The following chapters are thought as free education for those of you out there, who have to deal with EMC and electronics design.
We wrote down some of the most essential theoretical knowledge, which you need for mastering EMC. If you wish that we add a specific topic or if you have any questions: Please write us email@example.com.
Table Of Content
- 2 Frequency vs. Wavelength
- 3 Time Domain vs. Frequency Domain
- 4 Radio-Frequency Parameters
1 The Decibel
The decibel is defined as a ratio of two quantities, typically power, voltage or current. In the field of EMC, the decibel must be understood. Here the most important points:
Gain [dB] and Loss [dB].
Let's have a look at the amplifier or damping network below.
The power, voltage and current gain of this network can be expressed in [dB] as [1.1]:
If R1 and R2 are equal (typically 50Ω), then the following term is equal 0:
And we can write the following for power/voltage/current gain:
Points to remember when it comes to gain calculations in decibel:
Amplification. If P2 is bigger than P1, the gain value in [dB] is positive. This means if there is an amplification, the power gain in [dB] is positive.
Damping. If P2 is smaller than P1, the gain value in [dB] is negative. This means if there is a power loss, the power gain in [dB] is negative.
Cut-off Frequency. At the cut-off frequency, the output power (P2) is half the input power (P1). And the power/voltage/current gains are all -3dB.
Ratio to [dB]. If power increases by factor 2, the power/voltage/current gains increase by +3dB. If power increases by factor 10, the power/voltage/current gains increase by +10dB. To get an overview, have a look at our power/voltage/current ratios table below.
Absolute Levels [dBm, dBµV, dBµA].
The most common absolute power, voltage and current levels in EMC are [dBm], [dBµV] and [dBµA]. They are calculated like this [1.1]:
The most common absolute power, voltage and current levels in EMC are [dBm], [dBµV] and [dBµA]. They are calculated like this:
For example, a negative dBm-value means that the power level is <1mW. 0dBm = 1mW and a dBm-value bigger than 0 means that the power is higher than 1mW. The same for dBµV and dBµA: 0dBµV = 1µV (0dBµA = 1µA), a negative dBµV-value means <1µV, and a positive dBµV-value means >1µV.
Conversion formulas and tables.
Find below tables and formulas to convert between the different dB-units and from linear ratio values to dB-values and vice versa. In addition, you can download a Excel sheet, which contains conversion calculations and more [1.2].
2 Frequency & Wavelength
It is fundamental that professional engineers in the field of EMC understand the basics about signal frequency f and wavelength λ. We therefore summarized the topic as follows:
EMC and Frequency.
In general, EMC issues occur with signals of frequency f > 9kHz. This is the reason why most EMC Standards do not consider signals with f < 9kHz.
Conducted Emission. Conducted emissions tend to occur at f < 30MHz.
Radiated Emission. Radiated emissions tend to occur at f > 30MHz.
The frequency f of a sinusoidal signal and its wavelength λ [m] have the following relationship [2.1]:
Whereas v [m/sec] is the propagation velocity of the signal and f [Hz] the frequency of the signal.
Wavelength: Blank Wires vs. Cables & PCB traces.
It is important to understand that the signal propagation velocity v depends on the transport medium through which the electromagnetic field is traveling. Therefore, the same signal with the same frequency f has a different wavelength λ in a blank wire (surrounded by air) than in a cable or PCB trace (surrounded by an insulation). The wavelength λ [m] is calculated the following way [2.1]:
where v is the signal propagation velocity in [m/sec], c is the speed of light (3E8 [m/sec]), f is the frequency of the sinusoidal signal in [Hz], εr is the relative permittivity  and μr is the relative permittivity  of the media through which the electromagnetic field is propagating. VF is called the velocity factor.
Wavelength in a blank wire. The wavelength λ of a signal with frequency f in a blank wire (or antenna surrounded by air) depends only on the speed of light c and the signal frequency f (v=c, because εr = 1 and μr =1 and therefore VF=1) [2.1]:
Wavelength in cables and PCB traces. The wavelength λ of a signal with frequency f in an insulated copper wire or a cable or a Printed Circuit Board (PCB) trace is [2.1]:
Where c is the speed of light (3E8 [m/sec]), f is the signal frequency [Hz], εreff the effective dielectric constant (relative permittivity)  through which the electromagnetic wave is propagating. The effective dielectric constant εreff is defined as the uniform equivalent dielectric constant for a transmission line, even in presence of different dielectrics (e.g. FR-4 and air for a microstrip line, see picture below).
The relative permeability μr is assumed to be equal 1 for cables and PCBs, because the insulation materials are non-magnetic. Thus, the velocity factor VF depends primarily on the effective relative permittivity (also called effective dielectric constant) εreff of the insulation or PCB material.
The calculation of the effective dielectric constant εreff  depends on the insulation material and the geometry of the transmission line (e.g. ribbon cable, microstrip, coplanar waveguide, etc.), because the amount of the electric field lines in the different medias depend on the geometry of the transmission line (e.g. see the microstrip line below).
The Excel sheet below contains a calculator for calculating the effective dielectric constant εreff  (effective permittivity) for some of the most common transmission lines:
PCB-traces. Microstrip, stripline, coplanar waveguide with reference plane.
Cables. Ribbon cable, twisted pair.
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 c [3E8m/sec]: VF=v/c. Thus, the smaller the velocity factor VF , the smaller the wavelength λ [m]. The table below shows the approximate velocity factors for different insulation and PCB materials and different transmission line types [2.2, 2.3, 2.4].
The table below shows some rough approximations of wavelengths in different conductors (cables, PCBs) compared to free-space (air). Possible assumptions for wavelength λ in PCBs (FR-4) and cables are:
PCB. λPCB≈0.5*λair (assumption εreff ≈3.0...4.5 → VF≈0.5).
Cables. λCable≈0.7*λair (assumption εreff ≈1.5...3.0 → VF≈0.7).
3 Time vs. Frequency Domain
Electrical signals - periodic or non-periodic - can be measured in the time domain (e.g. with an oscilloscope) or in the frequency domain (e.g. with a spectrum analyzer). This means that an electrical signal can be described either in the time domain or in the frequency domain. The time domain representation helps you to determine the signal integrity (ringing, reflection), whereas the frequency domain representation helps you to determine at which frequencies a signal potentially leads to radiated emissions.
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=2𝝅f (notation: x(t)○─●X(ω)). 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.
The picture below shows a representation of a square wave signal (1V amplitude) with the sum of only 4 harmonic sine waves and a direct current (DC) component of 0.5V. For the representation of an ideal square wave, an indefinite number of sine waves would be necessary (because the rise- and fall-time of an ideal square wave is 0sec).
The different variants of the Fourier analysis are presented in the list below. All variants have in common that they present the necessary math for converting a signal from the time to the frequency domain and vice versa: X(ω)●─○x(t). Every Fourier analysis variant has its own field of application.
Fourier series. Time signal = continuous, periodic. Frequency spectrum = discrete. Formulas for the calculation of the complex Fourier coefficients cn of a signal x(t) are presented below [3.1]:
T is the period length of the signal x(t) in [sec], f0 is the first harmonic of the signal x(t) in [Hz], ω0 is the first harmonic of the signal x(t) in [rad], j=√(-1) is the complex number and n stands for the n-th harmonic frequency [0, 1, 2, ...].
Fourier transform. Time signal = continuous, non-periodic. Frequency spectrum = continuous. Formulas of the Fourier transform of a signal x(t) (left) and the inverse-Fourier transform (right) are given below [3.2]:
X(ω) is the Fourier transform (spectrum) of x(t), ω is the frequency in [rad], j=√(-1) is the complex number and t the time in [sec].
Discrete Fourier transform (DFT). Time signal = discrete, periodic. Frequency spectrum = discrete. The DFT is also applied to non-periodic signals in the time domain by periodically continuing the non-periodic signals in order to make them computable with the DFT. The DFT is by far the most common method of modern Fourier analysis. The Fast Fourier Transformation (FFT) is a fast algorithm for calculating the DFT (in case the bock length N is a power of two) [3.3].
X is the spectrum of x and X[k] is the k-th sample of the spectrum at ωk. 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].
Discrete-time Fourier transform (DTFT). Time signal = discrete, non-periodic. Frequency spectrum = continuous. The DTFT can be viewed as the form of the DFT when its length N approaches infinity [3.4].
X is the spectrum of x. 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]. ῶ denotes the continuous normalized radian frequency variable [-π...+π]. ῶ is the product of the radian frequency ω [rad] and the sampling time Ts.
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 [3.5].
X is the Laplace transform of x. s=σ+jω is a complex variable with a real part σ and a imaginary part with the radian frequency ω [rad]. In case of σ=0, the Laplace transform reduces to the Fourier transform.
Z-transform. The Z-transform is the time-discrete counterpart to the Laplace transform with z = e^(sTsampling) = e^((σ+jω)Tsampling) [3.6].
X is the Z-transform of x. z is a complex variable. 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].
Spectra of digital waveforms.
Waveforms of primary importance in electronic circuits are clock and data signals. 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]. The two pictures below show an extract of a digital waveform and the corresponding Fourier series [3.7].
The frequency spectrum envelope of a trapezoid-waveform signal compared to an ideal square wave signal is shown below. It can be seen, that there is no -40dB drop-off for the ideal square wave. Instead, the frequency spectrum envelope of an ideal square wave drops constantly with -20dB/decade.
The picture below compares the frequency spectrum envelope curves of two trapezoid-waveforms with different pulse-width tpw [sec], but with identical periodicity T [sec], rise- and fall-time tr and tf [sec] and amplitude A [V]. It can be seen how the duty-cycle D=tpw/T of a digital signal influences the frequency spectrum.
The picture below compares the frequency spectrum envelope curves of two trapezoid-waveforms with different rise- and fall-time tr and tf [sec], but with identical periodicity T [sec], pulse-width tpw [sec] and amplitude A [V]. It can be seen that a reduction of the rise- and fall-time leads to lower amplitudes of the higher frequencies in the spectrum (and therefore to potentially lower radiated emissions).
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 [W] in the frequency component is bigger than 50% of the power in an ideal square wave's signal with the same amplitude A [V] and duty-cycle D=tpw/T [%]. A drop in 50% of the power [W] is the same as a drop of 70% in amplitude [V] or a drop of 3dB.
The rule of thumb for calculating the bandwidth [Hz] - or the highest significant sine wave frequency - of a trapezoid digital signal is [3.8]:
Where t10%-90% is the rising- or fall-time (whichever is smaller) from 10% to 90% of the slope of a digital signal in [sec].
The picture below shows the frequency spectrum envelop curves of an ideal square wave (t10%-90%=0) and a real waveform (trapezoid with t10%-90%≠0). The bandwidth of the real waveform can be found at f=0.35/t10%-90%. Note: It is assumed that there is no ringing in the real waveform. In case of ringing, the frequency spectrum envelope for f>1/(πt10%-90%) would not drop off with -40dB/decade.
4 Radio-Frequency Parameters
This section introduces some of the most common Radio-Frequency (RF) parameters used in the field of EMC:
Reflection Coefficient Γ.
We speak of matched impedances in case the load impedance Zload is the complex conjugate of the source impedance Zsource. In radiated emission and immunity EMC testing, it is important to understand the term matching and how to quantify it. All receiver and/or transmitter antennas must be matched to their receiver and/or transmitter equipment impedance (typical Z0 = 50Ω).
The reflection coefficient Γ (=s11 in case of 1-port networks!) is defined as [4.4]:
All variables are complex numbers. Vforward is the forward voltage wave to the load and Vreflection is the reflected voltage wave by the load. Zsource is the complex source impedance and Zload is the complex load impedance. Zsource is typically the characteristic impedance Z0 or the transmission line impedance. The reflection coefficient Γ is often given in [dB]:
VSWR  means Voltage Standing Wave Ratio. The VSWR expresses the ratio of maximum and minimum voltage of a standing wave pattern on a transmission line. The VSWR can be calculated by using the reflection coefficient from above [4.5]:
The return loss [dB] is the dB-value of the loss of power in the signal reflected (Preflected) by a discontinuity in a transmission line or due to a impedance mismatch. Return loss [dB] is the negative value of the reflection coefficient Γ in [dB] [4.6].
Insertion Loss [dB].
The term Insertion Loss (IL) is generally used for describing the amount of power loss due to the insertion of one or several of the following components (passive 2-port networks):
Transmission Line (cable, PCB trace)
The insertion loss (IL) represents the power ratio in [dB] of the power P1 and the power P2 of the picture above. P1 is the power, which would be transferred to the load in case the source is directly connected to the load. The power P2 represents the power which is transferred to the load in case the Passive 2-Port Network is inserted between the source and the load [4.2], [4.3].
Impedance Matching Summary.
The table below shows how to convert between VSWR , return loss [dB] and the reflection coefficient . Z0 is the "system impedance" (typical Z0 = 50Ω or Z0 = 75Ω).
In order to give you an idea what a good match means in terms of VSWR, reflection coefficient or return loss: we summarized all the values in this table below.
5 Transmission Lines
In EMC, it is essential to understand transmission lines and when to apply the concept of transmission lines. Why? Because once a signal interconnection line exceeds a critical length lcritical [m], impedance matching (Zsource, Zline, Zload) MUST be considered to prevent reflections and ringing, thus preventing unwanted radiated emissions and bad signal quality - a signal integrity topic [5.2].
This chapter introduces you to transmission lines and their most important properties.
What is a transmission line?
A transmission line is a series of conductors, often but not necessarily two, used to guide electromagnetic energy form one place to the other [5.3]. It's that simple. The more complicated part is the math behind it (Maxwell's equations), because we do no longer consider a conductor as a lumped element (e.g. a simple R-L series circuit to emulate a electrical interconnection). 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 its characteristic impedance Z0.
Here some common transmission line geometries [5.3]:
Coax. The electromagnetic energy is propagating through the dielectric (mostly PTFE, because of its low loss and stable εr for many frequencies) between the center conductor and the inside surface of the outer conductor (shield) of a coaxial cable.
Microstrip. Transmission line where the signal conductor is on the top or bottom layer of a PCB with an adjusted return path conductor (e.g. ground plane or power supply plane).
Stripline. Transmission line where the signal conductor is embedded between two signal return path conductors (e.g. ground or power supply plane) on a PCB.
Balanced line. Two conductors of the same size and shape with equal impedance to ground and all other conductors (e.g. ethernet cable).
Waveguide. A waveguide consists of a single hollow conductor used to guide the electromagnetic energy. Waveguides are used in the gigahertz frequency range and they cannot pass direct currents (DC) signals.
When to consider a signal path as a transmission line?
Every signal interconnection is a transmission line. However, it is not necessary to threat every signal path as transmission line. Rules of thumb - if a conductor should be treated as transmission line or not - are mentioned below: one rule of thumb for the frequency domain and one for the time domain.
Frequency domain. A common rule of thumb, when working in the frequency domain, is the following [5.3]:
Interconnection length llong ≥ λ/10. Consider the signal path as transmission line, in order to minimize signal distortions and ringing due to reflections and in order to minimize radiated emissions and electromagnetic interference (EMI).
Interconnection length lshort < λ/10. Consider the signal path as simple conductor. If an interconnection length l [m] is short with respect to the signal wavelength λ [m], it is good practice that the interconnection is considered as simple conductor with lumped-element parameters (e.g. resistor R [Ω] in series with inductance L [H]).
When determine 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-times (rather than the fundamental frequency). The rule of thumb for calculating the bandwidth [Hz] - or the highest significant sine wave frequency - of a rectangular digital signal is [5.1]:
Where t10%-90% is the rising- or fall-time (whichever is smaller) from 10% to 90% of the slope of a digital signal in [sec]. The corresponding minimum wavelength λmin [m] is
Where v [m/sec] is the propagation velocity and fmax [Hz] is the highest significant sinusoidal frequency in a digital signal. The frequency domain approximation for the critical length lcritical-fd [m] can be calculated with respect to the digital signal rise-/fall-time t10%-90%:
Where c [m/sec] is the speed of light (3E8m/sec), fmax [Hz] is the highest significant sinusoidal frequency in the digital signal, t10%-90% [sec] is the rising- or fall-time (whichever is smaller) from 10% to 90% of the slope of the digital signal and εreff  is the effective dielectric constant (relative permittivity) through which the electromagnetic wave is propagating.
Time domain. There is also a rule of thumb for the time domain [5.3]:
t10-90% ≤ 2*tpd. If the rise- or fall-time t10-90% [sec] of a digital signal is smaller than twice the propagation delay tpd [sec] (along the signal line), then the signal path should be considered as transmission line, in order to minimize signal distortions and ringing due to reflections and in order to minimize radiated emissions and electromagnetic interference (EMI).
t10-90% > 2*tpd. If the rise- and fall-time t10-90% [sec] of a digital signal are bigger than twice the propagation-delay tpd [sec] of the signal across the conductor, the signal path may be considered as simple conductor.
The time domain approximation for the critical length lcritical-td [m] can be calculated with respect to the digital signal rise-/fall-time t10%-90%:
Where c [m/sec] is the speed of light (3E8m/sec), t10%-90% [sec] is the rising- or fall-time (whichever is smaller) from 10% to 90% of the slope of the digital signal and εreff  is the effective dielectric constant (relative permittivity) through which the electromagnetic wave is propagating.
Critical length lcritical. The two rules above lead to similar values for lcritical [m], where the frequency domain lcritical-fd is smaller than the time domain lcritical-td. It is therefore recommended to go with the rule of thumb of the frequency domain: Consider an interconnection as transmission line if it is longer than λ/10.
The characteristic impedance Z0, also called the wave impedance, is an important property of a transmission line. First, have a look at the equivalent circuit of a transmission line (the distribution parameter model of a two-conductor transmission line) [5.3]:
The picture shows transmission line segments of an infinitely small length dz in [m]. The parameters are defined per unit length (e.g [m]):
R' = Resistance per unit length [Ω/m]
L' = Inductance per unit length [H/m], mostly in [mH/m] or [nH/m]
C' = Capacitance per unit length [F/m], mostly in [pF/m]
G' = Conductance per unit length [S/m]
The formula for the characteristic impedance Z0 of a transmission line is defined as [5.3]:
where ω is the angular frequency [rad/sec] and all other parameters are explained above. In practice, it is often adequate to describe transmission lines as lossless (R' = 0, G' = 0). In that case, the equivalent circuit is simplified to this:
The formula for the characteristic impedance Z0 of lossless transmission lines is defined as [5.3]:
Enough theory, let's think about how to calculate the characteristic impedance for your projects.
Cables. In case your cable contains signals which require impedance matching (e.g. CAN or Ethernet), you should only use cables which match with the required impedance (e.g. 100Ω or 50Ω). In general, it is not necessary to calculate the characteristic impedance of cables, this is already done for you by the cable manufacturer.
PCBs. Because every PCB design is individually configurable (type of dielectric material, thickness of copper, PCB trace geometry etc.), it is usually necessary to calculate the characteristic impedance by yourself. Therefore, you should know the following data:
Dielectric media. Thickness, dielectric constant εr and loss tangents tan(δ) of the PCB substrate. Have a look at our PCB stackup and dielectric material collection here.
Trace. PCB trace width and geometry, copper thickness, height above reference planes, distance to copper-pour (and separation distance for differential lines).
Solder resist. For PCB traces at top or bottom, you must know the thickness and dielectric constant of the solder resist.
There are plenty of free online and offline tools available for calculating the characteristic impedance of arbitrary PCB traces or signal pairs. However, be aware that these free tools usually use approximations and the results are only accurate in a certain parameter range. If you need accurate values use a solver, which calculates the transmission line impedance based on Maxwell's Equations (e.g. HyperLynx or Si8000). To help you even more, we wrote down some information on PCB stackups and properties of PCB materials.
The propagation constant γ [1/m] is an important property of a transmission line. It describes the attenuation and phase shift of the signal as it propagates through the transmission line. Propagation constant γ is calculated like this [5.3]:
where R'=resistance per unit length [Ω/m], L'=inductance per unit length [H/m], C'=capacitance per unit length [F/m], G'=conductance per unit length [S/m], ω=angular frequency [rad] of the signal, μ=μrμ0, μ0=4πE-7 [H/m] μr=relative permeability , ε=εrε0, ε0=8.85E-12 [F/m] εr=relative permittivity.
Propagation constant γ in can also be written as [5.3]:
where the real part α in [1/m] is the attenuation constant and the imaginary part β in [rad/m] is the phase constant. α and β are calculated like this [5.3]:
For a lossless line (R'=0, G'=0, dielectric losses = 0) we get:
No attenuation α [1/m]:
Phase shift β [rad/m]:
As already mentioned, the lossless line model is good enough in most cases (up to several 100MHz). However, with increasing frequency (1GHz and higher) high frequency losses may not be neglected anymore. High frequency losses are a result of [5.3]:
Ohmic Loss. Resulting from the resistance of the conductors. Ohmic losses are a function of the frequency f as well as of the geometry of the conductor (skin-effect!).
Dielectric Loss. Resulting from the dielectric material absorbing energy from the propagating electric field (heating the material). Dielectric losses are a function of frequency f, the dissipation factor (tan(𝛿)) and the dielectric constant (εr ).
NOTE: Dielectric losses do NOT depend on the geometry of the transmission line, only on the dielectric material.
6 Near-Field vs. Far-Field
EMI often happens due to electromagnetic radiation (see coupling paths). In order to lower the coupling of these kind of interference, it is important to understand the near-field and far-field.
Electromagnetic Field Characteristics.
An electromagnetic field is characterized by three things [6.4]:
Source. The antenna (e.g. dipole/horn/loop antenna, cable, PCB trace).
Media. The media surrounding the source (e.g. air, plastics, metal).
Distance. The distance between the source and the observation point. Close (compared to the wavelength λ) to the source, the field properties are determined primarily by the source characteristics (wave impedance Zw). Far from the source, the field depends mainly on the medium through which the field is propagating.
Therefore, the space around a source of radiation can be split into two regions: The near field and the far field.
For EMC emission measurements, it is important to know if a certain measurement takes place in the near-field or the far-field. Different probes and antennas have to be used accordingly.
Near-field. Special near-field probes are used to measure the H- or E-field.
Far-field. Log-periodic, biconical or horn antennas are used to measure the EM-field.
Before we dive into the near- and far-field topic, we have to talk about the wave impedance. The wave impedance is the characteristic of a medium (air, transmission line dielectric medium etc.) in which a wave propagates.
Why is it important to know the characteristic impedance? Because the ratio of reflected and transmitted amplitude of the wave at an interface, from one medium to another, is determined by the characteristic impedance of the two media.
For ANY electromagnetic wave, the wave impedance ZW in [Ω] is defined as [6.1]:
E is the electric field in [V/m] and H is the magnetic field in [A/m].
The characteristic impedance Z0 [Ω] of a medium (NOT a transmission line, here we talk about electromagnetic waves traveling through a medium like air, insulator, metal shield, etc.) is defined as [6.1]:
ω=angular frequency [rad/sec] of the signal, μ=permeability of the medium [H/m], ε=dielectric constant of the medium [F/m], σ=conductivity of the medium [S/m].
For insulators and air (σ << jωε) the characteristic impedance is independent of the signal frequency [6.1]:
μ=permeability of the medium [H/m], ε=dielectric constant of the medium [F/m], μ0=permeability of free-space 4πE-7 [H/m], μr=relative permeability of the media , ε0=dielectric constant of free-space 8,85E-12 [F/m], εr=relative dielectric constant of the media.
For free-space (and approximately for air) are: μr=1, εr=1. Therefore, the characteristic impedance of free-space (air) is defined as [6.1]:
Wave Impedance vs. Distance.
The graph below shows the wave impedance ZW [Ω] in dependency of the distance d [m] to the source (normalized with λ/(2*π)) for an ideal Hertzian dipole (an infinitesimal element of length dl). The graph is a simplification and it should just illustrate how the magnetic field H and electric field E differ in their wave impedance ZW in the near field and that the electromagnetic field (EM-field) has a constant wave impedance ZW [6.1].
Near-Field to Far-Field Boundary.
The electromagnetic field around an antenna can be divided in three regions [6.2]:
- Radiating near-field
The regions depend on the maximum linear dimension of the antenna D [m] and the wavelength of the signal λ [m]:
D<λ/2. Electrically small antennas. The reactive near-field is significant. For electrically small antennas, the radiating near-field and the far-field are minimal if they exist at all.
D>λ/2. Electrically large antennas. All three regions are significant: the reactive near-field, the radiating near-field and the far-field.
In the near field, the characteristic impedance depends primarily on the source and the electric and magnetic fields have to be considered separately (because the ratio of E/H is not constant).
The electric field E is predominant in these cases [6.3]:
The source voltage is high compared to the source current (E/H > 377Ω).
The source impedance is high (e.g. dipole or straight wire antennas).
As a consequence of a predominant E-field:
The wave impedance near the antenna is high.
E-field attenuates with a rate of 1/d^3 in the near field (d = distance to source).
H-field attenuates with a rate of 1/d^2 in the near field (d = distance to source).
The magnetic field H is predominant in these cases [6.3]:
The source voltage is low compared to the source current (E/H < 377Ω).
The source impedance is low (e.g. loop antennas).
As a consequence of a predominant H-field:
The wave impedance near the antenna is low.
E-field attenuates with a rate of 1/d^2 in the near field (d = distance to source).
H-field attenuates with a rate of 1/d^3 in the near field (d = distance to source).
As mentioned above, the near-field can be divided into the following two regions:
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 case of D<λ/2, the reactive near-field extends until the distance d [m] from the antenna by [6.2]:
where λ is the wavelength [m] of the signal. In case of D>λ/2, the reactive near-field extends until the distance d [m] from the antenna by [6.2]:
where λ is the wavelength [m] of the signal and D the maximum linear dimension of the antenna [m].
Radiating Near-Field (Fresnel Zone). 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. Energy is radiated as well as exchanged between the source and a reactive near-field. In case of D>λ/2, the radiating near-field extends until the distance d [m] from the antenna by [6.2]:
where λ is the wavelength [m] of the signal and D the maximum linear dimension of the antenna [m]. Beyond the radiating near-field, the far-field begins.
Far-Field (Fraunhofer Region).
In the far-field, the E- and H-fields move perpendicular (orthogonal) and in phase to each other and form a plane wave.
E- and H-field attenuate with a rate of 1/d in the far field (d = distance to source).
The wave impedance in free-space (air) is 377Ω.
There is usually only a far-field region in case of D>λ/2, where λ is the wavelength [m] of the signal and D the maximum linear dimension of the antenna [m]. However, if there is a far-field, it starts roughly at the following distance d [m] from the antenna [6.2]:
7 Antennas & Radiation
This chapter is a brief introduction to the topic of antennas and electromagnetic radiation. We skip the math intense part around the Maxwell Equations. The formulas and statements in this chapter are applicable to the far field / free-space (not the near field), matched impedances (of antennas and receiver/transmitter equipment) and matched polarization (of the electromagnetic waves and the antenna polarization).
Phil Smith (inventor of the Smith Chart) explained antennas like this: ”A component that matches the (transmission) line to space”.
Isotropic Radiator and Power Density S.
First, we need to introduce two terms: isotropic radiator and power density S [W/m^2]. An 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. S represents the power density of the field around an antenna at a given distance d. The power density S of a isotropic radiator is simply the radiated power Prad by the antenna [W] divided by the surface area of the sphere [m^2] with the distance d [m] from the center of the radiator [7.1].
The power density Sw [W/m^2] of a plane wave is defined as product of the E-field [V/m] and H-Field [A/m]. ZW is the wave impedance 120π = 377 [Ω].
Antenna Gain G.
In EMC testing, we use directional antennas (e.g. biconical dipole antennas, horn antennas) which have an antenna gain G. The antenna gain G  ([dB]=log10(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 SAME power input (this means the antenna efficiency factor η=Prad/Pt, which considers the antenna losses, is already taken into account in gain G) [7.2]:
The antenna gain Gi of an ideal halve-wave (λ/2) dipole is 1.64  (2.15 [dBi]), whereas the antenna gain Gd of an ideal halve-wave (λ/2) dipole is 1 (0 [dBd]) [7.2]:
The antenna gain compared to an isotropic radiator Gi and the antenna gain compared to a dipole antenna Gd are in the following relation to each other [7.2]:
Now, we are ready to calculate the often used terms: effective isotropic radiated power (EIRP, referred to an isotropic radiator) and effective radiated power (ERP, referred to a λ/2-dipole) [7.3]:
Pt = transmitter antenna input power [W], Gi = antenna gain referred to a isotropic radiator , Gd = antenna gain referred to an ideal λ/2-dipole . The power density S [W/m^2] at a distance d [m] from the transmitter antenna can now be written as:
Effective Aperture Ae.
Another useful parameter to know is the effective area of an antenna, also called effective aperture Ae [m^2]. The effective aperture represents the ratio of power Pr [W] (output power of the receiver antenna) to the power density SW [W/m^2] (power density of the plane wave which reaches the antenna) [7.2]:
The maximum effective aperture Aem [m^2] for any antenna can be calculated out of the signal wavelength λ [m] and the antenna gain referred to an isotropic radiator Gi . The maximum effective aperture Aem refers to the circumstance where the load impedance is the conjugate of the antenna impedance (matched impedances, which means that the maximum power transfer to the load takes place) and the incident wave and the polarization of the antenna are matched.
Antenna Factor AF.
The antenna factor AF [1/m] or [dB/m] is the term which is often needed during EMC emissions testing. AF is used to calculate the received field strength E [V/m] based on the measured voltage Vr [V] at the receiver's antenna terminals. We focus on the antenna factor of E-field antennas. Therefore the antenna factor is the ratio of the strength of the electric field E [V/m] at the antenna to the voltage Vr [V] at the terminals of the receiver antenna [7.4]:
Now, we have everything ready to express AF [1/m] or [dB/m] as a function of only the plane wave frequency f [Hz] and the receiver antenna gain Gir  (in case of matched impedances and aligned antenna to the polarization of the incident wave):
Vr = voltage at receiver antenna terminals [V], Zr = impedance of receiver [Ω] (typically 50 [Ω]), E = field strength of plane wave [V/m], λ = wavelength of plane wave signal, Gir = receiver antenna gain referred to an isotropic antenna , ZW = plane wave impedance [Ω] (377 [Ω] for far field), Aem = maximum effective aperture [m^2]
If we rearrange the formula from above and replace λ=c/f (c=3E8[m/sec]), we get AF as a function of f, Zr and Gir:
AF is often used in [dB/m]:
Replace the wavelength λ=c/f, where c=3E8 [m/sec]. Now, we can express AF [dB/m] as a function of signal frequency f in [MHz] (!), receiver antenna gain Gir, and impedance at receiver measurement instrument Zr [Ω]:
Antenna Formulas: Emission Testing.
We focus here on EMC emission testing like CISPR 11 or CISPR 32. 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 EMI receiver or spectrum analyzer (for [dBm] to [dBμV] conversion: look here).
When we set the receiver impedance Zr to 50 [Ω] and wavelength λ=c/f (c=3E8 [m/sec]), we get this for the receiver antenna factor AF:
The field strength E at the antenna can be calculated based on AF (which is frequency dependent!) and Vr (output voltage from the receiver antenna):
Let's consider the cable loss [dB] from the antenna to the measurement unit (EMI receiver, spectrum analyzer) and call the measured voltage at the receiver Vmeasure [dBμV]:
Antenna Formulas: Immunity Testing.
We focus here on EMC immunity testing like IEC 61000-4-3. In other words: the physical quantity of interest is the E-field [V/m] at a certain distance d [m] for a given transmitter antenna input power [W].
From the antenna fundamentals from above, we know how to calculate the power density S [W/m^2] at a distance d [m] for a transmitter antenna with an antenna gain Git  and the power Pt [W] at the transmitter antenna terminals:
Furthermore, we know the power density S of a plane wave in free space (far field characteristic impedance of Zw=120π [Ω]) is given by [7.1]:
If we combine these two formulas above, we can determine the field strength E [V/m] at a given distance d [m] from the antenna for a given transmitter antenna input power Pt [W]:
The required power input power of a transmitting antenna Pt [W] for achieving a desired field strengt E [V/m] at a given distance d [m] is:
Free-Space Path Loss (FSPL).
The term free-space loss (FSL) or free-space path loss (FSPL) is important to understand for radiated emission and immunity testing. The FSPL is the attenuation of the electromagnetic field (EM-field) between a transmitter and a receiver antenna. It is assumed that the space between the antennas is free of obstacles and a line-of-sight path through free-space. As mentioned at the beginning of this chapter: all the formulas are valid for the far field or in other words: free-space. In addition to that, the Friis transmission equation assumes matched impedances and matched polarizations of the transmit and receiver antennas. The Friis transmission equation is stated as [7.5]:
Pr = receiver antenna output power [W], Pt = transmitter antenna input power, Git = antenna gain of transmitter antenna, Gir = antenna gain of receiver antenna, λ = wave length [m] and d = distance between transmitter and receiver antennas [m]. The free-space path loss FSPL is now defined as:
d = distance between transmitter and receiver antennas [m], λ = wave length [m], c = speed of light 3E8[m/sec], f = frequency [Hz].
For radiated EMC testing, the term link budget is often used. If you want to calculate the received power Pr [dBm] for a given receiver-transmitter-setup (transmitter, frequency, antennas, distance, etc.), you calculate the link budget. In simple terms, this means:
Received Power Pr [dBm] = Transmitted Power [dBm] + Gains [dB] - Losses [dB].
If we rewrite the link budget formula above in a little more detailed way, we get:
Pr = receiver antenna output power [dBm], Pt = transmitter antenna input power [dBm], Lt = transmitter losses (coaxial cable, connectors, etc.) [dB], Git = transmitter antenna gain [dB], FSPL = free-space path loss [dB], Lmisc = miscellaneous losses (fading, polarization mismatch, etc.) [dB], Gir = receiver antenna gain [dB], Lr = receiver losses (coaxial cable, connectors, etc.) [dB].
Introduction and definitions of the term skin depth δ [m]:
Conductors: The skin depth δ [m] is defined as the distance from the conductor edge where the current density has fallen to 37% (37% = 1/e = 1/2.72) of the current density at the surface of the conductor J0 [A/m^2]. The current density Jd [A/m^2] at distance d [m] from the conductor surface is defined as [8.2]:
8 Skin Effect
The resistance per unit length [Ω/m] for direct current (RDC') and for alternating current (RAC' ) of any conductor can be written as [8.1]:
ρ = specific electrical resistivity of the conductor material [Ωm]. σ = specific electrical conductivity of the conductor material [S/m]. A = cross-sectional area of the conductor [m^2], Aeff = effective cross-sectional area of the conductor through which the current effectively flows [m^2].
For direct current (DC, 0 Hz), the area A is equal to the complete conductor diameter. However, for alternating current (AC) with frequency f, the magnetic fields produced by current in the conductor forces the current flow towards the outer surface of the conductor. The higher the signal frequency f, the smaller the cross-section Aeff through which the current effectively flows.
As a consequence of the skin effect, the resistance of a conductor increases with increasing frequency.
The diagram below shows RAC' of a round copper wire with outer diameter D=1mm and a copper PCB trace with width w=0.25mm and height h=0.035mm (1oz). The calculations for RAC' are approximations and ignore the return current path (proximity effect) and assume a single conductor surrounded only by air. However, t