Filtering

Abstract

In EMC, filtering helps to minimize emissions of a product and increase the product’s immunity against electromagnetic interference. This chapter presents the concepts of determining the right filter type and ensuring the proper components are chosen regarding power dissipation, noise current type, and high-frequency behavior.

Filters can be classified according to their attenuation in the frequency-domain:

• Low-pass filters

• High-pass filters

• Band-pass filters

• Band-stop filters

or according to active components involved or not:

• Passive filters

• Active filters

or according to the noise current type:

• Differential-mode noise filters

• Common-mode noise filters

or according to the suppression of transients:

• ESD filters

• Burst filters

• Surge filters

or according to the implementation in hardware or software:

• Analog filters (hardware)

• Digital filters (software)

In this chapter, all the filter types listed above are explained in further detail.

If everyone is thinking alike, then no one is thinking.

—Benjamin Franklin

15.1 Filter Characterization

A linear and time-invariant (LTI) filter can be characterized in the time- and the frequency-domain (Fig. 15.1):

• Time-domain. In the time-domain, a filter can be characterized by its impulse response h(t) or its step response s(t).

  Fig. 15.1

  

  A filter is typically described in the frequency-domain by its frequency response $\underline {H}(j\omega )$ (Bode plot) or in the time-domain its step response s(t)

• Frequency-domain. In the frequency-domain, a filter can be characterized by its frequency response $\underline {H}(j\omega )$ (Fourier transform, bode plot).

In this chapter, we focus on filters which are causal LTI-systems (except for the non-linear transient filters in Sect. 15.9 and some non-linear digital algorithms in Sect. 15.10). LTI-systems can be completely characterized by one of these functions [4]:

• Impulse response h(t)

• Step response s(t)

• Transfer function H(s)

Causal , linear , and time-invariant (LTI) means:

• Causal. A system is causal if the output depends only on present and past but not on future input values.

• Linear. A system is linear if the output y(t) is a linear map** of the input x(t). If a and b are constants, the output for a ⋅ x(t) is a ⋅ y(t) and b ⋅ y(t) for b ⋅ x(t). In addition, for linear systems, you can apply the principle of superposition (the output for x = a ⋅ x ₁ + b ⋅ x ₂ is y = a ⋅ y ₁ + b ⋅ y ₂).

• Time-invariant. A system is time-invariant when the output y(t) for a given input x(t) does not depend on whether x(t) is applied now or after a time T [sec]. If the output for x(t) is y(t), then the output for x(t − T) is y(t − T).

Note: real-world components show non-linear effects, e.g., due to current saturation I [A] (inductors, ferrites) or non-ideal high-frequency behavior (inductors, capacitors). Therefore, real-world filters are only linear for certain operating conditions. The non-linearities and other undesired effects of filter components are described in Chap. 11. In this chapter, we assume that the filter components are used in their linear operating region.

15.1.1 Time-Domain: Step Response

The goal of a filter characterization in the time-domain is to determine signal distortions and the time delay caused by the filter (Fig. 15.2).

Fig. 15.2

Time-domain characterization functions. (a) Impulse Dirac delta function δ(t). (b) Unit step function u(t)

If the input signal x(t) of an LTI-system is the Dirac delta δ(t) function:

$$\displaystyle \begin{aligned} \delta(t) = \begin{cases} +\infty & t = 0 \\ 0 & \text{otherwise,} \end{cases} \end{aligned} $$

(15.1)

then the output is defined as the impulse response h(t). The impulse response characterizes an LTI-system entirely, and for any given input signal x(t) the output y(t) can be calculated as [4]:

$$\displaystyle \begin{aligned} y(t) = h(t) * x(t) = \int_{-\infty}^{t}h(t-\tau)x(\tau)d\tau \end{aligned} $$

(15.2)

where

• y(t) = the output signal from an LTI-system in [V]

• h(t) = the impulse response of the LTI-system in [V]

• x(t) = the input signal to an LTI-system in [V]

• ∗ = convolution, a mathematical operation of two functions

In practice, the step response s(t) (and not the impulse response h(t)) is usually used to characterize a filter [4]:

$$\displaystyle \begin{aligned} u(t) = \begin{cases} 1 & t \geq 0 \\ 0 & t < 0. \end{cases} \end{aligned} $$

(15.3)

If the input is the unit step : x(t) = u(t), the output y(t) is called step response s(t). The step response s(t) can be calculated by integrating the impulse response h(t) and the impulse response h(t) by taking the derivative of the step response s(t) [4]:

$$\displaystyle \begin{aligned} s(t) &= \int_{-\infty}^{t}h(\tau)d\tau \end{aligned} $$

(15.4)

$$\displaystyle \begin{aligned} h(t) &= \frac{d}{dt}s(t) \end{aligned} $$

(15.5)

Figure 15.3 shows the step response of two different low-pass filters, where:

• Step signal amplitude u(t) = 1 V for $t>0\,\sec$

  Fig. 15.3

  

  Step response simulation of two 2nd-order low-pass filters [1]

• Low source impedance $\underline {Z}_{source} = R_{S1} = R_{S2} = {10}\,{\Omega }$

• High load impedance $\underline {Z}_{load} = R_{L1} = R_{L2} = 1\,\mathrm {M}\Omega$

• Filter type = 2nd order passive low-pass filter

• Filter topology = L-topology

The step responses in Fig. 15.3 show that even passive low-pass filters can cause a significant amount of ringing and time delay. Therefore, it is always necessary to also check the time-domain behavior of an EMI-filter, not just in the frequency-domain behavior.

15.1.2 Frequency-Domain: Frequency Response

The frequency response $\underline {H}(j\omega )$ (Fourier transform ) is the transfer function $\underline {H}(s)$ (Laplace transform , s = σ + jω) of a filter for a sinusoidal input signal, meaning s = jω. The frequency response $\underline {H}(j\omega )$ describes the attenuation in [dB] and the phase shift in [rad] or [^∘] as a function of frequency ω = 2πf.

The Fourier transform of the filter’s impulse response h(t) is the frequency response $\underline {H}(j\omega )$ of the filter [4]:

(15.6)

The frequency response characterizes a filter in the frequency-domain. For any given input signal $\underline {X}(j\omega )$, the output signal $\underline {Y}(j\omega )$ can be calculated as [4]:

$$\displaystyle \begin{aligned} \underline{Y}(j\omega) = \underline{H}(j\omega)\cdot\underline{X}(j\omega) \end{aligned} $$

(15.7)

where

• $\underline {Y}(j\omega ) =$ the output signal of the filter in [V]

• $\underline {H}(j\omega ) =$ the frequency response of the filter in [V]

• $\underline {X}(j\omega ) =$ the input signal of the filter in [V]

• ω = 2πf = angular frequency of the sinusoidal input signal in [rad/sec]

The frequency response $\underline {H}(j\omega )$ can also be calculated by using the Fourier transform of the step response $\underline {S}(j\omega )$:

(15.8)

$$\displaystyle \begin{aligned} &\underline{H}(j\omega) = j\omega \cdot \underline{S}(j\omega) \end{aligned} $$

(15.9)

In practice, the frequency response $\underline {H}(j\omega )$ is shown in the form of a Bode plot . Figure 15.4 presents the Bode plot of the identical filters of Fig. 15.3, which shows us a couple of interesting points:

Fig. 15.4

Frequency response simulation of two second-order low-pass filters [1]

• Ringing and resonance. It can be seen that the ringing in the time-domain matches the resonance in the frequency-domain at roughly f _r = 50 kHz.

• Order vs. attenuation and phase shift. The number of reactive components gives the order n of an analog filter; in other words, the sum of inductors L [H] and capacitors C [F] of the filter is equal to the filter order n. Because the filters are of second-order (n = 2), the attenuation after the cut-off frequency is n ⋅20 dB = 40 dB and the maximum phase shift is n ⋅−90 deg = −180 deg.

• Source and load impedance vs. filter performance. The filter performance (attenuation, phase shift) depends on the filter components themselves and the source and load impedances. Have a look at the CL low-pass filter in Fig. 15.4, where the source impedance R _S2 = 10 Ω builds a low-pass RC-filter with the first filter component C ₂ [F] which has a cut-off frequency of f _g = 1∕(2πR _S2 C ₂) = 160 kHz and the load impedance R _L2 = 1 M Ω builds a low-pass RL-filter with the filter component L ₂ [H] and a cut-off frequency of f _g = R _L2∕(2πL ₂) = 1.6 GHz.

15.2 Low-Pass Filters

Low-pass filters are the most common filters in the world of EMI and EMC. They reject undesired HF energy above a desired cut-off frequency f _c [Hz]. Figure 15.5 shows the frequency response of ideal low-pass filters (skin effect, stray capacitance, and other non-linearities are neglected). At the cut-off frequency f _c [Hz], the insertion loss is 3 dB and the phase shift is −45 deg. The attenuation for frequencies f > f _c is n ⋅20 dB and the maximum phase shift of a low-pass filter is n ⋅−90 deg, where n is the filter order.

Fig. 15.5

Bode plot of ideal RC low-pass filters of order n = 1, n = 2 and n = 3

In most cases, EMI-filters against radiated or conducted high-frequency (HF) interference are analog passive low-pass filters with resistors, capacitors, inductors, ferrites, or common-mode chokes as filter components. The filter order n is usually not higher than three (maximum phase shift = n ⋅90 deg = 270 deg) because of the signal distortion and because filters of order n ≥ 4 could easily lead to instability due to the large phase shift of 360 deg and more.

The filter performance (e.g., cut-off frequency f _c [Hz]) depends not only on the filter components themselves but also on the source impedance $\underline {Z}_{S}$ [Ω] and the load impedance $\underline {Z}_{L}$ [Ω]. Therefore, the input and output impedances of the filter should be matched in the useful frequency range and mismatched in the EMI noise frequency range (an overview is given in Fig. 15.24).

It is highly recommended to simulate EMI analog circuit board filters with the real-world source and load impedances and the non-ideal SPICE models of the filter elements. The SPICE simulations of the low-pass filters in Fig. 15.6 show that it is important to consider the non-ideal filter components (not the ideal components) because the non-ideal behavior of capacitors, inductors, and ferrite beads influence the filter performance significantly at high frequencies.

Fig. 15.6

Passive low-pass filter SPICE simulations with $\underline {Z}_{S} = {10}\,\Omega$ and $\underline {Z}_{L} = 1\,\mathrm {M}{\Omega }$. Tool = LTspice [1]. L = Würth 744766001 WE-GF, C = Würth 885012008026 NP0. (a) 1st order low-pass filter. (b) Second-order low-pass filter. (c) 3rd order low-pass filter

Fig. 15.7

Bode plot of ideal RC high-pass filters of order n = 1, n = 2, and n = 3

15.3 High-Pass Filters

High-pass filters do attenuate low-frequency (LF) signals and let high-frequency (HF) signals pass without attenuating them. Figure 15.7 shows the frequency response of ideal high-pass filters. In the field of EMC, high-pass filters are not very common, and we do not go into further detail about high-pass filters at this point.

Fig. 15.8

Bode plot of an ideal LC-band-pass filter. f _r [Hz] denotes the resonance frequency and f _c1 [Hz] and f _c2 [Hz] the cut-off frequencies

15.4 Band-Pass Filters

Band-pass filters do only pass signals of the desired frequency range with little to no insertion loss. Figure 15.8 shows the frequency response of an ideal band-pass filter. However, band-pass filters are very seldom used in the field of EMC and are not discussed in further detail here.

Fig. 15.9

Frequency response of an ideal passive LC band-stop filter. f _r [Hz] denotes the resonance frequency and f _c1 [Hz] and f _c2 [Hz] the cut-off frequencies

15.5 Band-Stop Filters

Band-stop filters do attenuate only signals within the desired frequency range and pass the others. Figure 15.9 shows the Bode plot of a passive band-stop filter with an ideal. Band-stop filters are not very common EMI-filters and therefore not further discussed here.

Fig. 15.10

Step and frequency response comparison of an active and a passive low-pass filter with pass-band up to f _c = 60 kHz and order n = 2 [1]

15.6 Active and Passive Filters

The filters in the previous sections were all passive filters . If operational amplifiers or transistors are part of a filter, it is called an active filter . In most cases, EMI-filters are passive filters because the advantages of active filters mentioned in Table 15.1 are often not needed for EMI-filter applications.

Table 15.1 Active vs. passive filters: advantages and disadvantages

Figure 15.10 compares two second-order passive and active low-pass filters. Although the active filter shows a narrower transition between the pass-band to the stop-band, the attenuation of the active filter at frequencies of f > 100 MHz is not as good as for the passive filter and the active filter shows significant ringing in the time-domain. This example should just illustrate that an active filter is not always better than a passive filter; it all depends on the application’s requirements.

Fig. 15.11

Differential-mode vs. common-mode noise currents

15.7 Differential- and Common-Mode Filters

Differential-mode noise and common-mode noise are in detail explained in Sects. 12.3 and 12.4 and compared in Fig. 15.11. The filter components and topology for differential- and common-mode filters are not identical. Therefore, it is necessary to know if you deal with differential-mode or common-mode noise or both.

Fig. 15.12

Example of a differential-mode LC-filter

15.7.1 Differential-Mode Filters

Differential-mode noise filters are filters between a power or signal line and its return current line (see Fig. 15.12). Typical differential-mode noise filter components are:

• X-Capacitors. See Fig. 15.14 and Sect. 11.3 on page 157.

  Fig. 15.13

  

  Common-mode filter with Y-capacitors and a common-mode choke

• Inductors. See Sect. 11.4 on page 164.

• PCB mount ferrite beads. See Sect. 11.5.2 on page 171.

15.7.2 Common-Mode Filters

Common-mode noise filters have the purpose to attenuate common-mode noise. An example of a common-mode filter is given in Fig. 15.13. Typical common-mode noise filter components are:

• Y-capacitors. See Fig. 15.14 and Sect. 11.3 on page 157.

  Fig. 15.14

  

  X-capacitors vs. Y-capacitors

  Fig. 15.15

  

  Example of an AC mains filter. Courtesy of Würth Elektronik GmbH

• Common-mode chokes. See Sect. 11.6 on page 173.

• Cable mount ferrite beads. See Sect. 11.5.1 on page 169.

15.8 Mains Supply Filters

Special care must be taken regarding the safety requirements of filter components of (public) mains supply filters (Fig. 15.15). This is especially true for the X- and Y-capacitors:

• X-capacitors. X-capacitors are capacitors between line (L) and neutral (N) of the AC mains. A failure (short circuit of the X-capacitor) could result in fire.

  Fig. 15.16

  

  ESD transient pulse. (a) ESD current pulse form. (b) Approximate maximum ESD-pulse energy

• Y-capacitors. Y-capacitors are capacitors between line (L) to ground and between neutral (N) to ground of the AC mains. The capacitance of a Y-capacitor must be low enough so that the maximum allowed leakage current through the Y-capacitor is below the required limit!

The classification of X- and Y-capacitors according to IEC 60384-14 [3] is shown in Table 11.1 on page 158.

The applicable safety standards of a product must be considered during the design of a mains supply filter. Examples are:

• IEC 61010-1. Safety requirements for electrical equipment for measurement, control, and laboratory use—Part 1: General requirements [5].

• IEC 62368-1. Audio/video, information and communication technology equipment—Part 1: Safety requirements [2].

• IEC 62477-1. Safety requirements for power electronic converter systems and equipment—Part 1: General [6].

15.9 Transient Suppression Filters

A distinction is made between these three types of high-voltage transients (Table 15.2):

• Electrostatic discharge (ESD). Low energy pulses, very short rise-time.

  Table 15.2 Overview of high-voltage transient EMC testing parameters

• Electrical fast transients (EFT). Bursts of low energy pulses, short rise-time.

• Lightning surge pulses (surge). High energy pulses, medium rise-time.

The rise-time and the pulse energy are the most important parameters of voltage transients. Therefore, the pulse forms and energy are presented in Figs. 15.16 (ESD), 15.17 (EFT), and 15.18 (surge). The pulse energies were calculated by considering the maximum tolerances of the test equipment (timing, test voltage). The maximum pulse energy for ESD was calculated by neglecting the internal losses and by just considering the energy E [J] stored in the capacitor of the ESD generator:

$$\displaystyle \begin{aligned} E_{ESD} = \frac{1}{2}CV_p^2 = \frac{{150}\,\mathrm{pF}}{2}V_p^2 \end{aligned} $$

(15.10)

Fig. 15.17

Single EFT transient pulse. (a) Single EFT-pulse form (open-circuit voltage). (b) Approximate maximum single EFT-pulse energy

Fig. 15.18

Surge transient pulse. (a) Surge-pulse form (open-circuit voltage). (b) Approximate maximum 1.2/50 µs surge-pulse energy (line-to-line)

Fig. 15.19

High-voltage transient suppressor filter stage

where

• C = 150 pF = capacitance of the ESD generator’s energy storage capacitor

• V _p = ESD test voltage in [V]

The maximum EFT- and surge-pulse energy E [J] was calculated by approximating the pulse energy with the following formula [8]:

$$\displaystyle \begin{aligned} E_{EFT,surge} = \left(\frac{1}{3}\cdot\frac{V_p^2}{R}\cdot t_1\right)+\left(\frac{V_p^2\cdot\tau}{-2\cdot R}\right)\cdot\left(e^{-\frac{2(t_3-t_1)}{\tau}}-1\right) \end{aligned} $$

(15.11)

where

• V _p = EFT/surge peak test voltage in [V]

• R = load resistance of the EFT/surge test generator in [Ω]

• t ₁ = time to peak voltage in [sec]

• t ₂ = time until voltage has fallen off to 50 % of peak voltage in [sec]

• t ₃ = time to negligible voltage in [sec]

• $\tau = -(t_2-t_1)/\ln (0.5) =$ exponential rate of decay in [sec]

As ESD- and EFT-pulses show similar rise-times [nsec] and pulse energies [mJ], thus, they can usually be filtered with the same filter components. However, dealing with surge-pulses means dealing with potentially more than 1000 times higher energy compared to ESD or EFT transients. Thus, surge filters do usually need components with high energy absorption capability.

A transient filter stage consists typically of two components (see Fig. 15.19):

• Series impedance $\underline {\mathbf {Z}}_{\mathbf {1}}$. The series impedance limits the transient current through the shunt element and the circuit. Typical series protection components are: resistors , ferrites , or the series resistance and inductance of the conductor itself.

  Fig. 15.20

  

  Digital filter with ADC. f _c = cut-off frequency of analog anti-aliasing low-pass filter f _s = 1∕T _s = sampling-frequency of the ADC

• Shunt impedance $\underline {\mathbf {Z}}_{\mathbf {2}}$. The shunt element limits the transient voltage across the circuit. Typical shunt protection components are clam** devices like TVS diodes (see Sect. 11.8.2) and varistors (see Sect. 11.8.1) or crowbar devices like thyristors or gas discharge tubes (see Sect. 11.9), or simply capacitors (see Sect. 11.3).

In the following, we focus on the selection of an appropriate shunt protection element $\underline {Z}_2$ [Ω]. The following points must be considered when choosing a voltage protection shunt element:

• V _operation < V _standoff. The maximum operating voltage must be smaller than the standoff voltage of the shunt element. Up to the standoff voltage (reverse working voltage), the shunt element does not conduct any significant amount.

• V _clamp < V _damage, V _trigger < V _damage. With the maximum clam** voltage or the trigger voltage, the maximum voltage across the shunt element is meant. This voltage must be lower than the voltage at which the circuit to be protected experiences a defect or malfunction.

• Energy absorption: E _pulse < E _absorption. The protective elements must absorb the energy (voltage, current) of the expected transient pulse without any damage.

• Signal integrity: C _shunt < C _max. The protective elements must not interfere with the useful signal (e.g., due to high capacitance or leakage current) and the signal integrity must be guaranteed.

• Lifetime: n _expected < n _max. The protective elements must guarantee to withstand the minimum number of expected transient events n during the lifetime, and the protective devices’ degradation must be tolerable.

15.10 Digital Filters

Up to this point, all presented filters in this chapter were analog filters . This section is dedicated to digital filters , which have a couple of advantages (Table 15.3).

Table 15.3 Digital vs. analog filters

Here are some fundamental points about digital filters (Fig. 15.20):

• Nyquist-Shannon sampling theorem .¹ In order to convert an analog signal x(t) with the bandwidth B [Hz] (highest frequency in the signal) to a digital signal x[n], a minimum sampling frequency of f _s > (2 ⋅ B) [Hz] is required.

  Fig. 15.21

  

  Concept of a median filter

• Aliasing. Aliasing happens when the Nyquist-Shannon sampling theorem is violated. In order to prevent aliasing, an analog anti-aliasing low-pass filter is placed in front of every ADC . This low-pass filter typically has a cut-off frequency f _c [Hz] that is much lower than half of the sampling frequency f _s [Hz]: f _c ≪ f _s∕2.

• FIR-filters. FIR-filters have finite impulse responses (FIR) h[n] and do not have feedback coefficients. Linear moving-average filters are FIR filters.

• IIR-filter. IIR-filters have an infinite impulse response (IIR) h[n], use feedback coefficients, and provide more design freedom than FIR-filters. Digital filters with Butterworth, Bessel, or Chebyshev behavior require a digital IIR-filter.

Even though the usage of digital filters in EMC is limited, they can help improve robustness, especially when it comes to voltage transient immunity. Here are some typical examples:

• Debounce filter. Analog or digital inputs are read multiple times and checked for unexpected state changes. For example, it could be determined that an input state is only valid when it is stable over a certain period (debounce time).

• Analog sensor sanity check. When reading an analog sensor input, e.g., a pressure or a flow rate, the sensor value could be checked if it is within a reasonable range. If not, the sensor value could either be reread, or the system could throw a warning.

• Input states sanity check. In the case of multiple sensors (e.g., position sensors along a linear drive with a slider), a sanity check could be performed (e.g., it is only possible that one single position sensor along the linear drive detects the slider). In the event of an implausible result, a new status query could be triggered, or the system could enter a safe state.

• Median filter. Median filters are non-linear filters that provide a reliable way to get rid of short spikes and dips in an input signal. The filter takes an array of N values, sorts it in ascending order, and returns the value in the middle of the sorted array. Figure 15.21 shows the conceptual workflow.

  Fig. 15.22

  

  Concept of a spike-remover

• Spike-remover. A spike remover detects a spike in an array of data and replaces the spike data with valid data points. Figure 15.22 shows the principle of a single data point spike-spike remover.

  Fig. 15.23

  

  Analog filters: Time-domain vs. frequency-domain

15.11 Summary

• Filter characterization. EMI-filters should always be characterized by their step response (time-domain) and their frequency response (frequency-domain) (Fig. 15.23).

  • Time-domain. Determine distortion and time delay caused by the EMI-filter.

    Fig. 15.24

    

    EMI low-pass filter overview

  • Frequency-domain. Determine attenuate [dB] and phase shift of the EMI-filter.

• High-frequency filters. Filters against radiated and conducted HF emissions and immunity are in most cases passive low-pass filters like shown in Fig. 15.24.

• Transient suppression filters. Typically non-linear clam** devices (TVS diodes, varistors) or capacitors are used as protective shunt devices. Figure 15.25 summarizes the most important points when selecting a high-voltage transient suppressor device.

  Fig. 15.25

  

  Selecting a protective shunt device for high-voltage transient suppression

• Digital filters. Filters implemented in software and firmware are helpful to improve EMC immunity. Examples are debounce filters, input state sanity checks, sensor value sanity checks, spike-remove filters, and median filters (Fig. 15.26).

  Fig. 15.26

  

  Digital filters: time-domain vs. frequency-domain