Problems

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Problems

• Sections 16.2 and 16.3 Circuit Element Models and Circuit Analysis
• 16.1 The current in an RLC circuit is described by

$$\frac{d^2i}{dt^2} + 10\frac{di}{dt} + 25i = 0$$

If

$$i(0) = 7$$

A and $di(0)/dt = 0$ , find $i(t)$ for $t > 0$ .

16.2 The differential equation that describes the voltage in an RLC network is

$$3\frac{d^2i}{dt^2} + 15\frac{di}{dt} + 12i = 0$$

Given that v(0) = 0, dv(0)/dt = 6 mA∕s, obtain i(t).

16.3 The natural response of an RLC circuit is described by the differential equation

$$\frac{d^2v}{dt^2} + 2\frac{dv}{dt} + v = 0$$

for which the initial conditions are v(0) = 350 V and dv(0)/dt = 0. Solve for v(t).

• 16.4 If R = 20 Ω, L = 0.6 H, what value of C will make an RLC series circuit:
  • (a) overdamped?
  • (b) critically damped?
  • (c) underdamped?

16.9 Which of the following equations is called the state equation?

(a)

$$\dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{z}$$

\n(b) $\mathbf{y} = \mathbf{C}\mathbf{x} + \mathbf{D}\mathbf{z}$
\n(c) $\mathbf{H}(s) = \mathbf{Y}(s)/\mathbf{Z}(s)$
\n(d) $\mathbf{H}(s) = \mathbf{C}(s\mathbf{I} - \mathbf{A})^{-1}$

16.10 A single-input, single-output system is described by the state model as:

B

$$\dot{x}_1 = 2x_1 - x_2 + 3z$$

\n

$$\dot{x}_2 = -4x_2 - z$$

\n

$$y = 3x_1 - 2x_2 + z$$

Which of the following matrices is incorrect?

(a)

$$A = \begin{bmatrix} 2 & -1 \\ 0 & -4 \end{bmatrix}$$

(b) $B = \begin{bmatrix} 3 \\ -1 \end{bmatrix}$
(c) $C = \begin{bmatrix} 3 & -2 \end{bmatrix}$ (d) $D = 0$

Answers: 16.1b, 16.2d, 16.3c, 16.4b, 16.5b, 16.6a, 16.7b, 16.8b, 16.9a, 16.10d.

16.5 The responses of a series RLC circuit are

$$v_c(t) = [30 - 10e^{-20t} + 30e^{-10t}]u(t)$$ $$i_L(t) = [40e^{-20t} - 60e^{-10t}]u(t)$$

mA

where vC(t) and iL(t) are the capacitor voltage and inductor current, respectively. Determine the values of R, L, and C.

16.6 Design a parallel RLC circuit that has the characteristic equation

$$s^2 + 100s + 10^6 = 0.$$

16.7 The step response of an RLC circuit is given by

$$\frac{d^2i}{dt^2} + 2\frac{di}{dt} + 5i = 30$$

Given that i(0) = 18 A and di(0)/dt = 36 A/s, solve for i(t).

16.8 A branch voltage in an RLC circuit is described by

$$\frac{d^2v}{dt^2} + 4\frac{dv}{dt} + 8v = 120$$

If the initial conditions are v(0) = 0 = dv(0)/dt, find v(t).

16.9 A series RLC circuit is described by

$$L\frac{d^2i(t)}{dt} + R\frac{di(t)}{dt} + \frac{i(t)}{C} = 15$$

Find the response when L = 0.5H, R = 4Ω, and C = 0.2 F. Let i(0⁻) = 7.5 A and [di(0⁻)/dt] = 0. 16.10 The step responses of a series RLC circuit are

$$V_c = 40 - 10e^{-2000t} - 10e^{-4000t} \text{ V}, t > 0$$ $$i_L(t) = 3e^{-2000t} + 6e^{-4000t} mA, t > 0$$

• (a) Find C.
  • (b) Determine what type of damping is exhibited by the circuit.
• 16.11 The step response of a parallel RLC circuit is

$$v = 10 + 20e^{-300t} (\cos 400t - 2 \sin 400t) \text{V}, t \ge 0$$

when the inductor is 50 mH. Find R and C.

16.12 Determine i(t) in the circuit of Fig. 16.35 by means of the Laplace transform.

Figure 16.35

For Prob. 16.12.

16.13 Using Fig. 16.36, design a problem to help other students better understand circuit analysis using Laplace transforms.

• 16.14 Find i(t) for t > 0 for the circuit in Fig. 16.37. Assume is(t) = [6(t) + 3δ(t)]mA.

16.15 For the circuit in Fig. 16.38, calculate the value of R needed to have a critically damped response.

Figure 16.38

16.16 The capacitor in the circuit of Fig. 16.39 is initially uncharged. Find v0(t) for t > 0.

Figure 16.39

16.17 If is(t) = 7.5e −2t u(t) A in the circuit shown in Fig. 16.40, find the value of io(t).

Figure 16.40

For Prob. 16.17.

16.18 Find v(t), t > 0 in the circuit of Fig. 16.41. Let vs = 12 V.

Figure 16.41

For Prob. 16.18.

16.19 The switch in Fig. 16.42 moves from position A to position B at t = 0 (please note that the switch must connect to point B before it breaks the connection at A, a make before break switch). Find v(t) for t > 0.

For Prob. 16.19.

16.20 Find i(t) for t > 0 in the circuit of Fig. 16.43.

16.21 In the circuit of Fig. 16.44, the switch moves (make before break switch) from position A to B at t = 0. Find v(t) for all t ≥ 0.

16.22 Find the voltage across the capacitor as a function of time for t > 0 for the circuit in Fig. 16.45. Assume steady-state conditions exist at t = 0⁻.

Figure 16.45 For Prob. 16.22.

16.23 Obtain v(t) for t > 0 in the circuit of Fig. 16.46.

Figure 16.46 For Prob. 16.23.

16.24 The switch in the circuit of Fig. 16.47 has been closed for a long time but is opened at t = 0. Determine i(t) for t > 0.

Figure 16.47 For Prob. 16.24.

16.25 Calculate v(t) for t > 0 in the circuit of Fig. 16.48.

Figure 16.48 For Prob. 16.25.

16.26 The switch in Fig. 16.49 moves from position A to position B at t = 0 (please note that the switch must connect to point B before it breaks the connection at A, a make before break switch). Determine i(t) for t > 0. Also assume that the initial voltage on the capacitor is zero.

Figure 16.49 For Prob. 16.26.

16.27 Find v(t) for t > 0 in the circuit in Fig. 16.50.

16.28 For the circuit in Fig. 16.51, find v(t) for t > 0.

Figure 16.55 For Prob. 16.32.

16.33 Using Fig. 16.56, design a problem to help other students understand how to use Thevenin’s theorem (in the s-domain) to aid in circuit analysis.

Figure 16.56 For Prob. 16.33.

16.30 Find vo(t), for all t > 0, in the circuit of Fig. 16.53.

16.31 Obtain v(t) and i(t) for t > 0 in the circuit in Fig. 16.54.

Figure 16.54 For Prob. 16.31.

16.34 Solve for the mesh currents in the circuit of Fig. 16.57. You may leave your results in the s-domain.

Figure 16.57 For Prob. 16.34.

16.35 Find vo(t) in the circuit of Fig. 16.58.

Figure 16.58 For Prob. 16.35.

16.36 Refer to the circuit in Fig. 16.59. Calculate i(t) for t > 0.

Figure 16.59

For Prob. 16.36.

16.37 Determine v for t > 0 in the circuit in Fig. 16.60.

For Prob. 16.37.

16.38 The switch in the circuit of Fig. 16.61 is moved from position a to b (a make before break switch) at t = 0. Determine i(t) for t > 0.

Figure 16.61

For Prob. 16.38.

16.39 For the network in Fig. 16.62, find i(t) for t > 0.

Figure 16.62 For Prob. 16.39.

16.40 In the circuit of Fig. 16.63, find v(t) and i(t) for t > 0. Assume v(0) = 0 V and i(0) = 1.25 A.

Figure 16.63 For Prob. 16.40.

16.41 Find the output voltage vo(t) in the circuit of Fig. 16.64.

For Prob. 16.41.

16.42 Given the circuit in Fig. 16.65, find i(t) and v(t) for t > 0.

Figure 16.65 For Prob. 16.42.

16.43 Determine i(t) for t > 0 in the circuit of Fig. 16.66.

Figure 16.67

16.45 Find v(t) for t > 0 in the circuit in Fig. 16.68.

For Prob. 16.45.

16.46 Determine io(t) in the circuit in Fig. 16.69.

Figure 16.69

For Prob. 16.46.

16.47 Determine io(t) in the network shown in Fig. 16.70.

Figure 16.70

For Prob. 16.47.

Figure 16.71 For Prob. 16.48.

16.49 Find i0(t) for t > 0 in the circuit in Fig. 16.72.

Figure 16.72

For Prob. 16.49.

16.50 For the circuit in Fig. 16.73, find v(t) for t > 0. Assume that i(0) = 2 A.

Figure 16.73

For Prob. 16.50.

Figure 16.74

For Prob. 16.51.

16.52 Given the circuit shown in Fig. 16.75, determine the values for i(t) and v(t) for all t > 0.

Figure 16.75

For Prob. 16.52.

16.53 In the circuit of Fig. 16.76, the switch has been in position 1 for a long time but moved to position 2 at t = 0. Find:

(a) v(0⁺), dv(0⁺)/dt (b) v(t) for t ≥ 0.

For Prob. 16.44.

Problems 751

Figure 16.76

For Prob. 16.53.

16.54 The switch in Fig. 16.77 has been in position 1 for t < 0. At t = 0, it is moved from position 1 to the top of the capacitor at t = 0. Please note that the switch is a make before break switch; it stays in contact with position 1 until it makes contact with the top of the capacitor and then breaks the contact at position 1. Determine v(t).

16.55 Obtain i1 and i2 for t > 0 in the circuit of Fig. 16.78.

Figure 16.78 For Prob. 16.55.

16.56 Calculate io(t) for t > 0 in the network of Fig. 16.79.

16.57 (a) Find the Laplace transform of the voltage shown in Fig. 16.80(a). (b) Using that value of vs(t) in the circuit shown in Fig. 16.80(b), find the value of vo(t).

Figure 16.80 For Prob. 16.57.

16.58 Using Fig. 16.81, design a problem to help other students better understand circuit analysis in the s-domain with circuits that have dependent sources.

Figure 16.81 For Prob. 16.58.

16.59 Find vo(t) in the circuit of Fig. 16.82 if vx(0) = 10 V and i(0) = 5 A.

Figure 16.82

For Prob. 16.59.

For Prob. 16.60.

16.60 Find the response v(t) for t > 0 in the circuit in Fig. 16.83. Let R = 8 Ω, L = 2 H, and C = 125 mF.

16.61 Find the voltage vo(t) in the circuit of Fig. 16.84 by means of the Laplace transform. *

Figure 16.84

For Prob. 16.61.

16.62 Using Fig. 16.85, design a problem to help other

students better understand solving for node voltages by working in the s-domain.

Figure 16.85

For Prob. 16.62.

16.63 Consider the parallel RLC circuit of Fig. 16.86. Find v(t) and i(t) given that v(0) = 7.5 V and i(0) = −3 A.

Figure 16.86

For Prob. 16.63.

16.64 The switch in Fig. 16.87 moves from position 1 to position 2 at t = 0. Find v(t), for all t > 0.

Figure 16.87

For Prob. 16.64.

16.65 For the RLC circuit shown in Fig. 16.88, find the complete response if v(0) = 100 V when the switch is closed.

Figure 16.88

For Prob. 16.65.

* An asterisk indicates a challenging problem. For Prob. 16.69.

16.66 For the op amp circuit in Fig. 16.89, find v0(t) for t > 0. Take vs = 12 e⁻5*^t u*(t) V.

Figure 16.89

For Prob. 16.66.

16.67 Given the op amp circuit in Fig. 16.90, if v1(0⁺) = 2 V and v2(0⁺) = 0 V, find v0 for t > 0. Let R = 100 kΩ and C = 1 μF.

Figure 16.90

For Prob. 16.67.

16.68 Obtain V0/Vs in the op amp circuit in Fig. 16.91.

Figure 16.91

For Prob. 16.68.

16.69 Find I1(s) and I2(s) in the circuit of Fig. 16.92.

Figure 16.93

For Prob. 16.70.

16.71 For the ideal transformer circuit in Fig. 16.94, determine io(t).

Figure 16.94

For Prob. 16.71.

Section 16.4 Transfer Functions

16.72 The transfer function of a system is

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

Find the output when the system has an input of 14e⁻t∕³ u(t).

• 16.73 When the input to a system is a unit step function, the response is 120 cos 2tu(t). Obtain the transfer function of the system.
• 16.74 Design a problem to help other students better
• understand how to find outputs when given a transfer function and an input.
• 16.75 When a unit step is applied to a system at t = 0, its response is

y(t) = [6 + 0.75 e⁻3*^t* − e⁻2*^t* (3 cos 4t + 4.5 sin 4t)]u(t)

What is the transfer function of the system?

16.76 For the circuit in Fig. 16.95, find H(s) = Vo(s)∕Vs(s). Assume zero initial conditions.

Figure 16.95 For Prob. 16.76.

16.77 Obtain the transfer function H(s) = Vo∕Vs for the circuit of Fig. 16.96.

Figure 16.96

For Prob. 16.77.

16.78 The transfer function of a certain circuit is

$$H(s) = \frac{10}{s+1} - \frac{6}{s+2} + \frac{12}{s+4}$$

Find the impulse response of the circuit.

16.79 For the circuit in Fig. 16.97, find:

Figure 16.97

For Prob. 16.79.

16.80 Refer to the network in Fig. 16.98. Find the following transfer functions:

(a)

$$H_1(s) = V_o(s)/V_s(s)$$

\n(b) $H_2(s) = V_o(s)/I_s(s)$
\n(c) $H_3(s) = I_o(s)/I_s(s)$
\n(d) $H_4(s) = I_o(s)/V_s(s)$

$$v_{s} \xrightarrow{I_{s}} 1 \Omega \xrightarrow{1 H} 1 \Omega \xrightarrow{I_{0}}$$

\n

$$1 \Gamma \xrightarrow{1 \Gamma} 1 \Gamma \xrightarrow{1 \Omega} \xleftarrow{1} v_{0}$$

Figure 16.98

For Prob. 16.80.

16.81 For the op-amp circuit in Fig. 16.99, find the transfer function, T(s) = I(s)/Vs(s). Assume all initial conditions are zero.

Figure 16.99 For Prob. 16.81.

16.82 Calculate the gain H(s) = Vo∕Vs in the op amp circuit of Fig. 16.100.

• 16.83 Refer to the RL circuit in Fig. 16.101. Find:
  • (a) the impulse response h(t) of the circuit.
  • (b) the unit step response of the circuit.

Figure 16.101

• For Prob. 16.83.
• 16.84 A parallel RL circuit has R = 4 Ω and L = 1 H. The input to the circuit is is(t) = 1.4e^−t u(t) A. Find the inductor current iL(t) for all t > 0 and assume that iL(0) = −1.4 A.
• 16.85 A circuit has a transfer function

s a transfer function

$$H(s) = \frac{3(s + 4)}{(s + 1)(s + 2)^2}$$

Find the impulse response.

Section 16.5 State Variables

• 16.86 Develop the state equations for Prob. 16.12.
• 16.87 Develop the state equations for the problem you designed in Prob. 16.13.
• 16.88 Develop the state equations for the circuit shown in Fig. 16.102.

Figure 16.102 For Prob. 16.88.

16.89 Develop the state equations for the circuit shown in Fig. 16.103.

16.90 Develop the state equations for the circuit shown in Fig. 16.104.

For Prob. 16.90.

16.91 Develop the state equations for the following differential equation.

$$\frac{d^2y(t)}{dt^2} + \frac{6\ dy(t)}{dt} + 7y(t) = z(t)$$

16.92 Develop the state equations for the following differential equation. *

$$\frac{d^2y(t)}{dt^2} + \frac{7\,dy(t)}{dt} + 9y(t) = \frac{dz(t)}{dt} + z(t)$$

16.93 Develop the state equations for the following differential equation. *

$$\frac{d^3y(t)}{dt^3} + \frac{6 \ d^2y(t)}{dt^2} + \frac{11 \ dy(t)}{dt} + 6y(t) = z(t)$$

16.94 Given the following state equation, solve for y(t): *

$$\dot{\mathbf{x}} = \begin{bmatrix} -4 & 4 \\ -2 & 0 \end{bmatrix} x + \begin{bmatrix} 0 \\ 2 \end{bmatrix} u(t)$$ $$\mathbf{y}(t) = \begin{bmatrix} 1 & 0 \end{bmatrix} x$$

16.95 Given the following state equation, solve for y1(t) and y2(t). *

$$\dot{\mathbf{x}} = \begin{bmatrix} -2 & -1 \\ 2 & -4 \end{bmatrix} x + \begin{bmatrix} 1 & 1 \\ 4 & 0 \end{bmatrix} \begin{bmatrix} u(t) \\ 2u(t) \end{bmatrix}$$ $$\mathbf{y} = \begin{bmatrix} -2 & -2 \\ 1 & 0 \end{bmatrix} x + \begin{bmatrix} 2 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} u(t) \\ 2u(t) \end{bmatrix}$$

Section 16.6 Applications

16.96 Show that the parallel RLC circuit shown in Fig. 16.105 is stable.

Figure 16.105

For Prob. 16.96.

16.97 A system is formed by cascading two systems as shown in Fig. 16.106. Given that the impulse responses of the systems are

$$h_1(t) = 21e^{-t}u(t), \qquad h_2(t) = e^{-4t}u(t)$$

• (a) Obtain the impulse response of the overall system.
• (b) Check if the overall system is stable.

Figure 16.106

For Prob. 16.97.

16.98 Determine whether the op amp circuit in Fig. 16.107 is stable.

Figure 16.107

For Prob. 16.98.

16.99 It is desired to realize the transfer function

$$\frac{V_2(s)}{V_1(s)} = \frac{2s}{s^2 + 2s + 6}$$

using the circuit in Fig. 16.108. Choose R = 1 kΩ and find L and C.

Figure 16.108 For Prob. 16.99.

16.100 Design an op amp circuit, using Fig. 16.109, that will realize the following transfer function:

$$\frac{V_o(s)}{V_i(s)} = -\frac{s + 1000}{2(s + 4000)}$$

Choose C1 = 10 μF; determine R1, R2, and C2.

Figure 16.109

For Prob. 16.100.

16.101 Realize the transfer function

$$\frac{V_o(s)}{V_s(s)} = -\frac{s}{s+10}$$

using the circuit in Fig. 16.110. Let Y1 = sC1, Y2 = 1∕R1, Y3 = sC2. Choose R1 = 1 kΩ and determine C1 and C2.

Figure 16.110

For Prob. 16.101.

16.102 Synthesize the transfer function

ize the transfer function
\n

$$\frac{V_o(s)}{V_{in}(s)} = \frac{10^6}{s^2 + 100s + 10^6}$$

using the topology of Fig. 16.111. Let Y1 = 1∕R1, Y2 = 1∕R2, Y3 = sC1, Y4 = sC2. Choose R1 = 1 kΩ and determine C1, C2, and R2.