## Bonnie S. Heck, Edward W. Kamen

## Chapter 8

## Analysis of Continuous-Time Systems by Use of the Transfer Function - all with Video Answers

## Educators

Chapter Questions

For the following linear time-invariant continuous-time systems, determine if the system is stable, marginally stable, or unstable:

(a) $H(s)=\frac{s-4}{s^2+7 s+3}$

(b) $H(s)=\frac{s+3}{s^2+3}$

(c) $H(s)=\frac{2 s+3}{s^2+2 s-12}$

(d) $H(s)=\frac{3 s^3-2 s+6}{s^3+s^2+s+1}$

(e) $H(s)=\frac{4 s+8}{\left(s^2+4 s+13\right)(s+4)}$

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Consider the field-controlled dc motor given by the input/output differential equation

$$

L_f I \frac{d^3 y(t)}{d t^3}+\left(L_f k_d+R_f I\right) \frac{d^2 y(t)}{d t^2}+R_f k_d \frac{d y(t)}{d t}=k x(t)

$$

Assume that all the parameters $I, L_f, k_d, R_f$, and $k$ are strictly positive ( $>0$ ). Determine if the motor is stable, marginally stable, or unstable.

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Consider the model for the ingestion and metabolism of a drug defined in Problem 6.19. Assuming that $k_1>0$ and $k_2>0$, determine if the system is stable, marginally stable, or unstable. What does your answer imply regarding the behavior of the system? Explain.

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Determine if the mass-spring system in Problem 2.23 is stable, marginally stable, or unstable. Assume that $k_1, k_2$, and $k_3$ are strictly positive $(>0)$.

Samuel Hannah

Numerade Educator

Consider the single-eye system studied in Problem 2.35. Assuming that $T_e>0$, determine if the system is BIBO stable.

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For each of the linear time-invariant continuous-time systems with impulse response $h(t)$ given as follows, determine if the system is BIBO stable.

(a) $h(t)=\left[2 t^3-2 t^2+3 t-2\right][u(t)-u(t-10)]$

(b) $h(t)=\frac{1}{t}$ for $t \geq 1, h(t)=0$ for all $t<1$

(c) $h(t)=\sin 2 t$ for $t \geq 0$

(d) $h(t)=e^{-t} \sin 2 t$ for $t \geq 0$

(e) $h(t)=e^{-t^2}$ for $t \geq 0$

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Using the Routh-Hurwitz test, determine all values of the parameter $k$ for which the following systems are stable:

(a) $H(s)=\frac{s^2+60 s+800}{s^3+30 s^2+(k+200) s+40 k}$

(b) $H(s)=\frac{2 s^3-3 s+4}{s^4+s^3+k s^2+2 s+3}$

(c) $H(s)=\frac{s^2+3 s-2}{s^3+s^2+(k+3) s+3 k-5}$

(d) $H(s)=\frac{s^4-3 s^2+4 s+6}{s^5+10 s^4+(9+k) s^3+(90+2 k) s^2+12 k s+10 k}$

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Suppose that a system has the following transfer function:

$$

H(s)=\frac{8}{s+4}

$$

(a) Compute the system response to the inputs (i)-(iv). Identify the steady-state solution and the transient solution.

(i) $x(t)=u(t)$

(ii) $x(t)=t u(t)$

(iii) $x(t)=2(\sin 2 t) u(t)$

(iv) $x(t)=2(\sin 10 t) u(t)$

(b) Use MATLAB to compute the response numerically from $x(t)$ and $H(s)$. Plot the responses, and compare them with the responses obtained analytically in part (a).

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Consider three systems which have the following transfer functions:

(i) $H(s)=\frac{32}{s^2+4 s+16}$

(ii) $H(s)=\frac{32}{s^2+8 s+16}$

(iii) $H(s)=\frac{32}{s^2+10 s+16}$

For each system, do the following:

(a) Determine if the system is critically damped, underdamped, or overdamped.

(b) Calculate the step response of the system.

(c) Use MATLAB to compute the step response numerically. Plot the response, and compare it with the plot of the response obtained analytically in part (b).

Eric Mockensturm

Numerade Educator

A first-order system has the step response shown in Figure P8.10. Determine the transfer function.

Khoobchandra Agrawal

Numerade Educator

A second-order system has the step response shown in Figure P8.11. Determine the transfer function.

Khoobchandra Agrawal

Numerade Educator

Consider the mass-spring-damper system with the input/output differential equation

$$

M \frac{d^2 y(t)}{d t^2}+D \frac{d y(t)}{d t}+K y(t)=x(t)

$$

where $M$ is the mass, $D$ is the damping constant, $K$ is the stiffness constant, $x(t)$ is the force applied to the mass, and $y(t)$ is the displacement of the mass relative to the equilibrium position.

Figure Can't Copy

(a) Determine the pole locations for the cases (i) $M=1, D=50.4$, and $K=3969$; and (ii) $M=2, D=50.4$, and $K=3969$. Show the location of the poles on a pole-zero plot. Compute the natural frequency and the time constant for each of the cases. Which has the higher frequency of response? For which case does the transient response decay faster?

(b) Use MATLAB to compute the impulse response of the system for the two cases, and compare your results with the predictions made in part (a).

(c) Repeat parts (a) and (b) for the cases (i) $M=1, D=50.4$, and $K=15,876$ and (ii) $M=2, D=50.4$, and $K=15,876$.

Lainey Roebuck

Numerade Educator

Again consider the mass-spring-damper system in Problem 8.12. Let $M=1, D=50.4$, and $K=3969$.

(a) Compute the response to a unit step in the force.

(b) Compute the steady-state response to an input of $x(t)=10 \cos (20 \pi t) u(t)$.

(c) Compute the steady-state response to an input of $x(t)=10 \cos (2 \pi t) u(t)$.

(d) Use MATLAB to simulate the system with the inputs given in parts (a)-(c). Verify that your answers in parts (a)-(c) are correct by plotting them along with the corresponding results obtained from the simulation.

(e) Use the Mass-Spring-Damper demo available on the textbook Web page to simulate the system with the inputs given in parts (a)-(c), and compare the responses with those plotted in part (d). Change the damping parameter to $D=127$, and use the applet to simulate the step response. Sketch the response.

Lainey Roebuck

Numerade Educator

Consider the two systems given by the following transfer functions:

(i) $H(s)=\frac{242.5(s+8)}{(s+2)\left[(s+4)^2+81\right](s+10)}$

(ii) $H(s)=\frac{115.5(s+8)(s+2.1)}{(s+2)\left[(s+4)^2+81\right](s+10)}$

(a) Identify the poles and zeros of the system.

(b) Without computing the actual response, give the general form of the step response.

(c) Determine the steady-state value for the step response.

(d) Determine the dominant pole(s).

(e) Use MATLAB to compute and plot the step response of the system. Compare the plot with the answers expected in parts (b) to (d).

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For each of the circuits in Figure P8.15, compute the steady-state response $y_{s s}(t)$ resulting from the following inputs with zero initial conditions:

(a) $x(t)=u(t)$

(b) $x(t)=(5 \cos 2 t) u(t)$

(c) $x(t)=\left[2 \cos \left(3 t+45^{\circ}\right)\right] u(t)$

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Consider the mass-spring system in Problem 2.23. Assume that $M_1=1, M_2=10$, and $k_1=k_2=k_3=0.1$. Compute the steady-state response $y_{s s}(t)$ resulting from the following inputs with zero initial conditions:

(a) $x(t)=u(t)$

(b) $x(t)=(10 \cos t) u(t)$

(c) $x(t)=\left[\cos \left(5 t-30^{\circ}\right)\right] u(t)$

Lainey Roebuck

Numerade Educator

A linear time-invariant continuous-time system has transfer function $H(s)=2 /(s+1)$. Compute the transient response $y_{t r}(t)$ resulting from the input $x(t)=3 \cos 2 \mathrm{t}-4 \sin t, t \geq 0$, with zero initial conditions

Figure Can't Copy

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A linear time-invariant continuous-time system has transfer function

$$

H(s)=\frac{s^2+16}{s^2+7 s+12}

$$

Compute the steady-state and transient responses resulting from the input $x(t)=$ $2 \cos 4 t, t \geq 0$, with zero initial conditions.

Arpit Gupta

Numerade Educator

A linear time-invariant continuous-time system has transfer function

$$

H(s)=\frac{s^2+1}{(s+1)\left(s^2+2 s+17\right)}

$$

Compute both the steady-state response $y_{\mathrm{ss}}(t)$ and the transient response $y_{\mathrm{tr}}(t)$ when the input $x(t)$ is

(a) $x(t)=u(t)$, with zero initial conditions.

(b) $x(t)=\cos t, t \geq 0$, with zero initial conditions.

(c) $x(t)=\cos 4 t, t \geq 0$, with zero initial conditions.

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A linear time-invariant continuous-time system has transfer function

$$

H(s)=\frac{s+2}{(s+1)^2+4}

$$

The input $x(t)=C \cos \left(\omega_0 t+\theta\right)$ is applied to the system for $t \geq 0$ with zero initial conditions. The resulting steady-state response $y_{s s}(t)$ is

$$

y_{\mathrm{sx}}(t)=6 \cos \left(t+45^{\circ}\right), t \geq 0

$$

(a) Compute $C, \omega_0$, and $\theta$.

(b) Compute the Laplace transform $Y_{\mathrm{tr}}(s)$ of the transient response $y_{\mathrm{tr}}(t)$ resulting from this input.

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A linear time-invariant continuous-time system has transfer function $H(s)$ with $H(0)=3$. The transient response $y_{\mathrm{tr}}(t)$ resulting from the step-function input $x(t)=u(t)$ with zero initial conditions at time $t=0$ has been determined to be

$$

y_{\mathrm{tr}}(t)=-2 e^{-t}+4 e^{-3 t}, t \geq 0

$$

(a) Compute the system's transfer function $H(s)$.

(b) Compute the steady-state response $y_{\mathrm{ss}}(t)$ when the system's input $x(t)$ is equal to $2 \cos \left(3 t+60^{\circ}\right), t \geq 0$, with zero initial conditions.

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A linear time-invariant continuous-time system has transfer function $H(s)$. The input $x(t)=3(\cos t+2)+\cos \left(2 t-30^{\circ}\right), t \geq 0$, produces the steady-state response $y_{\mathrm{ss}}(t)=$ $6 \cos \left(t-45^{\circ}\right)+8 \cos \left(2 t-90^{\circ}\right), t \geq 0$, with zero initial conditions. Compute $H(1)$ and $H(2)$.

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Consider a second-order system in the form

$$

H(s)=\frac{\omega_n^2}{s^2+2 \zeta \omega_n s+\omega_n^2}

$$

Let $s \rightarrow$ j $\omega$ to obtain $H(\omega)$, and suppose that $0<\zeta<1$. Without factoring the denominator, find an expression for $|H(\omega)|$. To determine if a peak exists in $|H(\omega)|$, take the derivative of $|H(\omega)|$ with respect to $\omega$. Show that a peak exists for $\omega \neq 0$ only if $\zeta \leq 1 / \sqrt{2}$. Determine the height of the peak. What happens to the peak as $\zeta \rightarrow 0$ ?

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Sketch the magnitude and phase plots for the systems that follow. In each case, compute $|H(\omega)|$ and $\angle H(\omega)$ for $\omega=0, \omega=3-\mathrm{dB}$ points, $\omega=\omega_p$ and $\omega \rightarrow \infty$. Here, $\omega_p$ is the value of $\omega$ for which $|H(\omega)|$ is maximum. Verify your calculations by plotting the frequency response, using MATLAB.

(a) $H(s)=\frac{10}{s+5}$

(b) $H(s)=\frac{5(s+1)}{s+5}$

(c) $H(s)=\frac{s+10}{s+5}$

(d) $H(s)=\frac{4}{(s+2)^2}$

(e) $H(s)=\frac{4 s}{(s+2)^2}$

(f) $H(s)=\frac{s^2+2}{(s+2)^2}$

(g) $H(s)=\frac{4}{s^2+\sqrt{2}(2 s)+4}$

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Sketch the magnitude and phase plots for the circuits shown in Figure P8.25. In each case, compute $|H(\omega)|$ and $\angle H(\omega)$ for $\omega=0, \omega=3-\mathrm{dB}$ points, and $\omega \rightarrow \infty$.

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Repeat Problem 8.25 for the circuits in Figure P8.15.

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Consider the $R L C$ circuit shown in Figure P8.27. Choose values for $R$ and $L$ such that the damping ratio $\zeta=1$ and the circuit is a lowpass filter with approximate $3-\mathrm{dB}$ bandwidth equal to $20 \mathrm{rad} / \mathrm{sec}$; that is, $|H(\omega)| \geq(0.707)|H(0)|$ for $0 \leq \omega \leq 20$.

Figure Can't Copy

Khoobchandra Agrawal

Numerade Educator

A linear time-invariant continuous-time system has transfer function $H(s)$. It is known that $H(0)=1$ and that $H(s)$ has two poles and no zeros. In addition, the magnitude function $|H(\omega)|$ is shown in Figure P8.28. Determine $H(s)$.

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A linear time-invariant continuous-time system has transfer function $H(s)=K /(s+a)$, where $K>0$ and $a>0$ are unknown. The steady-state response to $x(t)=4 \cos t, t \geq 0$, is $y_{\mathrm{sx}}(t)=20 \cos \left(t+\phi_1\right), t \geq 0$. The steady-state response to $x(t)=5 \cos 4 t, t \geq 0$, is $y_{\mathrm{ss}}(t)=10 \cos \left(4 t+\phi_2\right), t \geq 0$. Here $\phi_1, \phi_2$ are unmeasurable phase shifts. Find $K$ and $a$.

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Using MATLAB, determine the frequency response curves for the mass-spring system in Problem 2.23. Take $M_1=1, M_2=10$, and $k_1=k_2=k_3=0.1$.

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Draw the asymptotic Bode plots (both magnitude and phase plots) for the accompanying systems. Compare your plots with the actual Bode plots obtained from MATLAB.

(a) $H(s)=\frac{16}{(s+1)(s+8)}$

(b) $H(s)=\frac{10(s+4)}{(s+1)(s+10)}$

Figure Can't Copy

(c) $H(s)=\frac{10}{s(s+6)}$

(d) $H(s)=\frac{10}{(s+1)\left(s^2+4 s+16\right)}$

(e) $H(s)=\frac{10}{(s+1)\left(s^2+s+16\right)}$

(f) $H(s)=\frac{1000(s+1)}{(s+20)^2}$

Amit Srivastava

Numerade Educator

A linear time-invariant continuous-time system has a rational transfer function $H(s)$ with two poles and two zeros. The frequency function $H(\omega)$ of the system is given by

$$

H(\omega)=\frac{-\omega^2+j 3 \omega}{8+j 12 \omega-4 \omega^2}

$$

Determine $H(s)$.

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Consider the three-pole Butterworth filter given by the transfer function

$$

H(s)=\frac{\omega_c^3}{s^3+2 \omega_c s^2+2 \omega_c^2 s+\omega_c^3}

$$

(a) Derive an expression for the impulse response $h(t)$ in terms of the $3-\mathrm{dB}$ bandwidth $\omega_c$ Plot $h(t)$ when $\omega_c=1 \mathrm{rad} / \mathrm{sec}$.

(b) Compare your result in part (a) with the impulse response of an ideal lowpass filter with frequency function $H(\omega)=p_2(\omega)$. Discuss the similarities and differences in the two impulse responses.

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Again consider the three-pole Butterworth filter defined in Problem 8.33.

(a) For the case when $\omega_c=2 \pi$, compute the output response of the filter when the input is $x(t)=u(t)-u(t-1)$ with zero initial conditions.

(b) Repeat part (a) for the case when $\omega_c=4 \pi$.

(c) Using MATLAB, plot the responses found in parts (a) and (b).

(d) Are the results obtained in part (c) expected? Explain.

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For the three-pole Butterworth filter with $\omega_c=1$, compute the output response $y(t)$ when the input $x(t)$ is

(a) $x(t)=1,-\infty<t<\infty$

(b) $x(t)=2 \cos t,-\infty<t<\infty$

(c) $x(t)=\cos \left(10 t+30^{\circ}\right),-\infty<t<\infty$

(d) $x(t)=2(\cos t)(\sin t),-\infty<t<\infty$

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Again, consider the three-pole Butterworth filter with $\omega_c=1$. The output response resulting from the input $x(t)=\cos 0.5 t,-\infty<t<\infty$, can be expressed in the form $y(t)=B \cos \left[0.5\left(t-t_d\right)\right],-\infty<t<\infty$, where $t_d$ is the time delay through the filter. The response resulting from the input $x(t)=\cos t,-\infty<t<\infty$, can be expressed in the form $y(t)=C \cos \left(t-t_d+\phi\right),-\infty<t<\infty$, where $\phi$ is the phase distortion resulting from the nonlinear phase characteristic of the filter. Compute $t_d$ and $\phi$.

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Repeat Problem 8.34 for the three-pole Chebyshev filter with transfer function

$$

H(s)=\frac{0.251 \omega_c^3}{s^3+0.597 \omega_c s^2+0.928 \omega_c^2 s+0.251 \omega_c^3}

$$

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Repeat Problem 8.35 for the three-pole Chebyshev filter with $\omega_c=1$.

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The objective of this problem is to design both a highpass and a bandpass filter, starting from the two-pole Butterworth filter with transfer function

$$

H(s)=\frac{\omega_c^2}{s^2+\sqrt{2} \omega_c s+\omega_c^2}

$$

(a) Design the highpass filter so that the $3-\mathrm{dB}$ bandwidth runs from $\omega=10$ to $\omega=\infty$.

(b) Design the bandpass filter so that the 3-dB bandwidth runs from $\omega=10$ to $\omega=20$.

(c) Using MATLAB, determine the frequency response curves of the filters constructed in parts (a) and (b).

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Design a three-pole Butterworth stopband filter with a stopband from $\omega=10$ to $\omega=$ $15 \mathrm{rad} / \mathrm{sec}$.

(a) Plot the frequency response curves for the resulting filter.

(b) From the magnitude curve plotted in part (a), determine the expected amplitude of the steady-state responses $y_{s s}(t)$ to the following signals: (i) $x(t)=\sin 5 t$, (ii) $x(t)=\sin 12 t$, and (iii) $x(t)=\sin 5 t+\sin 12 t$.

(c) Verify your prediction in part (b) by using MATLAB to compute and plot the response of the system to the signals defined in part (b). You may use 1 sim and integrate long enough for the response to reach steady state, or use Simulink. [Note: When simulating a continuous-time system to find the response, computers approximate the system as being discrete time. Therefore, when defining the signals $x(t)$ for a time vector $t=0: T: t f$, make sure that the time increment $T$ for which $x(t)$ is defined satisfies the Nyquist sampling theorem; that is, $2 \pi / T$ is at least twice the highest frequency in $x(t)$. See the comments in Problem 1.2 for further information.]

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Design a three-pole Chebyshev bandpass filter with a passband from $\omega=10$ to $\omega=$ $15 \mathrm{rad} / \mathrm{sec}$. Allow a 3-dB ripple in the passband.

(a) Plot the frequency response curves for the resulting filter.

(b) From the magnitude curve plotted in part (a), determine the expected amplitude of the steady-state responses $y_{s s}(t)$ to the following signals:(i) $x(t)=\sin 5 t$, (ii) $x(t)=\sin 12 t$, and (iii) $x(t)=\sin 5 t+\sin 12 t$.

(c) Verify your prediction in part (b) by using MATLAB to compute and plot the response of the system to the signals defined in part (b). (Consider the comment in Problem 8.41 regarding the selection of the time increment when using MATLAB.)

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Design a lowpass Butterworth filter with a bandwidth of $10 \mathrm{rad} / \mathrm{sec}$. Select an appropriate number of poles so that a $25-\mathrm{rad} / \mathrm{sec}$ sinusoidal signal is attenuated to a level that is no more than $5 \%$ of its input amplitude. Use MATLAB to compute and plot the response of the system to the following signals. (Consider the comments in Problem 8.41 regarding the selection of the time increment when using MATLAB.)

(a) $x(t)=\sin 5 t$

(b) $x(t)=\sin 25 t$

(c) $x(t)=\sin 5 t+\sin 25 t$

(d) $x(t)=w(t)$ where $w(t)$ is a random signal whose values are uniformly distributed between 0 and 1. (Use $x=$ rand $(201,1)$ to generate the signal for the time vector $t$ $=0: .05: 10$.) Plot the random input $x(t)$, and compare it with the system response.

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Design a highpass type 1 Chebyshev filter with a bandwidth of $10 \mathrm{rad} / \mathrm{sec}$. Select an appropriate number of poles so that a $5-\mathrm{rad} / \mathrm{sec}$ sinusoidal signal is attenuated to a level that is no more than $10 \%$ of its input amplitude and there is at most a $3-\mathrm{dB}$ ripple in the passband. Use MATLAB to compute and plot the response of the system to the signals that follow. (Consider the comment in Problem 8.41 regarding the selection of the time increment when using MATLAB.)

(a) $x(t)=\sin 5 t$

(b) $x(t)=\sin 25 t$

(c) $x(t)=\sin 5 t+\sin 25 t$

(d) $x(t)=w(t)$, where $w(t)$ is a random signal whose values are uniformly distributed between 0 and 1. (Use $x=$ rand (201,1) to generate the signal for the time vector $t=0: .05: 10$.) Plot the random input $x(t)$, and compare it with the system response.

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Repeat Problem 8.39 for the two-pole Chebyshev filter with transfer function

$$

H(s)=\frac{0.50 \omega_c^2}{s^2+0.645 \omega_c s+0.708 \omega_c^2}

$$

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