PROBLEMS 367

7.66 For the network in Fig. P7.66, choose C so the lime consta nt will be 120 I-L S for I > O.

t= 0 4 kO

Vs 23 kO 18 kO

Figure p].66

7.67 The di fferential equat ion that describes the current io( l ) in a network is

d' io(t) di,,{l ) --,- + 6-- + 4io( l ) = 0 dr dt

Find (a) the characteristi c equation of the network, (b) the network 's natural frequencies, and (c) the expression fori,,(I).

7.68 The terminal current in a network is described by the equation

,I'io(l ) di, (I) --,- + 8- - + 16i,,{ l) = 0

dr dl

Find (a) the characteristi c equation of the network, (b) the network's natura l frequencies, and (c) the equati on for i,,{I ).

7.69 The voltage 'V I( t) in a network is defined by the equation

d'v,(l ) dv,(t) - -,- + 4-- + 5v,(t) = 0

dr dr

Find (a) the characteristic equation of the network, (b) the circuit's natural frequencies, and (c) the expression for v ,(t) .

7.70 The output voltage of a circuit is described by the differ-ential equation

d'vo ( I) dvo( I) --, - + 8-- + 10v.,{ l ) = 0

dt ' dt

Find (a) the characteristic equation of the circuit, (b) the network's natural frequencies, and (c) the equation for V,,(I).

C 771 The voltage vl{ t) in a network is defined by the equulion d'1) ,(I ) [ dV,(I)] --,- + 2 - - + 5v,(1) = 0 dl - dl

Find (a) the characteristic equation of the network. (b) the circu it's natural frequencies. (el the expression for V,( I).

2kO

C

0 +

7 kO Vo( I)

0

7.72 The parameters for a parallel RLC circuit are R = I n , L = 1/ 2 H. and C = 1/ 2 F. Determine the type of damping exhibited by the circuit.

7.73 A series RLC circui t contains a res istor of R = 2 nand 0 a capacitor C = 1/ 2 F. Select the value of the induclOr so that the circuit is critically damped.

7.74 A parallel RLC circuit contains a resistor I? = I nand 0 an inductor L = 2H. Se lect the value of the capacitor so that the circui t is critically damped.

7.75 For the underdamped circuit shown in Fig. P7.75, 0 determine the voltage v( t) if the initial conditions on the storage clements arc iJO) I A and v,(O) = 10 V.

2H

I Figure P].75

+ +

V(t) 50 1 ,~ vc(O) 40 F

7.76 In the criticall y damped circuit shown in Fig. P7.76. the 0 initial conditions on the storage clements ure i,.(O) = 2 A and v,(O) = 5 V. Determine the voltage v( t ).

+ +

vc(o) 'r 001 F v(t) 100 r

Figure P7.76 7.77 Find v,.{t) for t > 0 in the circuit in Fig. P7 .77 . ()

t = 0

+

1 A 80 0.04 F

Figure P7.77

CHAP TE R 7 F IRS T - A N D SECON D - ORDE R TR ANS IEN T CI R CU IT S

() 7.78 Find V,(I) for I > 0 in the circuit in Fig. P7.78 if v,.(O) = O.

7.80 Find V,,(/) for I > 0 in the circuit in Fig. P7.80 and plot the response including the time interval just prior to moving the switch.

I ~ 0 1 k!l 100mH 2 k!l 1 mH

+

1 ~F vdl) + 12V 1 k!l Vo(l)

Figure P7.78 Figure P7.80

Find 'Vo(l ) for I > 0 in the circuit in Fig. P7.79 and plot the response including the time interval just prior to clos-ing the switch.

7.81 In the circuit shown in Fig. P7.8 1, fi nd V( I ) > O.

12V +

Figure P7.79

+

1 H 10!l 'Vo(l) Figure P7.81

~ H 5

20

7.82 Find ve(r) for I > 0 in the circui t in Fig. P7.82 and plot the response including the time interv'll just prior to moving the switch.

I ~ 0 1 mH

+

4 kO 1 k!l Vo(l)

Figure P7.82

7.83 Find V,,( / ) for I > 0 in the network in Fig. P7.83 and plot the response including the time interval just prior to moving the switch.

10 k!l Sk!l I ~ 0

100 V 10 k!l 2 kO

100 mH

Figure P7.83

1 T2 F

+

V(I)

PROBLEMS 369

0 7.84 Find vQ(t) for [ > 0 in the circuit in Fig. P7.S4 and plOi the response includi ng the lime interval j ust prior to closing the switch.

1.5 A 40

Figure P7.84

80

so

~ H 8

o 7.8 5 The switch in the circui t in Fig. P7.85 has been closed D for a lo ng time and is opened at I = O. Find ;(1) for Ji:! / > o.

/ ~ 0

50 50

50

i(/) + 20 V 0.04 F

1 H

10 V

Figure p].8S

e 7.86 The switch in (ile c ircuit in Fig P7.86 has been closed for a long time and is opened at I = O. Find ;(t ) for / > o.

so

12V +

i(/)

1.25 H

Figure P7.86

1.50

/ ~ 0

0.1 F

SV

/ ~ 0 o I

.lH 8

+

370 CHAPTER 7 FIRST AND SECOND ORDER TRANSIENT C I RC UITS

~ 7.87 Find V,,(I) fo r I > 0 in the circuit in Fig. P7.87 and plot the response including the time interval just prior to mov ing the switch.

7.88 Design a para lle l IILe eircllil with II ~ I ki1 0 that h

,/

7.92 Given the network in Fig. P7.92. use PSPICE to plot 'V"(I) over a 10-s interval starting al T = 0 using a IOO-nls step size .

. \ 2n -1~ 0 4n J

2H

12 V + 6n 2A t ,,'" - + 1 F 2n vo(t)

-

figure P7.92 7.93 Given the network in Fig. P7.93. use PSPICE to plot V"(I ) over a 10-s

interval starting at t :::: 0 usi ng a IOO-ms step size.

1 ~ 0 12 V 20 kn 10 kn

6V + ---{) 50 ",F +

10mH 5 kn Vo(t)

figure P7.93

7.94 Given (he network in Fig. P7.94a and the input voltage shown in Fig. P7.94b, use PSPICE to plot the voltage V(,( t ) over the interval 0 :5 ( :s 4 s using a 20-ms step size.

100 kn 1Mn +

Vil1(t) 2mH 0.1 ",F Vo(t)

0

(a) Vil1(t) V

10

o 1 (5) (b)

figure P7.94

PROBLEMS 371

372 CHAPTER 7 FIRST - AND SECOND-ORDER TRANSIENT CIRCU I TS

7.95 Given the network in Fig. P7.95a, and the input voltage shown in Fig. P7.95b, plot 'V1I(t) using PSPICE over the interval 0 :::s: ( :::s: 10 s using a IOO-ms-slep size.

+

Vin(r) 20 20 20 +

1 F ;o p;: ~2A ! 1 H 1 0 40 Vo t -

0

(al

10

o r (s) (bl

Figure P7.95

7.96 Given the network in Fig. P7.96a and the input in Fig. P 7.96b, lise PSPICE to plot vo( r) in the interval 0 :::::: r :5 4 5 using a 20-ms step size.

10 kO

1/

5 kO 100'~F + v,(r) r 10 mH ! 10 kO vo(r)

(al v,(r) V

10 r-- .--

o 2 3 4 r (s) (bl

Figure P7.96

(r)

T YP I CAL PR O B LE MS F OUND ON THE FE E XAM 373

TYPICAL PROBLEMS FOUND ON THE FE EXAM

7FE-1 In the circu it in Fig. 7PFE- I. the switch, which has been closed for a long time. opens at [ = O. Find the value of the capacitor voltage V,(I) a ll ~ 2s. a. 0.936 V b. 0.756 V c. 0.264 V d. 0.462 V

I ~ a 8 kO 6 kO

+

12 V 6 kO vc(t) 100 ~F 6 kO

Figure 7PFE-l

7FE-2 In the network in Fig. 7PFE2. the switch closes all = O. Find vo( t ) at I = Is. a. 5.62 V b. 1.57 V c . 4.25 V d. 3.79 V

12 kO 4kO

12 V 12 kO

~-----~O I ~ O __ _ _ ~ +

100 ~F vo(t)

Figure 7PFE-2 7FE-3 Assume that the switch in the network Fig. 7PFE-3 has been closed for

some time. At [ = 0 the switch opens. Determine the time required for the capacitor voltage to decay 10 one-half of its initially charged value. a. 0.4 16 s b. 0.6255 c. 0.235 s d. 0.143 s

12 kO

+

12V Vc( I) 100 ~F 6kO

Figure 7PFE-3

374 CHAP T E R 7 FI RST, AND SECONDORDER TRA N S I E NT C I RC U ITS

7FE ~4 Find the inductor current it( r) for r > 0 in the circui t in Fig. 7PFE-4. a. i 1.( I) = 3 - 2e- ,/6 A, 1 > a h. i,.(r ) = I + 2e- 21 / 3 A, r > 0 c. i,.(I) = 6 - e-'/6 A, 1 > a d. i 1.( I ) = 3 - e-2,/.\ A, I > a

t = 0 20

10 V 2 0 20 1 A

Figure 7PFE'4

7FES Find the inductor current il.(I) for 1 > a in the circuit in Fig. 7PFE5.

a. i1.( I) = 1.4 + 0.4e- " iJ A , 1 > 0 b. i 1.( I ) = 1.2 + 0.4e- " i3 A , 1 > 0 c. i,.( I) = 0.4 + 0.2e-" /3 A, 1 > 0 d. il,(I) = 2.4 + 0.6e- ', /3 A, 1 > 0

30

12V + 40 40

Figure 7PFE'5