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## Knowledge Base

The following chapters are thought of 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 info@academyofemc.com.

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• ###### 1  Decibel [dB]
• 2  Frequency vs. Wavelength
• 3  Time Domain vs. Frequency Domain

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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 are 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 0: And we can write the following for power/voltage/current gain: Points to remember when it comes to the calculation of gain and loss in decibel:

• Amplification. If P2 is bigger than P1, the gain value in [dB] is positive. This means if there is 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 an Excel sheet, which contains conversion calculations and more [1.2].   1 - Decibel
Gain and Loss
Absolute Levels
Unit Conversion
EMC an Frequency

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It is fundamental that professional engineers in the field of EMC understand the basics of signal frequency f [Hz] and wavelength λ [m]:

EMC and Frequency.

In general, EMC issues occur with signals of frequency > 9kHz. This is the reason why most EMC Standards do not consider signals with < 9kHz.

• Conducted Emission. Conducted emissions tend to occur at f < 30MHz.

• Radiated Emission. Radiated emissions tend to occur at f > 30MHz.

Wavelength: Calculation.

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 of signals traveling long blank wires vs. cables & PCB traces.

It is important to understand that the signal propagation velocity v [m/sec] depends on the transport medium through which the electromagnetic field is traveling. Therefore, the same signal with the same frequency f [Hz] has a different wavelength λ [m] in a blank wire (surrounded by air) than in a cable or PCB trace (surrounded by insulation material).​ 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 permeability 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 along 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 along 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 to 1.0 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 media depending 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 a 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). 2 - Wavelength
l calculation
l air vs. conductor

## 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, reflections), whereas the frequency domain representation helps you to determine at which frequencies a signal may lead to radiated emissions.

Fourier Analysis.

Fourier analysis is an integral transform and 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 integral transforms 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 integral transform 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 xx[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].  denotes the continuous normalized radian frequency variable [-π...+π].  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+ 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^(+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 in the time-domain and an amplitude frequency spectrum of a digital signal with its envelope curve [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 of two trapezoid waveforms with different pulse-width tpw [sec], but with identical periodicity T [sec], rise- and fall-time tr [sec] 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. However, it is remarkable that a clock signal with a 90% duty-cycle has a lower amplitude of the first harmonic than a clock signal with a 50% duty-cycle (of the same frequency [Hz] and with the same rise/fall-time [sec]). The clock signal with a 90% duty-cycle has more power, but this power adds to the DC component (0 Hz). The picture below compares the frequency spectrum of two trapezoid waveforms with different rise- and fall-time tr [sec] 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 at the higher frequencies in the spectrum (and therefore to potentially lower radiated emissions). In this example, an increase of the rise- and fall-time [sec] by factor 10, reduces the amplitude of the high-frequency harmonics ( f > 32MHz) also by factor 10 (20 dB). 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. 3 - Time/Frequency Domain
Fourier analysis
Digital waveforms spectra
Bandwidth of digital signals

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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 Z= 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.

VSWR means Voltage Standing Wave Ratio. The VSWR expresses the ratio of the maximum voltage Vmax [V] of a standing voltage wave pattern and the minimum voltage of a standing wave pattern Vmin [V] on a transmission line. A VSWR value of 1.0 means perfectly matched. A VSWR value of infinity means complete mismatch (100% of the forward wave is reflected). The VSWR can be calculated by using the reflection coefficient from above [4.5]:  Return Loss.

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 an impedance mismatch. A low RL value indicates that not much power is transferred to the load and is reflected instead. 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)

• Connector

• Passive Filter 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]. Scattering Parameters (S-Parameters).

Scattering Parameters} - also called S-parameters - are commonly used in high-frequency or microwave engineering to characterize a two-port circuit (see the picture below). The scattering parameters describe the relation of the power wave parts a1, b1, a2, and b2 that are transferred and reflected from a two-ports input and output. The physical dimension for the incident a and reflected b power waves is not Watt, it is √Watt. Generally speaking, the S-parameter sij is determined by driving port j with an incident wave of voltage Vj+ and measuring the outgoing voltage wave Vi- at port i. Considering the picture above, the four scattering parameters can be computed as follows: 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 Z= 50Ω or Z= 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. 4 - RF Parameters
Reflection coefficient
VSWR
Return loss
Insertion loss
Impedance matching summary
S-Paramters

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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, ZlineZload) 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 from 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 an electrical interconnection). Rather, we consider the signal conductor as a transmission line through which an electromagnetic field is moved from one point to another. Transmission lines are characterized or described by their characteristic impedance Z0.

Here are 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 a transmission line. Rules of thumb - if a conductor should be treated as a 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 a 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 lshortλ/10. Consider the signal path as a 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 a simple conductor