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14.47 For the resonator in Fig. 14.18(a), let node y be lifted from ground. If a signal source Vi is connected to node y and the output Vo is taken across the resonator, derive an expression for the transfer function Vo/Vi and show that it is a high pass with ω0 and Q given by Eqs. (14.39) and (14.40), respectively. Design the high-pass circuit to obtain ω0 = 106 rad/s and Q = 1. Use R = 1 kΩ.
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14.46 For each of the circuits in Fig. P14.46, find the transmission as ω approaches zero and as ω approaches ∞, and hence find the transmission zeros.
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14.45 For the LCR resonator of Fig. 14.18(a), find the change in ω0 that results from (a) increasing L by 1% (b) increasing C by 1% (c) decreasing R by 1%
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D 14.44 Design the LCR resonator of Fig. 14.18(a) to obtain poles with ω0 = 2 × 105 rad/s and Q = 4. Use R = 10 kΩ. 2 nF, 12.5 mH
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14.43 Analyze the circuit in Fig. 14.18(c) to determine its transfer function T(s) ≡ Vo(s)/Vi(s), and hence show that its poles are characterized by ω0 and Q of Eqs. (14.39) and (14.40), respectively.
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D *14.42 Design a low-pass filter to meet the following specifications: fp = 3.4 kHz, Amax = 1 dB, fs = 4kHz, Amin = 35 dB. The dc transmission is unity. (a) Find the required order of Chebyshev filter. What is the excess (above 35 dB) stopband attenuation obtained? (b) Find the poles and the transfer function. 10, 4 dB;
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14.41 Contrast the attenuation provided by a sixth-order Chebyshev filter at ωs = 2ωp to that provided by a Butterworth filter of equal order. For both, Amax = 1 dB. Sketch |T| for both filters on the same axes.
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14.40 Observe that Eq. (14.28) can be used to find the frequencies in the passband at which | T | is at its peaks and at its valleys. (The peaks are reached when the cos2[ ] term is zero, and the valleys correspond to the cos2[ ] term equal to unity.) Find these frequencies for a fifth-order filter. Peaks: 0.95ωp, 0.59ωp, 0; Valleys: ωp, 0.81ωp, 0.31ωp
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14.39 For a fifth-order Chebyshev filter with a unity transmission at dc and with a 1-dB passband ripple, find the attenuation realized at fs = 2fp. 45.3 dB
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14.38 On the same diagram, sketch the magnitude of the transfer function of a Butterworth and a Chebyshev low-pass filter of fifth order and having the same ωp and Amax. At the stopband edge, ωs, which filter gives greater attenuation?
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14.37 Sketch the transfer function magnitude for a low-pass Chebyshev filter of (a) sixth order and (b) seventh order.
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D 14.36 Design a Butterworth filter that meets the following low-pass specifications: fp = 10 kHz, Amax = 3 dB, fs = 20 kHz, Amin = 40 dB, and dc gain = 1. Find N, the poles, and T(s). What is the attenuation provided at 30 kHz? 7; Poles: ω0 = 2π × 104 rad/s, Q1 = 2.247, Q2 = 0.802, Q3 = 0.555, real pole at s = –2π × 104;
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14.35 Show that the order N of a Butterworth filter can be obtained from the approximate expression
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D 14.34 Determine the order N of the Butterworth filter for which Amax = 0.5 dB, Amin = 20 dB, and the selectivity ratio ωs/ωp = 1.7. What is the actual value of minimum stopband attenuation realized? If Amin is to be exactly 20 dB, to what value can Amax be reduced? 7; 23.15 dB, 0.25 dB
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14.33 Find the poles of a Butterworth filter having a 0.5-dB bandwidth of 103 rad/s and N = 6
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14.32 Calculate the value of attenuation obtained at a frequency 2 times the 3-dB frequency of a seventh-order Butterworth filter. 42.1 dB
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14.31 Find the transfer function of a second-order high-pass notch filter for which ωn = 1 rad/s, ω0 = 1.3 rad/s, Q = 3, and the high-frequency asymptotic gain is unity. Sketch | T(jω) | and give the value of the dc gain.
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D **14.30 (a) For a second-order notch function with ωn = ω0, show that for the attenuation to be greater than A dB over a frequency band BWa, the value of Q must satisfy
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D **14.29 (a) Show that |T| of a second-order bandpass function is geometrically symmetrical around the center frequency ω0. That is, the members of each pair of frequencies ω1 and ω2 for which
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14.28 Find the transfer function of a second-order high-pass filter with poles at and a high-frequency gain of unity. What are ω0 and Q of the poles?
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