# 2. The low-pass filter - PDF

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03_Filterschaltungen_GB :23 Uhr Seite Low-pass filter 2. The low-pass filter The low-pass filter is the most commonly used filter circuit in EMC. However, to improve understanding
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03_Filterschaltungen_GB :23 Uhr Seite Low-pass filter 2. The low-pass filter The low-pass filter is the most commonly used filter circuit in EMC. However, to improve understanding and to evaluate the effectiveness of the filter, some detailed observations may be helpful. The observation should be made on the basis of the two most commonly used low-pass circuit varieties (Figure 1). Fig. 1: Low-pass filters of first and second order. Laplace transformation The converted form of the Laplace transformation is used to improve the mathematical representation of filters or quadrupoles in the frequency domain. The so-called Laplace transformation in the complex variable domain results, with Frequency response the transformed function in the frequency domain, i.e. from a technical viewpoint the circuit s frequency response. So-called functions in the complex variable domain can be found using some mathematical rules from basic time domain functions. For example, for and for 03_Filterschaltungen_GB :23 Uhr Seite 131 Calculation table: 131 Both functions can be presented graphically (Figure 2). Fig. 2: Transfer function F x (p) against angular frequency. 03_Filterschaltungen_GB :23 Uhr Seite Attenuation Resonance point As p is to the first power in F 1 (p), the filter shows an increase in attenuation of 20 db/dec; the filter with the complex variable function F 2 (P) has, with a p of second power, an attenuation response of 40 db/dec. Shifts of the angular frequency axis depend on time constants. At ω 0 (ω 1 : F 1 (P)) there is an attenuation of 3 db for F 1 (P) and an attenuation of 6 db for F 2 (P). In contrast to the RC filter, the LC filter has a resonance point. The resistive losses (see part 1) of the inductor must be taken into account to describe the series resonance. This results in the equivalent circuit in Figure 3. Fig. 3: Simplified equivalent low-pass filter circuit taking resistive coil losses into account. Impedance The impedance Z(w) is given by Resonance occurs if the imaginary component is 0, i.e. It follows that: 03_Filterschaltungen_GB :23 Uhr Seite 133 Furthermore, at resonance 133 The amplitude and phase responses of the filter in Figure 3 are presented graphically in Figure 4. Amplitude response Phase response Fig. 4: Amplitude and phase response of the filter in Figure 3. In practice this means that the choke must have a constant high resistive impedance component over the required filter bandwidth to keep the resonant amplitude as low as possible and its bandwidth as broad as possible. both the useful and the noise signal frequencies should lie below the resonant frequency of the low-pass filter. quality factor and loss values of the capacitor generally play a lesser role if the resistive components of the choke are high. broadband, critical useful signals must lie within the linear phase response of the filter (far below resonance) to avoid distortion. additional resonance phenomena occur due to each additional parasitic element. source and sink impedances must also be taken into account with the filter properties. The result is e.g.: a network with source, filter and sink as in Figure 5. Resistive part of impedance Resonant frequency Capacitor Linear phase response Source and sink impedance 03_Filterschaltungen_GB :23 Uhr Seite Parasitic impedance Fig. 5: Example of a network, taking parasitic impedances of the components into account. 3. Filter circuitry LC low-pass Filters are frequency dependent voltage dividers, consisting of a combination of resistors or inductors and capacitors. Frequency dependence means a change in the electrical properties with frequency. The most commonly used filter, the LC low-pass, works on the basis that the impedance of the inductor rises with increasing frequency and the impedance of the capacitor falls with increasing frequency. This would solve most EMC problems in theory, were it not for some side-effects, which reduce the filter function, sometimes even negating it altogether. 3.1 Filter ground reference Weaknesses of filter reference grounds Ground One of the most important conditions for useful function of an LC filter is the capacitor s ground reference (Figure 1). Fig. 1: Circuit diagram and faulty construction of an LC low-pass filter. Every additional impedance in series with the capacitor, whether of parasitic origin inside the capacitor (see section Components, Capacitors), caused by layout or construction, reduces the effectiveness of the filter. Long connections between the capacitor and ground are additional unwanted series inductances regardless of whether the inductance comes from the connecting legs of the capacitor, the conductor tracks or bolts on the component group fixture. Designers and layout specialists are often faced with seemingly almost insoluble problems in this regard, as restrictions such as the space availability within the component group, number of con-
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