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systems analysis and design
The Analysis And Design Of Linear Circuits 7th Edition Roland E Thomas, Albert J Rosa, Gregory J Toussaint - Solutions
Consider the gain plot in Figure P12–54.(a) Find a transfer function corresponding to the straightline gain plot. Note that the magnitude of the actual frequency response must be exactly 5 at the geometric mean of the two cutoff frequencies (245 rad/s).(b) Use MATLAB to plot the Bode magnitude of
Consider the gain plot in Figure P12–55.(a) Find the transfer function corresponding to the straightline gain plot.(b) Use MATLAB to plot the Bode magnitude of the transfer function.(c) Design a circuit that will realize the transfer function found in part (a).(d) Use OrCAD to verify your circuit
Consider the gain plot in Figure P12–56.(a) Find the transfer function corresponding to the straightline gain plot.(b) Use MATLAB to plot the Bode magnitude of the transfer function.(c) Design a circuit that will realize the transfer function found in part (a).(d) Use OrCAD to verify your circuit
Consider the gain plot in Figure P12–57. The goal is to design a circuit that will result in the dashed curve shown on the plot.(a) Find the transfer function corresponding to the straightline gain plot.(b) Use MATLAB to plot the Bode magnitude of the transfer function.(c) Adjust the poles so
A Tunable Tank Circuit The RLC circuit in Figure P12–68 (often called a tank circuit)has R ¼ 4.7 kV, C = 680 pF, and an adjustable (tunable) L ranging from 64 to 640 mH.(a) Show that the circuit is a bandpass filter.(b) Find the frequency range (in Hz) over which the center frequency can be
Filter Design Specification Construct a transfer function whose gain response lies entirely within the non-shaded region in Figure P12–69. IT(jw)ldB 20 20 2 20 -20 -40 - (rad/s) 1 10 100 1000 10000 FIGURE P12-69
Chip RC Networks Integrated circuit (chip) RC networks are used at parallel data ports to suppress radio frequency noise. In a certain application, RF noise at 3.2MHz is interfering with a 4-bit parallel data signal operating at 1.1 MHz.Achip RC network is to be used to reduce the RF noise on the
Design Evaluation Your company issued a request for proposals listing the following design requirements and evaluation criteria.Design Requirements: Design a low-pass filter with a passband gain of 9 10% and a cutoff frequency of 90 10% krad/s. A sensor drives the filter input with a 1-kV source
Design Evaluation In a research laboratory, you need a bandpass filter to meet the following requirements:Design Requirements: Passband gain: 10 5%, B ¼ 10 krad/s 5%, v0 ¼ 5krad/s 2%, vCL ¼ 2krad/s 10%.Evaluation Criteria: Filter performance, parts count, use of standard parts, and cost.The
Design Evaluation In a cable service distribution station, you need a bandstop filter to meet the following requirements:Design Requirements: Passband gain: 10 5%, B ¼ 3.3 kHz 5%, f0 ¼ 500 Hz 2%, fCL ¼ 75 Hz 10%. Filter must interface with a 50-V source and a 500-V load.Evaluation Criteria:
Derive expressions for the Fourier coefficients of the periodic waveform in Figure P13–1. 5 v(t) (V) 1 ms FIGURE P13-1 250 s t(s)
Derive expressions for the Fourier coefficients of the periodic waveform in Figure P13–5. vs(t) (V) VA 0 To 62 FIGURE P13-5 -t(s)
Use the results in Figure 13–4 to calculate the Fourier coefficients of the square wave in Figure P13–6. Write an expression for the first four nonzero terms in the Fourier series. v(t) (V) 10 10 5 -10 t (ms) 10 FIGURE P13-6
Use the results in Figure 13–4 to calculate the Fourier coefficients of the shifted triangular wave in Figure P13–7.Write an expression for the first four nonzero terms in the Fourier series. v(t) (V) 15 -15 t (us) 100 200 FIGURE P13-7
Derive expressions for the Fourier coefficients of the periodic waveform in Figure P13–8.(a) Write an expression for the first four nonzero terms in the Fourier series.(b) Plot the spectrum of the Fourier coefficients an and bn. v(t) (V) 8s- 2 2 s -2 FIGURE P13-8 t(s)
Use the results in Figure 13–4 to calculate the Fourier coefficients of the full-wave rectified sine wave inFigure P13–10.UseMATLAB to verify your results. Write an expression for the first four nonzero terms in the Fourier series. v(t) (V) 170 FIGURE P13-10 t(s) 1 60
Find the Fourier series for the waveform in Figure 13–19. v(t) (V) VA To/2 2To/3 5T0/6 To 0 To/6 To/3 -VA FIGURE P13-19
The periodic pulse train in Figure P13–20 is applied to the RL circuit shown in the figure.(a) Use the results in text Figure 13–4 to find the Fourier coefficients of the input for VA ¼ 10V, T0 ¼ p ms; and T ¼T0=4:(b) Find the first four nonzero terms in the Fourier series of vo(t) for R ¼
The periodic triangular wave in Figure P13–21 is applied to the RC circuit shown in the figure. The Fourier coefficients of the input are:a0 ¼ 0 an ¼ 0 bn ¼ 8VAðnpÞ2 sin np 2If VA¼ 20V and T0 ¼ 2p ms, find the first four nonzero terms in the Fourier series of vO(t) for R ¼ 10 kV, and C ¼
The periodic sawtooth wave in Figure P13–22 drives the OP AMP circuit shown in the figure.(a) Use the results in text Figure 13–4 to find the Fourier coefficients of the input for VA ¼ 5V and T0 ¼ 4p ms.(b) Find the first four nonzero terms in the Fourier series of vO(t) for R1 ¼ 20 kV, R2
The periodic triangular wave in Figure P13–26 is applied to the RLC circuit shown in the figure.(a) Use the results in text Figure 13–4 to find the Fourier coefficients of the input for VA ¼ 10V and T0 ¼ 400p ms.(b) Find the amplitude of the first five nonzero terms in the Fourier series for
Find the rms value of the periodic waveform in Figure P13–40 and the average power the waveform delivers to a resistor. Find the dc component of the waveform and the average power carried by the dc component. What fraction of the total average power is carried by the dc component? What fraction
Repeat Problem 13–40 for the periodic waveform in Figure P13–41. vs(t) VA 0 To 2T0 FIGURE P13-41
The input to the circuit in Figure P13–45 is the voltage vsðtÞ ¼ 15 cosð2p2000tÞ þ 5 cos ð2p6000tÞ V Calculate the average power delivered to the 50-V load resistor. vs(t) 1 + 50 w 0.75 F 50 vo(t) FIGURE P13-45
The input in Figure P13–67 is v1(t)¼5ejtjV. Use Fourier transforms to find v2(t). V1 + 10 ww + 0.1 F V2 FIGURE P13-67
The input in Figure P13–69 is v1(t) ¼ 2 sgn(t) V. Use Fourier transforms to find v2(t). V1 1 H 1 uF m 1 + V2 + | FIGURE P13-69
The input in Figure P13–70 is v1(t) ¼ 10e4tu(t) V. Use Fourier transforms to find v2(t). V1 + 10 www 1 F 10 V2 FIGURE P13-70
The input in Figure P13–72 is v1(t) ¼ 5u(t) V. Use Fourier transforms to find v2(t). + V1 1 0.1 ww 15 ww FIGURE P13-72 + V2
The frequency response of a linear system is shown in Figure P13–80. Find the system impulse response h(t). 10 H(@) 5 -2 -B 0 + +2 FIGURE P13-80
The frequency response of a linear system is shown in Figure P13–81. Find the system impulse response h(t). - H(0) A 0 FIGURE P13-81 +B 3 00
The transfer function of an ideal bandpass filter is H(v)¼ 1 for 500 v 1000 rad/s. Use MATLAB to find the 1-V energy carried by the output signal when the input is x(t) ¼15 e1000tu(t). What percentage of the input signal energy.
Fourier Series from a Bode Plot The transfer function of a linear circuit has the straight-line gain and phase Bode plots in Figure P13–90. The first four terms in the Fourier series of a periodic input v1(t) to the circuit are v1ðtÞ ¼ 42 cosð100tÞ þ 14 cosð300tÞ þ 8:4 cosð500tÞþ 6
The current in a 100-Vresistor is i(t)¼4 u(t+2)þ 8 u(t) 4 u(t2). Find the total energy delivered to the resistor.
The impulse response of a filter is h(t) ¼ 100 e10tu(t).Find the l-V energy in the output signal when the input is x(t) ¼ d(t).
The impulse response of a filter is h(t) ¼ 100 e400tu(t).Find the l-V energy in the output signal when the input is x(t) ¼ 5 e100tu(t).
The transfer function of an ideal high-pass filter isH(v)¼1 for jvj 2 krad/s. The filter input signal is x(t) ¼ 10 e2000t u(t). Find the 1-V energy carried by the input signal and the percentage of the input energy that appears in the output.
Find the l-V energy carried by the signal Then find the percentage of the l-V energy carried in the frequency band jvj a.
Compute the 1-V energy carried by the signal f(t) ¼5e2.5tu(t).
Findthe 1-VenergycarriedbythesignalF(v)¼8/(v2þ16).
The impulse response of a linear system is h(t) ¼Aeatu(t) þ Aea(t)u(t), with a>0. Let A ¼ 10 and a ¼ 2.5 and use MATLAB to plot jH(v)j. On the same axes, plot jH(v)j for A ¼ 10 and a ¼ 5. Describe the system frequency response and the influence of the parameter a.
Spectrum of a Periodic Impulse Train A periodic impulse train can be written as xðtÞ ¼ T0 X1 n¼1 dðt nT0ÞFind the Fourier coefficients of x(t). Plot the amplitude spectrum and comment on the frequencies contained in the impulse train.
The impulse response of a linear system is h(t) ¼ A[d(t)sin(bt)/pt]. Let A ¼ 5 and b ¼ 2 and use MATLAB to plot jH(v)j. On the same axes, plot jH(v)j for A ¼ 5 and b ¼ 4.Describe the system frequency response and the influence of the parameterb. You may need to use the following equality in
The impulse response of a linear systemis h(t) ¼ A[d(t) aeatu(t)], with a>0. LetA¼ 5 and a ¼ 2 and useMATLAB to plot jH(v)j. On the same axes, plot jH(v)j for A ¼ 5 and a ¼ 4. Describe the system frequency response and the influence of the parameter a.
Power Supply Filter Design The input to a power supply filter is a full-wave rectified sine wave with f0 ¼ 60 Hz. The filter is a first-order low pass with unity dc gain. Select the cutoff frequency of the filter so that all of the ac components in the filter output are all less than 1% of the dc
The impulse response of a linear system is h(t) ¼ 2d(t) 4etu(t). UseMATLAB and Fourier transform techniques to find the output for an input x(t) ¼ sgn(t).
The impulse response of a linear system is h(t) ¼ e2jtj.Find the output for an input x(t) ¼ u(t).
Spectrum Analyzer Calibration A certain spectrum analyzer measures the average power delivered to a calibrated resistor by the individual harmonics of periodic waveforms. The calibration of the analyzer has been checked by applying a 1-MHz square wave and the following results reported f ðMHzÞ 1
The impulse response of a linear system is h(t)¼e2tu(t).Find the output for an input x(t) ¼ u(t).
The input in Figure P13–72 is v1(t) ¼ 10e4jtj V. Use Fourier transforms to find v2(t).
The input in Figure P13–70 is v1(t) ¼ 10 sgn(t) V. Use Fourier transforms to find v2(t).
The input in Figure P13–67 is v1(t) ¼ 10 sgn(t) V. Use Fourier transforms to find v2(t).
Use the frequency shifting property to show that FfcosðbtÞuðtÞg ¼ jv b2 v2þ p 2½dðv bÞ þ dðv þ bÞ
Use the frequency shifting property to prove the modulation property Fff ðtÞsinðbtÞg ¼ Fðv bÞ2j Fðv þ bÞ2j
Use the reversal property to show that F Aeajtj sgnðtÞn o¼2Ajv v2 þ a2
Given that the Fourier transform of f(t) is FðvÞ ¼ 1600ðjv þ 20Þðjv þ 40Þuse the integration property to find the waveform gðtÞ ¼Z t1 f ðxÞdx
Use the time-shifting property to find the inverse transforms of the following functions:(a) F1ðvÞ ¼ ½6pdðvÞ j3=vej3v(b) F2ðvÞ ¼ 9ej4v=ðjv þ 5Þ(c) F3ðvÞ ¼ 2 cosð5vÞ=jv
Use the duality property to find the inverse transforms of the following functions:(a) F1ðvÞ ¼ 3 cosð10vÞ(b) F2ðvÞ ¼ 6uðvÞ 3(c) F3ðvÞ ¼ 6ej3vj
Find the inverse transforms of the following functions:(a) F1ðvÞ ¼ 4pdðvÞ þ 4pdðv 2Þ þ 4pdðv 4Þ(b) F2ðvÞ ¼ 4pdðvÞ j2=v þ 4pdðv 2Þ(c) F3ðvÞ ¼ 2pdðvÞ j2=v
Find the Fourier transforms of the following waveforms:(a) f 1ðtÞ ¼ 5 cos½2pðt 3Þ(b) f 2ðtÞ ¼ 2ej4tsgnðtÞ
Find the Fourier transforms of the following waveforms:(a) f 1ðtÞ ¼ 2 sinðtÞ 2 cosðtÞ(b) f 2ðtÞ ¼ ð2=tÞ sinðtÞ 2 cosðtÞ
Find the Fourier transforms of the following waveforms:(a) f 1ðtÞ ¼ 2e2t uðtÞ 2 sgnðtÞ(b) f 2ðtÞ ¼ 2e2t uðtÞ 2uðtÞ
Find the Fourier transforms of the following waveforms:(a) f 1ðtÞ ¼ 2uðtÞ 2(b) f 2ðtÞ ¼ 2 sgnðtÞ 2uðtÞ(c) f 3ðtÞ ¼ sgnðtÞ 1
Find the inverse transforms of the following functions:(a) F1ðvÞ ¼ 1600 jvðjv þ 20Þðjv þ 40Þ(b) F2ðvÞ ¼v2 jvðjv þ 20Þðjv þ 40Þ
Find the inverse transforms of the following functions:(a) F1ðvÞ ¼ 5000ðjv þ 50Þðjv þ 100Þ(b) F2ðvÞ ¼ 40jvðjv þ 20Þðjv þ 40Þ
Virtual Keyboard Design Electronic keyboards are designed using the following equation that assigns particular frequencies to each of the 88 keys in a standard piano keyboard, f ðnÞ ¼ 440 ffiffiffi 2 12 pn49 Hz where n is the key number. There is a need for an amplifier that can pass middle C,
Use MATLAB and the inversion integral to find the inverse transform of the following function:FðvÞ ¼ sinðpv=4Þ½uðv þ 4Þ uðv 4Þ
Use MATLAB and the inversion integral to find the inverse transform of the following function:FðvÞ ¼ jp8½2uðv þ 2Þ 4uðvÞ þ 2uðv 2Þ
Use the inversion integral to find the inverse transform of the following function:FðvÞ ¼ 10p½uðv þ 1Þ uðv 1Þ
Use MATLAB and the defining integral to find the Fourier transform of the following waveform:f ðtÞ ¼ Acosðpt=4Þ½uðtÞ uðt 4Þ
Use the defining integral to find the Fourier transform of f(t) ¼ At[u(t) u(t 1)].
Use the defining integral to find the Fourier transform of f(t) ¼ A[u(t þ T0) u(tT0)].
Use the defining integral to find the Fourier transform of f(t) ¼ A[u(t) u(t 3)].
Find an expression for the average power delivered to a resistor R by a triangular wave voltage with amplitude VA and period T0. How many components of the Fourier series are required to account for 98% of the average power carried by the waveform?
Estimate the rms value of the periodic voltage vðtÞ ¼ VA h2 cosðv0tÞ þ 1 3cosð3v0tÞ 1 5cosð5v0tÞþ 1 7cosð7v0tÞ . . .i V
Repeat Problem 13–42 for a first-order high-pass filter with the same cutoff frequency and passband gain.
A first-order low-pass filter has a cutoff frequency of 600 rad/s and a passband gain of 20 dB. The input to the filter is v(t) ¼ 10 cos 300t þ 6 cos 1200t V. Find the rms value of the steady-state output.
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.
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 13–36.
Find the rms value of a triangular 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 13–36.
The voltage across a 50-V resistor is vðtÞ ¼ 60 þ 24 sinð200ptÞ 8 sinð600ptÞ þ 4:8 sinð1000ptÞV(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 waveform
The current through a 500-V resistor is iðtÞ ¼ 50 þ 36cosð120pt 30Þ 12cosð360pt þ 45ÞmA Find the rms value of the current and the average power delivered to the resistor.
A sawtooth wave with VA ¼ 10Vand T0 ¼ 20p ms drives a circuit whose transfer function is TðsÞ ¼ 100sðs þ 50Þ2 þ 4002(a) Find the amplitude of the first four nonzero terms in the Fourier series of the steady-state output. What term in the Fourier series tends to dominate the response?
An ideal time delay is a signal processor whose output is vO(t) ¼ vIN(tTD). Write an expression for vO(t) for TD ¼0.5 ms and vINðtÞ ¼ 10 þ 10 cosð2p500tÞ þ 2:5 cosð2p1000tÞþ 0:625 cosð2p4000tÞ V Discuss the spectral changes caused by the time delay.
The voltage across a 1000-pF capacitor is a triangular wave with VA ¼ 150Vand f0 ¼ 1 kHz. Construct plots of the amplitude spectra of the capacitor voltage and current.Discuss any differences in spectral content.
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 ¼ 8V and T0 ¼ 20p ms.Design your filter with a Q of 20.(b) Compare the magnitudes of the fundamental and of the fifth harmonic with
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 ¼ 8V and T0 ¼ 20p ms.Design your filter with a Q of 10.(b) Compare the magnitudes of the fundamental and of the fifth harmonic with
Repeat Problem 13–27 for T(s) ¼ 400/(s þ 400).
A sawtooth wave with VA ¼ 12Vand T0 ¼ 10p ms drives a circuit with a transfer function T(s) ¼ s/(s þ 400). 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 comment
(a) Design a passive low-pass RC filter to block the fundamental and all harmonics from a full-wave rectified sinusoidal waveform. Use the results of text Figure 13–4 to find the Fourier coefficients of the input for VA¼ 170V, T0 ¼16.6 ms.(b) Find the first four nonzero terms in the Fourier
Fourier Series and Fourier Transforms.Given a rectangular pulse as shown in Figure 13–4, with amplitude A, width T, and period T0, we can compute and plot the coefficients in the corresponding Fourier series. If we allow T0 to increase to infinity, the waveform is a single pulse and the Fourier
(a) Design a low-pass OP AMP circuit to pass only the fundamental and the next nonzero harmonic of a 20pms square wave. The gain of the OP AMP should be þ10.(b) Find the first four nonzero terms in the Fourier series of the output of your filter.
The periodic sawtooth wave in Figure P13–22 above drives the OP AMP circuit shown in the figure.(a) Use the results in text Figure 13–4 to find the Fourier coefficients of the input for VA ¼ 8V and T0 ¼ 20p ms(b) Find the first four nonzero terms in the Fourier series of i(t) for R1 ¼ 100
The equation for a full-wave rectified cosine is v(t) ¼ VA jcos(2pt/T0)j V.(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).
The first five terms in the Fourier series of a periodic waveform are vðtÞ ¼ 5 þ 25 hp 4cosð500tÞ 1 3cosð1000tÞ 1 15 cosð1500tÞ 1 35 cosð2500tÞi V(a) Find the period and fundamental frequency in rad/s and Hz. Identify the harmonics present in the first five terms.(b) Use MATLAB to
The first four terms in the Fourier series of a periodic waveform are vðtÞ ¼ 15 hsinð150ptÞ 1 9sinð450ptÞ þ 1 25 sinð750ptÞ 1 49 sinð1050ptÞi V(a) Findtheperiodandfundamentalfrequencyinrad/sandHz.Identify the harmonics present in the first four terms.(b) Identify the symmetry
The equation for a periodic waveform is vðtÞ ¼ VA½sinð4p=T0Þ þ jsinð4pt=T0Þj(a) Sketch the first two cycles of the waveform and identify a related signal in Figure 13–4.(b) Use the Fourier series of the related signal to find the Fourier coefficients of v(t).(c) Use MATLAB to sketch an
The equation for the first cycle (0 t T0) of a periodic pulse train is vðtÞ ¼ VA½uð3t T0Þ uð3t 2T0Þ V(a) Sketch the first two cycles of the waveform and identify a related signal in Figure 13–4.(b) Use the Fourier series of the related signal to find the Fourier coefficients of
A sawtooth wave has peak-to-peak amplitude of 24V and a fundamental frequency of 2 kHz. 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 original signal
The waveform f(t) is a 2-kHz triangular wave with a peak-to-peak amplitude of 8V. Use the results in Figure 13–4 to write an expression for the first four nonzero terms in the Fourier series of g(t) ¼ 4 þ f(t) and plot its amplitude spectrum. Use MATLAB to plot two periods of g(t) and an
A half-wave rectified sine wave has an amplitude of 339Vand a fundamental frequency of 50 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.
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