the signal line during the rise and/or fall time (whichever is smaller) *t*10−90% in [m]

*v* = propagation velocity of the signal along the signal line in [m/sec]

*t*10−90% = rise and/or fall time (whichever is smaller) from 10 to 90 % of the digital signal in [sec]

*<sup>c</sup>* <sup>=</sup> <sup>1</sup>*/(*√*μ*00*)* <sup>=</sup> <sup>2</sup>*.*<sup>998</sup> · 108 m/sec <sup>=</sup> speed of light


# *7.2.3 Critical Length* **lcritical**

The two rules above lead to similar values for the *critical length l*critical [m], where the frequency domain *l*critical*,*fd is smaller than the time domain *l*critical*,*td. It is therefore recommended to go with the rule of thumb of the frequency domain: Consider an interconnection as a transmission line if it is longer than:

$$l\_{\text{critical}} = \frac{\lambda\_{\text{min}}}{10} \tag{7.5}$$

where,

*λmin* = wavelength of the highest significant harmonic frequency in the signal (what significant means is described in Sect. 5.3) in [m]

# **7.3 Characteristic Impedance Z0**

# *7.3.1 Characteristic Impedance Z***<sup>0</sup>** *of Any Transmission Line*

The *characteristic impedance Z*<sup>0</sup> [] is the most important property of a transmission line. The characteristic impedance *Z*<sup>0</sup> [] cannot be measured with a simple DC ohmmeter, and it is defined as the ratio of voltage *V* + [V] to current *I* + [A] of a *single* traveling wave along a transmission line [5]:

$$\underline{Z}\_0 = \frac{\underline{V}^+}{\underline{I}^+} \tag{7.6}$$

where

*V* <sup>+</sup> = voltage of the forward wave (the <sup>+</sup> emphasizes that voltage and current travel in the same direction) in complex phasor form in [V]

*I* <sup>+</sup> = current of the forward wave (the <sup>+</sup> emphasizes that voltage and current travel in the same direction) in complex phasor form in [A]

Figure 7.2 shows the equivalent circuit model for a transmission line with two conductors. The so-called distributed parameter model divides the transmission line into infinitely small segments of length *dz* in [m]. The parameters of these segments are defined per-unit-length (e.g., per [m]):

$$R' = \text{Resistance per-unit-length [ $\Omega/m$ ]} \tag{7.7}$$

$$L' = \text{Inductance per-unit-length [H/m]} \tag{7.8}$$

$$\mathbf{C}' = \text{Capacitance per-unit-length [F/m]} \tag{7.9}$$

$$G' = \text{Conductance per-unit-length [S/m]} \tag{7.10}$$

The formula for the characteristic impedance *Z*<sup>0</sup> [] of a transmission line—based on the distributed parameter model—is defined as [3]:

$$\underline{Z}\_0 = \sqrt{\frac{R' + j\omega L'}{G' + j\omega C'}}\tag{7.11}$$

**Fig. 7.2** Distributed parameter model of a transmission line [3]

where,

*ω* = 2*πf* = angular frequency in [rad/sec] *<sup>j</sup>* <sup>=</sup> √−<sup>1</sup> <sup>=</sup> imaginary unit

The following subsections present formulas for the calculation of the characteristic impedance *Z*<sup>0</sup> [] for typical electrical transmission lines (the formulas are approximations; in case you need accurate results, you should use a solver that applies Maxwell's Equations presented in Appendix E):


# *7.3.2 Characteristic Impedance Z***<sup>0</sup>** *of Lossless Transmission Lines*

In practice, it is often adequate to describe transmission lines as lossless (*R*- = 0, *G*- = 0). In that case, the transmission line model can be simplified—like shown in Fig. 7.3—and the calculation of the characteristic impedance is reduced to:

$$Z\_0 = \sqrt{\frac{L'}{C'}}\tag{7.12}$$

where,

*L*- = inductance per-unit-length [H/m] *C*-= capacitance per-unit-length [F/m]

**Fig. 7.3** Distributed parameter model of a loss less transmission line [3]

**Fig. 7.4** Cross section of two parallel wires or twisted pair

# *7.3.3 Characteristic Impedance Z***<sup>0</sup>** *of Parallel Wires*

The characteristic impedance *Z*<sup>0</sup> [] of two parallel wires with identical diameter *D* [m]—according to Fig. 7.4—can be calculated as [6]:

$$Z\_0 = \frac{\eta\_0}{\pi \sqrt{\epsilon'\_{r,eff}}} \cosh^{-1} \left(\frac{d}{D}\right) \tag{7.13}$$

$$
\epsilon\_{r,eff}' = \epsilon\_{r2}' + 0.25 \left( \epsilon\_{r1}' - \epsilon\_{r2}' \right) \tag{7.14}
$$

where,

*η*<sup>0</sup> = 377 Ω = intrinsic impedance of free-space (vacuum) [] *d* = distance from center to center of the two wires in [m] *D* = diameter of the two wires in [m] - *r,eff* = effective dielectric constant (relative permittivity) through which the electromagnetic wave is propagating - *<sup>r</sup>*<sup>1</sup> = relative permeability of the insulation around the conductors


# *7.3.4 Characteristic Impedance Z***<sup>0</sup>** *of Twisted Pairs*

The characteristic impedance *Z*<sup>0</sup> [] of a twisted pair with identical diameter *D* [m]—according to Fig. 7.4—can be calculated as [6]:

$$Z\_0 = \frac{\eta\_0}{\pi \sqrt{\epsilon'\_{r,eff}}} \cosh^{-1} \left(\frac{d}{D}\right) \tag{7.15}$$

$$
\epsilon\_{r,eff}' = \epsilon\_{r2}' + q \left( \epsilon\_{r1}' - \epsilon\_{r2}' \right) \tag{7.16}
$$

$$q = 0.25 + 0.0004\theta \tag{7.17}$$

$$T = \frac{\tan\left(\theta\right)}{\pi d} \tag{7.18}$$

$$\theta = \tan^{-1}(T\pi d) \tag{7.19}$$

where,

*η*<sup>0</sup> = 377 Ω = intrinsic impedance of free space (vacuum) [] *d* = distance from center to center of the two wires in [m] *D* = diameter of the two wires in [m] - *r,eff* = effective dielectric constant (relative permittivity) through which the electromagnetic wave is propagating - *<sup>r</sup>*<sup>1</sup> = relative permeability of the insulation around the conductors - *<sup>r</sup>*<sup>2</sup> = relative permeability of the medium around the insulated conductors *θ* = angle between the twisted pair's center line and the twist in [rad] *T* = twists per length in [1/m]

# *7.3.5 Characteristic Impedance Z***<sup>0</sup>** *of Coaxial Cables*

The characteristic impedance *Z*<sup>0</sup> [] of a round coaxial cable—according to Fig. 7.5—can be calculated as [6]:

$$Z\_0 = \frac{\eta\_0}{2\pi\sqrt{\epsilon\_r'}}\ln\left(\frac{D\_{coax}}{D\_{core}}\right) \tag{7.20}$$

where,

*η*<sup>0</sup> = 377 Ω = intrinsic impedance of free space (vacuum) [] *Dcoax* = inner diameter of the coaxial cable shield in [m] *Dcore* = outer diameter of the core wire of the coaxial cable in [m] - *<sup>r</sup>* = dielectric constant of the dielectric between core and shield

**Fig. 7.5** Coaxial cable

**Fig. 7.6** Microstrip line

# *7.3.6 Characteristic Impedance Z***<sup>0</sup>** *of Microstrip Lines*

The characteristic impedance *Z*<sup>0</sup> [] of a microstrip line—according to Fig. 7.6 can be calculated as [6]:

$$Z\_0 = \frac{\eta\_0}{2\pi\sqrt{2}\sqrt{\epsilon\_r'+1}}\tag{7.21}$$

$$\ln\left(1+\frac{4h}{w'}\left(\frac{14+8/\epsilon\_r'}{11}\frac{4h}{w'}+\sqrt{\left(\frac{14+8/\epsilon\_r'}{11}\right)^2\left(\frac{4h}{w'}\right)^2+\frac{1+1/\epsilon\_r'}{2}\pi^2}\right)\right) \tag{7.22}$$

$$w' = w + \Delta w'\tag{7.23}$$

$$
\Delta w' = \Delta w \left(\frac{1 + 1/\epsilon\_r'}{2}\right) \tag{7.24}
$$

$$
\Delta w = \frac{t}{\pi} \ln \left( \frac{4e}{\sqrt{\left(\frac{t}{\pi}\right)^2 + \left(\frac{1/\pi}{w/t + 1.1}\right)^2}} \right) \tag{7.25}
$$

where,

*η*<sup>0</sup> = 377 Ω = intrinsic impedance of free space (vacuum) [] *h* = distance of the PCB trace above the reference (ground) plane in [m] *w* = width of the PCB trace in [m] *w*- = corrected PCB trace width due to thickness *t* in [m] *t* = PCB trace thickness in [m] - *<sup>r</sup>* = dielectric constant of the PCB dielectric material

# *7.3.7 Characteristic Impedance Z***<sup>0</sup>** *of Coplanar Waveguide with Reference Plane*

The characteristic impedance *Z*<sup>0</sup> [] of a coplanar waveguide with reference plane (ground) according to Fig. 7.7 can be calculated as [6]:

**Fig. 7.8** Stripline

$$Z\_0 = \frac{\pi 60 \,\Omega}{\sqrt{\epsilon'\_{r,eff}}} \frac{1}{\frac{K(k)}{K(k')} + \frac{K(k\_1)}{K(k'\_1)}}\tag{7.26}$$

$$k = \frac{w}{w + 2s} \tag{7.27}$$

$$k\_{\parallel} = \frac{\tanh\left(\frac{\pi w}{4h}\right)}{\tanh\left(\frac{\pi \left(w + 2s\right)}{4h}\right)}\tag{7.28}$$

$$k' = \sqrt{1 - k^2} \tag{7.29}$$

$$k\_1' = \sqrt{1 - k\_1^2} \tag{7.30}$$

$$
\epsilon\_{r,eff}^{'} = \frac{1 + \epsilon\_r \frac{K(k')}{K(k)} \frac{K(k\_1)}{K(k\_1')}}{1 + \frac{K(k')}{K(k)} \frac{K(k\_1)}{K(k\_1')}} \tag{7.31}
$$

where,

*K(x)* = elliptic integral of the first kind of *x* (*x* stands for *k*, *k*- , *k*1, *k*- 1) *h* = distance between the two reference (ground) planes in [m] *w* = width of the PCB trace in [m] *s* = space between the PCB trace and the coplanar reference plane in [m] - *r,eff* = effective dielectric constant of the PCB dielectric material

# *7.3.8 Characteristic Impedance Z***<sup>0</sup>** *of Centered Striplines*

The characteristic impedance *Z*<sup>0</sup> [] of a stripline—which is located in the middle of two reference planes according to Fig. 7.8—can be calculated as [6]:

$$Z\_0 = \frac{60\,\Omega}{\sqrt{\epsilon\_r'}} \ln\left(\frac{4h}{\pi K\_1}\right), \text{for } \frac{w}{h} \le 0.35\tag{7.32}$$

$$\mathbf{Z}\_{0} = \frac{94.15 \,\Omega}{\frac{w/h}{1 - t/h} + \frac{K\_2}{\pi}} \frac{1}{\sqrt{\epsilon\_r'}}, \text{for } \frac{w}{h} > 0.35\tag{7.33}$$

$$K\_{l} = \frac{w}{2} \left[ 1 + \frac{t}{\pi w} \left( 1 + \ln \left( \frac{4 \pi w}{t} \right) + 0.51 \pi \left( \frac{t}{w} \right)^{2} \right) \right] \tag{7.34}$$

$$K\_2 = \frac{2}{1 - t/h} \ln\left(\frac{1}{1 - t/h} + 1\right) - \left(\frac{1}{1 - t/h} - 1\right) \ln\left(\frac{1}{\left(1 - t/h\right)^2} - 1\right) \tag{7.35}$$

where,

*h* = distance between the two reference (ground) planes in [m] *t* = thickness of the PCB trace in [m] *w* = width of the PCB trace in [m] - *<sup>r</sup>* = dielectric constant of the PCB dielectric material

#### **7.4 Per-Unit-Length Inductance L and Capacitance C-**

Electrically short transmission lines (*l < λ/*10) can be modeled as lumpedelement *RLC* networks (Fig. 7.9). The following subsections present formulas for the calculation of the *per-unit-length inductance L*- [H/m] and the *per-unit-length capacitance C*-[F/m] for typical electrical transmission lines:


**Fig. 7.9** Lumped-element RLC-network examples of transmission lines. (**a**) Lumped *π*-model. (**b**) Lumped *T* -model

#### *7.4.1 L* *and C**of Two-Wire Lines*

For a setup with two parallel wires (e.g., ribbon cable with identical diameter *D* [m]), where the current flows in the opposite direction, the per-unit-length inductance *L*- [H/m] and the per-unit-length capacitance *C*-[F/m] are given as [4]:

$$L'\_{TwoWireLine} \approx \frac{\mu'\_{r,eff}\mu\_0}{\pi} \cosh^{-1}\left(\frac{d}{D}\right) \tag{7.36}$$

$$C\_{TwoWireLine}' \approx \frac{\pi \epsilon\_{r,eff}' \epsilon\_0}{\cosh^{-1}\left(\frac{d}{D}\right)}\tag{7.37}$$

where,

cosh−<sup>1</sup> <sup>=</sup> area hyperbolic cosine (inverse hyperbolic cosine) function [1, +∞] *μ*- *r,eff* = effective relative permeability of the environment of the two wires and

therefore the material through which the magnetic flux flows *<sup>μ</sup>*<sup>0</sup> <sup>=</sup> 12.57 ·10−<sup>7</sup> H/m <sup>=</sup> permeability of vacuum, absolute permeability - *r,eff* = effective relative permittivity (dielectric constant) of the environment of the two wires and, therefore, the material through which the electric field lines are formed

<sup>0</sup> <sup>=</sup> <sup>8</sup>*.*<sup>854</sup> · <sup>10</sup>−<sup>12</sup> F/m <sup>=</sup> permittivity of vacuum, absolute permittivity *d* = distance from center to center of the two wires in [m] *D* = diameter of the two wires in [m]

The effective relative permeability *μ*- *r,eff* is often equal 1.0, because the insulation material of cables and the environment around the lines is usually non-magnetic (e.g., plastics, aluminum, or copper).

The calculation of the effective relative permeability - *r,eff* is often complicated because there are different dielectric materials involved (e.g., cable insulation and air). The approximate effective dielectric constant - *r,eff* for a pair of wire—like shown in Fig. 7.10—is given in as [6]:


**Table 7.1** Unit-per-length inductance *L* and capacitance *C* of two-wire lines with PVC insulation around each wire

$$
\epsilon\_{r,eff}' \approx \epsilon\_{r2}' + 0.25 \left( \epsilon\_{r1}' - \epsilon\_{r2}' \right) \tag{7.38}
$$

where,


Based on Eqs. 7.36 and 7.37, Table 7.1a lists examples of per-unit-length inductance *L* and per-unit-length capacitance *C* of a two-wire line without an additional cable jacket (*r*<sup>2</sup> = 1). On the other hand, Table 7.1b lists the values for *L* and *C* with a two-wire line and a PVC cable jacket around the wires (*r*<sup>2</sup> = 3).

#### *7.4.2 L' and C' of Wires Above Ground Plane*

In the case of a wire above an infinite ground plane (Fig. 7.11), the per-unit-length loop inductance *L*- [H/m] and the per-unit-length capacitance *C*- [F/m] are given as [4]:

$$L'\_{WireAboveGnd Plane} \approx \frac{\mu'\_{r,eff}\mu\_0}{2\pi} \cosh^{-1}\left(\frac{2d}{D}\right) \tag{7.39}$$

$$C\_{WireAboveGndPlane}' \approx \frac{2\pi \epsilon\_{r,eff}' \epsilon\_0}{\cosh^{-1}\left(\frac{2d}{D}\right)}\tag{7.40}$$

where,

cosh−<sup>1</sup> <sup>=</sup> area hyperbolic cosine (inverse hyperbolic cosine) function [1, +∞] *μ*- *r,eff* = effective relative permeability of the environment of the wire and therefore the material through which the magnetic flux flows

*<sup>μ</sup>*<sup>0</sup> <sup>=</sup> 12.57 ·10−<sup>7</sup> H/m <sup>=</sup> permeability of vacuum, absolute permeability 

*r,eff* = effective relative permittivity (dielectric constant) between the wire and the ground plane and therefore the material through which the electric field lines are formed

<sup>0</sup> <sup>=</sup> <sup>8</sup>*.*<sup>854</sup> · <sup>10</sup>−<sup>12</sup> F/m <sup>=</sup> permittivity of vacuum, absolute permittivity *d* = distance from the center of the wire to the surface of the ground plane [m] *D* = diameter of the wire in [m]

The effective relative permeability *μ*- *r,eff* is usually equal 1.0, because all the media around the conductor are non-magnetic (e.g. plastics, copper), whereas the effective relative permittivity - *r,eff* between the wire and the ground plane is very difficult to calculate, as it depends very much on the distance of the wire to the ground plane, the thickness of the insulation around the wire, and the media between the wire and the ground plane. Table 7.2 gives an idea about the ranges of the perunit-length inductance *L*- [H/m] and the per-unit-capacitance *C*- [F/m] of a single wire above a ground plane in case the wire is close to the ground plane or far away.


**Table 7.2** Unit-per-length inductance *L* and capacitance *C* of a single PVC insulated wire above an infinite ground plane


**Fig. 7.12** Cross section of a coaxial cable

**Table 7.3** Unit-per-length inductance *L* and capacitance *C*of different coaxial cables


#### *7.4.3 L* *and C**of Coaxial Cables*

For a coaxial cable (Fig. 7.12), the per-unit-length loop inductance *L*- [H/m] and the per-unit-length capacitance *C*-[F/m] are given as [4]:

$$L'\_{coax} \approx \frac{\mu\_r \mu\_0}{2\pi} \ln\left(\frac{D\_{coax}}{D\_{core}}\right) \tag{7.41}$$

$$C\_{coax}' \approx \frac{2\pi\epsilon\_r\epsilon\_0}{\ln\left(\frac{D\_{coax}}{D\_{cor}}\right)}\tag{7.42}$$

where,

*μr* = relative permeability of the coaxial cable dielectric material in [1] *<sup>μ</sup>*<sup>0</sup> <sup>=</sup> 12.57 ·10−<sup>7</sup> H/m <sup>=</sup> permeability of vacuum, absolute permeability - *<sup>r</sup>* = relative permittivity of the coaxial cable dielectric material in [1] <sup>0</sup> <sup>=</sup> <sup>8</sup>*.*<sup>854</sup> · <sup>10</sup>−<sup>12</sup> F/m <sup>=</sup> permittivity of vacuum, absolute permittivity *Dcoax* = inner diameter of the coaxial cable shield in [m] *Dcore* = outer diameter of the core wire of the coaxial cable in [m]

Table 7.3 gives an idea about the ranges of the per-unit-length inductance *L*- [H/m] and the per-unit-capacitance *C*-[F/m] of typical coaxial cables.

#### *7.4.4 L* *and C**of PCB Traces*

It is not common to specify the per-unit-length inductance *L*- [H/m] and the perunit-length capacitance *C*- [F/m] for PCB traces. Instead, it is more common to specify the characteristic impedance *Z*<sup>0</sup> [] of the transmission line (PCB trace), which can be expressed as:

$$Z\_0 = \sqrt{\frac{L'}{C'}}\tag{7.43}$$

Another fundamental parameter is the propagation velocity *v* [m/sec] of a signal along a transmission line:

$$v = \frac{1}{\sqrt{L'C'}} = \frac{c}{\sqrt{\mu\_{r,eff}' \epsilon\_{r,eff}'}} \tag{7.44}$$

$$v = \frac{c}{\sqrt{\epsilon\_{r,eff}^{'}}}, \text{ because usually } \mu\_{r,eff}^{'} = 1.0\tag{7.45}$$

Thus, we can write:

$$L' = \frac{Z\_0}{v} = \frac{Z\_0 \sqrt{\epsilon'\_{r,eff}}}{c} \tag{7.46}$$

$$C' = \frac{1}{vZ\_0} = \frac{\sqrt{\epsilon'\_{r,eff}}}{cZ\_0} \tag{7.47}$$

where,

*L*- = per-unit-length inductance of the transmission line (e.g., PCB trace) [H/m] *C*- = per-unit-length capacitance of the transmission line (e.g., PCB trace) [F/m] *<sup>c</sup>* <sup>=</sup> <sup>1</sup>*/(*√*μ*00*)* <sup>=</sup> <sup>2</sup>*.*<sup>998</sup> · 108 m/sec <sup>=</sup> speed of light *μ*- *r,eff* = effective magnetic relative permeability through which the electromagnetic wave is propagating


Tables 7.4, 7.5 and, 7.6 present some example values of per-unit-length inductance *L*- [H/m] and per-unit-length capacitance *C*- [F/m] for microstrip lines, striplines, and coplanar waveguides with reference plane. All values are approximations.

Formulas for calculating the characteristic impedance *Z*<sup>0</sup> of some selected PCB structures can be found in the next Chap. 11.1.7.2.

**Table 7.4** Unit-per-length inductance *L* and capacitance *C* of mirostrip lines shown in Fig. 7.6. *t* = thickness of the PCB trace. *h* = distance of the PCB trace above the reference plane. *w* = width of the PCB trace


**Table 7.5** Unit-per-length inductance *L* and capacitance *C* of coplanar waveguides with a reference plane shown in Fig. 7.7. *h* = distance between the two reference (ground) planes. *w* = width of the PCB trace. *s* = space between the PCB trace and the coplanar reference plane


**Table 7.6** Unit-per-length inductance *L* and capacitance *C* of striplines shown in Fig. 7.8. *t* = thickness of the PCB trace. *h* = distance between the two reference planes. *w* = width of the PCB trace


### **7.5 Propagation Constant** *γ*

The *propagation constant γ* [1/m]—also called propagation factor—describes the attenuation and phase shift of the signal as it propagates through the transmission line. To be even more precise: the propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction (Fig. 7.13).

Let's imagine a sinusoidal voltage, current, electric field, or magnetic field which propagates in the direction of *z* [m] and which has amplitude of *A*<sup>0</sup> at its source and an amplitude of *Az* at the distance *z* [m] from the source. Then *Az* can be written as:

$$\underline{A}\_{\mathbb{Z}}(z) = \underline{A}\_0 \cdot e^{-\underline{\chi}z} \tag{7.48}$$

where,

*Az* = complex phasor of a sinusoidal voltage [V], current [A], electric field [V/m], or magnetic field [A/m] at the distance *z* away from the source

*A*<sup>0</sup> = complex phasor of a sinusoidal voltage [V], current [A], electric field [V/m], or magnetic field [A/m] at the source

*γ* = the complex propagation constant in [1/m]

*z* = distance traveled along a transmission line in [m]

The propagation constant *γ* [1/m] can be calculated based on the per-unit-length parameters of a transmission line [2]:

$$
\underline{\chi} = \sqrt{(R' + j\omega L')(G' + j\omega C')} = j\omega \sqrt{\underline{\xi\mu}} \tag{7.49}
$$

where,

*R*- = resistance per-unit-length in [/m] *L*- = inductance per-unit-length in [H/m] *C*-= capacitance per-unit-length in [F/m]

*G*-= conductance per-unit-length in [S/m]

*ω* = 2*πf* = angular frequency of the signal in [rad/sec]

*μ* = *μ*- − *jμ*-- = complex permeability of the media through which the electromagnetic wave is propagating in [H/m]

 = - − *j* -- = complex permittivity of the media through which the electromagnetic wave is propagating in [F/m]

More details about the complex permittivity and permeability can be found in Appendix Sects. D.5 and D.6.

Another notation of *γ* is given as [2]:

$$
\underline{\chi} = \alpha + j\beta \tag{7.50}
$$

where,

*α* = attenuation constant (or attenuation factor) in [1/m]

*β* = phase constant (or phase factor) in [rad/m]

This means that the transmission line Eq. 7.48 can be rewritten as:

$$
\underline{A}\_{\varepsilon} = \underline{A}\_0 \cdot e^{-\alpha \varepsilon} e^{-j\beta \varepsilon} \tag{7.51}
$$

In a general form, the *attenuation constant α* [1/m] and the *phase constant β* [rad/m] can be calculated like this [2]:

$$\alpha = \omega \left\lfloor \frac{(\epsilon' \mu' - \epsilon'' \mu'')}{2} \cdot \left( \sqrt{1 + \left( \frac{\epsilon' \mu'' + \epsilon'' \mu'}{\epsilon' \mu' - \epsilon'' \mu''} \right)^2} - 1 \right) \right\rfloor \tag{7.52}$$

$$\beta = \frac{2\pi}{\lambda} = \omega \left\lfloor \frac{(\epsilon'\mu' - \epsilon''\mu'')}{2} \cdot \left( \sqrt{1 + \left( \frac{\epsilon'\mu'' + \epsilon''\mu'}{\epsilon'\mu' - \epsilon''\mu''} \right)^2} + 1 \right) \right\rfloor \tag{7.53}$$

where,

*ω* = 2*πf* = angular frequency of the signal in [rad/sec] - = real part of the complex permittivity ( = - − *j* --) in [F/m] -- = imaginary part of the complex permittivity ( = - − *j* --) in [F/m] *μ*- = real part of the complex permeability (*μ* = *μ*- − *jμ*--) in [H/m] *μ*-- = imaginary part of the complex permeability (*μ* = *μ*- − *jμ*--) in [H/m]

The magnetic loss *μ*- can often be neglected (*μ*-- = 0) and the dielectric loss - can be written as [6]:

$$
\underline{\epsilon} = \epsilon' - j\epsilon'' = \epsilon' - j\frac{\sigma}{\omega} \tag{7.54}
$$

$$
\epsilon'' = \frac{\sigma}{\omega} \tag{7.55}
$$

where,


With *μ*-- = 0 and -- = *σ/ω*, the attenuation constant *α* [1/m] and the phase constant *β* [rad/m] can be calculated like this [6]:

$$\alpha = \omega \sqrt{\frac{\mu' \epsilon'}{2} \cdot \left(\sqrt{1 + \left(\frac{\sigma}{a\nu \epsilon'}\right)^2} - 1\right)}\tag{7.56}$$

$$\beta = \omega \sqrt{\frac{\mu' \epsilon'}{2} \cdot \left(\sqrt{1 + \left(\frac{\sigma}{\alpha \epsilon'}\right)^2} + 1\right)}\tag{7.57}$$

For a lossless line (*σ* = 0) we get [2]:

• No attenuation *α* [1/m]:

$$
\alpha = 0\tag{7.58}
$$

• Phase shift *β* [rad/m]:

$$
\beta = \omega \sqrt{R'C'} = \omega \sqrt{\epsilon' \mu'} \tag{7.59}
$$

where,

*ω* = 2*πf* = angular frequency of the signal in [rad/sec]


**Fig. 7.14** Transmission line input impedance *Zin(l)*

#### **7.6 Input Impedance of Transmission Lines**

#### *7.6.1 Input Impedance of Any Transmission Line*

The *characteristic impedance Z*<sup>0</sup> [] of a transmission line—as already discussed in Chap. 7.3—is the ratio of the amplitude of a *single* voltage wave to its current wave. Since most transmission lines also have a reflected wave, the characteristic impedance is generally not the impedance that is measured on the line. The impedance *Zin(l)* [] (Fig. 7.14) measured at a given distance *l* [m] from the load impedance *ZL* [] can be expressed as [6]:

$$\underline{Z}\_{n}(l) = \frac{\underline{V}(l)}{\underline{I}(l)} = \underline{Z}\_{0} \frac{\underline{Z}\_{L} + \underline{Z}\_{0} \tanh\left(\underline{\underline{\nu}}l\right)}{\underline{Z}\_{0} + \underline{Z}\_{L} \tanh\left(\underline{\underline{\nu}}l\right)} = \underline{Z}\_{0} \frac{\underline{Z}\_{L} \cosh\left(\underline{\underline{\nu}}l\right) + \underline{Z}\_{0} \sinh\left(\underline{\underline{\nu}}l\right)}{\underline{Z}\_{0} \cosh\left(\underline{\underline{\nu}}l\right) + \underline{Z}\_{L} \sinh\left(\underline{\underline{\nu}}l\right)}\tag{7.60}$$

where,

*Zin(l)* = impedance measured at distance *l* from the load *ZL* in [] *V (l)* = voltage at distance *l* from the load *ZL* in [V] *I (l)* = current at distance *l* from the load *ZL* in [V] *Z*<sup>0</sup> = the complex characteristic impedance of the transmission line in [] *ZL* = the complex load impedance in [] *γ* = *α* + *jβ* = the complex propagation constant in [1/m] *l* = the distance from where the input of the transmission line to the load in [m]

#### *7.6.2 Input Impedance of a Lossless Transmission Line*

For a *lossless transmission line*, the propagation constant is purely imaginary [6]:

$$
\alpha = 0\tag{7.61}
$$

$$
\underline{\chi} = j\beta = \frac{2\pi j}{\lambda} \tag{7.62}
$$

and the input impedance to a lossless transmission line can be calculated as:

$$\underline{Z}\_{ln}(l) = \underline{Z}\_0 \frac{\underline{Z}\_L + j\underline{Z}\_0 \tan\left(\beta l\right)}{\underline{Z}\_0 + j\underline{Z}\_L \tan\left(\beta l\right)} = \underline{Z}\_0 \frac{\underline{Z}\_L \cos\left(\beta l\right) + j\underline{Z}\_0 \sin\left(\beta l\right)}{\underline{Z}\_0 \cos\left(\beta l\right) + j\underline{Z}\_L \sin\left(\beta l\right)}\tag{7.63}$$

where,

*Zin(l)* = impedance measured at distance *l* from the load *ZL* in [] *Z*<sup>0</sup> = the complex characteristic impedance of the transmission line in [] *ZL* = the complex load impedance in [] *l* = the distance from where the input of the transmission line to the load in [m] *β* = 2*π/λ* = phase constant (or phase factor) in [rad/m]

# *7.6.3 Input Impedance of a Transmission Line At* **l =** *λ/***2**

When the distance from the input of the transmission line to the load is a multiple of *λ/*2 *(l* = *nλ/*2*)* and therefore *βl* = *nπ* (where *n* is an integer), the input impedance to the transmission line *Zin(l)* is equal the load impedance *ZL* [6]:

$$
\underline{Z}\_{ln}(l) = \underline{Z}\_{L} \tag{7.64}
$$

# *7.6.4 Impedance of a Transmission Line At* **l =** *λ/***4**

When the distance from the input of the transmission line to the load is a multiple of *λ/*4 *(βl* = *nπ/*2*)* and therefore *l* = *nλ/*4 (where *n* is an integer), the input impedance to the transmission line *Zin(l)* is [6]:

$$
\underline{Z}\_{ln}(l) = \frac{\underline{Z}\_0^2}{\underline{Z}\_L} \tag{7.65}
$$

This means in case *ZL* = 0, the transmission line input impedance becomes *Zin* = ∞ and vice versa.

#### *7.6.5 Impedance of a Matched Transmission Line*

In case the load impedance *ZL* is equal the characteristic impedance *Z*<sup>0</sup> of the transmission line (in other words: the load is matched), we can write:

$$
\underline{Z}\_{ln}(l) = \underline{Z}\_0 = \underline{Z}\_L \tag{7.66}
$$

#### *7.6.6 Impedance of a Shorted Transmission Line*

For the case of a shorted load *ZL* = 0, the input impedance *Zin(l)* is purely imaginary and a periodic function of position and wavelength *λ* [m] (frequency *f* [Hz]):

$$\underline{Z}\_{ln}(l) = j\underline{Z}\_0 \tan\left(\beta l\right) = j\underline{Z}\_0 \tan\left(\frac{2\pi l}{\lambda}\right) \tag{7.67}$$

#### *7.6.7 Impedance of an Open Transmission Line*

For the case of an open load *ZL* = ∞, the input impedance *Zin(l)* is purely imaginary and a periodic function of position and wavelength *λ* [m] (frequency *f* [Hz])

$$\underline{Z}\_{ln}(l) = -j\underline{Z}\_0 \cot(\beta l) = -j\underline{Z}\_0 \cot\left(\frac{2\pi l}{\lambda}\right) \tag{7.68}$$

#### **7.7 High-Frequency Losses**

The lossless line model is often accurate enough for frequencies up 100 MHz. However, above 100 MHz, the *high-frequency losses* may not be neglected anymore, and therefore the attenuation factor *α* [1/m] cannot be assumed to be zero. Highfrequency losses are a result of [3]:


Note: The dielectric loss does not depend on the geometry of the transmission line, only on the dielectric material.

#### **7.8 Loss Tangent tan***(δ)*

The electric and magnetic losses quantify a material's inherent dissipation of electromagnetic energy (e.g., heat). The *loss tangent* tan*(δ)* (Fig. 7.15) is used to express how lossy a medium or a transmission line is: a small tan*(δ)* means low loss.

The electric loss tangent—often called dielectric loss tangent or *dissipation factor Df* —is defined as [2]:

$$\tan(\delta\_e) = \frac{\epsilon''}{\epsilon'} = \frac{\text{loss current}}{\text{charging current}} \tag{7.69}$$

where,


The electrical conductivity *σ* [S/m] of a dielectric material is defined as [2]:

$$
\sigma = a\epsilon''\tag{7.70}
$$

and therefore the loss tangent can be written as function of *σ* [S/m]:

$$\tan(\delta\_e) = \frac{\epsilon''}{\epsilon'} = \frac{\sigma}{o\epsilon'} \tag{7.71}$$

The magnetic loss tangent tan*(δm)* is usually not of interest for traditional transmission lines because the materials involved are not magnetic. However, for the sake of completeness, here the definition of the magnetic loss tangent [2]:

**Fig. 7.15** Charging and loss current density

$$\tan(\delta\_m) = \frac{\mu''}{\mu'} \tag{7.72}$$

where,


#### **7.9 Balanced vs. Unbalanced Transmission Lines**

EMC design engineers should be well aware of the concept and benefits of *balanced transmission lines*. Here the differences between *balanced* and *unbalanced transmission lines*:


**Fig. 7.16** Balanced transmission line with common-mode noise

**Fig. 7.17** Unbalanced transmission line with noise on signal line

Balanced transmission lines are very robust against common-mode noise because common-mode signals will be canceled out at the receiver's side. Common-mode noise coupling is explained in Sect. 12.4. However, to prevent differential-mode noise coupling, the two signal conductors must be routed close to each other (e.g., twisting them). Differential-mode noise coupling is explained in Sect. 12.3.

Generally, *single-ended interfaces* should be sent over *unbalanced transmission lines*, and *differential signal interfaces* should be sent over *balanced transmission lines*. Examples:

	- **PCB data lines.** Microstrip lines (see Fig. 7.6), striplines (see Fig. 7.8), and coplanar waveguides (see Fig. 7.7)
	- **Cables and wires.** Coaxial cables (see Fig. 7.5) and multi-layer flat-ribbon or flat-flex cables with at least one solid ground plane
	- **PCB data lines.** Microstrip lines and striplines routed as differential pairs
	- **Cables and wires.** Twisted pair (shielded or unshielded, see Fig. 7.4), twinlead cables, flat-ribbon or flat-flex cables (one layer)

#### **7.10 Single-Ended vs. Differential Interfaces**

*Single-ended signal interfaces* consist of a single signal line and a reference potential. In contrast, *differential signal interfaces* consist of two complementary signals—a differential pair—and a reference potential (compare Figs. 7.18 and 7.19).

While Ethernet is a truly differential signal interface, most of the differential interfaces listed below are pseudo-differential, meaning that two single-ended digital signal sources are switched inverted relative to each other (around a common mode voltage level). The advantage of a pseudo-differential interface to a truly differential is that it doesn't need a *balun* (transformer which converts between a balanced signal and an unbalanced signal; see Sect. 11.7). The disadvantage of a pseudo-differential interface over a truly differential interface is that due to

**Fig. 7.18** Single-ended signal

**Fig. 7.19** Differential signal, where *VDM* [V] is the differential signal voltage and *VCM* [V] the common-mode voltage

slight timing differences at signal changes (from high to low and vice versa), a common-mode noise current is generated, which could lead to unintended radiated emissions (common-mode noise is the number one source for unintended radiation; see Sect. 9.9).

	- **Communication interfaces.** RS232, VGA video connectors, SCSI interfaces, VMEbus, PCI
	- **PCB data buses.** I 2C, SPI
	- **Digital signaling.** TTL, CMOS
	- **Communication interfaces.** Ethernet, USB, HDMI, DVI-D, Serial ATA, PCI Express
	- **Digital signaling.** LVDS, ECL, PECL, CML/SCL

#### **7.11 Summary**


$$d\_{\rm critical} \approx \frac{\lambda\_{\rm min}}{10} \tag{7.73}$$


**Table 7.7** Transmission line parameters for any media, lossless media (*σ* = 0), low-loss media good (--*/*- = *σ/(ω*- *) <<* 0*.*01*)*) and good conductors --*/*-*>* 100*)* [5]

where,

*l*critical = critical length in [m]

*λmin* = wavelength of the highest significant harmonic frequency in the signal in [m]

• **Characteristic impedance Z0.** The characteristic impedance *Z*<sup>0</sup> [] of a uniform transmission line is the ratio of the complex amplitudes of voltage *V* [V] and current *I* [A] of a single wave propagating along the line in the same direction (in the absence of reflections in the other direction).

$$\underline{Z}\_0 = \sqrt{\frac{R' + j\alpha L'}{G' + j\alpha C'}}\tag{7.74}$$

where,

*R*- = resistance per-unit-length in [/m] *L*- = inductance per-unit-length in [H/m] *C*- = capacitance per-unit-length in [F/m] *G*- = conductance per-unit-length in [S/m] *ω* = 2*πf* = angular frequency in [rad/sec] *<sup>j</sup>* <sup>=</sup> √−<sup>1</sup> <sup>=</sup> imaginary unit

For a lossless line (*R*- = 0, *G*-= 0), the characteristic impedance is reduced to:

$$Z\_0 = \sqrt{\frac{L'}{C'}}\tag{7.75}$$

	- *L*-= 1 µH/m = 1 nH/mm
	- *C*-= 50 pF/m = 50 fF/mm

$$\underline{\chi} = \alpha + j\beta = \sqrt{(R' + j\omega L')(G' + j\omega C')} = j\omega \sqrt{\underline{\xi\mu}} \tag{7.76}$$

where,

*α* = attenuation constant (or attenuation factor) in [1/m]

*β* = phase constant (or phase factor) in [rad/m]

*<sup>j</sup>* <sup>=</sup> √−<sup>1</sup> <sup>=</sup> imaginary unit


*μ* = *μ*- − *jμ*-- = complex permeability of the media through which the electromagnetic wave is propagating in [H/m]

 = - − *j* -- = complex permittivity of the media through which the electromagnetic wave is propagating in [F/m]


### **References**


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# **Chapter 8 Electromagnetic Fields**

*Thoroughly conscious ignorance is the prelude to every real advance in science.*

*—*James Clerk Maxwell

### **8.1 The Electromagnetic Field**

The *electromagnetic field* can be seen as the combination of an electric field −→*E* [V/m] and a magnetic field −→*H* [A/m] traveling in the same direction. Figure 8.1 shows an electromagnetic field in form of a plane wave, where the electric field and the magnetic field are perpendicular to each other.

The *Poynting vector*<sup>1</sup> −→*S* [W/m2] represents the direction of propagation of an electromagnetic wave and the complex power density vector of a radiated electromagnetic field in [W/m2]. −→*S* is defined as the cross product of the two vector fields −→*E* and −→*H* <sup>∗</sup> [1]:

$$
\overrightarrow{S} = \overrightarrow{E} \times \overrightarrow{H}^\* \tag{8.1}
$$

where,

−→*<sup>S</sup>* <sup>=</sup> the directional energy flux vector (the energy transfer per unit area per unit time) of an electromagnetic field in [W/m2]

−→*<sup>E</sup>* <sup>=</sup> electric field vector in [V/m] −→*<sup>H</sup>* <sup>∗</sup> <sup>=</sup> complex conjugate of the magnetic field vector −→*<sup>H</sup>* in [A/m]

Instead of working with complex field vectors, it is often much more practical to work with the average power density *Savg* [W/m2]. For a uniform plane wave

<sup>1</sup> The Poynting vector represents the directional energy flux (the energy transfer per unit area per unit time) of an electromagnetic field. The SI unit of the Poynting vector is the watt per square meter [W/m2]. It is named after its discoverer John Henry Poynting who first derived it in 1884.

R. B. Keller, *Design for Electromagnetic Compatibility–In a Nutshell*, https://doi.org/10.1007/978-3-031-14186-7\_8

(electromagnetic field in the far-field, where the wave impedance *Zw* [] is equal to the intrinsic impedance *η*) in a lossless medium (*r* = 1*.*0 and *μr* = 1*.*0), the calculation of the average power density *Savg* [W/m2] can be simplified to:

$$S\_{avg} = \frac{1}{2} |\overrightarrow{E} \mid \cdot |\overrightarrow{H}| = \frac{|\overrightarrow{E} \mid^2}{2\eta} = \frac{|\overrightarrow{H} \mid^2 \eta}{2} \tag{8.2}$$

where,

$$\left| \begin{array}{l} \overrightarrow{E} \,\right| = \text{amplitude (magnitude) of the electric field vector} \text{(phase in [V/m])} \end{array} \right| $$

−→*<sup>H</sup>* | = amplitude (magnitude) of the magnetic field vector/phasor in [A/m]


In Eq. 8.2 the factor 1*/*2 can be omitted for RMS values of −→*E* [V/m] and −→*H* [A/m]. For free space, *η* [] can be set to *η*<sup>0</sup> = 377 Ω. More details about the electromagnetic and other physical fields can be found in Appendix D.

### **8.2 Electromagnetic Field Characteristics**

Three parameters determine the electromagnetic field characteristics [4]:


properties are determined primarily by the source characteristics (low-impedance or high-impedance source). Far from the source, the field depends mainly on the medium through which the field is propagating. Therefore, the space around a radiation source (antenna) can be split into two regions: the near-field and the far-field.

More details about the near-field, the far-field, the wave impedance *Zw* [], and the intrinsic impedance *η* [] are presented in the following Sects. 8.3–8.5.

### **8.3 Near-Field vs. Far-Field**

As an EMC design engineer, there is no way around the near- and far-field topic. For example, for shielding, it is crucial whether the electromagnetic wave to be shielded is a near-field or a far-field wave (see Chap. 13). Moreover, for EMC emission troubleshooting, it is crucial to know if a measurement takes place in the near-field or the far-field because different probes and antennas have to be used accordingly:


The electromagnetic field around an antenna can be divided into three regions [1]:


The regions depend on the maximum linear dimension of the antenna *D* [m] and the wavelength of the signal *λ* [m] (see Fig. 8.2):


#### *8.3.1 Near-Field*

In the *near-field*—also called *Fresnel zone*—the wave impedance *Zw* [] depends primarily on the source, and the electric and magnetic fields have to be considered separately (because the ratio of |*E(z)*|*/*|*H(z)*| is not constant). Usually, there is

**Fig. 8.2** Approximate near-field to far-field boundary [1]. (**a**) The reactive near-field of electrically small antennas. (**b**) The near-/ and far-field of electrically large antennas

either the electric *E*-field or the magnetic *H*-field predominant in the near-field [5]:

#### • **Predominant E-field:**

	- The source voltage is low compared to the source current.
	- The source impedance is low (e.g., loop antennas, magnetic dipole antennas).
	- The wave impedance near the antenna is low.
	- *<sup>E</sup>*-field attenuates with a rate of 1*/d*<sup>2</sup> in the near-field (*<sup>d</sup>* <sup>=</sup> distance to source).
	- *<sup>H</sup>*-field attenuates with a rate of 1*/d*<sup>3</sup> in the near-field (*<sup>d</sup>* <sup>=</sup> distance to source).

As mentioned above, the near-field can be divided into the following two regions: the reactive near-field and the radiating near-field.

#### **8.3.1.1 Reactive Near-Field**

In the *reactive near-field*, energy is stored in the electric and magnetic fields very close to the source but not radiated from them. Instead, energy is exchanged between the signal source and the fields.

In the case of *D < λ/(*2*π )*—this means that the antenna is not a very effective radiator—the reactive near-field extends until the distance *d* [m] from the antenna by Balanis [1]:

$$d\_{reactive-near-field} \le \frac{\lambda}{2\pi} \tag{8.3}$$

In the case of *D > λ/(*2*π )*—where there is the chance that the antenna is an effective radiator—the reactive near-field extends until the distance *d* [m] from the antenna by:

$$d\_{reactive-near-field} \le 0.62 \sqrt{\frac{D^3}{\lambda}}\tag{8.4}$$

where,

*λ* = wavelength of the sinusoidal signal in [m] *D* = maximum linear dimension of the antenna in [m]. *D* = *l* for a wire antenna.

#### **8.3.1.2 Radiating Near-Field**

If a *radiating near-field* exists, then it is defined as the region between the reactive near-field and the far-field. In the radiative or radiating near-field, the angular field distribution depends on distance *d* from the antenna, unlike in the far-field where it does not depend on the distance. In addition, in the radiating near-field, the radiating power density is greater than the reactive power density.

If the antenna has a maximum dimension *D* [m] that is small compared to the wavelength *λ* [m], the radiating near-field region may not exist.

In the case of *D > λ/(*2*π )*, the radiating near-field begins after the reactive nearfield region has ended (*dreactive*−*near*−*f ield* ) and ends where the far-field begins [1]:

$$d\_{radiating-near-field} = 0.62 \sqrt{\frac{D^3}{\lambda}} \dots \frac{2D^2}{\lambda} \tag{8.5}$$

where,

*λ* = wavelength of the sinusoidal signal in [m]. *D* = maximum linear dimension of the antenna in [m]. *D* = *l* for a wire antenna.

### *8.3.2 Far-Field*

In the *far-field*—also called *Fraunhofer region*—the *E*- and *H*-fields move perpendicular (orthogonal) and in phase to each other and form a plane wave.


There is usually only a far-field region in the case of *D > λ/(*2*π )*, where *λ* [m] is the wavelength of the signal and *D* [m] is the maximum linear dimension of the antenna. However, if there is a far-field, it starts roughly at the following distance *d* [m] from the antenna [1]:

$$d\_{far-field} > \frac{2D^2}{\lambda} \tag{8.6}$$

where,

*λ* = wavelength of the sinusoidal signal in [m]

*D* = maximum linear dimension of the antenna in [m]. *D* = *l* for a wire antenna.

#### **8.4 Intrinsic Impedance** *η*

#### *8.4.1 Intrinsic Impedance of Any Media*

The *intrinsic impedance η* [] is a property of a medium, and it influences the electromagnetic waves that are traveling through that medium. The intrinsic impedance is dependent on the conductivity, permittivity and permeability of the medium. It is a complex number and defined as [6]:

$$\underline{\eta} = \sqrt{\frac{\underline{\mu}}{\underline{\epsilon}}} = \sqrt{\frac{\mu'-j\mu''}{\epsilon'-j\epsilon''}} = \sqrt{\frac{\mu''+j\mu'}{\epsilon''+j\epsilon'}} = \sqrt{\frac{\mu''\omega+j\omega\mu'}{\epsilon''\omega+j\omega\epsilon'}}\tag{8.7}$$

where,

*μ* = *μr μ*<sup>0</sup> = *μ*- − *jμ*-- = complex permeability of the medium in [H/m] = *r* <sup>0</sup> = - − *j* -- = complex permittivity of the medium in [F/m] *ω* = 2*πf* = angular frequency of the signal in [rad/sec] *<sup>j</sup>* <sup>=</sup> √−<sup>1</sup> <sup>=</sup> imaginary unit

#### *8.4.2 Intrinsic Impedance of Magnetic Lossless Media*

If we neglected the magnetic losses (*μ*-- = 0) and set the dielectric loss to -- = *σ/ω*, we can write the intrinsic impedance of a medium like this [2]:

$$
\underline{\eta} = \sqrt{\frac{j\omega\mu'}{\sigma + j\omega\epsilon'}}\tag{8.8}
$$

where,

*ω* = 2*πf* = angular frequency of the signal in [rad/sec] *μ*- = *μ*- *rμ*<sup>0</sup> = permeability of the magnetic lossless medium in [H/m] *σ* = specific conductance of the magnetic lossless medium in [S/m] - = - *<sup>r</sup>* <sup>0</sup> = permittivity (dielectric constant) of the dielectric lossless medium in [F/m] *<sup>j</sup>* <sup>=</sup> √−<sup>1</sup> <sup>=</sup> imaginary unit

#### *8.4.3 Intrinsic Impedance of Lossless Insulators*

The calculation of the intrinsic impedance *η* [] can even be more simplified for good insulators and lossless media (*σ jω*- ) [2]:

$$
\eta = \sqrt{\frac{\mu'}{\epsilon'}} \tag{8.9}
$$

where,

*μ*- = *μ*- *rμ*<sup>0</sup> = permeability of the lossless media in [H/m] - = - *<sup>r</sup>* <sup>0</sup> = permittivity (dielectric constant) of the lossless media in [F/m]

#### *8.4.4 Intrinsic Impedance of Free-Space*

For free space—where *μ*- *<sup>r</sup>* = 1*.*0 and - *<sup>r</sup>* = 1*.*0—the intrinsic impedance is [6]:

$$
\eta\_0 = \sqrt{\frac{\mu\_0}{\epsilon\_0}} \approx \text{377} \,\Omega\tag{8.10}
$$

where,

*<sup>μ</sup>*<sup>0</sup> <sup>=</sup> 12.57 ·10−<sup>7</sup> H/m <sup>=</sup> permeability of vacuum, absolute permeability <sup>0</sup> <sup>=</sup> <sup>8</sup>*.*<sup>854</sup> · <sup>10</sup>−<sup>12</sup> F/m <sup>=</sup> permittivity of vacuum, absolute permittivity

#### *8.4.5 Intrinsic Impedance of Good Conductors*

For waves traveling through (not along: through, e.g., through a shield) good conductors—where *σ* 0*ω*—the intrinsic impedance is [6]:

$$\underline{\eta} = (1+j)\sqrt{\frac{\alpha\mu'}{2\sigma}} = (1+j)\frac{1}{\delta\sigma} \tag{8.11}$$

where,

*ω* = 2*πf* = angular frequency of the signal in [rad/sec] *μ*- = *μ*- *rμ*<sup>0</sup> = permeability of the conductive material in [H/m] *σ* = specific conductance of the conductive material in [S/m] *δ* = 1*/ πf μ*- *rμ*0*σ* = skin depth in [m]

# **8.5 Wave Impedance Zw**

#### *8.5.1 Wave Impedance of Any Wave*

Generally, the ratio of the *E*-field [V/m] to the *H*-field [A/m]—for any electromagnetic wave—is the *wave impedance Zw* []. Because of the vectorial character of the fields, the ratio must be defined in terms of the corresponding *x*-component *Zwx* [] and *y*-component *Zwy* [] of the *E(z)* [V/m] and *H(z)* [A/m] fields [3]:

$$\underline{Z}\_{w\boldsymbol{x}}(\boldsymbol{z}) = \frac{\left[\overrightarrow{\boldsymbol{E}}\,\,(\boldsymbol{z})\right]\_{\boldsymbol{x}}}{\left[\overrightarrow{\boldsymbol{H}}\,\,(\boldsymbol{z}) \times \overrightarrow{\boldsymbol{z}}\right]\_{\boldsymbol{x}}} = \frac{\underline{E}\_{\boldsymbol{x}}(\boldsymbol{z})}{\underline{H}\_{\boldsymbol{y}}(\boldsymbol{z})}\tag{8.12}$$

$$\underline{Z}\_{wy}(z) = \frac{\left[\overrightarrow{E}\,\,(z)\right]\_y}{\left[\overrightarrow{H}\,\,(z) \times \overrightarrow{z}\right]\_y} = -\frac{\underline{E}\_y(z)}{\underline{H}\_x(z)}\tag{8.13}$$

where,

−→*E (z)* <sup>=</sup> electric field vector traveling in *<sup>z</sup>*-direction in [V] −→*E (z) <sup>x</sup>* <sup>=</sup> *Ex (z)* <sup>=</sup> electric field *<sup>x</sup>*-component of the wave traveling in *<sup>z</sup>* direction as complex phasor in [V] −→*E (z) <sup>y</sup>* <sup>=</sup> *Ey (z)* <sup>=</sup> electric field *<sup>y</sup>*-component of the wave traveling in *<sup>z</sup>* direction as complex phasor in [V] −→*H (z)* <sup>=</sup> magnetic field vector traveling in *<sup>z</sup>*-direction in [A/m]

−→*H (z)* <sup>×</sup> −→*<sup>z</sup> <sup>x</sup>* <sup>=</sup> *Hy (z)* <sup>=</sup> magnetic field *<sup>y</sup>*-component of the wave traveling in *<sup>z</sup>* direction as complex phasor in [A/m] −→*H (z)* <sup>×</sup> −→*<sup>z</sup> <sup>y</sup>* <sup>=</sup> *Hx (z)* <sup>=</sup> magnetic field *<sup>x</sup>*-component of the wave traveling in *<sup>z</sup>* direction as complex phasor in [A/m] −→*<sup>z</sup>* <sup>=</sup> unity vector in *<sup>z</sup>*-direction

The wave impedance of the *x*-component *Zwx (z)* and *y*-component *Zwy (z)* can also be written with respect to the intrinsic impedance of the medium *η* and the amplitudes *E*0*x*<sup>+</sup> of the forward wave of the electric field *x*-component, *E*0*y*<sup>+</sup> of the forward wave of the electric field *y*-component, *E*0*x*<sup>−</sup> of the backward wave of the electric field *x*-component, and *E*0*y*<sup>−</sup> of the backward wave of the electric field *y*-component [3]:

$$\underline{Z}\_{w\boldsymbol{x}}(\boldsymbol{z}) = \frac{\underline{E}\_{\boldsymbol{x}}(\boldsymbol{z})}{\underline{H}\_{\boldsymbol{y}}(\boldsymbol{z})} = \underline{\eta} \frac{E\_{0\boldsymbol{x} + \boldsymbol{e}}e^{-\underline{\boldsymbol{\chi}}\boldsymbol{z}} + E\_{0\boldsymbol{x} - \boldsymbol{e}}e^{\underline{\boldsymbol{\chi}}\boldsymbol{z}}}{E\_{0\boldsymbol{x} + \boldsymbol{e}}e^{-\underline{\boldsymbol{\chi}}\boldsymbol{z}} - E\_{0\boldsymbol{x} - \boldsymbol{e}}e^{\underline{\boldsymbol{\chi}}\boldsymbol{z}}} \tag{8.14}$$

$$\underline{Z}\_{w\text{y}}(z) = -\frac{\underline{E}\_{\text{y}}(z)}{\underline{H}\_{\text{x}}(z)} = -\underline{\eta}\frac{E\_{0\text{y}+e}e^{-\underline{\chi}z} + E\_{0\text{y}-e}\underline{\nu}^{\underline{\chi}}}{E\_{0\text{y}+e}e^{-\underline{\chi}z} - E\_{0\text{y}-e}\underline{\nu}^{\underline{\chi}}}\tag{8.15}$$

where,


#### *8.5.2 Wave Impedance vs. Distance*

Figure 8.3 shows the wave impedance |*Zw*| [] in dependency of the distance *d* [m] from the radiation source (normalized to the near-/far-field boundary). The graph is a simplification, and it should illustrate that magnetic field antennas and electric field antennas have different wave impedances in the near-field and that the electromagnetic field in the far-field has a constant wave impedance (for free space: *Zw* = *η*<sup>0</sup> ≈ 377 Ω).

The formulas in Sect. 8.5 showed that the general calculation of the wave impedance *Zw* is complicated. However, this changes if we solely focus on the

**Fig. 8.3** Wave impedance |*Zw*| vs. distance *d* from source (antenna)

far-field: in the far-field, the wave impedance is equal to the intrinsic impedance of the material through which the wave is propagating (*Zw* = *η*).

In the near-field, the wave impedance depends primarily on the source. Electromagnetic waves are generated by two types of sources:


#### *8.5.3 Wave Impedance in the Near-Field of E-Field Antennas*

The wave impedance *Zwe* [] in the near-field of *E*-field antennas is approximately [5]:

#### 8.5 Wave Impedance *Zw* 105

$$\underline{Z}\_{we} \approx -j\frac{\underline{\eta}}{\beta d} = -j\frac{\underline{\eta}\lambda}{2\pi d} \tag{8.16}$$

where,

*Zwe* = wave impedance in the near-field of an *E*-field antenna in [] *η* = intrinsic impedance of media where wave is propagating through in [] *β* = 2*π/λ* = phase constant (or phase factor) of the sinusoidal wave in [rad/m] *λ* = wavelength of the sinusoidal wave in [m] *d* = distance from the *E*-field antenna in [m]

In case of free space, where *<sup>η</sup>* <sup>=</sup> *<sup>η</sup>*<sup>0</sup> <sup>=</sup> <sup>√</sup>*μ*0*/*<sup>0</sup> and *<sup>λ</sup>* <sup>=</sup> <sup>1</sup>*/(*√*μ*00*f )*, the wave impedance in the near-field of *E*-field antennas can be simplified to [4]:

$$|Z\_{we}| \approx \frac{1}{2\pi f \epsilon\_r' \epsilon\_0 d} = \frac{1}{2\pi f \epsilon\_0 d} \tag{8.17}$$

where,


#### *8.5.4 Wave Impedance in the Near-Field of H-Field Antennas*

The wave impedance *Zwm* [] in the near-field of *H*-field antennas is approximately [5]:

$$\underline{Z}\_{wm} \approx -j\underline{\eta}\beta d = -j\frac{\underline{\eta}2\pi d}{\lambda} \tag{8.18}$$

where,

*Zwm* = wave impedance in the near-field of a *H*-field antenna in [] *η* = intrinsic impedance of media where wave is propagating through in [] *β* = 2*π/λ* = phase constant (or phase factor) of the sinusoidal wave in [rad/m] *λ* = wavelength of the sinusoidal wave in [m] *d* = distance from the *H*-field antenna in [m]

In case of free space, where *<sup>η</sup>* <sup>=</sup> *<sup>η</sup>*<sup>0</sup> <sup>=</sup> <sup>√</sup>*μ*0*/*0) and *<sup>λ</sup>* <sup>=</sup> <sup>1</sup>*/(*√*μ*00*f )*, the wave impedance in the near-field of *H*-field antennas can be simplified to [4]:

$$|Z\_{wm}| \approx 2\pi f \,\mu\_r' \mu\_0 d = 2\pi f \,\mu\_0 d \tag{8.19}$$

where,


#### *8.5.5 Wave Impedance of Plane Waves*

For plane waves—meaning electromagnetic waves in the far-field—the wave impedance *Zw* [] is equal to the intrinsic impedance *η* [] of the medium:

$$\underline{Z}\_w = \underline{\eta} \tag{8.20}$$

In case of free space, the wave impedance is simply:

$$Z\_w = \eta\_0 = \sqrt{\frac{\mu\_0}{\epsilon\_0}} \approx 377 \,\Omega\tag{8.21}$$

where,

*<sup>μ</sup>*<sup>0</sup> <sup>=</sup> 12.57 ·10−<sup>7</sup> H/m <sup>=</sup> permeability of vacuum, absolute permeability <sup>0</sup> <sup>=</sup> <sup>8</sup>*.*<sup>854</sup> · <sup>10</sup>−<sup>12</sup> F/m <sup>=</sup> permittivity of vacuum, absolute permittivity

#### **8.6 Summary**

	- **Electrically small antennas D** *< λ/(***2***π )***.** Roughly speaking, only the reactive near-field is significant for electrically small antennas. It extends from the antenna to the distance *dreactive*−*near*−*f ield* :

$$d\_{reactive-near-field} = 0 \dots \frac{\lambda}{2\pi} \tag{8.22}$$

where,

*λ* = wavelength of the sinusoidal signal in [m]


**Table 8.1** Comparison of two criteria (Rayleigh and Maxwell) for the nearfield to far-field transition for various frequencies *f* [Hz] and maximum antenna dimensions *D* [m]

– **Electrically large antennas D** *> λ/(***2***π )***.** For electrically large antennas, all three regions are significant.

$$d\_{reactive-near-field} = 0 \dots 0.62 \sqrt{\frac{D^3}{\lambda}} \tag{8.23}$$

$$d\_{radiating-near-field} = 0.62 \sqrt{\frac{D^3}{\lambda} \dots \frac{2D^2}{\lambda}} \tag{8.24}$$

$$d\_{far-field} > \frac{2D^2}{\lambda} \tag{8.25}$$

where,

*λ* = wavelength of the sinusoidal signal in [m]

*D* = maximum linear dimension of the antenna in [m]. *D* = *l* for a wire antenna.

• **Intrinsic impedance** *η***.** The intrinsic impedance *η* depends only on the properties of the material (conductivity, permeability, permittivity), and for plane waves, the wave impedance *Zw* is equal to the intrinsic impedance.

$$
\underline{\eta} = \sqrt{\frac{\underline{\mu}}{\underline{\epsilon}}} \tag{8.26}
$$

where,


$$
\underline{Z}\_w = \underline{\eta} \tag{8.27}
$$

where,

*η* = intrinsic impedance of the material through which the electromagnetic wave is propagating in []

And for plane waves in free space, we get:

$$Z\_w = \eta\_0 = \sqrt{\frac{\mu\_0}{\epsilon\_0}} = 377 \,\Omega \tag{8.28}$$

where,

*<sup>μ</sup>*<sup>0</sup> <sup>=</sup> 12.57 ·10−<sup>7</sup> H/m <sup>=</sup> permeability of vacuum, absolute permeability <sup>0</sup> <sup>=</sup> <sup>8</sup>*.*<sup>854</sup> · <sup>10</sup>−<sup>12</sup> F/m <sup>=</sup> permittivity of vacuum, absolute permittivity

#### **References**


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# **Chapter 9 Antennas**

*Antennas do not match transmission line impedances to wave impedances. If they did, a* 377 Ω *transmission line would radiate all its power without requiring an antenna. Describing an antenna as an impedance transformator is similar to telling a child that babies are delivered by storks. Both are explanations that may satisfy a certain audience, but neither is consistent with the actual physics.*

*—* Dr. Todd Hubing

# **9.1 Antennas and EMC Testing**

An *antenna* is an interface between radio waves propagating through space and electric currents moving along metal conductors. It can act as a receiver (converting electromagnetic waves from space into voltages and currents in a conductor) or as a transmitter (converting voltages and currents into electromagnetic waves).

In EMC compliance testing, antennas have two functions:


Moreover, a distinction is made between *E*-field and *H*-field antennas.


And last but not least, antennas are divided into near-field and far-field antennas:

• **Near-field antennas.** Dedicated *near-field antennas* are called near-field probes (see Fig. 9.3). Electric and magnetic near-field probes are often used during EMC troubleshooting to identify unintentional radiators on PCBAs or unintended

**Fig. 9.1** *E*-field antenna examples by Rohde & Schwarz. From left to right: double-ridged waveguide horn antenna R&S®HF907, log-periodic antenna R&S®HL223, biconical antenna R&S®HK116E, and ultralog antenna R&S®HL562E

**Fig. 9.2** *H*-field antenna examples by Rohde & Schwarz. From left to right: active loop antenna R&S®HFH2-Z2E, triple-loop antenna R&S®HM020E

**Fig. 9.3** EMC near-field probes R&S®HZ-15 with oscilloscope R&S®RTO2000 by Rohde & Schwarz. *H*-field near-field probes have loops, *E*-field probes not

radiation through slots of enclosures. Magnetic field immunity tests are usually performed in the near-field of a loop antenna (e.g. at *f* = 50 Hz or 60 Hz).

• **Far-field antennas.** *Far-field antennas* are used for EMC compliance emission and immunity tests (see Fig. 9.1). In some cases, far-field antennas are also used for near-field radiated immunity tests (radiated fields in close proximity [6]).

#### **9.2 Isotropic Radiator**

A spherical *isotropic radiator* (isotropic antenna) radiates equally in all directions (from one single point) and therefore has no directivity. Such an antenna does only exist in theory.

Let us define *S* [W/m2] as the power density of the field around an antenna at a given distance *d* [m]. The average *power density S* [W/m2] of an isotropic radiator is simply the radiated power *Prad* [W] by the antenna divided by the surface area of the spherical surface [m2] at a distance *d* [m] from the center of the radiator [11]:

**Fig. 9.4** Signal strength of a spherical isotropic radiator vs. an antenna with directivity (gain)

where,

*Prad* <sup>=</sup> radiated power by an isotropic radiator in [m2] *d* = distance from the isotropic radiator to the spherical surface in [m]

#### **9.3 Antenna Directivity D**

The *directivity D* of an antenna is defined as the ratio of the radiation intensity in a given direction from the antenna to the radiation intensity averaged over all directions. Thus, the directivity is an antenna parameter that measures the degree to which the emitted radiation is concentrated in a single direction. Figure 9.4 shows the radiation pattern (power density) of a transmitting antenna with directivity compared to an isotropic radiator. Directivity is also defined for receiving antennas. The directivity when receiving is equal to the directivity when transmitting.

#### **9.4 Antenna Gain G**

Every real-world antenna has a directivity—meaning: it does not radiate equally in every direction. This leads us to the term *antenna gain*. The antenna gain *G* is a key performance number that combines the antenna's directivity *D* and electrical efficiency *e*<sup>0</sup> [2]:

$$G = e\_0 \cdot D \tag{9.2}$$

where,

*e*<sup>0</sup> = radiation efficiency of antenna (dimensionless)

*D* = directivity of antenna (without any losses, dimensionless)

For a loss-free antenna: *G* = *D*. The antenna gain *G* is defined as the ratio of the power radiated in the desired direction of an antenna compared to the power radiated from a reference antenna (e.g., isotropic radiator or dipole) with the identical power input (this means the antenna's electrical efficiency factor *Prad /Pt* , which considers the antenna losses, is already taken into account in gain *G*) [11]. The antenna gain with reference to an isotropic radiator *Gi* or to a *λ/*2-dipole antenna *Gd* is defined as:

$$G\_l = \frac{\text{Maximum radiation intensity of actual antenna}}{\text{Radiation intensity of isotropic antenna with same input power}} \qquad (9.3)$$

$$G\_d = \frac{\text{Maximum radiation intensity of actual antenna}}{\text{Radiation intensity of } \lambda/2 \text{-dipole antenna with same input power}} \qquad (9.4)$$

where,

*Gi* = gain of an antenna with reference to an isotropic radiator (dimensionless) *Gd* = gain of an antenna with reference to a *λ/*2-dipole antenna (dimensionless)

Antenna gains are often given in [dBi] and [dBd]:

$$G\_l \text{[dBi]} = 10 \log\_{10}(G\_l) \tag{9.5}$$

$$G\_d \text{[dBd]} = 10 \log\_{10}(G\_d) \tag{9.6}$$

The antenna gains of loss-free *λ/*2-dipoles and *λ/*4-monopoles compared to an isotropic radiator are [2]:

$$G\_{l, \lambda/2-dipole} = 1.64 \text{ loss-free antenna} \tag{9.7}$$

$$G\_{l, \lambda/4-monopole} = 3.28 \text{ loss-free antenna} \tag{9.8}$$

$$G\_{l, \lambda/2 - d \mid pole} \text{[dBi]}, \text{[dBd]} = 10 \log\_{10}(1.64) = 2.15 \,\text{dBi} = 0 \,\text{dBd} \tag{9.9}$$

$$G\_{l, \lambda/4-monopole} \text{[dBi]}, \text{[dBd]} = 10 \log\_{10}(3.28) = 5.15 \,\text{dBi} = 3 \,\text{dBd} \qquad (9.10)$$

Now, we are ready to calculate the often used terms: *effective isotropic radiated power* (EIRP) and effective radiated power (ERP) [9]:

$$\text{EIRP} = P\_l G\_l = 1.64 \cdot \text{ERP} \tag{9.11}$$

$$\text{ERP} = P\_l G\_d = \frac{\text{EIRP}}{1.64} \tag{9.12}$$

where,

**Fig. 9.5** Transmitting and receiving antennas in free-space

EIRP = RMS input power required to lossless isotropic radiator to give the identical maximum power density far from the antenna as the actual antenna in [W]

ERP = RMS input power required to lossless *λ/*2-dipole to give the identical maximum power density far from the antenna as the actual antenna in [W]

*Pt* = power input to the transmitting antenna in [W]

*Gi* = antenna gain compared to isotropic radiator (dimensionless)

*Gd* = antenna gain compared to *λ/*2-dipole (dimensionless)

Assuming line of sight and free space (Fig. 9.5), the maximum power density *Smax* [W*/*m2] at a distance *d* [m] from the transmitting antenna (in the direction of the transmitting antenna's main-lobe and in the far-field) can be written as:

$$S\_{\text{max}} = \frac{\text{EIRP}}{4\pi d^2} \tag{9.13}$$

where,

EIRP = RMS input power required to lossless isotropic radiator to give the identical maximum power density far from the antenna as the actual antenna in [W]

*d* = distance from the transmitting antenna (in the far-field) in [m]

The term *realized antenna gain Grealized* does not only include the antenna efficiency (ohmic losses, etc.) but also the impedance mismatch loss and is given as [1]:

$$G\_{realized} = G\_l \cdot \left(1 - |\underline{\varGamma}|^2\right) \tag{9.14}$$

where,

*Gi* = antenna gain, compared to a spherical isotropic radiator (dimensionless) |*Γ* | = magnitude of the complex reflection coefficient between the signal generator or amplifier and the antenna (dimensionless)

# **9.5 Effective Aperture A***<sup>e</sup>*

The effective area of an antenna is also called effective aperture *Ae* [m2]. The *effective aperture* represents the ratio of power *Pr* [W] (output power of the receiving antenna) to the power density *Sr* [W*/*m2] of the electromagnetic wave at the receiving antenna [2]:

$$A\_e = \frac{P\_r}{S\_r} \tag{9.15}$$

where,

*Pr* = RMS output power at the receiving antenna terminals in [W] *Sr* <sup>=</sup> power density *Sr* at the receiving antenna in [W*/*m2]

The *maximum effective aperture Aem* [m2] for any antenna is [2]:

$$A\_{em} = \frac{\lambda^2}{4\pi} G\_l \tag{9.16}$$

where,

*λ* = wavelength of the sinusoidal incident wave—which must be matched with the receiving antenna's polarization—in [m]

*Gi* = antenna gain of the receiving antenna (dimensionless)

The antenna gain *Gi* can be expressed as a function of the maximum effective aperture *Aem* [m2] and the wavelength *λ* [m] of the incident wave:

$$G\_l = \frac{4\pi A\_{em}}{\lambda^2} \tag{9.17}$$

#### **9.6 Antenna Factor AF**

The *antenna factor* AF is a function of the signal's frequency *f* [Hz] and is used to calculate the received field strength *E* [V/m] or *H* [A/m] based on the measured voltage *Vr* [V] at the receiving antenna's terminals [11]:

$$\text{AF}\_E \text{ [1/m]} = \frac{E}{V\_r} \tag{9.18}$$

$$\text{AF}\_{H} \text{ [S/m]} = \frac{H}{V\_{r}} \tag{9.19}$$

where,

**Fig. 9.6** Simplified equivalent circuit of a receiving antenna with open load circuit voltage *V* <sup>0</sup> [V], antenna impedance *Za* [], receiver impedance *Zr* [], and voltage at the receiving antenna terminals *V <sup>r</sup>* [V]

*E* = RMS value of the electric field strength at the receiving antenna in [V/m] *H* = RMS value of the magnetic field strength at the receiving antenna in [S/m] *Vr* = RMS value of the voltage at the receiving antenna's terminals in [V]

In case of matched impedances at the receiving antenna's terminals (Fig. 9.6), free space, a plane wave, and aligned receiving antenna to the polarization of the incident wave, we can write:

$$P\_r = S\_r \cdot A\_{em} \tag{9.20}$$

$$\frac{V\_r^2}{\text{Re}(\underline{Z}\_r)} = \frac{E^2}{\eta} \cdot \frac{\lambda^2}{4\pi} G\_{lr} \tag{9.21}$$

In case of free space and a plane wave, we can set *λ* = *c/f* and *η* = *η*<sup>0</sup> = 120*π* Ω in Eq. 9.21 and get:

$$\text{AF}\_E^2 = \frac{E^2}{V\_r^2} = \frac{\eta}{\text{Re}(\underline{Z\_r})} \cdot \frac{4\pi}{\lambda^2} \cdot \frac{1}{G\_{ir}} \tag{9.22}$$

$$\text{AF}\_{E} = \frac{E}{V\_{r}} = \sqrt{\frac{480\pi^{2}}{c^{2}}} \cdot \frac{f}{\sqrt{\text{Re}(\underline{\mathbf{Z}}\_{r})G\_{lr}}} \tag{9.23}$$

where,


*λ* = wavelength of the sinusoidal incident plane wave in [m] *Gir* = antenna gain of the receiving antenna (dimensionless)

In case of *Zr* = 50 Ω, we can write:

$$\text{AF}\_E \text{ [1/m]} = \frac{9.73}{\lambda \sqrt{G\_{lr}}} \tag{9.24}$$

where,

AF*<sup>E</sup>* = antenna factor of an *E*-field antenna in case of a plane wave (far-field) in free-space, a 50 Ω-system and matched polarization of the incident *E*-field and the antenna in [1/m]

*λ* = wavelength of the sinusoidal incident plane wave in [m]

*Gir* = antenna gain of the receiving antenna (dimensionless)

The antenna factor AF is often given in decibel:

$$\text{AF}\_E\text{ [dB(1/m)]} = 20\log\_{10}\left(\text{AF}\_E\text{ [1/m]}\right) \tag{9.25}$$

$$\text{AF}\_{H}\text{ [dB(S/m)]} = 20\log\_{10}\left(\text{AF}\_{H}\text{ [S/m]}\right) \tag{9.26}$$

#### **9.7 Ground Plane Reflection and Multipath Effect**

Assuming a transmitting and a receiving antenna placed above a reflective ground plane, where the receiving antenna and the ground plane are in the *far-field* of the transmitting antenna (see Fig. 9.7). The received field at the receiver antenna is the sum of the direct wave −→*<sup>E</sup> <sup>d</sup>* [V/m] and the reflected wave −→*<sup>E</sup> <sup>r</sup>* [V/m]:

$$
\overrightarrow{E}\_{sum} = \overrightarrow{E}\_d + \overrightarrow{E}\_r \tag{9.27}
$$

**Fig. 9.7** Ground plane and receiving antenna are in the far-field of the transmitter

where,

−→*<sup>E</sup> sum* <sup>=</sup> *<sup>E</sup>*-field sum of direct and reflected waves at receiving antenna in [V/m] −→*<sup>E</sup> <sup>d</sup>* <sup>=</sup> *<sup>E</sup>*-field at the receiving antenna from direct path in [V/m] −→*<sup>E</sup> <sup>r</sup>* <sup>=</sup> *<sup>E</sup>*-field at the receiving antenna from reflected path in [V/m]

This phenomena is called *multipath effect*. The field strength |*Esum*| [V/m] at the receiving antenna depends on the distances the waves have traveled, the polarization of the antennas, the reflection coefficient *Γ* of the ground plane, the attenuation *α* [1/m] of the media, and the frequency *f* [Hz] of the signal. Assuming that the ground plane and the receiving antenna are in the far-field of the transmitting antenna (so that *Rrad* is not influenced by the ground plane), a perfectly conducting ground plane (*Γ* = +1 or −1 [2]) and a lossless media (*α* = 0), the reflective field strength |*Er*| [V/m] could be as high as the field strength of the direct wave |*Ed* | [V/m]. However, it could also happen that both fields cancel each other out at the receiving antenna. This means:

$$|E\_{sum,max}| \approx \mathcal{Z}|E\_d|\tag{9.28}$$

$$|E\_{sum,min}| \approx 0\,\text{V/m}\tag{9.29}$$

#### **9.8 Intended Antennas**

In EMC, the term *intended antenna* denotes an antenna for generating or receiving a specific electric, magnetic, or electromagnetic field in a predefined setup.

#### *9.8.1 E-Field Antennas and Emission Testing*

*Radiated emission* EMC standards like CISPR 11 [8], CISPR 32 [3], or FCC Part 15 [4] specify the radiated emission limits in [µV/m] or [dBµV/m]. Therefore, *Efield antennas* are used as receiving antennas (Fig. 9.8). In other words, the physical quantity of interest is the *E*-field [dBµV/m], and the measured physical quantity is the voltage *Vmeasure* [dBµV] at the measurement equipment (e.g., spectrum analyzer or EMI receiver). The field strength of the *E*-field at the antenna can be calculated like this:

$$E = \mathbf{A}\mathbf{F}\_E + \text{losses} + V\_{measure} \tag{9.30}$$

where,

*E* = RMS field strength of the *E*-field at the antenna in [dBµV/m] AF*<sup>E</sup>* = antenna factor of an *E*-field antenna in [dB(1/m)]

**Fig. 9.8** Example of an EMC emission and immunity testing *E*-field antenna. (**a**) Wideband biconilog antenna R&S®HL562E and frequency range from 30 MHz to 6 GHz. (**b**) Antenna factor of the wideband biconilog antenna R&S®HL562E by Rohde & Schwarz

Losses = losses of all cables and connectors from the receiver antenna to the measurement equipment (e.g., spectrum analyzer) in [dB]

*Vmeasure* = measured RMS voltage at the measurement equipment (e.g., spectrum analyzer) in [dBµV]

#### *9.8.2 E-Field Antennas and Immunity Testing*

The physical quantity of interest for many *radiated immunity* EMC standards (like IEC 61000-4-3 [5]) is the *E*-field at a certain distance *d* (e.g., *E* = 10 V/m at *d* = 10 m). Therefore, *E-field antennas* like bilogical, log-periodic, or horn antennas are used—basically the identical antennas as for *E*-field emission testing.

From the antenna fundamentals from above, we know how to calculate the maximum power density *Smax* [W/m2] at a distance *d* [m] for a transmitting antenna in free space with antenna gain *Git* and power *Pt* [W] at the transmitting antenna terminals:

$$S\_{\text{max}} = \frac{\text{EIRP}}{4\pi d^2} = \frac{P\_l G\_{ll}}{4\pi d^2} \tag{9.31}$$

where,


Furthermore, we know the power density *S* [W/m2] of a plane wave in free space (far-field: wave impedance *Zw* [] is equal characteristic impedance *η*<sup>0</sup> = 120*π* Ω) is given as [11]:

$$S = \frac{E^2}{\eta\_0} = \frac{E^2}{120\pi \,\Omega} \tag{9.32}$$

where,

*E* = RMS value of the electric field strength in [V/m] *η*<sup>0</sup> = 120*π* Ω = intrinsic impedance of free-space in []

If we combine Eqs. 9.31 and 9.32, we can determine the field strength *E* [V/m] at a given distance *d* [m] from the antenna for a given transmitting antenna input power *Pt* [W]:

$$E = \frac{\sqrt{30P\_lG\_{li}}}{d} = \frac{\sqrt{30P\_l10^{\frac{G\_{li}\left[\text{dBi}\right]}{10}}}}{d} \tag{9.33}$$

where,


Thus, the required input power *Pt* [W] of a transmitting antenna for achieving a desired field strength *E* [V/m] at a given distance *d* [m] is (in free space, no reflective ground plane):

$$P\_l = \frac{(E \cdot d)^2}{30 \cdot G\_{li}} = \frac{(E \cdot d)^2}{30 \cdot 10^{\frac{G\_{li} \cdot \text{[dBl]}}{10}}} \tag{9.34}$$

where,


#### *9.8.3 H-Field Antennas and Emission Testing*

Magnetic *radiated emission* EMC standards like MIL-STD-461 (RE101, *f* = 30 Hz to 100 kHz) require *H-field loop antennas* as receiving antennas (Fig. 9.9). In case the antenna factor is given in [dB(S/m)], the field strength of the *H*-field at the antenna can be calculated like this:

$$H = \text{AF}\_{H} + \text{losses} + V\_{measure} \tag{9.35}$$

where,

*H* = RMS field strength of the *H*-field at the antenna in [dBµA/m]

AF*<sup>H</sup>* = antenna factor of an *H*-field antenna in [dB(S/m)]


In case the antenna factor for the receiving loop antenna is given in [dBpT/µV], the field strength of the *H*-field at the antenna can be calculated like this:

$$H = (\text{AF}\_H + \text{losses}) \cdot V\_{measure} \tag{9.36}$$

where,

*H* = RMS field strength of the *H*-field at the antenna in [dBpT]

AF*<sup>H</sup>* = antenna factor of an *H*-field antenna in [dBpT/µV]


**Fig. 9.9** Example of an emission testing *H*-field antenna. (**a**) A.H. Systems Inc. SA-560 passive loop antenna. (**b**) Coil factor of the passive loop antenna SAS-560 by A.H. Systems. The passive loop antenna SAS-560 was specifically designed for tests according to MIL-STD-461 RE101

#### *9.8.4 H-Field Antennas and Immunity Testing*

Radiated magnetic field EMC immunity standards, like IEC 61000-4-8 [7] (*f* = 50 Hz and 60 Hz) or MIL-STD-461 (RS101, *f* = 30 Hz to 100 kHz) require *induction coils* as transmitting antennas. In contrast to the *E*-field EMC tests where the identical antennas are used for immunity and emission tests—the *H*-field immunity tests specify dedicated antennas, which are different from the *H*-field emission measurements.

Two important properties of an induction coil for *H*-field immunity tests are:


**Fig. 9.10** *H*-field homogeneity of a 1×1 m square loop antenna [7]. (**a**) 3 dB—area in the plane to the loop antenna. (**b**) 3 dB—area orthogonal to the loop antenna

generated by an induction coil of given dimensions and the corresponding current *I* [A] value. The field is measured at the center of the coil plane without the EUT.

The manufacturer of an induction coil for magnetic immunity tests specifies in the datasheet of the coil antenna:


The magnetic field strength *H* [A/m] at the position of interest (relative to the induction coil) can be calculated like this:

$$H = c\_f \cdot I \tag{9.37}$$

where,

*H* = RMS value of the magnetic field in the area of interest in [A/m]


### **9.9 Unintended Antennas**

Current loops, PCB structures, cables, or slots may act as *unintended antennas*. As a result, they receive or emit electromagnetic radiation unintentionally.

#### *9.9.1 E-Field from Differential-Mode Currents in Small Loops*

Let us assume a setup like shown in Fig. 9.11 where the following is given [11]:


Given the points above, the maximum electric field *EDM,max* [V/m] caused by a differential-mode current *I DM* [A] can be approximated as [11]:

$$E\_{DM,max} \approx \begin{cases} 1.32 \cdot 10^{-14} \frac{I\_{DM} f^2 A}{d} & \text{without reflecting ground plane} \\ 2.64 \cdot 10^{-14} \frac{I\_{DM} f^2 A}{d} & \text{with reflecting ground plane (see Fig. 9.7)} \end{cases} \tag{9.38}$$

where,

*EDM,max* = maximum RMS field strength radiated by electrically small current loop with area *A* and differential current *IDM* in [V/m]

*IDM* = RMS value of the differential-mode current in [A] *f* = frequency of the sinusoidal current signal in [Hz] *<sup>A</sup>* <sup>=</sup> area of the current loop in [m2]

*d* = distance to the center of the current loop in [m]

**Fig. 9.11** Radiated emissions *EDM* [V/m] due to differential-mode current *IDM* [A] around a current loop with area *A* [m2]. The direction of the electric field *EDM* [V/m] is parallel to the wires

Reducing unintended radiation emissions due to a small differential-mode current loop gives the following options:


Differential-mode currents do not radiate efficiently, and in practice, radiated emissions due to differential-mode currents are seldom an issue. Usually, the common-mode currents along cables are the root cause for high radiated emissions.

### *9.9.2 E-Field from Common-Mode Currents in Short Cables*

Let us assume a setup like shown in Fig. 9.12 where the following is given [11]:


Given the points above, the maximum electric field *ECM,max* [V/m] caused by the common-mode current *ICM* [A] can be approximated as [11]:

$$E\_{CM,\max} \approx \begin{cases} 0.63 \cdot 10^{-6} \frac{l\_{CM}fl}{d} & \text{without reflecting ground plane} \\ 1.26 \cdot 10^{-6} \frac{l\_{CM}fl}{d} & \text{with reflecting ground plane (see Fig. 9.7)} \end{cases} \tag{9.39}$$

where,

*ECM,max* = maximum RMS field-strength radiated by electrically short cable of length *l* and common-mode current *ICM* in [V/m]

*ICM* = RMS value of the common-mode current in [A] *f* = frequency of the sinusoidal current signal in [Hz] *l* = length of the conductor or transmission line in [m] *d* = distance to the center of the cable in [m]

Because common-mode currents are normally not required for system operation, the first rule is to avoid them. Once the common-mode currents are here: reduce them. One of the most common options to reduce common-mode currents is adding a clamp ferrite over the cable or wires, which are causing unintended radiated emissions. More information about cable mount ferrite beads can be found in Sect. 11.5.1 on page 169.

#### *9.9.3 Maximum E-Field from Common-Mode Signals*

Section 9.9.2 does only consider short cables (antennas) compared to the wavelength *λ* [m]. Therefore, these cables are not very effective radiators, which does not mean that they will never lead to failed EMC emission tests. Nonetheless, things get even worse when cable lengths *l* [m] match a multiple of a quarter wavelength *nλ/*4 [m]. This section presents simple approximations for the worst-case emissions from given common-mode voltages and currents.

#### **9.9.3.1 Generic Calculation of Maximum E-Field**

Let us assume a sinusoidal signal with a RMS voltage *VRMS* [V] that drives a resonant structure like a *λ/*4-monopole antenna. As a result, the current distribution along the antenna has a RMS value of *IRMS* [A]. Assuming a lossless antenna (*Rloss* = 0 Ω), where the antenna input impedance *Zin* [] is:

$$\underline{Z}\_{ln} = R\_{ln} + jX\_{ln} = R\_{loss} + R\_{rad} + jX\_{ln} = R\_{rad} + jX\_{ln},\tag{9.40}$$

we can calculate the radiated power *Prad* [W] of this lossless antenna as [11]:

$$P\_{rad} = \frac{V\_{RMS}^2}{R\_{rad}} = I\_{RMS}^2 R\_{rad} \tag{9.41}$$

Combining Eqs. 9.34 and 9.41 for a lossless transmitter (where directivity *D* is equal the antenna gain *Gi*), we get:

$$P\_{rad} = \frac{V\_{RMS}^2}{R\_{rad}} = I\_{RMS}^2 R\_{rad} = \frac{E^2}{\eta\_0} \frac{4\pi d^2}{G\_l} \tag{9.42}$$

This leads us to the maximum radiated field-strength *Emax* [V/m] at the resonance of a lossless antenna with matched impedance, matched polarization, and line-of-sight path:

$$E\_{max} \approx \sqrt{\frac{V\_{RMS}^2 \cdot \eta\_0 \cdot G\_l}{R\_{rad} \cdot 4\pi d^2}} = \sqrt{\frac{I\_{RMS}^2 \cdot R\_{rad} \cdot \eta\_0 \cdot G\_l}{4\pi d^2}}\tag{9.43}$$

where,

*Emax* = maximum *E*-field at distance *d* for matched impedances, matched polarization, line-of-sight path, and a lossless antenna driven by *VRMS* in [V/m] *VRMS* = RMS voltage that drives the lossless antenna in [V] *IRMS* = RMS value of the current distribution along the lossless antenna in [A] *Rrad* = radiation resistance of the lossless antenna (function of frequency) in [] *η*<sup>0</sup> = 120*π* Ω = intrinsic impedance of free space in [] *Gi* = *D* = antenna gain of the lossless antenna (dimensionless) *d* = distance from the antenna, where *Emax* is measured in [m]

#### **9.9.3.2 Maximum E-Field for a Given Common-Mode Current**

Assuming a common-mode current *ICM* [A] flows along a resonant structure (e.g., a cable). When in resonance (and only then!), the maximum electric field-strength *Emax* [V/m] can be approximated based on Eq. 9.43 as:

$$E\_{\max} \approx \sqrt{\frac{R\_{rad} \cdot \eta\_0 \cdot G\_l}{4\pi}} \frac{I\_{CM}}{d} \tag{9.44}$$

$$E\_{\max} \approx \begin{cases} 60 \frac{I\_{CM}}{d} & \text{without reflecting ground plane} \\ 120 \frac{I\_{CM}}{d} & \text{with reflecting ground plane (see Fig. 9.7)} \end{cases} \tag{9.45}$$

where,

*Emax* = maximum RMS *E*-field at distance *d* for matched impedances, matched polarization, line-of-sight path, and a resonant structure driven by *ICM* in [V/m] *ICM* = RMS common-mode current that flows along the resonant structure in [A] *Rrad* = radiation resistance in [] (*λ/*2-dipole: 73 Ω, *λ/*4-monopole: 36*.*5 Ω) *η*<sup>0</sup> = 120*π* Ω = intrinsic impedance of free-space in [] *Gi* = antenna gain (*λ/*2-dipole, 1.64; *λ/*4-monopole, 3.28) *d* = distance from the antenna, where *Emax* is measured in [m]

#### **9.9.3.3 Maximum E-Field for a Given Common-Mode Voltage**

Assuming a common-mode voltage *V CM* [V] that drives a resonant structure (e.g., a cable). When in resonance (and only then!), the maximum electric field-strength *Emax* [V/m] can be approximated based on Eq. 9.43 as:

$$E\_{\max} \approx \sqrt{\frac{\eta\_0 \cdot G\_i}{R\_{rad} \cdot 4\pi}} \frac{V\_{CM}}{d} \tag{9.46}$$

$$E\_{\max} \approx \begin{cases} 0.8 \frac{V\_{CM}}{d} & \lambda/2 \text{-dipole without reflecting ground plane} \\ 1.6 \frac{V\_{CM}}{d} & \lambda/2 \text{-dipole with reflecting ground plane (see Fig. 9.7),} \\ 1.6 \frac{V\_{CM}}{d} & \lambda/4 \text{-monopole without reflecting ground plane} \\ 3.2 \frac{V\_{CM}}{d} & \lambda/4 \text{-monopole with reflecting ground plane (see Fig. 9.7)} \end{cases} \tag{9.47}$$

where,

*Emax* = maximum RMS *E*-field at distance *d* for matched impedances, matched polarization, line-of-sight path, and a resonant structure driven by *VCM* in [V/m] *VCM* = RMS common-mode voltage that drives the resonant structure in [V] *Rrad* = radiation resistance in [] (*λ/*2-dipole, 73 Ω; *λ/*4-monopole, 36*.*5 Ω) *η*<sup>0</sup> = 120*π* Ω = intrinsic impedance of free space in [] *Gi* = antenna gain (*λ/*2-dipole, 1.64; *λ/*4-monopole, 3.28) *d* = distance from the antenna, where *Emax* is measured in [m]

**Fig. 9.13** Unintended antenna structure: cable connected to a PCBA on one end and attached to an infinite ground plane at the other end. An antenna-mode (common-mode) voltage *V AM* [V] drives an antenna-mode current *I AM* [A] through a cable

### *9.9.4 Maximum E-Field from PCB with Attached Cable*

Let us assume a setup like shown in Fig. 9.13 where the following is given:


The maximum RMS value of the electric field *EAM,max* [V/m] measured in a anechoic chamber (with a reflecting ground plane) caused by an antenna-mode signal with *V AM* [V] and *I AM* [A] that drives the board-cable structure as an antenna can be approximated like this [12–14]:

$$E\_{AM,max} \approx \begin{cases} \frac{60 \cdot I\_{AM,max}}{d} \cdot \frac{2}{\sin\left(\sqrt{2}\right)} & \text{when } I\_{cable} \le \frac{\lambda}{2} \\\\ \frac{60 \cdot I\_{AM,max}}{d} \cdot \frac{2}{\sin\left(\sqrt{\frac{\lambda}{I\_{cable}}}\right)} & \text{when } I\_{cable} > \frac{\lambda}{2} \end{cases} \tag{9.48}$$

where,

$$I\_{AM,max} = \frac{V\_{AM}}{\frac{R\_{min}}{k\_{board}k\_{cable}}}\tag{9.49}$$

and

$$k\_{board} = \begin{cases} \sin\left(\frac{2\pi \cdot l\_{board}}{\lambda}\right) & \text{when } l\_{board} \le \frac{\lambda}{4} \\ 1.0 & \text{otherwise} \end{cases} \tag{9.50}$$

$$k\_{cable} = \begin{cases} \sin\left(\frac{2\pi \cdot l\_{cable}}{\lambda}\right) & \text{when } l\_{cable} \le \frac{\lambda}{4} \\ 1.0 & \text{otherwise} \end{cases} \tag{9.51}$$

$$l\_{board} = \frac{1 + \frac{2L}{W}}{\frac{2L}{W}} \cdot \sqrt{L^2 + W^2} \tag{9.52}$$

*VAM* = antenna-mode RMS voltage driving the monopole antenna (cable) in [V] *IAM,max* = highest antenna-mode RMS current that exists on the cable in [A] *Rmin* = 36*.*5 Ω = radiation resistance of a resonant *λ/*4-monopole antenna in [] *lboard* = effective board length in [m] *lcable* = cable length in [m] *L* = length of the PCBA (board) in [m] *W* = width of the PCBA (board) in [m] *kboard* = impact of board size *lboard* on *Rmin* in case *lboard* ≤ *λ/*4 *kcable* = impact of cable length *lcable* on *Rmin* in case *lcable* ≤ *λ/*4 *d* = distance to the radiating structure (PCBA, cable) in [m] *f* = frequency of the sinusoidal signal *V AM* in [Hz] *λ* = wavelength of the sinusoidal signal *V AM* in [m] *<sup>c</sup>* <sup>=</sup> <sup>2</sup>*.*<sup>998</sup> · 108 m/sec <sup>=</sup>speed of light

#### **9.10 Free-Space Path Loss**

The term free-space loss, free-space path loss factor, or *free-space path loss* (FSPL) refers to the attenuation of the electromagnetic field between a transmitting and a receiving antenna in case the space between the antennas is free of obstacles and a line-of-sight path through free-space.

A way to calculate the FSPL is given by the *Friis transmission equation*, which assumes matched impedances, matched polarization of the transmit, and receive antennas and that the receiving antenna is in the far-field [11]:

$$\frac{P\_r}{P\_l} = G\_{lt} G\_{lr} \left(\frac{\lambda}{4\pi d}\right)^2\tag{9.53}$$

$$\text{FSPL} = \left(\frac{4\pi d}{\lambda}\right)^2\tag{9.54}$$

$$\text{FFPL [dB]} = 20 \log\_{10} \left( \frac{4 \pi d}{\lambda} \right) = 20 \log\_{10} \left( \frac{4 \pi df}{c} \right) \tag{9.55}$$

where,

*Pr* = output power at the receiving antennas terminals in [W] *Pt* = input power to the transmitting antenna in [W] *Gir* = antenna gain of the transmitting antenna (dimensionless) *Gir* = antenna gain of the receiving antenna (dimensionless) *λ* = wavelength of the sinusoidal electromagnetic wave in [m] *d* = distance between the transmitting and receiving antennas in [m] *f* = frequency of the sinusoidal electromagnetic wave in [Hz] *<sup>c</sup>* <sup>=</sup> <sup>1</sup>*/(*√*μ*00*)* <sup>=</sup> <sup>2</sup>*.*<sup>998</sup> · 108 m/sec <sup>=</sup> speed of light

# **9.11 Link Budget**

The *link budget* is an accounting of all of the power gains and losses that a signal experiences along a transmission path (from transmitter to receiver). If you want to calculate the received power *Pr* [dBm] for a given transmitter-receiver setup (transmitter, frequency, antennas, distance, etc.), you calculate the link budget. In simple terms, this means:

Received power *Pr* [dBm] = Transmitted power *Pt* [dBm] + Gains [dB] - Losses [dB]

If we rewrite the link budget formula above in a little more detailed way, we get:

$$P\_r = P\_l - L\_l + G\_{lt} - \text{FSPL} - L\_{misc} + G\_{lr} - L\_r \tag{9.56}$$

where,


*Lt* = transmitter losses (coaxial cable, connectors, etc.) in [dB]

FPSL = free-space path loss in [dB]


# **9.12 Summary**

• **Antennas Types and EMC** There are dedicated antennas for the near-field and the far-fields and for the *E*- and *H*-fields (Table 9.1).




# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 10 Skin Effect**

*Nothing in life is to be feared, it is only to be understood. Now is the time to understand more, so that we may fear less.*

*—*Marie Curie

# **10.1 Skin Depth**

Introduction and definitions of the term skin depth *δ* [m] (Fig. 10.1):

• **Conductors.** The skin depth *δ* [m] is defined as the distance from the conductor surface where the current density has fallen (caused by the skin effect) to 37% <sup>=</sup> 1/e <sup>=</sup> 1/2.72 of the current density at the surface of the conductor *<sup>J</sup>*<sup>0</sup> [A/m2]. The current density *Jd* [A/m2] at distance *d* [m] from the conductor surface is defined as [2]:

$$|J\_d| = |J\_0| \cdot e^{-\frac{d}{\delta}} \tag{10.1}$$

where:


#### **Fig. 10.1** Skin depth *δ*

$$|E\_d| = \text{Re}(|E\_0| \cdot e^{-\frac{\gamma d}{2}}) = \text{Re}(|E\_0| \cdot e^{-\alpha d} \cdot e^{-j\beta d}) = |E\_0| \cdot e^{-\alpha d} = |E\_0| \cdot e^{-\frac{d}{\delta}} \tag{10.2}$$

$$|H\_d| = \text{Re}(|H\_0| \cdot e^{-\frac{\gamma}{2}d}) = \text{Re}(|H\_0| \cdot e^{-\alpha d} \cdot e^{-j\beta d}) = |H\_0| \cdot e^{-\alpha d} = |H\_0| \cdot e^{-\frac{d}{\delta}} \tag{10.3}$$

where:


From Eqs. 10.2 and 10.3, we know that the skin depth *δ* [m] is defined as the inverse of the attenuation constant *α* [1/m] [1]:

$$\delta = \frac{1}{\alpha} = \frac{1}{\alpha \sqrt{\frac{\left(\epsilon' \mu' - \epsilon'' \mu'' \right)}{2} \cdot \left(\sqrt{1 + \left(\frac{\epsilon' \mu'' + \epsilon'' \mu'}{\epsilon' \mu' - \epsilon'' \mu''}\right)^2} - 1\right)}} \tag{10.4}$$

where:

*ω* = 2*πf* = angular frequency of the signal in [rad/sec] - = real part of the complex permittivity ( = - − *j* --) in [F/m] -- = imaginary part of the complex permittivity ( = - − *j* --) in [F/m] *μ*- = real part of the complex permeability (*μ* = *μ*- − *jμ*--) in [H/m] *μ*-- = imaginary part of the complex permeability (*μ* = *μ*- − *jμ*--) in [H/m]

**Fig. 10.2** Attenuation of current density *J* [A/m2], electric field strength *E* [V/m], and magnetic field strength *H* [A/m] due to skin effect

For good conductors (with *σ ω* and - *<sup>r</sup>* = 1*.*0) with negligible magnetic losses (*μ*--= 0), we can write [2]:

$$\delta = \sqrt{\frac{2}{\omega \mu\_r' \mu\_0 \sigma}} = \sqrt{\frac{2}{\omega \mu\_r' \mu\_0 \sigma}} = \frac{1}{\sqrt{\pi f \mu\_r' \mu\_0 \sigma}} \tag{10.5}$$

where:

*δ* = skin depth in [m] *ω* = 2*πf* = angular frequency of the signal in [rad/sec] *μ*- *<sup>r</sup>* = relative permeability of the material through which the signal current is flowing or through which the electromagnetic wave is traveling in [1] *<sup>μ</sup>*<sup>0</sup> <sup>=</sup> <sup>4</sup>*<sup>π</sup>* · <sup>10</sup>−<sup>1</sup> H/m <sup>=</sup> 12.57 ·10−<sup>7</sup> H/m <sup>=</sup> permeability of vacuum

*σ* = specific conductance of the material through which the signal current is flowing or through which the electromagnetic wave is traveling in [S/m]

Figure 10.3 presents some example values of skin depths for silver, copper, gold, aluminum, nickel, iron, and stainless steel 316. The following points have to be considered when calculating the skin depth for ferromagnetic metals (like nickel, iron):


**Fig. 10.3** Skin depth of some example metals. Permeability *μ*- *<sup>r</sup>* is assumed to be constant. (**a**) Table with the assumed electrical conductivity *σ* and permeability *μ*- *<sup>r</sup>*. (**b**) Skin depth *δ* as a function of frequency *f*

#### **10.2 DC vs. AC Resistance**

The resistance per-unit-length [/m] for direct current (*R*- *DC*) and for alternating current (*R*- *AC*) of any conductor can be written as [2]:

$$R\_{DC}' = \frac{\rho}{A} = \frac{1}{\sigma A} \tag{10.6}$$

$$R'\_{AC} = \frac{\rho}{A\_{eff}} = \frac{1}{\sigma A\_{eff}}\tag{10.7}$$

where:

*ρ* = specific electrical resistivity of the conductor material in [m]

*σ* = specific conductance of the conductor material in [S/m]

*<sup>A</sup>* <sup>=</sup> cross-sectional area of the conductor in [m2]

*Aeff* = effective cross-sectional area of the conductor through which the current effectively flows in [m2]

For direct current (DC, 0 Hz), the cross-sectional area *Aeff* [m2] through which the DC current flows is equal to the conductor cross-sectional area *A* [m2]. However, for high-frequency AC current with frequency *f* [Hz], the magnetic field—produced by current in the conductor—forces the current flow toward the outer surface of the conductor, and as a consequence of that, the current density increases exponentially from the core of the conductor toward the conductor's surface. The higher the signal frequency *f* [Hz], the smaller the cross section *Aeff* [m2] through which the current effectively flows.

The accurate calculation of *Aeff* [m2] is difficult but can be reasonably approximated by assuming that the current is uniformly distributed over the skin depth *<sup>δ</sup>* [m]. In case of *<sup>D</sup> <sup>δ</sup>*, the approximate *Aeff W ire* [m2] of a round conductor (circular wire) with diameter *D* [m] can be calculated as follows:

$$A\_{eff\,Wire} \approx A - A\_{no\,Curr} = \frac{D^2}{4}\pi - \frac{(D - 2\delta)^2}{4}\pi = \left(D\delta - \delta^2\right)\pi \qquad (10.8)$$

where:

*<sup>A</sup>* <sup>=</sup> cross-sectional area of the round conductor with diameter *<sup>D</sup>* in [m2] *AnoCurr* = the approximate cross-sectional area of the conductor where no current

flows (as a consequence of the skin effect) in [m2]

*δ* = skin depth in [m]

*D* = diameter of round conductor in [m]

For a PCB trace according to Fig. 10.4, the effective area *Aeff P CBtrace* [m2] can be approximately calculated like this:

**Fig. 10.5** Increased resistance due to skin effect: PCB copper trace (width *w* = 0*.*25 mm, height *h* = 35 µm = 1 oz) vs. round copper wire (diameter *D* = 1 mm)

$$A\_{effPCBtrace} \approx A - A\_{noCurr} = wh - (w - 2\delta)\left(h - 2\delta\right) = 2\delta\left(w + h\right) - 4\delta^2 \tag{10.9}$$

where:

*<sup>A</sup>* <sup>=</sup> cross-sectional area of the PCB trace with width *<sup>w</sup>* and height *<sup>h</sup>* in [m2] *AnoCurr* = the approximate cross-sectional area of the conductor where no current

flows (as a consequence of the skin effect) in [m2]

*δ* = skin depth in [m] *w* = width of the PCB trace in [m] *h* = height or thickness of the PCB trace in [m]

Figure 10.5 shows *R*- *AC* [/m] of a PCB trace versus a round copper wire. The calculations for *R*- *AC* [/m] are approximations and ignore the return current path (proximity effect) and assume a single conductor surrounded by air only. However, the diagram gives an idea of how the skin effect influences the resistance at higher frequencies.

#### **10.3 Surface Resistance**

The *surface resistance* (or *sheet resistance*) *Rs* [/square] for a thin layer with height *h* [m]—like shown in Fig. 10.6—is defined as [4]:

$$R\_s = \frac{1}{\sigma h} \tag{10.10}$$

where:

*σ* = specific conductance of the conductor material in [S/m]

**Fig. 10.6** Sheet (thin layer) of height of *h*, width *w*, and length *l*. Current flow is parallel to length *l* (not perpendicular to the sheet)

*h* = height or thickness of a thin conductor layer [m]

For a good conductor (*σ ω*- ), with negligible magnetic losses *μ*-- = 0 and at high frequency (skin depth *δ* much smaller than conductor cross-sectional dimensions), the thickness of the conductor sheet *h* [m] can be set to the skin depth *δ* [m], and the surface resistance *Rs* [/square] is defined as [3]:

$$R\_s = \frac{1}{\sigma \delta} = \sqrt{\frac{a\mu'}{2\sigma}}\tag{10.11}$$

where:

*σ* = specific conductance of the conductor material in [S/m] *δ* = skin depth in [m] *ω* = 2*πf* = angular frequency of the signal in [rad/sec] *μ*- = *μ*- *rμ*<sup>0</sup> = ability to store energy in a medium when an external magnetic field is applied = real part of the complex permeability (*μ* = *μ*- − *jμ*--) in [H/m]

At high frequencies, where the current flows at the surface of the conductor (skin effect), the resistance of a conductor with perimeter *p* [m] and length *l* [m] can be approximated as:

$$R \approx R\_s \frac{l}{p} = \frac{l}{\sigma \delta p} \tag{10.12}$$

where:

*Rs* = surface resistance of the conductor material in [/square] *l* = length of the conductor in [m] *p* = perimeter of the conductor in [m] *δ* = skin depth in [m]

#### **10.4 Summary**

• **Skin Depth** *δ* **of Good Conductor.**

$$\delta = \frac{1}{\sqrt{\pi f \mu\_r' \mu\_0 \sigma}}\tag{10.13}$$

where:

*μ*- *<sup>r</sup>* = relative permeability of the conductor material in [1] *<sup>μ</sup>*<sup>0</sup> <sup>=</sup> <sup>4</sup>*<sup>π</sup>* · <sup>10</sup>−<sup>1</sup> H/m <sup>=</sup> 12.57 ·10−<sup>7</sup> H/m <sup>=</sup> permeability of vacuum *σ* = specific conductance of the conductor material in [S/m]


$$R\_s = \frac{1}{\sigma \delta} = \sqrt{\frac{\omega \mu'}{2\sigma}}\tag{10.14}$$

where:


#### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 11 Components**

*We will make electricity so cheap that only the rich will burn candles.*

—Thomas A. Edison

# **11.1 Conductors**

# *11.1.1 Definition of a Conductor*

First, let us give a criterion for an electrical *conductor* and an insulator (dielectric material) [16]:

• **Conductor.** *σ/ ω*- <sup>2</sup> <sup>1</sup>*.*<sup>0</sup>

$$\text{\textbullet \text{\textbullet}} \text{\textbullet} \text{\textbullet} \text{\textbullet} \text{\textbullet} \text{\textbullet} \text{\textbullet} \text{\textbullet} \text{\textbullet}$$

where:

*σ* = specific conductance of the material in [S/m] *ω* = 2*πf* = angular frequency of the sinusoidal signal in [rad/sec] - = - *<sup>r</sup>* <sup>0</sup> = permittivity of the material in [F/m]

# *11.1.2 Conductor Equivalent Circuits*

Figure 11.1 shows that the equivalent circuit of a single conductor is a simple *RL*circuit, where the self-inductance *L* [H] is split into an internal and an external self-inductance:

$$L = L\_{\text{lin}} + L\_{\text{ext}} = \frac{\Phi\_{\text{lin}}(t)}{i(t)} + \frac{\Phi\_{\text{ext}}(t)}{i(t)} \tag{11.1}$$

145

**Fig. 11.2** Internal and external *E*- and *H*-field of a single round conductor with uniform current distribution (DC) in a homogeneous environment

where:

*L* = total partial inductance of a single conductor in [H] *Lint* = internal inductance of a single conductor in [H] *Lext* = external inductance of a single conductor in [H] *Φint(t)* = internal magnetic flux of conductor caused by the current *I* in [Wb] *Φext(t)* = external magnetic flux of conductor caused by the current *I* in [Wb] *i(t)* = current through the conductor in [A]

Figure 11.2 shows that a single conductor has an internal *H*-field [A/m], but the internal *E*-field [V/m] is zero (nonexistent). Thus, a single conductor has an internal inductance *Lint* [H] (linked to the internal *H*-field) but no internal capacitance (because of the missing *E*-field). In general, the internal inductance *Lint* [H] is much smaller than the external inductance *Lext* [H]. This is especially valid for high frequencies (more details in Sect. 11.1.4).

At increasing signal frequency *f* [Hz], a conductor changes its impedance *Z(f )* []. Sections 11.1.3–11.1.7 present the calculations of the high-frequency equivalent circuit parameters of the models shown in Fig. 11.1 (single conductor) and Fig. 11.3 (conductor and its return current path). Here is an explanation of the equivalent lumped-element circuit models:



**Fig. 11.4** Surface current at high frequency

#### *11.1.3 Resistance of a Conductor*

The *resistance* of a conductor depends primarily on the electrical conductivity *σ* [S/m] of the conductor's material and the signal frequency *f* [Hz] (skin effect; see Fig. 10.5).

In the case of DC or low frequencies, where the radius *r* [m] of a round conductor or the height *h* [m] of a PCB trace is much smaller than the skin depth *δ* [m], the resistance of a conductor can be calculated to:

$$R\_{LF} = R\_{LF}' \cdot l = \frac{1}{\sigma A} \cdot l,\text{ at low-frequency} \tag{11.2}$$

where:

*RLF* = low-frequency (LF) resistance of a conductor of length *l* in [] *σ* = specific conductance of the conductor material in [S/m] *<sup>A</sup>* <sup>=</sup> cross-sectional area of the conductor in [m2] *l* = length of the conductor in [m]

For high frequencies, the skin effect kicks in, and the current starts to flow close to the conductor's surface (see Figs. 11.4 and 11.5). Therefore, when the crosssectional area—through which the current flows—is mainly determined by the skin depth *δ* [m], the resistance of the conductor can be approximately calculated as [2]:

$$R\_{HF} \approx \frac{R\_s}{p} \cdot l = \frac{l}{\sigma \delta p} = \frac{l}{p} \sqrt{\frac{\pi f \mu'}{\sigma}}, \text{ at high-frequency} \tag{11.3}$$

where:

*RH F* = high-frequency (HF) resistance of a conductor of length *l* in []

*Rs* = 1*/(σ δ)* = surface resistance of the conductor material (read more about the term surface resistance in Sect. 10.3) in [/square]

*μ*-= *μrμ*<sup>0</sup> = permeability of the conductor in [H/m]

*f* = frequency of the sinusoidal signal in [Hz] *p* = perimeter/circumference of the cross section of the conductor in [m] *l* = length of the conductor in [m]

For a circular wire (*p* = *Dπ*) or a rectangular conductor (*p* = 2*wh*), the highfrequency resistance can be approximated as:

$$R\_{HF,round} \approx R\_s \frac{l}{D\pi} = \frac{l}{\sigma \delta D \pi}, \text{ at high-frequency} \tag{11.4}$$

$$R\_{HF, rectangular} \approx R\_s \frac{l}{2wh} = \frac{l}{\sigma \delta 2wh}, \text{ at high-frequency} \tag{11.5}$$

where,

*RH F ,round* = high-frequency (HF) resistance of a round conductor with diameter *D* [m] and length *l* [m] in []

*RH F ,rectangular* = high-frequency (HF) resistance of a rectangular conductor with a of width *w* [m] and height *h* [m] in []

*Rs* = 1*/(σ δ)* = surface resistance of the conductor material in [/square]

*D* = diameter of the round conductor in [m]

*w* = width of rectangular conductor in [m]

*h* = height or thickness of rectangular conductor in [m]

#### *11.1.4 Internal Inductance of a Conductor*

The *internal self-inductance Lint* [H] is present due to the magnetic flux *Φint* [Wb] through the conductor itself. The internal self-inductance is usually much smaller than the external inductance (*Lint Lext*), because normally the conductor diameter is smaller than the loop area of the magnetic flux outside the conductor. In addition, the internal inductance is frequency dependent and decreases by factor <sup>√</sup>*<sup>f</sup>* (once the skin effect comes into play, see Fig. 11.6).

The internal self-inductance for DC and low-frequency *Lint,LF* of round conductors (e.g., wires) and rectangular conductors (e.g., PCB traces with width *w* and

thickness/height *h*) can be approximated as [9]:

$$L\_{int,LF,round} \approx \frac{\mu'}{8\pi} \cdot l = \\$0\,\text{nH/m} \cdot l,\text{ at low-frequency}\tag{11.6}$$

$$L\_{\rm int, LF, rectangular} \approx \frac{\mu'}{8} \frac{wh}{(w+h)^2} \cdot l,\text{ at low-frequency} \tag{11.7}$$

where:

*μ*- = *μ*- *rμ*<sup>0</sup> = *μ*<sup>0</sup> = permeability of the conductor material in [H/m] *w* = width of rectangular conductor in [m] *h* = height or thickness of rectangular conductor in [m] *l* = length of the conductor in [m]

Equation 11.6 shows that the internal inductance of a wire does not depend on the diameter *D* [m] of the wire. For rectangular conductors, there exist no closed-form solutions for the internal inductance [9].

The internal self-inductance for high frequency can be approximated as [2, 16]:

$$L\_{Int, HF} \approx \frac{R\_s}{\omega p} \cdot l = \frac{l}{\omega \sigma \delta p} = \frac{R\_{HF}}{\omega}, \text{ at high-frequency} \tag{11.8}$$

where:

*Rs* = 1*/(σ δ)* surface resistance of the conductor material in [/square] *RH F* = high-frequency (HF) resistance in [] *l* = length of the conductor in [m] *ω* = 2*πf* = angular frequency of the sinusoidal signal in [rad/sec] *p* = perimeter/circumference of the cross section of the conductor in [m] *σ* = specific conductance of the conductor material in [S/m] *δ* = skin depth [m]

### *11.1.5 Internal Impedance of a Conductor*

Every conductor has an *internal impedance* which can be modeled as a lumpedelement *RL*-model (see Fig. 11.1) with resistance *R* and internal inductance *Lint* :

$$\underline{Z}\_{\rm int} = \mathcal{R} + j\omega L\_{\rm int} = \left(\mathcal{R}' + j\omega L'\_{\rm int}\right) \cdot l \tag{11.9}$$

where:

*R* = *R*- *l* = electrical resistance of the conductor in [] *Lint* = *L*- *int l* = internal self-inductance of the conductor in [H] *ω* = 2*πf* = angular frequency of the sinusoidal signal in [rad/sec] *l* = length of the conductor in [m] *<sup>j</sup>* <sup>=</sup> √−<sup>1</sup> <sup>=</sup> imaginary unit

For DC and low frequencies (below skin effect), the internal impedance of a conductor is:

$$\underline{Z}\_{\text{int},LF} = \mathcal{R}\_{LF} + jL\_{\text{int},LF}, \text{ at low frequency} \tag{11.10}$$

where:

*RLF* = LF resistance according to Eq. 11.2 in [] *Lint,LF* = LF internal inductance according to Eqs. 11.6 and 11.7 in [H] *<sup>j</sup>* <sup>=</sup> √−<sup>1</sup> <sup>=</sup> imaginary unit

Figure 11.7 shows that once the skin effect comes into play, the following happens:

**Fig. 11.7** Internal impedance of a conductor vs. frequency


Therefore, for the internal impedance of a conductor at high frequency, we can write:

$$\underline{Z}\_{\text{int},HF} = (1+j)\,R\_{HF} = (1+j)\,\frac{l}{p}\sqrt{\frac{\pi f \mu'}{\sigma}},\text{ at high-frequency} \tag{11.11}$$

where:

*RH F* = HF resistance according to Eq. 11.3 in [] *l* = length of the conductor in [m] *p* = perimeter/circumference of the cross section of the conductor in [m] *f* = frequency of the sinusoidal signal in [Hz] *μ*- = *μrμ*<sup>0</sup> = permeability of the conductor in [H/m] *σ* = specific conductance of the conductor material in [S/m] *<sup>j</sup>* <sup>=</sup> √−<sup>1</sup> <sup>=</sup> imaginary unit

#### *11.1.6 External Inductance of a Single Conductor*

The *external inductance Lext* [H] is present due to the flux outside (external) of the conductor. It is to be mentioned that *Lext* is much bigger than *Lint* because the conductor diameter is usually much smaller than the loop area of the magnetic flux.

The self-inductance of a single round conductor *Lround* [H] and a single rectangular conductor *Lrectangular* [H], like shown in Fig. 11.8, in a surrounding with *μr* = 1, can be approximated as [16]:

$$L\_{round} \approx 2 \cdot 10^{-8} \cdot l \left[ \ln \left( \frac{4l}{D} \right) - 1 + \frac{D}{2l} + \frac{\mu\_r' T(x)}{4} \right] \tag{11.12}$$

**Fig. 11.8** Round conductor with diameter *D* (e.g., wire) and rectangular conductor with width *w* and height or thickness *h*

$$L\_{rectangular} \approx 2 \cdot 10^{-5} \cdot l \left[ \ln \left( \frac{2l}{w+h} \right) + 0.25049 + \frac{w+h}{3l} + \frac{\mu\_r' T(\mathbf{x})}{4} \right] \tag{11.13}$$

where:

*l* = length of the round conductor in [m] *D* = diameter of the round conductor in [m] *w* = width of rectangular conductor in [m] *h* = height or thickness of rectangular conductor in [m] *μ*- *<sup>r</sup>* = relative magnetic permeability of the round conductor *T (x)* = <sup>0</sup>*.*873011+0*.*00186128*<sup>x</sup>* <sup>1</sup>−0*.*278381*x*+0*.*127964*x*<sup>2</sup> <sup>=</sup> compensation of the inductance for AC effects *<sup>x</sup>* <sup>=</sup> *Dπ*2*μ*- *f σ μ*- = *μ*- *rμ*<sup>0</sup> = permeability of the conductor material in [H/m] *f* = frequency of the sinusoidal signal in [Hz] *σ* = specific conductance of the conductor material in [S/m]

Even more simplified (neglecting *T (x)*, *μr* = 1), we can write [16]:

$$L\_{round} \approx 2 \cdot 10^{-8} \cdot l \left[ \ln \left( \frac{4l}{D} \right) - 0.75 \right] \tag{11.14}$$

$$L\_{rectangular} \approx 2 \cdot 10^{-5} \cdot l \left[ \ln \left( \frac{2l}{w+h} \right) + 0.5 + 0.2235 \left( \frac{w+h}{l} \right) \right] \tag{11.15}$$

Remember that these formulas above for *Lround* [H] and *Lrectangular* [H] do only give you the *partial self-inductance* of a dedicated conductor with length *l*. When looking at a current loop, which consists of different conductor parts, all partial selfinductance and mutual inductances of all the conductor parts which eventually form the current loop have to be considered.

#### *11.1.7 Conductor with Return Current Path*

If you do not only consider a single conductor but also the return current path, we have a transmission line and the following happens:

• **Resistance.** At high frequencies—where currents flow primarily on the conductor surface,—the interaction between the forward and its return current leads to a nonuniform current distribution on the surface of conductors [9]. This effect is called the *proximity effect* and has a slight influence on the HF resistance *RH F* []. The proximity effect will not be discussed further, and the proximity effect is ignored in all considerations throughout this book.


#### **11.1.7.1 Conductor Length** *< (λ/***10***)*

In case the length *l* [m] of the conductor is shorter than a tenth of the wavelength *λ* [m], the conductor and its return current path can be modeled with lumped *RLC*-elements (see Fig. 11.3). However, in case the signal line is longer than *λ/*10, reflections and ringing may occur and the signal line should not be modeled with a simple *RLC*-circuit. Instead, the characteristic impedance *Z*<sup>0</sup> [] and the distributed parameter model are more appropriate to do so (see Sect. 7.3).

The calculations of the resistance *R* [] and the per-unit-length resistance *R*- [/m] of a conductor at low and high frequencies are presented in Sect. 11.1.3. The inductance *L* [H] and the capacitance *C* [F] of a conductor and its return current path depend on several parameters: the geometry of the conductor (and its return current path), the distance *d* [m] between the forward and backward current, the permeability *μ*- = *μ*- *rμ*<sup>0</sup> of the material(s) through which the magnetic flux flows, and the permittivity - = - *<sup>r</sup>* <sup>0</sup> of the material(s) through which the electric field lines are formed. Formulas for calculating the per-unit-length inductance *L*- [H/m] and capacitance *C*-[F/m] are presented in Sect. 7.4.

#### **11.1.7.2 Conductor Length** *> (λ/***10***)*

If the conductor length *l* [m] is longer than a tenth of the signal wavelength (*l > λ/*10), it is no longer enough to model the signal path as a simple lumped-element circuit with *R* [], *L* [H], and *C* [F]. Instead, the conductor and its return current path should be modeled as transmission line according to the distributed parameter model (according to Fig. 7.2, with *R*- [/m], *L*- [H/m], *C*- [F/m], and *G*- [S/m]) and with the characteristic impedance *Z*<sup>0</sup> [].

Formulas for calculating the characteristic impedance *Z*<sup>0</sup> [] of common transmission lines is presented in Sect. 7.3.

#### **11.2 Resistors**

There are, roughly said, three basic classes of passive *resistors*. All of them are designed for different applications and have their own advantages and drawbacks: **Fig. 11.9** Simplified high-frequency equivalent circuit of a resistor [3, 4]


The impedance *Z* [] of an ideal resistor is:

$$\underline{Z} = \underline{R} \tag{11.16}$$

The simplified high-frequency equivalent circuit for resistors is presented in Fig. 11.9. The resistor's parasitic inductance *Ls* [H] and capacitance *Cp* [F] values depend on the resistor material type, the mechanical construction (SMT, axial leaded), and the nominal resistance value (low or high resistance). Be aware that the equivalent circuit model in Fig. 11.9 does not include the parasitic effects of the resistor's mounting parts (PCB pads). The impedance *Z* of a nonideal resistor can be written as:

$$\underline{Z} = \frac{R + j\omega L\_s}{1 - \omega^2 L\_s C\_p + j\omega RC\_p} \tag{11.17}$$

where:

*R* = nominal value of the resistor in [] *Ls* = parasitic series inductance in [H] *Cp* = parasitic parallel (shunt) capacitance in [F] *ω* = 2*πf* = signal frequency in [rad/s]

Besides the parasitic elements of *Ls* [H] and *Cp* [F], the frequency response of a resistor also depends on the resistor's nominal value itself. Figure 11.10a shows the simplified frequency response for resistors with small nominal values and Fig. 11.10b for large resistor values.

Nowadays, film resistors are widely used because they are inexpensive and accurate. Figure 11.11 shows a simulation of the frequency behavior of different types of film resistors. With increasing frequency, the parasitic parallel (shunt) capacitance *Cp* [F] and the parasitic series inductance *Ls* [H] lead to nonideal frequency response. The plots in Fig. 11.11 show that for frequencies *f >* 100 MHz, only surface-mount technology (SMT) resistors should be used (axial leaded resistors are not suitable for high-frequency applications due to the wired terminals). Above

**Fig. 11.10** Impedance of a resistor at high frequencies (simplified). (**a**) Small resistor values (e.g. *<* 100 ). (**b**) Large resistor values (e.g. *>* 100 )

1 GHz, dedicated high-frequency thin-film resistors show acceptable performance, whereas thick-film resistors often fail because they have higher parasitic inductance and capacitance compared to thin-film resistors.

**Fig. 11.11** Impedance |*Z*|*/R* vs. frequency *f* [Hz] of different film resistors (simulation, skin effect not considered). (**a**) 50 film resistors. (**b**)1k film resistors

# **11.3 Capacitors**

*Capacitors* can be categorized by its dielectric material:


Moreover, capacitors can be categorized according to their application:

• **Decoupling.** Decoupling or bypass capacitors are placed close to the power supply pins of integrated circuits (IC) to provide a stable supply voltage to the ICs. In addition, decoupling capacitors help prevent voltage drops or spikes


**Table 11.1** Safety capacitor classification [15]


on the power supply line in case of a sudden change of current drawn by the integrated circuit. Usually, MLCCs with X7R or X5R dielectric are used as decoupling capacitors.


The impedance *Z* [] of an ideal capacitor is:

$$\underline{Z} = \frac{1}{j\omega C} \tag{11.18}$$

Figure 11.12 shows a simplified equivalent circuit model of a nonideal capacitor. A capacitor is not a pure capacitance *C* [F]. The series inductance *Ls* (ESL) is caused by the leads and the internal structure. The leads and internal losses also cause the series resistance *Rs* (ESR). The parallel resistor *Rp* [] represents the nonideal dielectric material (leakage current) and has typically a value of 10 G or more [8]. The impedance *Z* [] of a nonideal capacitor can be written as (with neglected *Rp* []):

$$\underline{Z} = \frac{1 - \alpha^2 L\_s C + j\omega R\_s C}{j\omega C} \tag{11.19}$$

**Fig. 11.13** Impedance of a capacitor at high frequencies (simplified)

where:

*C* = nominal capacitance in [F] *Rs* = parasitic series resistance in [] *Ls* = parasitic series inductance in [H] *ω* = 2*πf* = signal frequency in [rad/s]

Figure 11.13 shows the frequency response (impedance vs. frequency) of a real, nonideal capacitor. The capacitance dominates from DC to the serial resonant frequency *fr* [Hz] of *Ls* [H] and *C* [F]. At the capacitor's resonance frequency *fr* [Hz], the impedance of a capacitor is equal to *Rs* (ESR). However, the inductance *Ls* [H] dominates for frequencies higher than the resonant frequency, and the capacitor becomes an inductor. *Rp* [] is usually neglected when it comes to high-frequency response calculations of capacitors. Note: the ESR is not simply a resistance that can be measured with an ohmmeter. The ESR represents the ohmic and dielectric losses of a capacitor.

Terms that are often used in connection with the nonideal behavior of capacitors are dissipation factor (DF), loss tangent tan *(δe)* (Fig. 11.14), and quality factor *Q*, and they are defined like this:

$$\tan\left(\delta\_e\right) = \frac{\text{ESR}}{X\_C} = \frac{1}{\mathcal{Q}}\tag{11.20}$$

$$\text{DF} = \tan\left(\delta\_e\right) = \frac{1}{\mathcal{Q}}\tag{11.21}$$

$$\mathcal{Q} = \frac{X\_C}{\text{ESR}} = \frac{1}{\tan(\delta\_\ell)} = \frac{1}{\text{DF}} \tag{11.22}$$

**Fig. 11.15** Different form factors of capacitors. Courtesy of Würth Elektronik GmbH

where:

*δe* = loss angle of the capacitor in [rad] tan *(δe)* = loss tangent ESR= equivalent series resistor of the capacitor in [] *XC* = 1*/ (ωC)* = reactance of the capacitor in [] DF= dissipation factor *Q* = quality factor

The higher the dissipation factor (DF) and the loss tangent tan *(δe)*, the higher the losses of a capacitor.

Figure 11.15 presents form factors of various capacitors, and Fig. 11.16 compares the impedance behavior of C0G capacitors.

For capacitor values *<*100 µF, multilayer ceramic capacitors (MLCCs) are in most cases the best option. MLCCs have excellent small ESR and ESL and are inexpensive, accurate, robust, stable long term, and available in small packages (e.g., in the SMT packages 0201, 0402). However, MLCCs have also their drawbacks which must be considered [10]:

	- Class I. Dielectrics like C0G (NP0) and U2J are ultra-stable across a wide temperature range from −55 ◦C to +125 ◦C. Table 11.2 shows the codes of class I MLCCs.
	- Class II. Dielectrics like X7R and X5R have a much higher dielectric constant - *<sup>r</sup>* than class I dielectrics, but the capacitance can vary as much as ±15 %

**Fig. 11.16** Impedance |*Z*| [] vs. frequency *f* [Hz] of C0G SMT capacitors in a 0402 case: 1 pF, 10 pF, 100 pF, 1 nF. Courtesy of Würth Elektronik GmbH

**Table 11.2** Class I ceramic capacitor types. For example, C0G: ±30 ppm/K, U2J: −750 ppm/K ±120 ppm/K [10]


over the range of −55 ◦C to +85 ◦C (X5R) or to 125 ◦C. Table 11.3 shows the codes of class II and class III MLCCs.



**Table 11.3** Class II and class III ceramic capacitor types [10]. For example, X7R: ±15 % from −55 ◦C to +125 ◦C, Z5U: 22 % to −56 % from +10 ◦C to +85 ◦C

**Fig. 11.17** DC bias voltage derating of MLCCs. (**a**) MLCC DC bias voltage derating vs. dielectric material. Exact values vary. (**b**) MLCC DC bias voltage derating vs. case size. Exact values vary

**Fig. 11.18** Typical aging of MLCCs. Exact values can vary

**Fig. 11.19** Typical capacitance derating of MLCCs due to signal frequency. Exact values vary

of capacitance when an AC voltage within a reasonable range is applied and a decrease for small AC voltage amplitudes (e.g., −10 % at 10 mV). If a high enough AC voltage is applied, eventually, it will reduce capacitance just as a DC voltage. Capacitance decreases more quickly with smaller case sizes (see Fig. 11.17b).


The conclusion from the points above is that whenever a stable capacitance *C* [F] is needed, use a class I MLCC capacitor (e.g., NP0, C0G for filter applications). Another important point to remember is that the capacitance *C* [F] of class II and class III MLCCs is a function of temperature *T* [ ◦C], DC voltage *V* [V], time (aging) *t* [h], and frequency *f* [Hz]. Nonetheless, class II and class III MLCCs have their use cases. For example, X7R capacitors are commonly used for decoupling on PCB designs because X7R capacitors provide a higher nominal capacitance *C* [F] than class I capacitors for the same package size.

For capacitor values *>* 100 µF, tantalum, aluminum electrolyte, or polymer electrolyte capacitors are the most popular. However, we do not discuss any other capacitor types than MLCCs because the trend is clearly toward MLCCs due to their small size, high reliability, long lifetime, and low ESL at relatively low costs.

#### **11.4 Inductors**

*Inductors* can be categorized in:


The impedance *Z* [] of an ideal inductor is:

$$\underline{Z} = j a L \tag{11.23}$$

The simplified equivalent circuit model of a nonideal inductor is shown in Fig. 11.20. The ideal inductance *L* [H] is influenced by the winding capacitance *Cp* [F] and the ohmic series resistance of the winding conductor *Rs* []. *Rp* [] represents the core losses and can be neglected for nonmagnetic cores (air instead of a magnetic core).

The impedance *Z* [] (Fig. 11.21) of a nonideal inductor can be written as:

$$\underline{Z} = \frac{R\_s + j\omega L}{1 - \alpha^2 L C\_p + j\omega R\_s C\_p}, R\_p \text{ neglected} \tag{11.24}$$

**Fig. 11.20** Simplified high-frequency equivalent circuit of an inductor [4, 5]. (**a**) Air core inductor. (**b**) Magnetic core inductor

**Fig. 11.21** Impedance of an inductor at high frequencies (simplified)

$$\underline{Z} = \frac{R\_s + j\omega L}{1 + R\_s/R\_p - \omega^2 L C\_p + j\omega \left(L/R\_p + R\_s C\_p\right)}, \; R\_p \; considered \qquad (11.25)$$

where:

*L* = nominal self-inductance in [H] *Cp* = parasitic parallel (shunt) capacitance in [F] *Rs* = series resistance of the windings in [] *Rp* = parallel resistance representing the magnetic losses in [] *ω* = 2*πf* = signal frequency in [rad/s]

Self-inductance *L* [H] is defined as the ratio of the total magnetic flux *Φ(t)* [Wb] to the current *i(t)* [A], which causes that magnetic flux. Assuming a magnetic lossless material through which the magnetic flux flows (*μ*- = *μ*- *rμ*0, *μ*-- = 0), we can write:

$$L = \frac{\Phi(t)}{i(t)} = \frac{\overrightarrow{B} \cdot \overrightarrow{A}}{i(t)} = \frac{\left(\mu' \overrightarrow{H}\right) \cdot \overrightarrow{A}}{i(t)}\tag{11.26}$$

where:

*Φ(t)* = magnetic flux produced by the current *i(t)* through inductor in [Wb] *i(t)* <sup>=</sup> current through the inductor in [A] −→*<sup>B</sup>* <sup>=</sup> magnetic flux density vector in [T] −→*<sup>H</sup>* <sup>=</sup> magnetic field vector in [A/m] −→*<sup>A</sup>* <sup>=</sup> vector area through which the magnetic flux flows in [m2]

**Fig. 11.22** Impedance |*Z*| [] vs. frequency *f* [Hz] of the filter inductor Würth Elektronik 7847121060. Courtesy of Würth Elektronik GmbH

*μ*- = *μ*- *rμ*<sup>0</sup> = magnetic permeability of the material through which the magnetic field flows [H/m]

Equation 11.26 shows that the self-inductance *L* [H] is proportional to the core material's relative magnetic permeability *μ*- *<sup>r</sup>*. It is important to note that *μ*- *<sup>r</sup>* and therefore *μ*and the self-inductance *L* [H] are a function of:


As an example for the nonideal behavior of inductors, Fig. 11.22 presents the frequency-dependent impedance |*Z*| [] and the current-dependent inductance *L* [H] of a rod-inductor.

With the introduction of the complex magnetic permeability (Fig. 11.23):

**Fig. 11.23** Typical frequency response of an inductor and its complex magnetic permeability *μ* = *μ*- − *jμ*-and its impedance |*Z*| [5]

**Fig. 11.24** Equivalent circuit of an inductor when considering magnetic losses only and ignoring ohmic losses and winding capacitance

$$
\underline{\mu} = \mu' - j\mu'' \tag{11.27}
$$

where:

 $\mu' = \mu\_r'$  $\mu\_0 = \text{real part of the complex permeability in [H/m]}$  $\mu'' = \mu\_r''$  $\mu\_0 = \text{imaginary part of the complex permeability in [H/m]}$  $\mu\_j = \sqrt{-1}$ 

we can separate the ideal self-inductance *L* [H] and the magnetic losses of the core material *Rml* []. This concept can be applied to all types of inductors and ferrites. Now we can write:

$$\underline{Z} = j\omega \underline{\mu} L\_0 = j\omega L\_0 \left(\mu' - j\mu''\right) = R\_{ml} + jX\_{ind} \tag{11.28}$$

$$R\_{ml} = \omega L\_0 \mu^{\prime\prime}, \text{ magnetic loss component} \tag{11.29}$$

$$X\_{ind} = \omega L\_0 \mu', \text{ inductance component} \tag{11.30}$$

where:

*μ* = *μ*- − *jμ*-- = complex magnetic permeability of the inductor in [H/m] *ω* = 2*πf* = signal frequency in [rad/s]

*L*<sup>0</sup> = inductance of an air inductor of the same construction and field distribution, without core material (*μ* = *μ*0*, μ*- *<sup>r</sup>* = 1) in [H]

*Rml* = magnetic losses modeled as series resistor to the inductance in [] *Xind* = inductive reactance in []

Figure 11.24 shows the equivalent circuit of an inductor when ohmic losses *Rs* [] and capacitive parasitics *Cp* [F] are ignored. The loss tangent tan *(δm)* of this **Fig. 11.25** Loss angle *δm* of an inductor

setup is given as:

$$\tan\left(\delta\_m\right) = \frac{R\_{ml}(f)}{X\_{ind}(f)} = \frac{\mu''(f)}{\mu'(f)}\tag{11.31}$$

where:

*Rml(f )* = magnetic losses modeled as series resistor to the inductance in [] *Xind (f )* = inductive reactance in []

Figure 11.23 and Eq. 11.31 illustrate that the inductance *L* [H] is frequencydependent. Thus, the loss tangent tan *(δm)* (Fig. 11.25) is also frequency-dependent.

When considering all losses, the quality factor *Q* of an inductor is defined as:

$$\mathcal{Q} = \frac{X\_L}{R\_L} = \frac{\omega L}{R\_L} = \frac{1}{\tan \left( \delta\_m \right)} \tag{11.32}$$

where:

*XL* = reactance of the inductor (considering also parasitic elements) in [] *RL* = ohmic and magnetic losses of the inductor in [] *ω* = 2*πf* = signal frequency in [rad/s] tan *(δm)* = loss tangent for loss angle *δm*

An ideal inductor would have *Q* = ∞ (lossless). However, for EMC applications, a high *Q*-factor is not always desired in the first place (except when acting as a filter) as this means that the inductor has low losses and a high self-inductance *L* [H]. In most cases, EMC applications require inductors with high losses over a wide frequency range and a *Q*-factor typically *Q <* 3 in the frequency range of interest [5]. This leads us directly to the next Sect. 11.5 about ferrite beads.

### **11.5 Ferrites Beads**

*Ferrite beads* are magnetic components that play a crucial role in suppressing high-frequency noise and preventing electronic units from undesired radiation. Simply described, ferrite beads are electrical conductors surrounded by a magnetic material (ferrite). Ferrites are ceramic ferrite magnetic materials made from iron oxide (Fe2O3) and oxides of other metals. There are no eddy-current core losses (like known from iron cores) because the ferrites are made of ceramics and are electrically nonconductive. Therefore, ferrite materials can be used to provide selective attenuation of high-frequency signals.

Inductors and ferrite beads are both inductive components. However, there are some differences:


The behavior of ferrite beads is often described as frequency-dependent resistance because, in the frequency range of interest, the impedance *Z* [] of ferrite beads is dominated by its resistive part. Different types of ferrite beads can be found on the market. The two most common form factors are:


### *11.5.1 Cable Mount Ferrite Beads*

With *cable mount ferrite beads*, the ferrite material is placed around a cable, a wire, or a group of conductors. Figure 11.26 shows typical examples of cable mount ferrite beads. In order to achieve optimal attenuation, the air gap between the cable and the surrounding ferrite bead should be minimal.

Figure 11.27 shows the simplified high-frequency equivalent circuit of a cable mount ferrite bead. The self-inductance *L* [H] and the magnetic losses depend on the signal frequency *f* [Hz]. The losses are modeled as resistance *R* []. The bead material can be characterized by the complex magnetic permeability [13]:

**Fig. 11.26** Cable mount ferrite beads. Courtesy of Würth Elektronik GmbH

**Fig. 11.27** Simplified high-frequency equivalent circuit of a cable mount ferrite bead [13]

$$
\underline{\mu}(f) = \mu'(f) - j\mu''(f) \tag{11.33}
$$

where:

*μ*- = *μ*- *rμ*<sup>0</sup> = real part of the complex permeability of the bead material (refers to the stored magnetic energy) in [H/m]

*μ*-- = *μ*-- *rμ*<sup>0</sup> = imaginary part of the complex permeability of the bead material (refers to the magnetic losses in the bead material) in [H/m] *<sup>j</sup>* <sup>=</sup> √−<sup>1</sup>

Figure 11.23 shows that *μ*- *(f )* and *μ*--*(f )* are both shown as functions of frequency *f* [Hz]. The complex impedance *Z* [] of a cable mount ferrite bead can be written as [13]:

$$\underline{Z} = j\omega L\_{bead} = j\omega \underline{\mu} K = j\omega \left(\mu'-j\mu''\right) K \tag{11.34}$$

$$=\underbrace{\omega\mu\_{0}\mu\_{r}^{\prime\prime}(f)K}\_{R(f)} + j\underbrace{\omega\mu\_{0}\mu\_{r}^{\prime}(f)K}\_{L(f)}\tag{11.35}$$

where:

*ω* = 2*πf* = signal frequency in [rad/s] *μ*-- = *μ*-- *rμ*<sup>0</sup> = imaginary part of the complex permeability in [H/m] *μ*- = *μ*- *rμ*<sup>0</sup> = real part of the complex permeability in [H/m] *K* = some constant depending on the bead dimensions in [m]

Typical cable mount ferrite beads can be expected to give impedances of order 100 for frequencies *>*100 MHz. Figure 11.28 presents the impedance |*Z*| [] of a snap ferrite as a function of signal frequency *f* [Hz]. The impedance of the ferrite bead depends on:

**Fig. 11.28** Impedance |*Z*| [] vs. frequency *f* [Hz] of the snap ferrite bead Würth Elektronik 74271733. Courtesy of Würth Elektronik GmbH


#### *11.5.2 PCB Mount Ferrite Beads*

Figure 11.29a shows some common form factors of SMT *chip ferrite beads*, whereas Fig. 11.29b shows the inner structure. Chip ferrite beads consist of a conductor surrounded by a ferromagnetic material (typical NiZn).

The simplified high-frequency equivalent circuit of a chip ferrite bead is shown in Fig. 11.30. *RDC* [] represents the DC series resistance, *L* [H] represents the inductance, *Cp* [F] represents the parasitic capacitance, and *RAC* [] represents the intended losses at high frequencies. Figure 11.31 shows a typical frequency response of a PCB mount ferrite bead:

**Fig. 11.29** SMT chip ferrite beads. Courtesy of Würth Elektronik GmbH. (**a**) Different form factors of SMT chip ferrite beads. (**b**) Inner structure of chip ferrite beads. Left: WE-CBF. Right: WE-CBF HF (optimized for high-frequency applications due to low *Cp*)


The nonideal behavior of PCB mount ferrite beads comprises the following point:


**Fig. 11.31** Impedance |*Z*| [] vs. frequency *f* [Hz] of the EMI suppression chip ferrite bead Würth Elektronik 74269244182. Courtesy of Würth Elektronik GmbH

#### **11.6 Common-Mode Chokes**

A *common-mode choke* consists of two coils wound around a common ferromagnetic core (sometimes even more than two coils). The differential signal current flows through one coil and back through the second one (see Fig. 11.34b). Commonmode chokes in EMC are primarily used to block common-mode currents and, therefore, to eliminate unintended radiated electromagnetic emissions.

Figure 11.33d presents the high-frequency equivalent circuit of a commonmode choke, where *Rs* [] represents the ohmic losses, *L* [H] represents the self-inductance of each coil, *M* [H] represents the mutual inductance of the coils, *Rp* [] represents the magnetic core losses, *Cp* [F] represents the parasitic capacitance between the turns of the coils, and *C*<sup>12</sup> [F] represents the parasitic capacitance between the two coils.

When ignoring the parasitic capacitances and the ohmic and magnetic losses, the coil impedances of an ideal common-mode choke can be written as [13]:

$$\underline{Z}\_{1} = \frac{\underline{V}\_{1}}{\underline{I}\_{1}} = \frac{j\alpha L\_{1}\underline{I}\_{1} + j\alpha M\_{12}\underline{I}\_{2}}{\underline{I}\_{1}}\tag{11.36}$$

$$\underline{Z}\_2 = \frac{\underline{V}\_2}{\underline{I}\_2} = \frac{j\alpha L\_2 \underline{I}\_2 + j\alpha M\_{21} \underline{I}\_1}{\underline{I}\_2} \tag{11.37}$$

**Fig. 11.32** Nonideal behavior of ferrite beads. Courtesy of Würth Elektronik GmbH [1]. (**a**) Impedance |*Z*| [] vs. DC bias current *I* [A] of Würth WE-CBF 742861160. (**b**) Impedance |*Z*| [] vs. temperature *T* [ ◦C] of Würth WE-CBF 742792040

where:

*Z*<sup>1</sup> = impedance of common-mode choke coil 1 in [] *Z*<sup>2</sup> = impedance of common-mode choke coil 2 in [] *V* <sup>1</sup> = voltage over coil 1 in [V] *V* <sup>2</sup> = voltage over coil 2 in [V] *I* <sup>1</sup> = current through coil 1 in [A] *I* <sup>2</sup> = current through coil 2 in [A]

**Fig. 11.33** Common-mode choke. (**a**) Common-mode impedances of a common-mode choke. (**b**) Differential-mode impedances of a common-mode choke. (**c**) Common-mode choke symbol. (**d**) High-frequency equivalent circuit of a common-mode choke

*L*<sup>1</sup> = self-inductance of coil 1 in [H] *L*<sup>2</sup> = self-inductance of coil 2 in [H] *M*<sup>12</sup> = mutual inductance at coil 1 due to coupling with coil 2 in [H] *M*<sup>21</sup> = mutual inductance at coil 2 due to coupling with coil 1 in [H] *ω* = 2*πf* = signal frequency in [rad/s]

If we now assume that the common-mode current *ICM* [A] splits up equally through both coils of the common-mode choke (*I*1*CM* = *I*2*CM* = *ICM/*2) and that the self-inductance of the coils is equal to the mutual inductance (*L*<sup>1</sup> = *L*<sup>2</sup> = *M*<sup>12</sup> = *M*21; see Fig. 11.34a), the common-mode impedance of each coil is:

$$\underline{Z}\_{1CM} = j\omega (L+M) = j\omega 2L \tag{11.38}$$

$$\underline{Z}\_{2CM} = j\omega (L+M) = j\omega 2L \tag{11.39}$$

where:

*Z*1*CM* = ideal common-mode impedance of common-mode choke coil 1 in [] *Z*2*CM* = ideal common-mode impedance of common-mode choke coil 2 in [] *L* = self-inductance of a single coil of the common-mode choke in [H] *M* = mutual inductance between the coils of the common-mode choke in [H] *ω* = 2*πf* = signal frequency in [rad/s]

**Fig. 11.34** Common-mode choke models for common-mode currents *ICM* vs. differential-mode currents *IDM* [13]. Parasitic capacitances (*Cp*, *C*12) and losses (*Rs Rp*) were ignored. (**a**) *ICM* = common-mode current. (**b**) *IDM* = differential-mode current

Equations 11.38 and 11.39 show why common-mode chokes have a high impedance for common-mode currents. On the other hand, an ideal common-mode choke (*L*<sup>1</sup> = *L*<sup>2</sup> = *M*<sup>12</sup> = *M*21; see Fig. 11.34b) would not influence the differentialmode current at all:

$$\underline{Z}\_{1DM} = j\omega (L - M) = 0 \tag{11.40}$$

$$
\underline{Z}\_{2DM} = j\omega (L - M) = 0 \tag{11.41}
$$

where:


Figure 11.35 shows the different form factors of common-mode chokes, and Fig. 11.36 presents a typical frequency response of a common-mode choke. Common-mode chokes do influence not only the common-mode signals but also the differential-mode signals (see Fig. 11.36).

The intentional behavior of common-mode chokes can be summarized like this:


**Fig. 11.35** Different form factors of common-mode chokes as mains power line filters or lowvoltage signal interfaces. Courtesy of Würth Elektronik GmbH

**Fig. 11.36** Impedance |*Z*| [] vs. frequency *f* [Hz] of the common-mode line filter 744230121 from Würth. Courtesy of Würth Elektronik GmbH

The nonideal (and therefore unintended) behavior of common-mode chokes is summarized here:


# **11.7 Baluns**

A *balun* (a contraction of balanced-unbalanced) is a component that is used to convert between a balanced signal and an unbalanced signal. A balun is a bidirectional component and can be used either way (from balanced to unbalanced and vice versa). There are two main modes of baluns:


In addition to a distinction between voltage and current baluns, there is also a distinction between isolation transformers (Fig. 11.37a) and autotransformers (Fig. 11.37b). Typical balun applications are:


**Fig. 11.37** Baluns. (**a**) Balun in voltage mode. (**b**) Autotransformer baluns are electrically nonisolating and operate with only one winding. (**c**) Balun in current mode

**Fig. 11.38** 10BASE-T1 and 100BASE-T1 Single Pair Ethernet (SPE) interface with galvanic isolation of 1*.*5 kV and balun WE-STST 74930000 from Würth. Courtesy of Würth Elektronik GmbH

The applications and use cases for baluns are very broad. Therefore, it is difficult to give a conclusive set of properties to watch out for. The first thing to decide is if a voltage or a current balun or an autotransformer balun with only one coil is needed. If you are unsure which type of balun you need, you could look for an application note that describes your application best and start with the solution described there.

#### **11.8 Clamping Devices**

*Voltage clamping devices* —like varistors (Sect. 11.8.1) and TVS diodes (Sect. 11.8.2)—commonly block the current flow up to a specified voltage. Once the voltage reaches that clamping voltage, the voltage across the voltage clamping device does not increase anymore. In other words, the voltage is *clamped*. Voltage clamping devices are commonly used to protect circuits from transient pulses like:


Figure 11.46 shows how a voltage clamping device limits the transient voltage across its pins.

#### *11.8.1 Varistors*

*Varistors* are voltage-dependent resistors (VDR), and they are typically used as overvoltage protection devices against bursts (EFTs) and surges, which arrive at the victim via the public mains supply or the DC supply port. Varistors belong to the category of clamping devices.

The high-frequency equivalent circuit for varistors is presented in Fig. 11.39. The series inductance *Ls* [H] represents the varistor's terminal inductance (approximately 1 nH/mm), the series resistor *Rs* [] represents the varistor's terminal resistance (a few [m]), the parallel capacitance *Cp* [F] represents the intergranular capacitance (10 pF up to 10 nF), the parallel resistor *Rp* [] represents the

**Fig. 11.41** Varistor (VDR). (**a**) Schematic symbol of varistors. The letter U is often used instead of the letter V. (**b**) Current *I* [A] vs. voltage *V* [V] of ZnO and SiC varistors. ZnO VDRs belong to the category of metal oxide varistors (MOV) and are the most common.

intergranular resistance (several [M]), and finally *RV ar* [] represents the ideal varistor (0 *Ω* to ∞) (Fig. 11.40).

Figure 11.41b presents a varistor's nonlinear and bidirectional current-voltage characteristic, and Fig. 11.42 presents extracted data from a varistor's datasheet. At the breakdown voltage *VBR* [V] or varistor voltage *VV ar* [V], the varistor changes its state from insulating to conducting. Usually, the breakdown voltage is specified for 1 mA DC current. Ideally, a VDR would block every current as long as the voltage across the VDR is smaller than the breakdown voltage *VBR* [V], and as soon it is larger than *VBR* [V], the voltage would not increase and an infinite amount of current could flow through the VDR.

Unfortunately, we do not live in an ideal world, and VDRs have some points to be considered when using them in a design:


**Fig. 11.42** Varistor 820442711E from Würth for 230 VAC applications. Courtesy of Würth Elektronik GmbH

level you need for your product: 1, 2, 3, or 4). Figure 11.40 shows different form factors and case sizes of VDRs.


current can be extracted from the current-to-voltage characteristic curve for any given voltage (see Fig. 11.42).

• **Aging.** As shown in Fig. 11.42, VDRs show an aging effect based on the number of current pulses they experience.

#### *11.8.2 TVS Diodes*

*Transient-voltage-suppression diodes* (TVS diodes) are typically used as overvoltage protection devices against ESD pulses, bursts (EFTs), and surges (if the current through the diode is limited). TVS diodes belong to the category of clamping devices. Figure 11.43a presents different form factors of TVS diodes.

Compared to varistors, TVS diodes have a shorter response time and a lower clamping voltage *Vclmap* [V] (for the same breakdown voltage *VBR* [V]). On the other side, TVS diodes do endure less pulse current *IP P* [A] compared to varistors. TVS diodes are available in two configurations:


Figure 11.44 presents a typical current vs. voltage curve of a bidirectional TVS diode. The most essential parameters of TVS a diode are:

**Fig. 11.43** TVS diodes. (**a**) Different form factors of TVS diodes. Courtesy of Würth Elektronik GmbH. (**b**) Left: unidirectional TVS diode. Right: bidirectional TVS diode

**Fig. 11.45** TVS diode 824012823 from Würth for high-speed interfaces like USB 3.1 (10 GB/s) or HDMI 2.0. Courtesy of Würth Elektronik GmbH


### **11.9 Crowbar Devices**

*Crowbar devices* commonly block the current flow up to a specific voltage. Once that voltage is exceeded, the crowbar device changes suddenly into a low impedance state, the voltage across the crowbar device gets low, and the current through the crowbar device increases and is only limited by the external circuit. After that, when the current through the crowbar device falls below a certain threshold, the crowbar device resets itself. Examples of crowbar devices are *TVS thyristors* (see Fig. 11.47b) and *gas discharge tubes* (see Fig. 11.47c). Crowbar devices are mainly used to protect circuits from high-energy transient pulses like:


Compared to voltage clamping devices (see Sect. 11.8), the crowbar devices can handle much more energy, but the downside is that they do not turn on as fast as voltage clamping devices. Figure 11.46 compares the principles of voltage clamping vs. crowbar behavior (Fig. 11.47).

We do not go further into detail about crowbar devices because for many products on the market, the presented clamping devices—like varistors and TVS diodes, in combination with other passive filter components like capacitors,—are most practical and economical to use. However, gas discharge tubes are very popular for lightning protection of antennas and telecommunication products.

**Fig. 11.47** Schematic symbols of typical crowbar devices. (**a**) Thyristor symbol. (**b**) Symbols of dedicated TVS thyristors, also called thyristor surge protection device (TSPD). (**c**) Symbols of a gas discharge tubes (GDT) with two electrodes and three electrodes

#### **11.10 Summary**

Table 11.4 summarizes the components presented in this chapter, and Table 11.5 provides a detailed comparison of transient suppression components.

**Table 11.4** Summary of the high-frequency response of conductors, resistors, capacitors, inductors, ferrite beads, and common-mode chokes



**Table 11.5**Summary of transient suppressor devices and their typical properties and applications [5,

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

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# **Chapter 12 Noise Coupling**

*The only difference between success and failure is the ability to take action.*

—Alexander Graham Bell

# **12.1 Coupling Paths**

One of the most important concepts to understand in EMC is the concept of *coupling paths*. To start off, let's see what parts are involved when electromagnetic interference (EMI) happens and why focusing on coupling paths is so important:


Figure 12.1 shows us that if there are issues with EMI and EMC, one has the following three options to overcome these issues:

#### 1. **Reduce the noise level of the noise source.**


**Fig. 12.2** EMI noise coupling paths


Let us have a closer look. There are different kinds of coupling paths, some of them are conductive (galvanic), and some are nonconductive (radiated). Figure 12.2 shows the four different types of coupling:


**Fig. 12.3** Conductive coupling examples. (**a**) Shared path to ground. (**b**) Shared power supply

#### 4. **Radiated coupling** (electromagnetic field coupling)

The four coupling types are discussed in Sects. 12.1.1–12.1.4.

#### *12.1.1 Conductive Coupling*

*Conductive coupling* can occur when two or more circuits share a common path or common conductor and, therefore, a common impedance *Zcom* []. This shared path leads typically to ground or another reference plane (see Fig. 12.3a). Why could a common conductor path lead to EMC problems or a failed EMC test case? Here is an example: if one of the two circuits in Fig. 12.3a experiences an ESD, burst, or surge pulse, a high current may flow for a short time through a common conductor segment and introduces a noise voltage to the other circuit, which may lead to a malfunction. This is the reason why it is crucial having low-impedance earth/ground planes. A low-impedance ground plane helps to minimize the conductive coupling. In addition, a common path often leads to bad signal integrity as well.

Suppose that circuit 1 in Fig. 12.3a drives some power electronics and circuit 2 is a sensitive measurement circuit. The high current of the power electronics (noise current) introduces a noise voltage *V <sup>n</sup>* [V] in the said common impedance *Zcom* [] and will lead to interference on the measurement circuit 2. Another example is shown in Fig. 12.3b, where two circuits share a power supply. In the case that circuit 2 is switched off and *I* <sup>2</sup> [A] suddenly drops to 0 A, voltage *V* <sup>1</sup> [V] will increase by *ΔV* <sup>1</sup> = *I* <sup>2</sup> *Zline*−*supply* <sup>+</sup> *Zline*−*GND* .

Generally speaking, the noise voltage *V <sup>n</sup>* [V] introduced to the victim circuit is proportional to the common impedance *Zcom* []:

$$
\underline{V}\_n = \underline{I}\_n \cdot \underline{Z}\_{com} \tag{12.1}
$$

where:

*I <sup>n</sup>* = current of the noise source in [A] *Zcom* = common impedance of noise source and victim in []

Equation 12.1 states that the noise voltage *V <sup>n</sup>* [V] is independent of the frequency *f* [Hz] of the noise signal. However, it must be mentioned that the common impedance *Zcom(f )* [] is a function of frequency, and it is to be expected that |*Zcom(f )*| [] will increase with increasing frequency (skin effect, inductance *L* [H] of *Zcom*). Note: whether the amplitude of the noise voltage |*V <sup>n</sup>*| [V] is acceptable or not must be decided under consideration of the useful signal's amplitude (signal-tonoise ratio; see Sect. 6.6).

Some possible measures against conductive coupling are given in the list below. However, be aware that not every option is feasible in any case. The best option has to be decided for every case individually, as there is always a trade-off between project timelines, budget, and hardware costs.


#### *12.1.2 Capacitive Coupling*

*Capacitive coupling* can occur if there is a coupling capacitance *C* [F] between two circuits (see Fig. 12.4). The field of concern for the capacitive coupling is the electric *E*-field. Thus, capacitive coupling is a *near-field coupling* (near-field; see Sect. 8.3), which means that the noise source and the victim, which receives the noise, are closely located to each other.

It is assumed that the capacitive coupling discussed in this section is a weak coupling. That means that the noise coupling is a one-way effect from the noise source to the victim circuit (negligible back reaction from victim to source).

Why could capacitive coupling lead to problems during EMC testing? For example, several wires are together in the same cable, which means that each wire is capacitively coupled to the other wires inside that cable (the capacitance is the

**Fig. 12.4** Capacitive coupling between two circuits. (**a**) Physical representation. (**b**) Equivalent circuit

bigger, the longer the cable, and the closer the wires to each other). One of the wires drives the reset signal of a controller system. During EMC testing, an ESD pulse happens to a connector pin that is connected to a wire in that cable. Suppose there are no measures against ESD at the connector pin. In that case, this pulse may discharge through one of the cable's wires and couple capacitively into the other wires in the same cable and potentially resets the controller (if the controller reset signal is not appropriately filtered).

Capacitive coupling happens due to a noise source *v*1*(t)* [V] with high *dv*1*/dt* [V/s] where a noise current *in(t)* [A] is coupled via a stray capacitance *C*<sup>12</sup> [F] to the victim. The faster the voltage change *dv*1*/dt* [V/s] and the larger the stray capacitance *C*<sup>12</sup> [F], the higher the noise current *in(t)* [A] through *C*<sup>12</sup> [F] [6, 7]:

$$i\_n(t) = C\_{12} \frac{dv\_1}{dt} \tag{12.2}$$

The noise voltage *V <sup>n</sup>* [V] introduced to the victim circuit of Fig. 12.4 due to a capacitive coupling can be calculated like this [6]:

$$\underline{V}\_n = \frac{j\omega\left(\frac{C\_{12}}{C\_{12} + C\_{2G}}\right)}{j\omega + \frac{1}{\mathcal{R}(C\_{12} + C\_{2G})}} \underline{V}\_1 \tag{12.3}$$

where:

*V <sup>n</sup>* = noise voltage introduced to the victim (circuit 2) in [V] *V* <sup>1</sup> = voltage of the noise source (circuit 1) in [V] *C*1*<sup>G</sup>* = total capacitance of the noise source circuit to ground in [F] *C*2*<sup>G</sup>* = total capacitance of the victim circuit to ground in [F] *C*<sup>12</sup> = total couple/stray capacitance between circuit 1 and circuit 2 in [F] *R* = load resistor of the victim circuit in [] *ω* = 2*πf* = angular frequency of the sinusoidal noise signal in [rad/sec]

In most cases, *R* [] is of much lower impedance than the parallel impedances of *C*<sup>12</sup> and *C*2*<sup>G</sup>* [6]:

$$R \ll \frac{1}{\frac{1}{1/(j\omega C\_{12})} + \frac{1}{1/(j\omega C\_{2G})}} = \frac{1}{j\omega \left(C\_{12} + C\_{2G}\right)}\tag{12.4}$$

If the condition of Eq. 12.4 is given, we can simplify the noise voltage *V <sup>n</sup>* [V] calculation to [6]:

$$
\underline{V}\_n = j\omega RC\_{12} \underline{V}\_1 \text{ , in case of relatively low } R \tag{12.5}
$$

$$v\_n(t) = i\_n(t) \cdot R = C\_{12} \frac{dv\_{\parallel}}{dt} \cdot R \text{ , in case of relatively low } R \tag{12.6}$$

where:

*V <sup>n</sup>* = noise voltage introduced to the victim (circuit 2) in [V] *vn(t)* = noise voltage introduced to the victim (circuit 2) in [V] *I <sup>n</sup>* = noise current through coupling capacitance in [A] *in(t)* = noise current through coupling capacitance in [A] *V* <sup>1</sup> = voltage of the noise source (circuit 1) in [V] *C*<sup>12</sup> = total couple/stray capacitance between circuit 1 and circuit 2 in [F] *R* = load resistor of the victim circuit in [] *ω* = 2*πf* = angular frequency of the sinusoidal noise signal in [rad/sec]

In case the resistance *R* [] in circuit 2 to ground is large compared to the parallel impedances of *C*<sup>12</sup> and *C*2*G*, such that [6]:

$$R \gg \frac{1}{\frac{1}{1/(j\omega C\_{12})} + \frac{1}{1/(j\omega C\_{2G})}} = \frac{1}{j\omega \left(C\_{12} + C\_{2G}\right)}\tag{12.7}$$

then Eq. 12.4 reduces to a capacitive voltage divider [6]:

$$
\underline{V}\_n = \left(\frac{C\_{12}}{C\_{12} + C\_{2G}}\right) \underline{V}\_1 \text{, in case of relatively large } R \tag{12.8}
$$

where:

*V <sup>n</sup>* = noise voltage introduced to the victim (circuit 2) in [V] *V* <sup>1</sup> = voltage of the noise source (circuit 1) in [V] *C*<sup>12</sup> = total couple/stray capacitance between circuit 1 and circuit 2 in [F] *C*2*<sup>G</sup>* = total capacitance of the victim circuit to ground in [F]

This means for a very large *R* [], the conductive noise coupling is independent of the frequency *ω* [rad/sec] and the noise voltage is of larger amplitude than when *R* [] is small. The frequency response in Fig. 12.5 confirms that.

Given all this information, we can now define some possible measures against a capacitive coupling (assuming that the voltage level *V* <sup>1</sup> [V] of the noise generator and the victim circuit load *R* [] cannot be changed):

**Fig. 12.5** Frequency response of capacitively coupled noise voltage |*V <sup>n</sup>*|

	- **Shielding.** Adding a shield around the noise source or the victim reduces the coupling capacitance to a minimum (given that the shield does cover the complete noise source and/or victim). Shielding against capacitive coupling is already achieved by connecting only one end of the shield to ground [6].
	- **Spatial separation.** If possible, the source of noise and the victim circuit could be spatially separated from one another.
	- **Change conductor orientation.** Let us say the noise source circuit and the victim circuit have conductors parallel to each other. If feasible, a change of the orientation to an angle of 90° between the wires would reduce the coupling capacitance.

### *12.1.3 Inductive Coupling*

Inductive coupling can occur if there is a mutual inductance *M* [H] between two or more circuits. The field of concern for the inductive coupling is the magnetic *H*-field. Thus, inductive coupling is a *near-field coupling* (near-field; see Sect. 8.3), which means that the noise source and the victim, which receives the noise, are closely located to each other.

It is assumed that the inductive coupling discussed in this section is a weak coupling. That means that the noise coupling is a one-way effect from the noise source to the victim circuit (negligible back reaction from victim to source).

How could inductive coupling lead to problems during EMC testing? For example, inductive coupling could lead to issues during EMC testing when high currents flow through cables and PCB traces, e.g., during surge testing where currents up to several [kA] could occur. Large currents often induce voltages in neighboring circuits, which inadvertently disrupts them. These induced voltages are due to inductive coupling.

Inductive coupling happens due to a noise current *i*1*(t)* [A] that changes over time *di*1*/dt* [A/s] and induces a noise voltage *vn(t)* [V] in a nearby circuit via a mutual inductance *M*<sup>12</sup> [H]. The faster the current change *di*1*/dt* [A/s] and the larger the mutual inductance *M*<sup>12</sup> [H], the higher the noise voltage *vn(t)* [V] induced into the victim circuit [6]:

$$v\_n(t) = M\_{12} \frac{di\_1}{dt} \tag{12.9}$$

The noise voltage *V <sup>n</sup>* [V] induced to the victim circuit shown in Fig. 12.6c due to an inductive coupling can be calculated like this [6]:

**Fig. 12.6** Inductive coupling between two closely spaced circuits. (**a**) Physical representation. (**b**) Equivalent circuit (**c**) Induced noise depends on the area enclosed by the distributed victim circuit. Magnetic field of magnetic flux density *B* [T] cuts the area *A* [m2] at an angle *θ* [rad]. (**d**) Frequency response of inductive coupled noise voltage |*V <sup>n</sup>*| [V]

$$\underline{V}\_n = j\omega |B||A|\cos\left(\theta\right) = j\omega\Phi\_{12} = j\omega M\_{12}|\underline{I}\_1|\tag{12.10}$$

where:

*V <sup>n</sup>* = induced noise voltage to the victim (circuit 2) in [V] |*B*| = magnetic flux density caused by the noise source current (circuit 1) in [T] <sup>|</sup>*A*| = enclosed loop area of the victim circuit (circuit 2) in [m2] *<sup>θ</sup>* <sup>=</sup> angle between the magnetic flux vector −→*<sup>B</sup>* and the area vector −→*<sup>A</sup>* in [rad] *Φ*<sup>12</sup> = magnetic flux coupled into the victim circuit area loop (circuit 2) in [Wb] *M*<sup>12</sup> = *Φ*12*/*|*I* <sup>1</sup>| = mutual inductance between circuit 1 and circuit 2 in [H] *I* <sup>1</sup> = noise source current (circuit 1) in [A] *ω* = 2*πf* = angular frequency of the sinusoidal noise signal in [rad/sec]

Equations 12.9 and 12.10 show the parameters that influence the induced noise voltage *V <sup>n</sup>* [V] to the victim. Suppose that the amplitude of the noise current cannot be changed, we still have the following parameters left, which are all directly proportional to the amplitude of the noise voltage: frequency *ω* [rad/sec], the magnetic flux density *B* [T], the area *A* [m2], and the angle *θ* [rad].

Given all this information, we can now define some possible measures against an inductive coupling (assuming that the amplitude of the noise current *I* <sup>1</sup> [A] cannot be changed):


### *12.1.4 Electromagnetic Coupling*

*Electromagnetic coupling* is a *far-field coupling* (far-field; see Sect. 8.3). This means that the noise source and the victim are located far from each other compared to the wavelength *λ* [m]. The field of concern for radiated coupling is the electromagnetic field (*EM*-field; see Sect. 8), where the *H*-field and the *E*-field travel perpendicular to each other, and their amplitudes both fall off with the factor 1/*d*, where *d* [m] is the distance to the radiating source.

The electromagnetic coupling also plays an important role during EMC tests, e.g., during the radiated immunity test according to IEC 61000-4-3, where the EUT is placed in the far-field of the radiating *E*-field test antenna (80 MHz to 6 GHz, typically *d* = 3 m), and, moreover, during radiated emission testing according to CISPR 32, where the EUT is the radiator and the receiving antenna is an *E*-field antenna placed in the far-field (30 MHz to 6 GHz, typically *d* = 3 m or *d* = 10 m).

In the case of an electrical apparatus or installation, any structure may be a good antenna for electromagnetic radiation, e.g., wires, cables, circuit loops or structures on PCBAs, or even larger structures like metal chassis. The best option to develop a robust product against EMI, which produces very low unintended radiation, is always achieved by considering EMC right from the beginning of the project. However, once a product fails EMC testing, there are several possible measures to reduce electromagnetic coupled emissions and increase immunity against electromagnetic fields. Note: because most antennas are reciprocal, a measure against unintended radiation also helps to increase immunity and vice versa.

Here are some typical options on how to reduce unwanted electromagnetic coupling:


#### **12.2 Differential-Mode vs. Common-Mode**

In EMC, a distinction is made between *common-mode* and *differential-mode* noise currents. Both noise currents could lead to unintended radiation, as shown in Sect. 9.9. Sections 12.3 and 12.4 show how the coupling paths presented above can cause differential-mode and common-mode noise. In practice, things are more complicated than stated in this section because coupling mechanisms often happen in cascades and interfere with each other. Nonetheless, you get an idea of where to focus when designing an electrical system or electronic circuit.

**Fig. 12.7** Differential-mode vs. common-mode

Before we jump into the topic of common-mode and differential-mode noise, let us define the terms differential-mode and common-mode in Sects. 12.2.1 (voltages) and 12.2.2 (currents).

#### *12.2.1 Differential-Mode vs. Common-Mode Voltage*

Let us have a look at Fig. 12.7, and let us define the differential-mode and commonmode voltages:


For a transmission line with two conductors—like shown in Fig. 12.7,—the *imbalance factor h* is defined as [4]:

$$\underline{h} = \frac{\underline{Z}\_1}{\underline{Z}\_1 + \underline{Z}\_2} \tag{12.11}$$

where:

*h* = imbalance factor or current divisor factor of a transmission line *Z*<sup>1</sup> = characteristic impedance of conductor 1 with respect to ground in [] *Z*<sup>2</sup> = characteristic impedance of conductor 2 with respect to ground in []

The imbalance factor *h* is used for calculating the common-mode voltage *V CM* [V] of any transmission line for a given differential-mode voltage *V DM* [V] [4]:

$$
\underline{V}\_{DM} = \underline{V}\_1 - \underline{V}\_2 \tag{12.12}
$$

$$
\underline{V}\_{CM} = \underline{h} \cdot \underline{V}\_1 + (1 - \underline{h}) \cdot \underline{V}\_2 \tag{12.13}
$$

where:

*V DM* = differential-mode voltage between conductor 1 and 2 in [V] *V CM* = common-mode voltage to ground in [V] *V* <sup>1</sup> = voltage of conductor 1 with respect to ground in [V] *V* <sup>2</sup> = voltage of conductor 2 with respect to ground in [V] *h* = imbalance factor or current divisor factor of a transmission line

#### *12.2.2 Differential-Mode vs. Common-Mode Current*

The difference between *differential-mode* **Differential-Mode Current** and *common-mode* current:


Let us have a look at Fig. 12.7. The *imbalance factor h* defined in Eq. 12.11 can be used for calculating the differential-mode and common-mode current of any transmission line [4]:

$$
\underline{I}\_{DM} = (1 - \underline{h}) \cdot \underline{I\_1} - \underline{h} \cdot \underline{I\_2} \tag{12.14}
$$

$$
\underline{I}\_{CM} = \underline{I}\_1 + \underline{I}\_2 \tag{12.15}
$$

where:

*I DM* = differential-mode current in [A] *ICM* = common-mode current through conductor 1 and 2 in [A] *I* <sup>1</sup> = current through conductor 1 in [A] *I* <sup>2</sup> = current through conductor 2 in [A] *h* = imbalance factor or current divisor factor of a transmission line

#### **12.3 Differential-Mode Noise Sources**

*Differential-mode noise* sources cannot be found on any bill of material (BOM) or any schematic. These noise sources are unwanted and unintended. Section 9.9.1

**Fig. 12.8** Differential-mode noise current [8]

**Fig. 12.9** How conductive coupling causes differential-mode noise (simplified)

shows how small differential-mode current loops could lead to unwanted radiated emissions.

A not conclusive list of possible differential-mode interference noise sources is given here and then shortly explained in the following sections:


### *12.3.1 Conductive Coupling to Differential-Mode Noise*

Two or more circuits share a common current path (e.g., a common return current path), and the voltage drop of one of these circuits introduces a differential-mode noise voltage *V nDM* [V] in another circuit(s) and vice versa (see Fig. 12.9). The calculation of the differential-mode noise voltage *V nDM* [V] and more details about conductive coupling—also named *common impedance coupling*—are given in Sect. 12.1.1.

#### *12.3.2 Capacitive Coupling to Differential-Mode Noise*

Assuming a noise current *I nDM* [A] is coupled capacitively to one of the two lines of a differential signal. This noise current causes differential-mode noise voltage

**Fig. 12.10** How capacitive coupling to a single conductor causes differential-mode noise (simplified)

**Fig. 12.11** How inductive coupling causes differential-mode noise voltage (simplified)

*V nDM* [V] and could therefore lead to problematic interference on that circuit or unintended RF emissions (see Fig. 12.10). The calculation of the differential-mode noise current *I nCM* [A] and more details about the *capacitive coupling* are given in Sect. 12.1.1.

#### *12.3.3 Inductive Coupling to Differential-Mode Noise*

Let's imagine a magnetic field which induces a noise voltage *V <sup>n</sup>* [V] to a differential signal current loop which leads to a differential-mode noise current *In* [A] in that circuit (see Fig. 12.11). This induced noise voltage and the noise current could lead to, e.g., problematic interference on that circuit or unintended RF emissions. The calculation of the induced differential-mode noise voltage *V <sup>n</sup>* [V] and more details about inductive coupling are given in Sect. 12.1.3.

#### *12.3.4 Common-to-Differential-Mode Conversion*

A common-mode noise current *I nCM* [A] can be converted to a differential-mode noise voltage *V nDM* [V] in the case of unbalanced lines. Unbalanced lines mean that

**Fig. 12.12** How mode conversion (due to unbalanced lines) causes differential-mode noise voltage (simplified)

the lines are not 100 % balanced;—in other words, the forward and return current lines of a differential signal have [3]:


In both cases, a common-mode current leads to a differential-mode voltage (see Fig. 12.12). This type of conversion is called *common-to-differential-mode conversion*.

Electronics designers should especially ensure that signal lines at the input of high-gain instrument amplifiers are well balanced. Otherwise, even a tiny commonmode noise current *I nCM* [A] could cause a significant noise voltage *V nDM* [V] at the output of the high-gain amplifier.

#### **12.4 Common-Mode Noise Sources**

*Common-mode noise* sources cannot be found on any bill of material (BOM) or any schematic. These noise sources are unwanted and unintended. Common-mode currents are the number 1 source of unintentional radiated emissions (Fig. 12.13). Sections 9.9.2, 9.9.3, and 9.9.4 show how common-mode currents in cables could lead to unintended radiated emissions.

A not conclusive list of possible common-mode interference noise sources is given here and then shortly explained in the following sections:


**Fig. 12.13** Common-mode noise current [8]

**Fig. 12.14** Concurrent capacitive coupling on a pair of wire causes common-mode noise currents, which could lead to high radiated emissions (simplified)

#### *12.4.1 Capacitive Coupling to Common-Mode Noise*

Let us imagine that noise currents are coupled capacitively (and equally) onto two differential signal lines, which form a current loop. As a consequence, the coupled common-mode current *I nCM* [A] could lead to radiated or conducted emission issues.

Figure 12.14 shows a simplified setup of how electric near-field coupling could cause common-mode noise. In addition, Sect. 12.1.2 presents the calculation of the coupled noise current and which actions can be taken to reduce it. Finally, how a differential signal leads to common-mode noise with electric field coupling is presented in the following paper [1], where the coupling is called voltage-driven mechanism.

#### *12.4.2 Electromagnetic Coupling to Common-Mode Noise*

A cable or a part of an electronic circuit acts as an antenna and receives electromagnetic radiation, e.g., from radio stations or smartphones, etc (Fig. 12.15). The receiving structure is in the far-field of the transmitter. With this coupling, an unintended noise current is brought to an electrical circuit and could lead to interference. Details about the electromagnetic coupling and how to reduce it can be found in Sect. 12.1.4.

**Fig. 12.15** Electromagnetic coupling that leads to a common-mode current (simplified)

**Fig. 12.16** Noisy reference point that leads to a common-mode current (simplified)

#### *12.4.3 Reference Point Noise to Common-Mode Noise*

A noisy reference point (ground) or a voltage potential difference of spatially divided circuits and systems could cause a common-mode noise current *I nCM* [A] and, therefore, high radiated emissions. Figure 12.16 shows in a simplified form a setup with a noisy reference point. The noise source voltage *V nCM* [V] itself could be caused by a current-driven differential-to-common-mode conversion [1].

#### *12.4.4 Differential-to-Common-Mode Conversion*

Any *change* in electrical balance to ground of a transmission line converts some power of the differential-mode signal into a common-mode signal [4]. This *differential-to-common-mode conversion* happens if—and only if—there is a change in balance of the transmission line [2]. Because this circumstance often causes high radiated emissions and failed EMC tests, the differential-to-common-mode conversion will be analyzed in the following by applying the *imbalance difference modeling* method (IDM [2, 5]).

**Fig. 12.17** Differential-to-common-mode conversion due to imbalance of a transmission line (change of characteristic impedances to ground)

Consider a transmission line that experiences a sudden change in the impedance of its lines to ground. This is typically the case when the transmission line leads from a PCB to a cable: in the connector, the individual conductors of the transmission line experience a sudden change in the characteristic impedance to ground (change in balance or imbalance). Let us look at Fig. 12.17, where a differential-mode signal experiences a sudden change in impedance to ground (at the imbalance interface). This sudden change leads to a *common-mode voltage V AM* [V] and *common-mode current I AM* [A], which we call antenna-mode voltage and antenna-mode current because they could lead to unintended radiation. We defined the *imbalance factor h* in Eq. 12.11. On the left side of the imbalance interface, conductors 1 and 2 have an imbalance factor of *hL* = *Z*1*L/(Z*1*<sup>L</sup>* + *Z*2*L)*, and on the right side of the imbalance interface, the conductors have the imbalance factor of *hR* = *Z*1*R/(Z*1*<sup>R</sup>* +*Z*2*R)*. The common-mode voltage *V AM* [V] and current *I AM* [A] that are developed due to the imbalance change are given as [4, 9]:

$$
\underline{V}\_{AM} = \underline{V}\_{DM} \cdot \Delta \underline{h} \tag{12.16}
$$

$$
\underline{I}\_{AM} = \frac{\underline{V}\_{AM}}{\underline{Z}\_{AM}}\tag{12.17}
$$

where:

*V DM* = differential-mode voltage at the imbalance interface in [V] *V AM* = common-mode voltage developed at the imbalance interface in [V] *Δh* = *hL* − *hR* = change of imbalance factor *h* at the imbalance interface *I AM* = common-mode current through conductor 1 and 2 in [A] *ZAM* = input impedance of antenna that is formed by conductors 1 and 2 in []

Imbalance difference modeling (IDM) is a powerful method for quickly calculating the worst-case common-mode voltage driving a cable (as an antenna). As a worst-case, it can be assumed an imbalance factor change of *Δh* = 0*.*5 (perfectly balanced to completely unbalanced and vice versa), which means that the developed common-mode voltage is half the differential-mode amplitude:

$$|\underline{V}\_{AM}| = |\underline{V}\_{DM}| \cdot 0.5\tag{12.18}$$

In EMC, the theoretical maximum field strength *Emax* [V/m] for a given setup at distance *d* [m] in the far-field is of primary interest. Therefore, here is a way for a worst-case estimation of the radiated emissions of a digital signal due to transmission line imbalance (e.g., change from an unbalanced microstrip line to a balanced ribbon cable):


$$V\_{AM} = 0.5 \cdot V\_{DM}$$

3. **Worst-case radiated emissions EAM***,***max**. Use Eq. 9.48 on page 130 to calculate the estimated worst-case radiated emissions *Emax* [V/m] for the given distance *d* [m] (far-field) and the antenna-mode voltage *VAM* [V] (current, voltage, and field-strength are RMS values):

$$E\_{AM,max} \approx \begin{cases} \frac{60 \cdot I\_{AM,max}}{d} \cdot \frac{2}{\sin\left(\sqrt{2}\right)} & \text{when } l\_{cable} \le \frac{\lambda}{2} \\\frac{60 \cdot I\_{AM,max}}{d} \cdot \frac{2}{\sin\left(\sqrt{\frac{\lambda}{l\_{cable}}}\right)} & \text{when } l\_{cable} > \frac{\lambda}{2} \end{cases} \tag{12.19}$$

where:

$$
\underline{I}\_{AM,max} = \frac{\underline{V}\_{AM}}{\frac{36.5 \text{ } \Omega}{k\_{board}k\_{cable}}} \tag{12.20}
$$

$$k\_{board} = \begin{cases} \sin\left(\frac{2\pi \cdot l\_{board}}{\lambda}\right) & \text{when } l\_{board} \le \frac{\lambda}{4} \\ 1.0 & \text{otherwise} \end{cases} \tag{12.21}$$

$$k\_{cable} = \begin{cases} \sin\left(\frac{2\pi \cdot l\_{cable}}{\lambda}\right) & \text{when } l\_{cable} \le \frac{\lambda}{4} \\ 1.0 & \text{otherwise} \end{cases} \tag{12.22}$$

$$l\_{board} = \frac{1 + \frac{2L}{W}}{\frac{2L}{W}} \cdot \sqrt{L^2 + W^2} \tag{12.23}$$

*VAM* = common-mode RMS voltage driving the monopole antenna (cable) in [V] *IAM,max* = highest common-mode RMS current that exists on the cable in [A] *Rmin* = 36*.*5 = radiation resistance of a resonant *λ/*4-monopole antenna in [] *lboard* = effective board length in [m] *lcable* = cable length in [m] *L* = length of the PCBA (board) in [m] *W* = width of the PCBA (board) in [m] *kboard* = impact of board size *lboard* on *Rmin* in case *lboard* ≤ *λ/*4 *kcable* = impact of cable length *lcable* on *Rmin* in case *lcable* ≤ *λ/*4 *d* = distance to the radiating structure (PCBA, cable) in [m] *λ* = wavelength of the sinusoidal signal *V AM* in [m]

### **12.5 Summary**

Table 12.1 presents a summary of EMI noise coupling paths.


**Table 12.1** EMI noise coupling path summary

# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

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# **Chapter 13 Shielding**

*Action expresses priorities.*

—Mahatma Gandhi

In the field of EMC, shields are used to:


The shielding theory presented in this book is based on the accepted shielding theory for electromagnetic waves, initially proposed by Schelkunoff ((1943) Electromagnetic waves. D. van Nostrand Company Inc, New York, pp 303–312) in 1943. The formulas in this chapter are approximations for shields with high electrical conductivity. Before we jump into the theory of shielding, here are two practical pieces of advice:


# **13.1 Shielding Theory**

*Shielding* of electromagnetic waves is usually achieved by:


#### **13.2 Shielding Effectiveness**

Let us have a look at Fig. 13.2 and let us define a few terms. The *shielding effectiveness* (SE) describes how well a shield attenuates an incident wave (electrical field strength *Ei* [V/m], magnetic field strength *Hi* [A/m]) when propagating through a shield. After passing through a shield, the remaining wave has a field strength of *Et* [V/m] and *Ht* [A/m]. The reflected wave has field strength *Er* [V/m] and *Hr* [A/m]. Shielding effectiveness is now defined as [4]:

$$\text{SE}\_{E,dB} = 20 \log\_{10} \left| \frac{E\_l}{E\_l} \right| \tag{13.1}$$

$$\text{SSE}\_{H,dB} = 20 \log\_{10} \left| \frac{H\_l}{H\_l} \right| \tag{13.2}$$

where:


The shielding effectiveness of the *E*-field and the *H*-field compared:


The shielding effectiveness is the sum of *reflection loss RdB* [dB], *absorption loss AdB* [dB], and the *multiple-reflection loss correction MdB* [dB] (Fig. 13.9).

$$\text{SE}\_{dB} = \underbrace{\text{R}\_{dB}}\_{\begin{array}{c} \text{Reflection} \\ \text{I} \end{array}} + \underbrace{\text{A}\_{dB}}\_{\begin{array}{c} \text{Absorption} \\ \text{loss} \end{array}} + \underbrace{\text{M}\_{dB}}\_{\begin{array}{c} \text{Multiple} \\ \text{reflection} \end{array}} \tag{13.3}$$

Assuming a uniform plane wave arriving perpendicular to a solid shield, which does not have any slots or apertures, we can write [4]:

$$R\_{dB} = 20\log\_{10}\left|\frac{\left(\underline{Z}\_w + \underline{\eta}\_s\right)^2}{4\underline{Z}\_w\underline{\eta}\_s}\right|\tag{13.4}$$

$$A\_{dB} = 20\log\_{10}\left|e^{at}\right|\tag{13.5}$$

$$M\_{dB} = 20\log\_{10}\left|1 - \left(\frac{\underline{Z}\_w - \underline{\eta}\_s}{\underline{Z}\_w + \underline{\eta}\_s}\right)^2 e^{-2\alpha t}\right|\tag{13.6}$$

where:

*RdB* = reflection loss of both boundaries of the shield (attenuation due to reflection of the incident wave at the boundaries of the shield, *E*-fields are primarily reflected at the first boundary—when entering the shield, whereas *H*-fields are primarily reflected at the second boundary—when leaving the shield) in [dB]


*Zw* = *η*<sup>0</sup> ≈ 377 wave impedance of the incident wave in []

*ηs* = intrinsic impedance of the shield material in []

*t* = thickness of the shield in [m]

*α* = attenuation coefficient of the shield material in [1/m]

Looking at the equations above, the following can be said:

	- *R* dominates at lower frequencies because the absorption loss *A* is low.
	- *R* in the near-field differs with changing radiation source impedance.
	- *R* in the near-field is different from *R* in the far-field.
	- *A* dominates at higher frequencies, where the skin depth is small.
	- *A* is identical for near-field and far-field radiation.
	- *M* can be neglected for good conducting (|*ηs*| |*Zw*|) and thick shields (*t δ*), where *AdB <* 15 dB.
	- *M* must be considered for thin shields (*t δ*), where multiple reflections inside the shield reduce the shielding effectiveness.

# **13.3 Far-Field Shielding**

Figure 13.3 presents the absorption, reflection, and multiple-reflection loss correction of a solid shield in the *far-field*. It can be seen how the reflection loss dominates for low frequencies and the absorption loss for high frequencies.

# *13.3.1 Reflection Loss R for Far-Field Shielding*

The reflection loss *RdB* [dB] of a solid shield with good conductivity (|*ηs* | |*η*0|) in the far-field can be approximated as:

**Fig. 13.3** Shielding effectiveness of a solid aluminum shield with *t* = 0*.*1 mm in the far-field

$$\mathcal{R} \approx \left| \frac{\eta\_0}{4\underline{\eta}\_s} \right| = \sqrt{\frac{\sigma}{4\pi f \mu\_r' \varepsilon\_0}}\tag{13.7}$$

$$\mathcal{R}\_{dB} \approx 20 \log\_{10} \left| \frac{\eta\_0}{4 \underline{\eta}\_s} \right| = 20 \log\_{10} \left[ \sqrt{\frac{\sigma}{4 \pi f \mu\_r' \varepsilon\_0}} \right] \tag{13.8}$$

where:

*R* = reflection loss for a plane wave and a solid shield with good conductivity *η*<sup>0</sup> = *μ*0*/ε*<sup>0</sup> ≈ 377 = intrinsic impedance of free-space in [] |*ηs* |=|*(*1 + *j )πf μ*0*μ*- *r/σ*| = 2*πf μ*0*μ*- *r/σ* = shield impedance in [] *σ* = specific conductance of the shield material in [S/m] *f* = frequency of the sinusoidal plane wave in [Hz] *μ*- *<sup>r</sup>* = relative magnetic permeability of the shield material *<sup>ε</sup>*<sup>0</sup> <sup>=</sup> <sup>8</sup>*.*<sup>854</sup> · <sup>10</sup>−<sup>12</sup> F/m <sup>=</sup> permittivity of vacuum, absolute permittivity

#### *13.3.2 Absorption Loss A for Far-Field Shielding*

The absorption loss *AdB* [dB] of a solid shield barrier of thickness *t* [m] and of good conductivity (*α* = 1*/δ*) in the far-field can be approximated as:

$$A \approx e^{at} = e^{t/\delta} \tag{13.9}$$

$$A\_{dB} \approx 20 \log\_{10} e^{t/\delta} = 8.7 \cdot \frac{t}{\delta} = 8.7 \cdot t \cdot \sqrt{\pi f \mu\_0 \mu\_r' \sigma} \tag{13.10}$$

where:

*A* = absorption loss for a plane wave and a solid shield with good conductivity *α* = 1*/δ* = attenuation coefficient of the shield material in [1/m]

*t* = thickness of the shield in [m] *δ* = 1*/ πf μ*0*μ*- *rσ* = skin depth for a wave with frequency *f* in [m] *f* = frequency of the sinusoidal plane wave in [Hz] *μ*- *<sup>r</sup>* = relative magnetic permeability of the shield material *<sup>μ</sup>*<sup>0</sup> <sup>=</sup> 12.57 ·10−<sup>7</sup> H/m <sup>=</sup> permeability of vacuum, absolute permeability *σ* = specific conductance of the shield material in [S/m]

# *13.3.3 Multiple-Reflection Loss Correction M for Far-Field Shielding*

The multiple-reflection loss correction *MdB* [dB] must be considered for thin shields, where the absorption loss *AdB* [dB] is below 15 dB. For a plane wave in the far-field and a solid shield barrier of good conductivity, *MdB* [dB] can be approximately calculated as [3]:

$$M \approx 1 - e^{-2t/\delta} \tag{13.11}$$

$$M\_{dB} \approx 20\log\_{10}\left(1 - e^{-2t/\delta}\right) \tag{13.12}$$

where:

*M* = multiple-reflection loss correction for a plane wave and a thin solid shield with good conductivity

*t* = thickness of the shield in [m] *δ* = 1*/ πf μ*0*μ*- *rσ* = skin depth for a wave with frequency *f* in [m]

#### **13.4 Near-Field Shielding**

Figure 13.4 shows that the reflection loss *RdB* [dB] for electric dipole antennas in the *near-field* is much higher compared to the far-field and compared to magnetic dipoles in the near-field.

Comparing *near-field* vs. *far-field* shielding, we can say:


**Fig. 13.4** Reflection loss *RdB* [dB] for various wave impedances *Zw* []

• **Multiple-reflection loss correction.** *MdB* [dB] is affected by the wave impedance *Zw* [] (see Eq. 13.6). However, the influence of the wave impedance on *MdB* [dB] is only minor. Therefore, in most cases, it can be assumed that *MdB* [dB] is not affected significantly by the wave impedance.

#### *13.4.1 Near-Field Shielding of Electric Sources*

To calculate the approximate absorption loss *AdB* [dB] and multiple-reflection loss correction *MdB* [dB], Eqs. 13.10 and 13.12 can be used. *RdB,E* [dB] is the reflection loss when shielding a near-field wave from an electric dipole (predominant *E*-field) and it is approximately:

$$\mathcal{R}\_e \approx \left| \frac{\underline{Z}\_{we}}{4\underline{\eta}\_s} \right| \tag{13.13}$$

$$R\_{dB,e} \approx 20 \log\_{10} \left| \frac{\underline{Z}\_{we}}{4 \underline{\eta}\_s} \right| = 244 + 10 \log\_{10} \left( \frac{\sigma}{f^3 \varepsilon\_{rw}^{\prime 2} \mu\_{rs}^{\prime} d^2} \right) \tag{13.14}$$

where:


*f* = frequency of the sinusoidal plane wave in [Hz]

*μ*- *rs* = relative magnetic permeability of the shield material

*ε*- *rw* = relative electric permittivity of the medium through which the near-field wave is propagating (can be assumed to be 1.0 in most cases)

*d* = distance from the source antenna (electric dipole) to the shield surface in [m]

The near-field to far-field boundary and wave impedances are discussed in Sects. 8.3 and 8.5.

#### *13.4.2 Near-Field Shielding of Magnetic Sources*

To calculate the approximate absorption loss *AdB* and multiple-reflection loss correction *MdB* [dB], Eqs. 13.10 and 13.12 can be used. *RdB,M* [dB] is the reflection loss when shielding a near-field wave from a magnetic dipole (predominant *H*-field) and it is approximately:

$$\mathcal{R}\_m \approx \left| \frac{\underline{Z}\_{wm}}{4\underline{\eta}\_s} \right| \tag{13.15}$$

$$R\_{dB,m} \approx 20\log\_{10}\left|\frac{\underline{Z}\_{wm}}{4\underline{\eta}\_s}\right| = -63 + 10\log\_{10}\left(\frac{\sigma f d^2 \mu\_{rw}^2}{\mu\_{rs}'}\right) \tag{13.16}$$

where:


*d* = distance from the source antenna (electric dipole) to the shield surface in [m]

Note: Eq. 13.16 is an approximation and assumes |*Zwm*||*ηs* |. In case of a negative reflection loss *RdB,m*, set *RdB,m* = 0 and *MdBm* = 0. In case *RdB,m* is positive and near zero, Eq. 13.16 is in slight error [3].

The near-field to far-field boundary and wave impedances are discussed in Sects. 8.3 and 8.5.

#### *13.4.3 Low-Frequency Magnetic Field Shielding*

Low-frequency (*f <* 100 kHz) magnetic field waves are the most difficult waves to shield because the absorption loss is low due to the low-frequency *f* [Hz] and the reflection loss is low due to the low wave impedance *Zw* []. Figure 13.5 illustrates this problem for a solid 1 mm aluminum shield in the far-field and the near-field.

Therefore, two additional methods for *shielding low-frequency magnetic fields* are presented here:


#### **13.5 Slots and Apertures**

*Slots and apertures* can cause considerable leakage if their largest dimension is *l > (λ/*10*)*. In addition, they are efficient radiators (yes, radiators!) when their maximum linear dimension *l* [m] is equal to *λ/*2 [2]. Thus, if a slot or aperture has a linear dimension of *l* ≥ *λ/*2, the shielding effectiveness SE*dB* [dB] can be assumed to be 0 dB.

At high frequencies (*f >* 1 MHz), slots and apertures are usually of more concern than the attenuation or reflection loss of a shield, because the sum of attenuation and reflection loss of any reasonably thick (*t δ*) and conductive shield (|*ηs* | |*Zw*|) is higher than 100 dB (see Fig. 13.5). In other words, in cases where the intrinsic SE of a shielding material is high, the SE is defined by the slots and apertures of the shield.

#### *13.5.1 Single Aperture*

If we assume a shield with high intrinsic SE, so that the SE is defined by the slots and apertures of the shielding enclosure, the SE can be calculated based on the maximum linear dimension *l* [m] (which must be less than *λ/*2). In the case of one single aperture, we can write [3]:

**Fig. 13.5** Approximate shielding effectiveness of a solid aluminum shield with thickness *t* = 1 mm. (**a**) In the far-field. (**b**) In the near-field of an electric dipole at distance *d* = 10 mm. (**c**) In the near-field of a magnetic dipole at distance *d* = 10 mm

$$\text{SE}\_{dB} \approx 20 \log\_{10} \left( \frac{\lambda}{2l} \right) \tag{13.17}$$

where:

*λ* = wavelength of the sinusoidal wave in [m] *l* = maximum linear dimension of one single aperture (*l < λ/*2) in [m]

#### *13.5.2 Multiple Apertures*

If there is not only one aperture but multiple apertures—as shown in Fig. 13.8b the shielding effectiveness will be reduced even further. The shielding effectiveness of a linear array of equally and closely spaced apertures *n* of maximum length *l* [m]—where the total array length *larray* [m] is less than *λ/*2—is approximately [3]:

$$\text{SE}\_{dB} \approx 20 \log\_{10} \left( \frac{\lambda}{2l} \right) - 20 \log\_{10} \sqrt{n} = 20 \log\_{10} \left( \frac{\lambda}{2l\sqrt{n}} \right) \tag{13.18}$$

where:

*λ* = wavelength of the sinusoidal wave in [m] *l* = maximum linear dimension of one single aperture with *l < λ/*2 in [m] *n* = number of closely spaced, identical apertures in an array with *larray < λ/*2

In the case of a multidimensional array of *m* rows (and *m<n*; see Fig. 13.8c), the additional rows (second, third, . . .) will not reduce the SE significantly [3].

**Fig. 13.8** Apertures in shields. (**a**) Single aperture of maximum linear dimension *l* [m]. (**b**) Linear array of identical and closely spaced apertures. (**c**) Multidimensional array of identical and closely spaced apertures

Thus, the SE of a multidimensional array of equal-sized apertures is the SE of one single hole, minus the SE reduction of the first row of *n* apertures (see Eq. 13.18, Fig. 13.10).

Apertures located on different surfaces, which all look in different directions, do not decrease the overall shielding effectiveness because they radiate in different directions [3]. Thus, it is a good idea to distribute apertures around the surface of a shielding enclosure.

**Fig. 13.9** Shielding

material

**Fig. 13.11** Apertures shaped to form a waveguide. (**a**) Circular aperture with waveguide. Left: cross-sectional view. Right: frontal view. (**b**) Rectangular aperture with waveguide. Left: crosssectional view. Right: frontal view

#### *13.5.3 Waveguide Below Cutoff*

The shielding effectiveness of an aperture can be improved by adding a depth to the aperture. Figure 13.11 shows apertures, which have been extended to *waveguides*. The concept is as follows: the waveguide has a defined cutoff frequency *fc* [Hz], and when a wave with frequency *f fc* enters the waveguide, it will be attenuated. Above the cutoff frequency *fc* [Hz], the waveguide is very efficient at passing all frequencies and thus does not improve the SE of the aperture anymore.

First, we have to determine the cutoff frequency *fc* [Hz] of the waveguide aperture, which is given for free-space (air, where *v* ≈ *c*) as [3, 4]:

$$f\_{c,cwg} \approx \frac{1.8412 \cdot v}{\pi D} = \frac{1.8412 \cdot c}{\pi D} \tag{13.19}$$

$$f\_{c,rwg} \approx \frac{v}{2w} = \frac{c}{2w} \tag{13.20}$$

where:

*fc,cwg* = waveguide cutoff frequency of a circular waveguide in [Hz] *fc,rwg* = waveguide cutoff frequency of a rectangular waveguide in [Hz] *v* = velocity of the wave in [m/sec] *<sup>c</sup>* <sup>=</sup> <sup>1</sup>*/(*√*μ*0*ε*0*)* <sup>=</sup> <sup>2</sup>*.*<sup>998</sup> · 108 m/sec <sup>=</sup> speed of light *w* = largest inner side length (width) of rectangular waveguide in [m]

*D* = inner diameter of circular waveguide in [m]

Now, we can calculate the shielding effectiveness SE*dB* of *n* identical and closely spaced waveguide apertures in case the wave has a frequency *f fc* [4] [3]:

$$\text{SE}\_{dB, cwg} \approx 20 \log\_{10} \left( \frac{\lambda}{2l\sqrt{n}} \right) + 32 \frac{t}{w} \tag{13.21}$$

$$\text{SE}\_{dB, rwg} \approx 20 \log\_{10} \left( \frac{\lambda}{2l\sqrt{n}} \right) + 27.3 \frac{t}{D} \tag{13.22}$$

where:

SE*dB,cwg* = shielding effectiveness of a circular waveguide aperture in [dB] SE*dB,rwg* = shielding effectiveness of a rectangular waveguide aperture in [dB] *λ* = wavelength of the sinusoidal wave in [m] *l* = maximum linear dimension of one single aperture (*l < λ/*2) in [m] *w* = largest inner side length (width) of rectangular waveguide in [m] *D* = inner diameter of circular waveguide in [m] *t* = depth of the waveguide in [m] *n* = number of closely spaced, identical apertures in an array with *larray < λ/*2 (set *n* = 1 in case of a single aperture)

#### **13.6 Grounding of Shields**

First, a disclaimer: this section is about solid shielding enclosures and not about shielded cables. Grounding of cable shields is discussed in Sect. 13.7.4.

A solid *shielding enclosure* does not need to be grounded to be an effective shield for electromagnetic waves from outside of the enclosure to the inside, and vice versa [3]. However, metallic shields are often grounded for safety reasons and to protect the inside from electrostatic charges.

In case cables are leaving or entering the shield, consider these points:


Consider the galvanic series to prevent corrosion when connecting dissimilar metals for grounding (e.g., aluminum and steel; see Table H.2). This is especially important in the case of safety grounds.

### **13.7 Cable Shields**

# *13.7.1 Transfer Impedance* **Z***<sup>t</sup>*

The *transfer impedance Zt* [/m] is a property of *shielded cables*, and it is used to measure the shielding effectiveness of cable shields. The transfer impedance describes the magnetic coupling between the noise current *I* [A] along the shield to the induced voltage *V* [V] to the inner conductor per length *l* [m]. The smaller the transfer impedance *Zt* [/m], the more effective the shield.

Figure 13.12 shows the schematic diagram of a transfer impedance measurement setup of a cable shield. *Zt* [/m] can be calculated as [6]:

$$\underline{Z}\_{l} = \frac{\underline{V}}{l \cdot \underline{I}}\tag{13.23}$$

where:

*V* = measured voltage between the center conductor and the inner surface of the shield in [V]

*I* = test current in the shield in [A]

*l* = length of the shielded cable under test in [m]

The transfer impedance *Zt(f )* [/m] is a function of frequency *f* [Hz]. At low frequencies *f* [Hz], *Zt* ·*l* is equal the direct current resistance *RDC* [] of the shield. At high frequencies (*>* 1 MHz), *Zt* [/m] of a solid shield decreases because of the skin effect (current flows at outside layer of the shield), whereas for braided shields, *Zt* [/m] increases at high frequencies (see Fig. 13.13). Remember: a low transfer impedance *Zt* [/m] means high shielding effectiveness.

#### *13.7.2 Cable Shielding Against Capacitive Coupling*

Figure 13.14 shows a setup of two closely spaced circuits, where circuit 1 is the noise source and circuit 2 is the victim. The noise is capacitively coupled from circuit 1 to circuit 2, where circuit 2 is shielded and the shield is grounded. Whether

**Fig. 13.12** Transfer impedance *Zt* [/m] measurement circuit

**Fig. 13.13** Typical normalized transfer impedance of a solid shield and a braided shield

the shield is grounded at both ends or just at one end does not matter in this case, because the shielding effectiveness against capacitively coupled noise does not depend on a closed-loop shield current; the shielding effectiveness against capacitively coupled noise does mainly depend on the impedance of the shield to ground (the lower the impedance, the better), the value of the stray coupling capacitance *C*<sup>12</sup> [F] (the lower, the better), and the victim's impedance |*Z*2| (the lower, the better).

In the case of a setup like shown in Fig. 13.14, where the shield is grounded with very low impedance, the noise voltage *V <sup>n</sup>* [V] can be calculated as:

*V <sup>n</sup>* = *V* <sup>1</sup> <sup>1</sup> <sup>1</sup> *<sup>Z</sup>*<sup>2</sup> <sup>+</sup>*jωC*2*S*+*jωC*2*<sup>G</sup>* 1 *jωC*<sup>12</sup> <sup>+</sup> <sup>1</sup> <sup>1</sup> *<sup>Z</sup>*<sup>2</sup> <sup>+</sup>*jωC*2*S*+*jωC*2*<sup>G</sup>* = *V* <sup>1</sup> *jωC*12*Z*<sup>2</sup> <sup>1</sup> <sup>+</sup> *jωZ*2*(C*2*<sup>G</sup>* <sup>+</sup> *<sup>C</sup>*2*S)* (13.24)

where:

*V* <sup>1</sup> = voltage of the noise source (circuit 1) in [V] *Z*<sup>2</sup> = impedance of the victim circuit to ground in [] *C*1*<sup>G</sup>* = capacitance of circuit 1 (noise source) to ground (neglected here, because *C*1*<sup>G</sup>* is parallel to the ideal voltage source *V* 1) in [F] *C*2*<sup>G</sup>* = capacitance of circuit 2 (victim) to ground in [F]

**Fig. 13.15** Frequency response of capacitively coupled noise voltage |*V <sup>n</sup>*| [V] to a shielded cable (where the shield is a good conductor and grounded)


Equation 13.24 and Fig. 13.15 lead us to the conclusion that effective shielding against capacitive coupling is all about minimizing the coupling stray capacitance *C*<sup>12</sup> [F]. Thus, the following two points are important to remember when it comes to shielding against electric field coupling:


#### *13.7.3 Cable Shielding Against Inductive Coupling*

Figure 13.16 shows a setup of two closely spaced circuits, where circuit 1 is the noise source and circuit 2 is the victim. The noise is inductively coupled from circuit 1 to circuit 2, where circuit 2 is shielded, and the shield is grounded at both ends. If the shield is made of a nonmagnetic material and the shield would only be connected to ground at one end, there would be no shielding effect, because the

**Fig. 13.16** Simplified circuit representation of cable shielding against inductive coupling

shielding effectiveness against inductively coupled noise depends on the induced shield current *I <sup>S</sup>* [A] (in the case of a nonmagnetic shield material).

The shielding effect due to the induced shield current is explained like this:


The induced noise voltage *V* <sup>12</sup> [V] from circuit 1 (noise source) to circuit 2 (victim) is:

$$
\underline{V}\_{12} = j a \mathcal{M}\_{12} \underline{I}\_1 \tag{13.25}
$$

where:

*M*<sup>12</sup> = *Φ*12*/ i*1*(t)* = mutual inductance between circuit 1 and circuit 2 in [H] *I* <sup>1</sup> = current of the noise source (circuit 1) in [A] *ω* = 2*πf* = angular frequency of the sinusoidal noise signal in [rad/sec]

The induced shield voltage *V* <sup>1</sup>*<sup>S</sup>* [V] from circuit 1 (noise source) to the shield which must be grounded at both ends—is:

$$
\underline{V}\_{1\,\mathrm{S}} = j\alpha \mathcal{M}\_{1\,\mathrm{S}} \underline{I}\_{1} \tag{13.26}
$$

where:

*M*1*<sup>S</sup>* = *Φ*1*S/ i*1*(t)* = mutual inductance between circuit 1 and the shield in [H] *I* <sup>1</sup> = current of the noise source (circuit 1) in [A] *ω* = 2*πf* = angular frequency of the sinusoidal noise signal in [rad/sec]

If we model the shield as simple *RL*-series-circuit, we can write the shield current *I <sup>S</sup>* [A], induced by the noise current *I* <sup>1</sup> [A] from circuit 1, as:

$$\underline{I}\_{\mathcal{S}} = \frac{\underline{V}\_{1\mathcal{S}}}{R\_{\mathcal{S}} + j\omega L\_{\mathcal{S}}} \tag{13.27}$$

where:

*V* <sup>1</sup>*<sup>S</sup>* = induced shield voltage to shield due to *I* <sup>1</sup> in circuit 1 (noise source) in [V] *RS* = resistance of the shield current loop in [] *LS* = inductance of the shield current loop in [H] *ω* = 2*πf* = angular frequency of the sinusoidal noise signal in [rad/sec]

The mutual inductance *MS*<sup>2</sup> [H] between the shield and any inner conductor is equal to the shield inductance *LS* [H] [3]. Therefore, the induced compensation voltage *V <sup>S</sup>*<sup>2</sup> [V] to the shielded conductor 2 due to the shield current *I* <sup>1</sup> [A] can be written as:

$$
\underline{V}\_{S2} = j\omega \underline{M}\_{S2} \underline{I}\_{\underline{S}} = j\omega \underline{L}\_{\underline{S}} \underline{I}\_{\underline{S}} \tag{13.28}
$$

where:

*M*1*<sup>S</sup>* = mutual inductance between the shield and any inner conductor in [H] *LS* = inductance of the shield current loop in [H] *I <sup>S</sup>* = induced shield current due to current *I* <sup>1</sup> in circuit 1 in [A] *ω* = 2*πf* = angular frequency of the sinusoidal noise signal in [rad/sec]

If the mutual inductance *M*<sup>12</sup> [H] of circuit 1 to circuit 2 is equal to the mutual inductance *M*1*<sup>S</sup>* [H] of circuit 1 to the shield around circuit 2 (because the loop area and orientation of the shield and the inner conductors are nearly identical), we can calculate the total induced noise voltage *V <sup>n</sup>* [V] in circuit 2 (victim) as:

$$\underline{V}\_{n} = \underline{V}\_{12} - \underline{V}\_{S2} = j\omega\mathcal{M}\_{12}\underline{I}\_{1} - j\omega\mathcal{M}\_{S2}\underline{I}\_{S} = j\omega\left(\mathcal{M}\_{12}\underline{I}\_{1} - L\_{S}\underline{I}\_{S}\right) \tag{13.29}$$

$$= j\omega\left(\mathcal{M}\_{12}\underline{I}\_{1} - L\_{S}\frac{\underline{V}\_{1S}}{R\_{S} + j\omega L\_{S}}\right) = j\omega\left(\mathcal{M}\_{12}\underline{I}\_{1} - \frac{j\omega\mathcal{M}\_{1S}\underline{I}\_{1}}{R\_{S}/L\_{S} + j\omega}\right)$$

$$\dot{\omega} = j\omega \underline{I}\_1 \left( M\_{12} - \frac{j\omega M\_{12}}{R\_S/L\_S + j\omega} \right) = j\omega M\_{12} \underline{I}\_1 \left( \frac{R\_S/L\_S}{R\_S/L\_S + j\omega} \right) \tag{13.31}$$

**Fig. 13.17** Frequency response of inductively coupled noise voltage |*V <sup>n</sup>*| [V] to a shielded cable vs. an unshielded cable (where the shield is a good conductor, nonmagnetic, and grounded at both ends)

where:

*M*<sup>12</sup> = *Φ*12*/ i*1*(t)* = mutual inductance between circuit 1 and circuit 2 in [H] *I* <sup>1</sup> = current of the noise source (circuit 1) in [A] *RS* = resistance of the shield current loop in [] *LS* = inductance of the shield current loop in [H] *ω* = 2*πf* = angular frequency of the sinusoidal noise signal in [rad/sec]

Equation 13.31 and Fig. 13.17 lead us to the conclusion that shielding against magnetic field coupling is all about minimizing the mutual inductance *M*<sup>12</sup> [H] and minimizing the shield current loop resistance *RS* [] (given the shield is connected at both ends of the cable and is of nonmagnetic material):


Equation 13.31 shows that in the case of nonmagnetic shielding material (*μ*- *<sup>r</sup>* = 1), there is no shielding effect for low-frequency magnetic fields and that the shielding effectiveness increases with increasing frequency (in case the shielding is grounded at both ends).

#### *13.7.4 Grounding of Cable Shields*

To be most effective, cable shields should be connected to the outside of the shielded enclosure if available or to circuit ground otherwise. The connection(s) of the shield to chassis/ground should be of low impedance. The most effective termination is a 360◦ connection of the shield. This means that the shield is contacted around the entire circumference to ground (e.g., with a conductive cable clamp). Avoid pigtail ground connections because they are of high impedance at high frequencies and noise can be picked up by pigtails.

Question: connecting a cable shield to ground at both ends of the cable or only at one end? Answer: it depends:

• **Shield grounded at one end.** Grounding the shield at only one end prevents lowfrequency (*f <* 100 kHz) noise current to flow along the shield, which could couple noise to the inner conductors. However, at higher frequencies, a current flow cannot be prevented effectively due to the stray capacitance of the shield to other conductors and ground.

In addition, grounding at only one end leads to the fact that the shield could act as an antenna for high frequencies where the shield is an effective antenna (e.g., at *l* = *λ/*4). Therefore, shield grounding at only one end is only recommended for low-frequency applications with short cables, where the primary goal is the protection against electric *E*-fields.

• **Shield grounded at both ends.** Whenever in doubt, connect the cable shield at both ends to chassis/ground. Shield grounding at both ends is recommended for high-frequency (*f >* 100 *kHz*), digital circuit applications and when the cable length *l* [m] is longer than *l > λ/*20, where *λ* [m] is the wavelength of the highest significant frequency of the functional signal inside the cable. In addition, grounding at both ends acts as an additional measure against magnetic *H*-field coupling, because the magnetic field induces a current to the shield, which induces a voltage in the shielded conductor, which ideally cancels out the induced noise voltage (see Sect. 13.7.3).

In the case of a solid shield, noise currents with frequency *f >* 1 *MHz* flow primarily at the shield's outer surface. This effect leads to the circumstance that a coaxial cable acts like a triaxial cable (a cable with two shields), where the signal return current flows at the inner surface of the shield and the noise current at the outer surface. This means that there is nearly any common impedance coupling in coaxial cables for frequencies above *f >* 1 *MHz*.

• **Hybrid shield grounding.** Figure 13.18 shows the principle of the *hybrid cable shield grounding*. This method of grounding enables a selective current flow through the shield. For high-frequency signals, both sides are connected to ground, which enables the high-frequency current to flow through the shield, and due to the skin effect, the high-frequency current flows at the outer surface of the shield. Thus, the inner conductor is protected against high-frequency

**Fig. 13.19** Proposed workflow for designing a shielded enclosure

magnetic fields. On the other hand, low-frequency signals (DC or mains power frequency signals of 50 Hz or 60 Hz) are blocked by the capacitor *CG* [F] in Fig. 13.18. Hybrid cable shield grounding is a good option to minimize DC and low-frequency shield currents in case of high voltage differences at both ends of the cable while maintaining a good SE against high-frequency noise.

#### **13.8 Summary**

Figure 13.19 presents a proposal for the design workflow of a shielded enclosure. Table 13.1 compares cable shield terminations at one end vs. both ends.


# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 14 Grounding**

*Of all the conductors used in interconnecting electronics, the most complex is ironically the one that generally gets the least attention: the ground.*

—Henry W. Ott

# **14.1 Types of Ground**

In EMC, a distinction is made between these types of ground:


More ground types and their symbols are mentioned in Sect. 14.2.

# **14.2 Ground Symbols**

As mentioned above in Sect. 14.3, a distinction is made between different ground types. The standard IEC 60417 [1] defines the symbols for safety, functional, and chassis ground. These ground symbols are presented in Fig. 14.1.

**Fig. 14.1** Earth and ground symbols according to IEC 60417. (**a**) No. 5017. Earth, ground. To identify an earth (ground) terminal in cases where neither the symbol 5018 nor 5019 is explicitly required [1]. (**b**) No. 5018. Functional earthing/grounding (US). To identify a functional earthing (grounding) terminal, for example, of a specially designed earthing (grounding) system to avoid causing malfunction of the equipment [1]. (**c**) No. 5019. Protective earth/ground (US). To identify any terminal which is intended for connection to an external conductor for protection against electric shock in case of a fault or the terminal of a protective earth (ground) electrode [1]. (**d**) No. 5020. Frame or chassis. To identify the frame or chassis terminal [1]. (**e**) No. 5021. Equipotentiality. To identify the terminals which, when connected together, bring the various parts of an equipment or of a system to the same potential, not necessarily being the earth (ground) potential, e.g., for local bonding [1]. (**f**) No. 6032. Do not connect to protective earth/ground (US). To indicate that conductive parts shall not be connected to protective earth (ground), e.g., on electrical equipment with conductive parts inside an insulating enclosure [1]. (**g**) No. 6092. Class II equipment with functional earthing/grounding (US). To identify class II equipment (appliances with double insulation and no protective conductor) with functional earthing (grounding) [1]

#### **14.3 EMC Grounding Philosophy**

A ground is often thought to be equipotential, meaning there is no voltage difference along a ground conductor or a ground plane. However, this is not true and there is a voltage drop across the *Zg* [] when current flows along a ground conductor or a ground plane. This voltage drop could lead to *common impedance coupling* (see Sect. 12.1.1). The ground impedance can be written as (Fig. 14.2):

$$\underline{Z}\_{\rm g} = R\_{\rm g} + j\omega L\_{\rm g} \tag{14.1}$$

where:

*Rg* = ground resistance in [] *Lg* = ground inductance in [H] *ω* = 2*πf* = angular frequency of the sinusoidal signal in [rad/sec]

Another point to remember is that the radiated emission due to differential-mode current is proportional to the current loop area *A* [m2] (see Sect. 9.9.1). Thus, care must be taken that the current loop area *A*, formed by the signal current flowing along the signal conductor and through ground back to the signal source, is as small as possible.

The EMC grounding philosophy includes the following points:


#### **14.4 Return Current Path on PCB Ground Planes**

Current does always take the path of the least impedance *Z* []. Therefore, in case of a PCB signal trace above a solid ground plane (see Fig. 14.3), we can state the following:

• **At low frequencies** (around *f <* 100 kHz), the signal return current path in the *ground plane* flows more or less straight from via 2 back to via 1. This is because at low frequencies, the resistance *Rg* [] dominates the impedance *Zg* = *Rg* +*jωLg* of the *return current path*, and the lowest resistance from via 2 to via 1 is a direct line.

**Fig. 14.3** Signal return currents in a solid PCB ground plane. (**a**) At low frequencies (around *f <* 100 kHz). (**b**) At high frequencies (around *f >* 1 MHz)

• **At high frequencies** (around *f >* 1 MHz), the signal return current in the ground plane flows directly below the respective PCB trace. This is because the inductive reactance *X* = 2*πf Lg* starts to dominate the impedance *Zg* = *Rg* + *jωLg* of the return current path at high frequencies and the inductive reactance is lowest when the return current flows directly underneath the forward current PCB trace.

The points above lead to the conclusion that a solid ground plane should not be split up underneath a high-speed data line because in such a case, the signal return current would have to flow around that gap in the ground plane. This would lead to increased differential current loop area and therefore unintended emissions (see Sect. 9.9.1). Generally speaking, a solid ground plane without any gaps does often lead to the lowest unintended emissions.

Bear in mind that a digital signal—like a clock or data signal—consists of many harmonics where the amplitudes of the high-frequency harmonics primarily depend on the signal's rise- and fall-time (see Sect. 5.2). Given this, the return current path of a digital signal on a PCB is not identical for all its harmonics: the higher the frequency, the closer does the harmonic signal follow the line underneath the forward current path of the digital signal.

#### **14.5 Grounding of Systems**

A system in the context of *grounding of systems* means, e.g., a PCBA with multiple circuits placed on the PCB or an electric device which consists of several PCBAs and/or electronic modules and units. We call these circuits, PCBAs, modules, and units in the following just circuits. The grounds of these circuits can be connected in many different ways. Basically, there are three variants [2]:


**Fig. 14.4** Single-point ground systems. Ideally: *Zg*<sup>1</sup> = *Zg*<sup>2</sup> = *Zg*<sup>3</sup> = 0, *Cstray* = 0F. (**a**) Daisy-chain ground system. (**b**) Star ground system

#### *14.5.1 Single-Point Ground Systems*

*Single-point ground systems* are only useful for low frequencies (*<*100 kHz). It is nearly impossible to implement a single-point ground system at high frequencies (*<*1 MHz) because high-frequency currents start to flow through the stray capacitances *Cstray* [F], thus converting a single-ground system into an unintended and, therefore, not optimal multi-ground system.

Figure 14.4 shows two types of single-point ground systems:


#### *14.5.2 Multipoint Ground Systems*

The concept of a *multi-ground system* —like shown in Fig. 14.5—is usually applied for high-frequency systems. At high frequencies (*>*1 MHz), the ground impedance

**Fig. 14.5** Multipoint ground system. Ideally: *Zg*<sup>1</sup> = *Zg*<sup>2</sup> = *Zg*<sup>3</sup> = 0

**Fig. 14.6** Hybrid ground system. Low-frequency (LF) signals flow to the left to the single-ground point and high-frequency (HF) signals through the capacitors *Cg*1, *Cg*2, and *Cg*<sup>3</sup>

*Zg* = *Rg* + *jωLg* is primarily dominated by the ground inductance *Lg* [H]. This means that the ground connections of the circuits to ground should be as short as possible with low impedance (e.g., with multiple vias). For PCBs, a multipoint ground system is best achieved with one or several solid and uninterrupted ground plane(s).

#### *14.5.3 Hybrid Ground Systems*

*Hybrid ground systems* provide different paths to ground for low-frequency currents *ILF* [A] and high-frequency currents *I H F* [A]. Figure 14.6 shows a typical hybrid ground system, which provides a single-point ground system for low-frequency signals and a multipoint ground system for high-frequency signals.

Hybrid grounding can also be applied to cable shields (see Sect. 13.7.4), where one end of the cable shield is connected to ground with low impedance and the other end is connected via a capacitor. A hybrid grounded cable shield could provide reasonable protection against inductive coupling of HF magnetic fields and at the same time prevent LF currents from flowing along the cable shield.

# **14.6 Ground Loops**

A *ground loop* is a current loop formed by ground conductors and ground itself, like shown in Fig. 14.7. Ground loops could lead to the following interference problems:


Figure 14.8 compares how ground loop interference does affect single-ended (unbalanced) and differential signal (balanced) interfaces. Well-balanced differential interfaces (e.g., LVDS) are more robust against ground loop coupling than single-ended interfaces (e.g., CMOS) because the noise voltage *V <sup>n</sup>* [V] affects both signal lines of a differential interface and is canceled out. A list of digital singleended and differential signal interfaces can be found in Sect. 7.10 on page 90.

In many cases, ground loops do not lead to EMC problems. Meaning, the EMC design engineers should not avoid ground loops at any cost. However, EMC design engineers should be aware of ground loops and how to deal with them if they lead to an EMC issue. Ground loops are primarily a problem for low-frequency applications (*f <* 100 kHz) because the size of ground loops is usually large. Thus, they have a large inductance, which means that they represent a high impedance for HF signals. A typical example of an EMC issue due to a ground loop is the 50*/*60 Hz hum coupled into audio systems.

If ground loops are a problem, the following are options for minimizing their impact:

**Fig. 14.8** Ground loop noise coupling: single-ended vs. differential interfaces. *V <sup>g</sup>* [V] could be caused by a voltage potential difference between the ground connections of circuits 1 and 2, or *V <sup>g</sup>* [V] could be induced by a magnetic field. (**a**) Ground loop noise coupling in case of a single-ended (unbalanced) interface. The noise voltage *V <sup>n</sup>* [V] does directly interfere with the useful signal. (**b**) Ground loop noise coupling in case of a differential (balanced) interface. *V <sup>n</sup>* [V] does add to both differential signal lines likewise, and the noise voltage is canceled out

	- **Transformers, photocouplers.** If the interface between circuit 1 and 2 in Fig. 14.8 is realized with a transformer or a photocoupler (see Fig. 14.9a,b), the low-frequency ground current path is interrupted. However, HF currents may still flow through the stray capacitance of the transformer or photocoupler.
	- **Common-mode chokes.** If a common-mode choke is added to the interface between circuits 1 and 2 (see Fig. 14.9c), the HF ground loop currents through the signal ground conductor are attenuated. However, at very low frequencies of *f <* 10 kHz and very high frequencies of *f >* 1 GHz, the attenuation of the ground loop currents due to a common-mode choke is limited because of its relatively low inductance *L* [H] and its stray capacitance *C* [F].
	- **Fiberglass.** Using a fiberglass communication system between circuits 1 and 2—like shown in Fig. 14.9d—breaks the ground loop for currents of any frequency *f* [Hz]. However, fiberglass communication systems are costly.

**Fig. 14.9** How to break ground loops. (**a**) A transformer can be used to break the ground loop for low-frequency ground loop currents. (**b**) A photocoupler can be used to break the ground loop for low-frequency ground loop currents. (**c**) A common-mode choke can be used to break the ground loop for high-frequency ground loop currents. (**d**) A fiberglass communication system can be used to break the ground loop for noise currents for any frequency *f* [Hz]

# **14.7 Summary**

	- Do always consider ground as a return current path.
	- Minimize current loop area between forward and return current path.
	- Consider and minimize common impedance coupling in the ground path.
	- Signals with *f <* 100 kHz: the return current path takes the path of the least resistance and takes the shortest way possible through the ground plane.
	- Signals with *f >* 1 MHz: the return current path takes the path of the least inductance and flows back to the source directly underneath the PCB signal trace of the forward current.

#### • **Grounding of systems**.

	- Ground loop currents are typically caused by:
		- · Voltage potential difference in the ground system.
		- · Magnetic fields coupling to the ground current loop.
	- Reducing the ground impedance reduces the ground noise voltage caused by ground loop currents.
	- Single-point and hybrid ground systems help reduce ground loop currents.
	- Breaking the ground loop eliminates any ground noise current.

# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 15 Filtering**

*If everyone is thinking alike, then no one is thinking.*

—Benjamin Franklin

# **15.1 Filter Characterization**

A linear and time-invariant (LTI) filter can be characterized in the time- and the frequency-domain (Fig. 15.1):


In this chapter, we focus on filters which are causal *LTI-systems* (except for the non-linear transient filters in Sect. 15.9 and some non-linear digital algorithms in Sect. 15.10). LTI-systems can be completely characterized by one of these functions [4]:


*Causal*, *linear*, and *time-invariant* (LTI) means:


**Fig. 15.2** Time-domain characterization functions. (**a**) Impulse Dirac delta function *δ(t)*. (**b**) Unit step function *u(t)*

Note: real-world components show non-linear effects, e.g., due to current saturation *I* [A] (inductors, ferrites) or non-ideal high-frequency behavior (inductors, capacitors). Therefore, real-world filters are only linear for certain operating conditions. The non-linearities and other undesired effects of filter components are described in Chap. 11. In this chapter, we assume that the filter components are used in their linear operating region.

#### *15.1.1 Time-Domain: Step Response*

The goal of a filter characterization in the time-domain is to determine signal distortions and the time delay caused by the filter (Fig. 15.2).

If the input signal *x(t)* of an LTI-system is the *Dirac delta δ(t)* function:

$$\delta(t) = \begin{cases} +\infty & t = 0 \\ 0 & \text{otherwise,} \end{cases} \tag{15.1}$$

then the output is defined as the *impulse response h(t)*. The impulse response characterizes an LTI-system entirely, and for any given input signal *x(t)* the output *y(t)* can be calculated as [4]:

$$\mathbf{y}(t) = h(t) \* \mathbf{x}(t) = \int\_{-\infty}^{t} h(t - \tau) \mathbf{x}(\tau) d\tau \tag{15.2}$$

where

**Fig. 15.3** Step response simulation of two 2nd-order low-pass filters [1]

*y(t)* = the output signal from an LTI-system in [V] *h(t)* = the impulse response of the LTI-system in [V] *x(t)* = the input signal to an LTI-system in [V] ∗ = convolution, a mathematical operation of two functions

In practice, the *step response s(t)* (and not the impulse response *h(t)*) is usually used to characterize a filter [4]:

$$u(t) = \begin{cases} 1 & t \ge 0 \\ 0 & t < 0. \end{cases} \tag{15.3}$$

If the input is the *unit step*: *x(t)* = *u(t)*, the output *y(t)* is called step response *s(t)*. The step response *s(t)* can be calculated by integrating the impulse response *h(t)* and the impulse response *h(t)* by taking the derivative of the step response *s(t)* [4]:

$$s(t) = \int\_{-\infty}^{t} h(\tau)d\tau\tag{15.4}$$

$$h(t) = \frac{d}{dt}s(t)\tag{15.5}$$

Figure 15.3 shows the step response of two different low-pass filters, where:


The step responses in Fig. 15.3 show that even passive low-pass filters can cause a significant amount of ringing and time delay. Therefore, it is always necessary to also check the time-domain behavior of an EMI-filter, not just in the frequencydomain behavior.

#### *15.1.2 Frequency-Domain: Frequency Response*

The *frequency response H(jω)* (*Fourier transform*) is the transfer function *H(s)* (*Laplace transform*, *s* = *σ* + *jω*) of a filter for a sinusoidal input signal, meaning *s* = *jω*. The frequency response *H(jω)* describes the attenuation in [dB] and the phase shift in [rad] or [◦] as a function of frequency *ω* = 2*πf* .

The Fourier transform of the filter's impulse response *h(t)* is the frequency response *H(jω)* of the filter [4]:

$$h(t) \smile \bullet \underline{H}(j\omega) = \int\_{-\infty}^{\infty} h(t)e^{-j\omega t}dt\tag{15.6}$$

The frequency response characterizes a filter in the frequency-domain. For any given input signal *X(jω)*, the output signal *Y (jω)* can be calculated as [4]:

$$
\underline{Y}(j\omega) = \underline{H}(j\omega) \cdot \underline{X}(j\omega) \tag{15.7}
$$

where

*Y (jω)* = the output signal of the filter in [V] *H(jω)* = the frequency response of the filter in [V] *X(jω)* = the input signal of the filter in [V] *ω* = 2*πf* = angular frequency of the sinusoidal input signal in [rad/sec]

The frequency response *H(jω)* can also be calculated by using the Fourier transform of the step response *S(jω)*:

$$\mathbf{s}(t) \boxdot \mathbf{\underline{s}}(j\omega) \tag{15.8}$$

$$
\underline{H}(j\omega) = j\omega \cdot \underline{S}(j\omega) \tag{15.9}
$$

In practice, the frequency response *H(jω)* is shown in the form of a *Bode plot*. Figure 15.4 presents the Bode plot of the identical filters of Fig. 15.3, which shows us a couple of interesting points:


**Fig. 15.4** Frequency response simulation of two second-order low-pass filters [1]

**Fig. 15.5** Bode plot of ideal *RC* low-pass filters of order *n* = 1, *n* = 2 and *n* = 3

filter with the first filter component *C*<sup>2</sup> [F] which has a cut-off frequency of *fg* = 1*/(*2*πRS*2*C*2*)* = 160 kHz and the load impedance *RL*<sup>2</sup> = 1 M builds a low-pass *RL*-filter with the filter component *L*<sup>2</sup> [H] and a cut-off frequency of *fg* = *RL*2*/(*2*πL*2*)* = 1*.*6 GHz.

#### **15.2 Low-Pass Filters**

*Low-pass filters* are the most common filters in the world of EMI and EMC. They reject undesired HF energy above a desired cut-off frequency *fc* [Hz]. Figure 15.5 shows the frequency response of ideal low-pass filters (skin effect, stray capacitance, and other non-linearities are neglected). At the cut-off frequency *fc* [Hz], the insertion loss is 3 dB and the phase shift is −45 deg. The attenuation for frequencies *f >fc* is *n* · 20 dB and the maximum phase shift of a low-pass filter is *n* · −90 deg, where *n* is the filter order.

In most cases, EMI-filters against radiated or conducted high-frequency (HF) interference are analog passive low-pass filters with resistors, capacitors, inductors, ferrites, or common-mode chokes as filter components. The filter order *n* is usually not higher than three (maximum phase shift = *n* · 90 deg = 270 deg) because of the signal distortion and because filters of order *n* ≥ 4 could easily lead to instability due to the large phase shift of 360 deg and more.

The filter performance (e.g., cut-off frequency *fc* [Hz]) depends not only on the filter components themselves but also on the source impedance *ZS* [] and the load impedance *ZL* []. Therefore, the input and output impedances of the filter should be matched in the useful frequency range and mismatched in the EMI noise frequency range (an overview is given in Fig. 15.24).

It is highly recommended to simulate EMI analog circuit board filters with the real-world source and load impedances and the non-ideal SPICE models of the filter elements. The SPICE simulations of the low-pass filters in Fig. 15.6 show that it is important to consider the non-ideal filter components (not the ideal components) because the non-ideal behavior of capacitors, inductors, and ferrite beads influence the filter performance significantly at high frequencies.

#### **15.3 High-Pass Filters**

*High-pass filters* do attenuate low-frequency (LF) signals and let high-frequency (HF) signals pass without attenuating them. Figure 15.7 shows the frequency response of ideal high-pass filters. In the field of EMC, high-pass filters are not very common, and we do not go into further detail about high-pass filters at this point.

#### **15.4 Band-Pass Filters**

*Band-pass filters* do only pass signals of the desired frequency range with little to no insertion loss. Figure 15.8 shows the frequency response of an ideal band-pass filter. However, band-pass filters are very seldom used in the field of EMC and are not discussed in further detail here.

#### **15.5 Band-Stop Filters**

*Band-stop filters* do attenuate only signals within the desired frequency range and pass the others. Figure 15.9 shows the Bode plot of a passive band-stop filter with an ideal. Band-stop filters are not very common EMI-filters and therefore not further discussed here.

**Fig. 15.6** Passive low-pass filter SPICE simulations with *ZS* = 10 and *ZL* = 1 M. Tool = LTspice [1]. *L* = Würth 744766001 WE-GF, *C* = Würth 885012008026 NP0. (**a**) 1st order low-pass filter. (**b**) Second-order low-pass filter. (**c**) 3rd order low-pass filter

#### **15.6 Active and Passive Filters**

The filters in the previous sections were all *passive filters*. If operational amplifiers or transistors are part of a filter, it is called an *active filter*. In most cases, EMI-filters

**Fig. 15.7** Bode plot of ideal *RC* high-pass filters of order *n* = 1, *n* = 2, and *n* = 3

**Table 15.1** Active vs. passive filters: advantages and disadvantages


are passive filters because the advantages of active filters mentioned in Table 15.1 are often not needed for EMI-filter applications.

Figure 15.10 compares two second-order passive and active low-pass filters. Although the active filter shows a narrower transition between the pass-band to the stop-band, the attenuation of the active filter at frequencies of *f >* 100 MHz is not

#### 15.7 Differential- and Common-Mode Filters 253

**Fig. 15.10** Step and frequency response comparison of an active and a passive low-pass filter with pass-band up to *fc* = 60 kHz and order *n* = 2 [1]

**Fig. 15.11** Differential-mode vs. common-mode noise currents

as good as for the passive filter and the active filter shows significant ringing in the time-domain. This example should just illustrate that an active filter is not always better than a passive filter; it all depends on the application's requirements.

#### **15.7 Differential- and Common-Mode Filters**

Differential-mode noise and common-mode noise are in detail explained in Sects. 12.3 and 12.4 and compared in Fig. 15.11. The filter components and topology for differential- and common-mode filters are not identical. Therefore, it is necessary to know if you deal with differential-mode or common-mode noise or both.

**Fig. 15.12** Example of a differential-mode *LC*-filter

**Fig. 15.13** Common-mode filter with Y-capacitors and a common-mode choke

#### *15.7.1 Differential-Mode Filters*

*Differential-mode noise filters* are filters between a power or signal line and its return current line (see Fig. 15.12). Typical differential-mode noise filter components are:


#### *15.7.2 Common-Mode Filters*

*Common-mode noise filters* have the purpose to attenuate common-mode noise. An example of a common-mode filter is given in Fig. 15.13. Typical common-mode noise filter components are:


**Fig. 15.15** Example of an AC mains filter. Courtesy of Würth Elektronik GmbH

#### **15.8 Mains Supply Filters**

Special care must be taken regarding the safety requirements of filter components of (public) mains supply filters (Fig. 15.15). This is especially true for the X- and Y-capacitors:


The classification of X- and Y-capacitors according to IEC 60384-14 [3] is shown in Table 11.1 on page 158.

The applicable safety standards of a product must be considered during the design of a mains supply filter. Examples are:


#### **15.9 Transient Suppression Filters**

A distinction is made between these three types of high-voltage transients (Table 15.2):


The rise-time and the pulse energy are the most important parameters of voltage transients. Therefore, the pulse forms and energy are presented in Figs. 15.16 (ESD), 15.17 (EFT), and 15.18 (surge). The pulse energies were calculated by considering the maximum tolerances of the test equipment (timing, test voltage). The maximum pulse energy for ESD was calculated by neglecting the internal losses and by just considering the energy *E* [J] stored in the capacitor of the ESD generator:

$$E\_{ESD} = \frac{1}{2}CV\_p^2 = \frac{150\,\text{pF}}{2}V\_p^2\tag{15.10}$$



**Fig. 15.16** ESD transient pulse. (**a**) ESD current pulse form. (**b**) Approximate maximum ESDpulse energy

**Fig. 15.17** Single EFT transient pulse. (**a**) Single EFT-pulse form (open-circuit voltage). (**b**) Approximate maximum single EFT-pulse energy

**Fig. 15.18** Surge transient pulse. (**a**) Surge-pulse form (open-circuit voltage). (**b**) Approximate maximum 1.2/50 µs surge-pulse energy (line-to-line)

where

*C* = 150 pF = capacitance of the ESD generator's energy storage capacitor *Vp* = ESD test voltage in [V]

The maximum EFT- and surge-pulse energy *E* [J] was calculated by approximating the pulse energy with the following formula [8]:

$$E\_{EFT,surge} = \left(\frac{1}{3} \cdot \frac{V\_p^2}{R} \cdot t\_1\right) + \left(\frac{V\_p^2 \cdot \tau}{-2 \cdot R}\right) \cdot \left(e^{-\frac{2(t\_3 - t\_1)}{\tau}} - 1\right) \tag{15.11}$$

where

*Vp* = EFT/surge peak test voltage in [V] *R* = load resistance of the EFT/surge test generator in [] *t*<sup>1</sup> = time to peak voltage in [sec] *t*<sup>2</sup> = time until voltage has fallen off to 50 % of peak voltage in [sec] *t*<sup>3</sup> = time to negligible voltage in [sec] *τ* = −*(t*<sup>2</sup> − *t*1*)/* ln*(*0*.*5*)* = exponential rate of decay in [sec]

As ESD- and EFT-pulses show similar rise-times [nsec] and pulse energies [mJ], thus, they can usually be filtered with the same filter components. However, dealing with surge-pulses means dealing with potentially more than 1000 times higher energy compared to ESD or EFT transients. Thus, surge filters do usually need components with high energy absorption capability.

A transient filter stage consists typically of two components (see Fig. 15.19):


In the following, we focus on the selection of an appropriate shunt protection element *Z*<sup>2</sup> []. The following points must be considered when choosing a voltage protection shunt element:


# **15.10 Digital Filters**

Up to this point, all presented filters in this chapter were *analog filters*. This section is dedicated to *digital filters*, which have a couple of advantages (Table 15.3).



#### **Table 15.3** Digital vs. analog filters

**Fig. 15.20** Digital filter with ADC. *fc* = cut-off frequency of analog anti-aliasing low-pass filter *fs* = 1*/Ts* = sampling-frequency of the ADC

<sup>1</sup> The original Nyquist-Shannon sampling theorem [7]: If a function *f (t)* contains no frequencies higher than *B* [Hz], *f (t)* is completely determined by giving its ordinates at a series of points spaced 1*/(*2*B)* seconds apart.

Even though the usage of digital filters in EMC is limited, they can help improve robustness, especially when it comes to voltage transient immunity. Here are some typical examples:


**Fig. 15.21** Concept of a median filter

**Fig. 15.22** Concept of a spike-remover

# **15.11 Summary**

	- **Time-domain.** Determine distortion and time delay caused by the EMI-filter.
	- **Frequency-domain.** Determine attenuate [dB] and phase shift of the EMIfilter.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 16 EMC Design Guidelines**

*Everything should be made as simple as possible, but no simpler.*

—Albert Einstein

# **16.1 Most Common EMC Issues in Practice**

The guidelines in this chapter are biased by the author's own experience, gathered during many years of electronics development, firmware programming, and systems engineering. This should be mentioned here as a kind of a disclaimer and that you, the reader, can better understand how to weigh and classify the guidelines presented here.

From the author's own experience, these are the most common EMC issues during product development:


• **Electrostatic discharge.** Grounding, grounding, grounding. ESD testing is performed according to IEC 61000-4-2. Every electrically floating part of a product increases the chance of failing the ESD-test because it increases the chance that undesired currents—caused by ESD-pulses—flow along wires and PCB signal lines and eventually lead to damaged integrated circuits or disturbed signals.

# **16.2 Guideline # 1: Never Route Signals Over Split Reference Planes**

#### **Goal = reduced radiated emissions.**

Do not route signals over split return signal reference planes (GND, power planes)! Never! This leads to unnecessary large current loops (as the current return cannot flow directly underneath the forward current, Fig. 16.1) and large current loops tend to lead to high radiated emission values.

**Rule of thumb:** Whenever in doubt, do not split return current reference planes (GND, power planes). Go with a solid-filled reference plane instead. There must be a good reason for splitting planes! There should always be at least one solid reference plane closely adjacent to high-frequency signals (*f >* 1 MHz).

### **16.3 Guideline # 2: Always Consider the Return Current**

#### **Goal = reduced radiated emissions and common-impedance coupling.**

Always consider the return current! Always! And with the return current in mind, minimize the common return current paths of high-current circuits (e.g., motors) and sensitive circuits (e.g., unamplified sensor signals) because these common paths lead to common-impedance coupling (see Sect. 12.1.1). In addition, try to minimize the loop area *A* [m2] between the forward and return current of high-frequency signals because these loops may lead to unintended radiated emissions (see Sect. 9.9.1).

**Hint:** Assuming a high-frequency signal*s*<sup>1</sup> (*f >* 1 MHz) along a PCB trace on layer 1 and a reference plane (e.g., GND or power plane) on layer 2, the return current of the high-frequency signal *s*<sup>1</sup> on the reference plane prefers to flow directly under the trace of *s*<sup>1</sup> because the high-frequency return current always takes the path of the least inductance (see Sect. 14.4).

**Fig. 16.1** Signal and its return current path in case of split ground planes

# **16.4 Guideline # 3: Decoupling—Use Low-Inductance Capacitors AND Planes**

#### **Goal = reduced radiated emissions and common-impedance coupling.**

*Decoupling* is important! (Fig. 16.2) Always consider decoupling! **Hint:** Decoupling depends on the number of PCB layers and stackup:


### **16.5 Guideline # 4: Use Ground Planes on PCB for Shielding**

#### **Goal = reduced interference on circuit board.**

Use solid-filled reference planes (e.g., GND or power supply planes) on a PCB to separate noise signals (e.g., motor signals) from sensitive signals (e.g., unamplified sensor signals). The reference plane will act as a shield and will lower electromagnetic interference (EMI). However, be aware that such a copper shield will

**Fig. 16.2** PCB power-supply decoupling. (**a**) PCB with closely spaced power supply and GND planes (global decoupling). (**b**) PCB without closely spaced power supply and GND planes (local decoupling)

primarily be effective for *E*-fields at any frequency and high-frequency *H*-fields (*f >* 1 MHz), but not for low-frequency *H*-fields because copper has low relative permeability *μ*- *<sup>r</sup>* and does not act as a shield against low-frequency magnetic fields.

# **16.6 Guideline # 5: Route High-Frequency Signals Adjacent to a Plane**

#### **Goal = reduced radiated emissions.**

High-frequency signals are always to be routed close to an adjacent solid reference plane (GND or power supply plane, which acts in this case as a high-frequency ground). For single-layer designs, use guard traces close to the high-frequency signals where the return currents can flow or GND copper fill.

#### **16.7 Guideline # 6: Control Rise- and Fall-Time**

#### **Goal = reduced radiated emissions.**

Increase the rise- and fall-time of any digital signal (especially clock signals) as far as possible. A short rise- and fall-time means larger signal bandwidth and, therefore, more high-frequency content in a digital signal, which could lead to radiated emissions or reflections in case of a long transmission line (compared to the wavelength of the high-frequency content).

**Rule of thumb:** Add a series resistor (typically 33 ) close to the driver's output to reduce the rise- and fall-time.

**Rule of thumb:** The highest significant frequency content *fmax* [Hz] in a digital signal does not depend on the first harmonic (fundamental frequency). Instead, it depends on the rise- and fall-time:

$$f\_{\text{max}} \approx \frac{0.35}{t\_{10\% - 90\%}}\tag{16.1}$$

where

*t*10%−90% = rise- and/or fall-time (whichever is smaller) from 10 to 90 % of the slope of a digital signal in [sec]

**Rule of thumb:** Every PCB trace of length longer than *λ/*10 [m] should be considered a transmission line and no longer as a simple interconnection (where *λ* [m] is the wavelength). This means that such a trace should be laid out with controlled impedance *Z*<sup>0</sup> []. In other words, avoid impedance changes or discontinuities along the PCB trace, as these impedance changes or discontinuities could lead to reflections or ringing. Reflections and ringing affect the signal integrity and lead to increased electromagnetic radiation.

#### **16.8 Guideline # 7: Keep Clock lines As Short As Possible**

#### **Goal = reduced radiated emissions.**

To paraphrase the honorable Henry W. Ott: "Get paranoid about clock routing!" [4]. Take care about clock lines and their return current paths and keep them as short as possible!

# **16.9 Guideline # 8: Fill Top and Bottom Layers with Circuit GND**

#### **Goal = reduced radiated emissions.**

Fill the top and bottom layer of a PCB with a solid ground plane around the signals (copper area) and metalize the PCB edges (Fig. 16.3). This helps minimize radiated emission because the filled GND areas at the top and bottom layers shield innerlayer signals and prevent radiation. Moreover, the filled copper areas help maintain a low impedance return current path and short current loops.

Important: do not forget to place a grid of ground stitching vias throughout the whole PCB (otherwise, some small copper islands will start to radiate and you will get more radiation than without the solid copper fill)! This is very important! The rule of thumb in this section below presents a method of determining the maximum distance between stitching vias.

In addition, plated PCB outside edges (connected to circuit GND) help prevent the inner PCB layers from radiating. The plated PCB edges also help increase the cooling efficiency of a PCB because there is an additional copper surface where heat exchange can occur. The additional costs for metalized PCB edges are low.

**Rule of thumb:** When filling top and bottom layers with ground (copper pour), it is best practice to add a grid of ground stitching vias over the whole PCB. Otherwise, some small GND copper areas would tend to radiate! The distance between these vias within that grid depends on the highest frequency *fmax* [Hz] on the PCB. Given a signal with wavelength *λ* [m], it is a rule of thumb that a stub or trace of length *l* ≥ *λ/*10 starts to become a problem (regarding radiation) and a trace of length *l < λ/*20 will not be a problem (in between *λ/*10 and *λ/*20 is a gray area [4]). Therefore, the distance between the vias should be shorter than *λ/*10 of *fmax* [Hz].


**Table 16.1** *λ/*10 for a given frequency *fmax* [Hz] and PCB material with *ε*- *r*

The wavelength *λ* [m] of a sinusoidal signal running through a PCB signal trace is according to Eq. 4.2:

$$
\lambda = \frac{c}{f \cdot \sqrt{\varepsilon\_r'}} \tag{16.2}
$$

where

*<sup>c</sup>* <sup>=</sup> <sup>1</sup>*/(*√*μ*0*ε*0*)* <sup>=</sup> <sup>2</sup>*.*<sup>998</sup> · 108 m/s <sup>=</sup> speed of light *f* = frequency of a sinusoidal signal in [Hz] *ε*- *<sup>r</sup>* = relative permeability of the PCB material (typical *ε*- *<sup>r</sup>* = 4*.*5 for FR4)

However, how to determine *fmax* [Hz] or *λ/*10 [m], respectively? Usually, the highest frequency *fmax* [Hz] on a board can be found in the digital signals, e.g., the clock signals:

$$f\_{\text{max}} = \frac{0.35}{t\_{10\% - 90\%}}\tag{16.3}$$

where

• *t*10%−90% = rise- and/or fall-time (whichever is smaller) from 10 to 90 % of the slope of a digital signal in [sec]

Table 16.1 shows example values of high-frequency digital signals rise-/fall-time, its corresponding highest frequency content *fmax* [Hz] and *λ/*10-values (as mentioned above, the recommended distance between the vias of the grid of vias is *< λ/*10).

# **16.10 Guideline # 9: Add Stitching Vias Around High-Speed Signal Vias**

#### **Goal = reduced radiated emissions.**

Imagine the following scenario, a high-speed signal changes from one plane to another plane of a PCB. In order to minimize ground bounce, the return current path impedance should be minimized [1]. There are these two options, depending on the return current path:


# **16.11 Guideline # 10: Add a Capacitor Close to Every Pin of a Connector**

#### **Goal = reduced radiated emissions, increased immunity.**

Filtering of signals directly at the connector is very important! This helps increase ESD immunity to a PCB, reduce radiated emissions, and increase immunity to coupled burst signals on IO-cables. Every signal or power supply line which enters or leaves a PCB needs a filter element, e.g., a ceramic capacitor. One side of the capacitor should be connected close to the connector pin, the other pin tied to the ground plane with low inductance. Table 16.2 proposes some common capacitor values depending on the signal's data rate.

**Hint:** Signal lines which leave a device (e.g., a connector which people can touch with their hands) is exposed to ESD (±2 kV, ±4 kV, ±6 kV, ±8 kV). In this case,

**Fig. 16.4** Return current path in case of a reference plane change [1]


**Table 16.2** Capacitors of IO-filters vs. data rate (rule of thumb)

use capacitors with a high voltage rating (e.g., *>* 250 V, depending on capacitance and ESD test voltage and other components involved, e.g., like ferrite beads between connector pin and capacitor or conductor length).

# **16.12 Guideline # 11: Connect Circuit GND to Chassis at IO Area**

#### **Goal = reduced radiated emissions, increased immunity.**

Bound a circuit GND to chassis at the area where a cable leaves/enters the chassis. Connect it with very low impedance! It is important that GND and chassis have the same potential in the IO area:


# **16.13 Guideline # 12: Lay Cables Along Chassis (GND/Earth)**

#### **Goal = reduced radiated emissions.**

Whenever possible, lay cables constantly along the chassis. This keeps the electromagnetic field generated by the voltages and currents in the cable at a minimum radiation level. The word *constantly* is essential because when laying out cables constantly along a chassis, there is no change in electrical balance (a change in electrical balance leads to common-mode currents) [3].

# **16.14 Guideline # 13: Don't Use Cable Shield as Signal Conductor for Low-Frequency Signals**

#### **Goal = increased immunity.**

The cable shield should not be one of the signal conductors for low-frequency signals because of the potential interference with the ground loop current in the shield.

**Hint:** This rule does not apply for high-frequency signals, where the signal return current and the noise current are separated by the skin effect within the shield (return current of the high-frequency signal flows on the surface of the inner side of the shield and the noise current on the outer surface).

# **16.15 Guideline # 14: Cable Shield Grounding on Only One End for Low-Frequency Signals**

#### **Goal = increased immunity to electric-fields.**

It may lead to problems when laying both ends of a cable shield to ground for lowfrequency signals because ground loop currents in the shield could interfere with the signals inside the cable.

**Rule of thumb:** For shielding of low-frequency (*<*100 kHz) signals:


If you are in control of only one side of the cable shield (because at the other end of the cable is an unknown device from various manufacturers), then lay the cable shield to ground with low inductance (no pigtails, use a 360◦ shield clamp) or implement a hybrid ground. A hybrid ground is a compromise for shielding against high-frequency signals, while minimizing low-frequency ground loop currents in the shield (see Sect. 13.7.4 and Fig. 16.5): Lay the cable shield to ground with a resistor (to reduce the ground loop current in the shield for low-frequency signals) and add a parallel capacitor to that resistor (to allow high-frequency signals to flow through the cable shield).

**Fig. 16.5** Hybrid cable shield grounding

# **16.16 Guideline # 15: Cable Shield Grounding on Both Ends for High-Frequency Signals**

#### **Goal = increased immunity, reduced magnetic-field emissions.**

It is a must to lay both ends of high-frequency signal cable shields to ground with low inductance (no pigtails, use a 360◦ shield clamp or the like). **Rule of thumb:** For high-frequency (*>*1 MHz) signals:


# **16.17 Guideline # 16: Minimize Loop Area of Signals in Cables**

#### **Goal = increased immunity to magnetic-fields.**

The best way to protect a signal from magnetic fields is to reduce the current loop area *A* [m2] [4]. Minimizing the loop area *A* [m2] in case of a cable means twisting the wires of the forward and return current or using neighbor conductors for the forward and return current in flat ribbon cables.

### **16.18 Guideline # 17: Avoid Electrical Balance Changes**

#### **Goal = reduced radiated emissions.**

A change from electrically balanced to unbalanced and vice versa (mode conversion) is called an electrical balance change (balanced and unbalanced transmission lines are topic in Sect. 7.9). Electrical balance changes lead to common-mode currents. And we do not want common-mode currents! Common-mode currents are often the cause for unintended radiated emissions. Thus, mode conversions should be avoided. The maximum common-mode voltage *VCM,max* [V] generated due to

**Fig. 16.6** Good and bad examples of PCB-to-cable-interconnections. (**a**) Good examples. (**b**) Bad examples

**Fig. 16.7** Good and bad examples of how to filter balanced and unbalanced signal lines. (**a**) Good examples. (**b**) Bad examples

mode change is *VCM,max* = *ΔhmaxVDM* = 0*.*5*VDM* [2], where *h* is the imbalance factor (Sect. 12.2.1) and *Δhmax* is the maximum imbalance factor change.

Avoiding electrical balance changes for cables connected to a PCB means:


Avoiding electrical balance changes for filters means:


**Rule of thumb:** Send single-ended signals over unbalanced transmission lines and differential signals over balanced transmission lines. Balanced and unbalanced transmission lines are explained in Sect. 7.9 and single-ended and differential signal interfaces in Sect. 7.10.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Appendix A Constants**

*For the sake of brevity, we will always represent this number 2.718281828459. . . by the letter e.*

—Leonhard Euler

This chapter contains useful physical and mathematical *constants* for EMC design engineers.


# **Appendix B Complex Numbers**

*The shortest path between two truths in the real domain passes through the complex domain.*

—Jacques Hadamard

This chapter does shortly introduce the topic of *complex numbers*, which is commonly used in the field of electrical engineering.

#### **B.1 Definition and Notation**

Complex numbers and variables are noted with an underline throughout this book. The complex number *c* has the form:

$$
\underline{c} = a + jb \tag{\text{B.I}}
$$

where

*a* = Re*(c)* = real number, real part of the complex number *c b* = Im*(c)* = real number, imaginary part of the complex number *c <sup>j</sup>* <sup>=</sup> √−<sup>1</sup>

#### **B.2 Cartesian Complex Plane**

A complex number *c* can be seen as a point in a Cartesian coordinate system with the coordinates (a,b). The horizontal axis presents the real part and the vertical axis the imaginary part (see Fig.B.1). The Cartesian form is noted as:

$$
\underline{c} = a + jb \tag{\text{B.2}}
$$

**Fig. B.1** Cartesian complex plane

**Fig. B.2** Polar complex plane

where

*a* = Re*(c)* = real part of *c* = horizontal coordinate *b* = Im*(c)* = imaginary part of *c* = vertical coordinate

#### **B.3 Polar Complex Plane**

A complex number *c* can be represented by the polar form with a defined amplitude |**c**| and angle *φ* [rad]: the polar coordinates (|*c*|, *φ*) (see Fig. B.2). The polar form by using the Euler's formula—is also known as phasor notation and can be written as:

$$\underline{c} = |\underline{c}| \cdot e^{j\phi} \tag{\text{B.3}}$$

where

<sup>|</sup>*c*| = <sup>√</sup> *a*<sup>2</sup> + *b*<sup>2</sup> = magnitude or amplitude or absolute value of *c φ* = arctan 2*(b, a)* = angle or phase of *c e* = 2*.*71828 = Euler's number *<sup>j</sup>* <sup>=</sup> √−<sup>1</sup>

#### **B.4 Complex Conjugate**

The complex conjugate of *c* = *a* +*j b* is noted as *c*<sup>∗</sup> in this book and it is defined as:

$$
\underline{c}^\* = a - jb \tag{\text{B.4}}
$$

#### **B.5 Mathematical Operations with Complex Numbers**

If we define two complex numbers as *c*<sup>1</sup> = *a*<sup>1</sup> + *j b*<sup>1</sup> and *c*<sup>2</sup> = *a*<sup>2</sup> + *j b*2, we can write:

$$k \cdot \underline{c}\_{\text{l}} = k \cdot (a\_{\text{l}} - jb\_{\text{l}}) = ka\_{\text{l}} - jkb\_{\text{l}} \tag{B.5}$$

$$
\underline{c}\_1 + \underline{c}\_2 = a\_1 + jb\_1 + a\_2 + jb\_2 = (a\_1 + a\_2) + j(b\_1 + b\_2) \tag{B.6}
$$

$$
\underline{c}\_1 - \underline{c}\_2 = a\_1 + jb\_1 - (a\_2 + jb\_2) = (a\_1 - a\_2) + j(b\_1 - b\_2) \tag{B.7}
$$

$$\underline{c}\_{1} \cdot \underline{c}\_{2} = (a\_{1} + jb\_{1}) \cdot (a\_{2} + jb\_{2}) = (a\_{1}a\_{2} - b\_{1}b\_{2}) + j(a\_{1}b\_{2} + a\_{2}b\_{1})\tag{B.8}$$

#### **B.6 Impedances in the Complex Plane**

Figure B.3 presents the impedance of resistors *R*, inductors *L*, and capacitors *C* in the Cartesian complex plane.

# **Appendix C Vectors**

*The laws of nature are but the mathematical thoughts of God.* —Euclid, father of geometry

This chapter does shortly introduce the topic of the *Euclidean vector geometry*. Euclidean vector geometry is needed for electromagnetism, where we have to deal with vector fields.

# **C.1 Vector Definition**

A vector is a directed line segment with two properties: a direction and a magnitude. Figure C.1 shows a vector −→*<sup>A</sup>* with its projection vectors −→*Ax* , −→*Ay* , −→*Az* in a rectangular coordinate system with the unit vectors −→*ax* , −→*ay* , −→*az* [1].

# **C.2 Sum of Vectors**

The sum of vectors −→*A* and −→*B* is shown in Fig. C.2. Vector addition is commutative [1]; this means:

$$
\overrightarrow{A} + \overrightarrow{B} = \overrightarrow{B} + \overrightarrow{A} \tag{C.1}
$$

#### **C.3 Vector Multiplication with Scalars**

When multiplying vector −→*A* with scalar *k* (a scalar has only a magnitude but no direction), vector −→*A* changes its magnitude. Moreover, in the case of a negative scalar, vector −→*A* also changes its direction (direction is reversed, see Fig.C.3) [1].

#### **C.4 Vector Dot Product**

The *dot product*—also named scalar product—is commutative, and it is defined as [1]:

$$
\begin{array}{c}
\overrightarrow{A} \cdot \overrightarrow{B} = \overrightarrow{B} \cdot \overrightarrow{A} = |\overrightarrow{A}| |\overrightarrow{B}| \cos(\phi\_{AB})\end{array}
\tag{C.2}
$$

where


#### C.5 Vector Cross Product 285

**Fig. C.4** The projections of used for the dot product


The result of the vector dot product is a scalar (Fig.C.4). The scalar product can be interpreted as product of the magnitude of vector | −→*<sup>A</sup>* <sup>|</sup> and the magnitude of the projection of vector −→*B* onto vector −→*A* or the scalar product can be interpreted as product of the magnitude of vector | −→*<sup>B</sup>* <sup>|</sup> and the magnitude of the projection of vector −→*A* onto vector −→*B* .

### **C.5 Vector Cross Product**

The vector cross product of vectors −→*A* and −→*B* is defined as [1]:

$$
\overrightarrow{A} \times \overrightarrow{B} = -\overrightarrow{B} \times \overrightarrow{A} = |\overrightarrow{A}| |\overrightarrow{B}| \sin(\phi\_{AB}) \overrightarrow{a\_n} \tag{C.3}
$$

where


The direction of the vector cross product −→*<sup>A</sup>* <sup>×</sup> −→*<sup>B</sup>* is defined with the right-hand rule (Fig.C.5): point the thumb into the direction of the first vector of the cross product, then point the forefinger into the direction of the second vector of the cross product, and finally, the middle finger points into the direction of the cross product −→*an* . The right-hand rule for −→*<sup>Z</sup>* <sup>=</sup> −→*<sup>X</sup>* <sup>×</sup> −→*<sup>Y</sup>* is shown in Fig. C.6.

**Fig. C.5** The cross product of two vectors

**Fig. C.6** The right-hand rule illustrated on the 200 francs Swiss banknote [2]

#### **References**


# **Appendix D Electromagnetism**

*Faraday is, and must always remain, the father of that enlarged science of electromagnetism.*

—James Clerk Maxwell

This chapter contains helpful content and definitions for EMC design engineers on the subject of *electromagnetism*. The most basic electromagnetic definitions and physical quantities are presented.

## **D.1 Voltage**

*Voltage* is the difference in electric potential between two points in an electric circuit or system. In case of sinusoidal voltage waves, there is often the complex phasor notation used [4]:

$$\underline{V} = |V|e^{j(\omega t + \phi)} = \hat{V}e^{j(\omega t + \phi)}\tag{D.l}$$

where *ω* = 2*f π* in [rad/sec], |*V* | = *V*ˆ = magnitude/peak/amplitude of the voltage *V* in [V], *t* is the time in [sec], and *φ* is an arbitrary angle in [rad].

**[V].** The SI<sup>1</sup> unit of an electric potential difference is *V* Volts [V]. [dBV] <sup>=</sup> 20 log10 *<sup>V</sup>* 1 V . Decibel value of voltage *V* [V] with reference to 1 V. [**dB**µ**V**] <sup>=</sup> 20 log10 *<sup>V</sup>* 1µ V . Decibel value of voltage *V* [V] with reference to 1µ V.

<sup>1</sup> SI = The International System of Units is the modern form of the metric system. It is the only system of measurement with an official status in nearly every country in the world.

R. B. Keller, *Design for Electromagnetic Compatibility–In a Nutshell*, https://doi.org/10.1007/978-3-031-14186-7

#### **D.2 Current**

An electric *current* is a stream of charged particles, such as electrons or ions, moving through an electrical conductor (e.g., wire or PCB trace) or space (e.g., air). In the case of sinusoidal current waves, there is often the complex phasor notation used [4]:

$$\underline{I} = |I|e^{j(\alpha t + \phi)} = \hat{I}e^{j(\alpha t + \phi)}\tag{D.2}$$

where *ω* = 2*f π* in [rad/sec], |*I* | = ˆ*I* = magnitude/peak/amplitude of the current *I* in [A], *t* is the time in [sec], and *φ* is an arbitrary angle in [rad].

**[A].** The SI unit of the electric current *I* is Ampere [A]. **[dBA]** <sup>=</sup> 20 log10 *<sup>I</sup>* 1 A . Decibel value of current *I* [A] with reference to 1 A. [**dB**µ**A**] = 20 log10 *<sup>I</sup>* 1µ A . Decibel value of current *I* [A] with reference to 1µ A.

#### **D.3 Impedance**

Electrical *impedance Z* [] is the electrical property of a device (e.g., resistor, filter, antenna) which describes how voltage and current interact in that device. The impedance is defined as a complex number with a real part *R* (resistance) and an imaginary part *X* (reactance).

$$
\underline{Z} = \mathcal{R} + jX \tag{\mathcal{D}.3}
$$

The unit of *R* and *X* is Ohm []. In the case of direct current, the reactance *X* [] is zero and therefore *Z* = *R*. Ohm's law states:

$$\underline{Z} = \frac{\underline{V}}{\underline{I}} \tag{\text{D.4}}$$

where *V* [V] is the complex voltage across a device with impedance *Z* [] and *I* [A] the complex current through impedance *Z*. For direct current (DC), Ohm's law can be simplified to:

$$R = \frac{V}{I} \tag{D.5}$$

[]. Impedance *Z* is a complex number with the derived SI unit Ohm [].

#### **D.4 Power**

*Power P* [W] is the amount of energy transferred or converted per unit time. Electric power *P* [W] is the rate (per unit time) at which an electric circuit transfers electrical energy *E* [Ws, J]. Electric power is a function of voltage *V* [V] and current *I* [A]:

$$P = V\_{RMS} \cdot I\_{RMS} \tag{D.6}$$

where *VRMS* is the root mean square voltage in [V] and *IRMS* the root mean square current in [A]. The active electrical power *P* [W] that is converted in a component with an impedance of resistance *R* [] can be calculated like this:

$$P = \frac{V\_{RMS}^2}{R} = I\_{RMS}^2 \cdot R \tag{D.7}$$

In the phasor domain—with *<sup>V</sup>* <sup>=</sup> *V e*<sup>ˆ</sup> *j (ωt*+*φV )* and *<sup>I</sup>* <sup>=</sup> <sup>ˆ</sup>*I ej (ωt*+*φI )* —the average power is defined as [4]:

$$P = \frac{1}{2} \operatorname{Re}(\underline{V} \cdot \underline{I}^\*) \tag{D.8}$$

where *I* <sup>∗</sup> [A] is the complex conjugate of *I* [A] and the factor <sup>1</sup> <sup>2</sup> appears because *V* [V] and *I* [I] represent peak values.

**[W].** The SI unit of power is Watt [W] or Joule per second [J/s]. [**dBW**] = 20 log10 *<sup>P</sup>* 1 W . Decibel value of power *P* [W] with reference to 1 W. [**dBm**] = 20 log10 *<sup>P</sup>* 1 mW . Decibel value of power *P* [W] with reference to 1 mW.

#### **D.5 Complex Permittivity**

The *permittivity ε* [F/m] is a measure of the electric polarizability of a dielectric. Materials with high permittivity polarize more in response to an applied electric field −→*E* [V/m] than materials with low permittivity. Thus, a material with a large dielectric constant (permittivity) is capable of storing more energy. The electric displacement field −→*D* [C/m2] represents the distribution of electric charges in a given medium resulting from the presence of an electric field −→*E* [4]:

−→*<sup>D</sup>* <sup>=</sup> *<sup>ε</sup>* −→*E* (D.9)

Permittivity *ε* [F/m] is not a constant, as it can vary with the frequency of the field applied, the field strength of the applied field, humidity, temperature, and other parameters.

The relative permittivity *εr* is defined as the ratio of the absolute permittivity *ε* [F/m] to the permittivity of vacuum *ε*<sup>0</sup> [F/m]:

$$
\underline{\varepsilon\_r} = \frac{\underline{\varepsilon}}{\varepsilon\_0} \tag{D.10}
$$

The relative permittivity is directly related to the electric susceptibility *χe* by [3]:

$$
\underline{\chi}^{\epsilon} = \underline{\varepsilon}^{\prime} - 1 \tag{D.11}
$$

otherwise written as:

$$
\underline{\varepsilon} = \underline{\varepsilon}^{\prime} \varepsilon^{0} = (1 + \underline{\chi}^{\prime}) \cdot \varepsilon^{0} \tag{\text{D.12}}
$$

Permittivity can be written as a complex value [2]:


where *ε* represents the ability to store energy in a medium when an external electric field is applied and *ε*- the energy loss—the energy dissipated when a medium is under the influence of an external electric field.

The product of the signal's angular frequency *ω* [1/rad] and the material's loss factor *ε*-- = *ε*-- *<sup>r</sup> ε*<sup>0</sup> [F/m] is equivalent to the specific conductance of the material *σ* [S/m] [5]:

$$
\sigma = \omega \varepsilon'' \tag{\text{D.14}}
$$

$$
\varepsilon'' = \frac{\sigma}{\omega} \tag{\text{D.15}}
$$

This means that for an ideal, lossless insulator (*σ* = 0): *ε*-- = 0 and *ε* = *ε*-= *εrε*0.

#### **D.6 Complex Permeability**

*Permeability* µ [H/m] is the measure of magnetization that a material obtains in response to an applied magnetic field. In electromagnetism, the term magnetic field is used for two distinct but closely related vector fields denoted by the symbols −→*H* [A/m] (magnetic field) and −→*B* [T] (magnetic flux density). However, −→*H* [A/m] and −→*B* [T] differ in how they account for magnetization and the relation between these two magnetic field vectors is given as [4]:

$$
\overrightarrow{B} = \underline{\mu}\,\overrightarrow{H}\tag{\text{D.16}}
$$

Permeability is not a constant, as it can vary with frequency of the applied magnetic field, the field strength of the applied field, humidity, temperature, and other parameters.

The relative permeability µ*<sup>r</sup>* is the ratio of the absolute permeability µ [H/m] to the permeability of vacuum µ<sup>0</sup> [H/m]:

$$
\underline{\mu}\_r = \frac{\underline{\mu}}{\mu\_0} \tag{D.17}
$$

Relative permeability is directly related to magnetic susceptibility *χm* by [3]:

$$
\underline{\chi}\_m = \underline{\mu}\_r - 1 \tag{D.18}
$$

otherwise written as:

$$
\overline{\mu} = \overline{\mu}^\* \mu\_0 = (1 + \overline{\chi\_m}) \cdot \mu\_0 \tag{D.19}
$$

Permeability can be written as complex value [2]:

$$\underline{\underline{\mu}} = \underbrace{\underline{\underline{\mu}}}\_{\begin{array}{c} \underline{\text{complex}} \\ \underline{\text{Complex}} \\ \underline{\text{reducible}} \\ \end{array}} \cdot \underbrace{\underline{\underline{\mu}\_{0}}}\_{\begin{array}{c} \underline{\text{Vacuum}} \\ \underline{\text{permeability}} \\ \end{array}} = (\underbrace{\underline{\underline{\mu'}}\_{\begin{array}{c} \underline{\text{Rau}} \\ \underline{\text{Rau}} \\ \end{array}} - j \underbrace{\underline{\underline{\mu'}}\_{\begin{array}{c} \underline{\text{Vacuum}} \\ \underline{\text{Ima}} \\ \end{array}}) \cdot \underbrace{\underline{\underline{\mu}\_{0}}}\_{\begin{array}{c} \underline{\text{Vacuum}} \\ \underline{\text{Vacuum}} \\ \end{array}} = \mathfrak{\mu'} - j \,\mathfrak{\mu''} $$

where µ- [H/m] represents the ability to store energy in a medium when an external magnetic field is applied and µ-- [H/m] the energy loss (the energy dissipated when a medium is under the influence of an external magnetic field).

#### **D.7 Electric Field**

An *electric field* is a physical field that surrounds each electric charge and exerts a force on all other charges in the field. The electrostatic force is a vector quantity, and so is the electric field −→*E* [V/m]. The electric field −→*E* [V/m] at a specific position is equal to the electrostatic force −→*F* [N] (Coulomb force) which is a vanishingly small positive test charge *q* [C] held at that point experiences [4]:

$$
\overrightarrow{E} = \frac{\overrightarrow{F}}{q} \tag{\text{D.21}}
$$

where the unit of −→*E* is volt per meter [V/m], −→*F* is in Newton [N], and *q* is in Coulomb [C].

**[V/m].** The derived SI unit for an electric field *E* is Volts per meter [V/m].


#### **D.8 Electric Flux**

The *electric flux ΦE* [Vm] through a surface is the surface integral of the normal component of the electric field −→*E* [V/m] over that surface −→*A* [m2]. Electric flux is proportional to the total number of electric field lines going through a surface. If the electric field is uniform, the electric flux through a surface of vector area −→*A* [m2] is [4]:

$$\Phi\_E = \overrightarrow{E} \cdot \overrightarrow{A} = |E| \cdot |A| \cdot \cos \theta \tag{D.22}$$

where <sup>|</sup>*E*<sup>|</sup> [V/m] is the magnitude of the electric field (the electric field vector −→*<sup>E</sup>* ), <sup>|</sup>*A*<sup>|</sup> [m2] is the area of the surface, and *<sup>θ</sup>* [rad] is the angle between the electric field vector and the normal (perpendicular) of −→*A* [m2].

**[Vm].** The derived SI unit of electric flux *ΦE* is the [Vm].

#### **D.9 Electric Flux Density**

In a dielectric material, the presence of an electric field −→*E* [V/m] causes the bound charges in the material (atomic nuclei and their electrons) to slightly separate, inducing a local electric dipole moment. The *electric flux density* −→*D* [C/m2]—also called *electric displacement field*—is defined as [4]:

$$
\overrightarrow{D} = \varepsilon\_0 \overrightarrow{E} + \overrightarrow{P} = \varepsilon\_0 (1 + \chi\_e) \overrightarrow{E} = \varepsilon\_0 \varepsilon\_r' \overrightarrow{E} \tag{D.23}
$$

where *<sup>ε</sup>*<sup>0</sup> [F/m] is the vacuum permittivity (permittivity of free space), −→*<sup>E</sup>* [V/m] is the electric field intensity, −→*P* [C/m2] is the polarization density vector field (or electric polarization, or simply polarization) that expresses the density of permanent or induced electric dipole moments in a dielectric material, *χe* is the electric susceptibility of the material (a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applied electric field), and *ε*- *<sup>r</sup>* is the complex relative permittivity of the material (dielectric constant). For vacuum: *χe* = 0 and therefore *ε*- *<sup>r</sup>* = 1.

[**C**/**m2**]. The derived SI unit for an electric displacement field *D* is coulombs per square meter [C/m2].

#### **D.10 Displacement Current Density**

The *displacement current density* −→*<sup>J</sup> <sup>D</sup>* <sup>=</sup> *<sup>∂</sup>* −→*D /∂t* [A/m2] is defined as the rate of change of the electric flux density −→*D* [C/m2]. In other words, the displacement current is the partial derivative in time of the electric flux density −→*<sup>J</sup> <sup>D</sup>* <sup>=</sup> *<sup>∂</sup>* −→*D /∂t* [A/m2]. The displacement current density −→*<sup>J</sup> <sup>D</sup>* [A/m2] has the same unit like the electric current density −→*J* [A/m2], and it is a source of the magnetic field −→*H* [A/m] just as the electric current *I* [A] is. However, the *displacement current density* is not an electric current *I* [A] of moving charges but a time-varying electric field *∂* −→*E /∂t*. The displacement current density is defined as [4]:

$$
\overrightarrow{J}\_D = \frac{\partial \overrightarrow{D}}{\partial t} = \underbrace{\varepsilon\_0 \frac{\partial \overrightarrow{E}}{\partial t}}\_{\begin{subarray}{c}Displacement\\ current \end{subarray}} + \underbrace{\frac{\partial \overrightarrow{P}}{\partial t}}\_{\begin{subarray}{c} Performance\\ current \end{subarray}} \tag{D.24}
$$

where *<sup>ε</sup>*<sup>0</sup> [F/m] is the vacuum permittivity (permittivity of free space), −→*<sup>E</sup>* [V/m] is the electric field vector, and −→*P* [C/m2] is the polarization vector of the medium.

The term *(ε*0*∂* −→*E /∂t)* refers to the displacement current density −→*<sup>J</sup> <sup>D</sup>* [A/m2] in a vacuum, and it does not necessarily come from any actual movement of charges. However, it has an associated magnetic field −→*H* [A/m], just as a current *I* [A] due to charge motion. Some authors apply the name *displacement current* to the term *(ε*0*∂* −→*E /∂t)* by itself.

The term *(∂*−→*P /∂t)* comes from the change in polarization of the individual molecules of the medium (e.g., a dielectric material). Polarization results when, under the influence of an applied electric field −→*E* [V/m], the charges in molecules have moved from a position of exact cancellation. The positive and negative charges in molecules separate, causing an increase in the state of polarization −→*P* [C/m2]. A changing state of polarization corresponds to charge movement and so is equivalent to a current, hence the term *polarization current*.

[**A/m2**]. The derived SI unit for the displacement current density *JD* is ampere per square meter [A/m2].

### **D.11 Polarization Density**

The polarization density −→*P* [C/m2]—also called electric polarization or simply polarization—is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. An external electric field −→*E* [V/m] that is applied to a dielectric material causes a displacement of bound charged elements. These are elements which are bound to molecules and are not free to move around the material. Positive charged elements are displaced in the direction of the field, and negative charged elements are displaced opposite to the direction of the field. The molecules may remain neutral in charge, yet an electric dipole moment forms.

In general, the dipole moment −→*p* [Cm] changes from point to point within the dielectric. Hence, the polarization density −→*P* [C/m2] of a dielectric inside an infinitesimal volume *dV* [m3] with an infinitesimal dipole moment *d*−→*p* [Cm] is:

$$
\overrightarrow{P} = \frac{d\overrightarrow{p}}{dV} \tag{D.25}
$$

[**C/m2**]. The derived SI unit for the polarization density *P* is coulombs per square meter [C/m2].

#### **D.12 Electric Current Density**

The *current density* vector −→*J* [A/m2] is defined as a vector whose magnitude is the electric current *I* [A] per cross-sectional area *A* [m2] at a given point in space, its direction being that of the motion of the positive charges. The electric current *I* [A] can be defined by the integral of the electrical current density [4]:

$$
\underline{I} = \iint\_A \, \overrightarrow{J} \cdot d\, \overrightarrow{a} \tag{D.26}
$$

where \*\* *<sup>A</sup>* is the surface integral over the surface area *<sup>A</sup>* [m2], −→*<sup>J</sup>* [A/m2] is the current density vector which points through that cross-sectional *A* [m2], and *d*−→*a* [m2] is the differential cross-sectional area vector.

The current density −→*J* [A/m2] in materials with finite electrical resistance is directly proportional to the electric field −→*E* [V/m] in that medium. The proportionality constant is the specific conductance *σ* [S/m, 1/(m)] or resistivity *ρ* [m] of the material [4]:

$$
\overrightarrow{J} = \sigma \overrightarrow{E} = \frac{\overrightarrow{E}}{\rho} \tag{D.27}
$$

[**A/m2**]. The derived SI unit for the electric current density *J* is ampere per square meter [A/m2].

#### **D.13 Magnetic Field**

A *magnetic field* −→*H* [A/m] is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. The magnetic −→*H* [A/m] field is defined [4]:

$$
\overrightarrow{H} = \frac{1}{\mu\_0} \cdot \overrightarrow{B} - \overrightarrow{M} \tag{D.28}
$$

where <sup>µ</sup><sup>0</sup> <sup>=</sup> 12.57 ·10−<sup>7</sup> H/m is the vacuum permeability, −→*<sup>B</sup>* [T] is the magnetic flux density vector field, and −→*M* [A/m] is the magnetization vector (how strongly a region of material is magnetized). In a vacuum, the −→*B* [T] and −→*H* [A/m] fields are related through the vacuum permeability: −→*B /*µ<sup>0</sup> <sup>=</sup> −→*<sup>H</sup>* ; but in a magnetized material, the terms differ by the material's magnetization −→*M* [A/m] at each point. The relation of the magnetic field −→*H* [A/m] and the magnetic flux density −→*B* [T] is often described with the following formula:

$$
\overrightarrow{\overline{H}} = \frac{1}{\mu\_0 \cdot \underline{\mu}\_r} \cdot \overrightarrow{B} \tag{D.29}
$$

where µ*<sup>r</sup>* is the complex relative permeability of the material which is penetrated by the magnetic flux *ΦM* [Wb].

**[A/m].** The derived SI unit for a magnetic field *H* is Ampere per meter [A/m].


#### **D.14 Magnetic Flux**

The *magnetic flux ΦM* [Wb] through a surface is the surface integral of the normal component of the magnetic field −→*B* [T] over that surface. It is analogous to electric current *I* [A] in an electric circuit. If the magnetic field is uniform, the magnetic flux passing through a surface of vector area −→*A* [m2] is [4]:

$$\Phi\_M = \overrightarrow{B} \cdot \overrightarrow{A} = |B| \cdot |A| \cdot \cos \theta \tag{D.30}$$

where |*B*| [T] is the magnitude of the magnetic field (the magnetic flux density vector −→*<sup>B</sup>* ), <sup>|</sup>*A*<sup>|</sup> [m2] is the area of the surface, and *<sup>θ</sup>* [rad] is the angle between the magnetic field vector and the normal (perpendicular) of −→*A* [m2].

**[Wb].** The derived SI unit of magnetic flux *ΦM* is the Weber [Wb].

#### **D.15 Magnetic Flux Density**

The number of magnetic flux lines *ΦM* [Wb] that go through the unit area perpendicular to the magnetic field −→*H* [A/m] is called *magnetic flux density* −→*B* [T]. The relation of the magnetic flux density −→*B* [T] and the magnetic field −→*H* [A/m] is often described with the following formula [4]:

$$
\overrightarrow{\mathcal{B}} = \mu\_0 \left( \overrightarrow{\mathcal{H}} + \overrightarrow{\mathcal{M}} \right) \tag{D.31}
$$

where <sup>µ</sup><sup>0</sup> [H/m] is the vacuum permeability and −→*<sup>M</sup>* [A/m] is the magnetization vector. In a vacuum, −→*B* [T] and −→*H* [A/m] are proportional, with the multiplicative constant depending on the physical units. Inside a material, they are different and not proportional to each other (non-linear dependency). A different notation for the magnetic flux density vector −→*B* [B] is:

$$
\overrightarrow{\mathcal{B}}' = \mu\_0 \underline{\mu}\_r \overrightarrow{\mathcal{H}} \tag{D.32}
$$

where µ*<sup>r</sup>* is the complex relative permeability of the material which is penetrated by the magnetic flux.

**[T].** Tesla is a derived SI unit of the *magnetic induction* (magnetic flux density). One Tesla is equal to one Weber per square meter [Wb/m2]. According to the Lorentz force law (see Appendix E.5), a particle carrying a charge of one Coulomb [C] that moves perpendicularly through a magnetic field of one Tesla [T] at a speed of one meter per second [m/sec] experiences a force with magnitude of one Newton [N].

$$\text{T} = \frac{\text{V} \cdot \text{s}}{\text{m}^2} = \frac{\text{N}}{\text{A} \cdot \text{m}} = \frac{\text{J}}{\text{A} \cdot \text{m}^2} = \frac{\text{H} \cdot \text{A}}{\text{m}^2} = \frac{\text{Wb}}{\text{m}^2}$$

where T = Tesla, V = Volt, s = second, m = meter, N = Newton, A = Ampere, J = Joule, H = Henry, and Wb = Weber. 1 T = 10 000 G.

**[mT]** <sup>=</sup> <sup>1</sup>·10−3T <sup>=</sup> 1 mT <sup>=</sup> 10 G. **[pT]** <sup>=</sup> <sup>1</sup>·10−12T <sup>=</sup> 1 pT <sup>=</sup> <sup>1</sup>·10−<sup>8</sup> G. **[dBpT]** <sup>=</sup> 20 log10 *<sup>B</sup>* 1 pT  . Decibel value of the magnetic flux density *B* [T] with reference to 1 pT.

**[G].** The Gauss [G] is the unit of measurement of magnetic induction, also known as magnetic flux density. 1 G <sup>=</sup> <sup>1</sup>·10−4T <sup>=</sup> <sup>0</sup>*.*1 mT

#### **D.16 Electromagnetic Field**

The *electromagnetic field* can be regarded as the combination of an electric field −→*E* [V/m] and a magnetic field −→*H* [A/m]. The electric field −→*E* [V/m] is caused by static charges, whereas the magnetic field −→*H* [A/m] is caused by moving charges (currents). How charges and currents interact with the electromagnetic field is described by Maxwell's equations and the Lorentz force law (see Appendix E.5). The *Poynting vector*<sup>2</sup> −→*S* [W/m2] represents the direction of propagation of an electromagnetic wave. −→*S* [W/m2] is defined as the cross product of the two vector fields −→*E* [V/m] and −→*H* <sup>∗</sup> [A/m] [1]:

$$
\overrightarrow{S} = \overrightarrow{E} \times \overrightarrow{H}^\* \tag{D.33}
$$

where −→*H* <sup>∗</sup> [A/m] is the complex conjugate of −→*H* [A/m]. For applications of timevarying fields, it is often desirable to find the *average power density* −→*<sup>S</sup> av* [W/m2]. The time average Poynting vector −→*<sup>S</sup> av* [W/m2] can be written as [1]:

$$\overrightarrow{\boldsymbol{S}}\_{av} = \frac{1}{2} \operatorname{Re} \left( \overrightarrow{\boldsymbol{E}} \times \overrightarrow{\boldsymbol{H}}^\* \right) \tag{\text{D.34}}$$

where the factor <sup>1</sup> <sup>2</sup> appears because the −→*<sup>E</sup>* [V/m] and −→*<sup>H</sup>* [A/m] fields represent peak values.

For a uniform plane wave (electromagnetic field in the far-field: wave impedance *Zw* [] is equal intrinsic impedance *η* []) in a lossless medium (*εr* = 1 and <sup>µ</sup>*<sup>r</sup>* <sup>=</sup> 1) the calculation of the average power density *Sav* [W/m2] can be simplified to:

$$S\_{av} = \frac{1}{2} |\overrightarrow{E} \mid \cdot |\overrightarrow{H}| = \frac{|\overrightarrow{E} \mid ^2}{2|\underline{\eta}|} = \frac{|\overrightarrow{H} \mid ^2 |\underline{\eta}|}{2} \tag{D.35}$$

<sup>2</sup> The Poynting vector represents the directional energy flux (the energy transfer per unit area per unit time) of an electromagnetic field. The SI unit of the Poynting vector is the watt per square meter [W/m2]. It is named after its discoverer John Henry Poynting who first derived it in 1884.

where | −→*<sup>E</sup>* <sup>|</sup> [V/m] is the amplitude (magnitude) of the electric field, <sup>|</sup> −→*<sup>H</sup>* <sup>|</sup> [A/m] is the amplitude of the magnetic field, and |*η*| [] is the intrinsic impedance of where the wave is traveling through. The factor <sup>1</sup> <sup>2</sup> can be omitted for RMS values of −→*<sup>E</sup>* [V/m] and −→*<sup>H</sup>* [V/m]. For free-space <sup>|</sup>*η*<sup>|</sup> can be set to *<sup>η</sup>*<sup>0</sup> <sup>=</sup> <sup>377</sup> .


### **References**


# **Appendix E Maxwell's Equations and Lorentz's Force Law**

*The special theory of relativity owes its origins to Maxwell's equations of the electromagnetic field.*

—Albert Einstein

This chapter presents Maxwell's equations and Lorentz's force law. These formulas are the foundation of classical electromagnetism.

#### **E.1 Gauss' Law for Electricity**

Electric field lines diverge from a positive charge and converge on a negative charge; the charge is the source of the electric field. The electric flux *ΦE* [Vm] out of any closed surface is proportional to the total charge *q* [C] enclosed within the surface *A* [m2] [1]:

$$\underbrace{\oint\_{A} \overrightarrow{E} \cdot d\overrightarrow{a}}\_{Elecric} = \frac{q}{\varepsilon\_{0}}\tag{E.1}$$

where

, *<sup>A</sup>* <sup>=</sup> closed surface integral over the closed surface area *<sup>A</sup>* (e.g., a sphere) −→*<sup>E</sup>* <sup>=</sup> electric field vector in [V/m] *<sup>d</sup>*−→*<sup>a</sup>* <sup>=</sup> infinitesimal small surface area vector in [m2] *q* = electric charge within the closed surface *A* in [C] *<sup>A</sup>* <sup>=</sup> closed surface area around the charge *<sup>q</sup>* in [m2] *<sup>ε</sup>*<sup>0</sup> <sup>=</sup> <sup>8</sup>*.*<sup>854</sup> · <sup>10</sup>−<sup>12</sup> F/m <sup>=</sup> permittivity of vacuum

#### **E.2 Gauss' Law for Magnetism**

Magnetic field lines do not diverge, and the field of the magnetic flux density is source-free; there are no magnetic monopoles. The net magnetic flux *ΦB* [Wb] out of any closed surface *A* [m2] is zero [1]:

$$\underbrace{\oint\_{A} \overrightarrow{B} \cdot d \, \overrightarrow{a}}\_{\begin{subarray}{c} Magnetic \\ flux \end{subarray}} = 0 \tag{E.2}$$

where

, *<sup>A</sup>* <sup>=</sup> closed surface integral over the closed surface area *<sup>A</sup>* (e.g., a sphere) −→*<sup>B</sup>* <sup>=</sup> magnetic flux density vector in [T] *<sup>d</sup>*−→*<sup>a</sup>* <sup>=</sup> infinitesimal small surface area vector in [m2] *<sup>A</sup>* <sup>=</sup> closed surface area in [m2]

#### **E.3 Faraday's Law**

The line integral of the electric field −→*E* [V/m] around a closed loop *L* [m] is equal to the negative of the rate of change of the magnetic flux *ΦB* [Wb] through the area enclosed by the loop. This line integral is equal to the generated voltage or electromotive force emf [V] in the loop. Faraday's law is the basis for electric generators [1]:

$$\oint\_{L} \overrightarrow{E} \cdot d \, \overrightarrow{l} \, ^{\circ} = -\underbrace{\frac{d}{dt} \underbrace{\iint\_{A} \overrightarrow{B} \cdot d \, \overrightarrow{a}}\_{\textit{Magnetic}}}\_{\textit{flux}} = \text{emf} \tag{E.3}$$

where

, *<sup>L</sup>* <sup>=</sup> closed line integral along a the closed loop *<sup>L</sup>* (e.g., a circle) −→*<sup>E</sup>* <sup>=</sup> electric field vector in [V/m] *d* −→*<sup>l</sup>* <sup>=</sup> infinitesimal small vector along the closed loop in [m] \*\* *<sup>A</sup>* <sup>=</sup> surface integral over the surface area *<sup>A</sup>* (which is enclosed by *<sup>L</sup>*) −→*<sup>B</sup>* <sup>=</sup> magnetic flux density vector in [T] *<sup>d</sup>*−→*<sup>a</sup>* <sup>=</sup> infinitesimal small surface area vector in [m2] *L* = closed loop distance with length in [m]

*<sup>A</sup>* <sup>=</sup> surface area (which is enclosed by *<sup>L</sup>*) in [m2]

emf = electromotive force: the open circuit output voltage developed between two terminals of an electric generator/battery (source) in [V]

### **E.4 Ampére's Law**

In the case of static electric field −→*E* [V/m] (meaning *∂* −→*E /∂t* <sup>=</sup> 0), the line integral of the magnetic flux density −→*B* [T] around a closed loop *L* [m]—which encloses an area *A* [m2]—is proportional to the conduction electric current *IC* [A] flowing through the loop area [1]:

$$\oint\_{L} \overrightarrow{B} \cdot d\overrightarrow{l} = \mu\_{0} \iint\_{A} \left( \overrightarrow{J\_{C}} + \overrightarrow{J\_{D}} \right) \cdot d\overrightarrow{d} \tag{E.4}$$

$$I = \iint\_{A} \left(\mu\_0 \overrightarrow{\mathcal{J}}\,^\*\!\!/ \_C + \mu\_0 \frac{\partial \cdot \overrightarrow{D}}{\partial t}\right) \cdot d\,\overrightarrow{d} \tag{E.5}$$

$$=\iint\_{A} \left(\mu\_{0}\overrightarrow{\mathcal{J}}\,^{\circ}\_{C} + \mu\_{0}\varepsilon\_{0}\frac{\partial\overrightarrow{E}}{\partial t}\right) \cdot d\overrightarrow{\mathcal{A}}\tag{E.6}$$

$$I = \iint\_A \mu\_0 \, \overrightarrow{J} \, \_C \cdot d \, \overrightarrow{d} + \mu\_0 \varepsilon\_0 \cdot \frac{d}{dt} \underbrace{\iint\_A \overrightarrow{E} \cdot d \, \overrightarrow{d}}\_{} \tag{E.7}$$

$$
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$$=\mu\_0 \underline{I}\_C + \mu\_0 \varepsilon\_0 \frac{d\Phi\_E}{dt} \tag{E.8}$$

$$=\mu\_0 \left(\underline{I}\_C + \underline{I}\_D\right) \tag{E.9}$$

where

, *<sup>L</sup>* <sup>=</sup> closed line integral along a the closed loop *<sup>L</sup>* (e.g., a circle) −→*<sup>B</sup>* <sup>=</sup> magnetic flux density vector in [T] *d* −→*<sup>l</sup>* <sup>=</sup> infinitesimal small vector along the closed loop in [m] \*\* *<sup>A</sup>* <sup>=</sup> surface integral over the surface area *<sup>A</sup>* (which is enclosed by *<sup>L</sup>*) −→*<sup>J</sup> <sup>C</sup>* <sup>=</sup> conduction current density vector in [A/m2] −→*<sup>J</sup> <sup>D</sup>* <sup>=</sup> displacement current density vector in [A/m2] <sup>µ</sup><sup>0</sup> <sup>=</sup> 12.57 ·10−<sup>7</sup> H/m <sup>=</sup> permeability of vacuum *<sup>ε</sup>*<sup>0</sup> <sup>=</sup> <sup>8</sup>*.*<sup>854</sup> · <sup>10</sup>−<sup>12</sup> F/m <sup>=</sup> permittivity of vacuum *∂* −→*E /∂t* <sup>=</sup> change of electric field per time in [(V/m)/sec]


#### **E.5 Lorentz's Force Law**

The Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge *q* [C] due to electromagnetic fields −→*E* [V/m] and −→*B* [T]. A particle of charge *q* [C] moving with a velocity −→*v* [m/sec] in an electric field −→*E* [V/m] and a magnetic field with the magnetic flux density −→*B* [T] experiences a force of [1]:

$$
\overrightarrow{F} = \underbrace{q \overrightarrow{E}}\_{\begin{subarray}{c}Electic\\frac{e}{2}\end{subarray}} + \underbrace{q \overrightarrow{v} \times \overrightarrow{B}}\_{\begin{subarray}{c}Magnetic\\frac{e}{2}\end{subarray}} \tag{E.10}
$$

where

−→*<sup>F</sup>* <sup>=</sup> force vector which is experienced by charge *<sup>q</sup>* in [N] *<sup>q</sup>* <sup>=</sup> electric charge of the (moving) particle in [C] −→*<sup>E</sup>* <sup>=</sup> electric field vector in [V/m] −→*<sup>B</sup>* <sup>=</sup> magnetic flux density vector in [T] −→*<sup>v</sup>* <sup>=</sup> velocity vector in [m/sec]

#### **Reference**

1. Clayton R. Paul and Syed A. Nasar. *Introduction to electromagnetic fields*. McGraw-Hill, 1982.

# **Appendix F Integral Transforms**

*Fourier is a mathematical poem.*

—Lord Kelvin

This chapter presents the most popular integral transforms in a compact format. Integral transforms are used to transform time-domain signals to the frequency domain or control systems and filter design.

#### **F.1 Fourier Series**

The *Fourier series* is applied to continuous, periodic signals. The frequency spectrum is discrete. Formulas for the calculation of the complex Fourier coefficients *cn* of a signal *x(t)* are presented below [2]:

$$\chi(t) = \sum\_{n = -\infty}^{\infty} \left( c\_n \cdot e^{j\alpha\_{001}} \right) \tag{5.1}$$

$$c\_n = \frac{1}{T} \cdot \int\_0^T \left( \mathbf{x}(t) \cdot e^{-j\alpha\_0 nt} \cdot dt \right) \tag{\text{F.2}}$$

 $a\_0 = 2\pi f\_0$ ,  $f\_0 = \frac{1}{T}$ 

Il luach a fhe:  $\div$  uma  $\omega(t) = t$ .  $\Pi$ 

where *T* [sec] is the period length of the signal *x(t)*, *f*<sup>0</sup> [Hz] is the first harmonic of the signal *x(t)*, *<sup>ω</sup>*<sup>0</sup> [rad] is the first harmonic of the signal *x(t)*, *<sup>j</sup>* <sup>=</sup> √−1, and *<sup>n</sup>* stands for the *n*-th harmonic frequency [0*,* 1*,* 2*, ...*].

#### **F.2 Fourier Transform**

The *Fourier transform* is applied to continuous, non-periodic signals. The frequency spectrum is continuous. Formulas of the Fourier transform of a signal *x(t)* and the inverse-Fourier transform are given below [2]:

$$X(\omega) = \int\_{-\infty}^{\infty} \left( \chi(t) \cdot e^{-j\omega t} \cdot dt \right) \tag{F.3}$$

$$\mathbf{x}(t) = \frac{1}{2\pi} \cdot \int\_{-\infty}^{\infty} \left( X(\omega) \cdot e^{j\omega t} \cdot d\omega \right) \tag{\text{F.4}}$$

where *X(ω)* is the Fourier transform (spectrum) of *x(t)*, *ω* is the frequency in [rad], *<sup>j</sup>* <sup>=</sup> √−1, and *<sup>t</sup>* the time in [sec].

#### **F.3 Discrete Fourier Transform (DFT)**

The *discrete Fourier transform* is applied to discrete, periodic signals. The frequency spectrum is discrete. The DFT is also applied to non-periodic signals in the time domain by periodically continuing the non-periodic signals to make them computable with the DFT. The DFT is by far the most common method of modern Fourier analysis. The *Fourier Transform!Fast Fourier Transformation* (FFT) is a fast algorithm for calculating the DFT (if the block length *N* is a power of two) [1].

$$X[k] = \sum\_{n=0}^{N-1} \left( \mathbf{x}[n] \cdot e^{-j\frac{2\pi f\_k}{N}kn} \right), k = 0, 1, 2, \dots, N-1 \tag{\text{F.S.}}$$

$$\text{tr}[n] = \frac{1}{N} \cdot \sum\_{k=0}^{N-1} \left( X[k] \cdot e^{j\frac{2\pi f\_k}{N}kn} \right), n = 0, 1, 2, \dots, N - 1 \tag{5.6}$$

 $w\_k = \frac{2\pi f\_s}{N}$ ,  $f\_s = \frac{1}{T\_s}$ 

where *X* is the spectrum of *x* and *X*[*k*] is the *k*-th sample of the spectrum at *ωk*. *x*[*n*] is the amplitude of the *n*-th sample of the discrete-time signal, which is sampled with the sampling time *Ts* [sec] and the sampling frequency *fs* [Hz]. In the digital signal processing literature, *fs* [Hz] is often set to one and the equations for the DFT and the IDFT are simplified like this:

$$X[k] = \sum\_{n=0}^{N-1} \left( \mathbf{x}[n] \cdot e^{-j\frac{2\pi}{N}kn} \right), k = 0, 1, 2, \dots, N-1 \tag{\text{F.7}}$$

$$\ln[n] = \frac{1}{N} \cdot \sum\_{k=0}^{N-1} \left( X[k] \cdot e^{j\frac{2\pi}{N}kn} \right), n = 0, 1, 2, \dots, N - 1 \tag{F.8}$$

$$\alpha\_k = \frac{2\pi}{N}k, f\_s = 1, T\_s = 1$$

$$\textbf{F4} \quad \textbf{Discrete-Time Fourier Transform (DTFT)}$$

The *discrete-time Fourier transform* is applied to discrete, non-periodic signals. The frequency spectrum is continuous. The DTFT can be viewed as the form of the DFT when its length *N* approaches infinity [1].

$$X(\widetilde{\omega}) = \sum\_{n = -\infty}^{\infty} \left( \mathbf{x}[n] \cdot e^{-j\widetilde{\alpha}n} \right) \tag{\text{F.9}}$$

$$\chi[n] = \frac{1}{2\pi} \int\_{-\pi}^{\pi} \left( X(\widetilde{\boldsymbol{\omega}}) \cdot e^{j\widetilde{\boldsymbol{\omega}} \boldsymbol{n}} \cdot d\widetilde{\boldsymbol{\omega}} \right) \tag{F.10}$$

$$\widetilde{\boldsymbol{\omega}} = \boldsymbol{\omega} T\_s, \; f\_s = \frac{1}{T\_s}$$

*Ts*

where  $X$  is the spectrum of  $x$  and  $x[n]$  is the signal amplitude of the  $n$ -th sample of the discrete-time signal, which is sampled with the sampling time  $T\_s$  [sec] and the sampling frequency  $f\_s$  [Hz].  $\widetilde{\omega}$  denotes the continuous normalized radian frequency variable  $[-\pi, \dots + \pi]$ .  $\widetilde{\omega}$  is the product of the radian frequency  $\omega$  [rad] and the sampling time  $T\_s$  [sec].

#### **F.5 Laplace Transform**

The *Laplace transformation* belongs—like the Fourier analyses—to the group of integral transformations. It is mentioned here for the sake of completeness. The Laplace transform is used for system analysis (e.g., control systems, filters), whereas the Fourier transform is used for signal analysis [1].

$$X(\mathbf{s}) = \mathcal{L}^{\rho} \{ \mathbf{x}(t) \} = \int\_{-\infty}^{\infty} \left( \mathbf{x}(t) \cdot e^{-st} \cdot dt \right) \tag{\text{F.11}}$$
 
$$\dots$$

$$s = \sigma + jw$$

where *X* is the Laplace transform of *x*, *t* [sec] the time (real variable) and *s* = *σ* + *jω* is a complex variable with a real part *σ* and an imaginary part with the radian frequency *ω* [rad]. In case of *σ* = 0, the Laplace transform reduces to the Fourier transform.

#### **F.6 Z-Transform**

The unilateral *Z-transform* starts at *n* = 0 (and not at *n* = −∞, since signals are typically defined to begin at time *n* = 0, and since filters are often assumed to be causal) and is the time-discrete counterpart to the Laplace transform with *<sup>z</sup>* <sup>=</sup> *<sup>e</sup>sTs* <sup>=</sup> *<sup>e</sup>(σ*+*jω)Ts* [1].

$$X[z] = \sum\_{n=0}^{\infty} \left( \mathbf{x}[n] \cdot z^{-n} \right) \tag{\text{F.12}}$$

$$z = e^{sT\_s}, s = \sigma + j\omega, f\_s = \frac{1}{T\_s}$$

where *X* is the Z-transform of *x*, *z* is a complex variable, and *x*[*n*] is the *n*-th sample of the discrete-time signal, which is sampled with the sampling time *Ts* [sec] and the sampling frequency *fs* [Hz].

#### **References**


# **Appendix G Material Properties**

*You cannot be sure that you are right unless you understand the arguments against your views better than your opponents do.*

—Milton Friedman

This chapter contains comprehensive data on relevant materials for EMC design engineers. This comprises metals with their conductivity and relative permeability (important properties for shielding applications) and plastics with their relative permittivity. In addition, the relevant components of a PCB are presented and explained.

# **G.1 Conductor Properties**

Metals and their properties play an important role in EMC. For example, it is necessary to know the *specific conductance σ* [S/m] or *resistivity ρ* [/m] to calculate the skin depth. The values in Table G.1 applies for DC (0 Hz) and room temperature (25 ◦C).

In addition to the electrical conductivity *σ* [S/m], the relative *permeability* µ- *r* of materials is another important property, e.g., when it comes to shielding of lowfrequency (*f <* 100 kHz) magnetic fields. The relative magnetic permeability µ- *<sup>r</sup>* of a material tells us how much better this material is able to "conduct" the magnetic flux, or in other words, how big the flux-density *B* [T] in a material is compared to vacuum (where µ- *<sup>r</sup>* = 1) for a given field strength *H* [A/m].

Our primary interest lies in *ferromagnetic materials* (soft magnetic) because they can be used for shielding low-frequency magnetic fields. Let us have a look at how magnetic materials are classified:


	- **Soft Magnetic.** Coercivity *Hc* [A/m] is low (typically *<* 1 kA/m). Example applications: shielding, transformers, and ferrite cores.
	- **Hard Magnetic.** Coercivity *Hc* [A/m] is high (typically *>* 10 kA/m). Example application: permanent magnets.

The data for µ- *<sup>r</sup>* in Table G.1 applies to room temperature (25 ◦C) and DC (0 Hz). Please note that µ- *<sup>r</sup>* is current *I* [A], temperature *T* [ ◦C], and frequency *f* [Hz] dependent. µ- *<sup>r</sup>* of a material may increase or decrease with increasing temperature, until a certain temperature (curie temperature) where µ- *<sup>r</sup>* = 1. With increasing signal frequency and increasing current, the value of µ- *<sup>r</sup>* is getting smaller. For example, Mumetal has a µ- *<sup>r</sup>* of over 10'000 at *f* = 0 Hz, but similar to steel at *f* = 20 kHz!

#### G.1 Conductor Properties 309


#### **Table G.1** Metals and their properties [1, 2]

#### **G.2 Insulator Properties**

To calculate the signal wavelength in a cable or another electric structure surrounded by an insulator (plastics), the *dielectric constant* (dielectric permittivity) *ε*- *<sup>r</sup>* must be known. Therefore, Tables G.2 and G.3 present the *ε*- *<sup>r</sup>* of common plastics and other *insulators*. The *ε*- *<sup>r</sup>* data apply to room temperature of 25 ◦C (unless otherwise noted).


**Table G.2** Plastics and their dielectric constants *ε*- *<sup>r</sup>* [2]

#### G.3 PCB Materials 311


**Table G.3** Miscellaneous insulators and their dielectric constants *ε*- *<sup>r</sup>* [2]

# **G.3 PCB Materials**

The dielectric constant (relative permittivity *ε*- *<sup>r</sup>*) of a *printed circuit board* (PCB) base material determines the characteristic impedance of the PCB traces and is, therefore, an important parameter in the field of EMC and signal integrity. Besides the dielectric constant *ε*- *<sup>r</sup>*, the loss tangent tan *(δ)* (also called dissipation factor Df) of a PCB material is also of interest because it influences the loss at high frequencies (together with the ohmic loss, which increases with increasing frequency due to the skin effect).

The arrangement of the copper and insulation layers of a PCB is called the PCB layer stackup, or just stackup. Let us have a look at an example of a PCB stackup

**Fig. G.2** Example of a PCB stackup with layers

of a six-layer board in Fig. G.2. These are the following materials that one must consider when calculating the characteristic impedance of PCB traces:

	- 0*.*5 oz = 0*.*7 mils = 17*.*5µ m
	- 1 oz = 1*.*4 mils = 35µ m
	- 2 oz = 2*.*8 mils = 70µ m

The circuit traces are etched into the copper layers before the PCB is laminated together (with adhesive, heat, and pressure).


are interleaved to build up the required number of layers. The cores are all etched individually and then sandwiched together with layers of prepreg on the top and bottom and bonding the two cores together. Cores are typically made out of FR-4 substrates because FR-4 has a good price-to-performance ratio regarding low dissipation factor, low variation of *ε*- *<sup>r</sup>* over a wide frequency range, and moisture absorption.

Here are some tips regarding PCB layer stackups:


# **References**


# **Appendix H Galvanic Series**

*Learn what is to be taken seriously and laugh at the rest.*

—Hermann Hesse

Knowing the galvanic series is important when designing interconnections between dissimilar metals to bond shields or safety earth connections. If you do not care which metals you use, the interconnections may corrode within a short time, and the shielding effect or safety function is not given anymore.

# **H.1 Corrosion**

Why does *corrosion* happen and recommendations for connections of dissimilar metals:

	- 1. The two metals have a galvanic incompatibility (voltage difference to high). A difference of hundreds of millivolts is likely to result in galvanic corrosion, but only a few tens of millivolts is unlikely to be a problem.
	- 2. An electrolyte (e.g., water, moisture) is present.
	- 3. The two metals have an electrical conducting connection.
	- **Harsh** (outdoor, high humidity, salt laden, military). Choose metals where electrode potential difference is ≤0*.*15 V.
	- **Normal** (non-temperature/humidity controlled, consumer product, indoor). Choose metals where electrode potential difference is ≤0*.*3 V.
	- **Controlled** (temperature/humidity controlled, indoor). Choose metals where electrode potential difference is ≤0*.*5 V.

### **H.2 Corrosion Prevention**

Three methods for corrosion prevention of dissimilar metal connections:


#### **H.3 Standard Electrode Potential Series**

Table H.1 shows *standard electrode potentials* (E◦) for various metals. The standard electrode potential is defined by measuring the potential relative to a standard hydrogen electrode (SHE) using 1 mol solution at 25 ◦C at the pressure of 1 atm.

An electrode potential series can be derived for metals in any electrolyte solution. Be careful! The real-world corrosion rate depends on the solution conditions like electrolyte concentration, pH, flow rate, aeration, temperature, humidity, and pressure. Therefore, it is common to use the seawater electrode potential Table H.2 rather than the standard hydrogen electrode potential Table H.1.

**Table H.1** Standard electrode potentials [5]


#### **H.4 Seawater Galvanic Series**

The seawater *galvanic series* is often used to approximate the probable galvanic effects in other environments for which there are no data. For example, from the standard electrode potentials shown in Table H.1, it can be seen that aluminum (Al) should behave anodically toward zinc (Zn) and presumably would retard the corrosion of zinc in a usual coupled situation. However, the reverse is true, as can be seen from the established galvanic series of metals in seawater in Table H.2.

In Table H.2, metals are grouped. All metals, alloys, and platings of the same group have similar electro-motive forces (EMF) within 0*.*05 V when coupled with a saturated calomel electrode in seawater at room temperature. All members of a group, regardless of metallurgical similarity or dissimilarity, are considered


**Table H.2** Seawater electrode potentials [1, 2]

compatible. Compatible couples between groups have been specified in the table (green areas) based on a potential difference of 0*.*25 V maximum.

In Table H.3, the galvanic series of selected metals in seawater is presented. This series can be used as a reference to minimize galvanic corrosion when selecting metals that will be in direct contact. Generally said, the closer the metals in the series, the less galvanic corrosion is expected. In a galvanic couple, the metal higher in the series represents the anode and will corrode preferentially in the environment to the cathode, which is lower in the series.



#### **References**


# **Appendix I Triboelectric Effect**

*What we know is a drop, what we don't know is an ocean.*

—Isaac Newton

The *triboelectric effect* is the physical effect that leads to electrically charged materials after they are separated from a different material with which they were in contact. Rubbing two materials with each other increases the contact between their surfaces, hence the triboelectric effect.

Most everyday static electricity is triboelectric. The polarity and strength of the charges produced differ according to the materials, surface roughness, temperature, humidity, strain, and other properties. These static charges are why electrical apparatus and installations have to be tested for their immunity to *electrostatic discharges* (ESD).

### **I.1 How Electrostatic Charges Are Produced**

When two or more non-conductive materials with different affinities for electrons make and break contact, electrostatic voltage potentials up to several ten thousand volts may result. The reason why this happens is that electrons can be exchanged between materials on contact. Materials with weakly bound electrons tend to lose them, while materials with sparsely filled outer shells tend to gain them. Every time a material with weakly bound electrons and a material with the tendency to gain electrons make contact, electrons spend more time on the material with the higher affinity for electrons. When conductive materials are separated, electrons quickly return to the material they initially came from. However, when the materials are not conductive, the separation of the materials may occur before the electrons are back to the material where they initially came from. This electron exchange leads to a positively charged material (the one that lost electrons) and a negatively charged material.

In practice, the magnitude of charge depends heavily on the relative humidity rH [%] of air. In a humid environment, the slight amount of moisture on the surface of a non-conductive material can lead to a rise of conductivity high enough to prevent significant charge separation. Therefore, the larger the electrical charges are, the lower the relative humidity. At an optimal relative humidity of 50 %–55 %, the conductivity of the material surfaces is increased to such an extent that electrical charges can be dissipated without any problems [1]. However, bear in mind that the often prevailing opinion that moist air conducts electricity better than dry air does not apply. It is the moisture on the surface of materials, which prevents static electricity.

#### **I.2 Triboelectric Series**

The triboelectric series is a list of materials ranking from the most positive (materials that tend to lose electrons, Fig. I.1) to the most negative (materials that tend to gain electrons).

**Fig. I.1** Triboelectric series [2]

# **References**


# **Appendix J Country-Specific EMC Regulations**

*If a product cannot be legally sold because of its inability to comply with the governmental EMC requirements, the fact that it may perform a function that gives it significant sales potential is unimportant!*

—Dr. Clayton R. Paul

Every government issues its own EMC regulations (laws, directives) for its country. This chapter lists the legal statutory basis and requirements regarding EMC for different countries and customs unions.

The author can accept no responsibility or liability for the content of this chapter and throughout this book. All information provided in this chapter and throughout the book are provided as is, with no warranties, and should not be used as a reference for compliance testing. Please make sure to test your product according to the recognized EMC standards at an accredited EMC laboratory.

### **J.1 Australia and New Zealand**

The Regulatory Compliance Mark (RCM, Fig.J.1) is the compliance mark for Australia and New Zealand. By the end of March 2016, the RCM replaced the former compliance marks called A-Tick (telecommunications equipment requirements) and C-Tick (EMC requirements).


**Fig. J.1** Regulatory Compliance Mark (RCM). The mandatory compliance mark for Australia and New Zealand

**Fig. J.2** ANATEL label is mandatory for radio- and telecommunications equipment in Brazil

> bility (EMC), electromagnetic energy (EME), electromagnetic radiation (EMR), electrical safety [13].


# **J.2 Brazil**

The ANATEL label (Fig.J.2) is mandatory for radio- and telecommunications equipment in Brazil. It consists of the ANATEL logo and a numerical code HHHHH-AA-FFFFF [36], where:


**Fig. J.3** The ISED label of an Apple iPhone 6s


# **J.3 Canada**

The Innovation, Science and Economic Development (ISED) certification number is made up of a Company Number (CN) assigned by ISED's CEB, followed by the Unique Product Number (UPN). For example, IC: CN-UPN. The CN can only be obtained from the Certification and Engineering Bureau (CEB) of ISED. The registration number (Fig.J.3) can be presented electronically (on the device's display) or physically on a product label. An FCC test report will be accepted by ISED if it is less than 1 year old [27].


# **J.4 China**

The China Compulsory Certificate mark (CCC) is a mandatory mark for products sold on the Chinese market (Fig.J.4).

• **Effective countries.** China.

**Fig. J.4** The China Compulsory Certificate mark (CCC)

**Fig. J.5** The Eurasian Conformity mark (EAC)


# **J.5 Eurasian Economic Union**

The Eurasian Conformity (EAC) mark (Fig.J.5) is the product conformity mark of the Eurasian Economic Union (EAEU, former Eurasian Customs Union (EACU)).


**Fig. J.6** The CE conformity mark

# **J.6 European Union**

The Conformité Européenne (CE) mark (Fig.J.6) is the product conformity mark of the European Union (EU) [11].

	- **EMC Directive (EMCD).** Directive 2014/30/EU [42] of the European Parliament and of the Council of 26 February 2014 on the harmonization of the laws of the Member States relating to electromagnetic compatibility. Be aware that the EMC directive can be overruled by other EU directives or regulations [2014/30/EU article 2, paragraph 3]. Examples are:

**Automotive.** Regulation (EC) 661/2009. The Guide for the EMCD [21] states that vehicles and equipment subject to type approval under UNECE Regulation 10 are totally excluded from the scope of the EMCD. The Commission Regulation (EU) 2019/543 of 3 April 2019 amending Annex IV to Regulation (EC) 661/2009 refers for electromagnetic compatibility to the United Nations Economic Commission for Europe Regulation 10 (UNECE R 10 ).

**Radio equipment.** Directive 2014/53/EU. The RED overrules the EMCD according to the introductory "whereas"-text, point (8). However, in article 3, paragraph 1b, the RED calls for the same protection regarding EMC as the EMCD. Generally spoken, regarding EMC testing for equipment under RED, the same harmonized EMC standards—as for EMCD—plus the ETSI EN 301 489-x series have to be applied.

**Medical.** Regulation (EU) 2017/745. The Medical Device Regulation (MDR) overrules the EMCD with its article 1, paragraph 11.

**In vitro diagnostics.** Regulation (EU) 2017/746. The In vitro Diagnostics Regulation (IVDR) overrules the EMCD with its article 1, paragraph 5.





covered by the New Approach Directives. It is forbidden to affix the CE marking to products not covered by at least one of the New Approach Directives [1].


An important concept of EMC standards and product compliance within the EU is the presumption of conformity. Products that proved compliance with the Harmonized Standards (published via EU Official Journal (OJEU) [2]) benefit from a presumption of conformity with the corresponding essential requirements (directives and regulations) of the applicable legislation [40]. Harmonized Standards are European standards adopted upon a request made by the European Commission to apply EU harmonization legislation.

Application of harmonized or other standards is voluntary. The manufacturer can always apply other technical specifications to meet the requirements (but will carry the burden of demonstrating that these technical specifications answer the needs of the essential requirements, more often than not, through, a process involving a third-party conformity assessment body—in other words, reversal of the burden of proof) [40].


Besides the Harmonized EMC Standards, there are is also sector-specific EMC standards like the ED-14D by the European Organization for Civil Aviation Equipment (EUROCAE). The ED-14D standard defines the Environmental Conditions and Test Procedures for Airborne Equipment. ED-14D is harmonized with the US RTCA DO-160 standard [32].

• **Guidance.**


# **J.7 India**

The Indian Standards Institution (ISI) mark (Fig.J.8) is the product conformity mark of India, and the Telecommunication Engineering Centre (TEC) logo is mandatory for telecommunication equipment.


**Fig. J.8** Left: Telecommunication Engineering Centre (TEC) logo. Right: Indian Standards Institution (ISI) mark


### **J.8 Japan**

The PSE Category A label (diamond) is the Japanese conformity marking for Specific Products. The PSE Category B label (round) is the Japanese conformity mark for Non-Specific Products. PSE stands for Product Safety Electrical Appliance & Material. The Ministry of Economy, Trade and Industry (METI) decides which products belong to Category A (specified products) and which products belong to Category B (non-specified products). The difference for Category A (diamond) to Category B (round) is that Category A products must pass an additional conformity assessment by a Registered Conformity Assessment Body [28].

The Voluntary Control Council for Interference (VCCI) marking is the voluntary Japanese conformity marking for Information Technology Equipment (ITE) based on the CISPR recommendations (Fig.J.9).

• **Effective countries.** Japan.

**Fig. J.9** The Japanese compliance labels for Specified Electrical Appliances and Materials (Category A, diamond), Non-Specified Electrical Appliances and Materials (Category B, round), and the Voluntary Control Council for Interference marking (VCCI)

**Fig. J.10** The Korean Certification (KC) mark

	- **PSE Category A (diamond).** Mandatory for Specified Products [28].
	- **PSE Category B (round).** Mandatory for Non-Specified Products [28].
	- **VCCI.** Voluntary control of radio disturbances emitted from Information Technology Equipment (ITE).

# **J.9 Republic of Korea**

The Korean Certification (KC) marking (Fig.J.10) is the product conformity mark of the Republic of Korea (South Korea).


# **J.10 Switzerland**

The CH mark is the Swiss conformity mark. Switzerland also accepts products with the CE mark (according to the agreement between the European Community and the Swiss Confederation on mutual recognition in conformity assessment [6]) (Fig.J.11).


**Fig. J.11** The Swiss conformity (CH) mark

**Fig. J.12** The BSMI mark is the conformity mark of Taiwan


# **J.11 Taiwan**

The Commodity Inspection Mark (Fig.J.12), also named Bureau of Standards, Metrology and Inspection (BSMI) Mark or Product Safety Mark, is the product conformity mark of Taiwan for Information Technology Equipment (ITE), audio and video devices, appliances, and wireless products.


**Fig. J.13** The TSE Mark is the Turkish conformity mark

# **J.12 Turkey**

According to the EMCD Guide about the EMC Directive (EMCD) 2014/30/EU [21], the EMCD also applies to Turkey. Therefore, Turkey accepts the CE mark if the declaration of conformity for the CE mark is translated into Turkish.

For radio equipment, the used radio frequency determines how to get the approval. If the equipment uses harmonized frequencies (EU, Turkey), the RED 2014/53/EU (2014/53/AB) can be applied. In the case of non-harmonized frequencies (frequencies that are not allowed or regulated in EU), a Turkish BTK type approval is required.

The TSE Mark is the Turkish conformity mark (Fig.J.13) for electronic products, indicating compliance of a product to relevant Turkish Standards (TSE) and permits using the TSE mark upon the product and/or packaging. In case a product has the CE conformity, the TSE mark is optional.

The TSEK Mark (Certificate of Conformity to the TSE Criteria) is the Turkish conformity mark if there are no relevant Turkish Standards available and your product is not CE certified. Such products are subject to the technical conditions accepted by the Turkish Standards Institute based on the conditions of Turkey and/or other countries.

	- EMC Directive (EMCD). ELEKTROMANYET˙IK UYUMLULUK YÖNET-MEL˙IG˘ ˙I (2014/30/AB).
	- Radio Equipment Directive (RED). TELS˙IZ EK˙IPMANLARI YÖNET-MEL˙IG˘ ˙I TASLAGI (2014/53/AB). ˘

**Fig. J.14** The United Kingdom Conformity mark (UKCA mark) is the mandatory conformance mark in Great Britain

# **J.13 United Kingdom**

The United Kingdom (UK) left the EU on 1 February 2020. The UKCA mark (Fig.J.14) is the new mandatory conformity mark in Great Britain (England, Wales, and Scotland) to indicate conformity of a product to the UK legislation.

Special rules apply for Northern Ireland. The Northern Ireland Protocol came into force on 1 January 2021. For as long as it is in force, Northern Ireland will align with relevant EU rules relating to the placing on the market of manufactured goods [19].

From 1 January 2023, the UKCA marking must, in most cases, be affixed directly to the product [20]. You can still use the CE marking until then. However, the CE marking is only valid in Great Britain for areas where GB and EU rules remain the same. If the EU changes its rules and you CE mark your product on the basis of those new rules you will not be able to use the CE marking to sell in Great Britain, even before 31 December 2022 [20].


### **J.14 United States of America**

The Federal Communications Commission (FCC) mark (Fig.J.15) is the product conformity mark of the United States of America (USA).

• **Effective countries.** United States of America (USA).

**Fig. J.15** The Federal Communications Commission (FCC) mark


The Code of Federal Regulations (CFR), Title 47 (Telecommunications), Chapter I (Federal Communication Commission) contains the following parts, which can be seen as the United States (US) EMC regulations:


One difference between the US regulations and the EU regulations is that the US regulations (FCC, Title 47, Chapter I) also specify the limits in the law, e.g., the conducted limits of unintentional radiators 47 CFR 15.107. Whereas in the EU directives, this is not the case and the test limits are specified in the EMC standards issued by the IEC and/or ISO organizations. On the other side, the testing methods for the US are mostly specified by the standards organization (e.g., ANSI, IEEE, or AAMI). Here are the most important EMC standards for the US:

– **IEEE/ANSI C63.4** American National Standard for Methods of Measurement of Radio-Noise Emissions from Low-Voltage Electrical and Electronic Equipment in the Range of 9 kHz to 40 GHz. ANSI C63.4, conducted emission testing, conducting ground plane, digital equipment, electric field measurement, line impedance stabilization network, low-voltage electrical equipment, low-voltage electronic equipment, magnetic field measurement, normalized site attenuation, radiated emission testing, radio-noise emissions, radio-noise power, site attenuation, unintentional radiators.


become international standards and were therefore withdrawn. However, there are still active SAE EMC standards remaining, which were not (yet) adopted by ISO and CISPR.

– **MIL-STD.** MIL-STDs are US military standards. Here the two most important United States Department of Defense (DoD) EMC standards:

**MIL-STD-461.** Requirements for the Control of Electromagnetic Interference Characteristics of Subsystems and Equipment. MIL-STD-461 contains emission and immunity requirements. Unlike FCC or IEEE/ANSI standards, MIL-STD-461 is not a legally binding EMC standard. MIL-STD-461 is a contractual standard, where test limits and emission levels can be negotiated and waivers are possible. This standard is best suited for items that have the following features: electronic enclosures that are no larger than an equipment rack, electrical interconnections that are discrete wiring harnesses between enclosures, and electrical power input derived from prime power sources. MIL-STD-461 should not be directly applied to items such as modules located inside electronic enclosures or entire platforms. The principles in MIL-STD-461 may be useful as a basis for developing suitable requirements for those applications.

**MIL-STD-464.** Electromagnetic Environmental Effects (E3) Requirements for Systems. MIL-STD-464 is the DoD EMC standard which establishes electromagnetic environmental effects (E3) interface requirements and verification criteria for airborne, sea, space, and ground systems, including associated ordnance. MIL-STD-464 is applied at system or platform level.

– **RTCA DO-160** contains Environmental Conditions and Test Procedures for Airborne Equipment and is the minimum standard for the environmental testing of commercial avionics hardware. It is published by the Radio Technical Commission for Aeronautics (RTCA). Like the MIL-STD standards, RTCA DO-160 is a contractual standard (not legally binding). The European Organization for Civil Aviation Equipment (EUROCAE) works jointly with RTCA in the development of standards and publishes an identical standard called EUROCAE ED-14D [32].

# **References**


# **Appendix K EMC Test Setups and Equipment**

*Testing leads to failure, and failure leads to understanding.*

—Burt Rutan

This chapter presents typical EMC test setups and equipment. As an EMC design engineer, it is important to understand the EMC test equipment and setups. In particular:


All this information gives guidance on how to design specific filters, shield barriers, or power supplies.

The author can accept no responsibility or liability for the content of this chapter and throughout this book. All information provided in this chapter and throughout the book are provided as is, with no warranties, and should not be used as a reference for compliance testing. Additionally, the test setups presented in this chapter and through this book are not legally binding. Please make sure to test your product according to the recognized EMC standards at an accredited EMC laboratory.

# **K.1 Pre-compliance vs. Compliance Testing**

Before we jump into the topic of compliance EMC test setups and equipment, here is a summary and comparison between pre-compliance and compliance testing:

• **Pre-compliance testing and troubleshooting.** *Pre-compliance EMC testing* takes place at not accredited EMC test sites. Such pre-compliance testing is typically performed by the product development team itself with the goal to identify obvious weak spots of the product regarding EMC *before* the product will be tested at an accredited EMC test laboratory. Troubleshooting is often part of the EMC pre-compliance tests. How to create your own troubleshooting kit is wonderfully described in this book: [17].

Typical measurement and test equipment for pre-compliance tests and troubleshooting are:

	- **EMI receivers.** *EMI receivers* are very similar to ordinary *spectrum analyzers* with some essential differences, e.g., regarding pre-selection filter, preamplifier, and quasi-peak detector (specified in CISPR 16-1-1).
	- **Fully or semi-anechoic chambers.** Radiated emission and immunity tests take place usually in fully or semi-anechoic chambers.
	- **Certified test equipment.** Certified test equipment is required for compliance testing. The specifications for certified test equipment (voltage and current generators, antennas, filters, test chambers) and their calibration is given in the respective EMC standards.

# **K.2 RF Emission: CISPR 11**

The IEC/CISPR 11, EN 55011 (Industrial, scientific and medical (ISM) radiofrequency equipment—electromagnetic disturbance characteristics—limits and methods of measurement) is about: conducted and radiated emissions of signals in the frequency range of 9 kHz to 400 GHz.

### *K.2.1 CISPR 11: Applicability*

Equipment covered by other CISPR product and product family emission standards are excluded from the scope of CISPR 11. CISPR 11 applies to industrial, scientific, and medical electrical equipment operating in the frequency range 0 Hz to 400 GHz and to domestic and similar appliances designed to generate and/or use locally radio-frequency energy. CISPR 11 standard covers emission requirements related to radio-frequency (RF) disturbances in the frequency range of 9 kHz to 400 GHz. Measurements need only be performed in frequency ranges where limits are specified.

CISPR 11 and CISPR 14: Induction cooking is the scope of CISPR 14-1 and CISPR 14-2. Microwave ovens are primarily the scope of CISPR 11 and CISPR 14- 2 (however, be aware of multi-function equipment which may also be the scope of CISPR 14-1, e.g., for click measurement).

For ISM Radio-Frequency (RF) applications in the meaning of the definition found in the United Nations specialized agency for information and communication technologies (ITU) Radio Regulations [15], CISPR 11 covers emission requirements related to radio-frequency disturbances in the frequency range of 9 kHz to 18 GHz. Requirements for ISM RF lighting equipment and Ultra Violet (UV) radiators operating at frequencies within the *ISM frequency bands* as defined by the ITU Radio Regulations are contained in CISPR 11. To clarify things, here are some points about ISM bands:


#### *K.2.2 CISPR 11: Group 1, 2*

There are two groups of ISM equipment defined in CISPR 11 [14]:

	- Laboratory equipment
	- Medical electrical equipment
	- Scientific equipment
	- Semiconductor-converters


**Fig. K.2** ISM frequency bands by ITU [2]

	- Microwave-powered UV irradiating apparatus
	- Microwave lighting apparatus
	- Industrial induction heating equipment operating at frequencies above 9 kHz
	- Dielectric heating equipment Industrial microwave heating equipment
	- Arc welding equipment
	- Microwave ovens
	- Medical electrical equipment
	- Electric welding equipment
	- Electro-discharge machining (EDM) equipment
	- Demonstration models for education and training
	- Battery chargers and power supplies—wireless power transfer (WPT) mode

# *K.2.3 CISPR 11: Class A, B*

There are two classes of ISM equipment defined in CISPR 11 [14] (Table K.1):



**Table K.1** CISPR 11 groups and classes

# *K.2.4 CISPR 11: Test Setup*

Test location [14]:


Measuring distance for radiated emissions [14]:

	- *x* [m] is the nearest distance between the exterior wall of the building in which the equipment is situated and the boundary of the user's premises in each measuring direction.
	- *a* = 2*.*5 for frequencies lower than 1 MHz.
	- *a* = 4*.*5 for frequencies equal to or higher than 1 MHz.

the floor which, including its cables, fits in an imaginary cylindrical test volume of 1*.*2 m in diameter and 1*.*5 m height (to ground plane).

Consistent with typical applications of the equipment under test, the level of the disturbance shall be maximized by varying the configuration of the equipment.

# *K.2.5 CISPR 11: Test Limits*

Be aware that CISPR 11 limits do not apply for ISM frequency bands (defined by ITU). Outside of ITU-designated ISM bands, the limits for the disturbance voltage and radiation disturbance in CISPR 11 apply.

# **K.2.5.1 CISPR 11: Test Limits at Test Site**

There are limits for conducted and radiated emissions specified in CISPR 11. Here are the limits for measurements performed on a test site (not in situ).

### **CISPR 11 Group 1 [14]:**


# **CISPR 11 Group 2 [14]:**


**Fig. K.4** Example of a typical unified test set up for floor standing equipment suitable for measurement of conducted as well as radiated disturbances. Further examples of typical arrangements of the EUT and associated peripherals are given in CISPR 16-2-3 and CISPR 16-2-1 [14]

**Fig. K.5** CISPR 11 CE limits for mains ports at test site [14]

**Fig. K.6** CISPR 11 CE voltage limits for DC ports for group 1 at test site [14]

**Fig. K.7** CISPR 11 CE current limits for DC ports for group 1 at test site [14]

**Fig. K.8** CISPR 11 RE limits for group 1 at test site [14]

**Fig. K.9** CISPR 11 RE limits for group 2 at test site [14]

#### **K.2.5.2 CISPR 11: Test Limits In Situ**

Under in situ conditions, an assessment of conducted disturbances is not required. **CISPR 11 Group 1 [14]:**


#### **CISPR 11 Group 2 [14]:**


**Fig. K.10** CISPR 11 magnetic field RE limits for group 1 in situ [14]

**Fig. K.11** CISPR 11 electric field RE limits for group 1 in situ [14]

**Fig. K.12** CISPR 11 magnetic field RE limits for group 2 in situ [14]

**Fig. K.13** CISPR 11 electric field RE limits for group 2 in situ [14]

# **K.3 RF Emission: CISPR 32**

The IEC/CISPR 32, EN 55032 (Electromagnetic compatibility of multimedia equipment—Emission requirements) is about: conducted and radiated emissions of signals in the frequency range of 9 kHz to 400 GHz. CISPR 32 replaced CISPR 13 and the popular CISPR 22 in March 2017.

# *K.3.1 CISPR 32: Applicability*

CISPR 32 applies to multimedia equipment (MME) and having a rated RMS AC or DC supply voltage not exceeding 600 V. Equipment within the scope of CISPR 13 or CISPR 22 is within the scope of CISPR 32. MME intended primarily for professional use is within the scope of CISPR 32. The radiated emission requirements in CISPR 32 are not intended to be applicable to the intentional transmissions from a radio transmitter as defined by the ITU, nor any spurious emissions related to these intentional transmissions. Equipment for which emission requirements in the frequency range covered by CISPR 32 are explicitly formulated in other CISPR publications (except CISPR 13 and CISPR 22) are excluded from the scope of this publication. CISPR 32 does not contain requirements for in situ assessment (in other words, the tests have to be done in an EMC test laboratory). The objectives of CISPR 32 publication are [3]:


CISPR 32 is sometimes referenced by other product and product family standards, outside of the scope defined above.

# *K.3.2 CISPR 32: Class A, B*

There are two classes of Information Technology Equipment (ITE) defined in CISPR 32 [3]:

• **Class A** (higher emission limits, industrial): Devices must have emissions that are below the limits of Class A, but the emissions exceed the limits of Class B. Class A devices shall have a warning notice in their manual (e.g., "Warning! This is a Class A device. This device may cause radio interference in residential areas; in this case, the operator may be required to take appropriate measures.").

	- No permanent location (e.g., battery-powered devices)
	- Telecommunication terminal equipment
	- Personal computers

#### *K.3.3 CISPR 32: Limits*

There are limits for conducted and radiated emissions specified in CISPR 32. They can be found in the pictures below. The limits for quasi-peak and average must not be exceeded by Class A devices (blue) or Class B devices (orange), respectively.

There are also the radiated emission limits for 3 m measurement distance added, which are in general 20 · log10*(*10 m*/*3 m*)* = 10*.*5 dB higher compared to the specified limits for 10 m distance. If the measurement takes place at 1 m distance (1 m between Equipment Under Test (EUT) and antenna), the limits are 20 · log10*(*10 m*/*1 m*)* = 20 dB higher than the 10 m limits.


#### **K.4 Harmonic Currents: IEC 61000-3-2**

The IEC 61000-3-2 (Limits for harmonic current emissions—equipment input current ≤ 16 A per phase) is a basic EMC publication which deals with the limitation of harmonic currents injected into the public mains supply system. It specifies limits of harmonic components of the input current which can be produced by equipment tested under specified conditions. The objective of IEC 61000-3-2 is to set limits for harmonic emissions of equipment within its scope so that, with due allowance for the emissions from other equipment, compliance with the limits ensures that harmonic disturbance levels do not exceed the compatibility levels defined in IEC 61000-2-2.

Professional equipment that does not comply with the requirements of IEC 61000-3-2 can be permitted to be connected to certain types of low voltage supplies if the instruction manual contains a requirement to ask the supply utility for

**Fig. K.14** CISPR 32 CE limits for mains ports [3]

permission to connect. Recommendations concerning this aspect are contained in IEC 61000-3-12.

Figure K.19 shows an example current curve of a lightning equipment with dimming control. The current curve shows a phase shift and harmonic distortions.

### *K.4.1 IEC 61000-3-2: Applicability*

IEC 61000-3-2 applies to:


**Fig. K.15** CISPR 32 CE voltage limits for telecom/LAN ports [3]

**Fig. K.16** CISPR 32 CE current limits for telecom/LAN ports [3]

**Fig. K.17** CISPR 32 RE limits up to 1 GHz [3]

**Fig. K.18** CISPR 32 RE limits above 1 GHz [3]

#### *K.4.2 IEC 61000-3-2: Class A, B, C, D*

There are four classes of equipment defined [5]:

	- Balanced three-phase equipment.

**Fig. K.19** Public mains supply current distortion example [5]

	- Personal computers and personal computer monitors.
	- Television receivers.
	- Refrigerators and freezers having one or more variable-speed drives to control compressor motor(s).

# *K.4.3 IEC 61000-3-2: Test Setup*

Figure K.20 shows an example of a single-phase measurement circuit. The measurement equipment shall be compliant with IEC 61000-4-7:



#### *K.4.4 IEC 61000-3-2: Limits*

The following categories of equipment are no limits specified in IEC 61000-3-2:


Before we have a look at the limits, have a look at Eqs. K.1 and K.2. According to IEC 61000-3-2, the total harmonic current (THC) is equal to the total RMS value of the harmonic current components of orders 2 to 40:

$$\text{THC} = \sqrt{\sum\_{h=2}^{40} I\_h^2} \tag{K.1}$$

According to IEC 61000-3-2, the total harmonic distortion (THD) is defined as the ratio of the RMS value of the sum of the harmonic components (in this context, harmonic current components *Ih* [A] of orders 2 to 40) to the RMS value of the fundamental component *I*<sup>1</sup> [A], expressed as:

$$\text{THD} = \sqrt{\sum\_{h=2}^{40} \left(\frac{I\_h}{I\_1}\right)^2} \tag{K.2}$$


IEC 61000-3-2 defines different limits depending on the equipment class:

	- Rated power ≥ 5 W and ≤ 25 W.

Harmonic currents shall not exceed the power-related limits of Class D (column 2 in Table K.4).

The third harmonic current, expressed as a percentage of the fundamental current, shall not exceed 86% and the fifth harmonic current shall not exceed 61%. In addition, the waveform of the input current shall be such that it reaches the 5% current threshold before or at 60◦, has its peak value before or at 65◦, and does not fall below the 5% current threshold before 90◦, referenced to any zero crossing of the fundamental supply voltage. The current threshold is 5% of the highest absolute peak value that occurs in the measurement window, and the phase angle measurements are made on the cycle that includes this absolute peak value. Components of current with frequencies above 9 kHz shall not influence this evaluation (a filter similar to the one described in IEC 61000-4-7:2002 may be used).

The THD (formula can be found above) shall not exceed 70%. The thirdorder harmonic current, expressed as a percentage of the fundamental current, shall not exceed 35%, the fifth-order current shall not exceed 25%, the seventh-order current shall not exceed 30%, the ninth- and eleventhorder currents shall not exceed 20%, and the second-order current shall not exceed 5%.

– Rated power *>*25 W.

**Luminaires.** For luminaires with incandescent lamps and built-in phase control dimming having a rated power greater than 25 W, the harmonics of the input current shall not exceed the limits of Class A equipment.

**Any other lightning equipment.** For any other lighting equipment having a rated power greater than 25 W, the harmonics of the input current shall not exceed the relative limits specified for Class C in Table K.3.

• **Class D.** Limits for Class D equipment are shown in Table K.4.


**Table K.4** IEC 61000-3-2 class D limits [5]


# **K.5 Flicker: IEC 61000-3-3**

IEC 61000-3-3 (limitation of voltage changes, voltage fluctuations, and flicker in public low-voltage supply systems, for equipment with rated current ≤ 16 A per phase and not subject to conditional connection) is a Product Family Standard about the limitation of voltage fluctuations and flicker impressed on the public low-voltage system. It specifies limits of voltage changes that may be produced by equipment tested under specified conditions and gives guidance on assessment methods.

The definition of flicker by IEC is as follows: Impression of unsteadiness of visual sensation induced by a light stimulus whose luminance or spectral distribution fluctuates with time [1].

Flicker can be considered a symptom resulting from the modulation of a load and its effect on its terminal voltage. While connected to a voltage source with finite impedance, any load modulation will cause voltage fluctuations on the supply line.

#### *K.5.1 IEC 61000-3-3: Applicability*

IEC 61000-3-3 applies to electrical and electronic equipment having an input current equal to or less than 16 A per phase, intended to be connected to public low-voltage distribution systems of between 220 V and 250 V line to neutral at 50 Hz, and not subject to conditional connection [6].


*Zref* [] is the internationally agreed reference source impedance for low-voltage supply networks.

#### *K.5.2 IEC 61000-3-3: Test Setup*

The test setup consists of [6] (Fig. K.21):

	- *RA* = 0*.*24 *, jXA* = 0*.*15 at 50 Hz
	- *RN* = 0*.*16 *, jXN* = 0*.*10 at 50 Hz

**Fig. K.21** Flicker test setup [6]

### *K.5.3 IEC 61000-3-3: Limits*

First, flicker limits are mainly based on the subjective perception of flicker in the light of 230 V/60 W incandescent lamp, which is caused by fluctuations in the supply voltage. Second, there are some terms to explain [6]:


Commonly known as *d* values, voltage change characteristics constitute three separate parameters—*d(t)*, *dc*, *dmax* (Figs. K.22 and K.23):


The IEC 61000-3-3 limits are as follows:


**Fig. K.22** Flicker voltage fluctuation terms [6]

	- 4% without additional conditions
	- 6% for equipment that is:

switched manually

switched automatically more than twice a day and is also fitted with a delayed restart not less than a few tens of seconds. Alternatively a manual restart after a power supply interruption.

**Fig. K.23** Flicker *Pst* = 1−curve for rectangular and equidistant voltage changes [6]. The concept behind this curve: If the performance of the EUT lies under the *Pst* = 1−curve, then the disturbance to the supply network is deemed to be acceptable in the short term. Remark: 1200 voltage fluctuations per minute lead to a 10 Hz flicker

– 7% for equipment that is:

attended while in use

switched on automatically or intended to be switched on automatically no more than twice per day. Must also be fitted with a delayed restart of not less than a few tens of seconds (or manual restart) after a power supply interruption.

#### **K.6 ESD: IEC 61000-4-2**

The IEC/EN 61000-4-2 EMC standard is about immunity testing (measurement techniques) against electrostatic discharge (ESD). This standard is part of the Basic EMC Publications.

IEC 61000-4-2 specifies immunity to *electrostatic discharge* from operating personnel and neighboring objects. The standard defines requirements, test methods, and test levels. The purpose of IEC 61000-4-2 is to establish a general and reproducible basis for determining the performance of electrical and/or electronic equipment when exposed to ESD. It also includes discharges of static electricity from persons to objects near the equipment under test.

### *K.6.1 IEC 61000-4-2: Test Setup*

There are two test methods defined, whereas the first one is the most commonly applied because post-installation ESD tests may significantly decrease the mean time between failure (MTTF) of the EUT:


The test setup described in the following refers to type conformity test performed in the laboratory. The following environmental conditions must be met during ESD testing [8]:


The test setup comprises the ESD test generator, the equipment under test (EUT), and its auxiliary equipment (AE) necessary to perform direct and indirect application of discharges to the EUT as applicable, in the following manner [8]:


Typical test setups for tabletop devices and floor-mounted apparatuses (with earth connection) are shown in Figs. K.24 and K.25.

The ESD generator emulates an electrical discharge of the human body. The storage capacity chosen for the ESD generator is a capacity of 150 pF, which is representative of that of the human body. Furthermore, a discharge resistance of 330 represents the source resistance of the human body when holding a metallic object in hand (such as a key or a tool). Figure K.26 shows the simplified circuit of an ESD generator. In the case of an air discharge test, the discharge switch, which is used for contact discharge, shall be closed.

**Fig. K.24** ESD tabletop devices test setup [8]

**Fig. K.25** ESD tabletop devices test setup [8]

#### *K.6.2 IEC 61000-4-2: Test Levels*

The voltage levels are increased gradually until the maximum severity level selected is reached. Discharges to the EUT (Fig. K.27) and coupling planes are performed at a minimum of 1 sec intervals for each polarity (Tables K.5 and K.6).

Table K.7 helps determine which test level to apply for specific relative humidity levels and materials involved. Test levels are specified in the corresponding Product (Family) EMC Standard or Generic EMC Standard.

### **K.7 RF Radiated Immunity: IEC 61000-4-3**

The IEC/EN 61000-4-3 EMC standard is about immunity testing (measurement techniques) against radiated, radio-frequency electromagnetic fields. This standard is part of the Basic EMC Publications.

The object of IEC 61000-4-3 is to establish a common reference for evaluating the immunity of electrical and electronic equipment when subjected to radiated, radio-frequency electromagnetic fields. The test method documented in this part of IEC 61000 describes a consistent method to assess the immunity of an equipment or

**Table K.5** ESD discharge current waveform parameters [8]



**Table K.7** Instructions for selecting the ESD immunity test level [8]


system against a defined phenomenon. This part deals with immunity tests related to the protection against RF electromagnetic fields from any source. Particular considerations are devoted to the protection against radio-frequency emissions from digital radiotelephones and other RF emitting devices.

#### *K.7.1 IEC 61000-4-3: Test Setup*

Tests have to be conducted in a shielded anechoic chamber (due to the high field strength) (Fig. K.28). The distance between the RF generation antenna and the equipment under test (EUT) is typically 3 m.Testing should be performed in a configuration as close as possible to the actual conditions in which the EUT will be used. A metallic grounding plane is not required, but the EUT should be placed on a table or support made of a non-metallic, non-conductive material (0*.*8 m table height for tabletop equipment, 0*.*05 m to 0*.*15 m support height for floor-standing equipment). Low dielectric constant materials such as polystyrene are recommended for non-conducting tables and supports. Materials like wood can become reflective when exposed to higher frequencies.

Here is a list of the most essential test equipment needed [9]:


**Fig. K.28** IEC 61000-4-3 RF immunity test setup [9]


### *K.7.2 IEC 61000-4-3: Test Levels*

IEC 61000-4-3 does not suggest that a single test level is applicable over the entire frequency range. Rather more, it should be tested for the appropriate test level for each frequency range (Table K.8).

Testing for general-purpose RF immunity covers the 80 MHz to 1000 MHz frequency range and should be performed without any gaps.

For mobile communication and other higher frequency devices, tests should be performed in the 800 MHz to 960 MHz and 1.4 GHz to 6.0 GHz frequency ranges. Tests in these ranges do not need to be applied continuously over the entire range, and the ranges may be limited to specific frequencies for compliance with specific operating bands in the country the product will be sold in.

The specific product standards establish the most appropriate level for each frequency range. The test levels and the frequency bands are selected in accordance with the electromagnetic radiation environment to which the EUT can be exposed when finally installed. The electromagnetic environment for radiated immunity according to IEC 61000-4-3 is divided into the following classes [9]:



### **K.8 EFT: IEC 61000-4-4**

The IEC/EN 61000-4-4 EMC standard is about immunity testing (measurement techniques) against repetitive electrical fast transients (EFT), also called bursts. This standard is part of the Basic EMC Publications.

IEC 61000-4-4 defines immunity requirements, test setups, test procedures, test equipment (and their calibration and verification), and ranges of test levels related to repetitive electrical fast transients (EFTs, bursts). In reality, bursts originate from switching transients: interruption of inductive loads, relay contact bounce, etc.

The object of IEC 61000-4-4 is to establish a common and reproducible reference in order to evaluate the immunity when subjected to bursts on the following ports [10]:


#### *K.8.1 IEC 61000-4-4: Test Setup*

Floor standing EUTs and equipment designed to be mounted in other configurations, unless otherwise mentioned, shall be placed on a ground reference plane and shall be insulated from it with an insulating support with a thickness of 0*.*1 m ± 0*.*05 m including non-conductive castors. Tabletop equipment and equipment normally mounted on ceilings or walls as well as built-in equipment shall be tested with the EUT located 0*.*1 m ± 0*.*01 m above the ground reference plane. Figure K.29 shows locations for supply line coupling (A) and location for signal lines coupling (B).

The simplified circuit diagram of the generator is given in Fig. K.30. The effective output impedance of the generator shall be 50 . Here is the explanation of the components:


The burst generator output signal consists of periodic burst pulse packages (75 pulses per package). The burst period is 300msec and the repetition period is either 10µs (100 kHz) or 200µs (5 kHz) (Fig. K.31).

**Fig. K.30** EFT voltage bursts test generator [10]

#### *K.8.2 IEC 61000-4-4: Test Levels*

The use of 5 kHz repetition frequency is traditional (Table K.9). However, 100 kHz is closer to reality. Product committees should determine which frequencies are relevant for specific products or product types. For example, with some products, there may be no clear distinction between power ports and signal ports, in which case it is up to product committees to make this determination for test purposes.

**Fig. K.31** EFT burst pulse packages [10]

**Table K.9** IEC 61000-4-4 EFT (bursts) voltage test levels [10]


Here the respective electromagnetic environmental classes for the test levels above [10]:


# **K.9 Surge: IEC 61000-4-5**

The IEC/EN 61000-4-5 EMC standard is about immunity testing (measurement techniques) against surges (high energy, high voltage pulses). This standard is part of the Basic EMC Publications.

IEC 61000-4-5 defines immunity requirements, test setups, test procedures, test equipment (and their calibration and verification), and ranges of test levels related to unidirectional surges caused by over-voltages from switching and lightning transients. Direct injections of lightning currents, i.e., direct lightning strikes, are not considered in this standard.

The object of IEC 61000-4-5 is to establish a common reference for evaluating the immunity of electrical and electronic equipment when subjected to surges.

#### *K.9.1 IEC 61000-4-5: Test Setup*

The test setup comprises the following equipment [13]:


Two types of surge pulse generators are specified in IEC 61000-4-5.


The simplified circuit diagrams of the 1.2/50µs and the 10/700µs hybrid surge generators are given in Figs. K.32 and K.33. The effective output impedance shall be 2 ± 10% for the 1.2/50µs generator and 40 ± 10% for the 10/700µs generator. Here are the explanation of the components:


**Fig. K.32** 1.2/50µs-surge voltage test generator [13]

**Fig. K.33** 10/700µs-surge voltage test generator [13]

• *S*2. Switch (closed if external matching resistors are attached, 10/700µs only).

The selection of the output (or source) impedance of the surge generator depends on:


Some countries (e.g., the USA) refer to other non-IEC standards that specify lower source impedances (more stringent testing). The EMC standard IEC 61000- 4-5 defines the following source impedances [13]:


Figures K.34 and K.35 show the 1.2/50µs generator and the 10/700µs generator open circuit voltage and short circuit current waveforms.

#### *K.9.2 IEC 61000-4-5: Test Levels*

Table K.10 shows the defined test levels of IEC 61000-4-5. Devices and installations connected to the public mains must have the minimum immunity level of:


The maximum peak current for each test level can be found in Table K.11: The selection of the test level should be based on the installation conditions. Unless otherwise specified in the product and product family standard, the table below should be used for this purpose. Here are the defined installation classes of IEC 61000-4-5 [13]:


**Fig. K.35** 10/700µs-surge generator waveforms. (**a**) Open circuit voltage waveform [13]. (**b**) Short circuit current waveform [13]

**Table K.10** IEC 61000-4-5 surge voltage test levels [13]




**Table K.11** IEC 61000-4-5 surge test maximum currents [13]

### **K.10 RF Conducted Immunity: IEC 61000-4-6**

The IEC/EN 61000-4-6 EMC standard is about immunity testing (measurement techniques) against conducted disturbances induced by radio-frequency fields (RF, 9 kHz to 80 MHz). This standard is part of the Basic EMC Publications. Equipment not having at least one conducting wire and/or cable (such as mains supply, signal line, or earth connection) which can couple the equipment to the disturbing RF fields is excluded from the scope of this publication. No tests are required in the 9 kHz–150 kHz range.

The objective of IEC 61000-4-6 is to establish a common reference for evaluating the functional immunity of electrical and electronic equipment when subjected to conducted disturbances induced by RF fields. Typical RF transmitters (sources of emissions) are:


It is assumed that electromagnetic field disturbances may act on the whole length of cables connected to the installed equipment. The dimensions of the disturbed equipment are assumed to be small compared to the wavelengths involved. The in-going and out-going cables and wires (e.g., mains, communication lines, interface cables) behave as passive receiving antenna networks (they can be several wavelengths long). The susceptible equipment is exposed to currents flowing through the equipment between those cable networks. Cable systems connected to equipment are assumed to be in resonant mode (*λ*/4, *λ*/2 dipoles) and are represented by coupling and decoupling devices having a common-mode impedance of 150 with respect to a ground reference plane.

#### *K.10.1 IEC 61000-4-6: Test Setup*

The effect of conducted disturbing signals, induced by electromagnetic radiation, is tested by injecting the signal via particular Coupling/Decoupling Networks (CDNs) to the cabling. All cables exiting the EUT shall be supported at a height of at least 30 mm above the reference ground plane (Figs. K.36 and K.37).

The IEC 61000-4-6 test method subjects the EUT to a source of disturbance comprising electric and magnetic fields, simulating those coming from intentional RF transmitters. The disturbing fields (*E* [V/m] and *H* [A/m]) are approximated by the electric and magnetic near-fields resulting from the voltages and currents caused by the test setup. The use of coupling and decoupling devices to apply the disturbing signals to one cable at a time, while keeping all other cables non-excited, can only approximate the real situation where disturbing sources act on all cables simultaneously, with a range of different amplitudes and phases.

The appropriate coupling of the disturbing signal to the cables connected to the EUT can be achieved by coupling and decoupling devices. Here are the different types of coupling devices:

	- M: Power line/mains connection.
	- C: Shielded coaxial cable.
	- T: Unshielded balanced signal pairs.
	- AF: Unshielded and unbalanced signal cables.
	- S: Shielded signal cables.
	- **Current clamps.** Establish an inductive coupling to the cable connected to the EUT.
	- **EM clamps.** Establish a capacitive and an inductive coupling to the cable connected to the EUT.

**Fig. K.37** IEC 61000-4-6 RF signal test generator [11]

**Table K.12** IEC 61000-4-6 conducted RF immunity test levels [11]


# *K.10.2 IEC 61000-4-6: Test Levels*

The EUT is subjected to an electromotive force (EMF) of 1 V, 3 V, or 10 V from 150 kHz to 80 MHz (Table K.12). This frequency range is 80% amplitude modulated (AM) with a 1 kHz sine wave. The RF signal generator provides the modulated frequency at a step rate of 1% of fundamental to the RF signal. The selection of the test level depends on the electromagnetic environment where the EUT will be used in. The immunity test levels of IEC 61000-4-6 are divided into the classes below [11]:


# **K.11 Magnetic Field Immunity: IEC 61000-4-8**

The IEC/EN 61000-4-8 relates to the immunity requirements of equipment, only under operational conditions, to radiated magnetic disturbances at power frequencies 50 Hz and 60 Hz. This standard is part of the Basic EMC Publications (according to IEC Guide 107). IEC 61000-4-8 defines:


IEC 61000-4-8 does not consider disturbances due to capacitive or inductive coupling in cables or other parts of the installation. However, other IEC standards dealing with conducted disturbances cover these aspects. The object of IEC 61000- 4-8 is to establish a common and reproducible basis for evaluating the performance of electrical and electronic equipment for household, commercial, and industrial applications when subjected to magnetic fields at power frequency (continuous and short-duration field).

In reality, the source of the magnetic fields is typically the power line current flowing in conductors, or occasionally, transformers in nearby equipment. IEC 61000-4-8 distinguishes between the following two cases of interference due to lowfrequency magnetic fields (50 Hz, 60 Hz):


Other types of magnetic fields are subject to further standards:


### *K.11.1 IEC 61000-4-8: Applicability*

IEC 61000-4-8 applies to equipment used at different locations [12]:


The applicability of IEC 61000-4-8 to equipment installed in these different locations is determined by the presence of the following phenomenon:

• The continuous magnetic field test is applicable to all equipment intended for public or industrial low-voltage networks or electrical installations.

• The short-term (pulsed) magnetic field test is typically applicable to equipment installed in places with a particularly harsh electromagnetic environment.

# *K.11.2 IEC 61000-4-8: Test Setup*

The test setup is simple and consists of the following parts:


The method for testing the immunity to magnetic fields is to produce a controlled magnetic field of known field strength by driving a large coil with a test generator, and placing the equipment in the center of the coil, thereby subjecting the equipment to the magnetic field. Figure K.38 shows the principle circuit diagram of a test generator: One or several induction coils generate the magnetic field. Over the complete volume of the EUT, the magnetic field strength must not fluctuate more than ±3 dB. The EUT dimensions must be equal or smaller than the testing volume. Below is a list of typical loop coil test antennas. However, other designs are possible.


Testing takes place in three orthogonal orientations (by rotating the antenna by 90◦). The EUT has to be placed on a 10 cm insulating support (e.g., dry wood) over the ground reference plane. Some example test setups are shown in Fig. K.39.

### *K.11.3 IEC 61000-4-8: Test Levels*

The required immunity test level for a specific product is specified in the corresponding EMC product standard, EMC family standard, or the generic EMC standard (Table K.13). The immunity test levels are specified in [A/m]. In free space, a magnetic field strength of 1 A/m corresponds to a magnetic flux density of 1*.*26µT. The selection of a test level depends on the expected operating environment of the equipment under test (EUT). Therefore, an appropriate test level for an EUT should be chosen based upon the magnetic field strengths the EUT is likely to encounter in its typical operating environment. The environments in which equipment can operate are divided into five classes:

**Table K.13** IEC 61000-4-8 magnetic field immunity test levels [12]



# **K.12 AC Dips: IEC 61000-4-11**

The IEC/EN 61000-4-11 EMC standard is about immunity testing (measurement techniques) of voltage dips, short interruptions, and voltage variations. This standard is part of the Basic EMC Publications (according to IEC Guide 107). IEC 61000- 4-11 defines the immunity test methods and test levels for electrical and electronic equipment connected to low-voltage power supply networks for:


Voltage dips or short interruptions are typically caused by faults in the public mains network. Voltage variations are typically the result of varying loads connected to the public mains.

**Fig. K.40** IEC 61000-4-11 test setup example [7]

### *K.12.1 IEC 61000-4-11: Applicability*

This standard applies to electrical and electronic equipment having a rated input current not exceeding 16A per phase for connection to 50 Hz or 60 Hz AC networks. It does not apply to electrical and electronic equipment for connection to DC or 400 Hz AC networks.

#### *K.12.2 IEC 61000-4-11: Test Setup*

The equipment under test (EUT) must be connected with the shortest specified power cable of the EUT. Figure K.40 shows one of several possible test setups.

#### *K.12.3 IEC 61000-4-11: Test Levels*

The test level voltages in IEC 61000-4-11 are stated relative to the rated equipment voltage (UT). The change between UT and the changed voltage is abrupt. The step can start and stop at any phase angle on the mains voltage. The following test voltage levels (in % UT) are used: 0%, 40%, 70%, and 80%, corresponding to dips with residual voltages of 0%, 40%, 70%, and 80%.

The required immunity test level for a specific product is specified in the corresponding EMC product standard, EMC family standard or in the generic EMC standard. The selection of a test level depends on the expected operating environment of the equipment under test (EUT). The environments in which equipment can operate are divided into three classes [7]:

	- a major part of the load is fed through converters
	- welding machines are present
	- large motors are frequently started
	- loads vary rapidly.

The test levels for voltage dips and interruptions are given in the tables below. The preferred test levels and durations given in Tables K.14 and K.15 consider the information given in IEC 61000-2-8. The test levels for voltage variations are only optional. Therefore, they are not mentioned here. Some example graphs of voltage dips and short interruptions are shown in Figures K.41, K.42, and K.43.


**Table K.14** IEC 61000-4-11 AC voltage dips test levels [7]


**Table K.15** IEC 61000-4-11 AC voltage interruptions test levels [7]

**Fig. K.41** IEC 61000-4-11 70% voltage dip for 25 periods

**Fig. K.42** IEC 61000-4-11 40% voltage dip RMS

# **K.13 Generic Immunity Levels for Residential Environments: IEC 61000-6-1**

The IEC 61000-6-1 is a Generic Standard for immunity (conducted, radiated) in residential, commercial, and light-industrial environments (indoor and outdoor). The immunity requirements cover the frequency range 0 Hz to 400 GHz.

**Fig. K.43** IEC 61000-4-11 short voltage interruption

IEC 61000-6-1 applies to:


# **K.14 Generic Immunity Levels for Industrial Environment: IEC 61000-6-2**

The IEC 61000-6-2 (Generic standards—immunity standard for industrial environments) is a generic standard for immunity (conducted, radiated) in industrial environments (indoor and outdoor). The immunity requirements cover the frequency range 0 Hz to 400 GHz.

IEC 61000-6-2 applies to:


# **K.15 Generic Emission Limits for Residential Environments: IEC 61000-6-3**

The IEC 61000-6-3 is a Generic Standard for emissions (conducted, radiated) in residential environments (indoor and outdoor). The emission requirements cover the frequency range 0Hz to 400GHz.

IEC 61000-6-3 applies to:


# **K.16 Generic Emission Limits for Industrial Environment: IEC 61000-6-4**

The IEC 61000-6-4 is a Generic Standard for emissions (conducted, radiated) in industrial environments (indoor and outdoor). The emission requirements cover the frequency range 9 kHz to 400 GHz.

IEC 61000-6-4 applies to:


# **K.17 Generic Emission Limits for Light-Industrial Environment: IEC 61000-6-8**

The IEC 61000-6-8 is a Generic Standard for emissions (conducted, radiated) in light-industrial environments (indoor and outdoor). The emission requirements cover the frequency range 0 Hz to 400 GHz.

IEC 61000-6-8 applies to:

	- is defined as professional equipment
	- is professionally installed and maintained
	- is not intended to be used in residential locations.

Otherwise, IEC 61000-6-3 applies to electrical and electronic equipment intended for use at commercial and light-industrial locations that do not satisfy these restrictions.

• Equipment for which no relevant dedicated product standard or product family emission standard exists.

Examples of commercial or light-industrial locations are:


# **References**


# **Appendix L American Wire Gauge (AWG)**

*One who makes no mistakes makes nothing at al.*

—Giacomo Casanova

This chapter presents conversions of different wire diameter units like [mm] or [mm2] to AWG (Table L.1).


**Table L.1** American Wire Gauge (AWG)

# **Appendix M Rules of Thumb**

*Sometimes an OKAY answer NOW! is more important than a good answer LATE.*

—Eric Bogatin

This chapter contains useful rules of thumb and simplified, easy-to-remember formulas for every EMC design engineer.

#### **M.1 Air Breakdown Voltage**

# **V** ≈ **3 kV/mm**

Figure M.1 shows that the breakdown voltage for air is around 5 kV/mm at sea level and 3 kV/mm at 5000 m altitude. To avoid unexpected voltage breakdowns, the air breakdown voltage is usually assumed to be 3 kV/mm.

#### **M.2 Amplitude of Square Wave Harmonics**

Assumption: ideal square wave with 0 sec rise-/fall-time and 50% pulse duration (see Fig. M.2).

$$V\_{n,RMS} \approx \frac{4}{\pi} \cdot \frac{1}{n} \cdot \frac{V\_{pp}}{2\sqrt{2}} \approx \frac{0.45 V\_{pp}}{n} \tag{\text{M.1}}$$

where

*Vn,RMS* = RMS value of the *n*-th harmonics sine wave in [V] *Vpp* = peak-to-peak value of the ideal square wave in [V]

403

**Fig. M.1** Paschen's original curves show the breakdown voltage of air as a function of pressure [1]

**Fig. M.2** Ideal square wave with its first seven harmonics

*n* = odd harmonics number [1, 3, 5, ...], even harmonics have an amplitude of 0 V for an ideal square wave signal

#### **M.3 Antenna Input Power for Desired E-Field Strength**

$$P\_l = \frac{(E \cdot d)^2}{30 \cdot G\_{li}} = \frac{(E \cdot d)^2}{30 \cdot 10^{\frac{G\_{li}[\text{dBi}]}{10}}} \tag{\text{M.2}}$$

where


*d* = distance from the transmitting antenna in [m]

#### **M.4 Bandwidth of Digital Signals**

**<sup>B</sup>** <sup>≈</sup> **<sup>0</sup>***.***<sup>35</sup> t10**%−**90**%

*B* [Hz] is the highest significant frequency in the spectrum of the digital signal and *t*10%−90% [sec] the minimum rise/fall-time of the digital signal from 10% to 90%.

#### **M.5 Critical Length of Transmission Lines**

**lcritical** *> λ/*10

A transmission line, which is longer than a tenth of the signals wavelength *λ*, should no longer be modeled with lumped *RLC*-elements, but instead with its characteristic impedance *Z*<sup>0</sup> []. More details in Sect. 7.2.3 on page 66.

#### **M.6 Characteristic Impedance of Wires**

**Z0** ≈ 100 to 150

Typical characteristic impedance *Z*<sup>0</sup> [] of a pair of wire is 100 to 150 [3].

# **M.7 Capacitance Per-Unit-Length**

# **C'** ≈ **50 fF/mm**

Assumption: small wire (e.g., AWG26) or narrow PCB trace (e.g., 0*.*25 mm), isolated with plastics, forward and return path close to each other (*<* 1 mm). More details in Sect. 7.4 on page 73.

# **M.8 Inductance Per-Unit-Length**

# **L'** ≈ **1 nH/mm**

Assumption: small wire (e.g., AWG26) or narrow PCB trace (e.g., 0*.*25 mm). The value represents the loop inductance per-unit-length: internal, external, and mutual inductances of forward and return current path. More details in Sect. 7.4 on page 73.

# **M.9 CMOS Input Impedance**

# **Rin** *>* **1G**

**Cin** ≈ **5pF**

The input capacitance *Cin* [F] of integrated CMOS circuits is often used in combination with an external output resistor *Rext* [] close to the driver's output for controlling the rise- and fall-time of digital signals (see Fig. M.3).

**Fig. M.3** *Rout* [] is used to control the voltage transients

# **M.10 E-Field from Differential-Mode Currents in Small Loops**

*EDM,max* ≈ ⎧ ⎨ ⎩ <sup>1</sup>*.*<sup>32</sup> · <sup>10</sup>−<sup>14</sup> *IDM <sup>f</sup>* <sup>2</sup>*<sup>A</sup> <sup>d</sup>* without reflecting ground plane <sup>2</sup>*.*<sup>64</sup> · <sup>10</sup>−<sup>14</sup> *IDM <sup>f</sup>* <sup>2</sup>*<sup>A</sup> <sup>d</sup>* with reflecting ground plane (Fig. 9.7 on page 114 ) (M.3)

where

*EDM,max* = maximum RMS field-strength measured in the far-field and radiated by an electrically a small current loop with area *A*, circumference *< λ/*4 and driven by the differential current *IDM* in [V/m] (Sect. 9.9.1 on page 121) *IDM* = RMS value of the differential-mode current in [A] *f* = frequency of the sinusoidal current signal in [Hz] *<sup>A</sup>* <sup>=</sup> area of the current loop in [m2] *d* = distance to the center of the current loop in [m]

#### **M.11 E-Field from Common-Mode Currents in Short Cables**

$$E\_{CM,max} \approx \begin{cases} 0.63 \cdot 10^{-6} \frac{I\_{CM}fl}{d} & \text{without reflecting ground plane} \\ 1.26 \cdot 10^{-6} \frac{I\_{CM}fl}{d} & \text{with reflecting ground plane (Fig. 9.7 on page 114)} \end{cases} \tag{14.4}$$

where

*ECM,max* = maximum RMS field-strength measured in the far-field and radiated by an electrically short cable of length *l < λ/*4 and driven by the common-mode current *ICM* in [V/m] (Sect. 9.9.2 on page 122 )

*ICM* = RMS value of the common-mode current along the cable in [A]

*f* = frequency of the sinusoidal current signal in [Hz]

*l* = length of the conductor or transmission line in [m]

*d* = distance to the center of the cable in [m]

# **M.12 E-Field from Common-Mode Current with Cable at Resonance**

$$E\_{\max} \approx \begin{cases} 60 \frac{I\_{CM}}{d} & \text{without reflecting ground plane} \\ 120 \frac{I\_{CM}}{d} & \text{with reflecting ground plane (Fig. 9.7 on page 114)} \end{cases} \tag{M.5}$$

where

*Emax* = maximum RMS *E*-field at distance *d* for matched impedances, matched polarization, line-of-sight path, and a resonant structure (that is resonant at the frequency of *ICM* ) driven by *ICM* in [V/m]

*ICM* = RMS common-mode current that flows along the resonant structure in [A] *d* = distance from the antenna, where *Emax* is measured in [m]

# **M.13 E-Field from Common-Mode Voltage with Cable at Resonance**

$$E\_{\max} \approx \begin{cases} 0.8 \frac{V\_{CM}}{d} & \lambda/2 \text{-dipole without reflecting ground plane} \\ 1.6 \frac{V\_{CM}}{d} & \lambda/2 \text{-dipole with reflecting ground plane (Fig. 9.7 on page 114 )}, \\ 1.6 \frac{V\_{CM}}{d} & \lambda/4 \text{-monopole without reflecting ground plane} \\ 3.2 \frac{V\_{CM}}{d} & \lambda/4 \text{-monopole with reflecting ground plane (Fig. 9.7 on page 114 )} \end{cases} (1.6)$$

where

*Emax* = maximum RMS *E*-field at distance *d* for matched impedances, matched polarization, line-of-sight path, and a resonant structure (that is resonant at the frequency of *VCM* ) driven by *VCM* in [V/m]

*VCM* = RMS common-mode voltage that drives the resonant structure in [V] *d* = distance from the antenna, where *Emax* is measured in [m]

#### **M.14 E-Field Strength for Given Antenna Power**

$$E = \frac{\sqrt{30P\_tG\_{it}}}{d} = \frac{\sqrt{30P\_t10^{\frac{G\_{it}\|\text{dBi}\|}{10}}}}{d} \tag{\text{M.7}}$$

where


*d* = distance from the transmitting antenna in [m]

#### **M.15 Galvanic Corrosion**

To prevent galvanic corrosion of the connection of dissimilar metals, the galvanic series voltage difference of the dissimilar metals should be ≤0*.*5 V for controlled environments and ≤0*.*15 V for harsh environments [2]. More details in Chap. H on page 311.

#### **M.16 Return Current on PCB Planes**

The return current path on PCBs with a solid reference plane [3]:


#### **M.17 Via Capacitance and Inductance**

**Cvia** <sup>≈</sup> **0.4 pF**<sup>1</sup> **Lvia** <sup>≈</sup> **1.2 nH**<sup>2</sup>

#### **References**


<sup>1</sup> *Cvia* <sup>≈</sup> <sup>0</sup>*.*0555*ε*- *rhd*<sup>1</sup> *<sup>d</sup>*2−*d*<sup>1</sup> , where *<sup>h</sup>* <sup>=</sup> thickness of PCB in [mm], *<sup>d</sup>*<sup>1</sup> <sup>=</sup> diameter of the via pad in [mm], *d*<sup>2</sup> = diameter of the anti-pad (void area between the pad and the copper plane) in [mm] and *Cvia* is the approximate via capacitance in [pF] [4].

<sup>2</sup> *Lvia* <sup>≈</sup> *<sup>h</sup>* 5 · <sup>1</sup> <sup>+</sup> ln 4*h d* , where *<sup>h</sup>* <sup>=</sup> thickness of PCB in [mm], *<sup>d</sup>* <sup>=</sup> inner diameter of via in [mm] and *Lvia* is the approximate via inductance in [nH] [4].

# **Index**

#### **Symbols**

1 dB compression point (P1dB), 56

#### **A**

Abbreviations, xxiii Acronyms, xxiii American Wire Gauge (AWG), 401 Ampére's law, 301 Analog-To-Digital Converter (ADC), 259 ANATEL label, 326 Antenna factor, 116 Antennas, 111 *λ/*2-dipoles, 114 *λ/*4-monoples, 114 antenna gain, 113 coil factor, 123 directivity, 113 *E*-field antennas, 111, 119, 120 effective aperture, 116 EIRP, 114 ERP, 114 far-field antennas, 112 H-field antenna, 122 induction coil, 123 intended antennas, 119 isotropic radiator, 112 maximum effective aperture, 116 near-field antennas, 111 realized antenna gain, 115 unintended antennas, 125 Attenuation constant, 83

#### **B**

Balanced transmission line, 89 Balun, 90, 178 Bandwidth, 45 Bandwidth of digital signals, 45, 405 Bode plot, 248 Breakdown voltage air, 403 BSMI mark, 334 Burst filters, 256 Bursts, 377

#### **C**

Cable, 37 Cable shields, 225 Capacitance of vias, 409 Capacitance per-unit-length, 406 Capacitive coupling, 192, 202, 204 Capacitors, 157, 258 aging, 163 decoupling, 157 frequency dependence, 163 multilayer ceramic capacitors, 157 safety capacitors, 158 temperature derating, 160 voltage derating, 161 Causal system, 245 CB scheme, 21 CCC mark, 327 CE mark, 329 CH mark, 333 Characteristic impedance, 69, 85 Characteristic impedance of wires, 405

© The Author(s) 2023 R. B. Keller, *Design for Electromagnetic Compatibility–In a Nutshell*, https://doi.org/10.1007/978-3-031-14186-7

CISPR 11, 344 CISPR 32, 357 Clamping devices, 180 Common impedance coupling, 191, 201, 236, 266 Common-mode, 198 Common-mode chokes, 173 Common-mode current, 200, 203, 206 Common-mode noise, 203 Common-mode to differential-mode conversion, 203 Common-mode voltage, 199, 206 Complex conjugate, 281 Complex numbers, 279 Complex permeability, 290 Complex permittivity, 289 Compliance marks, 19 Compliance testing, 344 Conducted immunity, 4, 388 Conducted immunity testing, 384 Conducted susceptibility, 4 Conductive coupling, 191 Conductivity, 307 Conductor, 145, 258 Conductor equivalent circuits, 145 Conformity assessment, 9 Constants, 277 Coplanar wave guide, 38 Corrosion, 315 Coupling capacitive coupling, 202 conductive coupling, 201 electromagnetic coupling, 197 inductive coupling, 195 Coupling path capacitive coupling, 192 Coupting paths, 189 Critical transmission line length, 405 Crowbar devices, 185 Curie temperature, 166 Current, 288 Current density, 294

#### **D**

Date Of Withdrawal (DOW), 331 Decibel, 24 Decoupling, 157, 267 global decoupling, 267 local decoupling, 267 Design guidelines, 265 Dielectric constant, 277, 289, 310 Differential-mode, 198 Differential-mode noise, 200

Differential-mode to common-mode conversion, 205 Differential-mode voltage, 199 Dirac delta function, 246 Directivity, 113 Displacement current, 293 Displacement current density, 293 Dissipation factor, 88 Distributed parameter model, 69

#### **E**

EAC mark, 328 Effective aperture, 116 Electric current density, 294 Electric displacement field, 292 Electric field, 291 Electric field coupling, 192 Electric flux, 292 Electric flux density, 292 Electrical Fast Transients (EFT), 256, 377 Electromagnetic Compatibility (EMC), 1 emission, 2 immunity, 2 susceptibility, 2 Electromagnetic coupling, 197, 205 Electromagnetic field, 95, 297 Electromagnetic Interference (EMI), 1 Electromagnetic Pulse (EMP), 4, 115 Electromagnetism, 287 Electromotive Force (EMF), 300 Electrostatic Discharge (ESD), 4, 321, 369 EMC compliance, 10 EMC design guidelines, 265 EMC Directive (EMCD), 10, 329 EMC regulations, 10, 325 EMC standards, 11 basic EMC publications, 16 EMC emission standards, 14 EMC immunity standards, 14 EMC product family standards, 17 EMC product standards, 17 generic EMC standards, 17 performance criteria, 15 EMC test lab, 21 EMC testing, 343 CB EMC test scheme, 21 emission testing, 33 immunity testing, 33 EMI receiver, 344 Emission conducted emission, 3 radiated emission, 3

Index 413

Emission testing, 119, 122, 344, 357 Equivalent Series Inductance (ESL), 157 Equivalent Series Resistance (ESR), 157 ESD filters, 256 ETSI, 13, 330 Euclidean geometry, 283 Euler's number, 277 Eurasian economic union 13g, 328 European Committee for Standardization (CEN), 330 European Committee for Electrotechnical Standardization (CENELEC), 13, 330 European Telecommunications Standards Institute (ETSI), 13, 329, 330 European Union (EU), 329

#### **F**

Far-field, 100, 118, 214, 216 near-field to far-field boundary, 98 Far-field coupling, 197 Faraday's law, 300 Fast Fourier Transform (FFT), 304 Federal Communications Commission (FCC), 336 Ferrite beads, 169 Ferrites, 169, 258 cable mount ferrite beads, 169 chip ferrite beads, 171 ferrite beads, 169 Ferromagnetic materials, 307 Filter characterization, 245 Filtering, 245 Filters, 245 active filters, 251 analog filters, 259 anti-aliasing filter, 259 band-pass filters, 250 common-mode filters, 254 differential-mode filters, 254 digital filters, 259 FIR filters, 259 high-pass filters, 250 low-pass filters, 249 passive filters, 251 transient suppression filters, 256 Flicker, 3, 365 Fourier analysis, 41 Fourier series, 42 Fourier transform, 42, 248, 304, 403 DFT, 42, 304 discrete fourier transform, 304 discrete-time fourier transform, 305 DTFT, 305

FFT, 43, 304 fourier series, 303 FR-4, 312 Fraunhofer region, 100 Free-Space Path Loss (FSPL), 131 Frequency, 34 Frequency response, 245, 248 Frequency-domain, 41 Fresnel zone, 97 Friis transmission equation, 131

#### **G**

Galvanic corrosion, 409 Galvanic series, 317 Gas Discharge Tubes (GDT), 185 Gas discharge tubes (GDT), 258 Gauss' law for electricity, 299 Gauss' law for magnetism, 300 Ground bounce, 271 Ground loops, 241 Ground planes, 237 Ground system categories, 238 Grounding, 235 chassis ground, 235 daisy-chain ground system, 239 functional ground, 235 ground symbols, 235 grounding of cable shields, 231 grounding of shielding enclosures, 224 hybrid cable shield grounding, 231 safety ground, 235 signal ground, 235 single-point ground systems, 239 star ground system, 239

#### **H**

Harmonic currents, 358 Harmonics amplitude, 403 Harmonised standards, 330 Hybrid cable grounding, 273 Hybrid ground systems, 240

#### **I**

IEC 61000-3-2, 358 IEC 61000-3-3, 365 IEC 61000-4-11, 392 IEC 61000-4-2, 369 IEC 61000-4-3, 373 IEC 61000-4-4, 377 IEC 61000-4-5, 380 IEC 61000-4-6, 384

IEC 61000-4-8, 388 Imbalance Difference Modeling (IDM), 205 Imbalance factor, 199, 200, 206 Immunity testing, 120, 123 Impedance, 288 Impedance matching, 61 *s*11, 50 reflection coefficient, 49 return loss, 51 S-parameters, 53 VSWR, 50 Impulse response, 245, 246 Indian Standards Institution (ISI) mark, 331 Inductance external inductance, 152 internal self-inductance, 149 loop inductance, 154 mutual inductance, 147 partical inductance, 153 Inductance of vias, 409 Inductance per-unit-length, 406 Inductive coupling, 195, 202 Inductors, 164 Curie temperature, 166 current saturation, 166 frequency dependence, 166 Innovation, Science and Economic Development (ISED), 327 Insertion loss, 52 Insulators, 310 Integrad transform, 303 Intended antenna, 119 Internal impedance, 151 internal inductance, 149 resistance, 148 International Electrotechnical Commission (IEC), 11 International Electrotechnical Commission for Electrical Equipment (IECEE), 21 International Telecommunication Union (ITU), 34 frequency bands, 34 frequency spectrum, 34 Intrinsic impedance, 100 Intrinsic impedance of vacuum, 277 In-Vitro Diagnostics Regulation (IVDR), 329 ISED certification number, 327 ISM frequency bands, 345 Isotropic radiator, 112

#### **K**

Korean Certification (KC) mark, 333

#### **L**

Laplace transform, 43, 248, 305 Life cycle costs, 5 Linear and Time-Invariant (LTI) system, 245 Linear system, 245 Link budget, 132 Lorentz force, 302 Loss tangent, 88

#### **M**

Magnetic field, 295 Magnetic field coupling, 195 Magnetic flux, 295 Magnetic flux density, 296 Magnetic induction, 296 Magnetic permeability, 277, 307 Material properties, 307 Maxwell's equations, 299 Medical Device Regulation (MDR), 329 Microstrip line, 38 Mode conversion, 203, 205 Multilayer Ceramic Capacitors (MLCCs), 157 Multipath effect, 119 Multi-point ground systems, 239 Mutual inductance, 147

#### **N**

Near-field, 97, 216 near-field to far-field boundary, 98 radiating near-field, 99 reactive near-field, 99 Near-field coupling, 192, 196 Noise coupling, 190 Noise factor, 55 Noise figure, 55 Nyquist-Shannon sampling theorem, 259

#### **O**

One-port circuit, 50

#### **P**

PCB laminates, 312 PCB materials, 311 Permeability, 290, 307 Permeability of vacuum, 277 Permittivity, 289, 310 Permittivity of vacuum, 277 Phase constant, 83 Polarization, 294 Polarization current, 293 Polarization density, 294

Index 415

Power, 289 Power density, 112, 297 Poynting vector, 95 Pre-compliance testing, 343 Presumption of conformity, 331 Printed Circuit Board (PCB), 37, 311 PCB trace, 37 Propagation constant, 82 Proximity effect, 153 PSE label, 332

#### **R**

Radiated coupling, 197 Radiated emissions common-mode signals, 130 maximum e-field from common-mode signals, 127 short cables, 126 small circuit loops, 125 Radiated immunity, 4 Radiated immunity testing, 373 Radiated susceptibility, 4 Radio Equipment Directive (RED), 329 RCM mark, 325 Reactance, 288 Reflection coefficient, 49 Reflection parameter, 61 Regulatory Compliance Mark (RCM), 325 Resistance, 148, 288 Resistivity, 307 Resistors, 154, 258 Resolution bandwidth, 58 Return current path, 237 Return loss, 51 Ringing, 46 Rules of thumb, 403

#### **S**

Safety, 21 Sampling frequency, 259 Scattering parameters, 53 Self-inductance, 165 Sheet resistance, 140 Shielding, 212 absorption loss, 213 cable shields, 225 grounding of cable shields, 231 grounding of shielding enclosures, 224 hybrid cable shield grounding, 231

low-frequency magnetic field shielding, 219 multiple-reflection loss, 213 reflection loss, 213 slots and apertures, 219 transfer impedance, 225 waveguide below cutoff, 223 Shielding effectiveness, 212 Signal-To-Noise Ratio (SNR), 55 Skin depth, 135 S-parameters, 53 Specific conductance, 307 Spectrum analyzers, 57, 344 average detector, 61 center frequency, 58 EMI receivers, 58 peak detector, 61 quasi-peak detector, 61 resolution bandwidth, 58 RMS detector, 61 span, 58 start frequency, 58 stop frequency, 58 sweep time, 58 video bandwidth, 58 Speed of light, 277 SPICE simulation, 251 Standard electrode potentials, 316 Step response, 247 Stitching vias, 271 Surface resistance, 140 Surge, 380 Surge filters, 256 Symbols, xxvii

#### **T**

Technical Committee 77 (TC 77), 11 Thyristors, 185, 258 Time-domain, 41 Time-invariant system, 245 Transfer impedance, 225 Transient filters capacitors, 258 clamping devices, 180 crowbar devices, 185 gas discharge tubes, 185, 258 TVS diodes, 183, 258 TVS thyristors, 185, 258 varistors, 180, 258 Transient suppression filters, 256 Transient-Voltage-Suppression Diodes (TVS diodes), 183, 258 Transmission line input impedance, 85

Transmission lines, 38, 65 attenuation constant, 83 balanced line, 66 balanced transmission line, 89, 90 characteristic impedance, 69, 85 coaxial cable, 65 critical length, 68 dielectric losses, 87 differential signal interfaces, 90 differential signaling interfaces, 90 dissipation factor, 88 distributed parameter model, 69 high-frequency losses, 87 loss tangent, 88 lossless transmission line, 70, 85 microstrip line, 65 ohmic losses, 87 per-unit-length capacitance, 75 per-unit-length inductance, 75 phase constant, 83 propagation constant, 82 single-ended interfaces, 90 stripline, 65 unbalanced transmission line, 89, 90 waveguide, 66 Triboelectric effect, 321 Turkish Standards (TSE) mark, 335 Twisted pair, 38 Two-port circuit, 53

#### **U**

Unbalanced transmission line, 89 The United Kingdom Conformity mark (UKCA mark), 336

United Nations Economic Commission for Europe Regulation 10 (UNECE R 10), 329 Units, xxvii Unit step, 247

#### **V**

Varistors, 180, 258 Vector geometry, 283 cross product, 285 dot product, 284 Vector network analyzer, 54 Velocity factor, 38 Via capacitance, 409 inductance, 409 Video bandwidth, 58 Voltage, 287 Voltagedips, 392 Voltage-Dependent Resistors (VDR), 180, 258 Voltage Standing Wave Ratio (VSWR), 50, 61

#### **W**

Wave impedance, 102 far-field, 106 near-field, 104, 105 Waveguide, 66, 223 Wavelength, 34, 37

#### **Z**

Z-transform, 43, 306