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Problems Section 6.2 Capacitors

6.6

6.1

If the voltage across a 7.5-F capacitor is 2te3t V, find the current and the power.

6.2

A 50-mF 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 current of 4 sin 4t A flows through a 5-F capacitor. Find the voltage v(t) across the capacitor given that v(0)  1 V.

6.5

The voltage across a 4-mF capacitor is shown in Fig. 6.45. Find the current waveform.

v (t) V 10

0

0

2

4

6

8

t (ms)

6

8

10

12 t (ms)

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 7 0 when current 5t mA flows through it.

6.8

A 4-mF capacitor has the terminal voltage vb

2

4

−10

v(t) V 10

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

50 V, Ae100t  Be600t V,

t0 t 0

If the capacitor has an initial current of 2 A, find: −10

(a) the constants A and B,

Figure 6.45

(b) the energy stored in the capacitor at t  0,

For Prob. 6.5.

(c) the capacitor current for t 7 0.

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The current through a 0.5-F capacitor is 6(1  et) 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. v (t) (V)

6.15 Two capacitors (25 mF and 75 mF) 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 30 mF. Calculate the value of C. a

16

C 0

1

2

3

14 F

t (s)

4

80 F

Figure 6.47 For Prob. 6.10. b

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). i(t) (mA)

Figure 6.50 For Prob. 6.16. 6.17 Determine the equivalent capacitance for each of the circuits of Fig. 6.51. 12 F

4F

15 10

6F

3F

5 0

2

−5

6

4

t (s)

8

4F (a)

−10

6F

Figure 6.48 For Prob. 6.11.

5F

6.12 A voltage of 30e2000t V appears across a parallel combination of a 100-mF capacitor and a 12- resistor. Calculate the power absorbed by the parallel combination.

4F

(b) 3F

6F

2F

6.13 Find the voltage across the capacitors in the circuit of Fig. 6.49 under dc conditions. 4F

C1

+ v1 −

3F

50 Ω

10 Ω

40 Ω

2F

(c) 20 Ω + −

60 V

Figure 6.51 + v2 −

For Prob. 6.17. C2

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

Figure 6.49 For Prob. 6.13.

Section 6.3 Series and Parallel Capacitors 6.14 Series-connected 20-pF and 60-pF capacitors are placed in parallel with series-connected 30-pF and 70-pF capacitors. Determine the equivalent capacitance.

Ceq

Figure 6.52 For Prob. 6.18.

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40 F

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

10 F

10 F

35 F

80

5 F 20 F

12

40

15 F

a

15 F

20

50 12

10

30

a

b

Figure 6.56

b

For Prob. 6.22. 60

Figure 6.53

6.23 Using Fig. 6.57, design a problem that will help other students better understand how capacitors work together when connected in series and in parallel.

For Prob. 6.19. 6.20 Find the equivalent capacitance at terminals a-b of the circuit in Fig. 6.54.

C1 V

a

1 F

1 F

+ −

C3

C2

C4

Figure 6.57 For Prob. 6.23.

2 F

2 F

6.24 For the circuit in Figure 6.58, determine (a) the voltage across each capacitor and (b) the energy stored in each capacitor.

2 F

60 F

3 F

3 F

3 F

90 V + −

3 F

20 F

30 F

80 F

14 F

Figure 6.58 For Prob. 6.24.

b

Figure 6.54

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

For Prob. 6.20. 6.21 Determine the equivalent capacitance at terminals a-b of the circuit in Fig. 6.55.

v1 

C2 vs, C1  C2

v2 

C1 vs C1  C2

assuming that the initial conditions are zero. 5 F

6 F

4 F

C1

a 2 F

3 F

12 F

b

Figure 6.55

vs

+ −

+ v1 − + v2 −

C2

is

For Prob. 6.21. (a)

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

Figure 6.59 For Prob. 6.25.

(b)

i1

i2

C1

C2

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(b) For two capacitors in parallel as in Fig. 6.59(b), show that the current-division rule is i1 

C1 is, C1  C2

i2 

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

C2 is C1  C2

assuming that the initial conditions are zero. 6.26 Three capacitors, C1  5 mF, C2  10 mF, and C3  20 mF, are connected in parallel across a 150-V source. Determine:

is (mA)

6 F

90 is 0

(a) the total capacitance,

2 t (s)

1

(b) the charge on each capacitor,

Figure 6.62

(c) the total energy stored in the parallel combination.

For Prob. 6.30.

6.27 Given that four 4-mF capacitors can be connected in series and in parallel, find the minimum and maximum values that can be obtained by such series/parallel combinations.

+ vo (t) −

3 F

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

*6.28 Obtain the equivalent capacitance of the network shown in Fig. 6.60. is (mA) 30 40 F

50 F

30 F

0

10 F

20 F

1

2

3

5

4

t

−30

Figure 6.60

i1

For Prob. 6.28. 6 F

is

6.29 Determine Ceq for each circuit in Fig. 6.61.

i2 + v −

4 F

Figure 6.63

C

For Prob. 6.31. C eq

C

C C

C

6.32 In the circuit of Fig. 6.64, let is  50e2t mA and v1(0)  50 V, v2(0)  20 V. Determine: (a) v1(t) and v2(t), (b) the energy in each capacitor at t  0.5 s.

(a)

C

C

C eq 12 F C (b)

Figure 6.61

C

+ is

For Prob. 6.29.

Figure 6.64 * An asterisk indicates a challenging problem.

For Prob. 6.32.

v1

– 20 F

v2

+ –

40 F

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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.

6.41 The voltage across a 2-H inductor is 20 (1  e2t) 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. v(t) (V)

5F + 45 V −

10

a 3F

2F 0

b

Figure 6.65

Figure 6.67

For Prob. 6.33.

For Prob. 6.42.

Section 6.4 Inductors 6.34 The current through a 10-mH inductor is 10et2 A. Find the voltage and the power at t  3 s. 6.35 An inductor has a linear change in current from 50 mA to 100 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, across the inductor for 0 6 t 6 p p200 s, and the energy stored at t  200 s.

1

3

2

t

5

4

6.43 The current in an 80-mH inductor increases from 0 to 60 mA. How much energy is stored in the inductor? *6.44 A 100-mH inductor is connected in parallel with a 2-k resistor. The current through the inductor is i(t)  50e400t 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 in the inductor at t  0. 6.45 If the voltage waveform in Fig. 6.68 is applied to a 10-mH inductor, find the inductor current i(t). Assume i(0)  0. v (t)

6.38 The current through a 40-mH inductor is

5

i(t)  b

0, te2t A,

t 6 0 t 7 0 0

Find the voltage v(t).

1

2

t

6.39 The voltage across a 200-mH inductor is given by –5

v(t)  3t2  2t  4 V

for t 7 0.

Determine the current i(t) through the inductor. Assume that i(0)  1 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 6.68 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. 2Ω

i(A) 10 3A 0

4Ω

+ vC −

2F

0.5 H 5Ω

2

4

6

t (ms)

Figure 6.66

Figure 6.69

For Prob. 6.40.

For Prob. 6.46.

iL

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6.47 For the circuit in Fig. 6.70, calculate the value of R that will make the energy stored in the capacitor the same as that stored in the inductor under dc conditions. R

6.52 Using Fig. 6.74, design a problem to help other students better understand how inductors behave when connected in series and when connected in parallel.

160 F 2Ω

5A

L4 4 mH

L2

Figure 6.70

Leq

For Prob. 6.47. 6.48 Under steady-state dc conditions, find i and v in the circuit in Fig. 6.71. i

5 mA

L3

L1

L5

L6

Figure 6.74 For Prob. 6.52.

2 mH

30 kΩ

+ v −

6 F

20 kΩ

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

Figure 6.71 For Prob. 6.48.

Section 6.5 Series and Parallel Inductors

6 mH

6.49 Find the equivalent inductance of the circuit in Fig. 6.72. Assume all inductors are 10 mH.

8 mH

a 5 mH

12 mH

8 mH 6 mH 4 mH b 8 mH

10 mH

Figure 6.75 For Prob. 6.53.

Figure 6.72 For Prob. 6.49. 6.50 An energy-storage network consists of seriesconnected 16-mH and 14-mH inductors in parallel with series-connected 24-mH and 36-mH inductors. Calculate the equivalent inductance.

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

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

9H 10 H

10 mH 60 mH 25 mH

12 H 4H

20 mH

a

6H

b 30 mH a

Figure 6.73

Figure 6.76

For Prob. 6.51.

For Prob. 6.54.

b

3H

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

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 6 t 6 6 s.

L

i(t)

L Leq L

L

2

L 0 (a)

1

2

3

4

5

6

t

Figure 6.80 For Prob. 6.58.

L L

L

L

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

L Leq

v1 

(b)

Figure 6.77

L1 vs, L1  L2

v2 

L2 vs L1  L2

assuming that the initial conditions are zero.

For Prob. 6.55.

(b) For two inductors in parallel as in Fig. 6.81(b), show that the current-division principle is 6.56 Find Leq in the circuit of Fig. 6.78.

i1 

L2 is, L1  L 2

i2 

L1 is L1  L 2

assuming that the initial conditions are zero. L

L

L

L1

L

L

L

+ v − 1 L

L

vs

+ v2 −

+ −

is

L2

i1

i2

L1

L2

L eq (a)

Figure 6.78

(b)

Figure 6.81

For Prob. 6.56.

For Prob. 6.59.

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

i

2 4H

6.60 In the circuit of Fig. 6.82, io(0)  2 A. Determine io(t) and vo(t) for t 7 0.

di dt io (t)

+−

a

+

L eq 3H

5H

4e–2t A

b

Figure 6.79

Figure 6.82

For Prob. 6.57.

For Prob. 6.60.

3H

5H

−

vo

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

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 7 0,

i1

(b) v just after the switch has been moved to position B,

i2

(c) v(t) long after the switch is in position B.

4 mH +

vo –

is

20 mH 4Ω

6 mH

B

t=0A

i L eq

Figure 6.83

12 V

For Prob. 6.61.

+ –

0.5 H

+ v –

5Ω

6A

Figure 6.86 6.62 Consider the circuit in Fig. 6.84. Given that v(t)  12e3t mV for t 7 0 and i1(0)  10 mA, find: (a) i2(0), (b) i1(t) and i2(t).

For Prob. 6.64.

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)  50e200t mV, t 0, find:

25 mH +

i1(t)

i2(t)

v(t)

20 mH

60 mH

(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.

Figure 6.84 For Prob. 6.62.

i(t) + Black box v

i1

i2

5H

20 H

t=0

−

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

Figure 6.87 For Prob. 6.65. + vo –

i1(t)

i2(t)

2H

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).

i2(t) (A) 4

i1(t) (A) 3

Section 6.6 Applications 0

Figure 6.85 For Prob. 6.63.

3

6 t (s)

0

2

4

6 t (s)

6.67 An op amp integrator has R  50 k and C  0.04 mF. If the input voltage is vi  10 sin 50t mV, obtain the output voltage.