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systems analysis and design

The Analysis And Design Of Linear Circuits 8th Edition Roland E. Thomas, Albert J. Rosa, Gregory J. Toussaint - Solutions

15-2 The Ideal Transformer (Sect. 15–4)Given a circuit containing ideal transformers:(a) Find specified voltages, currents, powers, and equivalent circuits.(b) Select the turns ratio to meet prescribed conditions.
15-1 Mutual Inductance (Sects. 15–1, 15–2, and 15–3)Given the current through or voltage across two coupled inductors, find other currents or voltages.
13–54 Power Supply Filter Design The input to a power supply filter is a full-wave rectified sine wave with f0 = 50 Hz. The filter is a first-order low pass with unity dc gain. Select the cutoff frequency of the filter so that the ac components in the filter output are all less than 1% of the dc
13–51 Repeat Problem 13–50 if T = T0=8.
13–50 Find an expression for the average power delivered to a resistor R by a rectangular pulse voltage with amplitude VA, period T0, and pulse width T = T0=4. How many components of the Fourier series are required to account for 98% of the average power carried by the waveform?
13–47 Repeat Problem 13–46 for a first-order high-pass filter with the same cutoff frequency and passband gain.
13–46 A first-order low-pass filter has a cutoff frequency of 1 krad/s and a passband gain of 40 dB. The input to the filter is vSðtÞ = 20 cos 500t + 12 cos 1500t V:Find the rms value of the steady-state output.
13–43 Use MATLAB to find the rms value of a half-wave rectified sine wave. Find the fraction of the total average power carried by the dc component plus the first three nonzero ac components in the Fourier series.
13–42 Find the rms value of a parabolic wave. Find the fraction of the total average power carried by the first three nonzero ac components in the Fourier series. Compare with the results found in Problem 13–40.
13–41 Find the rms value of a sawtooth wave. Find the fraction of the total average power carried by the first three nonzero ac components in the Fourier series. Compare with the results found in Problem 13–40.
13–40 Find the rms value of a square wave. Find the fraction of the total average power carried by the first three nonzero ac components in the Fourier series.
13–39 The voltage across a 50-Ω resistor is vðtÞ = 60 + 24 sinð200πtÞ−8 sinð600πtÞ + 4:8 sinð1000πtÞ V(a) Find expressions for the current through the resistor and the power dissipated by the resistor.(b) Find the average of the power expression by integrating over one period of the
13–35 An ideal time delay is a signal processor whose output is vOðtÞ = vINðt−TDÞ. Write an expression for vOðtÞ for TD =0:5ms and vINðtÞ = 10 + 10 cos ð2π500tÞ + 2:5 cos ð2π1000tÞ +0:625 cos ð2π4000tÞ V Discuss the spectral changes caused by the time delay.
13–34 The voltage across a 1500-pF capacitor is a triangular wave with VA = 250 V and f0 = 1 kHz. Construct plots of the amplitude spectra of the capacitor voltage and current. Discuss any differences in spectral content.
13–33 Design a notch RLC filter to block the third harmonic of a triangular wave.(a) Use the results in Figure 13–4 to find the Fourier coefficients of the input for VA = 12 V and T0 =10π ms. Design your filter with a Q of 20.(b) Compare the magnitudes of the fundamental and of the fifth
13–32 Design a tuned RLC filter to pass the third harmonic of a triangular wave.(a) Use the results in Figure 13–4 to find the Fourier coefficients of the input for VA = 15 V and T0 =10π ms. Design your filter with a Q of 10.(b) Compare the magnitudes of the fundamental and of the fifth
13–31 Repeat Problem 13–30 for TðsÞ = 200=ðs + 200Þ.
13–30 Asawtooth wave with VA = 10 Vand T0 =20π ms drives a circuit with a transfer function TðsÞ = s=ðs + 200Þ. Find the amplitude of the first five nonzero terms in the Fourier series of the steady-state output. Construct plots of the amplitude spectra for the input and output waveforms and
13–28 (a) Design a passive low-pass RC filter to block the fundamental and all harmonics froma full-wave rectified sinusoidal waveform. Use the results in Figure 13–4 to find the Fourier coefficients of the input for VA =170V, T0 =16:6 ms.(b) Find the first four nonzero terms in the Fourier
13–27 (a) Design a low-pass OP AMP circuit to pass only the fundamental and the next nonzero harmonic of a 20π ms square wave. The gain of the OP AMP should be +5.(b) Find the first four nonzero terms in the Fourier series of the output of your filter.(c) Validate your design using Multisim. The
13–23 An RC series circuit is driven by the following periodic source:vsðtÞ = 10 cos 10 kt + 5 cos 30 kt + 3:33 cos 50 kt V(a) Find the output taken across the capacitor when R=50 Ωand C =5 μF.(b) Simulate the output using Multisim and compare it to the input when C varies from 0.1 to 10 μF.
13–21 The equation for a full-wave rectified cosine is v t ð Þ=VAjcosð2πt=T0ÞjV.(a) Sketch v(t) for −T0 ≤ t ≤ T0.(b) Compute the Fourier coefficients for v(t).(c) Use the Fourier coefficients to plot an estimate for v(t).
13–16 A sawtooth wave has peak-to-peak amplitude of 5 V and a fundamental frequency of 100 Hz. Use the results in Figure 13–4 to write an expression for the first four nonzero terms in the Fourier series and plot the amplitude spectrum of the signal. Use MATLAB to plot two periods of the
13–15 The waveform f(t) is a 10-kHz triangular wave with a peakto-peak amplitude of 15 V. Use the results in Figure 13–4 to write an expression for the first four nonzero terms in the Fourier series of gðtÞ = 4 + f ðtÞ and plot its amplitude spectrum.Use MATLAB to plot two periods of g(t)
13–14 A half-wave rectified sine wave has an amplitude of 169 V and a fundamental frequency of 60 Hz. Use the results in Figure 13–4 to write an expression for the first four nonzero terms in the Fourier series. Use MATLAB to plot the amplitude spectrum of the signal.
13–2 A sine wave has an amplitude of 15 V, a radian frequency of 1000 rad/s, and a phase shift of 45. Find the Fourier series expression for this waveform.
13–1 Find the first four terms of the Fourier series of the square wave waveform shown in Figure P13–4.
Digital signal processing uses samples of an analog waveform, as contrasted with analog processing, which operates on the entire waveform. Sampling refers to the process of selecting discrete values of a time-varying analog waveform for further processing. By far, the most common method of sampling
The Fourier series serves as an introduction to the concept that a signal can be described by a spectrum that gives the distribution of amplitudes (and sometimes phases) of the sinusoidal components in a waveform. Radio, television, cell phones, satellite communication, and radar systems must
Derive an expression for the average power delivered to a resistor by a sawtooth voltage of amplitude VA and period T0. Then calculate the fraction of the average power carried by the dc component plus the first three ac components.
Derive an expression for the first three nonzero terms in the Fourier series of the steady–state output voltage in Example 13–8.
Design a first-order low pass filter to allow only the fundamental of the square wave of Design Example 13–7 to pass with an attenuation of ≤ 3 dB. Verify your results by calculating the magnitude of the fundamental and of the third harmonic.
Design a series RLC tuned filter to pass only the third harmonic of a 5-V 200 kHz square wave. Show that only the third harmonic is the dominant frequency at the output of the filter. Validate the design using Multisim.
Find the first four nonzero terms of the Fourier series in Eq. (13–15) for VA =25V,R=50 Ω,L=40 μH,ω0 = 1 Mrad=s: Use Multisim to simulate the circuit’s response using a sawtooth input and compare it to an input made up of the first four terms of the Fourier series representing the sawtooth
In this example, we use MATLAB to show that a truncated Fourier series approximates a periodic waveform. The waveform is a sawtooth with A= 10 and T0 = 2 ms.Calculate the Fourier coefficients of the first 20 harmonics and plot the truncated series representation of the waveform using the first 5
13-3 RMS Value and Average Power (Sect. 13–5)(a) Given a periodic waveform, find the rms value of the waveform and the average power delivered to a specified load.(b) Given the Fourier series of a periodic waveform, find the fraction of the average power carried by specified components and
13-2 Fourier Series and Circuit Analysis (Sect. 13–4)(a) Given a linear circuit with a periodic input waveform, find the Fourier series of a steady-state response.(b) Given a network function with a periodic input, find the amplitude and phase spectra of the steady-state output.
13-1 The Fourier Series (Sects. 13–1–13–3)(a) Given an equation or graph of a periodic waveform, derive expressions for the Fourier coefficients.(b) Given a0, an, and bn calculate the Fourier coefficients of a given periodic waveform.(c) Given a Fourier series of a periodic waveform,
The use of the decibel as a measure of performance pervades the literature and folklore of electrical engineering. The decibel originally came from the definition of power ratios in bels.The decibel (dB) is more commonly used in practice. The number of decibels is 10 times the number of bels:When
Show that the transfer function TðsÞ =V2ðsÞ=V1ðsÞ in Figure 12–12 has a high-pass gain characteristic. Construct the straight-line approximations to the gain and phase responses of the circuit V(s) Cs FIGURE 12-12 R V(8)
The circuit shown in Figure 12–12 has R=2:2 kΩ and C =0:33 μF. What is the gain of the circuit at ω = 1 krad=s in dB? Cs -13= V(s) FIGURE 12-12 R V(s)
(a) Show that the transfer function TðsÞ =V2ðsÞ=V1ðsÞ of the circuit in Figure 12–14(a) has a high-pass gain characteristic.(b) Select the element values to produce a passband gain of −4 and a cutoff frequency of 40 krad=s.(c) Use Multisim to simulate the design V(s) +1 R w -13 R www
Design a high-pass filter that has the following transfer function: T(s) = 200s S+5000
Your company issued a request for proposals listing the following design requirements and evaluation criteria.Design requirements call for a high-pass filter with a passband gain of unity and a cutoff frequency of 150 Hz10%. The filter input is driven by a sensor with a 50-Ω source
For each circuit in Figure 12–16, identify whether the gain response has low-pass or highpass characteristics and find the passband gain and cutoff frequency 10 WH 1 F 5 5 ww (a) 10 10 mH (b) F m 10 5 10 W 10 mH 5 (d) FIGURE 12-16
Consider the two circuits shown in Figure 12–17. Determine if each has a low-pass or highpass characteristic. Then design each to have a cutoff frequency of 10 krad=s and a gain of 200. The input resistance must be 1 kΩ. v(1) C R R www ww V(1) V(1) L R R www 12(t)
State whether the following transfer functions have low-pass or high-pass gain characteristics and find the passband gain and cutoff frequency (a) T(s) (b) T(s)= (c) T3(s): 1 10s-1+10-3 102 25s+103 20/s 50+20/s
Two first-order circuits in a cascade connection have the following transfer functionsWhat are the cutoff frequencies and the passband gain? Assume that the chain rule applies. T(s)=- S 20 2000 S and T2(s): = +1 S+40
There is a need for a design of a bandstop filter centered at 100 Hz that has the following transfer function 2%:A vendor has submitted the circuit of Figure 12–24, claiming that the design meets the specifications within 1%. Is the vendor’s claim accurate? T(s) = (s + 2005+410) (s+100) (s+4000)
Following the analysis pattern in Example 12–10, design a circuit that realizes the following transfer function. Use no resistor smaller than 1 kΩ. What are the passband gain and the cutoff frequencies of the filter? T(s)= 200(s2+200s+10) (s+100) (s+104)
(a) Construct the Bode plot of the straight-line approximation of the gain of the transfer function(b) Find the point at which the high-frequency gain falls below the dc gain T(s) = 12,500 (s+10) (s+50) (s+500)
You need to design a circuit that has the transfer function indicated by the straightline Bode plot shown in Figure 12–42.(a) Develop a transfer function TðsÞ from the straight-line graph.(b) Validate your results using MATLAB.(c) Design a cascade circuit that realizes the TðsÞ found in part
(a) Derive an expression for the straight-line approximation to the gain response of the following transfer function:(b) Find the straight-line gains at ω = 10, 30, and 100 rad=s.(c) Find the frequency at which the high-frequency gain asymptote falls below −20 dB.(d) Compare your answers in
(a) Construct the straight-line gain plot for the transfer function(b) Verify the solution using MATLAB. T(s)= 5000(s +100) s+400s +(500)
Construct a straight-line graph of the gain function of the following transfer function. Then use MATLAB to plot the actual Bode magnitude plot. 20s T(s) = $+2s+2500
A series RLC bandstop filter shown in Figure 12–33 has the following transfer functionDesign an RLC bandstop filter such that the rejected frequency is 25 krad=s and the bandwidth is 500 rad=s. The filter is connected to a Thévenin source with a 50-Ωsource resistance. Validate both the gain and
You are given a second-order low-pass filter with the following transfer function:Characterize the effects of changing ζ on the step response and the magnitude of the frequency response 10,000 T(s)=2+200s+10,000
There is a need for a passive notch filter at 100 rad=s. The narrower the notch, the better, but there should be minimal ringing of signals passing through. Shown below are the transforms of three filters submitted for consideration are shown below. Which would you recommend and why? s+10x+10,000
For t ≥ 0, the step response of a filter isIs this a low-pass, high-pass, bandpass, or bandstop filter? What is the passband gain and the approximate cutoff frequency? g(t)=-5e-40+25e-200r
The straight-line gain response of a filter is shown in Figure 12–56. What are the initial and final values of the step response? What is the approximate duration of the transient response? IT(jw)ldB 20 6 FIGURE 12-56 @ (rad/s) 20 100
A certain RLC series bandpass circuit has the following transfer function:Is it possible to alter a circuit element to keep the same center frequency while minimizing the ringing? 200s T(s)=$2 +200s+640,000
Modern piezoelectric pressure transducers are fabricated as an integrated circuit that includes both the sensing crystal and the signal-conditioning electronics. The step response of such a transducer takes the formwhere υTR (t) is the transducer output voltage caused by an abrupt change in
12–3 A certain low-pass filter has the Bode diagram shown in Figure P12–3.(a) How many dB down is the filter at 100 rad=s?(b) Estimate where the cutoff frequency occurs, then determine how many dB down is the filter at one octave after the cutoff frequency? 10 0.1 0.01 1 IT(jw)l 10 co (rad/s)
12–4 Find the transfer function TVðsÞ =V2ðsÞ=V1ðsÞ of the circuit in Figure P12–4.(a) Find the dc gain, infinite frequency gain, and cutoff frequency.Identify the type of gain response.(b) Sketch the straight-line approximations of the gain and phase responses.(c) Calculate the gain at
12–5 Find the transfer function TVðsÞ =V2ðsÞ=V1ðsÞ of the circuit in Figure P12–5.(a) Find the dc gain, infinite frequency gain, and cutoff frequency.Identify the type of gain response.(b) Sketch the straight-line approximation of the gain response.(c) Calculate the gain at ω=0:25 ωC,
12–6 Find the transfer function TVðsÞ =V2ðsÞ=V1ðsÞ of the circuit in Figure P12–6.(a) Find the dc gain, infinite frequency gain, and cutoff frequency. Identify the type of gain response.(b) Sketch the straight-line approximation of the gain response.(c) Calculate the gain at ω=0:1 ωC,
12–7 Find the transfer function TVðsÞ =V2ðsÞ=V1ðsÞ of the circuit in Figure P12–7.(a) Find the dc gain, infinite frequency gain, and cutoff frequency.Identify the type of gain response.(b) Sketch the straight-line approximation of the gain response.(c) Calculate the gain at ω=0:1 ωC,
12–12 You task is to connect themodules in Figure P12–12 so that the gain of the transfer function is 4 and the cutoff frequency of the filter is 500 rad=s when connected between the source and the load. Repeat if the cutoff frequency is 625 rad=s and the gain did not matter. Source 100 V(1)
12–13 Ayoung designer needed to design a low-pass filter with a cutoff of 1 krad=s and a gain of – 5. The filter is to fit as an interface between the source and the load. The designer was perplexed when no matter how the stages are connected the results are not what were expected. Explain the
12–18 Find the transfer function TVðsÞ =V2ðsÞ=V1ðsÞ of the circuit in Figure P12–18.(a) Find the dc gain, infinite frequency gain, and cutoff frequency. Identify the type of gain response.(b) Use MATLAB to plot the Bode magnitude gain response of the circuit.(c) What element value would
12–19 (a) Find the transfer function TVðsÞ =V2ðsÞ=V1ðsÞof the circuit in Figure P12–19.(b) What type of gain response does the circuit have?(c) What is the passband gain?(d) Design a suitable filter using practical, standard values of R and C from the inside rear cover so that the cutoff
12–22 The transfer function of a first-order circuit is(a) Identify the type of gain response. Find the cutoff frequency and the passband gain.(b) Use MATLAB to plot the magnitude of the Bode gain response.(c) Design a circuit to realize the transfer function.(d) Use Multisim to validate your
12–23 The transfer function of a first-order circuit is(a) Identify the type of gain response. Find the cutoff frequency and the passband gain.(b) Use MATLAB to plot the magnitude of the Bode gain response.(c) Design a circuit to realize the transfer function.(d) Use Multisim to validate your
12–25 A circuit has the following transfer functionUse MATLAB to plot the Bode diagram of the transfer function.From the plot, determine the following:(a) The nature of the filter, that is, LP, HP, BP, BR.(b) The center frequency in radians.(c) The cutoff frequencies.(d) The phase angles at ω=0,
12–27 A circuit has the following transfer functionUse MATLAB to plot the Bode diagram of the transfer function.From the plot determine the following:(a) The nature of the filter, that is, LP, HP, BP, BR?(b) The center frequency in radians.(c) The cutoff frequencies.(d) The phase angles at ω=0,
12–29 A circuit has the following transfer function:Use MATLAB to plot the Bode diagram of the transfer function.From the plot determine the following:(a) The nature of the filter, that is, LP, HP, BP, BR?(b) The cutoff frequency in radians.(c) The phase angles at ω=0, ω ! ∞, and ω = ωC?(d)
12–31 The circuit in Figure P12–31 produces a bandpass response for a suitable choice of element values. Identify the elements that control the two cutoff frequencies. Select the element values so that the passband gain is 100 and the cutoff frequencies are 1000 rad=s and 40 krad=s. Use
12–32 The circuit in Figure P12–32 produces a bandpass response for a suitable choice of element values. Identify the elements that control the two cutoff frequencies. Select the element values so that the passband gain is 100 and the cutoff frequencies are 1000 rad=s and 40 krad=s. Use
12–34 The circuit in Figure P12–34 produces a bandstop response for a suitable choice of element values.(a) Find the circuit’s transfer function.(b) Identify the elements that control the two cutoff frequencies.Select the element values so that the cutoff frequencies are 400 krad=s and 4000
12–38 A student needed to design a bandstop filter that was to block frequencies between 1000 rad=s and 10,000 rad=s with unity gain in the passbands. His design is shown in Figure P12–38. As a teaching assistant, you are required to grade his design. What grade would you assign and what
12–39 Determine the filter type for the circuit in Figure P12–39.Then findQ, B, ωC1, ωC2, and ω0. Is the circuit a narrow-band or a wide-band filter? 22 uF 100 mH m V(f) 390 www 90 V(1) 10 FIGURE P12-39
12–45 This problem looks at the effect of an inductor’s parasitic resistance on the circuit’s performance.(a) Consider the circuit in Figure P12–45. Find the circuit’s transfer function TðsÞ =V2ðsÞ=V1ðsÞ. Let RP = 0 (an inductor without a parasitic resistance) and find the transfer
12–50 Find the transfer function TVðsÞ =V2ðsÞ=V1ðsÞ for the bandpass circuit in Figure P12–50. Use MATLAB to visualize the Bode characteristics if R=50 Ω, L=50 μH, and C = 2000 pF. Design an active circuit to meet those same characteristics.Verify your design using Multisim. R w V(1) L
12–51 Show that the transfer function TVðsÞ =V2ðsÞ=V1ðsÞ of the circuit in Figure P12–51 has a bandstop filter characteristic. Derive expressions relating the notch frequency and the cutoff frequencies to R, L, and C. Then select values of R, L, and C so that the bandwidth is 10 krad=s
12–52 Figure P12–52 shows an RLC filter with an input current and an output voltage. The purpose of this problem is to determine the filter type using informal circuit analysis.Use the element impedances and basic analysis tools to find the magnitude of the output voltage jV2ðjωÞj at ω =
12–53 Aprofessor gave the following quiz to his students:Look at Figure P12–53. Each curve represents the voltage across an individual element in a series RLC circuit. Identify which curve belongs to which element, namely, R, L, C, or V1. Then explain how there can be two voltages greater than
12–54 The transfer function TVðsÞ =V2ðsÞ=V1ðsÞ for a particular circuit is(a) Construct the straight-line Bode plot of the gain and phase of the transfer function. Use the straight-line plots to estimate the amplitude and phase of the steadystate output for v1ðtÞ = 10 cos 20t V, v1ðtÞ =
12–55 The transfer function TVðsÞ =V2ðsÞ=V1ðsÞ for a particular circuit is(a) Construct the straight-line Bode plot of the gain and phase of the transfer function. Use the straight-line plots to estimate the amplitude and phase of the steady-state output for v1ðtÞ = 10 cos 50t V, v1ðtÞ
12–56 Find the transfer function TVðsÞ =V2ðsÞ=V1ðsÞ for the circuit in Figure P12–56.(a) Construct the straight-line Bode plot of the gain and phase of the transfer function. Use the straight-line plots to estimate the amplitude and phase of the steadystate output for v1ðtÞ = 10 cos 10t
12–57 Repeat Problem 12–56 using the circuit in Figure P12–57. + V(1) 1 F ww 100 1 FIGURE P12-57 V(1)
12–58 For the following transfer function(a) Construct the straight-line Bode plot of the gain. Is this a low-pass, high-pass, bandpass, or bandstop function? Estimate the cutoff frequency and passband gain.(b) Use MATLAB to plot the Bode magnitude of the transfer function.(c) Design a circuit
12–59 For the following transfer function(a) Construct the straight-line Bode plot of the gain. Is this a low-pass, high-pass, bandpass, or bandstop function? Estimate the cutoff frequency and passband gain.(b) Use MATLAB to plot the Bode magnitude of the transfer function.(c) Design a circuit
12–60 For the following transfer function(a) Construct the straight-line Bode plot of the gain. Is this a low-pass, high-pass, bandpass, or bandstop function? Estimate the cutoff frequency and passband gain.(b) Use MATLAB to plot the Bode magnitude of the transfer function.(c) Design a circuit
12–61 For the following transfer function(a) Construct the straight-line Bode plot of the gain. Is this a low-pass, high-pass, bandpass, or bandstop function? Estimate the cutoff frequency and passband gain.(b) Use MATLAB to plot the Bode magnitude of the transfer function.(c) Design a circuit
12–62 For the following transfer function TVðsÞ =V2ðsÞ=V1ðsÞ(a) Construct the straight-line Bode plot of the gain. Is this a low-pass, high-pass, bandpass, or bandstop function? Estimate the cutoff frequency(ies) and passband gain.(b) Use MATLAB to plot the Bode magnitude and phase of the
12–63 For the following transfer function TVðsÞ =V2ðsÞ=V1ðsÞ(a) Construct the straight-line Bode plot of the gain. Is this a low-pass, high-pass, bandpass, or bandstop function? Estimate the cutoff frequency(ies) and passband gain.(b) Use MATLAB to plot the Bode magnitude and phase of the
12–64 For the following transfer function TVðsÞ =V2ðsÞ=V1ðsÞ(a) Construct the straight-line Bode plot of the gain. Is this a low-pass, high-pass, bandpass, or bandstop function?(b) Use the straight-line plot to estimate the maximum gain and the frequency at which it occurs.(c) Use MATLAB to
12–65 Consider the gain plot in Figure P12–65.(a) Find the transfer function corresponding to the straightline gain plot.(b) Use MATLAB to plot the Bode magnitude of the transfer function.(c) Compare the straight-line gain and the actual gain atω = 10 and 100 rad=s.(d) Design a circuit to
12–66 Consider the gain plot in Figure P12–66.(a) Find the transfer function corresponding to the straightline gain plot.(b) Hand-draw a straight-line plot of the phase.(c) Use MATLAB to plot the Bode magnitude and phase of the transfer function.(d) Compare the straight-line gain and phase with
12–67 Consider the gain plot in Figure P12–67.(a) Find a transfer function corresponding to the straight-line gain plot. Note that the magnitude of the actual frequency response must be exactly 5 at the geometric mean of the two cutoff frequencies ð245 rad=sÞ.(b) Use MATLAB to plot the Bode

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