Questions

D **14.67 Repeat Problem 14.60 using the Tow–Thomas biquad of Fig. 14.26 to realize the second-order section in the cascade.

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D 14.66 Design the circuit of Fig. 14.26 to realize a low-pass notch filter with ω0 = 105 rad/s, Q = 10, dc gain = 1, and ωn = 1.3 × 105 rad/s. Use C = 10 nF and r = 20 kΩ.

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14.65 Starting from first principles, derive the transfer function Vo/Vi of the circuit in Fig. 14.26 and verify that it is the expression given by Eq. (14.70).

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D 14.64 Use the Tow–Thomas biquad of Fig. 14.26 to realize a high-pass notch filter with a notch frequency ωn and high-frequency gain of G. If a convenient value is selected for C, give expressions for the required values of R, R1,C1, R2, and R3 in terms of ω0, ωn, C, and G.

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14.63 Use the Tow–Thomas circuit in Fig. 14.25(b) to design a low-pass filter with f0 = 10 kHz, Q = 20, and dc gain = 10 V/V, and input resistance = 1 kΩ. Specify all component values.

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14.62 (a) Consider the KHN biquad in Fig. 14.24(a), with the three outputs Vhp, Vbp, and Vlp summed by the summer in Fig. 14.24(b). Using the transfer functions in Eqs. (14.59), (14.62), and (14.63), show that the transfer function realized at the output Vo is

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D 14.61 Design the KHN circuit of Fig. 14.24(a) to realize a bandpass filter with a center frequency of 3 kHz and a 3-dB bandwidth of 50 Hz. Use 10-nF capacitors. Give the complete circuit and specify all component values. What value of center-frequency gain is obtained? C = 10 nF, R = 5.31 kΩ, R1 = 10 kΩ, Rf = 10 kΩ, R2 = 1 kΩ, R3 = 119 kΩ, K = 1.983, gain = 119 V/V

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**14.60 Design a third-order low-pass filter whose |T| is equiripple in both the passband and the stopband (in the manner shown in Fig. 14.3, except that the response shown is for N = 5). The filter passband extends from ω = 0 to ω = 1 rad/s, and the passband transmission varies between 1 and 0.9. The stopband edge is at ω = 1.2 rad/s. The following transfer function was obtained using filter-design tables:

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D 14.59 Use the circuit in Fig. 14.22(b) to design a bandpass filter with f0 = 100 kHz and a 3-dB bandwidth of 1 kHz. The gain at f0 is required be 10 V/V. R = 2 kΩ, C = 796 pF, R6 = 200 kΩ

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D 14.58 Show that the circuit in Fig. 14.22(c) can be used to realize a high-pass notch filter (ωn < ω0) by selecting C62 = 0. Design the circuit to obtain ω0 = 105 rad/s, ωn = 0.8 × 105 rad/s, Q = 5, and a high-frequency gain of 5.

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D 14.57 Give the circuit for a high-pass filter based on the resonator of Fig. 14.21(b). Design the circuit to obtain ω0 = 103 rad/s, Q = 5, and a high-frequency gain of unity.

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D *14.56 Design a fifth-order Butterworth filter having a 3-dB bandwidth of 104 rad/s and a dc gain of 10 V/V. Use a cascade of two circuits of the type shown in Fig. 14.22(a) and a first-order op amp–RC circuit. Select appropriate component values.

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14.55 Starting from first principles and assuming ideal op amps, derive the transfer function of the circuit in Fig. 14.22(a).

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*14.54 Consider the Antoniou circuit of Fig. 14.20(a) with R5 eliminated, a capacitor C6 connected between node 1 and ground, and a voltage source V2 connected to node 2. Show that the input impedance seen by V2 is R2/s2C4C6R1R3. How does this impedance behave for physical frequencies (s = jω)? (This impedance is known as a frequency-dependent negative resistance, or FDNR.)

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14.53 Figure P14.53 shows a generalized form of the Antoniou circuit of Fig. 14.20(a). Here, R5 is eliminated and the other four components are replaced by general impedances Z1, Z2, Z3, and Z4.

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D 14.52 Design the circuit of Fig. 14.20(a) (utilizing suitable component values) to realize an inductance of (a) 10 H, (b) 1.0 H, and (c) 0.1 H. R1 = R2 = R3 = R5 = 10 kΩ; (a) C4 = 0.1 µF; (b) C4 = 10 nF; (c) C4 = 1 nF

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14.51 Consider the LCR resonator of Fig. 14.18(a) with node x disconnected from ground and connected to an input signal source Vx, node y disconnected from ground and connected to another input signal source Vy, and node z disconnected from ground and connected to a third input signal source Vz. Use superposition to find the voltage that develops across the resonator, Vo, in terms of Vx, Vy, and Vz.

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D 14.50 Modify the bandpass circuit of Fig. 14.19(c) to change its center-frequency gain from 1 to 0.5 without changing ω0 or Q.

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D 14.49 Show that the general notch circuit in Fig. 14.19(e) can be used to realize a high-pass notch (HPN) filter by making C2 = 0. (Recall that for the HPN case, ) Design the circuit for the case ω0 = 105 rad/s, ωn = 105/ rad/s, and Q = 2. Use R = 1 kΩ. Find all the component values, the dc gain, and the high-frequency gain.

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D *14.48 Show that the general notch circuit in Fig. 14.19(e) can be used to realize a low-pass notch (LPN) filter by making L2 = ∞. (Recall that for the LPN case, ωn > ω0.) Design the circuit to obtain rad/s, and Q = 2. Use R = 1 kΩ. Find all the component values, the dc gain, and the high-frequency gain.

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