Problems

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Problems

Section 6.2 Capacitors

• 6.1 If the voltage across a 7.5-F capacitor is 2te⁻3*^t* V, find the current and the power.
• 6.2 A 50-μF capacitor has energy w(t) = 10 cos2 377t J. Determine the current through the capacitor.
• 6.3 Design a problem to help other students better understand how capacitors work.
• 6.4 A voltage across a capacitor is equal to [2 – 2 cos(4t)] V and the current flowing through it is equal to 2 sin (4t) μA. Determine the value of the capacitance. Calculate the power being stored by the capacitor.
• 6.5 The voltage across a 4-μF capacitor is shown in Fig. 6.45. Find the current waveform.

For Prob. 6.5.

6.6 The voltage waveform in Fig. 6.46 is applied across a 55-μF capacitor. Draw the current waveform through it.

Figure 6.46

For Prob. 6.6.

• 6.7 At t = 0, the voltage across a 25-mF capacitor is 10 V. Calculate the voltage across the capacitor for t > 0 when current 5t mA flows through it.
• 6.8 A 4-mF capacitor has the terminal voltage

$$v = \begin{cases} 50 \text{ V}, & t \le 0 \\ Ae^{-100t} + Be^{-600t} \text{ V}, & t \ge 0 \end{cases}$$

If the capacitor has an initial current of 2 A, find:

• (a) the constants A and B,
• (b) the energy stored in the capacitor at t = 0,
• (c) the capacitor current for t > 0.

Problems 241

• 6.9 The current through a 0.5-F capacitor is 6(1 − e −t ) A. Determine the voltage and power at t = 2 s. Assume v(0) = 0.
• 6.10 The voltage across a 5-mF capacitor is shown in Fig. 6.47. Determine the current through the capacitor.

Figure 6.47

For Prob. 6.10.

6.11 A 4-mF capacitor has the current waveform shown in Fig. 6.48. Assuming that v(0) = 10 V, sketch the voltage waveform v(t).

Figure 6.48

For Prob. 6.11.

• 6.12 A voltage of 45e⁻2000*^t* V appears across a parallel combination of a 100-mF capacitor and a 12-Ω resistor. Calculate the power absorbed by the parallel combination.
• 6.13 Find the voltage across the capacitors in the circuit of Fig. 6.49 under dc conditions.

Figure 6.49

For Prob. 6.13.

Section 6.3 Series and Parallel Capacitors

6.14 Series-connected 20- and 60-pF capacitors are placed in parallel with series-connected 30- and 70-pF capacitors. Determine the equivalent capacitance.

6.15 Two capacitors (25 and 75 μF) are connected to a 100-V source. Find the energy stored in each capacitor if they are connected in:

(a) parallel (b) series

6.16 The equivalent capacitance at terminals a-b in the circuit of Fig. 6.50 is 20 μF. Calculate the value of C.

Figure 6.50

For Prob. 6.16.

6.17 Determine the equivalent capacitance for each of the circuits of Fig. 6.51.

Figure 6.51

For Prob. 6.17.

6.18 Find Ceq in the circuit of Fig. 6.52 if all capacitors are 4 μF.

6.19 Find the equivalent capacitance between terminals a and b in the circuit of Fig. 6.53. All capacitances are in μF.

Figure 6.53 For Prob. 6.19.

6.20 Find the equivalent capacitance at terminals a-b of the circuit in Fig. 6.54.

6.21 Determine the equivalent capacitance at terminals a-b of the circuit in Fig. 6.55.

6.22 Obtain the equivalent capacitance of the circuit in Fig. 6.56.

Figure 6.56

For Prob. 6.22.

Figure 6.57

For Prob. 6.23.

6.24 In the circuit shown in Fig. 6.58 assume that the capacitors were initially uncharged and that the current source has been connected to the circuit long enough for all the capacitors to reach steady state (no current flowing through the capacitors). Determine the voltage across each capacitor and the energy

Figure 6.58 For Prob. 6.24.

6.25 (a) Show that the voltage-division rule for two capacitors in series as in Fig. 6.59(a) is

$$v_1 = \frac{C_2}{C_1 + C_2} v_s$$

, $v_2 = \frac{C_1}{C_1 + C_2} v_s$

assuming that the initial conditions are zero.

Figure 6.59 For Prob. 6.25.

(b) For two capacitors in parallel as in Fig. 6.59(b), show that the current-division rule is

$$i_1 = \frac{C_1}{C_1 + C_2} i_s$$

, $i_2 = \frac{C_2}{C_1 + C_2} i_s$

assuming that the initial conditions are zero.

• 6.26 Three capacitors, C¹ = 5 μF, C² = 10 μF, and C³ = 20 μF, are connected in parallel across a 200-V source. Determine:
  • (a) the total capacitance,
  • (b) the charge on each capacitor,
  • (c) the total energy stored in the parallel combination.
• 6.27 Given that four 10-μF capacitors can be connected
• in series and in parallel, find the minimum and maximum values that can be obtained by such series/parallel combinations.
• 6.28 Obtain the equivalent capacitance of the network shown in Fig. 6.60. *

Figure 6.60 For Prob. 6.28.

6.29 Determine Ceq for each circuit in Fig. 6.61.

For Prob. 6.29.

6.30 Assuming that the capacitors are initially uncharged, find vo(t) in the circuit of Fig. 6.62.

6.31 If v(0) = 0, find v(t), i1(t), and i2(t) in the circuit of Fig. 6.63.

Figure 6.63 For Prob. 6.31.

6.32 In the circuit in Fig. 6.64, let is = 4.5e⁻2*^t* mA and the voltage across each capacitor is equal to zero at t = 0. Determine v1 and v2 and the energy stored in each capacitor for all t > 0.

For Prob. 6.32.

^* An asterisk indicates a challenging problem.

6.33 Obtain the Thevenin equivalent at the terminals, a-b, of the circuit shown in Fig. 6.65. Please note that Thevenin equivalent circuits do not generally exist for circuits involving capacitors and resistors. This is a special case where the Thevenin equivalent circuit does exist.

Figure 6.65

For Prob. 6.33.

Section 6.4 Inductors

• 6.34 The current through a 25-mH inductor is 10e⁻t∕² A. Find the voltage and the power at t = 3 s.
• 6.35 An inductor has a linear change in current from 100 mA to 200 mA in 2 ms and induces a voltage of 160 mV. Calculate the value of the inductor.
• 6.36 Design a problem to help other students better understand how inductors work.
• 6.37 The current through a 12-mH inductor is 4 sin 100t A. Find the voltage, and the energy stored at t = ___π ²⁰⁰s.
• 6.38 The current through a 40-mH inductor is

$$i(t) = \begin{cases} 0, & t < 0 \\ te^{-2t} \, \mathbf{A}, & t > 0 \end{cases}$$

Find the voltage v(t).

6.39 The voltage across a 50-mH inductor is given by

$$v(t) = [5e^{-2t} + 2t + 4] \text{ V} \qquad \text{for } t > 0.$$

Determine the current i(t) through the inductor. Assume that i(0) = 0 A.

6.40 The current through a 5-mH inductor is shown in Fig. 6.66. Determine the voltage across the inductor at t = 1, 3, and 5 ms.

Figure 6.66 For Prob. 6.40.

• 6.41 The voltage across a 2-H inductor is 20(1 − e −2t ) V. If the initial current through the inductor is 0.3 A, find the current and the energy stored in the inductor at t = 1 s.
• 6.42 If the voltage waveform in Fig. 6.67 is applied across the terminals of a 5-H inductor, calculate the current through the inductor. Assume i(0) = −1 A.

Figure 6.67

For Prob. 6.42.

• 6.43 The current in a 150-mH inductor increases from 0 to 60 mA (steady state). How much energy is stored in the inductor?
• 6.44 A 100-mH inductor is connected in parallel with a 2-kΩ resistor. The current through the inductor is i(t) = 35e⁻400*^t* mA. (a) Find the voltage vL across the inductor. (b) Find the voltage vR across the resistor. (c) Does vR(t) + vL(t) = 0? (d) Calculate the energy stored in the inductor at t = 0. *
• 6.45 If the voltage waveform in Fig. 6.68 is applied to a 25-mH inductor, find the inductor current i(t) for 0 < t < 2 seconds. Assume i(0) = 0.

Figure 6.68

For Prob. 6.45.

6.46 Find vC, iL, and the energy stored in the capacitor and inductor in the circuit of Fig. 6.69 under dc conditions.

For Prob. 6.46.

Figure 6.70

For Prob. 6.47.

6.48 Under steady-state dc conditions, find i and v in the circuit in Fig. 6.71.

For Prob. 6.48.

Section 6.5 Series and Parallel Inductors

6.49 Find the equivalent inductance of the circuit in Fig. 6.72. Assume all inductors are 40 mH.

Figure 6.72

For Prob. 6.49.

6.50 An energy-storage network consists of seriesconnected 16- and 14-mH inductors in parallel with series-connected 24- and 36-mH inductors. Calculate the equivalent inductance.

6.51 Determine Leq at terminals a-b of the circuit in Fig. 6.73.

Figure 6.73 For Prob. 6.51.

6.53 Find Leq at the terminals of the circuit in Fig. 6.75.

6.54 Find the equivalent inductance looking into the terminals of the circuit in Fig. 6.76.

6.55 Find Leq in each of the circuits in Fig. 6.77.

6.56 Find Leq in the circuit of Fig. 6.78.

For Prob. 6.56.

6.57 Determine Leq that may be used to represent the inductive network of Fig. 6.79 at the terminals. *

For Prob. 6.57.

6.58 The current waveform in Fig. 6.80 flows through a 3-H inductor. Sketch the voltage across the inductor over the interval 0 < t < 6 s.

Figure 6.80 For Prob. 6.58.

6.59 (a) For two inductors in series as in Fig. 6.81(a), show that the voltage division principle is

$$v_1 = \frac{L_1}{L_1 + L_2} v_s$$

, $v_2 = \frac{L_2}{L_1 + L_2} v_s$

assuming that the initial conditions are zero.

(b) For two inductors in parallel as in Fig. 6.81(b), show that the current-division principle is

$$i_1 = \frac{L_2}{L_1 + L_2} i_s$$

, $i_2 = \frac{L_1}{L_1 + L_2} i_s$

assuming that the initial conditions are zero.

For Prob. 6.59.

6.60 In the circuit of Fig. 6.82, io(0) = 2 A. Determine io(t) and vo(t) for t > 0.

For Prob. 6.60.

6.61 Consider the circuit in Fig. 6.83. Find: (a) Leq, i1(t), and i2(t) if is = 3e^−t mA, (b) vo(t), (c) energy stored in the 20-mH inductor at t = 1 s.

6.62 Consider the circuit in Fig. 6.84. Given that v(t) = 12e⁻3*^t* mV for t > 0 and i1(0) = −30 mA, find: (a) i2(0), (b) i1(t) and i2(t).

For Prob. 6.62.

6.63 In the circuit of Fig. 6.85, sketch vo.

6.64 The switch in Fig. 6.86 has been in position A for a long time. At t = 0, the switch moves from position A to B. The switch is a make-before-break type so that there is no interruption in the inductor current. Find:

(a) i(t) for t > 0,

(b) v just after the switch has been moved to position B, (c) v(t) long after the switch is in position B.

6.65 The inductors in Fig. 6.87 are initially charged and are connected to the black box at t = 0. If i1(0) = 4 A, i2(0) = −2 A, and v(t) = 50e⁻200*^t* mV, t ≥ 0, find:

(a) the energy initially stored in each inductor,

• (b) the total energy delivered to the black box from t = 0 to t = ∞,
• (c) i1(t) and i2(t), t ≥ 0, (d) i(t), t ≥ 0.

6.66 The current i(t) through a 20-mH inductor is equal, in magnitude, to the voltage across it for all values of time. If i(0) = 2 A, find i(t).

Section 6.6 Applications

6.67 An op amp integrator has R = 50 kΩ and C = 0.04 μF. If the input voltage is vⁱ = 10 sin 50t mV, obtain the output voltage. Assume that at t equal to zero, the output is equal to zero.

• 6.68 A 6-V dc voltage is applied to an integrator with R = 50 kΩ, C = 100 μF at t = 0. How long will it take for the op amp to saturate if the saturation voltages are +12 V and −12 V? Assume that the initial capacitor voltage was zero.
• 6.69 An op amp integrator with R = 4 MΩ and C = 1 μF has the input waveform shown in Fig. 6.88. Plot the output waveform.

6.70 Using a single op amp, a capacitor, and resistors of 100 kΩ or less, design a circuit to implement

$$v_o = -2 \int_0^t v_i(\tau) \, d\tau$$

Assume vo=0 at t=0.

6.71 Show how you would use a single op amp to generate

$$v_o = -\int_0^t (v_1 + 4v_2 + 10v_3) \, dt$$

If the inte grating capacitor is C = 5 μF, obtain the other component values.

6.72 At t = 1.5 ms, calculate vo due to the cascaded integrators in Fig. 6.89. Assume that the integrators are reset to 0 V at t = 0.

For Prob. 6.72.

6.73 Show that the circuit in Fig. 6.90 is a noninverting integrator.

Figure 6.90 For Prob. 6.73.

6.74 The triangular waveform in Fig. 6.91(a) is applied to the input of the op amp differentiator in Fig. 6.91(b). Plot the output.

For Prob. 6.74.

• 6.75 An op amp differentiator has R = 250 kΩ and C = 10 μF. The input voltage is a ramp r(t) = 7t mV. Find the output voltage.
• 6.76 A voltage waveform has the following characteristics: a positive slope of 20 V/s for 5 ms followed by a negative slope of 10 V/s for 10 ms. If the waveform is applied to a differentiator with R = 50 kΩ, C = 10 μF, sketch the output voltage waveform.

6.77 The output vo of the op amp circuit in Fig. 6.92(a) is shown in Fig. 6.92(b). Let Ri = Rf = 1 MΩ and C = 1 μF. Determine the input voltage waveform and sketch it. *

6.79 Design an analog computer circuit to solve for v(t), given the following equation and a value for f(t) and that v(0) = 0 V.

$$\left(\frac{dv(t)}{dt}\right) + 3vt = f t dt$$

6.80 Figure 6.93 presents an analog computer designed to solve a differential equation. Assuming f(t) is known, set up the equation for f(t).

Figure 6.93 For Prob. 6.80.

6.81 Design an analog computer to simulate the following equation to solve for v(t) (assume the initial conditions are zero):

$$(d^{3}v(t)/dt^{3}) + 3(dv(t)/dt) = 4f(t)$$

6.82 Design an op amp circuit such that

$$v_o = 10v_s + 2 \int v_s dt$$