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656 CHAPTER 12 VA R I A B L E - F R E Q U E N C Y N E T W O R K P E R F O R M A N C E the roots of N(s) cause H(s) to become zero and are called zeros of the function, and the roots of D(s) cause H(s) to become infinite and are called poles of the function. ■ ■ Bode plots are semilog plots of the magnitude and phase of a transfer function as a function of frequency. Straight-line approximations can be used to sketch quickly the magnitude characteristic. The error between the actual characteristic and the straight-line approximation can be calculated when necessary. The resonant frequency, given by the expression 0 = 1 1LC ■ The half-power, cutoff, or break frequencies are the frequencies at which the magnitude characteristic of the Bode plot is 1兾 12 of its maximum value. ■ The parameter values for passive circuit elements can be both magnitude and frequency scaled. ■ The four common types of filters are low-pass, high-pass, band-pass, and band rejection. ■ The bandwidth of a band-pass or band-rejection filter is the difference in frequency between the half-power points; that is, BW=HI-LO is the frequency at which the impedance of a series RLC circuit or the admittance of a parallel RLC circuit is purely real. ■ For series RLC circuits, Q = (1兾R) 1L兾C . For parallel RLC circuits, Q = R 1C兾L . For a series RLC circuit, BW=R/L. For a parallel RLC circuit, BW=1/RC. The quality factor is a measure of the sharpness of the resonant peak. A higher Q yields a sharper peak. PROBLEMS • 12.1 Determine the driving point impedance at the input terminals of the network shown in Fig. P12.1 as a function of s. 12.4 Find the driving point impedance at the input terminals of the circuit in Fig. P12.4 as a function of s. 1F R + 3 + vi(t) C vo(t) L 3 3 3 Zi 1H - - Figure P12.1 12.2 Determine the driving point impedance at the input terminals of the network shown in Fig. P12.2 as a function of s. R L + Figure P12.4 12.5 Determine the driving point impedance at the input terminals 1–2 or the network shown in Fig. P12.5 as a function of s, if (a) terminals 3–4 are open and (b) terminals 3–4 are stocked. + vi(t) C2 C1 vo(t) - 1 1 3 2 1F Z(s) 2 Figure P12.2 12.3 Determine the voltage transfer function Vo(s)兾Vi(s) as a function of s for the network shown in Fig. P12.3. Figure P12.5 12.6 Repeat Problem P12.5 with a 2- resistor connected between terminals 3 and 4 in Fig. P12.6. R1 1 + vi(t) + C R2 vo(t) 4 2H 1 3 2 1F Z(s) 2 L - Figure P12.3 - 2 Figure P12.6 2H 4 PROBLEMS 12.7 Determine the driving point impedance at the input terminals of the network shown in Fig. P12.7 as a function of s. 1H G(j) = 2 Z(s) 2 1 G(j) = 2 1F 4 2H vo(t) - Figure P12.8 12.9 Draw the Bode plot for the network function j4 + 1 j20 + 1 12.10 Draw the Bode plot for the network function H(j) = H(j) = (j + 1)(j + 10)2 1 12.8 Find the transfer impedance Vo(s)兾I s(s) for the network shown in Fig. P12.8. + 10j 12.17 Sketch the magnitude characteristic of the Bode plot for the transfer function Figure P12.7 iS(t) 12.16 Sketch the magnitude characteristic of the Bode plot for the transfer function 1F 2F 657 j (j + 1)(0.1j + 1) 12.11 Sketch the magnitude characteristic of the Bode plot for the transfer function 100(j) H(j) = (j + 1)(j + 10)(j + 50) 12.12 Draw the Bode plot for the network function 10j + 1 H(j) = j(0.01j + 1) 12.13 Sketch the magnitude characteristic of the Bode plot for the transfer function 20(0.1j + 1) H(j) = j(j + 1)(0.01j + 1) 12.14 Draw the Bode plot for the network function 100 H(j) = (j)2(j2 + 1) 12.15 Sketch the magnitude characteristic of the Bode plot for the transfer function 400(j + 2)(j + 50) G(j) = -2(j + 100)2 -2104 (j + 1)2(j + 10)(j + 100)2 12.18 Sketch the magnitude characteristic of the Bode plot for the transfer function -2 G(j) = (j + 1)3 12.19 Sketch the magnitude characteristic of the Bode plot for the transfer function +6.4 H(j) = (j + 1)(-2 + 8j + 16) 12.20 Sketch the magnitude characteristic of the Bode plot for the transfer function 10j + 1 H(j) = (j + 1)(0.1j + 1) 12.21 Sketch the magnitude characteristic of the Bode plot for the transfer function H(j) = 10(5j + 1) (100j + 1)(0.02j + 1) 12.22 Sketch the magnitude characteristic of the Bode plot for the transfer function H(j) = 0.1(2j + 1) j(0.1j + 1)(0.01j + 1) 12.23 Sketch the magnitude characteristic of the Bode plot for the transfer function H(j) = 0.5(10j + 1)(j + 1) j(0.1j + 1)(0.01j + 1)2 12.24 Sketch the magnitude of the Bode plot for the transfer function H(j) = 250(j + 10) (j)2(j + 100)2 12.25 Sketch the magnitude characteristic of the Bode plot for the transfer function G(j) = 10(j + 2)(j + 100) j(-2 + 4j + 100) 658 CHAPTER 12 VA R I A B L E - F R E Q U E N C Y N E T W O R K P E R F O R M A N C E 12.26 Sketch the magnitude characteristic of the Bode plot for the transfer function 12.30 Determine H(j) from the magnitude characteristic of the Bode plot shown in Fig. P12.30. +6.4(j) H(j) = (j + 1)(-2 + 8j + 64) –40 dB/dec 12.27 Find H(j) if its magnitude characteristic is shown in Fig. P12.27. –20 dB/dec –60 dB/dec |H| ±20 dB/dec –40 dB/dec 40 dB 0.2 –20 dB/dec 4 20 60 Figure P12.30 1 10 80 120 (rad/s) 12.31 The magnitude characteristic of a band-elimination filter is shown in Fig. P12.31. Determine H(j). Figure P12.27 12.28 Determine H(j) from the magnitude characteristic shown in Fig. P12.28. 0 dB –20 dB/dec 10 100 1000 ±20 dB/dec 10,000 (rad/s) Figure P12.31 20 dB –20 dB/dec 12.32 Given the magnitude characteristic in Fig. P12.32, find H(j). –20 dB/dec |H| 0.25 5 log 50 0 dB ±20 dB/dec ±20 dB/dec –40 dB/dec Figure P12.28 12.29 Determine H(j) from the magnitude characteristic of the Bode plot shown in Fig. P12.29. 4 10 20 100 600 (rad/s) Figure P12.32 12.33 Find H(j) if its magnitude characteristic is shown in Fig. P12.33. –20 dB/dec –40 dB/dec |H| 0 dB 40 dB –20 dB/dec –20 dB/dec –40 dB/dec –20 dB/dec –40 dB/dec 0.1 Figure P12.29 1 5 50 log 0.4 Figure P12.33 50 400 1000 (rad/s) 659 PROBLEMS 12.34 Find H(j) if its amplitude characteristic is shown in Fig. P12.34. |H| –20 dB/dec 0 dB –40 dB/dec –20 dB/dec 12.38 Given the network in Fig. P12.38, determine the change necessary in the variable resistor to cut the bandwidth of the circuit in half. What is the new bandwidth? 10 –20 dB/dec 20 0° V + – 500 F 20 mH 400 (rad/s) 1 812 60 Figure P12.34 Figure P12.38 12.35 Determine H(j) if its magnitude characteristic is shown in Fig. P12.35. –40 dB/dec |H| 12.39 A series RLC circuit resonates at 1000 rad兾s. If C = 20F, and it is known that the impedance at resonance is 2.4 , compute the value of L, the Q of the circuit, and the bandwidth. –20 dB/dec 0 dB –40 dB/dec (rad/s) 5 10 50 Figure P12.35 12.36 Find G(j) for the magnitude characteristic shown in Fig. P12.36. |G| i(t) R –20 dB/dec 20 dB 12.40 Given the series RLC circuit in Fig. P12.40, (a) derive the expression for the half-power frequencies, the resonant frequency, the bandwidth, and the quality factor for the transfer characteristic I兾Vin in terms of R, L, and C, (b) compute the quantities in part (a) if R = 10 , L = 50 mH, and C = 10 F. L =0.1 vin(t) ± – –20 dB/dec C –40 dB/dec 20 100 900 0.8 (rad/s) Figure P12.36 12.37 The series RLC circuit in Fig. P12.37 is driven by a variable-frequency source. If the resonant frequency of the network is selected as 0 = 1600 rad兾s, find the value of C. In addition, compute the current at resonance and at 0兾4 and 40. i(t) 10 mH Figure P12.40 12.41 A series resonant circuit has a Q of 120 and a resonant frequency of 10,000 rad兾s. Determine the half-power frequencies and the bandwidth of the circuit. 12.42 Given the network in Fig. P12.42, find 0, Q, max, and ∑Vo∑max. 1 24 cos (t+30°) V ± – + 2 6 cos t V ± – 10 F vo(t) - C Figure P12.37 2 mH Figure P12.42 660 CHAPTER 12 VA R I A B L E - F R E Q U E N C Y N E T W O R K P E R F O R M A N C E 12.43 A variable-frequency voltage source drives the network in Fig. P12.43. Determine the resonant frequency, Q, BW, and the average power dissipated by the network at resonance. 12.44 In the network in Fig. P12.44, the inductor value is 10 mH, and the circuit is driven by a variablefrequency source. If the magnitude of the current at resonance is 12 A, 0 = 1000 rad兾s, and L = 10 mH, find C, Q, and the bandwidth of the circuit. 100 R 50 mH 12 cos t V C 36 cos (t ±45°) V ± – ± – L 5 F Figure P12.44 Figure P12.43 12.45 The network in Fig. P12.45 operates at 200 c/s. Determine the manner in which the resistor bank should be connected to maximize the voltage Vo and determine the value of the maximum voltage (R1 = 12 , R2 = 6 , R3 = 4 ). R1 12 0° V + – R2 R3 500 μH + 50μF Vo – Figure P12.45 12.46 A series RLC circuit is driven by a signal generator. The resonant frequency of the network is known to be 1600 rad兾s and at that frequency the impedance seen by the signal generator is 5 . If C = 20 F, find L, Q, and the bandwidth. 12.47 A parallel RLC resonant circuit has a resistance of 200 . If it is known that the bandwidth is 80 rad兾s and the lower half-power frequency is 800 rad兾s, find the values of the parameters L and C. 12.48 A parallel RLC resonant circuit with a resonant frequency of 20,000 rad兾s has an admittance at resonance of 1 mS. If the capacitance of the network is 2 F, find the values of R and L. 12.49 A parallel RLC circuit, which is driven by a variable frequency 2-A current source, has the following values: R = 1 k, L = 100 mH, and C = 10 F. Find the bandwidth of the network, the half-power frequencies, and the voltage across the network at the half-power frequencies. 12.50 A parallel RLC circuit, which is driven by a variable frequency 2-A current source, has the following values: R = 1 k, L = 400 mH, and C = 10 F. Find the bandwidth of the network, the half-power frequencies, and the voltage across the network at the half-power frequencies. 12.51 Determine the parameters of a parallel resonant circuit that has the following properties: 0 = 2 Mrad兾s, BW = 20 rad兾s, and an impedance at resonance of 2000 . PROBLEMS 661 12.52 The source in the network in Fig. P12.52 is iS(t) = cos 1000t + cos 1500t A. R = 200 and C = 500 F. If 0 = 1000 rad兾s, find L, Q, and the BW. Compute the output voltage v0(t) and discuss the magnitude of the output voltage at the two input frequencies. + iS(t) R C vo(t) L - Figure P12.52 12.53 Consider the network in Fig. P12.53. If R = 1 k, L = 20 mH, C = 50 F, and RS = q , determine the resonant frequency 0 the Q of the network, and the bandwidth of the network. What impact does an RS of 10 k have on the quantities determined? RS R L C Figure P12.53 12.54 Determine the value of C in the network shown in Fig. P12.54 for the circuit to be in resonance. 12.56 Determine the new parameters of the network in Fig. P12.56 if new = 104old. 1 L=— H 2 8 6 C 4 cos 2t V 1 C=— F R=2 Z ± – 4 4H Figure P12.56 Figure P12.54 12.55 Determine the equation for the nonzero resonant frequency of the impedance shown in Fig. P12.55. 12.57 Determine the new parameters of the network shown in Fig. P12.57 if Z new = 104Z old . 1 L=— H 2 1 C=— F 8 C Z Figure P12.55 L R Z Figure P12.57 R=2 662 CHAPTER 12 VA R I A B L E - F R E Q U E N C Y N E T W O R K P E R F O R M A N C E 12.58 Sketch the magnitude characteristic of the Bode plot for the transfer function of the network in Fig. P12.58. What type of filter is this? 12.61 Determine what type of filter the network shown in Fig. P12.61 represents by determining the voltage transfer function. 400 Ω 1.8 kΩ R1 + vs(t) + – + vo(t) 50 μF 900 Ω L vi(t) – R2 vo(t) - Figure P12.58 12.59 Given the network in Fig. P12.59, sketch the magnitude characteristic of the transfer function V0 Gv(j) = (j) V1 Identify the type of filter. 10 H + vi(t) + - Figure P12.61 12.62 Given the network in Fig. P12.62, sketch the magnitude characteristic of the transfer function Vo Gv(j) = (j) V1 Identify the type of filter. + vo(t) 100 1 mF 1000 F + - + vi(t) - 10 H 12.60 Determine what type of filter the network shown in Fig. P12.60 represents by determining the voltage transfer function. R1 vo(t) - - Figure P12.59 100 Figure P12.62 12.63 Given the lattice network shown in Fig. P12.63, determine what type of filter this network represents by determining the voltage transfer function. R1 + C vi(t) R2 - + + vo(t) vi(t) - - C + vo(t) R1 - R2 Figure P12.60 Figure P12.63 12.64 The circuit in Fig. P12.64 is a dual-T notch filter. It has an advantage over the filter in Example 12.36 in that it contains no inductors, which tend to be bulky and heavy. Derive the transfer function for this filter and verify your work for the component values C = 100 nF and R = 1590 . 2C R R + vi(t) - Figure P12.64 + C R —— 2 C vo(t) - PROBLEMS 12.65 Given the network in Fig. P12.65, find the transfer function Vo (j) V1 ± – vi(t) 12.69 An OTA with a transconductance of 1 mS is required. A 5-V supply is available, and the sensitivity of gm to IABC is 20. (a) What values of IABC and RG do you recommend? (b) If RG has a tolerance of +5%, what is the possible range of gm in the final circuit? 12.66 Determine the value of L in the circuit in Fig. P12.66 that will place the network in resonance. 2F 18 6 L 3 Figure P12.66 12.67 Determine the voltage transfer function and its magnitude characteristic for the network shown in Fig. P12.67 and identify the filter properties. R2 R1 + vi(t) - C1 + - Figure P12.68 Figure P12.65 + 10 cos t V – C - - 12 + vo(t) R2 vo(t) C - ± – + vS(t) + R2 R1 12.68 Repeat Problem 12.67 for the network shown in Fig. P12.68. R1 and determine what type of filter the network represents. + 12.70 The OTA and 5-V source described in Problem 12.69 are used to create a tranconductance of 2.5 mS. (a) What resistor value is required? (b) If the input voltage to the amplifier is vin(t) = 1.5 cos(t)V, what is the output current function? 12.71 A particular OTA has a maximum transconductance of 5 mS with a range of 6 decades. (a) What is the minimum possible transconductance? (b) What is the range of IABC? (c) Using a 5-V power supply and resistor to set IABC, what is the range of values for the resistor and the power it consumes? 12.72 A circuit is required that can double the frequency of a sinusoidal voltage. (a) If vin(t) = 1 sin (t) V, show that the multiplier circuit in Fig. P12.72 can produce an output that contains a sinusoid at frequency 2. (b) We want the magnitude of the double-frequency sinusoid to be 1 V. Determine values for RG and RL if the transconductance range is limited between 10 S and 10 mS. C2 RG + vo(t) vin - IABC ± gm – RL + vo - Figure P12.67 663 Figure P12.72 664 CHAPTER 12 VA R I A B L E - F R E Q U E N C Y N E T W O R K P E R F O R M A N C E 12.73 The automatic gain control circuit in Fig. 12.73 is used to limit the transconductance, io兾vin. (a) Find an expression for vo in terms of vin, RG, and RL. (b) Express the asymptotic transconductance, io兾vin, in terms of RG and RL at vin = 0 and as vin approaches infinity. Given RL and RG values in the circuit diagram, what are the values of the asymptotic transconductance? (c) What are the consequences of your results in (b)? (d) If vin must be no more than Vcc for proper operation, what is the minimum transconductance for the functional circuit? 12.78 Design a band-pass filter with a low cutoff frequency of approximately 4535 Hz and a high cutoff frequency of approximately 5535 Hz. 12.79 Given the circuit in Fig. P12.67, design a second-order band-pass filter with a center frequency gain of -5, 0 = 50 krad兾s, and a BW = 10 krad兾s. Let C1 = C2 = C and R1 = 1 k. What is the Q of this filter? Sketch the Bode plot for the filter. Use the ideal op-amp model. 12.80 Referring to Example 12.39, design a notch filter for the tape deck for use in Europe, where power utilities generate at 50 Hz. VCC 4V 12.81 An engineer has proposed the circuit shown in Fig. P12.81 to filter out high-frequency noise. Determine the values of the capacitor and resistor to achieve a 3-dB voltage drop at 23.16 kHz. 10 k RG 4V ± – 12.77 Design a high-pass filter with a half-power frequency between 159 and 161 kHz. R IABC C vin ± – io RL vo 1 k Figure P12.81 12.82 For the high-pass active filter in Fig. P12.82, choose C, R1, and R2 such that Ho = 5 and fc = 3 kHz. Figure P12.73 12.74 In Fig. P12.74, Vx is a dc voltage. The circuit is intended to be a dc wattmeter where the output voltage value equals the power consumed by RL in watts. (a) The gm - IABC sensitivity is 20 S兾A. Find RG such that Ix兾I1 = 104. (b) Choose RL such that 1 V at Vo corresponds to 1 W dissipated in RL. C ± – vS(t) ± – + 10 k vo(t) R2 R1 - 5V ± – Ix RG 100 Rsense Figure P12.82 12.83 For the low-pass active filter in Fig. P12.83, choose R2 and C such that Ho = -7 and fc = 10 kHz. I1 + RL Vx - R2 ± – + C RL Vo - 1 k – ± Figure P12.74 12.75 Design a low-pass filter with a cutoff frequency between 15 and 16 kHz. 12.76 Design a low-pass filter using one resistor and one capacitor that will produce a 4.24-volt output at 159 Hz when 6 volts at 159 Hz are applied at the input. vS(t) ± – + vo(t) - Figure P12.83 PROBLEMS 12.84 The second-order low-pass filter shown in Fig. P12.84 has the transfer function Vo (s) = V1 -R3 1 a b R1 R2R3C1C2 s2 + 12.85 Given the second-order low-pass filter in Fig. P12.85, design a filter that has Ho = 100 and fc = 5 kHz. Set R1 = R3 = 1 k, and let R2 = R4 and C1 = C2. Use an op-amp model with Ri = q, R0 = 0, and A = (2)105. s 1 1 1 1 a + + b + C1 R1 R2 R3 R2R3C1C2 C1 Design a filter with Ho = -10 and fc = 500 Hz, assuming that C1 = C2 = 10 nF and R1 = 1 k. R1 R1 - Figure P12.84 R3 R4 – ± + - – ± C1 C2 vo(t) R2 + v1(t) R2 – ± ± vS(t) – C2 R3 665 + vo(t) - Figure P12.85

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