REVIEW QUESTIONS PROBLEMS. Figure For Prob Figure For Probs and CHAPTER 14 FrequencyResponse 633

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1 HAPTE 14 Frequencyesponse The Touch-Tone telephone and the crossover network are two typical applications of filters. The Touch-Tone telephone system employs filters to separate tones of different frequencies to activate electronic switches. The crossover network separates signals in different frequency ranges so that they can be delivered to different devices such as tweeters and woofers in a loudspeaker system. EVIEW QUESTIONS 14.1 A zero of the transfer function 10(s 1) H(s) = (s 2)(s 3) is at 10 1 (c) 2 (d) On the Bode magnitude plot, the slope of the pole 1/(5 jω) 2 is 20 db/decade 40 db/decade (c) 40 db/decade (d) 20 db/decade 14.3 On the Bode phase plot, the slope of [1 j10ω ω 2 /25] 2 is 45 /decade 90 /decade (c) 135 /decade (d) 180 /decade 14.4 How much inductance is needed to resonate at 5 khz with a capacitance of 12 nf? 2652 H H (c) H (d) mh 14.5 The difference between the half-power frequencies is called the: quality factor resonant frequency (c) bandwidth (d) cutoff frequency 14.6 In a series circuit, which of these quality factors has the steepest curve at resonance? Q = 20 Q = 12 (c) Q = 8 (d) Q = In a parallel circuit, the bandwidth B is directly proportional to. True False 14.8 When the elements of an circuit are both magnitude-scaled and frequency-scaled, which quality is unaffected? resistor resonant frequency (c) bandwidth (d) quality factor 14.9 What kind of filter can be used to select a signal of one particular radio station? lowpass highpass (c) bandpass (d) bandstop A voltage source supplies a signal of constant amplitude, from 0 to 40 khz, to an lowpass filter. The load resistor experiences the maximum voltage at: dc 10 khz (c) 20 khz (d) 40 khz Answers: 14.1b, 14.2c, 14.3d, 14.4d, 14.5c, 14.6a, 14.7b, 14.8d, 14.9c, 14.10a. POBEMS Section 14.2 Transfer Function 14.ind the transfer function / of the circuit in Fig Obtain the transfer function / of the circuit of Fig v i (t) (t) v i (t) (t) Figure For Prob Figure For Probs and

2 634 PAT 2 A ircuits 14.3 Given the circuit in Fig , determine the transfer function H( s) = (s )/(s ). If = 40 k and = 2 µf, specify the locations of the poles and zeros of H( s) Obtain the transfer function H(ω ) = I o /I s of the circuits shown in Fig H i s 20 Ω i o 0.25 F Figure For Prob Find the transfer function H(ω ) = / of the circuits shown in Fig i s v x Ω 0.5v x i o 2.5 H Figure For Prob epeat Prob for the circuits in Fig Figure For Prob Section 14.3 Figure For Prob The Decibel Scale 14.7 alculate H(ω) if H db equals 0.05 db 6.2 db (c) db 14.8 Determine the magnitude (in db) and the phase (in degrees) of H(ω) at ω = 1ifH(ω) equals (c) 10jω 2 jω (d) 3 1 jω 6 2 jω Section 14.4 Bode Plots 14.9 A ladder network has a voltage gain of 10 H(ω) = (1 jω)(10 jω) Sketch the Bode plots for the gain Sketch the Bode plots for 10 jω H(ω) = jω(2 jω) onstruct the Bode plots for G(s) = s 1 s 2 (s 10), s = jω Draw the Bode plots for 50(jω 1) H(ω) = jω(ω 2 10jω 25) onstruct the Bode magnitude and phase plots for 40(s 1) H(s) = (s 2)(s 10), s = jω Sketch the Bode plots for s G(s) = (s 2) 2 (s 1), s = jω

3 HAPTE 14 Frequencyesponse Draw Bode plots for (s 2) 2 G(s) = s(s 5)(s 10), s = jω A filter has s H(s) = s 2 10s 100 Sketch the filter s Bode magnitude and phase plots Sketch Bode magnitude and phase plots for N(s) = 100(s2 s 1) (s 1)(s 10), s = jω onstruct the straight-line approximate plots and the exact plots onstruct Bode plots for 10jω(1 jω) T(ω) = (10 jω)(100 10jω ω 2 ) Find the transfer function H(ω) with the Bode magnitude plot shown in Fig H (db) db/decade Section 14.5 Series esonance A series network has = 2k, = 40 mh, and = 1 µf. alculate the impedance at resonance and at one-fourth, one-half, twice, and four times the resonant frequency Design a series circuit that will have an impedance of 10 at the resonant frequency of ω 0 = 50 rad/s and a quality factor of 80. Find the bandwidth Design a series circuit with B = 20 rad/s and ω 0 = 1000 rad/s. Find the circuit s Q For the circuit in Fig , find the frequency ω for which v(t) and i(t) are in phase. i(t) v(t) 20 Figure For Prob Figure 14.7or Prob v (rad/s) The Bode magnitude plot of H(ω) is shown in Fig Find H(ω). H db/decade 40 db/decade Figure For Prob v (rad/s) The Bode phase plot of G(ω) of a network is depicted in Fig Find G(ω). Section 14.6 Parallel esonance Design a parallel resonant circuit with ω 0 = 10 rad/s and Q = 20. alculate the bandwidth of the circuit A parallel resonant circuit with quality factor 120 has a resonant frequency of rad/s. alculate the bandwidth and half-power frequencies It is expected that a parallel resonant circuit has a midband admittance of S, quality factor of 80, and a resonant frequency of 200 krad/s. alculate the values of,, and. Find the bandwidth and the half-power frequencies ework Prob if the elements are connected in parallel For the tank circuit in Fig , find the resonant frequency. f v (rad/s) I o cos vt 40 mh 50 Ω 1 mf Figure For Prob Figure For Probs and

4 636 PAT 2 A ircuits 14.3or the circuits in Fig , find the resonant frequency ω 0, the quality factor Q, and the bandwidth B. 2 Ω i(t) 2 3 H v(t) 6 Ω 0.4 F 20 mh 2 kω 3 mf 6 mf Figure For Prob For the network illustrated in Fig , find the transfer function H(ω) = (ω)/i(ω), the magnitude of H at ω 0 = 1 rad/s. Figure For Prob alculate the resonant frequency of each of the circuits in Fig I Figure For Probs , 14.61, and Section 14.7 Passive Filters Show that the circuit in Fig is a lowpass filter. alculate the corner frequency f c if = 2mHand = 10 k Find the transfer function /V s of the circuit in Fig Show that the circuit is a lowpass filter. (c) v s 0.25 F Figure For Prob For the circuit in Fig , find: the resonant frequency ω 0 Z in (ω) Z in 9 mf 20 mh 0. Figure 14.8or Prob Determine the cutoff frequency of the lowpass filter described by H(ω) = 4 2 jω10 Find the gain in db and phase of H(ω) at ω = 2 rad/s Determine what type of filter is in Fig alculate the corner frequency f c. 200 Ω Figure For Prob In the circuit of Fig , i(t) = 10 sin t. alculate the value of such that v(t) = sin t V. Find. v i (t) 0. (t) Figure For Prob *An asterisk indicates a challenging problem.

5 HAPTE 14 Frequencyesponse Obtain the transfer function of a highpass filter with a passband gain of 10 and a cutoff frequency of 50 rad/s In a highpass filter with a cutoff frequency of 100 khz, = 40 mh. Find Design a series type bandpass filter with cutoff frequencies of 10 khz and 11 khz. Assuming = 80 pf, find,, and Q Determine the range of frequencies that will be passed by a series bandpass filter with = 10, = 25 mh, and = 0.4 µf. Find the quality factor Show that for a bandpass filter, Section Ω 4 mf 4 Ω 1 mh Figure For Prob Active Filters Find the transfer function for each of the active filters in Fig sb H(s) = s 2 sb ω0 2 where B = bandwidth of the filter and ω 0 is the center frequency. Similarly, show that for a bandstop filter, v i s 2 ω0 2 H(s) = s 2 sb ω Determine the center frequency and bandwidth of the bandpass filters in Fig v i V s Figure For Probs and V s The filter in Fig has a 3-dB cutoff frequency at 1 khz. If its input is connected to a 120-mV variable frequency signal, find the output voltage at: 200 Hz 2 khz (c) 10 khz Obtain the transfer function of the active filter in Fig What kind of filter is it? Figure For Prob f f The circuit parameters for a series bandstop filter are = 2k, = 0., = 40 pf. alculate: the center frequency the half-power frequencies (c) the quality factor Find the bandwidth and center frequency of the bandstop filter of Fig v i i i Figure For Prob

6 638 PAT 2 A ircuits A highpass filter is shown in Fig Show that the transfer function is ( H(ω) = 1 ) f jω i 1 jω It must provide a steady-state output of (t) = 10 sin(2π 10 8 t 180 ) V for an input v s (t) = 4 sin(2π 10 8 t) V. f v i f v s i Figure For Prob Figure For Prob A general first-order filter is shown in Fig Show that the transfer function is A second-order active filter known as a Butterworth filter is shown in Fig Find the transfer function /. Show that it is a lowpass filter. 4 H(s) = s (1/ 1)[ 1 / 2 3 / 4 ], 3 4 s 1/ 2 s = jω What condition must be satisfied for the circuit to operate as a highpass filter? (c) What condition must be satisfied for the circuit to operate as a lowpass filter? 2 v s Figure For Prob Design an active lowpass filter with dc gain of 0.25 and a corner frequency of 500 Hz Design an active highpass filter with a high-frequency gain of 5 and a corner frequency of 200 Hz Design the filter in Fig to meet the following requirements: It must attenuate a signal at 2 khz by 3 db compared with its value at 10 MHz. Figure For Prob Section 14.9 Scaling Use magnitude and frequency scaling on the circuit of Fig to obtain an equivalent circuit in which the inductor and capacitor have magnitude 1Hand 1 respectively What values of K m and K f will scale a 4-mH inductor and a 20-µF capacitor to 1Hand2F respectively? alculate the values of,, and that will result in = 12 k, = 40 µh, and = 300 nf respectively when magnitude-scaled by 800 and frequency-scaled by A series circuit has = 10, ω 0 = 40 rad/s, and B = 5 rad/s. Find and when the circuit is scaled: in magnitude by a factor of 600, in frequency by a factor of 1000, (c) in magnitude by a factor of 400 and in frequency by a factor of edesign the circuit in Fig so that all resistive elements are scaled by a factor of 1000 and all

7 HAPTE 14 Frequencyesponse 639 frequency-sensitive elements are frequency-scaled by a factor of efer to the network in Fig Find Z in (s). Scale the elements by K m = 10 and K f = 100. Find Z in (s) and ω 0. Z in (s) 5 Ω 4 Ω Figure 14.9or Prob V 2 H o For the circuit in Fig , draw the new circuit after it has been scaled by K m = 200 and K f = Obtain the Thevenin equivalent impedance at terminals a-b of the scaled circuit at ω = 10 4 rad/s. Section Frequency esponse Using PSpice Obtain the frequency response of the circuit in Fig using PSpice. 4 kω 1 mf Figure For Prob kω Use PSpice to provide the frequency response (magnitude and phase of i) of the circuit in Fig Use linear frequency sweep from 1 to 10,000 Hz V 1 kω 1 kω 1 kω 0.5 mf 0.1 Figure For Prob I 1 mh a I x In the interval 0.1 <f <100 Hz, plot the response of the network in Fig lassify this filter and obtain ω F 2 Ω 0.5I x b Figure For Prob Scale the lowpass active filter in Fig so that its corner frequency increases from 1 rad/s to 200 rad/s. Use a 1-µF capacitor. 2 Ω Figure For Prob Figure For Prob Use PSpice to generate the magnitude and phase Bode plots of in the circuit of Fig V 2 H Figure For Prob F Obtain the magnitude plot of the response in the network of Fig for the frequency interval 100 <f <1000 Hz.

8 640 PAT 2 A ircuits 50 Ω 10 mf The crossover circuit in Fig is a lowpass filter that is connected to a woofer. Find the transfer function H(ω) = (ω)/ (ω). 1 0 A 10 Ω 20 Ω 4 mh Tweeter Amplifier Woofer Figure For Prob i Speakers Obtain the frequency response of the circuit in Fig (see Practice Problem 14.10). Take 1 = 2 = 100, = 2 mh. Use 1 <f <100,000 Hz. 14.7or the tank circuit of Fig , obtain the frequency response (voltage across the capacitor) using PSpice. Determine the resonant frequency of the circuit Using PSpice, plot the magnitude of the frequency response of the circuit in Fig Section Applications The resonant circuit for a radio broadcast consists of a 120-pF capacitor in parallel with a 240-µH inductor. If the inductor has an internal resistance of 400, what is the resonant frequency of the circuit? What would be the resonant frequency if the inductor resistance were reduced to 40? A series-tuned antenna circuit consists of a variable capacitor (40 pf to 360 pf) and a 240-µH antenna coil which has a dc resistance of 12. Find the frequency range of radio signals to which the radio is tunable. Determine the value of Q at each end of the frequency range. V i 1 Figure For Prob The crossover circuit in Fig is a highpass filter that is connected to a tweeter. Determine the transfer function H(ω) = (ω)/ (ω). Amplifier i 1 Figure For Prob Speakers V o Tweeter Woofer OMPEHENSIVE POBEMS A certain electronic test circuit produced a resonant curve with half-power points at 432 Hz and 454 Hz. If Q = 20, what is the resonant frequency of the circuit? In an electronic device, a series circuit is employed that has a resistance of 100, a capacitive reactance of5k, and an inductive reactance of 300 when used at 2 MHz. Find the resonant frequency and bandwidth of the circuit In a certain application, a simple lowpass filter is designed to reduce high frequency noise. If the desired corner frequency is 20 khz and = 0.5 µf, find the value of In an amplifier circuit, a simple highpass filter is needed to block the dc component while passing the time-varying component. If the desired rolloff frequency is 15 Hz and = 10 µf, find the value of Practical filter design should allow for source and load resistances as shown in Fig et = 4k and = 40-nF. Obtain the cutoff frequency when: s = 0, =, s = 1k, = 5k.

9 HAPTE 14 Frequencyesponse 641 V s s A low-quality factor, double-tuned bandpass filter is shown in Fig Use PSpice to generate the magnitude plot of (ω). 40 Ω 0.2 mf Figure 14.10or Prob The circuit in Fig is used for a lead compensator in a system design. Obtain the transfer function of the circuit. 1 0 V 1.24 mh 2 mf 4 Ω mh Figure For Prob From photoresistor output 2 To amplifier input Figure For Prob