Pan I in:uit. IlJl

HOMEWORK PROBLEMS

Section 4.1 Energy Storage Elements

4 .1 The current through a 0.5-11 inulIL:tor i\ gl\cn h) i L = 2c()\(3771 +rr / 6) . Write Ihe e rre.~'it)n for the

voltage .KIUSS the inuuctof.

4.2 The voltagc across a IOO-llF capac itor takes the fo l\o\\ing vu luc, all.:ulatc the e\pfl:~'I!ln for the current through the capucilOr 10 c.leh CilSC.

il. 1'( (t 1 = ..f.Ocos(20r - rr/2 ) V

b. I cCt) = 20 'in 1001 V

c. I'c(l ) = -60sin(gOI +:r/(}) V

d. I'c(l)=JO co,( IOOt +7T!-+ ) V

4.3 The current through a 25()-mll inductor tal,.e~ thc fn lh)\\ ing v'lll1c~ . C ti clilaic the e>-prc"loT1 fllr the vnltugc acro~, the inuuctllr in cach I.:asc .

a. i l (f) = '\,iI1251 A

b. i , (1) = lO cos ~()l A

c. i l .(!) = 25co,(1001 + ;r/3) .\

d. idtl = 20 sin( 101 - n/ 12) A

4.4 In the circuit ,hm\n 10 Figllfc P-+.4, let

for - :xl < I < 0

i (r)= J ~ lor 0 .::; I , I() :

1 10 fu r 10 ~ ~ I < '""

h nu the ncrg) stlJleu ill Ihe iIluliclor for -1.11 time .

tu

:!H

Figure P4.4

4.5 With refe rence ttl Prohlem 44. find Ihe energ) uel j\,ereu hy the sourcc for .111 time.

4.6 III the cir Ult shown in Figure N.'" leL

0 for - 'X, < ( "- 0

I fnrO::..,I < I().­

an ~I20 - 1 fo r I () ~ I < 20 s

() for 20, .:. 1 , '""C

r ind

a. The: ellergy ~tored in tht: inullctor for ali lill1t!

h. The energy dcll\crcd hy th~ ~ource for .tl i time

4.7 In the CI rcui t Sho\'11 in Figure P4.7. ler

for - x - I . 0

,,(I) = J ~ for 0 ::: I < 10 s

for 10 S .::; I < :x>

Find Ihe energy stored in Ih' capacitor for all time .

110

Figure P4.7

4.8 With reference to Problem -+.7 , li nd the energy uelivered hy the sou rce hlr all time.

4.9 In the circuit ,\auwn in f igure 1>4.7 let

0 for - ov < 1 < 0

for 0 t < 10 s

i(n = 120 -I tnr IO '::; I < :W .

0 for 20 s .::; t < :xl

Finu

u. rile energy storeu il the 'apaciLOr for all time

h The energ) uel i lereu by the ~ource for all time

4.10 Find the energy lored in each capacitor and II1U UI': tor. IInuer :-teady ·~!Ute conuiti ons. in lhe ..:i rclIi l ~h()\\n in Figure P..J. 10.

I F

Figure P4.10

4.11 Find the: energy ~t()red ttl eac h capacitor and inductor. under ~teauy-stale c() ndili ()n~. in the ci rcu it sh )wn in figure P4 . 11 .

Chapter 4 AC I etwork Analy~ i ~

12 V-=­

Figure P4.11

4.12 The plot o f time-dependent voltage is shown in hgure P4.12. The waveform is piecewise continuous. If this is the voltage ac ross a capacitor and C = 80 /-<F, determi ne the current through the capaci tor How can cun-ent flow '·th rough" a capacitorry

JI) 1~ I (illS)

Figure P4.12

4.13 The plot of a time-dependent vo ltage is shown in Figure P4. 12. The wavefornl is piecewi se continuous. rr th is IS the voltage across an inductor L = 35 mH, dctemline the current through the inductor f·- .sume the initial current is i d O) = O.

4.14 The voltage across an inductor plotted as a functioo of time is shown in Figure P4.14. If L = 0.75 mH, determine the current through the inductor at I=J5p.s.

10 15 10.1S)

- 1.9

Figure P4.14

4.15 If the waveform shown in Figure P4.IS is the voltnge across a capacitor plotted as a function of time

with

VPK = 20 V T = 40 p.s C = 680 nF

determine and plot the waveform for the current through the capacitor as a funct ion of time.

"..~ I IT I

Figure P4.1S

4.16 If the current through a 16-p.H inductor i zao at r = 0 and tbe voltage across the inductor (shown in Figure P4. 16) is

1<0

o < I < 20 p.sudf) = 1~t ' 1.2 nV t > 20/-<s

determine the current through the inductor at

t = 0 fJ- s.

dtl(nV)

1..2

20 40 I t>!S)

Figure P4.16

4 .17 Detennine and plot as a function of time tbe current through a compone nt if the voltage across it bas the waveform shown in Fi gure P4. 17 and the component is a

a. Resis tor R = 7 r2

b. CapaciLOr C = 0.5 ~(F

c. Inductor L = 7 mH

t(/I(

I ~ ­

F igure P4.17

4.18 If the pl ots shown ill Figure 1'4.18 are the voltage ac ros, and the current through an ldeal ca pacltol', determine the capacitance .

I I1- : 511-1 ;(1) (Al

Figure P4.18

4.19 [f the p lots shown in Fig ure P4 . 19 are the voltage

across and the current lhrough an idea l inducto r, de termine the induc tance.

1'(1) (V )

2

10 I~ I (ill S)

i(t) (A)

J

2

Figure P4.19

4.20 The voltage ac ross and the CUITent th rough a carac itor are shown in Figure P4.20. Dete rmine the value of the capacitance.

i,(1l (mAl

ISh 10 I (m s)

Figure P4.20

4.21 T he voltage across and the curren! through a car acitor are shown in Figure P4 .21. Determine thc va lue of the capacitance.

Part I Circuits 19:1

i,.(mA)

Figure P4.21

Section 4.2 Time-Dependent Signals

4.22 Fi nd the average and nllS value of x(t).

xU) = 2cos(wt) + 2.5

4.23 A controlled rec tifier circ uit is gene rating the

waveform of Fig ure P4.23 starting from a sinusoidal vo ltage of 110 V rms. F ind the average ancl rms voltag.e .

O utput w~ vc:form of co ntrolled rec lilier

'/LIC~)

/ \50

/ 2To7T 17-8 ~ n e> /\-50

\. 1(10 -

'\. )

- 150 ,() 4 Radians

Figure P4.23

4 .24 With reference to Proble m 4 .23, find the ang le 0 that corresponds to delivering exactly one·half of the

total available power in the waveform to a resistive

load.

4.25 Find the ratio between average and rms value of

the waveform of Figure P4. 15.

Figure P4.2S

IlJ4 Chapter 4 AC Network Analysis

4.26 Gi ven the cunent waveform shown in Figure

P4.26, find the power dls~i patcd by a I -Q resisto r.

i(f) (A)

/0

() , (s )

Figure P4.26

4.27 rind the ratio between average and rms \·aJue of the waveform of Figure P4.27.

F gure P4.27

4.28 Find the nllS value of the waveform shown in Figure P4.28.

Figure P4.28

4.29 Deten nine the nns (or effective) val ue of

V(I) = Vile + VAC = 50 + 70.7 cos(377t) V

Section 4.4: Phasor Solution of Circuits with Sinusoidal Excitation

4.30 If the current through and the voltage acros~ a component in an I c trjc circuit a re

i(t) = J7cos(wl - rr / 12) mA

V(I) = 3.5 eos(wl + 1.309) V

where w = 628.3 radls, d~termine

a, Whelher the component is a resislor. capacitor, or

inductor.

b. The value of the component in ohms. farad , or

he nrys.

4.31 De~cribe the sin usoidal (l'vcfo rm shown in FiglllL' P4.3 1, using time-depcndc!1l and phasnr notation.

.. «((Il l ( v)

Figure P4.31

4.32 Describe the si n llsoida l waveform show n in Figure

P4.32. using lime-dependent unci phasor notat ion.

i (w I) \rn A ) II ­

It wi ( ratl), , , , , , , , , I

, , I,, , , , ~ ..........--- r = 4 n1 S - ---;....'

Figure P4.32

4.33 Describe the sinusoidal wavefOnTl sl10wn in Figure P4.33. using time-dependent and pha.~or nOlation.

i ((dn (rnA) H

It '"' (raJ)

, '__._T =41ll.,--__>-'

Figure P4.33

4.34 The current through and the vollage acro. S ;J11

e lectrical component are

i (I) = ToCO " (OJ I . ~)

where

1" = 3 Ill A V,, =700 mV (V = 6.283 rad/s

195

a. Is the component inducti ve or capac it ive'l

b. Plot the instantaneoll s power p (t) as a function ot wI over the range 0 < (J) I ~1T ,

c . Deteml inl! the average po wer dis."lpatcd as ht!<l t In

lhe component.

d. Repeat pa rt ~ (b) and (c) ir the phase angie or the current is l'hanged tll 0°,

4 .35 Determine the e4ui v ~l cn t impedanc in tht: c ircuit shown in Pig ure P4.3.'i:

I ' , (r) = 7 cos 3,0001 ( ::'6 )

Rt = 2.3 k Q R2 = 1. 1 k Q

r = 190 mil C = 55 nF

Figure P4.35

4.36 Dete rmi ne the equiv<l lelll im pedance in the c ircuit shown in Figure P4.3S :

v, (I ) = 636 cos ( 3.0001 + 1; ) v

Nt = 3,3 kQ R2 = 22 I\.Q

L = 1,90 H C = 6 .8 nF

4.37 In the ci rc uit of Figure P4.37.

i , (t) = I"co~ (M + ~)

I" = 13 mA w = 1.000 rael /s

C = 0,5/-1 F

a. Stale, u\i ng phasnr nOI<l ti oll , the so urce curren t.

b, Determine the impcd,tnc l' or the capac itor.

c. si ng phasnr nOlation only and showing al l \\orb;, determine til voltage across the l'ap<lcitor. inclnd ing its polarity.

c

Figure P4.37

Part 1 Circuit,

4.38 D ' teml ine i I (t) in the e ir uit : hown 1Il Figure P4. J8 if

i l (J) = 14 I Acos ((vl + ~. 356 ) mA

; ,(1 ) = 5()~in «,;r _. () ,927) mA

(I) = 377 radh

z,

Figure P4.38

4 .39 Determine the 'urr ot through Z, in the circuit of Figure p~ .39.

V d = 1',2 = 1 70c()s ( ~77t) V

ZI = 5.4LO, 122 Q

Z2 = 2.3'<:::0 n Z, = J7 / 0, I92Q

z,

Figure P4.39

4 .40 Detem l ine the frC<lucncy so that the cUIl'ent 1; and the voltage V" in the Circuit of of Figure [ 4.40 arc in pha..~e .

l, = 13.000 + jM3 Q

R = 120 Q

L = J9 ml-l = 220 pF

C

Vi

R

L

Figure P4.40

+... I,

V"

4.41 In the eir ui t of Figure P4.40, determine the t'relj ucncy (0 , al which I, and " are in phase ,

Il)/i Chapter 4 AC Network Analys is

4 .42 The coil resistor in series with L models the

internal losses of an inductor in the circuit of Figure

P442. D ete rmine the current supplied by the source if

v,(r ) = V" cos(wt + 0)

Vo = 10 V w = 6 M rad js R, = 50 n R, = 40 n L = 20/.l.H C = US nF

IVs

e

Figure P4.42

4 .43 Using phasor techniques, solve for the current in the cit-cuit shown in Figure P443.

vJrl = 12 cos 31 V

Figure P4.43

4 .4 4 Using phasor techniques, so lve Jor the voltage v in the circuit shown in Figure P444.

3Q 3 H

Figure P4.44

+ v(I)

4 .4 5 So lve for II in the circuit shown in Figure P44S_

Figure P4.45

4.46 Solve for V2 in the circ uit shown in Figure P446.

Assume w = 2.

12Q 6H

V = 2) LOV

Figure P4.46

4 .47 With re fere nce to Problem 444, find the value of w for wbicb the current through the resistor is maximum.

4.48 Find the current througb the res istor in the circuit

shown in Fi gure P448.

isi- I) = I cos (200m )

Figure P4.48

4.49 Find VOl" (I) for the circu it shown in Figure P449.

+

Xc=lOkQ

Figure P4.49

4 .50 For the circuit shown in Figure P4 .S0, find the

impedanc ' Z , g iven w = 4 rad/s.

1I4 H

2Q

Figure P4.50

4.51 Find the admittance Y for the c ircuit shown in

F igure P4.51 when w = S rad/s.

}'- ] / IOF

Figure P4.51

Section 4.5: AC Circuit Analysis Methods

4.52 sing phasor techniques, sol ve for v in th e c ircuit shown in Figure P4 .S2.

9 Q H 1 H I

J6 W"" _W3'VCB: Figure P4.52

4.53 U ing pbasor technique~. solve for i in the circuit shown in Figure P4 .S3.

Figure P4.53

4.54 Determine the Thc vcnin equi valent circuit as see n by the load shown in Figure P4. 54 if

a.

b.

vsU)

P\(I)

= 10cos ( \, OOO!)

= 10 cos( I ,000. 0001)

L

1'S(I)

-I-

Source Filler

RS= R/ = 500 D L = 10 mH

Load

R=I W

F igure P4.54

Part J CirC llils

4.55 Find the Thevenin eq ui vale nt of the c ircuit shown in Fig ure P4.S5 as scen by the Ivad resistor.

Figure P4.55

4.56 Solve fo r idt) in til<' circuit of Figu re P4 .56. lI sing: phasor IcchniL[ues. if 1:, (1) = 2 co, 21, RI = 4 Q,

R. = 4 Q , L = 2 H , and C = j F.

Figure P4.56

4.57 Using mesh Cllrrent analysis, determi ne the curr ' nts i t (I) and i 2(1) in the ci rcuit shown in Figure P4.S7 .

\ 's (t) = 15 co, I 5001

..---... C = J ,UF

I!(I) ;

) ,,~ = IO()Q

Fig ure P4.57

4.58 Using node vol tage methodS, determine thc vol tages u, (t) and p. Ul in the c ircuit show n in Fi gul'e P4.5/; .

R,= 40£2 R2 = 10 D C =500j.lI-' L= 0.2 H

F igure P4.56

4.59 Thc circuit sho wn in Figure P4.S9 is H WI1eutstone bridge that will allow you to determine the reactance of an itldm;tOl' or a capac it llf. The: circuit i ~ adjusted by changing R I and R2 until Va" is l cro.

Ch apt r 4 /I.e Network A l1 a ly ~

<l. AS~lIming that the circuit i~ bn lanl:ed, thm is, Ihat II"" = 0, determille X~ in term~ oflhe circ uit elements .

b. If C i = 4, 7 liF, t, =0,098 JI. RI = 100 Q ,

R2 = I Q . 11 \ (1) = 24 sill (1 .000t), and 1',,/1 = 0, what i ~ the reac tance ol the unknown circ uit element'! Is it a capacitor or an jlldu l: tor~ Whu! i, its value?

c. What frequency should bc avoided by the source in thi, c ircuit, anu why"

\'.1'(1 ) :!:

Figure P4.S9

4.60 Compute the Thcveni ll impedance ~ecn by resistor R2 in Problem 4.56.

4.61 Compu te the j1(!veni n voltage seen by lhe inuucLan<;c L ill Problem -+.58.

4.62 Find the T hcvcni n equival ent eirl'ui t as seen fro m tcrm ina b a-b for the ci rcuit silo !l in Figure P4.62,

- jS ~ l

5 ':::- 300 V

f-~--oa

Figure P4.62

4.63 Compute the Th'venin vo llnge seen by resi~lor R~ in Problem -1.56 ,

4.64 Find the Norton equivalent ci rcui t seen by resi~tor Rl in Problem 4,56.

4.65 Wri te the two loop equ:ll io!ls required to ~olve for the loop currents in the circui t of Figure P4.65 in

a. Integral-di fferenti lll foml

b, Phasur form

Figure P4.65

4.66 Write the node equat ions requircu to ~o l ve for aJl voltages and cllrrent~ in the circuit of Figure P4.65. Assume all impedances and the two ~ollrce volt,lg s are J.. nowll .

4.67 In the circuit ),hown in Figure P4 ,67.

lid = 450 c S (ot

l ' ,1 = -l50cos wt V

A ~ollltion of till: circuit with thc grollnd at node e (IS

shown gives

7r V" = 450~0 V V'I = 4.+0'<::: - V

6 V, = 420'<::: - 3..+9 V

V". = 779 .5LO.098 V V,.,j = 153 ,9L1.2 V

V,.. = 230.6L 1. 875 V

Iflhe ground is IlOW moved from node e to node d, determinc VI, and V,,, .

z.

d

Figure P4.67

4.68 Determine 1'" in the circ uit or Figure P-l .68 if

U·, = 4co~ . (1 ,OUOI + ~6 ) V

L = 60 mH C = 12.5 I (F

RL = 120 Q

Figure P4.68

111<)Pan I Cir~uits

4.69 The mesh currents and source vo l lage~ in the ci reui t ~ hllwn in Figure P4.69 are

i l (I ) = 3.127 COS(wl - O.lS2 -)

i2(1) = 3 .914 co~ (wl - l. 7X) A

i3 (t) = 1.900 C()s(wl + O . (\5~) A

USI (t) = 130. 0 COS (WI + 0.176) V

v.d l) = IJO.Ocos(wl - 0.4J6) V

where (j) = 377.0 mel/s. Determine one o f the

followi ng: L,. C2, R1. or [ . .1. Figure P4.69