Problems

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Problems

Section 7.2 The Source-Free RC Circuit

7.1 In the circuit shown in Fig. 7.81

$$v(t) = 56e^{-200t} \,\text{V}, \quad t > 0$$ $$i(t) = 8e^{-200t} \text{ mA}, \quad t > 0$$

• (a) Find the values of R and C.
• (b) Calculate the time constant τ.
• (c) Determine the time required for the voltage to decay half its initial value at t = 0.

7.2 Find the time constant for the RC circuit in Fig. 7.82.

Figure 7.82

For Prob. 7.2.

For Prob. 7.3.

7.3 Determine the time constant for the circuit in Fig. 7.83.

7.4 The switch in Fig. 7.84 has been in position A for a long time. Assume the switch moves instantaneously from A to B at t = 0. Find v for t > 0.

For Prob. 7.4.

7.5 Using Fig. 7.85, design a problem to help other students better understand source-free RC circuits.

Figure 7.85

For Prob. 7.5.

7.6 The switch in Fig. 7.86 has been closed for a long time, and it opens at t = 0. Find v(t) for t ≥ 0.

For Prob. 7.6.

7.7 Assuming that the switch in Fig. 7.87 has been in position A for a long time and is moved to position B at t = 0, Then at t = 1 second, the switch moves from B to C. Find vC(t) for t ≥ 0.

Figure 7.87 For Prob. 7.7.

7.8 For the circuit in Fig. 7.88, if

$$v = 10e^{-4t}$$

V and $i = 0.2e^{-4t}$ A, $t > 0$

• (a) Find R and C.
• (b) Determine the time constant.
• (c) Calculate the initial energy in the capacitor.
• (d) Obtain the time it takes to dissipate 50 percent of the initial energy.

Figure 7.88

For Prob. 7.8.

7.9 The switch in Fig. 7.89 opens at t = 0. Find vo for t > 0.

Figure 7.89

For Prob. 7.9.

7.10 For the circuit in Fig. 7.90, find vo(t) for t > 0. Determine the time necessary for the capacitor voltage to decay to one-third of its value at t = 0.

Figure 7.90

For Prob. 7.10.

Section 7.3 The Source-Free RL Circuit

7.11 For the circuit in Fig. 7.91, find io for t > 0.

Figure 7.91 For Prob. 7.11.

Figure 7.92

For Prob. 7.12.

7.13 In the circuit of Fig. 7.93,

$$v(t) = 80e^{-10^{3}t} \text{V}, \quad t > 0$$ $$i(t) = 5e^{-10^{3}t} \text{mA}, \quad t > 0$$

• (a) Find R, L, and τ.
• (b) Calculate the energy dissipated in the resistance for 0 < t < 0.5 ms.

Figure 7.93

For Prob. 7.13.

Figure 7.94

• For Prob. 7.14.
• 7.15 Find the time constant for each of the circuits in Fig. 7.95.

For Prob. 7.15.

7.16 Determine the time constant for each of the circuits in Fig. 7.96.

For Prob. 7.16.

7.17 Consider the circuit of Fig. 7.97. Find vo(t) if i(0) = 15 A and v(t) = 0.

Figure 7.97 For Prob. 7.17.

7.18 For the circuit in Fig. 7.98, determine vo(t) when i(0) = 5 A and v(t) = 0.

Figure 7.98

For Prob. 7.18.

7.19 In the circuit of Fig. 7.99, find i(t) for t > 0 if i(0) = 5 A.

Figure 7.99 For Prob. 7.19.

7.20 For the circuit in Fig. 7.100,

and

$$i = 30e^{-50t}A, \qquad t > 0$$

v = 90 e⁻50*^t*

V

• (a) Find L and R.
• (b) Determine the time constant.
• (c) Calculate the initial energy in the inductor.
• (d) What fraction of the initial energy is dissipated in 10 ms?

Figure 7.100

For Prob. 7.20.

7.21 In the circuit of Fig. 7.101, find the value of R for which the steady-state energy stored in the inductor will be 2 J.

Figure 7.101

For Prob. 7.21.

7.22 Find i(t) and v(t) for t > 0 in the circuit of Fig. 7.102 if i(0) = 10 A.

For Prob. 7.22.

7.23 Consider the circuit in Fig. 7.103. Given that vo(0) = 10 V, find vo and vx for t > 0.

Figure 7.103 For Prob. 7.23.

Section 7.4 Singularity Functions

7.24 Express the following signals in terms of singularity functions.

7.25 Design a problem to help other students better understand singularity functions.

7.26 Express the signals in Fig. 7.104 in terms of singularity functions.

Figure 7.104 For Prob. 7.26.

7.27 Express v(t) in Fig. 7.105 in terms of step functions.

Problems 303

‒1 0 3 21 15 10 5 ‒10 ‒5 t v(t)

Figure 7.105

For Prob. 7.27.

7.28 Sketch the waveform represented by

$$i(t) = [r(t) - r(t-1) - u(t-2) - r(t-2)$$

$$r(t-3) + u(t)(t-4)]$$

A

7.29 Sketch the following functions:

(a)

$$x(t) = 10e^{-t}u(t - 1)$$

,
\n(b) $y(t) = 10e^{-(t-1)}u(t)$ ,
\n(c) $z(t) = \cos 4t\delta(t - 1)$

7.30 Evaluate the following integrals involving the impulse functions:

(a)

$$\int_{-\infty}^{\infty} 4t^2 \delta(t-1) dt$$

(b)

$$\int_{-\infty}^{\infty} 4t^2 \cos 2\pi t \delta(t-0.5) dt$$

7.31 Evaluate the following integrals:

(a)

$$\int_{-\infty}^{\infty} e^{-4t^2} \delta(t - 2) dt$$

(b)

$$\int_{-\infty}^{\infty} [5\delta(t) + e^{-t} \delta(t) + \cos 2\pi t \delta(t)] dt$$

7.32 Evaluate the following integrals:

(a)

$$\int_{1}^{t} u(\lambda) d\lambda$$

\n(b)

$$\int_{0}^{4} r(t-1) dt$$

\n(c)

$$\int_{1}^{5} (t-6)^{2} \delta(t-2) dt$$

• 7.33 The voltage across a 10-mH inductor is 45δ(t − 2)mV. Find the inductor current, assuming that the inductor is initially uncharged.
• 7.34 Evaluate the following derivatives:

(a)

$$\frac{d}{dt}[u(t-1)u(t+1)]$$

\n(b)

$$\frac{d}{dt}[r(t-6)u(t-2)]$$

\n(c)

$$\frac{d}{dt}[\sin 4tu(t-3)]$$

7.35 Find the solution to the following differential equations:

(a)

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

, $v(0) = -1$ V
(b) $2\frac{di}{dt} - 3i = 0$ , $i(0) = 2$

7.36 Solve for v in the following differential equations, subject to the stated initial condition.

(a)

$$dv/dt + v = u(t)$$

, $v(0) = 0$
(b) $2 dv/dt - v = 3u(t)$ , $v(0) = -6$

7.37 A circuit is described by

$$4\frac{dv}{dt} + v = 10$$

• (a) What is the time constant of the circuit?
• (b) What is v(∞), the final value of v?
• (c) If v(0) = 2, find v(t) for t ≥ 0.
• 7.38 A circuit is described by

$$\frac{di}{dt} + 3i = 2u(t)$$

Find i(t) for t > 0 given that i(0) = 0.

Section 7.5 Step Response of an RC Circuit

7.39 Calculate the capacitor voltage for t < 0 and t > 0 for each of the circuits in Fig. 7.106.

Figure 7.106 For Prob. 7.39.

7.40 Find the capacitor voltage for t < 0 and t > 0 for each of the circuits in Fig. 7.107.

For Prob. 7.40.

7.41 Using Fig. 7.108, design a problem to help other students better understand the step response of an RC circuit.

Figure 7.108

• For Prob. 7.41.
  • 7.42 (a) If the switch in Fig. 7.109 has been open for a long time and is closed at t = 0, find vo(t).
    • (b) Suppose that the switch has been closed for a long time and is opened at t = 0. Find vo(t).

For Prob. 7.42.

7.43 Consider the circuit in Fig. 7.110. Find i(t) for t < 0 and t > 0.

For Prob. 7.43.

7.44 The switch in Fig. 7.111 has been in position a for a long time. At t = 0, it moves to position b. Calculate i(t) for all t > 0.

Figure 7.111 For Prob. 7.44.

7.45 Find vo in the circuit of Fig. 7.112 when vs = 30u(t) V. Assume that vo(0) = 5 V.

Figure 7.112 For Prob. 7.45.

7.46 For the circuit in Fig. 7.113, is(t) = 5u(t). Find v(t).

Figure 7.113 For Prob. 7.46.

7.47 Determine v(t) for t > 0 in the circuit of Fig. 7.114 if v(0) = 0.

For Prob. 7.47.

Problems 305

7.48 Find v(t) and i(t) in the circuit of Fig. 7.115.

Figure 7.115

For Prob. 7.48.

7.49 If the waveform in Fig. 7.116(a) is applied to the circuit of Fig. 7.116(b), find v(t). Assume v(0) = 0.

Figure 7.116

For Prob. 7.49 and Review Question 7.10.

*7.50 In the circuit of Fig. 7.117, find ix for t > 0. Let R1 = R2 = 1 kΩ, R3 = 2 kΩ, and C = 0.25 mF.

Figure 7.117 For Prob. 7.50.

Section 7.6 Step Response of an RL Circuit

• 7.51 Rather than applying the shortcut technique used in Section 7.6, use KVL to obtain Eq. (7.60).
• 7.52 Using Fig. 7.118, design a problem to help other students better understand the step response of an RL circuit.

Figure 7.118 For Prob. 7.52.

7.53 Determine the inductor current i(t) for both t < 0 and t > 0 for each of the circuits in Fig. 7.119.

Figure 7.119 For Prob. 7.53.

7.54 Obtain the inductor current for both t < 0 and t > 0 in each of the circuits in Fig. 7.120.

For Prob. 7.54.

^* An asterisk indicates a challenging problem.

7.55 Find v(t) for t < 0 and t > 0 in the circuit of Fig. 7.121.

Figure 7.121

For Prob. 7.55.

7.56 For the network shown in Fig. 7.122, find v(t) for t > 0.

Figure 7.122

For Prob. 7.56.

*7.57 Find i1(t) and i2(t) for t > 0 in the circuit of Fig. 7.123.

Figure 7.123

For Prob. 7.57.

• 7.58 Rework Prob. 7.17 if i(0) = 10 A and v(t) = 20u(t) V.
• 7.59 Determine the step response vo(t) to is = 6u(t) A in the circuit of Fig. 7.124.

Figure 7.124 For Prob. 7.59.

7.60 Find v(t) for t > 0 in the circuit of Fig. 7.125 if the initial current in the inductor is zero.

Figure 7.125

For Prob. 7.60.

7.61 In the circuit in Fig. 7.126, is changes from 5 A to 10 A at t = 0; that is, is = 5u(−t) + 10u(t). Find v and i.

Figure 7.126

For Prob. 7.61.

7.62 For the circuit in Fig. 7.127, calculate i(t) if i(0) = 0.

Figure 7.127

For Prob. 7.62.

7.63 Obtain v(t) and i(t) in the circuit of Fig. 7.128.

Figure 7.128

For Prob. 7.63.

7.64 Determine the value of iL(t) and the total energy dissipated by the circuit from t = 0 sec to t = ∞ sec. The value of iin(t) is equal to [6 – 6u(t)] A.

7.65 If the input pulse in Fig. 7.130(a) is applied to the circuit in Fig. 7.130(b), determine the response i(t).

Section 7.7 First-order Op Amp Circuits

7.66 Using Fig. 7.131, design a problem to help other students better understand first-order op amp circuits.

For Prob. 7.66.

7.67 If v(0) = 10 V, find vo(t) for t > 0 in the op amp circuit in Fig. 7.132. Let R = 100 kΩ and C = 20 µF.

Figure 7.133

For Prob. 7.68.

7.69 For the op amp circuit in Fig. 7.134, find vo(t) for t > 0.

Figure 7.134

For Prob. 7.69.

7.70 Determine vo for t > 0 when vs = 20 mV in the op amp circuit of Fig. 7.135.

Figure 7.135 For Prob. 7.70.

• 7.71 For the op amp circuit in Fig. 7.136, suppose vs = 10u(t) V. Find v(t) for t > 0.

7.72 Find io in the op amp circuit in Fig. 7.137. Assume that v(0) = −2 V, R = 10 kΩ, and C = 10 µF.

Figure 7.137

For Prob. 7.72.

7.73 For the op amp circuit of Fig. 7.138, let R1 = 10 kΩ, Rf = 30 kΩ, C = 20 μF, and v(0) = 1 V. Find v0.

Figure 7.138

For Prob. 7.73.

7.74 Determine vo(t) for t > 0 in the circuit of Fig. 7.139. Let is = 10u(t) μA and assume that the capacitor is initially uncharged.

Figure 7.139

For Prob. 7.74.

7.75 In the circuit of Fig. 7.140, find vo and io, given that vs = 10[1 − e^−t ]u(t) V.

Figure 7.140 For Prob. 7.75.

Section 7.8 Transient Analysis with PSpice

• 7.76 Repeat Prob. 7.49 using PSpice or MultiSim.
• 7.77 The switch in Fig. 7.141 opens at t = 0. Use PSpice or MultiSim to determine v(t) for t > 0.

Figure 7.141

For Prob. 7.77.

7.78 The switch in Fig. 7.142 moves from position a to b at t = 0. Use PSpice or MultiSim to find i(t) for t > 0.

Figure 7.142 For Prob. 7.78.

7.79 In the circuit of Fig. 7.143, determine io(t).

Figure 7.143

• For Prob. 7.79.
• 7.80 In the circuit of Fig. 7.144, find the value of io for all values of 0 < t.

For Prob. 7.80.

7.81 Repeat Prob. 7.65 using PSpice or MultiSim.

Section 7.9 Applications

• 7.82 In designing a signal-switching circuit, it was found that a 100-µF capacitor was needed for a time constant of 3 ms. What value resistor is necessary for the circuit?
• 7.83 An RC circuit consists of a series connection of a 120-V source, a switch, a 34-MΩ resistor, and a 15-µF capacitor. The circuit is used in estimating the speed of a horse running a 4-km racetrack. The switch closes when the horse begins and opens when the horse crosses the finish line. Assuming that the capacitor charges to 85.6 V, calculate the speed of the horse.
• 7.84 A capacitor with a value of 10 mF has a leakage resistance of 2 MΩ. How long does it take the voltage across the capacitor to decay to 40% of the initial voltage to which the capacitor is charged? Assume that the capacitor is charged and then set aside by itself.
• 7.85 A simple relaxation oscillator circuit is shown in Fig. 7.145. The neon lamp fires when its voltage reaches 75 V and turns off when its voltage drops to 30 V. Its resistance is 120 Ω when on and infinitely high when off.
  • (a) For how long is the lamp on each time the capacitor discharges?
  • (b) What is the time interval between light flashes?

Figure 7.145 For Prob. 7.85.

7.86 Figure 7.146 shows a circuit for setting the length of time voltage is applied to the electrodes of a welding machine. The time is taken as how long it takes the capacitor to charge from 0 to 8 V. What is the time range covered by the variable resistor?

Figure 7.146 For Prob. 7.86.