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• Maicol Avendaño

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441

Problems

Problems Section 10.2

10.6 Determine Vx in Fig. 10.55.

Nodal Analysis

+ Vx

10.1 Determine i in the circuit of Fig. 10.50. 1Ω

i

2 cos 10t V + ‒

20 Ω

20 Ω

1F

0.2Vx

+ 60 0° V ‒

1Ω

1H

‒ j10 Ω

Figure 10.55 For Prob. 10.6.

Figure 10.50 For Prob. 10.1.

10.7 Use nodal analysis to find V in the circuit of Fig. 10.56.

10.2 Using Fig. 10.51, design a problem to help other students better understand nodal analysis. 2Ω

4 0° V + ‒

40 Ω ‒ j5 Ω

j4 Ω

+ Vo ‒

120 ‒15° V + ‒

j20 Ω

V

‒ j30 Ω

6 30° A

50 Ω

Figure 10.51

For Prob. 10.2. 10.3 Determine vo in the circuit of Fig. 10.52. 1 12

4Ω

16 sin 4t V

+ vo ‒

+ ‒

F

Figure 10.56 For Prob. 10.7.

2H

1Ω

2 cos 4t A

6Ω

10.8 Use nodal analysis to find current io in the circuit of Fig. 10.57. Let is = 6 cos(200t + 15°) A. 0.1 vo

Figure 10.52

io

For Prob. 10.3.

10.4 Compute vo(t) in the circuit of Fig. 10.53. ix

24 cos (4t + 45°) V + ‒

is

0.25 F

1H

0.5ix

1Ω

+ vo ‒

Figure 10.53

20 Ω

‒

50 μF

100 mH

Figure 10.57 For Prob. 10.8.

10.5 Find io in the circuit of Fig. 10.54. io

25 cos(4 × 103t) V + ‒

For Prob. 10.5.

+

10.9 Use nodal analysis to find vo in the circuit of Fig. 10.58.

For Prob. 10.4.

Figure 10.54

vo

40 Ω

2 kΩ

20 Ω

2 μF

0.25 H

50 μF

10 mH

io + ‒

10io

10 cos 103t V

+ ‒

Figure 10.58 For Prob. 10.9.

20 Ω

4io

30 Ω

+ vo ‒

442

Chapter 10

Sinusoidal Steady-State Analysis

10.10 Use nodal analysis to find vo in the circuit of Fig. 10.59. Let ω = 2 krad/s.

10.14 Calculate the voltage at nodes 1 and 2 in the circuit of Fig. 10.63 using nodal analysis. j4 Ω

2 µF

36 sin ωt A

+

+

vx

2 kΩ

0.1 vx 4 kΩ

50 mH

‒

20 30° A 1

vo ‒

2

‒ j2 Ω

10 Ω

‒ j5 Ω

j2 Ω

Figure 10.59 For Prob. 10.10.

10.11 Using nodal analysis, find io(t) in the circuit in Fig. 10.60.

0.25 F

Figure 10.63 For Prob. 10.14.

10.15 Solve for the current I in the circuit of Fig. 10.64 using nodal analysis.

2H

5 0° A 2Ω

1H io

8 sin (2t + 30°) V + ‒

0.5 F

j1 Ω

2Ω

cos 2t A

I 20 ‒90° V + ‒

Figure 10.60

‒ j2 Ω

4Ω

2I

For Prob. 10.11.

10.12 Using Fig. 10.61, design a problem to help other students better understand nodal analysis.

Figure 10.64 For Prob. 10.15.

10.16 Use nodal analysis to find Vx in the circuit shown in Fig. 10.65.

2io

j4 Ω

R2

+ V ‒ x

io R1

is

C

L

10.13 Determine Vx in the circuit of Fig. 10.62 using any method of your choice. ‒j2 Ω

For Prob. 10.13.

‒j3 Ω

3 45° A

For Prob. 10.16.

For Prob. 10.12.

Figure 10.62

5Ω

Figure 10.65

Figure 10.61

+ 40 30° V ‒

2 0° A

8Ω

+ Vx 3 Ω ‒

10.17 By nodal analysis, obtain current Io in the circuit of Fig. 10.66.

j6 Ω 10 Ω

j4 Ω 5 0° A

+ 100 20° V ‒ 3Ω

Figure 10.66 For Prob. 10.17.

Io

2Ω

1Ω

‒ j2 Ω

443

Problems

10.18 Use nodal analysis to obtain Vo in the circuit of Fig. 10.67 below. j6 Ω

8Ω + Vx ‒

4 45° A

2Ω

2Vx

j5 Ω

4Ω ‒ j1 Ω

‒ j2 Ω

+ Vo ‒

Figure 10.67 For Prob. 10.18.

10.22 For the circuit in Fig. 10.71, determine Vo∕Vs.

10.19 Obtain Vo in Fig. 10.68 using nodal analysis.

R1

j2 Ω 12 0° V

4Ω

+‒ + Vo ‒

2Ω

R2

Vs + ‒

‒ j4 Ω

C L

+ Vo ‒

0.2Vo

Figure 10.71 For Prob. 10.22.

Figure 10.68 For Prob. 10.19.

10.23 Using nodal analysis obtain V in the circuit of Fig. 10.72. 10.20 Refer to Fig. 10.69. If vs(t) = Vm sin ωt and vo(t) = A sin(ωt + ϕ), derive the expressions for A and ϕ.

j𝜔L

Vs + ‒

R + ‒

vs(t)

R

+ vo(t) ‒

C

L

+

1 j𝜔C

1 j𝜔C

‒

V

Figure 10.72 For Prob. 10.23.

Figure 10.69 For Prob. 10.20.

Section 10.3 10.21 For each of the circuits in Fig. 10.70, find Vo∕Vi for ω = 0, ω → ∞, and ω2 = 1∕LC.

Mesh Analysis

10.24 Design a problem to help other students better understand mesh analysis. 10.25 Solve for io in Fig. 10.73 using mesh analysis.

R

L

R

+ C

Vi ‒ (a)

Figure 10.70 For Prob. 10.21.

+

+

Vo

Vi

‒

‒

C 4Ω

+ L

Vo ‒

2H io

10 cos 2t V + ‒

(b)

Figure 10.73 For Prob. 10.25.

0.25 F

+ ‒

6 sin 2t V

444

Chapter 10

Sinusoidal Steady-State Analysis

10.26 Use mesh analysis to find current io in the circuit of Fig. 10.74.

10.29 Using Fig. 10.77, design a problem to help other students better understand mesh analysis.

1 µF

2 kΩ io 10 cos

103t

V + ‒

jXL1

+ 20 sin 103t V ‒

0.4 H

R3 R2

I1

R1

Figure 10.74

I2

jXL3

jXL2

For Prob. 10.26.

+‒

10.27 Using mesh analysis, find I1 and I2 in the circuit of Fig. 10.75.

‒ jXC

Vs

Figure 10.77 For Prob. 10.29.

j10 Ω

60 30° V + ‒

I1

40 Ω

+ 75 0° V ‒

I2

‒ j20 Ω

Figure 10.75

10.30 Use mesh analysis to find vo in the circuit of Fig. 10.78. Let vs1 = 120 cos(100t + 90°) V, vs2 = 80 cos 100t V.

For Prob. 10.27.

10.28 In the circuit of Fig. 10.76, determine the mesh currents i1 and i2. Let v1 = 10 cos 4t V and v2 = 20 cos(4t − 30°) V. 1Ω

1H

20 Ω vs1 + ‒

i1

50 µF

200 mH + vo ‒

10 Ω + v ‒ s2

1Ω

1H

Figure 10.78

1F v1 + ‒

300 mH

400 mH

+ v2 ‒

i2 1Ω

Figure 10.76

For Prob. 10.30.

10.31 Use mesh analysis to determine current Io in the circuit of Fig. 10.79 below.

For Prob. 10.28.

80 Ω + 50 120° V ‒

Figure 10.79 For Prob. 10.31.

‒ j40 Ω

Io

j60 Ω

‒ j40 Ω

20 Ω + ‒

30 ‒30° V

445

Problems

10.32 Determine Vo and Io in the circuit of Fig. 10.80 using mesh analysis.

10.38 Using mesh analysis, obtain Io in the circuit shown in Fig. 10.83. Io

j4 Ω

10 ‒30° A

2Ω

+ Vo ‒

3Vo

Io

‒ +

‒ j2 Ω

2 0° A

j2 Ω

2Ω

1Ω

Figure 10.80

+ ‒

‒ j4 Ω

10 90° V

1Ω

4 0° A

For Prob. 10.32.

Figure 10.83 For Prob. 10.38. 10.33 Compute I in Prob. 10.15 using mesh analysis. 10.39 Find I1, I2, I3, and Ix in the circuit of Fig. 10.84. 10.34 Use mesh analysis to find Io in Fig. 10.28 (for Example 10.10).

10 Ω

20 Ω

10.35 Calculate Io in Fig. 10.30 (for Practice Prob. 10.10) using mesh analysis. 10.36 Compute Vo in the circuit of Fig. 10.81 using mesh analysis. ‒ j3 Ω

j4 Ω 4 90° A

2Ω

2Ω

+ Vo ‒

‒ j15 Ω

I3

j16 Ω

Ix 12 64° V

I1

+ ‒

I2

‒ j25 Ω

8Ω

Figure 10.84 For Prob. 10.39.

2Ω

+ 12 0° V ‒

Section 10.4

Superposition Theorem

10.40 Find io in the circuit shown in Fig. 10.85 using superposition.

2 0° A

Figure 10.81

4Ω

For Prob. 10.36.

2Ω io

10.37 Use mesh analysis to find currents I1, I2, and I3 in the circuit of Fig. 10.82.

25 cos 4t V + ‒

+ 20 V ‒

1H

Figure 10.85 For Prob. 10.40.

I1 120 ‒90° V + ‒

I2

10.41 Find vo for the circuit in Fig. 10.86, assuming that is(t) = 2 sin (2t) + 3 cos (4t) A.

Z Z = 80 ‒ j35 Ω

‒ 120 ‒30° V +

Figure 10.82 For Prob. 10.37.

I3

Z

is(t)

Figure 10.86 For Prob. 10.41.

10 Ω

5H

+ vo ‒

446

Chapter 10

Sinusoidal Steady-State Analysis

10.42 Using Fig. 10.87, design a problem to help other students better understand the superposition theorem. Io

jXL V1 + ‒

10.46 Solve for vo(t) in the circuit of Fig. 10.91 using the superposition principle.

R2

6Ω + V ‒ 2

‒ jXC

R1

18 cos 3t V + ‒

1 12

2H + vo ‒

F

+ 15 V ‒

6 sin 2t A

Figure 10.91

Figure 10.87

For Prob. 10.46.

For Prob. 10.42.

10.43 Using the superposition principle, find ix in the circuit of Fig. 10.88.

1 8

10 cos(2t + 10°) A

F

3Ω

10.47 Determine io in the circuit of Fig. 10.92, using the superposition principle.

ix

1Ω

+ 20 cos(2t ‒ 60°) V ‒

4H

Figure 10.88

10 sin(t ‒ 30°) V

+ ‒

1 6

F

24 V

2H

‒+

io

2Ω

4Ω

2 cos 3t

Figure 10.92

For Prob. 10.43.

For Prob. 10.47.

10.44 Use the superposition principle to obtain vx in the circuit of Fig. 10.89. Let vs = 50 sin 2t V and is = 12 cos(6t + 10°) A.

10.48 Find io in the circuit of Fig. 10.93 using superposition. 20 μF

20 Ω vs + ‒

50 mF

20 Ω

+ vx ‒

is

150 cos 2000t V + ‒

io 40 mH

6 sin 4000t A

80 Ω

100 Ω

60 Ω

+ 72 V ‒

Figure 10.89 For Prob. 10.44.

Figure 10.93 For Prob. 10.48.

10.45 Use superposition to find i(t) in the circuit of Fig. 10.90.

i

+ 3 sin 4t V ‒

20 Ω 25 cos(20t + 15°) + ‒

300 mH

Figure 10.90 For Prob. 10.45.

Source Transformation

10.49 Using source transformation, find i in the circuit of Fig. 10.94.

20 Ω

8 cos(10t + 30°) V + ‒

Section 10.5

Figure 10.94 For Prob. 10.49.

i 20 Ω

5 mF 1H

447

Problems ‒ j5 Ω

10.50 Using Fig. 10.95, design a problem to help other students understand source transformation.

vs(t) + ‒

R2

C

8Ω

12 0° A

L

R1

a j10 Ω b

+ vo ‒

(b)

Figure 10.98 For Prob. 10.55.

Figure 10.95

10.56 For each of the circuits in Fig. 10.99, obtain Thevenin and Norton equivalent circuits at terminals a‑b.

For Prob. 10.50.

10.51 Use source transformation to find Io in the circuit of Prob. 10.42.

j4 Ω

6Ω

10.52 Use the method of source transformation to find Ix in the circuit of Fig. 10.96.

a ‒ j2 Ω 2 0° A

2Ω

‒ j2 Ω

j4 Ω

b

ix 60 0° V + ‒

(a)

4Ω

6Ω

5 90° A

30 Ω

‒ j3 Ω 120 45° V + ‒

Figure 10.96 For Prob. 10.52.

j10 Ω 60 Ω

a ‒ j5 Ω b

10.53 Use the concept of source transformation to find Vo in the circuit of Fig. 10.97. ‒ j3 Ω

4Ω 80 0° V + ‒

2Ω

j2 Ω

(b)

Figure 10.99 For Prob. 10.56.

j4 Ω

‒ j2 Ω

+ Vo ‒

Figure 10.97

10.57 Using Fig. 10.100, design a problem to help other students better understand Thevenin and Norton equivalent circuits. R1

‒ jXC

Vs + ‒

For Prob. 10.53.

10.54 Rework Prob. 10.7 using source transformation.

10.55 Find the Thevenin and Norton equivalent circuits at terminals a‑b for each of the circuits in Fig. 10.98.

25 30° V + ‒

10 Ω

jXL

Figure 10.100 For Prob. 10.57.

Section 10.6 Thevenin and Norton Equivalent Circuits

j20 Ω

a

10.58 For the circuit depicted in Fig. 10.101, find the Thevenin equivalent circuit at terminals a‑b. a

b b

Figure 10.101 For Prob. 10.58.

10 Ω ‒ j10 Ω

3 0° A

‒ j10 Ω

(a)

R2

30 90° V + ‒

448

Chapter 10

Sinusoidal Steady-State Analysis

10.59 Calculate the output impedance of the circuit shown in Fig. 10.102. ‒ j2 Ω

10.63 Obtain the Norton equivalent of the circuit depicted in Fig. 10.106 at terminals a-b.

10 Ω

10 Ω

50 mH a

+ Vo ‒ + 160 sin (200t + 60°) V ‒

j40 Ω

0.2Vo

500 µF b

Figure 10.106

Figure 10.102

For Prob. 10.63.

For Prob. 10.59.

10.60 Find the Thevenin equivalent of the circuit in Fig. 10.103 as seen from: (a) terminals a-b

10.64 For the circuit shown in Fig. 10.107, find the Norton equivalent circuit at terminals a-b.

(b) terminals c-d c

d j10 Ω

a

3 60° A 10 Ω

‒ j20 Ω

2 0° A

a

20 0° A + ‒

b

b ‒ j30 Ω

j80 Ω

Figure 10.107

Figure 10.103

For Prob. 10.64.

For Prob. 10.60.

10.61 Find the Thevenin equivalent at terminals a-b of the circuit in Fig. 10.104. 4Ω

10.65 Using Fig. 10.108, design a problem to help other students better understand Norton’s theorem.

a

Ix

vs(t)

R

+‒

io

‒ j3 Ω

15 0° A

40 Ω

60 Ω

10 Ω

1.5Ix

L

C1

C2

b

Figure 10.104

Figure 10.108

For Prob. 10.61.

For Prob. 10.65.

10.62 Using Thevenin’s theorem, find vo in the circuit of Fig. 10.105. 2vo +‒

10 Ω

10.66 At terminals a-b, obtain Thevenin and Norton equivalent circuits for the network depicted in Fig. 10.109. Take ω = 10 rad/s. 10 µF 12 cos ωt V ‒+

‒ j10 Ω

2 0° A

10 Ω

j5 Ω

j5 Ω

10 Ω

+ vo ‒

2 sin ωt A

+ vo ‒

10 Ω

1 2

H

a 2vo b

Figure 10.105 For Prob. 10.62.

Figure 10.109 For Prob. 10.66.

449

Problems

10.67 Find the Thevenin and Norton equivalent circuits at terminals a‑b in the circuit of Fig. 10.110.

R2 C R1

‒j5 Ω 12 Ω

13 Ω 90 30° V + ‒

a

vs + ‒

+ vo ‒

b j6 Ω

10 Ω

‒ +

Figure 10.113 For Prob. 10.70.

8Ω

10.71 Find vo in the op amp circuit of Fig. 10.114.

Figure 10.110 For Prob. 10.67.

+ ‒

10.68 Find the Thevenin equivalent at terminals a‑b in the circuit of Fig. 10.111.

io

+ vo 3

+ ‒

1 F 20

4io

Figure 10.111

‒

Figure 10.114 For Prob. 10.71.

b

10.72 Compute io(t) in the op amp circuit in Fig. 10.115 if vs = 4 cos(104t) V. 50 kΩ

For Prob. 10.68.

vs + ‒

Op Amp AC Circuits

10.69 For the integrator shown in Fig. 10.112, obtain Vo∕Vs. Find vo(t) when vs(t) = Vm sin ωt and ω = 1∕RC.

R

‒ +

+ ‒

io

1 nF

100 kΩ

Figure 10.115 For Prob. 10.72.

10.73 If the input impedance is defined as Zin = Vs∕Is, find the input impedance of the op amp circuit in Fig. 10.116 when R1 = 10 kΩ, R2 = 20 kΩ, C1 = 10 nF, C2 = 20 nF, and ω = 5000 rad/s.

C

vs + ‒

vo

10 kΩ

a

1 H vo ‒

Section 10.7

0.5 µF

+ 12 cos(2t + 30°) V ‒

2 kΩ

4Ω

3 sin10t V + ‒

+

C1

+ vo ‒

Figure 10.112 For Prob. 10.69.

Is

R1

Vs + ‒

C2

Zin

10.70 Using Fig. 10.113, design a problem to help other students better understand op amps in AC circuits.

R2

Figure 10.116 For Prob. 10.73.

+ + ‒-

Vo

450

Chapter 10

Sinusoidal Steady-State Analysis

10.74 Evaluate the voltage gain Av = Vo∕Vs in the op amp circuit of Fig. 10.117. Find Av at ω = 0, ω → ∞, ω = 1∕R1C1, and ω = 1∕R2C2.

10.76 Determine Vo and Io in the op amp circuit of Fig. 10.119.

io

20 kΩ C2

R2 R1

C1

Vs + ‒

‒ +

‒ j4 kΩ +

+ 6 30° V ‒

Vo ‒

+

‒ j2 kΩ

Vo

Figure 10.119

For Prob. 10.74.

For Prob. 10.76.

10.75 In the op amp circuit of Fig. 10.118, find the closed‑ loop gain and phase shift of the output voltage with respect to the input voltage if C1 = C2 = 1 nF, R1 = R2 = 100 kΩ, R3 = 20 kΩ, R4 = 40 kΩ, and ω = 2000 rad/s.

10.77 Compute the closed‑loop gain Vo∕Vs for the op amp circuit of Fig. 10.120.

R3 R1

R1

vs + ‒

C2 + ‒

vs + ‒

+ ‒

‒

Figure 10.117

C1

10 kΩ

R4

R2

+

C2

R2 + vo

C1

‒

+ vo

R3

‒

Figure 10.120 For Prob. 10.77.

‒

10.78 Determine vo(t) in the op amp circuit in Fig. 10.121 below.

Figure 10.118 For Prob. 10.75.

20 kΩ 10 kΩ

+ 10 sin(400t) V ‒

0.25 µF

0.5 µF + ‒ 10 kΩ

40 kΩ 20 kΩ

Figure 10.121 For Prob. 10.78.

vo

451

Problems 2Ω

10.79 For the op amp circuit in Fig. 10.122, obtain Vo.

6Ω

200 kΩ ‒j 100 kΩ ‒ + 200 0° µA

is

‒j 200 kΩ

4Ω

100 kΩ

‒ +

50 kΩ

8Ω 4 µF

10 mH

+ vo ‒

+ Vo Figure 10.125 ‒ For Prob. 10.83.

10.84 Obtain Vo in the circuit of Fig. 10.126 using PSpice or MultiSim.

Figure 10.122 For Prob. 10.79.

‒j2 Ω

10.80 Obtain vo(t) for the op amp circuit in Fig. 10.123 if vs = 12 cos(1000t − 60°) V.

0.1 µF ‒ +

vs + ‒

10 kΩ

‒ +

+ vo

Figure 10.123

10.85 Using Fig. 10.127, design a problem to help other students better understand performing AC analysis with PSpice or MultiSim.

‒ jXC

R2

AC Analysis Using PSpice

+

10.81 Use PSpice or MultiSim to determine Vo in the circuit of Fig. 10.124. Assume ω = 1 rad/s.

24 0° V + ‒

4 0° A

10 Ω

40 Ω j4 Ω

Vx

‒ jXL

Is

R4

Figure 10.127

25 Ω 30 Ω

+ R3 Vo ‒

0.25Vx

R1

For Prob. 10.80.

‒j2 Ω

+ Vo ‒

For Prob. 10.84.

‒

Section 10.8

2Ω

Figure 10.126

0.2 µF

20 kΩ

+ Vx ‒

1Ω

3 0° A

50 kΩ

2Vx

j4 Ω

For Prob. 10.85.

+ Vo ‒

10.86 Use PSpice or MultiSim to find V1, V2, and V3 in the network of Fig. 10.128. 8Ω V1

Figure 10.124 For Prob. 10.81.

j 10 Ω

60 30° V + ‒

10.82 Solve Prob. 10.19 using PSpice or MultiSim. 10.83 Use PSpice or MultiSim to find vo(t) in the circuit of Fig. 10.125. Let is = 2 cos(103t) A.

Figure 10.128 For Prob. 10.86.

‒ j4 Ω

V2

j 10 Ω

‒ j4 Ω

V3 4 0° A

452

Chapter 10

Sinusoidal Steady-State Analysis

10.87 Determine V1, V2, and V3 in the circuit of Fig. 10.129 using PSpice or MultiSim.

10.90 Figure 10.132 shows a Wien‑bridge network. Show that the frequency at which the phase shift between 1 the input and output signals is zero is f = ___ RC, 2π and that the necessary gain is Av = Vo∕Vi = 3 at that frequency.

j10 Ω ‒j4 Ω

V1

8Ω

4 0° A

2Ω

1Ω

V2

V3

‒ j2 Ω

j6 Ω

2 0° A R

R1

C Vi + ‒

Figure 10.129 For Prob. 10.87.

+ Vo ‒ C R2

R

10.88 Use PSpice or MultiSim to find vo and io in the circuit of Fig. 10.130 below.

4Ω

Figure 10.132 For Prob. 10.90.

20 mF

2H io

6 cos 4t V + ‒

0.5vo

+ ‒

4io

10 Ω

25 mF

+ vo ‒

Figure 10.130 For Prob. 10.88.

Section 10.9

10.91 Consider the oscillator in Fig. 10.133.

Applications

(a) Determine the oscillation frequency. (b) Obtain the minimum value of R for which oscillation takes place.

10.89 The op amp circuit in Fig. 10.131 is called an inductance simulator. Show that the input impedance is given by Vin Zin = ___ = jωLeq Iin where R1R3R4 Leq = ______ R2C

80 kΩ 20 kΩ

R1

R2 ‒ +

Figure 10.131 For Prob. 10.89.

R3

C ‒ +

R4

‒ + 0.4 mH 2 nF

Iin + V in ‒

10 kΩ

Figure 10.133 For Prob. 10.91.

R