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. We are aware that the list is not complete. If you wish that we add a specific topic or if you have any questions: Please write us email@example.com.
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:
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.
Factor to [dB]: If the power increases by factor 2, the power/voltage/current gains are increasing by +3dB. If the power increases by factor 10, the power/voltage/current gains are increasing by +10dB. To get an overview, have a look at our power ratios table.
Absolute Power, Voltage, Current Levels [dBm, dBµV, dBµA].
In EMC, it is common to use the decibel to calculate absolute power/voltage/current levels. Therefore, the denominator of the logarithmic calculation is replaced with a constant reference value P0 or V0 or I0 (typical 1mW, 1µV, 1µA):
The most common absolute power, voltage and current levels in EMC are 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 values to dB-values and vice versa.
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:
Whereas v [m/sec] is the propagation velocity of the signal and f [Hz] the frequency of the signal.
Wavelength: Air vs. Conductor.
It is important to understand that the signal propagation velocity v depends on the transport medium of the signal. Therefore, the same signal with the same frequency f has a different wavelength λ in free-space (vacuum, and approximately air) than in a conductor (cable, wire).
Wavelength in Free-Space. The wavelength λ of a signal with frequency f in free-space depends on the speed of light c (approximately 3 x 10E8 [m/sec]):
Wavelength in Conductors. The wavelength λ of a signal with frequency f in a copper wire or a cable cable or Printed Circuit Board (PCB) trace:
Where c is the speed of light (3 x 10E8 [m/sec]), f is the signal frequency [Hz], εr the relative permittivity , μr the relative permeability  and VF the velocity factor .
Due to the fact that the relative permeability μr is equal 1 for non-magnetic materials like copper, the velocity factor VF depends primarily on the relative permittivity (also called dielectric constant) εr of the insulator or PCB material.
The table below shows the calculated velocity factors VF for different insulation materials (PE, PTFE, PVC, PUR) and PCB materials.
The table below shows frequencies with their corresponding wavelengths in different conductors (cables, PCBs). Good assumptions for wavelength λ in PCBs (FR-4) and cables are:
FR-4: 0.5 x the wavelength λ compared to air. Assumed εr ≈4.5, VF≈0.5.
Cables: 0.6 x the wavelength λ compared to air. Assumed εr ≈2.8, VF≈0.6.
Wavelength of Digital Signals.
Let's imagine a typical digital signal with a rectangular waveform. When it comes to EMC, it is not the first harmonic (fundamental frequency) which is of interest, it is the highest frequency content in the signal, or let's say: the bandwidth. Because in EMC, you need to know if and up to which frequency a signal could generate emissions (and if you have to consider a signal interconnection as a transmission line).
The highest frequency content of a digital signal depends on its rise/fall-time. A good estimate is:
Where t10%-90% is the rising- or fall-time from 10% to 90% of the slope of a digital signal in [sec] and fknee is the maximum frequency content in [Hz].
3 Transmission Lines
In EMC, it is essential to understand transmission lines. Why? Because once a conductor (cable, PCB trace) is longer than a tenth of the signal- wavelength λ, it is NOT enough to only consider the signal conductor, it is necessary to also consider the return path and the signal propagation. Because in case the return path is ignored (large loops), or the characteristic impedance is not considered (impedance jumps): unwanted emissions may occur or bad signal quality (a signal integrity topic, not covered here).
This chapter introduces you to transmission lines and its most important properties (characteristic impedance, propagation constant, loss).
What is a transmission line?
A signal conductor together with its return path is called a transmission line. It's that simple. The more complicated part is the math behind it (Maxwell's equations), because we do no longer consider a conductor a lumped element (e.g. a simple R-L series circuit) where current flows. Rather more, we consider the signal conductor and its return path as transmission line through which an electromagnetic field is moved from one point to another. Transmission lines are characterized or described by its characteristic impedance.
Here some common transmission line geometries:
Coax: Coaxial cables.
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 on a PCB embedded between two signal return path conductors (e.g. ground or power supply plane).
Balanced Line: Two conductors of the same size and shape with equal impedance to ground and all other conductors (e.g. ethernet cable).
When to consider a signal path as a transmission line?
EVERY signal interconnection is a transmission line. However, if the interconnection is shorter than a tenth of the wavelength λ of the signal, it is good practice that the interconnection is considered as simple conductor with lumped-element parameters (e.g. serial resistor R and serial inductance L).
Interconnection < λ/10: Consider signal path as simple conductor.
Interconnection ≥ λ/10: Consider signal path as transmission line.
NOTE: For digital signals (e.g. clocks) it is important to determine the maximum frequency (and therefore the shortest wavelength λ) of the signal. Have a look here: How to determine the maximum frequency in a digital signal.
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):
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:
where ω is the angular frequency [rad] 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:
Enough of the theory and lets 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 and dielectric media 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, thickness (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, these free tools usually use approximations and the results are only accurate in a certain parameter range. It would be better to use a solver, which calculates the transmission line impedance based on Maxwell's Equations (e.g. HyperLynx or Si8000). The best free and open source solver, which delivers accurate results is the Multiple Dielectric Impedance Calculator (MDTLC, screen shot below).
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:
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:
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:
For a lossless line (σ=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 100 MHz). However, with increasing frequency (1 GHz and higher) high frequency losses may not be neglected anymore. High frequency losses are a result of:
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.
We don't go into more detail here (more in this book). However, we would like to introduce here the practical term: 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.
NOTE: The additionally introduced loss (the Insertion Loss), due to adding the said Passive 2-Port Network, can be introduced by intrinsic high-frequency losses (copper losses, dielectric loses) in the Passive 2-Port Network and/or impedance mismatches between the source and the Passive 2-Port Network or the Passive 2-Port Network and the load (Zload).
4 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 unterstand the near-field and far-field.
Electromagnetic Field Characteristics.
An electromagnetic field is characterized by three things:
Source. The antenna (e.g. dipole antenna, cable, PCB trace).
Media. The media surrounding the source (e.g. air, plastics).
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. 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.
Wave and Characteristic Impedance.
Before we dive into the near and far field topic, we have to talk about the wave impedance. The characteristic impedance, also the called wave impedance, is a 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:
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 media like air, insulator, metal shield) is defined as:
ω=angular frequency [rad] 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:
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:
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.
Near Field to Far Field Boundary.
In practice, the boundary between the near field (Fresnel region) and the far field (Fraunhofer region) does depend on the frequency (wavelength) and the geometry (dimensions) of the antenna. A realistic boundary dfarfield (distance to antenna) can be defined as the larger of
where D is the largest physical linear dimension of the antenna in [m] and λ the wavelength of the signal in [m]. Bear in mind that λ=c/f. The first formula for dfarfield is typically used for wire-type antennas, whereas the later is used for surface-type antennas.
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:
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:
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).
In the far field, the E- and H-fields move perpendicular (orthogonal) 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.
For EMC 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 to measure the H- or E-field.
Far field: Log-periodic, biconical or horn antenna to measure the EM-field.
5 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.
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):
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]):
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:
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):
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):
The effective aperture Ae [m^2] for any antenna can be calculated out of the signal wavelength λ [m] and the antenna gain Gi :
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:
Now, we have everything ready to express AF [1/m] or [dB/m] as a function of only the plane wave frequency f and the receiver antenna gain Gir:
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).
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 (changes over frequency!) 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:
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 Pt [W] for achieving a desired field strengt E [V/m] at a given distance d [m] is:
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.
The FSPL is derived from the Friis transmission equation:
Pr=receiver antenna power [W], Pt=transmitter antenna 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 loss  [dB] is now defined as:
d=distance between transmitter and receiver antennas [m], λ=wave length [m], c=3E8[m/sec], f=frequency [Hz].
Impedance Matching: VSWR and Return Loss.
We speak of matched impedances if 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).
VSWR and Return Loss [dB] are both used to quantify the impedance matching of a connection or two circuits.
IMPORTANT NOTE: We only talk about reflections of connections to a 1-port network load (e.g. like an antenna). Reflections considering the load as a 2-port network are more complicated, as there are also the reflections from the output port to be considered (which then propagate back to the input port of the 2-port network and make things even more complicated).
Before we go into detail about the VSWR, let us define the reflection coefficient Γ (=s11 in case of 1-port networks!):
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:
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].
Impedance Matching Summary.
The table below shows how to calculate VSWR  out of Return Loss [dB] or the reflection coefficient  and vice versa. Z0 is the "system impedance" (typical Z0 = 50Ω or Z0 = 75Ω) refers to Zsource in the picture above.
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.
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 input power [dBm], Pt=transmitter output power [dBm], Lt=transmitter losses (coax, connectors, ...) [dB], Git=transmitter antenna gain [dB], FSPL=free-space path loss [dB], Lmisc=miscellaneous losses (fading, polarization mismatch, ...) [dB], Gir=receiver antenna gain [dB], Lr=receiver losses (coax, connectors, ...) [dB].
6 Skin Effect
The resistance R' per unit length [Ω/m] of any conductor can be written as:
ρ = specific electrical resistivity of the conductor material [Ωm]. σ = specific electrical conductivity of the conductor material [S/m]. A = cross-sectional area over which the current 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 A of the current flow. As a consequence, the resistance of a conductor increases with increasing frequency.
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 Js [A/m^2]. The current density Jd [A/m^2] at distance d [m] from the conductor surface is defined as:
Shielding: Imagine an electromagnetic plane wave of field strength E0 and H0 entering an absorbing material (shield). The skin depth δ [m] is the distance an electromagnetic wave has to travel through that absorbing material until its field strength is reduced to 37% of E0 or H0 (37% = 1/e = 1/2.72). This means that the power of the plane electromagnetic wave is lowered by 20·log10(0.37) = 9dB after it traveled the distance δ [m]. The attenuation of an electromagnetic plane wave is defined like this:
Ed = remaining electric field [V/m] strength of a plane wave with field strength E0, after travelling distance d [m] through a medium with attenuation constant α [1/m]. Hd = remaining magnetic field strength [A/m] of a plane wave with field strength H0, after travelling distance d [m] through a medium with attenuation constant α [1/m]. Remember that γ = α+jβ is the so called propagation constant.
From above, you know that the skin depth δ [m] is defined as the inverse of the attenuation constant α [1/m]:
For good conductors (with σ>>ωε), the skin depth formula can be simplified to:
Below some example values of skin depths for silver, copper, gold, aluminum, nickel, iron and stainless steel 316.
NOTE: Important to know for calculation of skin depth of nickel, iron, stainless steel and any other ferromagnetic metal:
Relative permeability μr  depends on the specific material and alloy (therefore, be careful when reading our table and graphic below).
Relative permeability μr  depends on the frequency f [Hz] (we neglected that in the table and graphic below).
The purposes of shielding are:
Lower emission: Prevent radiated electromagnetic emissions of your product.
Increased immunity: Protect you product from external electromagnetic radiation.
We give you a quick overview here, so that you learn the most important concepts about shielding of electronic circuits, cables, devices and machines. Before we jump into the theory of shielding, here two practical advices:
Cables and wires. Every single signal which enters and/or leaves a shielded enclosure must be filtered or shielded. In case the cable is shielded: contact the cable shield 360º with the shielded enclosure.
Slots and apertures. Slots and apertures lower the shield effectiveness. In case the linear dimension l [m] of a slot/aperture is > λ/2, the shield is useless.
NOTE: All formulas in this chapter are approximations. It is assumed that the shield is made of a good conductor (characteristic impedance much smaller than in free space [377Ω]) and the shield thickness t [m] is much greater than the skin depth δ [m] of the signal frequency f [Hz] of interest.
Shielding Effectiveness (SE).
Shielding effectiveness describes how good a shield blocks an incident wave (electrical field strength Ei [V/m], magnetic field strength Hi [A/m]) from transmitting through the shield. After passing through the shield, the remaining wave has a field strength of Et and Ht. The reflected wave has field strength Er and Hr.
Shielding effectiveness can also be calculated like this:
R represents the reflection loss caused by reflection at the left and right surface, A represents the absorption loss of the wave as it proceeds through the shield, M represents the additional losses (multiple re-reflections and transmissions within the shield of thickness t [m]).
In general, it can be said that:
Reflection loss R dominates at lower frequencies.
Reflection loss R for a given shield differs from near field to far field.
Reflection loss R for a given shield in the near field differs with changing noise source impedance (high impedance sources vs. low impedance sources).
Absorption loss A dominates at higher frequencies.
Absorption loss A for a given shield is identical for near field and far field.
Multiple reflection loss M is 0 [dB] for good shields and can usually be neglected.
Far Field Shielding.
Let's assume that a shield is placed (without apertures) in the far field zone of the noise source (more detail about the near vs. far field here).
A good approximation for the absorption loss A [dB] of a shield in the far field is:
t represents the shield thickness [m], f represents the frequency [Hz], μr represents the relative permeability of the shield , σr represents the relative conductivity of the shield material to copper  (σCu = 5.8E7 S/m), δ represents the skin depth [m].
A good approximation for the reflection loss R [dB] of a shield in the far field is:
σ represents the conductivity of the shield material [S/m], σr represents the relative conductivity of the shield material to copper  (σCu = 5.8E7 S/m), ω and f represent the frequency [1/rad, Hz], μr represents the relative permeability of the shield , ε0 represents the relative permittivity (dielectric constant) .
Example. Shielding Effectiveness PCB (Far Field!).
The following pictures show the (far field!) shielding effectiveness of PCB ground planes with thickness 35 μm and 17.5 μm. It can be seen that a thinner ground plane shields as good as a thicker ground plane for frequencies below 10 MHz.
Example. Shielding Effectiveness Cable (Far Field!).
The following pictures show the (far field!) shielding effectiveness of a typical cable shield with thickness of 1.5 mil. It can be seen that the shielding effectiveness SE for silver and aluminum is better than compared to tin at frequencies f > 1MHz.
Near Field Shielding.
Let's assume a shield is placed at a distance d [m] in the near field of a noise source and you would like to know the shielding effectiveness. Calculating the shielding effectiveness in the near field area of a source is much more difficult, than in the far field. This is due to the fact, that the wave impedance in the near field is difficult to determine and it changes significantly with distance d (either with factor 1/d^2 or 1/d^3, more details here).
In order to calculate the shielding effectiveness in the near field of a source, it must be known if the source is a ...:
Electric field source. High wave impedance Zw, Hertzian electric dipole. Examples: wires, PCB traces, cables, spark gaps (e.g. DC motors).
Magnetic field source. Low wave impedance Zw, magnetic loop dipole. Examples: current loops, transformers, wireless charging devices.
From above, we know that the shielding effectiveness is the sum of reflection loss R [dB], absorption loss A [dB] and multiple-reflection loss M [dB]. In the following paragraphs the absorption and reflection loss of near field shielding are discussed in more detail.
The absorption loss A is unaffected by the type of source (near field, far field, electric/magnetic field source).
The reflection loss R is calculated differently to the far field and depends on the type of source.
Reflection Re [dB] loss of near field, electric field sources:
Reflection Rm [dB] loss of near field, magnetic field sources:
σr is the relative conductivity to copper , μr is the relative permeability , f is the signal frequency [Hz], d is the distance between source and shield [m].
The graphic below shows some interesting facts:
Reflection loss Re [dB] of electric field sources in near field increases with decreasing distance d [m] to the source. Re is considerably higher compared to the plane wave reflection loss R.
Reflection loss Rm [dB] of magnetic field sources in near field decreases with decreasing distance d [m] to the source. Rm is considerably lower compared to the plane wave reflection loss R.
Magnetic Field Shielding @ Low-Frequencies.
From above we learnt that the most difficult field to shield is a low-frequency magnetic field. This is due to the fact that the reflection loss R and the absorption loss A are low for low-frequency magnetic fields.
There are these two methods for shielding against low-frequency magnetic fields:
Shield with μr >> 1. Use of a low-reluctance shield material with a high magnetic permeability μr >> 1 (e.g. nickel, Mu-metal).
NOTE: The permeability decreases with increasing frequency f [Hz] and with increasing magnetic field strength H [A/m] or magnetic flux Φ [Wb] respectively.
Shorted turn. Use a loop conductor which is placed in the magnetic field. The induced current iind in the loop conductor will generate a counter magnetic field which will then lower the magnetic field in the vicinity of the loop.
Slots and Apertures.
Above, we always assumed a perfect solid shield. Slots and apertures are efficient radiators (yes, radiators!) when their maximum linear dimension (not area!) l [m] is equal to λ/2. Therefore, if a slot or aperture has a linear dimension l of λ/2, the shielding effectiveness SE = 0 [dB]. Shielding effectiveness of a single slot with maximum linear dimension l [m] (which is equal or less than λ/2) and for a signal with wavelength λ [m] is:
If there are multiple apertures, the shielding effectiveness will be reduced even more. We can calculate the shielding effectiveness of a linear (not multi-dimensional!) array of equally, closely spaced apertures n  of length l [m], where the total array length larray [m] is less than λ/2:
In case of a multi-dimensional array of m rows (and m<n !), only the n apertures of he first row have to be considered in case of shielding effectiveness reduction. In other words, the additional rows (2nd, 3rd, ...) will not lower the shield effectiveness significantly. The shielding effectiveness of a multi-dimensional array of equal size apertures will be the shielding effectiveness of one single hole, minus the shielding effectiveness reduction of the first row of n apertures (-20log10(√n)).
The shielding effectiveness reduction due to a linear array of n apertures (relative to a single aperture) is shown in the graph below. Be aware that the array length larray must be smaller than λ/2, otherwise the SE is 0 [dB].
Apertures located on different surfaces, which all look in different directions, do NOT decrease the overall shielding effectiveness, because they radiate in different directions.
There are, roughly said, three basic classes of resistors. All of them are designed for different applications and have their own advantages and drawbacks when it comes to high frequency:
Wirewound. For high power applications. High inductance L.
Film type. General purpose low-power resistors. Low inductance L.
Carbon composition. High energy surge applications. Low inductance L.
Here the equivalent circuit of a resistor with the parasitic elements C (between the solder pads and internally) and L (series lead inductance).
The following graph shows a typical frequency behavior (impedance vs. frequency) of an actual resistor. With increasing frequency, the parallel capacitance C starts to dominate and lowers the impedance of the resistor. At the resonant frequency of the parasitic capacitance C and the parasitic inductance L, the impedance reaches its minimum. For frequencies higher than the resonant frequency, the inductance dominates and the impedance starts to increase again.
Inductors can be categorized in:
Non-magnetic core. E.g. air inductors.
Magnetic core. E.g. ferrite beads (open loop magnetic core) and toroidal inductors (closed loop magnetic core inductors).
Non-ideal inductors have not only inductance, they have a series resistance (wire resistance) and a distributed capacitance between the windings. Here the equivalent circuit of an inductor:
The following graph shows a typical frequency behavior (impedance vs. frequency) of an actual inductor. At DC and low frequencies, the impedance is purely resisitive (Z = R). With increasing frequency, the impedance changes and the inductance dominates up to the parallel resonant frequency of L and C. For frequencies higher than the parallel resonant frequency, the capacitance dominates and the inductor becomes a capacitor.
Capacitors can be categorized by its dielectric material:
Ceramic. Multilayer ceramic capacitors (MLCCs) are the most widely used capacitors today. They have relatively low equivalent series inductance (ESL) and low equivalent series resistance (ESR). They are used up to several GHz (dielectric material C0G or NP0).
Electrolytic. Electrolytic aluminum and tantalum capacitor have high capacitance-to-volume ratio and they have quite high ESRs. They are usually used up to 25kHz...100kHz.
Paper. Film and paper capacitors have considerably lower ESR than electrolytic capacitors but still moderately large inductance. They are usually used up to several MHz.
An actual capacitor is not a pure capacitance C. The series inductance L (ESL) is from the leads and form the internal structure. The series resistor Rs (ESR) is also from the leads as well as from internal dielectric losses. The parallel resistor Rp represents the non-ideal dielectric material (leakage current).
The following graph shows a typical frequency behavior (impedance vs. frequency) of an actual capacitor. From DC up to the serial resonant frequency of L and C: the capacitance dominates. However, for frequencies higher than the resonant frequency, the inductance L dominates and the capacitor becomes an inductor.
9 Galvanic Series
Knowing the galvanic series is important when designing interconnections between different metals (e.g. for bonding of shields). If you don't take care which two different metals you use, the interconnection may corrode within short time (and the shielding effect is not given any more).
The galvanic series helps us to choose the right combination of metals for an interconnection, in terms of corrosion.
Corrosion. The less noble metal (anode) of an interconnection of two metals experiences galvanic corrosion, in case:
The two metals have a galvanic incompatibility (voltage difference too 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.
An electrolyte (e.g. water, moisture) is present.
The two metals have an electrical conducting connection.
Rate of corrosion. The rate of corrosion depends on the moisture of the environment, how far apart the metals are in the galvanic series (further apart leads to faster corrosion, because the ion transfer is faster) and other parameters like the type of electrolyte (pH, concentration, flow rate), temperature (rate of corrosion increased with higher temperatures), degree of aeration, humidity, pressure and even the geometry of the interconnection.
Recommendations. Here recommendations on how to choose metals depending on their potential difference.
Harsh (outdoor, high humidity, salt laden, military).
Choose metals where electrode potential difference is ≤0.15V.
Normal (non-temperature/humidity controlled, consumer product, indoor):
Choose metals where electrode potential difference is ≤0.25V.
Controlled (temperature/humidity controlled, indoor):
Choose metals where electrode potential difference is ≤0.50V.
The table below shows the standard electrode potential (E°), which is defined by measuring the potential relative to a standard hydrogen electrode (SHE) using 1mol solution at 25°C and at the pressure of 1atm.
ATTENTION: A galvanic 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 sea water electrode potential table, rather than the standard hydrogen electrode potential table!
The sea water 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 the table from above, 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 below from the established galvanic series of metals in sea water table.
In the table below, metals are grouped. All metals, alloys and platings of the same group have common electro-motive forces (EMF) within 0.05V when coupled with a saturated calomel electrode in sea water at room temperature. All members of a group regardless of metallurgical similarity or dissimilarity are considered compatible. Compatible couples between groups have been specified in the table (green areas) based on a potential difference of 0.25V maximum.
[Sources: MIL-DTL-14072F (2013-Aug-13); NASA-STD-6012 (2012-Mar-08)].
In the figure below, you see the galvanic series of selected metals in sea water. 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. [Sources: MIL-STD-889C (2016-Aug-22)].
Here some free information about of galvanic corrosion by the U.S. Army and NASA (free download, unlimited distribution). You will find there concrete and helpful data about metals and suitable coatings and treatments to prevent corrosion (e.g. MIL-STD-889).
10 Triboelectric Series
The triboelectric effect is the reason why certain materials get electrically charged when they are rubbed against each other (e.g. if you rub a balloon on your sweater or shirt). Therefore, the triboelectric effect is one of the causes for electrostatic charges, which then lead to electrostatic discharges (ESD).
Some materials tend to give electrons up easily, whereas others readily absorb them. The triboelectric series is a list of materials with a ranking from the most positive (materials that have tendency to lose electrons) to the most negative (materials that tend to gain electrons). The strength of the produced charge (also called static electricity) due to friction depends on the type of materials involved, their surface roughness, temperature, strain, humidity and other items. Therefore, the table below does not tell you something about the magnitude of the static charge, just its polarity.
[Source of the triboelectric series: Introduction to Electromagnetic Compatibility, Clayton R. Paul, 2. edition (2006)]
11 Material Properties
Find here the most important material properties and physical constants which you will need for your work as EMC design engineer. All values are typical values and are without warranties.
c = 2.998E8 [m/sec] ≈ 3E8 [m/sec]. Speed of light.
e = 1.602E−19 [C]. Elementary charge.
m = 9.109E−31 [kg]. Electron mass.
ε0 = 8.854E−12 [F/m]. Permittivity of vacuum, absolute permittivity.
μ0 = 4πE−7 [H/m] = 12.57E-7 [H/m]. Permeability of vacuum, absolute permeability.
Z0 = √(μ0/ε0) = 376.7303 [Ω] ≈377 [Ω]. Characteristic impedance of vacuum.
Metals - Electrical Conductivity σ & Resistivity ϱ.
The material properties like specific conductivity [S/m] and specific resistivity [Ω/m] are important factors when it comes to the calculation of e.g. the skin-depth. The values in the table below apply for zero frequency (0 Hz) and room temperature (25ºC).
[Sources: Reference data for engineers, Newnes, 9. edition (2002); Electrical resistivity and thermal conductivity of nine selected AISI stainless steels, C. Y. Ho and T.K. Chu, Cindas Report 45 (1977); Temperature dependence of electrical resistivity of metals, Lazarus Weiner and Premo Chiotti and H. A. Willhelm, Ames Laboratory (1952)].
Metals - Magnetic Permeability μr.
In the field of EMC, knowing the relative permeability μr  of materials is important when it comes to shielding of low-frequency (f<20kHz) magnetic fields. The relative magnetic permeability μr  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 μr = 1), for a given field strength H [A/m].
First of all, let us have a quick look how magnetic materials are classified. Our main interest lies in ferromagnetic materials (soft magnetic), because they can be used for shielding of low-frequency magnetic fields.
Anti-Ferromagnetic. μr  is slightly bigger than 1. The only pure metal which is anti-ferromagnetic is chromium (Cr).
Diamagnetic. μr  is slightly smaller than one. Diamagnetic materials are weakly repelled by a magnet. They can not be magnetized.
Paramagnetic. μr  is slightly bigger than one. Paramagnetic materials are weakly attracted by a magnet. They can not be magnetized.
Ferrimagnetic. μr  is bigger than 1, but much smaller compared to the μr of ferromagnets. Ferrimagnetic materials can be weakly magnetized ("weak magnets").
Ferromagnetic. μr  is much bigger than 1. Ferromagnetic materials can be magnetized and used as shielding material (against low-frequency magnetic fields) or for building permanent magnets. Ferromagnetic materials can be categorized by their coercivity (Hc [A/m]). A high coercivity of a material means that the external magnetic field must be high to change the polarization of the magnet.
Soft magnetic. Coercivity Hc [A/m] is low (typ. <1kA/m). Example applications: shielding, transformer and ferrite cores.
Hard magnetic. Coercivity Hc [A/m] is high (typ. >10kA/m). Example application: permanent magnets.
The μr  data in the table below apply to room temperature and direct current (DC, f=0Hz).
NOTE: μr is temperature and frequency dependent. μr of a material may increase or decrease with increasing temperature, until a certain temperature (curie temperature) where μr=1. With increasing signal frequency, the value of μr is getting smaller. E.g. Mumetal has a μr of over 10'000 at f=0Hz (DC), but similar to steel at f=20kHz!
The initial relative magnetic permeability μri  describes the relative permeability for low flux densities B [T]. The maximum relative magnetic permeability μrm  is usually by factor 2....5 (or even more) higher than μri . μrm is valid at a single point in the H-B-Hysteresis diagram (at this point the change of the magnetig field H [A/m] results in the biggest change in the flux density B [T]).
Insulators - Dielectric Constant εr & Loss Tangent tan(δ).
The dielectric constants (relative permittivity εr  or Dk ) in the table below should be considered representative for each material and they apply to room temperature (25ºC) and humidity (unless otherwiese noted).
[Sources: Reference Data For Engineers, 9. Edition (2002)].
PCB Materials - Dielectric Constant εr & Loss Tangent tan(δ).
The dielectric constant (relative permittivity εr or Dk ) of the printed circuit board (PCB) materials determine 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 εr, the loss tangent tan(δ)  or dissipation factor Df  of a PCB material is also of interest, because it influences the loss of high frequency signals (together with the resistive 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's have a look at an example of a PCB stackup of a 6 layer board.
Here the list of materials which you must consider when calculating the characteristic impedances of PCB traces:
Solder resist mask. The solder mask is a thin (usually green) layer that protects the copper conductors from oxidation and mechanical stress and helps to minimize the creation of short circuits by means of bridges formed by excess solder. The typical thickness of the solder mask (above the copper conductors) is 0.8mils = 20μm. The dissipation factor (loss tangents) is usually 0.025 @ 1GHz and the dielectric constant 3.3 to 3.8.
Copper layer. The copper layers consist of thin, rolled and annealed RA or electro-deposited ED copper of thickness:
0.5oz = 0.7mils = 17.5μm
1oz = 1.4mils = 35μm (standard)
2oz = 2.8mils = 70μm
The circuit traces are etched into the copper layers before the PCB is laminated together (with adhesive, heat and pressure).
Core. PCB cores are laminates (PCB base materials) with copper layers on both sides. The circuit traces are etched into their copper layers before the cooper clad laminate is glued together with the rest of the multilayer PCB. The distance between the two copper layers of a core has only little variation and the impedances can be controlled with high accuracy.
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 εr over a wide frequency range, moisture absorption.
Prepreg. Prepreg is the short word for preimpregnated. It is a flexible material, typically also containing woven glass, which is supplied to the PCB fabricator partially cured (not completely cooked). It is included between the rigid core layers in the layer stack during fabrication, and then heated to perform final curing, after which it becomes rigid, helping to join the core substrates of the finished PCB.
For boards that require 4 or more layers, core and prepreg layers 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, as well as 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 εr over a wide frequency range, moisture absorption.
Here some tips regarding PCB layer stackups:
Power decoupling plane. Design a PCB stackup with power supply plane and GND plane close together (<0.1 mm or 2...3 mils). This leads to an especially good decoupling at high frequencies (>1MHz).
Impedance controlled routing. Generally core is more reproducible than prepreg regarding thickness and dielectric constant. This means that controlled impedance layers should ideally be routed along the core material, rather than prepreg.
We summarized dielectric constants and loss tangents of different PCB materials from different vendors in the table below. If there are two things you should remember from this table, they are:
There are plenty of different PCB materials, dedicated for different applications.
FR-4 is not equal FR-4.
If the exact PCB data are unknown, we usually assume FR-4 with εr=4.5 and tan(δ)=0.015.