Chapter 7: Passive Filters

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Morris Barnett
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1 EETOMAGNETI OMPATIBIITY HANDBOOK 1 hater 7: Passive Filters 7.1 eeat the analytical analysis given in this chater for the low-ass filter for an filter in shunt with the load. The and for this filter are in series. 7.2 Inductors are often used in series with lines to block noise currents in lowimedance circuits. However, if the cable is carrying a fast digital signal, what is a ossible negative consequence of using this tye of filter? 7.3 An filter is to be used to filter out high-frequency noise. The source imedance is low (0.1 Ω), and the load imedance is low (10 Ω). Using a numerical ackage, design a filter to filter 10 MHz noise. The filter resonse should be at least 10 db down at 10 MHz relative to 100 khz. esonance eaks should not exist in the filter resonse from 1 khz to 100 MHz. The insertion loss of the filter at 100 khz should be less than 1 db. Plot the magnitude of the resonse in db from 1 khz to 100 MHz. The filter should use standard nonideal comonents. If these filter secifications cannot be met, state secifically why. eeat the entire analysis for a filter. Which filter is more aroriate for these imedances? 7.4 eeat Problem 7.3 for a source imedance that is high (1 kω) and a load imedance that is low (10 Ω). If these filter secifications cannot be met, state secifically why. 7.5 eeat Problem 7.3 for a source imedance that is high (1 kω) and a load imedance that is high (100 kω). If these filter secifications cannot be met, state secifically why. 7.6 Show that for the band-reject transfer function V ω + ω = V ω + jω BW + ω s 2 2 c 2 2 ( ) c the bandwidth and frequency resonse are BW and ω c, resectively. 7.7 A π filter (two shunt caacitors with a series inductor between them) is to be used to filter out high-frequency noise. The source imedance is low (0.1 Ω), and the load imedance is low (10 Ω). Using a numerical ackage, design a filter to filter 10 MHz noise. The filter resonse should be at least 10 db down at 10 MHz relative to 100 khz. esonance eaks should not exist in the filter resonse from 1 khz to 100 MHz. The insertion loss of the filter at 100 khz should be less than 1 db. Plot the magnitude of the resonse in db from 1 khz to 100 MHz. The π filter should use standard nonideal comonents. If these filter secifications cannot be met, state secifically why. eeat the entire analysis for a T filter (two series inductors with shunt caacitor between them). Which filter is more aroriate for these imedances? 7.8 eeat Problem 7.7 for a source imedance that is low (0.1 Ω) and a load imedance that is high (100 kω). If these filter secifications cannot be met, state secifically why. oyright 2002 by Kenneth. Kaiser, Version 08/17/04

2 2 EETOMAGNETI OMPATIBIITY HANDBOOK 7.9 eeat Problem 7.7 for a source imedance that is high (1 kω) and a load imedance that is low (10 Ω). If these filter secifications cannot be met, state secifically why A band-ass filter (arallel in shunt with the load) is to be used to ass a desired signal but reject nearby noise. The source imedance is low (0.1 Ω), and the load imedance is low (10 Ω). Using a numerical ackage, design a filter to ass a 10 MHz signal. The filter resonse should be at least 10 db down at 1 MHz and 100 MHz relative to 10 MHz. esonance eaks should not exist in the filter resonse for frequencies less than 1 MHz or greater than 100 MHz. esonance dis should not exist in the filter resonse from 1 MHz to 100 MHz. The insertion loss of the filter at 10 MHz should be less than 1 db. Plot the magnitude of the resonse in db from 1 khz to 500 MHz. The band-ass filter should use standard nonideal comonents. If these filter secifications cannot be met, state secifically why A band-reject filter (series in shunt with the load) is to be used to reject noise but ass nearby desirable signals. The source imedance is low (0.1 Ω), and the load imedance is low (10 Ω). Using a numerical ackage, design a filter to reject a 10 MHz noise signal. The filter resonse at 10 MHz should be at least 10 db down relative to 1 MHz and 100 MHz. esonance dis should not exist in the filter resonse for frequencies less than 1 MHz or greater 100 MHz (but less than 300 MHz). esonance eaks should not exist in the filter resonse between 1 MHz and 100 MHz (but less than 300 MHz). The insertion loss of the filter at 1 MHz and 100 MHz should be less than 1 db. Plot the magnitude of the resonse in db from 1 khz to 500 MHz. The band-reject filter should use standard nonideal comonents. If these filter secifications cannot be met, state secifically why Determine the transfer function (load voltage divided by the source voltage) for the filter given in Figure 1. Sketch its Bode magnitude lot including break frequencies, sloes, and amlitudes. Then, sketch the Bode magnitude lot of the filter s inut imedance. (Both of these lots should only be a function of the given comonent variables.) Determine any advantages and disadvantages of this filter comared to the same filter without. s Figure eeat Problem 7.12 for the filter given in Figure 2. oyright 2002 by Kenneth. Kaiser, Version 08/17/04

3 EETOMAGNETI OMPATIBIITY HANDBOOK 3 s Figure Design a low-ass video amlifier filter with a cutoff frequency of about 320 khz. The attenuation beyond this frequency should be about 60 db/decade. Assuming that the load is 1 kω, lot the inut imedance (magnitude and hase angle) of this filter (not including the source resistance) and the transfer voltage (magnitude and hase angle) of the filter (including the source resistance) from 10 khz to 10 MHz when the source resistance is equal to 10 Ω, 100 Ω, 1 kω, 10 kω, and 100 kω. Which of these source resistances rovides (over the assband) the flattest voltage resonse, most linear voltage hase resonse, the greatest voltage overshoot (in the frequency domain), the greatest voltage rile (in the frequency domain), and the fastest voltage transition at cutoff? 7.15E Determine the transfer function (load voltage divided by the source voltage) for the -tye filter shown in Figure 3. Sketch its Bode magnitude lot, ignoring any overshoot terms. Are the low frequencies boosted and the high frequencies attenuated? It is stated that the inut imedance is relatively constant with variation while the imedance seen by the load is quite variable with s variation. Determine whether this statement is ossibly true by numerically analyzing the equation for the inut and outut imedance. (For the outut imedance, assume that the source resistance is equal to s.) Figure E Determine the transfer function (load voltage divided by the source voltage) for the -tye filter shown in Figure 4. Sketch its Bode magnitude lot, ignoring any overshoot terms. Are the high frequencies boosted and the low frequencies attenuated? It is stated that the inut imedance is relatively constant with variation while the imedance seen by the load is quite variable with s variation. Determine whether this statement is ossibly true by numerically analyzing the equation for the inut and outut imedance. (For the outut imedance, assume that the source resistance is equal to s.) oyright 2002 by Kenneth. Kaiser, Version 08/17/04

4 4 EETOMAGNETI OMPATIBIITY HANDBOOK Figure It is commonly believed that the resonse of π filters are less sensitive to the source and load imedance when the source and load imedance are large. Numerically determine if and when this is true It is commonly believed that the resonse of Τ filters are less sensitive to the source and load imedance when the source and load imedance are small. Numerically determine if and when this is true Verify the Bode magnitude lots for the voltage transfer function and the inut imedance for the multile, filter # (rovided by your instructor) rovided in this chater. Since there are no inductors resent in these circuits, there is no overshoot. Therefore, do not neglect any overshoot-like terms. heck all break frequencies, sloes, and amlitudes Determine the Bode magnitude lots for the voltage transfer function and for the inut imedance for the filters given in Figure 5, Figure 6, and Figure 7. Since there are no inductors resent in these circuits, there is no overshoot. Therefore, do not neglect any overshoot-like terms. abel all break frequencies, sloes, and amlitudes. x y x y Figure 5 x y x y Figure 6 x x y y oyright 2002 by Kenneth. Kaiser, Version 08/17/04

5 EETOMAGNETI OMPATIBIITY HANDBOOK 5 Figure Derive the equations rovided in this chater in the high-q discussion for the series-to-arallel transformation of an circuit. Work with admittances instead of imedances Derive the results rovided in this chater in the high-q discussion for the seriesto-arallel transformation of an circuit Derive the results rovided in this chater in the high-q discussion for the arallel-to-series transformation of an circuit Plot the error versus Q associated with using the high-q aroximations for resistors, inductors, and caacitors for both the series-to-arallel and arallel-toseries transformations. Allow the Q to range from 1 to As stated in this chater, the Q of a network is defined as 2π times the ratio of the maximum energy stored to the energy lost each eriod of the excitation frequency. Starting with this energy definition, verify that the Q of circuit # (rovided by your instructor) in the Q table in this chater is the ratio of the reactance to the resistance of the imedance of the circuit It is stated in another book that increasing the resistance of a linear, assive circuit will reduce its Q. Find two counterexamles to show that this statement is not necessarily true If both the inductor and caacitor in the circuit shown in Figure 8 are modeled using a different resistor in shunt with each of these assive elements (as with the cases discussed in this chater), determine the ercent error between the desired versus actual overall Q of the circuit if the Q of the and are twenty times the desired Q. Do not assume that is small. Figure eeat Problem 7.27 but assume that both the inductor and caacitor are modeled using a different resistor in series with the and the The Q of an inductor is determined by measuring the resonse of the inductor (frequency corresonding to the maximum resonse and frequencies corresonding to the 3 db oints) when laced in arallel with a caacitor. The Q of the inductor measured in this manner is actually less than the actual Q of the inductor. Using an aroriate model for a real inductor, exlain why this is true One simle model for a crystal is shown in Figure 9. oyright 2002 by Kenneth. Kaiser, Version 08/17/04

6 6 EETOMAGNETI OMPATIBIITY HANDBOOK Figure 9 s What arameter is not included in this model? Given that the series and arallel resonant frequencies are, resectively, 1 s ωs =, ω = ωs 1+ s show that if the distance between these two resonances is ω = ω ωs, then ω ω ω 2 2 s s 2 = 2 ωs ωs If the load across the crystal is mainly caacitive in nature, how is the resonant frequency affected by this load? 7.31 By assuming that Q = ω > 5, determine aroximate exressions for the total Q of the circuit given in Figure 10 at both ω = 1 and at the true resonant frequency of the circuit. s Figure Design a band-ass filter with a center frequency of 10 khz and a bandwidth of about 800 Hz assuming that the source resistance is 10 Ω and the load resistance is 10 kω. The insertion loss of the filter at 10 khz should be less than 3 db. Plot the magnitude and hase of the voltage transfer function from 9 to 11 khz. Is the magnitude resonse flat over the given bandwidth? Is the hase resonse linear over the given bandwidth? Then, lot the inut and outut voltages versus time (on the same lot) for both of the following inut signals: oyright 2002 by Kenneth. Kaiser, Version 08/17/04

7 EETOMAGNETI OMPATIBIITY HANDBOOK 7 ( ) π ( 3 ) π ( 3 ) ( ) π ( 3 ) π ( 3 ) x t = 2cos t cos t + 70 x t = 2cos t cos t + 70 Exlain why the outut is or is not distorted in both cases Assume a crystal is modeled by the following arameters: = H, = F, = 8 Ω, = 4 F s Numerically verify all of the consequences given in this chater when a resistor, a caacitor, and an inductor are added in series and shunt with the crystal. The magnitude of the total imedance of the crystal with the external element should be lotted versus frequency for each case The relationshi between the hase and the frequency of a filter is given by θ = ωt d + φ When will hase distortion occur? Exlain Plot the magnitude resonse in db and the hase resonse of the multiole filter given in Figure 11. The frequency range of the lot should be at least an order of magnitude above and below the major cutoff or center frequency of the filter. Based on the filter descritions rovided in this chater and these lots, name this filter tye (e.g., low-ass Bessel). Provide your reasoning. 0.1 H 0.1 H + V s 50 Ω 50 F 50 F 80 F 50 Ω + V source load Figure eeat Problem 7.35 for the following transfer function: 1 H ( s) = where s = jω s s s s Determine the equation for the insertion loss of an filter. Simlify the exression Determine the equation for the insertion loss of a filter. Simlify the exression. oyright 2002 by Kenneth. Kaiser, Version 08/17/04

8 8 EETOMAGNETI OMPATIBIITY HANDBOOK 7.39 It is stated that for multisection filters, the load and source have less effect on the shae of the overall resonse as the number of sections increase. Determine whether this statement is reasonable by lotting and comaring the magnitude of the frequency resonse for a one-section, two-section, and three-section filter. For the one section filter let = 1 mh and = 1 µf. For the two section filter, let each section have the values = 1/2 mh and = 1/2 µf. For the three section filter, let each section have the values = 1/3 mh and = 1/3 µf. For each filter, lot the resonse for all nine ossible combinations of 1 Ω, 30 Ω, and 10 kω resistances for the source and load Physically exlain why a series circuit is caacitive for frequencies less than its resonant frequency and inductive for frequencies greater than its resonant frequency. Then, hysically exlain why a arallel circuit is inductive for frequencies less than its resonant frequency and caacitive for frequencies greater than its resonant frequency Show that at resonance the ortion of a series circuit aears like a short circuit and ortion of a arallel circuit aears like an oen circuit Plot, on the same set of axes, the total instantaneous energy in a series circuit versus time (over one comlete cycle) if = 30 Ω, = 1 mh, and = 0.01 µf for ω = ω o 10, ω = ωo, and ω = 10ω o. eeat this numerical analysis if = 3 Ω and = 300 Ω Numerically show when = 300 Ω, = 1 mh, and = 0.01 µf that the maximum value of the amlitude resonses of the resistor voltage, caacitor voltage, and inductor voltage for a series circuit are not the same and are equal to the values given by the equations given in this chater. 7.44E Show that the bandwidth of the caacitor s voltage amlitude resonse for a series circuit is aroximately equal to the bandwidth of the circuit s current resonse if the Q is high. First, assume that the maximum resonse occurs at resonance for the caacitor s voltage amlitude. Second, determine the amlitude of this resonse at resonance. Third, divide this maximum resonse at resonance by 2 and set it equal to the amlitude resonse, which is a function of the frequency. Fourth, solve for the frequencies corresonding to the 3 db oints. Fifth, subtract the real ositive uer 3 db frequency by the real ositive lower 3 db frequency to determine an exression for the bandwidth. Sixth, rewrite most of this bandwidth exression in terms of the Q at resonance: Q o = 1 Finally, assume that the Q at resonance is large and use the following aroximation valid when x is small: 1 1+ x 1+ x 2 oyright 2002 by Kenneth. Kaiser, Version 08/17/04

9 EETOMAGNETI OMPATIBIITY HANDBOOK E eeat the analysis given in Problem 7.44 for the inductor s voltage amlitude resonse for a series circuit eeat the analysis given in this chater for the 3-terminal caacitor but include a reasonable resistive term for each of the beaded leads. How does the erformance of the caacitor change? 7.47 eeat the analysis given in this chater for the 3-terminal caacitor but use a balanced three-terminal caacitor (two leads from both sides of the caacitor) Determine the bandwidth of a short-circuited one-quarter wavelength 75 Ω coaxial filter with a center frequency of 144 MHz Determine the bandwidth of an oen-circuited one-quarter wavelength 75 Ω coaxial filter with a center frequency of 144 MHz. oyright 2002 by Kenneth. Kaiser, Version 08/17/04

University of Twente

University of Twente Faculty of Electrical Engineering, Mathematics & Comuter Science Design of an audio ower amlifier with a notch in the outut imedance Remco Twelkemeijer MSc. Thesis May 008 Suervisors: