Section 14.5 Series Resonance

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Problems 665

Section 14.5 Series Resonance

• 14.25 A series RLC network has R = 2 kΩ, L = 40 mH, and C = 1μF. Calculate the impedance at resonance and at one-fourth, one-half, twice, and four times the resonant frequency.

14.26 Design a problem to help other students better understand ω0, Q, and B at resonance in series RLC circuits.

• 14.27 Design a series RLC resonant circuit with ω0 = 40 rad/s and B = 10 rad/s.
• 14.28 Design a series RLC circuit with B = 20 rad/s and ω0 = 1,000 rad/s. Find the circuit’s Q. Let R = 10 Ω.
• 14.29 Let vs = 20 cos(at) V in the circuit of Fig. 14.77. Find ω0, Q, and B, as seen by the capacitor.

Figure 14.77 For Prob. 14.29.

• 14.30 A circuit consisting of a coil with inductance 10 mH and resistance 20 Ω is connected in series with a capacitor and a generator with an rms voltage of 120 V. Find:
  • (a) the value of the capacitance that will cause the circuit to be in resonance at 15 kHz
  • (b) the current through the coil at resonance
  • (c) the Q of the circuit

Section 14.6 Parallel Resonance

• 14.31 Design a parallel resonant RLC circuit with ω⁰ = 100 krad/s and a bandwidth of 10 krad/s. Additionally what is the value of Q?
• 14.32 Design a problem to help other students better understand the quality factor, the resonant frequency, and bandwidth of a parallel RLC circuit.
• 14.33 A parallel resonant circuit with a bandwidth of 40 krad/s and the half-power frequencies are ω¹ = 4.98 Mrad/s and ω² = 5.02 Mrad/s, calculate the quality factor and resonant frequency.
• 14.34 A parallel RLC circuit has R = 100 kΩ, L = 100 mH, and a C = 10 μF. Determine the value of Q, the resonant frequency, and the bandwidth. If

R = 200 kΩ, how does that affect the values of Q, resonant frequency, and the bandwidth?

• 14.35 A parallel RLC circuit has R = 10 kΩ, L = 100 mH, and a resonant frequency of 200 krad/s. Calculate the value of C, the value of the quality factor, and the bandwidth.
• 14.36 It is expected that a parallel RLC resonant circuit has a midband admittance of 25 × 10⁻³ S, quality factor of 120, and a resonant frequency of 200 krad/s. Calculate the values of R, L, and C. Find the bandwidth and the half-power frequencies.
• 14.37 Rework Prob. 14.25 if the elements are connected in parallel.
• 14.38 Find the resonant frequency of the circuit in Fig. 14.78.

Figure 14.78 For Prob. 14.38.

14.39 For the “tank” circuit in Fig. 14.79, find the resonant frequency.

Figure 14.79

• 14.40 A parallel resonance circuit has a resistance of 2 kΩ and half-power frequencies of 86 kHz and 90 kHz. Determine:
  • (a) the capacitance
  • (b) the inductance
  • (c) the resonant frequency
  • (d) the bandwidth
  • (e) the quality factor
• 14.41 Using Fig. 14.80, design a problem to help other students better understand the quality factor, the resonant frequency, and bandwidth of RLC circuits.

Figure 14.80

For Prob. 14.41.

14.42 For the circuits in Fig. 14.81, find the resonant frequency ω0, the quality factor Q, and the bandwidth B.

For Prob. 14.42.

14.43 Calculate the resonant frequency of each of the circuits in Fig. 14.82.

Figure 14.82 For Prob. 14.43.

*14.44 For the circuit in Fig. 14.83, find:

(a) the resonant frequency ω⁰

(b) Zin(ω0)

Figure 14.83 For Prob. 14.44.

14.45 For the circuit shown in Fig. 14.84, find ω0, B, and Q, as seen by the voltage across the inductor.

Figure 14.84 For Prob. 14.45.

• 14.46 For the network illustrated in Fig. 14.85, find
  • (a) the transfer function H(ω) = Vo(ω)∕I(ω),

(b) the magnitude of H at ω0 = 1 rad/s.

Figure 14.85

For Probs. 14.46, 14.78, and 14.92.

Section 14.7 Passive Filters

• 14.47 Show that a series LR circuit is a low-pass filter if the output is taken across the resistor. Calculate the corner frequency fc if L = 2 mH and R = 10 kΩ.
• 14.48 Find the transfer function Vo∕Vs of the circuit in Fig. 14.86. Show that the circuit is a low-pass filter.

For Prob. 14.48.

14.49 Design a problem to help other students better understand low-pass filters described by transfer functions.

14.50 Determine what type of filter is in Fig. 14.87. Calculate the corner frequency fc.

Figure 14.87 For Prob. 14.50.

^* An asterisk indicates a challenging problem.

Problems 667

• 14.55 Determine the range of frequencies that will be passed by a series RLC band-pass filter with R = 10 Ω, L = 25 mH, and C = 0.4μF. Find the quality factor.
• 14.56 (a) Show that for a band-pass filter,

$$\mathbf{H}(s) = \frac{sB}{s^2 + sB + \omega_0^2}, \qquad s = j\omega$$

where B = bandwidth of the filter and ω0 is the center frequency.

(b) Similarly, show that for a band-stop filter,

$$\mathbf{H}(s) = \frac{s^2 + \omega_0^2}{s^2 + s + \omega_0^2}, \qquad s = j\omega$$

14.57 Determine the center frequency and bandwidth of the band-pass filters in Fig. 14.88.

Figure 14.88

For Prob. 14.57.

• 14.58 The circuit parameters for a series RLC bandstop filter are R = 250 Ω, L = 1 mH, C = 40 pF. Calculate:
  • (a) the center frequency
  • (b) the half-power frequencies
  • (c) the quality factor
• 14.59 Find the bandwidth and center frequency of the band-stop filter of Fig. 14.89.

Figure 14.89 For Prob. 14.59.

Section 14.8 Active Filters

• 14.60 Obtain the transfer function of a high-pass filter with a passband gain of 100 and a cutoff frequency of 40 rad/s.
• 14.61 Find the transfer function for each of the active filters in Fig. 14.90.

Figure 14.90

14.62 The filter in Fig. 14.90(b) 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:

(a) 200 Hz (b) 2 kHz (c) 10 kHz

14.63 Design an active first-order high-pass filter with

$$\mathbf{H}(s) = -\frac{100s}{s+10}, \qquad s = j\omega$$

Use a 1-μF capacitor.

14.64 Obtain the transfer function of the active filter in Fig. 14.91 on the next page. What kind of filter is it?

For Prob. 14.64.

14.65 A high-pass filter is shown in Fig. 14.92. Show that the transfer function is

Figure 14.92 For Prob. 14.65.

• 14.66 A “general” first-order filter is shown in Fig. 14.93.
  • (a) Show that the transfer function is

(a) Show that the transfer function is
\n

$$\mathbf{H}(s) = \frac{R_4}{R_3 + R_4} \times \frac{s + (1/R_1C)[R_1/R_2 - R_3/R_4]}{s + 1/R_2C},$$

\n

$$s = j\omega$$

• (b) What condition must be satisfied for the circuit to operate as a high-pass filter?
• (c) What condition must be satisfied for the circuit to operate as a low-pass filter?

14.67 Design an active low-pass filter with dc gain of 0.25 and a corner frequency of 500 Hz.

14.68 Design a problem to help other students better understand the design of active high-pass filters when specifying a high-frequency gain and a corner frequency.

14.69 Design the filter in Fig. 14.94 to meet the following requirements:

• (a) It must attenuate a signal at 2 kHz by 3 dB compared with its value at 10 MHz.
• (b) It must provide a steady-state output of vo(t) = 10 sin(2π × 10⁸ t + 180°) V for an input vs(t) = 4 sin(2π × 108 t) V.

Figure 14.94

For Prob. 14.69.

• *14.70 A second-order active filter known as a Butterworth filter is shown in Fig. 14.95.
  • (a) Find the transfer function Vo∕Vi.
  • (b) Show that it is a low-pass filter.

Figure 14.95

For Prob. 14.70.

Section 14.9 Scaling

14.71 Use magnitude and frequency scaling on the circuit of Fig. 14.79 to obtain an equivalent circuit in which the inductor and capacitor have magnitude 1 H and 1 F respectively.

14.72 Design a problem to help other students better understand magnitude and frequency scaling.

14.73 Calculate the values of R, L, and C that will result in R = 12 kΩ, L = 40μH, and C = 300 nF respectively when magnitude-scaled by 800 and frequency-scaled by 1000.

Problems 669

• 14.74 A circuit has R1 = 3 Ω, R2 = 10 Ω, L = 2H, and C = 1∕10 F. After the circuit is magnitude-scaled by 100 and frequency-scaled by 106 , find the new values of the circuit elements.
• 14.75 In an RLC circuit, R = 20 Ω, L = 4 H, and C = 1 F. The circuit is magnitude-scaled by 10 and frequency-scaled by 10⁵ . Calculate the new values of the elements.
• 14.76 Given a parallel RLC circuit with R = 5 kΩ, L = 10 mH, and C = 20μF, if the circuit is magnitude-scaled by Km = 500 and frequencyscaled by Kf = 105 , find the resulting values of R, L, and C.
• 14.77 A series RLC circuit has R = 10 Ω, ω0 = 40 rad/s, and B = 5 rad/s. Find L and C when the circuit is scaled:
  • (a) in magnitude by a factor of 600,
  • (b) in frequency by a factor of 1,000,
  • (c) in magnitude by a factor of 400 and in frequency by a factor of 10⁵ .
• 14.78 Redesign the circuit in Fig. 14.85 so that all resistive elements are scaled by a factor of 1,000 and all frequency-sensitive elements are frequency-scaled by a factor of 10⁴ .
• *14.79 Refer to the network in Fig. 14.96.
  • (a) Find Zin(s).
  • (b) Scale the elements by Km = 10 and Kf = 100. Find Zin(s) and ω0.

Figure 14.96

For Prob. 14.79.

• 14.80 (a) For the circuit in Fig. 14.97, draw the new circuit after it has been scaled by Km = 200 and Kf = 104 .
  • (b) Obtain the Thevenin equivalent impedance at terminals a-b of the scaled circuit at ω = 104 rad/s.

Figure 14.97 For Prob. 14.80.

14.81 The circuit shown in Fig. 14.98 has the impedance

The circuit shown in Fig. 14.98 has the impedance

$$Z(s) = \frac{1,000(s + 1)}{(s + 1 + j50)(s + 1 - j50)}, \qquad s = j\omega$$

Find:

• (a) the values of R, L, C, and G
• (b) the element values that will raise the resonant frequency by a factor of 103 by frequency scaling

Figure 14.98

For Prob. 14.81.

14.82 Scale the low-pass active filter in Fig. 14.99 so that its corner frequency increases from 1 rad/s to 200 rad/s. Use a 1-μF capacitor.

Figure 14.99

For Prob. 14.82.

14.83 The op amp circuit in Fig. 14.100 is to be magnitude-scaled by 100 and frequency-scaled by 105 . Find the resulting element values.

Figure 14.100

For Prob. 14.83.

Section 14.10 Frequency Response Using PSpice

14.84 Using PSpice or MultiSim, obtain the frequency response of the circuit in Fig. 14.101 on the next page.

Figure 14.101

14.85 Use PSpice or MultiSim to obtain the magnitude and phase plots of Vo∕Is of the circuit in Fig. 14.102.

Figure 14.102

For Prob. 14.85.

14.86 Using Fig. 14.103, design a problem to help other students better understand how to use PSpice to obtain the frequency response (magnitude and phase of I) in electrical circuits.

Figure 14.103

• For Prob. 14.86.
  • 14.87 In the interval 0.1 < f < 100 Hz, plot the response of the network in Fig. 14.104. Classify this filter and obtain ω0.

Figure 14.104 For Prob. 14.87.

14.88 Use PSpice or MultiSim to generate the magnitude and phase Bode plots of Vo in the circuit of Fig. 14.105.

For Prob. 14.88.

14.89 Obtain the magnitude plot of the response Vo in the network of Fig. 14.106 for the frequency interval 100 < f < 1,000 Hz.

Figure 14.106

For Prob. 14.89.

• 14.90 Obtain the frequency response of the circuit in Fig. 14.40 (see Practice Problem 14.10). Take R1 = R2 = 100 Ω, L = 2 mH. Use 1 < f < 100,000 Hz.
• 14.91 For the “tank” circuit of Fig. 14.79, obtain the frequency response (voltage across the capacitor) using PSpice or MultiSim. Determine the resonant frequency of the circuit.
• 14.92 Using PSpice or MultiSim, plot the magnitude of the frequency response of the circuit in Fig. 14.85.

Section 14.12 Applications

14.93 For the phase shifter circuit shown in Fig. 14.107, find H = Vo∕Vs.

Figure 14.107

For Prob. 14.93.

14.94 For an emergency situation, an engineer needs to make an RC high-pass filter. He has one 10-pF capacitor, one 30-pF capacitor, one 1.8-kΩ resistor, and one 3.3-kΩ resistor available. Find the greatest cutoff frequency possible using these elements.

14.95 A series-tuned antenna circuit consists of a variable capacitor (40 pF to 360 pF) and a 240-μH antenna coil that has a dc resistance of 12 Ω.

• (a) Find the frequency range of radio signals to which the radio is tunable.
• (b) Determine the value of Q at each end of the frequency range.

14.96 The crossover circuit in Fig. 14.108 is a low-pass filter that is connected to a woofer. Find the transfer function H(ω) = Vo(ω)∕Vi(ω).

For Prob. 14.96.