7.13 SUMMARYTable 7.13-1 summarizes the element equations for capaci-tors and inductors. (Notice that the voltage and currentreferred to in these equations adhere to the passive conven-tion.) Unlike the circuit elements we encountered in previouschapters, the element equations for capacitors and inductorsinvolve derivatives and integrals.Circuits that contain capacitors and/or inductors are able tostore energy. The energy stored in the electric field of acapacitor is equal to 1

2 Cv2 tð Þ, where v(t) is the voltage acrossthe capacitor. The energy stored in the magnetic field of ainductor is equal to 1

2 Li2 tð Þ, where i(t) is the current in theinductor.

Circuits that contain capacitors and/or inductors have mem-ory. The voltages and currents in that circuit at a particulartime depend not only on other voltages and currents at thatsame instant of time but also on previous values ofthose currents and voltages. For example, the voltage acrossa capacitor at time t1 depends on the voltage across thatcapacitor at an earlier time t0 and on the value of the capacitorcurrent between t0 and t1.A set of series or parallel capacitors can be reduced to anequivalent capacitor. A set of series or parallel inductors canreadily be reduced to an equivalent inductor. Table 7.13-2summarizes the equations required to do so.

Table 7.13-1 Element Equations for Capacitors and Inductors

CAPACITOR INDUCTOR

C

v(t)+ –

i(t)L

v(t)+ –

i(t)

i tð Þ ¼ Cddt

v tð Þ i tð Þ ¼ 1L

Z t

t0

v tð Þdtþ i t0ð Þ

v tð Þ ¼ 1C

Z t

t0

i tð Þdtþ v t0ð Þ v tð Þ ¼ Lddt

i tð Þ

vH

vL

t1 t2

vc(t) (V)

vc(t)

t (ms)

+–Vs = 5 V

Voltmeter

–

+

CR

t = t1

+ –vo

FIGURE 7.12-4 Using an operational amplifier integrator to measure an interval of time.

304 7. Energy Storage Elements

In the absence of unbounded currents, the voltage across acapacitor cannot change instantaneously. Similarly, in theabsence of unbounded voltages, the current in an inductorcannot change instantaneously. In contrast, the current in acapacitor and voltage across an inductor are both able tochange instantaneously.We sometimes consider circuits that contain capacitors andinductors and have only constant inputs. (The voltages of theindependent voltage sources and currents of the independentcurrent sources are all constant.) When such a circuit is atsteady state, all the currents and voltages in that circuit willbe constant. In particular, the voltage across any capacitorwill be constant. The current in that capacitor will be zero due

to the derivative in the equation for the capacitor current.Similarly, the current through any inductor will be constantand the voltage across any inductor will be zero. Conse-quently, the capacitors will act like open circuits and theinductors will act like short circuits. Notice that this situationoccurs only when all of the inputs to the circuit are constant.An op amp and a capacitor can be used to make circuits thatperform the mathematical operations of integration anddifferentiation. Appropriately, these important circuits arecalled the integrator and the differentiator.The element voltages and currents in a circuit containingcapacitors and inductors can be complicated functions oftime. MATLAB is useful for plotting these functions.

Table 7.13-2 Parallel and Series Capacitors and Inductors

SERIES ORPARALLEL CIRCUIT

EQUIVALENTCIRCUIT EQUATION

L1

L2

v(t)+ –

i(t)

C1

C2

v(t)+ –

i(t)

L1 L2

v(t)+ –

i(t)

v(t)+ –

i(t) C2C1

Leq

v(t)+ –

i(t)

Ceq

Ceq

v(t)+ –

i(t)

Leq

v(t)+ –

i(t)

v(t)+ –

i(t)

Leq ¼ 11L1

þ 1L2

Leq ¼ L1 þ L2

Ceq ¼ C1 þ C2

Ceq ¼ 11

C1þ 1

C2

PROBLEMS

Section 7.2 Capacitors

P 7.2-1 A 15-mF capacitor has a voltage of 5 V across it att ¼ 0. If a constant current of 25 mA flows through the capacitor,how long will it take for the capacitor to charge up to 150 mC?

Answer: t ¼ 3 ms

P 7.2-2 The voltage v(t) across a capacitor and current i(t) inthat capacitor adhere to the passive convention. Determine thecurrent i(t) when the capacitance is C ¼ 0:125 F, and thevoltage is v tð Þ ¼ 12 cos 2t þ 30-ð Þ V.

Hint: ddt

A cos ot þ yð Þ ¼ $A sin ot þ yð Þ ) ddt

ot þ yð Þ¼ $Ao sin ot þ yð Þ¼ Ao cos ot þ yþ p

2

* +* +

Answer: i tð Þ ¼ 3 cos 2t þ 120-ð Þ A

P 7.2-3 The voltage v(t) across a capacitor and current i(t) inthat capacitor adhere to the passive convention. Determine thecapacitance when the voltage is v tð Þ ¼ 12 cos 500t $ 45-ð Þ V andthe current is i tð Þ ¼ 3 cos 500t þ 45-ð Þ mA.

Answer: C ¼ 0:5 mF

Problem available in WileyPLUS at instructor’s discretion.

Problems 305

P 7.2-4 Determine v(t) for the circuit shown in FigureP 7.2-4a(t) when the is(t) is as shown in Figure P 7.2-4band vo 0$ð Þ ¼ $1 mV.

(b)(a)

1 2 3 4 5 6

–2

0

4

is v

+

–

2 pF

t (ns)

is ( A)µ

Figure P 7.2-4 (a) Circuit and (b) waveform of current source.

P 7.2-5 The voltage v(t) and current i(t) of a 1-F capacitoradhere to the passive convention. Also, v 0ð Þ ¼ 0 V andi 0ð Þ ¼ 0 A. (a) Determine v(t) when i tð Þ ¼ x tð Þ, where x(t)is shown in Figure P 7.2-5 and i(t) has units of A. (b) Determinei(t) when v tð Þ ¼ x tð Þ, where x(t) is shown in Figure P 7.2-5 andv(t) has units of V.

Hint:x tð Þ ¼ 4t $ 4when1 < t < 2,andx tð Þ ¼ $4t þ 12when2 < t < 3.

x

1

0

2

3

4

5

10 2 3 4 t (s)

Figure P 7.2-5

P 7.2-6 The voltage v(t) and current i(t) of a 0.5-F capacitoradhere to the passive convention. Also, v 0ð Þ ¼ 0 V andi 0ð Þ ¼ 0 A. (a) Determine v(t) when i tð Þ ¼ x tð Þ, where x(t)is shown in Figure P 7.2-6 and i(t) has units of A. (b) Determinei(t) when v tð Þ ¼ x tð Þ, where x(t) is shown in Figure P 7.2-6 andv(t) has units of V.

Hint: x tð Þ ¼ 0:2t $ 0:4 when 2 < t < 6.

x

0.2

0.0

0.4

0.6

0.8

1.0

20 4 6 8 t (s)

Figure P 7.2-6

P 7.2-7 The voltage across a 40-mF capacitor is 25 V att0 ¼ 0. If the current through the capacitor as a function of time isgiven by i tð Þ ¼ 6e$6t mA for t < 0, find v(t) for t > 0.

Answer: v tð Þ ¼ 50 $ 25e$6t V

P 7.2-8 Find i for the circuit of Figure P 7.2-8 if v ¼5 1 $ 2e$2tð Þ V.

+

–vi 200 kΩ10 µF

Figure P 7.2-8

P 7.2-9 Determine v(t) for t ( 0 for the circuit of FigureP 7.2-9a when is(t) is the current shown in Figure P 7.2-9b andv 0ð Þ ¼ 1 V.

is (A)

t (s)

(b)

2

–2

1 2 3

4

is (t) v(t)

(a)

0.5 F+

–

Figure P 7.2-9

P 7.2-10 Determine v(t) for t ( 0 for the circuit of FigureP 7.2-10a when v 0ð Þ ¼ $4 V and is(t) is the current shown inFigure P 7.2-10b.

306 7. Energy Storage Elements

is (A)2

0.25 0.5

t (s)

(b)

is (t)

2 Ω

0.1 F v(t)

(a)

+

–

Figure P 7.2-10

P 7.2-11 Determine i(t) for t ( 0 for the circuit of FigureP 7.2-11a when vs(t) is the voltage shown in Figure P 7.2-11b.

t (s)

vs (V)

2.01.51.00.50

(b)

20

5 F

(a)

5 Ω

i(t)

vs(t) +–

Figure P 7.2-11

P 7.2-12 The capacitor voltage in the circuit shown in FigureP 7.2-12 is given by

v tð Þ ¼ 12$ 10e$2t V for t ( 0

Determine i(t) for t > 0.

i(t)4 Ω

6 Ω

2 A

v(t)1

20 F+

–

Figure P 7.2-12

P 7.2-13 The capacitor voltage in the circuit shown inFigure P 7.2-13 is given by

v tð Þ ¼ 2:4þ 5:6e$5t V for t ( 0

Determine i(t) for t > 0.

v(t) i(t)2 mF 12 V100 Ω

20 Ω 400 Ω

+

–

+–

Figure P 7.2-13

P 7.2-14 The capacitor voltage in the circuit shown inFigure P 7.2-14 is given by

v tð Þ ¼ 10$ 8e$5t V for t ( 0

Determine i(t) for t > 0.

v(t)

i(t)

20 mF 12 V60 Ω12 Ω+

–

+–

Figure P 7.2-14

P 7.2-15 Determine the voltage v(t) for t > 0 for the circuitof Figure P 7.2-15b when is(t) is the current shown in FigureP 7.2-15a. The capacitor voltage at time t ¼ 0 is v 0ð Þ ¼ $12 V.

(a)

–2

2

4

108642–2–4 t (s)

is(t)(A)

(b)

v(t)is(t) F

+

–

1 3

Figure P 7.2-15 (a) The voltage source voltage. (b) The circuit.

P 7.2-16 The input to the circuit shown in Figure P 7.2-16is the current

i tð Þ ¼ 3:75e$1:2t A for t > 0

The output is the capacitor voltage

v tð Þ ¼ 4$ 1:25e$1:2t V for t > 0

Find the value of the capacitance C.

Problems 307

C

v(t)

i (t)

+ _

Figure P 7.2-16

P 7.2-17 The input to the circuit shown in Figure P 7.2-17is the current

i tð Þ ¼ 3e$25t A for t > 0

The initial capacitor voltage is vC 0ð Þ ¼ $2 V. Determine thecurrent source voltage v(t) for t > 0.

i(t)

v(t)+ _

vC(t)+ _

4 Ω 0.05 F

Figure P 7.2-17

P 7.2-18 The input to the circuit shown in Figure P 7.2-18is the current

i tð Þ ¼ 3e$25t A for t > 0

The output is the voltage

v tð Þ ¼ 9:6e$25t þ 0:4 V for t > 0

The initial capacitor voltage is vC 0ð Þ ¼ $2 V. Determine thevalues of the capacitance C and resistance R.

C

v(t)+ _

vC(t)+ _

i (t)

R

Figure P 7.2-18

P 7.2-19 The input to the circuit shown in Figure P 7.2-19 isthe voltage

v tð Þ ¼ 8þ 5e$10t V for t > 0

Determine the current i(t) for t > 0.

0.05 F

v(t)

4 Ω

+ –

i(t)

Figure P 7.2-19

P 7.2-20 The input to the circuit shown in Figure P 7.2-20is the voltage:

v tð Þ ¼ 3þ 4e$2t A for t > 0

The output is the current i tð Þ ¼ 0:3 $ 1:6e$2t V for t > 0Determine the values of the resistance and capacitance.

Answers: R ¼ 10 V and C ¼ 0:25 F

C

v(t)

+ –

i(t)

R

Figure P 7.2-20

P 7.2-21 Consider the capacitor shown in Figure P 7.2-21.The current and voltage are given by

i tð Þ ¼0:5 0 < t < 0:52 0:5 < t < 1:50 t > 1:5

8<

:

and v tð Þ ¼2t þ 8:6 0 ' t ' 0:5at þ b 0:5 ' t ' 1:5

c t ( 1:5

8<

:

where a, b, and c are real constants. (The current is given inamps, the voltage in volts, and the time in seconds.) Determinethe values of a, b, and c.

Answers: a ¼ 8 V/s; b ¼ 5:6 V, and c ¼ 17:6 V

C = 0.25 F

+i(t)

v(t)

_

Figure P 7.2-21

308 7. Energy Storage Elements

P 7.2-22 At time t ¼ 0, the voltage across the capacitorshown in Figure P 7.2-22 is v 0ð Þ ¼ $20 V. Determine thevalues of the capacitor voltage at times 1 ms, 3 ms, and 7 ms.

i (t)

i (t), mA

v(t)

+

–

2 4 7

2.5 µFt, (ms)

40

Figure P 7.2-22

Section 7.3 Energy Storage in a Capacitor

P 7.3-1 The current i through a capacitor is shown in FigureP 7.3-1. When v 0ð Þ ¼ 0 and C ¼ 0:5 F, determine and plotv(t), p(t), and w(t) for 0 s < t < 6 s.

i(A)

0.2

0.0

0.4

0.6

0.8

1.0

20 4 6 8 t (s)

Figure P 7.3-1

P 7.3-2 In a pulse power circuit, the voltage of a 10-mFcapacitor is zero for t < 0 and

v ¼ 5 1$ e$4000t! "V t ( 0

Determine the capacitor current and the energy stored in thecapacitor at t ¼ 0 ms and t ¼ 10 ms.

P 7.3-3 If vc(t) is given by the waveform shown in FigureP 7.3-3, sketch the capacitor current for $1 s < t < 2 s. Sketchthe power and the energy for the capacitor over the same timeinterval when C ¼ 1 mF.

20

–20

0–1

vc (V)

t (s)1 2

Figure P 7.3-3

P 7.3-4 The current through a 2-mF capacitor is 50 cos(10tþp/6) mA for all time. The average voltage across the capacitor iszero. What is the maximum value of the energy stored in thecapacitor? What is the first nonnegative value of t at whichthe maximum energy is stored?

P 7.3-5 A capacitor is used in the electronic flash unit of acamera. A small battery with a constant voltage of 6 V is used tocharge a capacitor with a constant current of 10 mA. How longdoes it take to charge the capacitor when C ¼ 10 mF? What isthe stored energy?

P 7.3-6 The initial capacitor voltage of the circuit shown inFigure P 7.3-6 is vc 0$ð Þ ¼ 3 V. Determine (a) the voltagev(t) and (b) the energy stored in the capacitor at t ¼ 0:2 sand t ¼ 0:8 s when

i tð Þ ¼3e5t A 0 < t < 1

0 t ( 1 s

(

Answers:

(a) 18e5t V; 0 ' t < 1(b) w 0:2ð Þ ¼ 6:65 J and w 0:8ð Þ ¼ 2:68 kJ

5 Ω

t = 0

+–

vc

+

–

v

0.2 F

i(t)

Figure P 7.3-6

P 7.3-7 (a) Determine the energy stored in the capacitor inthe circuit shown in Figure P 7.3-7 when the switch is closedand the circuit is at steady state. (b) Determine the energy storedin the capacitor when the switch is open and the circuit is atsteady state.

++–

–

v (t )2.2 mF

75 kΩ

75 kΩ12 V

Figure P 7.3-7

Section 7.4 Series and Parallel Capacitors

P 7.4-1 Find the current i(t) for the circuit of Figure P 7.4-1.

Answer: i tð Þ ¼ $1:2 sin 100t mA

Problems 309

6 cos 100t V 2 F3 Fµ

µ µ4 F

i(t)

+–

Figure P 7.4-1

P 7.4-2 Find the current i(t) for the circuit of Figure P 7.4-2.

Answer: i tð Þ ¼ $1:5e$250t mA

4 F5 + 3e–250t V 2 F4 Fµ µ

µµ4 F

i(t)

+–

Figure P 7.4-2

P 7.4-3 The circuit of Figure P 7.4-3 contains fiveidentical capacitors. Find the value of the capacitance C.

Answer: C ¼ 10 mF

14 sin 250t V

i(t) = 25 cos 250t mA

+–

CC C C

C

Figure P 7.4-3

P 7.4-4 The circuit shown in Figure P 7.4-4 contains sevencapacitors, each having capacitance C. The source voltage isgiven by

v tð Þ ¼ 4 cos 3tð ÞV

Find the current i(t) when C ¼ 1 F.

i(t)

v(t)

C C

C

CC

C

C

+–

Figure P 7.4-4

P 7.4-5 Determine the value of the capacitance C in thecircuit shown in Figure P 7.4-5, given that Ceq ¼ 8 F.

Answer: C ¼ 20 F

A

C

B

12 F 4 F

10 F 30 F

12 F

16 F

Ceq

Figure P 7.4-5

P 7.4-6 Determine the value of the equivalent capacitance Ceq,in the circuit shown in Figure P 7.4-6.

Answer: Ceq ¼ 10 F

a

b

Ceq

60 F 30 F

15 F

10 F

40 F

60 F

Figure P 7.4-6

P 7.4-7 The circuit shown in Figure P 7.4-7 consists ofnine capacitors having equal capacitance C. Determine thevalue of the capacitance C, given that Ceq ¼ 50 mF.

Answer: C ¼ 90 mF

C

C

Ceq

CC

C

C

C

CC

Figure P 7.4-7

P 7.4-8 The circuit shown in Figure P 7.4-8 is at steadystate before the switch opens at time t ¼ 0. The voltage v(t) isgiven by

310 7. Energy Storage Elements

v tð Þ ¼3:6 V

3:6e$2:5t V

for t ' 0

for t ( 0

,

(a) Determine the energy stored by each capacitor before theswitch opens.

(b) Determine the energy stored by each capacitor 1 s after theswitch opens.

The parallel capacitors can be replaced by an equivalentcapacitor.

(c) Determine the energy stored by the equivalent capacitorbefore the switch opens.

(d) Determine the energy stored by the equivalent capacitor 1 safter the switch opens.

18 V

t = 0

20 Ω5 Ω 60 mF 20 mFv(t)+

–

+

–

Figure P 7.4-8

P 7.4-9 The circuit shown in Figure P 7.4-9 is at steady statebefore the switch closes. The capacitor voltages are both zerobefore the switch closes v1 0ð Þ ¼ v2 0ð Þ ¼ 0ð Þ. The current i(t) isgiven by

i tð Þ ¼ 0 A2:4e$30t A

for t < 0for t > 0

,

(a) Determine the capacitor voltages v1(t) and v2(t) for t ( 0.(b) Determine the energy stored by each capacitor 20 ms after

the switch closes.

The series capacitors can be replaced by an equivalentcapacitor.

(c) Determine the voltage across the equivalent capacitor,þ on top, for t ( 0.

(d) Determine the energy stored by the equivalent capacitor20 ms after the switch closes.

5 Ω

25 Ω12 V

10 mF

40 mF

t = 0

i(t)

v1 (t)

v2 (t)

+

–

+

–+–

Figure P 7.4-9

P 7.4-10 Find the relationship for the division of currentbetween two parallel capacitors as shown in Figure P 7.4-10.

Answer: in ¼ iCn= C1 þ C2ð Þ; n ¼ 1; 2

C1 C2

i

i1 i2

Figure P 7.4-10

Section 7.5 Inductors

P 7.5-1 Nikola Tesla (1857–1943) was an American electricalengineer who experimented with electric induction. Teslabuilt a large coil with a very large inductance, shown in FigureP 7.5-1. The coil was connected to a source current

is ¼ 100 sin 400t A

so that the inductor current iL ¼ is. Find the voltage across theinductor and explain the discharge in the air shown in thefigure. Assume that L¼ 200H and the average dischargedistance is 2 m. Note that the dielectric strength of air is3* 106 V/m.

Figure P 7.5-1 Nikola Tesla sits impassively as alternating currentinduction coils discharge millions of volts with a roar audible 10miles away (about 1910).

P 7.5-2 The model of an electric motor consists of a seriescombination of a resistor and inductor. A current i tð Þ ¼ 4te$t Aflows through the series combination of a 10-V resistor and 0.1-Hinductor. Find the voltage across the combination.Answer: v tð Þ ¼ 0:4e$t þ 39:6te$t V

P 7.5-3 The voltage v(t) and current i(t) of a 1-H inductoradhere to the passive convention. Also, v 0ð Þ ¼ 0 V andi 0ð Þ ¼ 0 A.

(a) Determine v(t) when i tð Þ ¼ x tð Þ, where x(t) is shown inFigure P 7.5-3 and i(t) has units of A.

(b) Determine i(t) when v tð Þ ¼ x tð Þ, where x(t) is shown inFigure P 7.5-3, and v(t) has units of V.

# Everett Collection Historical/Alamy

Problems 311

x

1

0

2

3

4

5

10 2 3 4 t(s)

Figure P 7.5-3

Hint:x tð Þ ¼ 4t $ 4when1 < t < 2,andx tð Þ ¼ $4t þ 12when2 < t < 3.

P 7.5-4 The voltage v(t) across an inductor and current i(t) inthat inductor adhere to the passive convention. Determine thevoltage v(t) when the inductance is L ¼ 250 mH, and thecurrent is i tð Þ ¼ 120 sin 500t $ 30-ð Þ mA.

Hint: ddt

A sin ot þ yð Þ ¼ A cos ot þ yð Þ ) ddt

ot þ yð Þ¼ Ao cos ot þ yð Þ¼ Ao sin ot þ yþ p

2

* +* +

Answer: v tð Þ ¼ 15 sin 500t þ 60-ð Þ V

P 7.5-5 Determine iL tð Þ for t > 0 when iL 0ð Þ ¼ $2 mA forthe circuit of Figure P 7.5-5a when vs(t) is as shown in FigureP 7.5-5b.

(b)(a)

1 2 3

–1

4

vs

iL

5 mH

vs (mV)

t ( s)µ

+–

Figure P 7.5-5

P 7.5-6 Determine v(t) for t > 0 for the circuit of FigureP 7.5-6a when iL 0ð Þ ¼ 0 and is is as shown in Figure P 7.5-6b.

(a)

iL

4 mHisv

2 kΩ +

–

(b)0 83 5 7

0

–1

1

is (mA)

t (ms)

1

Figure P 7.5-6

P 7.5-7 The voltage v(t) and current i(t) of a 0.5-H inductoradhere to the passive convention. Also, v 0ð Þ ¼ 0 V, andi 0ð Þ ¼ 0 A.

(a) Determine v(t) when i tð Þ ¼ x tð Þ, where x(t) is shown inFigure P 7.5-7 and i(t) has units of A.

(b) Determine i(t) when v tð Þ ¼ x tð Þ, where x(t) is shown inFigure P 7.5-7 and v(t) has units of V.

Hint: x tð Þ ¼ 0:2t $ 0:4 when 2 < t < 6.

x

0.2

0.0

0.4

0.6

0.8

1.0

20 4 6 8 t (s)

Figure P 7.5-7

P 7.5-8 Determine i(t) for t ( 0 for the current of FigureP 7.5-8a when i 0ð Þ ¼ 25 mA and vs(t) is the voltage shown inFigure P 7.5-8b.

vs (t)

vs (V)

100 H

i(t) t(s)

(a) (b)

2

1

2 4 6 8 9

−2

−4

+–

Figure P 7.5-8

P 7.5-9 Determine i(t) for t ( 0 for the current of FigureP 7.5-9a when i 0ð Þ ¼ $2 A and vs(t) is the voltage shown inFigure P 7.5-9b.

312 7. Energy Storage Elements

vs (t)

vs (V)

5 H

i(t)

t(s)

(a) (b)

1 2 3−1

2

4

+–

Figure P 7.5-9

P 7.5-10 Determine i(t) for t ( 0 for the current of FigureP 7.5-10a when i 0ð Þ ¼ 1 A and vs(t) is the voltage shown inFigure P 7.5-10b.

vs(t)

vs (V)

2 H

i(t)

t(s)

(a) (b)

2 4 6

−1

2

+–

Figure P 7.5-10

P 7.5-11 Determine i(t) for t ( 0 for the circuit of FigureP 7.5-11a when i 0ð Þ ¼ 25 mA and vs(t) is the voltage shown inFigure P 7.5-11b.

vs(t)

vs (V)

200 H

i(t) t(s)

(a) (b)

1

1 2 3 5 6 7 8 9

−1

−2

+–

Figure P 7.5-11

P 7.5-12 The inductor current in the circuit shown inFigure P 7.5-12 is given by

i tð Þ ¼ 6þ 4e$8t A for t ( 0

Determine v(t) for t > 0.

12 V2 Ω

8 Ω 0.2 H

i(t)v(t)+ –

+–

Figure P 7.5-12

P 7.5-13 The inductor current in the circuit shown inFigure P 7.5-13 is given by

i tð Þ ¼ 5$ 3e$4t A for t ( 0

Determine v(t) for t > 0.

10 A 24 Ω 24 Ω24 Ω

4 H

v(t) i(t)+ –

Figure P 7.5-13

P 7.5-14 The inductor current in the circuit shown inFigure P 7.5-14 is given by

i tð Þ ¼ 3þ 2e$3t A for t ( 0

Determine v(t) for t > 0.

5 A

6 Ω

9 Ω 5 Hv(t)i(t)+

–

Figure P 7.5-14

P 7.5-15 Determine the current i(t) for t > 0 for the circuit ofFigure P 7.5-15b when vs(t) is the voltage shown in FigureP 7.5-15a. The inductor current at time t ¼ 0 is i 0ð Þ ¼ $12 A.

(a)

–2

2

4

108642–2–4 t (s)

(V)

vs(t)

(b)

i(t)

H+– 1 3

vs(t)

Figure P 7.5-15 (a) The voltage source voltage. (b) The circuit.

P 7.5-16 The input to the circuit shown in Figure P 7.5-16is the voltage

v tð Þ ¼ 15e$4t V for t > 0

Problems 313

The initial current in the inductor is i 0ð Þ ¼ 2 A. Determine theinductor current i(t) for t > 0.

2.5 H

v (t)

i(t)+ –

Figure P 7.5-16

P 7.5-17 The input to the circuit shown in Figure P 7.5-17is the voltage

v tð Þ ¼ 4e$20t V for t > 0

The output is the current

i tð Þ ¼ $1:2e$20t $ 1:5 A for t > 0

The initial inductor current is iL 0ð Þ ¼ $3:5 A. Determine thevalues of the inductance L and resistance R.

R

L

v (t)

iL(t)

i (t)

+ –

Figure P 7.5-17

P 7.5-18 The source voltage the circuit shown in FigureP 7.5-18 is v(t) = 8 e$ 400 t V after time t = 0. The initial inductorcurrent is iL(0) = 210 mA. Determine the source current i(t)for t> 0.

Answer: i(t) = 360 e$ 400t$ 190 mA for t> 0.

i L(t )

i (t )

+ –

200 Ω

50 mH

v (t )

Figure P 7.5-18

P 7.5-19 The input to the circuit shown in Figure P 7.5-19 isthe current

i(t) ¼ 5þ 2e$7t A for t > 0

The output is the voltage : v(t) ¼ 75$ 82e$7t V for t > 0

Determine the values of the resistance and inductance.

v (t )+ –

R L

i (t )

Figure P 7.5-19

P 7.5-20 Consider the inductor shown in Figure P 7.5-20.The current and voltage are given by

i tð Þ ¼5t $ 4:6 0 ' t ' 0:2at þ b 0:2 ' t ' 0:5

c t ( 0:5

8<

:

and v tð Þ ¼12:5 0 < t < 0:225 0:2 < t < 0:50 t > 0:5

8<

:

where a, b, and c are real constants. (The current is givenin amps, the voltage in volts, and the time in seconds.)Determine the values of a, b, and c.

Answers: a ¼ 10 A/s; b ¼ $5:6 A, and c ¼ $0:6 A

v (t)

i (t)

L = 2.5 H

+

–

Figure P 7.5-20

P 7.5-21 At time t ¼ 0, the current in the inductor shownin Figure P 7.5-21 is i 0ð Þ ¼ 45 mA. Determine the values of theinductor current at times 1 ms, 4 ms, and 6 ms.

v (t)

v (t), V

i (t)

2 4

250 mH

t, (ms)

20

+–

Figure P 7.5-21

P 7.5-22 One of the three elements shown in Figure P 7.5-22 is a resistor, one is a capacitor, and one is an inductor. Given

314 7. Energy Storage Elements

i tð Þ ¼ 0:25cos 2tð Þ A;

and va(t) ¼ $ 10 sin(2t) V, vb(t) ¼ 10 sin(2t) V, and vc(t) ¼10 cos(2t) V, determine the resistance of the resistor, thecapacitance of the capacitor, and the inductance of theinductor. (We require positive values of resistance,capacitance, and inductance.)

Answers: resistance ¼ 40 V, capacitance ¼ 0.0125 F, andinductance ¼ 20 H

va(t)

i (t)

+ – vb(t)

i (t)

+ –

vc(t)

i (t)

+ –

Figure P 7.5-22

P 7.5-23 One of the three elements shown in Figure P 7.5-23 is a resistor, one is a capacitor, and one is an inductor. Given

v tð Þ ¼ 24cos 5tð Þ V;

and ia tð Þ ¼ 3 cos 5tð Þ A; ib tð Þ ¼ 12 sin 5tð Þ A and ic tð Þ ¼$1:8 sin 5tð Þ A; determine the resistance of the resistor,the capacitance of the capacitor, and the inductance ofthe inductor. (We require positive values of resistance,capacitance, and inductance.)

v (t)

ia (t)

+ – v (t)

ib(t)

+ –

v (t)

i c(t)

+ –

Figure P 7.5-23

Section 7.6 Energy Storage in an Inductor

P 7.6-1 The current i(t) in a 100-mH inductor connectedin a telephone circuit changes according to

i tð Þ ¼0 t ' 04t 0 ' t ' 14 t ( 1

8<

:

where the units of time are seconds and the units of current areamperes. Determine the power p(t) absorbed by the inductorand the energy w(t) stored in the inductor.

Answers: p tð Þ ¼0 t ' 0

1:6t 0 < t < 10 t ( 1

8<

: and

w tð Þ ¼0 t ' 0

0:8t2 0 < t < 10:8 t ( 1

8<

:

The units of p(t) are W and the units of w(t) are J.

P 7.6-2 The current i(t) in a 5-H inductor is

i tð Þ ¼0 t ' 0

4 sin 2t t ( 0

,

where the units of time are s and the units of current are A.Determine the power p(t) absorbed by the inductor and theenergy w(t) stored in the inductor.

Hint: 2 cos Að Þ sin Bð Þ ¼ sin A þ Bð Þ þ sin A $ Bð Þ

P 7.6-3 The voltage v(t) across a 25-mH inductor used in afusion power experiment is

v tð Þ ¼0 t ' 0

6 cos 100t t ( 0

,

where the units of time are s and the units of voltage are V.The current in this inductor is zero before the voltage changesat t ¼ 0. Determine the power p(t) absorbed by the inductorand the energy w(t) stored in the inductor.

Hint: 2 cos Að Þ sin Bð Þ ¼ sin A þ Bð Þ þ sin A $ Bð Þ

Answer: p tð Þ ¼ 7:2sin200t W and w tð Þ ¼ 3:6 1 $ cos 200t½ , mJ

P 7.6-4 The current in an inductor, L ¼ 1=4 H, is i ¼ 4te$t Afor t ( 0 and i ¼ 0 for t < 0. Find the voltage, power, andenergy in this inductor.

Partial Answer: w ¼ 2t2e$2t J

P 7.6-5 The current through the inductor of a televisiontube deflection circuit is shown in Figure P 7.6-5 whenL ¼ 1=2 H. Find the voltage, power, and energy in theinductor.

Partial Answer:p ¼ 2t for 0 ' t < 1

¼ 2 t $ 2ð Þ for 1 < t < 2¼ 0 for other t

2

0 1 2

i (A)

t (s)

Figure P 7.6-5

Problems 315

Section 7.7 Series and Parallel Inductors

P 7.7-1 Find the current i(t) for the circuit of Figure P 7.7-1.

Answer: i tð Þ ¼ 15 sin 100t mA

6 cos 100t V +–

2 H

6 H 3 H

i(t)

Figure P 7.7-1

P 7.7-2 Find thevoltagev(t) for thecircuitofFigureP7.7-2.

Answer: v tð Þ ¼ $6e$250t mV

5 + 3e–250t A+

–

4 mH 4 mH

4 mH8 mHv(t)

Figure P 7.7-2

P 7.7-3 The circuit of Figure P 7.7-3 contains four identicalinductors. Find the value of the inductance L.

Answer: L ¼ 2:86 H

25 cos 250t V +–

L

L

L L

i(t) = 14 sin 250t mA

Figure P 7.7-3

P 7.7-4 The circuit shown in Figure P 7.7-4 contains seveninductors, each having inductance L. The source voltage isgiven by

v tð Þ ¼ 4 cos 3tð ÞV

Find the current i(t) when L ¼ 4 H.

L L

LL

L

L

L

i(t)

v(t)+–

Figure P 7.7-4

P 7.7-5 Determine the value of the inductance L in the circuitshown in Figure P 7.7-5, given that Leq ¼ 18 H.

Answer: L ¼ 20 H

A

B

20 H

25 H

20 H

60 H

10 H 30 H

L

Leq

Figure P 7.7-5

P 7.7-6 Determine the value of the equivalent inductance Leq

for the circuit shown in Figure P 7.7-6.

Answer: Leq ¼ 120 H

60 H

60 H

40 H

10 H

15 H 30 Ha

b

Leq

Figure P 7.7-6

P 7.7-7 The circuit shown in Figure P 7.7-7 consists of 10inductors having equal inductance L. Determine the value ofthe inductance L, given that Leq ¼ 12 mH.

Answer: L ¼ 35 mH

L

L

L L

L

L

L

L

L L

Leq

Figure P 7.7-7

P 7.7-8 The circuit shown in Figure P 7.7-8 is at steadystate before the switch closes. The inductor currents are bothzero before the switch closes i1 0ð Þ ¼ i2 0ð Þ ¼ 0ð Þ.

The voltage v(t) is given by

v tð Þ ¼ 0 V for t < 04e$5t V for t > 0

,

(a) Determine the inductor currents i1(t) and i2(t) for t ( 0.(b) Determine the energy stored by each inductor 200 ms after

the switch closes.

316 7. Energy Storage Elements

The parallel inductors can be replaced by an equivalentinductor.

(c) Determine the current in the equivalent inductor, directeddownward, for t ( 0.

(d) Determine the energy stored by the equivalent inductor200 ms after the switch closes.

24 Ω

12 Ω12 V

8 H 2 H+

−

i2(t)i1(t)

v(t)

t = 0

+–

Figure P 7.7-8

P 7.7-9 The circuit shown in Figure P 7.7-9 is at steadystate before the switch opens at time t ¼ 0. The current i(t) isgiven by

i tð Þ ¼ 0:8 A for t ' 00:8e$2t A for t ( 0

,

(a) Determine the energy stored by each inductor before theswitch opens.

(b) Determine the energy stored by each inductor 200 ms afterthe switch opens.

0.5 H

2 H

i(t) 5 Ω

15 Ω

12 V

t = 0

+–

Figure P 7.7-9

The series inductors can be replaced by an equivalentinductor.

(c) Determine the energy stored by the equivalent inductorbefore the switch opens.

(d) Determine the energy stored by the equivalent inductor200 ms after the switch opens.

P 7.7-10 Determine the current ratio i1/i for the circuitshown in Figure P 7.7-10. Assume that the initial currentsare zero at t0.

Answer:i1i¼ L1

L1 þ L2

i1

i2

L1

L2

i

Figure P 7.7-10

P 7.7-11 Consider the combination of circuit elementsshown in Figure P 7.7-11.

(a) Suppose element A is a 20-mF capacitor, element B is a5-mF capacitor, and element C is a 20-mF capacitor.Determine the equivalent capacitance.

(b) Suppose element A is a 50-mH inductor, element B is a30-mH inductor, and element C is a 20-mH inductor.Determine the equivalent inductance.

(c) Suppose element A is a 9-kV resistor, element B is a 6-kVresistor and element C is a 10-kV resistor. Determine theequivalent resistance.

Answers: (a) Ceq = 20 mF, (b) Leq = 16 mH, and (c) Req = 6 kV

A

B

b

C

a

Figure P 7.7-11

P 7.7-12 Consider the combination of circuit elementsshown in Figure P 7.7-12.

(a) Suppose element A is a 8-mF capacitor, element B is a16-mF capacitor, and element C is a 12-mF capacitor.Determine the equivalent capacitance.

(b) Suppose element A is a 20-mH inductor, element B is a5-mH inductor, and element C is an 8-mH inductor.Determine the equivalent inductance.

(c) Suppose element A is a 20-kV resistor, element B is a30-kV resistor, and element C is a 16-kV resistor. Deter-mine the equivalent resistance.

Answers: (a) Ceq = 8 mF, (b) Leq = 12 mH, and (c) Req = 28 kV

Problems 317

b

a

BA

C

Figure P 7.7-12

Section 7.8 Initial Conditions of Switched Circuits

P 7.8-1 Theswitch inFigureP7.8-1hasbeenopenfora longtime before closing at time t ¼ 0. Find vc(0

+) and iL(0+), thevalues of the capacitor voltage and inductor current immediatelyafter the switch closes. Let vc(1) and iL(1) denote the values ofthe capacitor voltage and inductor current after the switch hasbeen closed for a long time. Find vc(1) and iL(1).

Answers: vc 0þð Þ ¼ 12 V, iL 0þð Þ ¼ 0, vc 1ð Þ ¼ 4 V, andiL 1ð Þ ¼ 1 mA

12 V

8 kΩ

4 kΩ2 Fµ

25 mH+– vc(t)

iL(t) t = 0

+

–

Figure P 7.8-1

P 7.8-2 The switch in Figure P 7.8-2 has been open for a longtime before closing at time t ¼ 0. Find vc 0þð Þ and iL 0þð Þ, thevalues of the capacitor voltage and inductor current immedi-ately after the switch closes. Let vc(1) and iL(1) denote thevalues of the capacitor voltage and inductor current after theswitch has been closed for a long time. Find vc(1) and iL(1).

Answer: vc 0þð Þ ¼ 6 V, iL 0þð Þ ¼ 1 mA, vc 1ð Þ ¼ 3 V, andiL 1ð Þ ¼ 1:5 mA

12 V

6 kΩ

3 kΩ

25 mH+– 6 kΩ

iL(t)t = 0

2 Fµ vc(t)+

–

Figure P 7.8-2

P 7.8-3 Theswitch inFigureP7.8-3hasbeenopenfora longtime before closing at time t ¼ 0. Find vc 0þð Þ and iL 0þð Þ, thevalues of the capacitor voltage and inductor current immediately

after the switch closes. Let vc(1) and iL(1) denote the valuesof the capacitor voltage and inductor current after the switchhas been closed for a long time. Find vc(1) and iL(1).

Answers: vc 0þð Þ ¼ 0 V, iL 0þð Þ ¼ 0, vc 1ð Þ ¼ 8 V, andiL 1ð Þ ¼ 0:5 mA

12 V

8 kΩ

16 kΩ

25 mH+–

iL(t)t = 0

2 Fµ vc(t)+

–

Figure P 7.8-3

P 7.8-4 The switch in the circuit shown in Figure P 7.8-4 hasbeenclosedfora long timebefore itopensat time t = 0.Determinethe values of vR(0-) and vL(0-), the voltage across the 4-V resistorand the inductor immediately before the switch opens, and thevalues of vR(0+) and vL(0+), the voltage across the 4-V resistorand the inductor immediately after the switch opens.

+–

v L(t )

+

–

80 Ω 20 Ω

4 Ω

24 V

2.4 mH

+ –v R (t )

t = 0

Figure P 7.8-4

P 7.8-5 The switch in the circuit shown in Figure P 7.8-5 hasbeen open for a long time before it closes at time t = 0. Determinethe values of iR(0-) and iC(0-), the current in one of the 20-Vresistorsandin thecapacitor immediatelybefore theswitchcloses,and thevaluesof iR(0+)and iC(0+), thecurrent in that20-V resistorand in the capacitor immediately after the switch closes.

i C (t )

2.2 mF

t = 020 Ω

20 Ω

20 Ω

i R (t )

120 mA

Figure P 7.8-5

318 7. Energy Storage Elements

P 7.8-6 The switch in the circuit shown in Figure P 7.8-6has been open for a long time before it closes at timet = 0. Determine the values of vL(0-), the voltage across theinductor immediately before the switch closes, and vL(0+),the voltage across the inductor immediately after the switchcloses.

v L(t )+ –

t = 020 Ω

18 mH

120 mA

20 Ω

Figure P 7.8-6

P 7.8-7 The switch in the circuit shown in Figure P 7.8-7 hasbeen closed for a long time before it opens at time t = 0.Determine the values of iC(0-), the current in the capacitorimmediately before the switch opens, and iC(0+), the current inthe capacitor immediately after the switch opens.

i C (t ) 2.2 µF

t = 0

20 Ω20 V +–

Figure P 7.8-7

P 7.8-8 The circuit shown in Figure P 7.8-8 is at steady statewhen the switch opens at time t ¼ 0. Determine v1(0$),v1(0þ), i2(0$), i2(0þ), i3(0$), i3(0þ), v4(0$), and v4(0þ).

3 Ω

6 Ω50 mF

+

+

−

−

i3(t)v1(t)

6 H

12 V

v4(t)

t = 0i2(t)

+–

Figure P 7.8-8

*P 7.8-9 The circuit shown in Figure P 7.8-9 is at steady statewhen the switch opens at time t ¼ 0. Determine v1(0$),v1(0þ), i2(0$), and i2(0þ).

Hint: Modeling the open switch as an open circuit leads us toconclude that the inductor current changes instantaneously,which would require an infinite voltage. We can use a moreaccurate model of the open switch, a large resistance, to avoid theinfinite voltage.

i2(t)

50 mF

3 Ω 6 Ω

6 Ω

6 H

12 V

+

−

v1(t)

t = 0

+–

Figure P 7.8-9

P 7.8-10 The circuit shown in Figure P 7.8-10 is at steadystate when the switch closes at time t ¼ 0. Determine v1(0$),v1(0þ), i2(0$), and i2(0þ).

30 Ω

15 Ω

10 Ω 15 Ω24 V

50 mF+

−

v1(t)

i2(t)

3.5 H

t = 0

+–

Figure P 7.8-10

P 7.8-11 The circuit shown in Figure P 7.8-11 has reachedsteady state before the switch opens at time t ¼ 0. Determine thevalues of iL(t), vC(t), and vR(t) immediately before the switch opensand the value of vR(t) immediately after the switch opens.

Answers: iL 0$ð Þ ¼ 1:25 A; vC 0$ð Þ ¼ 20 V; vR 0$ð Þ ¼ $5 V;and vR 0þð Þ ¼ $4 V

iL(t)

vR(t)80 Ω

20 Ω

t = 0

+–

2 mF

0.125 H

25 V

+

–

+

–

vC(t)

4 Ω

Figure P 7.8-11

P 7.8-12 The circuit shown in Figure P 7.8-12 has reachedsteady state before the switch closes at time t = 0.

(a) Determine the values of iL(t), vC(t), and vR(t) immediatelybefore the switch closes.

Problems 319

(b) Determine the value of vR(t) immediately after the switchcloses.

30 Ω

t = 0

iL(t)

+

–

2 µF

0.125 H

35 V vC(t) vR(t)

+

–

40 Ω 40 Ω+–

Figure P 7.8-12

P 7.8-13 The circuit shown in Figure P 7.8-13 has reachedsteady state before the switch opens at time t¼ 0. Determine thevalues of iL(t), vC(t), and vR(t) immediately before the switchopens and the value of vR(t) immediately after the switch opens.

Answers: iL 0$ð Þ ¼ 0:4 A; vC 0$ð Þ ¼ 16 V; vR 0$ð Þ ¼ 0 V;and vR 0þð Þ ¼ $12 V

iL(t)

+

–

30 Ω

2 mF

20 Ω

40 Ω24 V

125 mHvC(t)

t = 0

vR(t)+–

+

–

Figure P 7.8-13

Section 7.9 Operational Amplifier Circuits andLinear Differential Equations

P 7.9-1 Design a circuit with one input, x(t), and one output,y(t), that are related by this differential equation:

12

d2

dt2y tð Þ þ 4

ddt

y tð Þ þ y tð Þ ¼ 52

x tð Þ

P 7.9-2 Design a circuit with one input, x(t), and one output,y(t), that are related by this differential equation:

12

d2

dt2y tð Þ þ y tð Þ ¼ $ 5

2x tð Þ

P 7.9-3 Design a circuit with one input, x(t), and one output,y(t), that are related by this differential equation:

2d3

dt3y tð Þ þ 16

d2

dt2y tð Þ þ 8

ddt

y tð Þ þ 10y tð Þ ¼ $4x tð Þ

P 7.9-4 Design a circuit with one input, x(t), and one output,y(t), that are related by this differential equation:

d3

dt3y tð Þ þ 16

d2

dt2y tð Þ þ 8

ddt

y tð Þ þ 10y tð Þ ¼ 4x tð Þ

Section 7.11 How CanWe Check . . . ?

P 7.11-1 A homework solution indicates that the current andvoltage of a 100-H inductor are

i tð Þ ¼

0:025 t < 1

$ t25

þ 0:065 1 < t < 3

t50

$ 0:115 3 < t < 9

0:065 t < 9

8>>>>><

>>>>>:

and

v tð Þ ¼

0 t < 1

$4 1 < t < 3

2 3 < t < 9

0 t > 9

8>>><

>>>:

where the units of current are A, the units of voltage are V, andthe units of time are s. Verify that the inductor current does notchange instantaneously.

P 7.11-2 A homework solution indicates that the current andvoltage of a 100-H inductor are

i tð Þ ¼

$ t200

þ 0:025 t < 1

$ t100

þ 0:03 1 < t < 4

t100

$ 0:03 4 < t < 9

0:015 t < 9

8>>>>>>><

>>>>>>>:

and

v tð Þ ¼

$1 t < 1

$2 1 < t < 4

1 4 < t < 9

0 t > 9

8>>><

>>>:

where the units of current are A, the units of voltage are V, andthe units of time are s. Is this homework solution correct?Justify your answer.

320 7. Energy Storage Elements

Design ProblemsDP 7-1 Consider a single-circuit element, that is, a singleresistor, capacitor, or inductor. The voltage v(t) and currenti(t) of the circuit element adhere to the passive convention.Consider the following cases:

(a) v tð Þ ¼ 4 þ 2e$3t V and i tð Þ ¼ $3e$3t A for t > 0(b) v tð Þ ¼ $3e$3t V and i tð Þ ¼ 4 þ 2e$3t A for t > 0(c) v tð Þ ¼ 4 þ 2e$3t V and i tð Þ ¼ 2 þ e$3t A for t > 0

For each case, specify the circuit element to be a capacitor,resistor, or inductor and give the value of its capacitance,resistance, or inductance.

DP 7-2 Figure DP 7-2 shows a voltage source and unspecifiedcircuit elements. Each circuit element is a single resistor,capacitor, or inductor. Consider the following cases:

(a) i tð Þ ¼ 1:131 cos 2t þ 45-ð Þ A(b) i tð Þ ¼ 1:131 cos 2t $ 45-ð Þ A

For each case, specify each circuit element to be a capacitor,resistor, or inductor and give the value of its capacitance,resistance, or inductance.

Hint: cos yþ fð Þ ¼ cos y cos f $ sin y sin f

i(t)

4 cos 2t V+ –

Figure DP 7-2

DP 7-3 Figure DP 7-3 shows a voltage source and unspecifiedcircuit elements. Each circuit element is a single resistor, capac-itor, or inductor. Consider the following cases:

(a) v tð Þ ¼ 11:31 cos 2t þ 45-ð Þ V(b) v tð Þ ¼ 11:31 cos 2t $ 45-ð Þ V

For each case, specify each circuit element to be a capacitor,resistor, or inductor and give the value of its capacitance,resistance, or inductance.

Hint: cos yþ fð Þ ¼ cos y cos f$ sin y sin f

v(t)+ –

4 cos 2t A

Figure DP 7-3

DP 7-4 A high-speed flash unit for sports photography requiresa flash voltage v 0þð Þ ¼ 3 V and

dv tð Þdt

####t¼0

¼ 24 V/s

The flash unit uses the circuit shown in Figure DP 7-4. Switch 1has been closed a long time, and switch 2 has been open a longtime at t ¼ 0. Actually, the long time in this case is 3 s.Determine the required battery voltage VB when C ¼ 1=8 F.

+

–v

1 Ω

t = 0

HC

Flashvoltage

+–

VB

Switch 2

t = 0

Switch 1

3 Ω

+– VB1 2

3Ω

Figure DP 7-4

DP 7-5 For the circuit shown in Figure DP 7-5, select a value ofR so that the energy stored in the inductor is equal to the energystored in the capacitor at steady state.

10 V

20 Ω

+–

R

1 Fµ

10 mH

Figure DP 7-5

Design Problems 321