answers to even-numbered problems

ANSWERS TO EVEN-NUMBERED PROBLEMS

Chapter 11-2

Decreases by 4 x 10-I ' J .1-4

2.5 A (0 < t < 6) ; -5 A (6 < t < 9) . Function is repetitive every9 seconds .

1-6

p(t) = 180 - 180 sin tat W. 147 J.1-8

P,, = 400 kW . Pa, = 20 kW .

1-10

(a) v, = v� - VB, v2 = vB - v,. . (b) VB = - 19 V, v, = 34 V.1-12

VP = VI ; V2 - (VI - V2) ; VR = VI - V2 + V3 ; VS = VI - V2 + V3 - V4-

1-14

(a) Delivers +400 W. (b) Delivers + 100 W. (c) Receives +550 W.(d) Receives +200 W.

1-16

No. 1 receives -700 W. No . 2 receives -650 W. No . 3 delivers +975 W. No . 4 delivers- 1575 W. No . 5 delivers -750 W.

1-18 75 MHz.

1-20 Nonlinear.1-22

(a) [26.4 +- 10 cos 10t - 25 sin 10t] A. (b) (- 16 + 4 cos 10t - 10 sin 10t) A.(c) ( - 22 - 5 .12 cos l0t +

12.8 sin l0t) A.1-24

v(t) = 0 when t < - 4 s ; -24e-3u+4I A when t > -4 s.1-26

9.02

x

10- I' J .

1-28

v < -8: 10 .7 V, 2.67 fl . - 8 < v < -4 : 4V, 0.571 fl . -4 < v < 0: open circuit .0<v<5:0.625fl .v>5 :6.43V, 1 .43 fl .

1-30

(a) v� = 10 V, v,. = 50 V . (b) vx = -97 V, v,. = -5 V.

1-32

i� source : 141 V, 707 W. ib source : - 189 V, -567 W. i,. source : -211 V, -2321 W .1,34

(a) 0 .5 mA . (b) 57 A. (c) -22 mA . (d) 11A.1-36

v, source : 29A, 493 W. v2 source : 0 A, 0 W . v3 source : - 13A, - 104 W.1-38

(a) 500 V (positive at Q), 25 fl . (b) 720 V (positive at P), 6 fl .(c) 4 A (out at P), 20 fl .- (d) 0.15 A (out at Q), 200 n.

Chapter 2

V2 = (RZIR,)v, ; v; = (R3lR7,)V7- .2-10

134.3 V .

2-12

G, = 6.83 S . Powers : 4.5 W, 0 .9 W, 61 .5 W, 8.1 W.2-14

2.22 W, 6 .67 kn .

2-16

(a) 1%. (b) 20%. (c) 0 .05% .2-18

G, = (G, + GZ) ; it = (-i~ + i,, - i, . - id) . v = ( iTIG,) . i, = (G,IG,)i,-.i2 = (GZIG,-)i,- .

2-20

142.5 A .

2-22

(a) v, = -58.1 V. i � delivers 1046 W; ib delivers 988 W; v, . delivers 2896 W.(b) v, = -203 V. i� delivers 2640 W; ib delivers 3452 W ; v,. delivers 2424 W.2-24 (a) v. - ( i a + ib)l(G � + Gb + K) . Power = -Kva .(b) v, = -i.,l[G,(I + K) + G2]. Power = -KG,V2 .

2-26 v, = [i�(I + KG b) + ib]l(G, + G,, + KG,Gb) .2-28

(a) R, = 10 .3 fl . i,,. = 9.69 A . (b) R, = 8 .29 fl . i, = 12 .1 A.

2-30

Using the same subscripts as the resistors : (a) i, = 9.69 A, i2 = 5 .61 A, i3 = 4.08 A,is = 1 .71 A, i 5 = 1 .28 A, ib = i 7 = 1 .09 A. (b) i, = i2 = 1 .72 A; i3 = i 4 = 3 .45 A;i s = i6 = 3 .45 A; i 7 = 6.90 A; iR = 12 .1 A; i9 = 5 .17 A; i, )) = 4.31 A; i � = 62 = 0.862 A.

2-32

Using the same subscripts as the resistors : i, = 1 .12 A; i2 = 0.875 A; i 3 = 0.854 A;i4 = 0.250 A; i 5 = 0.521 A; i6 = 0.625 A; i7 = i9 = 0.333 A; ix = 0.146 A.2-34

Percent error = 100[RLR,I(RLR,. + RLRS + R,RS)] .For 10% error : R,. = 9RLR5l(RL + RS ) . R,, = 54 kfl .

Chapter 33-2

(b) (i) A . (ii) E, H, and G .3-4

(a) -v� . (b) - v, + v,. - VD + VL .

(C) VK - VD .

3-6

V, = VK - VU ; V2 = VK - VC + Vg ; V3 = VK - VC ; V4 = VK ;

V5 = VK - V,J + VL ; V6 = VK - VC + Vg - V,,

1'7=

VK - VC + VJ .

Answers to Even-Numbered Problems

2-2 R, = 5.3 1Z . Powers : 11 .25 W, 15 .75 W, 6.75 W, 29.25 W, 11 .92 W .2-4 21 .5 M, 1 .31 W.

2-6 (a) I %. (b) 20%. (c) 50% . (d) 0.05% .

2-8 R, = (R, + RZ + R3); v,- = (V� + Vb + V,) ; i = VAT- v, = (R,IRr)VT ;

3-8

(a) ( G2 + G,)( VI - V2) + G3( VI - V6) + G4( VI - V5) + ( G5 + G6)( VI - V4)+ G7V, = -la.

(b) (GI + G2 + G3)(VI - V2) + (G4 + G5 + G6) (VI - V3) = - la + ib - i, - id - i,-

3-10

v, = - 1 .45 V; V2 = 1 .39 V; V3 = 0.866 V. Starting with Ga and going in alphabeticalorder of the subscripts, the branch currents and power dissipated in the conductances are: 5 .68 A(16 .1 W), 8.52 A (24 .2 W), 5.8 A (8 .41 W), 0.261 A (0.136 W), 0 .174 A (0.150 W),'0.0866 A(0.0750 W), 5.56 A (7.73 W) . Power delivered by the three current sources : 34.8 W, 11 .6 W,10 .4 W.

3-12

New voltages : v, = 11 .6 V, V2 = 14.4 V, v 3 = 13.87 V, VREF = 13 V. Other quantitiesnot affected .

3-14 G2/[(G,G2 + G IG S + G,G4 + G2G3 + G2G4)] .

3-16

Starting with G, and going in numerical order of the subscripts, the branch currents are:6 .67 A, 13 .3 A, 0 A, 5 .77 A, 19.2 A.

3-18

31 .3 A.

3-20

v� source : 1276 W; Vb source : 1253 W; i, source : 666 W.

3-22

Same as in (3-20) above.

3-24

v, = 6 .97 V; V2 = 4 .76 V; V3 = 6 .18 V . Starting with G. and proceeding in alphabeticalorder of the subscripts, the branch currents are: 5.58 A, 4.42 A, 0.618 A, 5.68 A, 3.81 A .

3-26

(a) (v2/v,) = (KG2 + G3)/(G3 + G4 ) . Let A = GIGS + G,G4 + G3G4+ G2G3 + G 2G4 (1 + K) . (b) (i2/i,) = -G4(KG2 + G3)/A . (c) (V,/l,) = (G 3 + G4)/0 .(d) (V 2 /l,) = [G, + G2 (l + K) + G3]/A-

3-28

Matrix elements are: (a) (G, + G2 + G6 + GS + G9), -(G, + G2), -(G8 + G9),-(G, + G2), (GI + G2 + G3 + G4 + G5), - (G4 + G5), - (Ga + G9), - (G4 + G5),(G5 + G4 + G7 + Gg + G9) . (b) (GQ + Gb + G,), - Ga , - Gb, - Ga, (Ga + Gb + Q.),- Gc, - Gb, - G,, (G . + Gb + G,) .

3-30

(a) v, = 2.94 V; V2 = 3 .31 V; v 3 = 3.06 V . (b) v, = 10/(G I + G2) ;V2 = 15 (G3 + G5)/A ; V3 = 1%314,where A = (G, + G2)[(G3 + G4 ) (G4 + G5) - GA-

3-34

(a) Matrix elements are : 2G ,, 0, -G,, -G,, 0, 2G,, -G,, -G,, -G,, -G 1 , (2G,+ G2), 0, -G,, - G1, 0, (2G, + G2)- (b) V2 = i,l(2G 2 ) . (c) (V I li,) = (G, + G2)/2G,G2 .

3-36

(a) (V142) = G2/0 ; (V 2/i, = (G 2 - K)IA, where A = G,(G2 + G3) + G2(K + G3) .(b) (V,/i2) = 0.121 S2, (V 2 /i,) = 0.0213 fl .

3-38

(a) 0 .163 fl, 0.163 fl, 0.463 S2 . (b) 1/(G, + G 2 ), (G3 + G5)/l1, (G3 + G4)/A,where A = [(G3 + G4)(G3 + G5) - G3'1-

3-40

(a) R; � = (G, + G2)[G3(R + 1) + G4 ] + G3G4/G,G 2[G3(P + 1) + G4 ] + G,G3G4 .R � = (G2 + G3)/G 2 [G 3 (R + 1) + G4 ] + G3G4 .(b) R ; �

=

(1 .9

+ 0.2 K) f. R, =

1 .6 fl .

3-42

R � = (G, + G; + GF)l[(G, + Gi)(G � + G,) + (1 + A) GFG � .

3-44

(a) 1000 (b) 2.

3-46

R, = 124 MfZ.

3-48 v� = A[(G, + G,)Vb - G,va]/[GI + GF (1 + A)] .

3-50

Use Eq . (3-92) with v, = (R2R3Vi, + RIR3Vi2 + R,R 2 Vi3 )/(R,R 2 + R,R3+ R2R 3 ) and G4 instead of G, .


81 5

Chapter 4

4-2

(a) i, = -0 .106 A ; i2 = 0.417 A ; i3 = - 1 .298 A . Currents in the other branches are:1 . 19 A in R 2 , 1 .61 A in R5 , and 1 .72 A in v� . 20.6 W delivered by v� , 21 .4 W delivered by v,, .VAB = 18 .8 V . (b) is = 0.636 A, ib = i, . = 0.454 A . Currents in the other branches are: 0 .182Ain R c , 1 .09 A in v� , and 0.636 in R� . 10 .9 W delivered by v.. v u, = 3 .62 V .

4-4

(volvbe) = [1 + (R2/R,)] .4-6

Same result as Eq . (3-93) in Chapter 3 .

4-8

v� delivers 1084 W; i b delivers -288 W . 520 W in R,, 22.7 W in R2, 163 W in R 3 ,and 90.8 W in R4 .

4-10 i� delivers (vbia) W ; vb delivers v b [(vblRL) - ia] W; R L consumes (v~Rc)W . vb receivespositive power (or delivers negative power) when RL > (vblia) .

4-12

i� ib i, . i,, i,, it i,

4-14

R,R . , = R 2R 3 . R, andR�, have no effect on the condition for balance .4-16 Open circuit : i, = 2v,/(R� + Rb) ; VAB = v,(Rb - R.)l(Rb + R�) .Short circuit : i, = v,(R. + R b)l2R,Rb . i s, = v,(Rb - R �)l2R �Rb .4-20

R, = Ru R bl(Ra + R b + R c ) ; R 2 = R o R,J(R o + R b + Rc) ;R 3 = Rb R,J(Ra + R b + R,.) .R� = (R,R 2 + R,R 3 + RZR 3 )/R 3 ; R b = (R,R2 + R,R 3 + RZR3)/R2 ;Rb = (R,R2 + R,R 3 + RZR3)/R, .

4-22 v 3 = - KZRZR3V,/{(R, + K 2R,)(R, + R 2 + R3)

4-24

v� delivers vaic W. i, delivers (R bic - v .) i, W ; Rb consumes (Rbi,. 2)W .i,. receives positive power when R b < ( VQ/ic) .

4-26

v � = R[(2/3) i~ + ib + (4/3) ic ] .4-32

Elements of the matrix are : (R, + R 2 + R3 + R4), -R 4 , -(R 2 + R 3), -R 4 ,(R 4 + R6 + R, + R s ), - (R6 + R,), - (R2 + R3), - (R6 + R,), (R2 + R3 + R6 + R2 + R5) .i, = 4.57 A, i2 = 2.57 A, i 3 = 2.86 A .4-34

Elements of the matrix are : (R, + R 2 + R 3 ), -(R 3 + KR 5 ), KR5 , -R 3 ,(R3 + R 4 + R, (1 + K)], -R5 (1 + K), 0, -R 5 , (R 5 + R6) .

4-36.

4.57 x 10-2 S, 2.57 x 10 -2 S, 2.86 x 10 -2 S .

4-38

Elements of the matrix are: (R, + R 2), -R2 , (K - R2), (R2 + R3) .

Chapter 5

5-2

VTh = v,(R2 - R,)/(R 2 + R,) . RTh = 2R,R 2/(R, + R2) .5-4

R; � = (20 + 32K)/(7 + 8K) .

5-6

(a) iN = 5 .74 A . (b) iN = - 12 .7A . (c) iN = 6.64 A . (d) iN = (R2 - R,)v,12R,R, .(e) iN = (R, - KR2)vs/(R, + R,) R 2 + RX, .

5-8

V Th = 4 .8 V, IN = 2 A, RTh = 2.4 l .

5-10

VTh = 10 V, RTh = 6 S:! . it = 1 .37 A, VL, = 1 .88 V .


+1 +1 0 0 -1 0 +1 00 +1 +1 0 0 -1 0 +10 0 +1 +1 +1 0 -1 0

+1 0 0 +1 0 +1 0 -1

5-12

(a) R = 0 . (b) Rb = - . (c) Ro = oo . (d) Rb = 0-5-14

(PLIP.,) = RLI(RL + R,) . (PUP S ) = 0 when RL = 0 and approaches 1 as R L ~ X. P, ismaximum when R L = 0 ; PL is maximum when R L = R, ; (PLIP.J is maximum when RL = X-

5-16 i s, . = i� + (i,.I2) + (vdR) .

5-18 vb = -Ria - 2Ri,. .

5-20

Both ratios = (R I + R2)18R,R2 .

5-22

Passive network: r,1 = 102 fl ; r12 = r21 = 107 fl ; r13 = r31 = -37 .9 fl ; r22 = 128 fl ;r23 = r32 = -51 .7 fl ; r33 = 34.5 fl . Data insufficient for active network.

Chapter 66-2

i(t) = CV� ,{(1 + m cos At) B cos Bt - mA sin At sin Bt}.

6-4

(a) v, .(t) - (1lb2C)[1 - (1 + bt)e - b'] . (b) w,.(t) = (1I2b4C)[I - (1 + bt)e-bx]2 .

p, . (t) = (Ilb2C)[ 1 - (I + bt)e-bx]te-b' .

6-6

(a) vL = -0.125 V (0 < t < 4) ; 0.25 V (4 < t < 6) . Repetitive .(b) wL = 0.025 (-2.5t + 10)2 0 < t < 4; 0.025 (5t - 20)2 (4 < t < 6) .(c) p(t) _ (0 .3125t - 1 .25) W (0 < t < 4) ; (1 .25t - 5) W (4 < t < 6) .

6-8

(a) i(t) _ (- 0. l t2 + 2t) A (0 < t < 10) ; (-1 .331 + 23.3) A (10 < t < 20);(4.44 x

10-2t2 - 3. l It + 41 .3) A (20 < t < 35); -13 .2 A (t > 35) .(b) i(t) = (-O.lt2 + 2t + 3)A (0 < t < 10) ; (-1 .331 + 26.3) A (10 < t < 20);(4 .44 1

10-2t 2 - 3.1 It + 44.3) A (20 < t < 35); - 10 .2 A (t > 35) .

6-10 x(t) = 75e- '°' + 25 .

6-12

(a) i(t) = 1�e-" . (b) i(t) = (BIA)(1 - e-^') .

6-14

R = 10 .5 kil C = 22 WF .

6-16

i,(t) _ -0.018e-2 .4t A (t > 0) . i2(t) = (1 .67 - 15e-2 .41 )MA (t > 0) .

6-18

v,.(t) _ [V� (I - k)12][1 - e -" I where ? = (K + 3)RC/2 .

6-20

(a) v,. increases to 9.93 V at t = 50 ms, decays to zero and increases again to 9 .93 V att = 200 ms and decays to zero . (b) v,. increases to 3.94 V at t = 50 ms, decays to 1 .45 V att = 150 ms, increases to 4.81 V at t = 200 ms, and decays to zero .

i, .(t) = 0.03e -1 .33, A . (c) v,.(t) = (45 - 20e-1 .331 ) V; i,.(t) = 0.0133e-1 .33' A .-3.13x 104,

-3.13x I Oar6-36

(a) iL(t) = 0.03e

A; VL(t) _

- 14.1 e

V.(b) iL(t) = (0 .2 - 0.2e-3.131104x) A ; VL(t) = 94e -3 .131104 f V .(c) iL = (0 .2 - 0.17e-3 .13x104,) A; vL(t) = 79.9e-3.13x1041 V .

6-40

R = 7 .36 fl; ; L = 92 mH; C = 6793 N.F .


81 7

6-22 v, I -- 80e -s ' V. v,.2 = 100 + 20e-s ' V .6-24 R = 2M. L = 39 .2 mH .

6-26 iL(t) = - 170 mA (t < 0) ; (-42.6 - 127 e -108x 1 `4) mA (t > 0) .

6-28 iL = -7A(t<0);(-2.5e-'4.3,-4.50)A(t>0) .i, = -4A(t<0) ;(-2 - 1 .43e-14.31) (t > 0) .

6-30 iL = (V�,IR)(I - e - ") (0 < t < tp); (V,�IR)(1 - e-'PI)e-"-'n" (t > t") .

6-32 R=4M.L= 111 H .

6-34 (a) v,.(t) = 25e - '-"' V ; i,(t) = -0.0167e-1'33' A. (b) v,.(t) = 45(1 - e -1 .33') V;

Chapter 7

(c) (-7.24 cos 50t + 49.0 sin 50t) V.7-8

(a) 36.0 cos (1001 + 56.3°) V. (b) 28 .0 cos (200t + 105°) V . (c) 10 cos 3001 V.7-10

(a) Amplitude = 2A cos ((~/2) . (b) Phase = (4)/2) . (c) Maximum when

= 0andminimumwhen ~ = 180° .

7-12

i(t) = 4.47 cos 105t A . v(t) = 112 cos 1051 . V . Pa, = 250 W.7-14

i(t)' _ -7 .2 x 10-2 sin (5 x 106t) A . p(t) = -0.432 sin 10 7t W.7-16

v(t) = 17 .8 cos (1 .26 x 104t) V. i(t) = -5.61 sin (1 .26 x 104t) A.p(t) = -50 sin (2.52 x 1041) W .

7-18

v (t) =

-30 sin 500t V . p(t) =

-60 sin 1000t W.7-20

i(t) = 20 cos 1000t A. v(t) _ -50 sin 10001 V. L = 2.5 mH .p (t) = -500 sin 2000t W.7-22

(a) Peak value increases by a factor K,2 and frequency is not affected for all three elements .Constant component for the resistor changes by a factor of K,2, but no change for the inductor orcapacitor . (b) Peak value unaffected for the resistor, decreases by K2 for the inductor, and increasesby K2 for the capacitor . Frequency increases by a factor K2 for all three elements . Constantcomponent not affected for all three elements . (c) No effect on any of the items .7-24

i(t) = 10 cos (2000t - 2 .86°) A.7-26

v(t) _ [I~l

G2 + 1/w2L2] cos (wt + arc tan 1/wLG) .7-28 Im = V�,/VR2 + w2L2 . tan

_ -(wLIR).7-30

G = 0.313 S . C = 1 .56 x 10-2 F .

7-32

(a) 1168 r/s . (b) 316 r/s . (c) 184 r/s .

7-34

p(t) = [19.9 + 42 .3 cos (5000t - 61 .9°)] W. Pave = 19.9 W-7-36

(a) Im = 4.62 A. p(t) = [500 + 577 cos (2wt - 30°)]W . (b) Im = 4.62 A .p(t) = 500 + 577 cos (2wt + 30°) W. (c) I�, = 4 A. p(t) = 500 + 500 cos 2wt.7-38

(a) 0.577 Vm. (b) 0 .745 I,� .

Chapter 8

8-2

(a) v1(t) = 10 cos (200t - 135°) V; (b) v2(t) = 10 sin (2001 - 75°) V;(c) v3(t) = 10 cos (200t - 180°) V.

8-4

(a) f,(t) = 2N/p-2+ q2 cos [bt + arc tan (qlp)] . (b) f2(t) = 2V/-n'-+a2 cos bt .(c) f(t) = 2

p2 + q2 cos [bt + r + arc tan (qlp)] . (d) f4(t) =

2

p2 + q2 cos bt .8-6

(a) h1 (t) = Im[22.4ej)500`+63 .4')] . (b) h2(t) = Im[14 .1e'(500`+9s.i°)] .(c) h3(t) = Im[18e'(500i-,46')] .

8-8

(a) l0/-30° V . (b) 24 .8/- 13° A .

8-10

(a) 5 cos (1001 + 36 .9°) A . (b) 8.06 cos (100t - 60.2°) A . (c) 3 cos (100t - 90°) A .8-12

(a) 2ej' 3 ' -30 ) A. (b) 16 sin (3t - 12°) V.8-14

(a) 47i + 8333 fidt + 14 x 10-3 (dildt) = 160 cos (3771) . (c) i(t) = 3 .21e'' 3111 ") A .(d) V(t) = 3.21e-''377`+20°) A. (e) i(t) = 3 .21 cos (3771 + 20°) A .8-16

i(t) = cos (2t - 97°)A .


7-2 (a) 477 HZ . (b) i(t) = 1 .5 cos (3000t - 90°) A.

7-4 i,(t) = 20 cos 524t A; i2(t) = 16 cos (524t + 120°) A .7-6 (a) (- 104 cos 100t + 60 sin 100t) V. (b) (26.0 cos 450t + 15 sin 4501) A .

8-18

(a) (v/R) + C(dvldt) = Io cos wt. (c) V = RIJ(1 + jwRC) . (d) R = 0.463 fl,C = 0.125 F . (e) v(t) = 0.756 cos (50t - 70.9°) V .

8-20

iL + LC(d 2ildt2) = I.cos (tl2NIL-Q. iL(t) = (4/3)1, � cos (t/2NL) .8-22

(a) 166/-73 .6 ° fl, 47 dl, -160 fl . (b) 48 .9/- 16° l, 47 fl, - 13 .5 fl .(c) 52.5/26 .5 ° fl, 47 fl, 23.4 fl . (d) 254/79.3 ° f, 47 fl, 250 dl .

8-24

(a) 300L° V, 4000/-90° V, 2000° V, 1000/-90° V, 3041/

80.5° V . (b) 790 r/s .(c) 856 r/s . (d) 730 r/s .

8-26

8-28

8-30

8-32

8-34

8-36X =B =

(a) w = (11RC) . I = (0.707°)(VS/R) . (b) w = 0.378/RC . I = (0.353L°)(Vs/R) .

C = 100 p,F . Zx = 22.4/26.4' fl . 20 .1 fl in series with 10 mH .

Z, = 100L° f . 48 .5 fl in series with 0.875 H .

(a) Y = 0.125/-80° S . (b) I = 5 x 10'/- 113 ° A . (c) V = 0.962/

30.5° V .

(a) 110 p,F . (b) 90 liF . (c) 140 p,F . (d) 60 VF .

(a) R = 10 f, X = 4 .12 dl, G = 0.0856 S, B = -0 .0353 S . (b) R = 5 .95 dl,-4.91 f, G = 0.1 S, B = 0 .0824 S . (c) R = 0.684 dl, X = 1 .880 11, G = 0.171 S,-0.470 S . (d)R=0.12 fl, X= -0.16 fl, G=3 S,B=4S .

(a) w = (G/C) . V = (0.707/-45°)(Is/G) (b) 2.64(G/C) . V = (0.353/-69 .3°)(I,/G) .8-38

8-40

(a) C = 250 p,F . (b) Y, = j1.75 S . 1750 wF .

8-42

8-44

8-46

8-50

8-52

8-54

8-56

8-58pf =

Chapter 9

9-2

(a) -j4.11 fl . (b) l0.OL° fl . (c) 3.60/-55 .6' fl .

9-4

(a) 4.83/10 .2' Sl . (b) Multiply each R and each L by 25 and divide each C by 25 .

C = 0.188 p,F .

G = R/[R2 + (wL - 1/WC)2 ] . B = -(wL - 1/wC)/[R 2 + (wL - 1/WC)21 .

(a) 6.67 r/s . (b) 3.33 rls . (c) G = 0.002 S, B = 0.004 S .

R = 18 .7 fl, L = 0.054 H .

Z, = 7.07/-45° f . L = 50 mH.

(a) 25 .2 mH or "19 .4 mH . (b) 83 .1 p,F or 155 wF . (c) 1 .04 mH or 0.487 mH .

133 kW, 117 kVAR.

(a) 2/-60° kVA in Z, . 2.66/41 .4° kVA in Z2 . (b) 3000 W, 28 VAR, 3000 VA .1 . (c) 2 .12 fl in series with 0.0586 mH .

(c) R remains unaffected . Multiply each L and C by 0 .2 .

9-6

(wL) = (R13) .

9-8

9-10

9-12

V, = 1592/15 .3' V . Z�, = 9.42L° n. P av = 67.3 kW . p f = 0.5 (current lagging) .

Zr = 8/67.6' fl .

(3.5 + j2 .5)

-(0 .5 + j2.5)

[

-(0.5 + j2.5)

(2 .5 + j1 .5)]

9-14

V, = (wLI,)l[wLG + j(w2LC + K - 1)1 . V 2 = V,(1 - K) .

9-16

Vu delivers 1000 W. VB delivers -500 W. V, . delivers 500 W.

9-18 Y ; a = [GI(YF + YL) + YF(9. + YL)]l(YF + YL) .

9-20 (8 + j2.5)

-(5 + j2 .5)

[ -(5 + j2.5)

(7 + j3 .5)]

9-22

1, = 2.84/55 .4' A . 1 2 = 5 .68/125' A .

Answers to Even-Numbered Problems 819

9-24

1� delivers 470 W. Ib delivers 2243 W . I,. delivers 1431 W.

9-26 Z; n _ [R, (Z, + ZL ) + Z,(ZL - KR,)]l(Z, + ZL).

9-28

V,. = 2 .59/- 129° V .

9-30

V� = 348/-57 .4° V .

9-32 (I,)V.,) = (,)2C2R/[w2C2R 2 - 3jwRC - 1] .

9-34

(a) i�(t) = 3 .85ej(21+46 .1 ') + 4.62ej)4` +25 .7°) A.(b) i o(t) = 3.85e -j(21+46.1*) + 4.62e-i(41+25 .7') A .(c) i�(t) = 3 .85 cos (2t + 46.1°) + 4.62 cos (4t + 25 .7°) A .

9-36

(a) IH(j(o)l _

(w 4 + 61w 2 + 100)/(w4 + 5w 2 + 4) . 9('w) = arc tan [9w/(10 - co2)]- arc tan [3w/(2 - W2)] . (b) JH(jw)I = 103 (1 + w2)(106w 2 + 1)/(w2 + 106 ) . e('w) = arctan (w) + arc tan (1/10 3w) - arc tan (103/w) . (c) IH(jw)l =

9(w2 + 1)/(w4 + 9w2) .9(jw) = 7r + arc tan w + arc tan (3/w)

9-38

(V2/V,) = 1/[(1 - 0.125w 2 + j0 . 125w] . Amplitude response starts at I at w = 0, reachesa peak value of 2.86 at 2.74 r/s, and approaches zero as w ---> -. Phase response starts at 0° atw = 0, becomes - 45° at 2.37 r/s, and approaches zero as w -> x.9-42 (V2/V,) = (1 - K)l(2 + jwRC) . 0.186(1 - K)/ - 68.2° at w = (5/RC) ;0.354(1 - K) -45° at w = (2/RC) ; 0.447(1 - K)L26.6° at w = RC;0.485(1 - K)/- 14 ° at w = (1/2RC) ; 0 .498(1 - K)//--5 .7 ° at w = (1/5RC) .

9-44 (VA;) = A;,�/Jjyjj . (V 2/I 1 ) = (V,/I 2 ) = -j(1/WL2)/jjYII,where Ilyll = [G2 + j(WC - 1/wL 2 )][G, - j(1/wL, + I/wL2)] + (1/w2L,L2) .

9-46

(a) VTh = 28.5L° V; ZTh = 1 .46LL ° Q. (b) VTh = 10.9/-35 .2 - V ;ZTh = 0.752/76.3' S2 .

9-48

(a) AA': VT,, = [(0 .91 + 0.182K) - j0.364] V; ZTh = (3 .09 - 0.454K) +j0 . 182(K - 1)fl . BB' : VTh = (4 - 2K)/[(8 - K) + j2] V ; ZTn = 2(6 + j2)/[(8 - K) + j2]d2 . (b) AA' : VTh = - 10K/[3 + j(0.8K - 0.2)] V . Z Th = 0 . BB' : VTh = (3 + jl)(1 + K) V ;ZTh = (0.3 + j0.1)(1 + K) fl .

9-50

ZTh = jwLR,l[(R, - K + jwL] .

9-52

(a) AA:' IN = [(10 + 2K - j4]/[(34 - 5K) + j (2K - 2)] A. BB' ;IN = (2 - K)/(6 + j2) A. (b) BB': IN = 10 A.

9-54

VTh =

100/ - 30. V . ZTh = 50 .1 5/ 3 .1 - fl .

9-56

(a) R = 0.6 S?, X = 0.2 fl . P~ = 209 W . (b) R = 0 .565 fl, X = 0.283 fl .Pmax = 208 W.

9-58

i (t) _ (2/3)N/-C/L V, cos (t/2V"L-C- + 90°) + V2 cos (2t/LC - 90 ° ) A.

9-60

l,(t) = 0.4471, cos (tl2RC + 63 .4 ° ) + 0.7071 2 cos (tIRC + 45°)+ 0.8921 3 cos (2t/RC + 26 .6°) A . iR(t) = to + 0 .8941, cos (tl2RC - 26.6°)+ 0 .7071 2 cos (tIRC - 45 ° ) + 0.44613 cos (2t1RC - 63.4 ° ) A .

9-62

v�(t) = 104 + 53.3 cos (500 t - 90 .1 °) + 66.7 cos (1000t + 92°) V .

9-64

v,.(t) = 415 cos (1000t + 90°) + 104 cos (2000t - 90°) V .

Chapter 10


10-2 (a) 4.74 mWb. (b) 2 .53 x 10 5 .

10-4 (a) 1 .28 x 10-5 Wb. (b) 0.938 x 108 .

10-6 1 .43 A .

10-8 3 .55 mWb.

10-10

(a) Mesh 1 : [R, + j(wL, - 1/wC)]I, - j(wM - 1/WC)IZ = Vs . Mesh 2 :-j (WM - 1/WC)(I, + [R Z + R3 + j(wLZ - 1/WC)]I 2 = 0 . (b) Loop 1 : [R, + jw(L, + LZ =2M)]I, + [R, + jw(L, - M)]I Z = Vs . Loop 2 : [R, + jw(L, - M)]I, + [R, + R z + R 3 +j(wL, - 1/WC)]IZ = V.s .

10-12

(a) VTn = 88.5/-81 .9' V . ZTn = 66.8L° fl .(b) VTh = 75 .6/6.34° V . ZTn = 20.8L° fl .

10-14

(a) 12.3 mJ . (b) 9 .42 mJ .

10-16

See answers to Problem 10-12 .

10-18

(a) 1 .92/ -50.3° A; 61 .3 W . (b) 0 .625 /

14.5° A; 30.2 W .

10-20

Z;~ = R(1 - a)2/(1 + ja 2wRC), where a = (N,INz) .

10-22

(a) Vub = 398L° V . (b) V�,. = 398/-30° V . (c) Vb� = 230/-120° V .(d) V, .u = 398° V. (e) V,b = 398L° V.

10-24

(a) 1. = 5 .02/41 .6 ° A . I b = 5 .02/-78 .4° A . I, . = 5 .02L° A.(b) 252 W. (c) 756 W.

10-26 Z2 = 3Z, .

10-28

(a) 1� = 3 .26° A. Ib = 11 .4/-75° A . I,. = 8.97° A.(b) Z�b consumes 722 W, Zn ,. 1083 W, and Z, .a 468 W. (c) 2273 W.

10-30

Phase sequence a-b-c : (a) I, = 13 .9 ° A ; IZ = 10 .4L:: 75° A ; 1 3 = 13 .9 / - 168° A .(b) 1� = 27 .8/15 .2' A ; I b = 17 .8/-126° A ; 1, = 17 .8° A. (c) Z, : 483 W. Z z : 809 W.Z 3 : 1451 W. (d) 2741 W . Phase sequence a-c-b : (a) 1, = 13 .9° A. I z = 10.4 165° A .1 3 = 13 .9/-48.4 ° A . (b) IQ = 15 .3L°A . Ib = 23 .3/- 176° A . 1, . = 23 .2 -34 .3° A .Answers to (c) and (d) are not affected by the phase sequence .

10-32

(a) Y = 4 .17 x 10-Z/-60° S . 48 .1 0 in parallel with 88 .2 mH . (b) C = 56 .7(c) Same average power as before . Apparent power = 20 kVa .

10-34

(a) Reading of W, = (3V"3V. 2/21ZL I) cos (8 - 30°) . Reading of Wz = (3v/ V,n z/21ZLI)cos (6 + 30°) . (c) 191 > 60° . (d) 0° .

10-36

W, reads 20 kW . Wz reads 25 kW . Total average power = 45 kW . Total reactive power- 5 kVAR (current lagging) . Total apparent power = 45.3 kVA .

Chapter 11

11-2

(a) L = 0.507 p,H . (b) I R = l OL° A . I L = 62 .8/ - 90° A . Ic = 62 .8L°A .

11-4

iG = 25 cos 1000t + 20 cos 2000t + 15 cos 3000t mA . iL = 50 cos (1000t - 90°)+ 20 cos (2000t - 90°) + 10 cos (3000t - 90 ° ) mA . is = 12 .5 cos (1000t + 90°)+ 20 cos (2000t + 90°) + 22 .5 cos (3000t + 90°) mA . is = 45 .1 cos (1000t - 56.3°)+ 20 cos (2000t) + 19 .5 cos (3000t + 39.8° ) mA .

11-6

(a)G=0.01 S.C=50p,F.L=0.2mH .(b) W, . = 50 x 10-6 [l + cos (2 x 1041)] J . WL = 50 x IO-'[I - cos (2 x 10 4 t)] J .(C) Wmax = 10-4 J .

11-8

(b) w = 1/

(LC - G2Lz/2) .

11-10

(a) Co = 1/wo2L . (b) C = (1/o2L) ± (N/3G/w~) .

11-12

(a) G = 0.0223 S . L = 4.56 VH . C = 88.9 nF . (b) iG = 3.34 cos (1 .57 x 10°t) A .iL -- 20 .9 cos(1 .57 x 1Obt - 90 °) A . is = 20 .9 cos (1 .57 x 1Obt + 90°) A .(c) wL = 5 x 10'[1 - cos (3 .14 x 1O b t)] J . W,. = 5 x 10'[1 + cos (3 .14 x 1O bt)] J .(d) wR = 10-3 J/cycle .

11-16

(a) BW = 200 r/s . Half-power at 124 r/s and 324 r/s . (b) Y = 0 .01 S. (c) 0.0141L° S .


821

11-18

(a) G = 2 x 10-4 S. L = 0 .319 mH. C = 1 .27 nF . (b) Half-power at (1 .495 x 10 6)r/s and (1 .652 x 10 6) r/s . (c) wT = 3.96 x 10-9 J. (d) wR = 2.5 x 10 -9 Rcycle .

11-22

(a) BW = 1 .25 x 105 r/s . Half-power at 6.262 x 106 r/s and 6.387 x 10 6 r/s .(b) R = I M SZ .

11-24

(a) 6324 r/s . (b) G = 2 x 10 -4 S; L = 50 mH; C = 500 nF . (c) 1, . = 790° mA.I, �;, = 792/ -86 .4° mA . (d) BW = 400 r/s . Half-power at : 6128 r/s and 6528 r/s .

11-26

RL = 0.305 f. L = 4.86 wH . C = 3 .18 nF .

11-32

(a) L = 0 .507 I,LH . (b) VR = IOL° V. VL = 1 .59

° V. Vc = 1 .59/-90 ° V.

11-34

(a) R = 100 S2 . L = 0.5 H. C = 20 nF . (b) wL = 1 .25 x 10`11 + cos(2 x 10°t)] J . w, = 1 .25 x 10-3 [1 - cos (2 x 1040] J. (c) WT = 2.5 x 10' J.11-36 (b) w =

N/-1/LC - RZ/2Lz11-38

(a) R = 44 .7 fl . L = 0.178 mH. C = 2.28 nF . (b) VR = 15010° V.VL = 93810° V. V, . = 938/-90° V . (c) wT = 5 x 10 -°[1 + cos (1 .57 x IO6t)] J.w, . = 5 x 10'[1 - cos (1 .57 x 1O6t)] J. (d) wR = 1.01 x 10-3 J/cycle .

11-40

(a) BW = 100 r/s . Half-power at : 1951 r/s and 2051 r/s . (b) Zm, � = 50 S2 .(c) 70 .7/ :t45' fl .

11-42

(a) R = 5 W. L = 31 .8 mH . C = 12.7 pF . (b) Half-power at : 1 .494 x 106 r/s and1 .651 x 10 4 r/s . (c) WT = 6 .36 x 10' J . (d) wR = 4 x 10-R J/cycle .

Chapter 12

12-2

(a)f(t) = 5e;'cv' e"oa + 5e-j30' e-J1001 . (b) f2(t) = 5e"5' e)-5-j377)i + 5e-j15' e,-

(e) f,(t) = 5e 3' + 5e - ". (d)f4(t) = 5e" - 5e-3r .

12-4

(a) Yin = 6.5(s2 + 7.08s +4.92)/( s3 +

17sz +

16s).(b) Y; � = 0.5(s4 + 5 .67sz +

1 .33)/( s' + 4.33s) . (c) Y; � = 3(s +

1)/s .(d) Y; � _ [(4 - 3KR)/3R][s + 1/RC(4 - 3KR)]l(s + 1/3RC) .12-6

(a) 0.154( s3 +

17sz +

16s)/(s= + 7 .08s + 4.92) . (b) 6.67s/( S4 + 5 .67s= +

1 .33) .(c) 2. (d) I/[C(4 - 3KR)][s + 1/(4 - 3KR)RC] .12-8

(a) Zeros at s =

-0.781, - 6 .30; poles at s = 0, -1, - 16 .(b) Zeros at s =

±jO.495, ± j2.33; poles at s = 0, ±j2.08. (c) Zero at s =

- I ;pole at s = 0 .(d) Zero at s = - 1/RC(4 - 3KR) ; pole at s = -1/3RC .

12-10

(a) Zeros at s = 0, - 1, - 16; poles at s =

-0.781, - 6 .30 . (b) Zero at s = 0; polesat s = ±jO.495, ±j2.33. (c) no critical frequencies . (d) pole at s = - 1/(4 - 3KR)RC .12-12

Series : Any pole of Z, or Zz is a pole of the total impedance. Any zero common to bothZ, and Z2 is a zero of the total impedance. Parallel: Any zero of Z, or Z: is a zero of the totalimpedance. Any pole common to both Z, and Zz is a pole of the total impedance.12-16 (V�lV,) = (AIR,C,)(s + 1/R2C2)/(s + 1/R,C,)[s + (1 + A)/R,C=] .


5+j377t) .

12-18

The asymptotic plots of the functions are as follows . (a) Horizontal at -40 dB up to 100r/s and downward at -20 dB/decade after that . (b) Upward at a slope of + 20 dB/decade from w= 0 to 100 r/s and levels off at a constant value of 0 dB after that . (c) Horizontal at -80 dB upto 100 r/s and downward at -40 dB/decade after that . (d) Upward at a slope of +40 dB/decadeup to 100 r/s and horizontal at 0 dB after that .

12-20

The asymptotic plots of the functions are as follows . (a) Horizontal at 40 dB to 0.01 r/s,upward at 20 dB/decade from 0.01 to 0.1 r/s, horizontal at 60 dB from 0.1 r/s to I r/s, and upwardat 20 dB/decade from 1 to 10 r/s, and horizontal at 80 db after that . (b) Upward at 20 dB/decadeto 10 r/s, horizontal at 20 dB from 10 r/s to 100 r/s, and downward at -20 dB/decade after that .12-22

(a) Single-stage asymptotic plot : Upward at 20 dB/decade to 2000 r/s, horizontal at 80 dB

from 2000 r/s to 104 r/s, and downward at -20 dB/decade after that . (b) For the three-stageamplifier, upward at 60 dB/decade to 2000 r/s, horizontal at 240 dB from 2000 to 104 r/s, anddownward at -60 dB/decade after that . (c) 240 dB . (d) Bandwidth = 1000 r/s (approx) .12-24

(a) HI(s) = 10"s/(s + 103)(s + 106 ) . (b) HZ(s) = (s + 103)/(s + 100) .12-26

(a) - 128° . (b) -2.95 dB .

12-28

(b) -246° . (c) 36.5 dB . (d) -164° at 0 dB; -40 dB at -180° .

12-32

(a) w� = 50.2 r/s . Q., = 5.02 . BW = 10 r/s . R = 10 S2 . C = 396 p,F .(b) Same as in (a) except that G = 3 .96 x

10-'S .

Chapter 13

13-2

Elements of the z matrix are: (a) 1 .8/(2 .9 + 0.8K), 0 .8/(2 .9 + 0 .8K),(0 .8 - 0.4K)/(2 .9 + 0.8K), 1/(2 .9 + 0.8K) . (b) 14(1 - K)/(3 - K), 5(1 - K)/(3 - K),(5 - 7.5K)/(3 - K), (8.75 - 5K)l(3 - K) .

13-4

Elements of the z matrix are: -K4/K 3 , 1/K3 , (K2K3 - K,K4)/K3 , K,/K3 .

13-10

(a) Elements of the z matrix : 0.311 /22.3 ° 51,, 0.194/-29.2° SZ, 0.969/-29° fl,1 .38/15 .8 - S2 . (b) V�/VS = 0.745/-47.Y . Z� = 1 .36/25 .9' .

13-12

Elements of the y matrix are: (a) (R, + R2)/R,R2, - IIR 2 , (K - 2), (R2 + R3)/R2R3 .(b) (4K - 7)/26(K - 1), -0.154, (4 - 6K )/26(K - 1), 0 .431 .

13-14

T model elements : 6.08 d::, 0 .963 fl, -0.296 1Z, 2 .371, . Pi model elements :-0.20 S, 0.65 S, 4.10 S, - 1 .6V, .

13-16

(a) Elements of the y matrix : gbb(gbe + gee)l(geb + gee + gee),l(gi- (gebge, )1(91',b + Bee + 9bc), (gm - gae)gi

b + gee + ge .),9ce + [ge,(g. + gev + gee)l(geb + gee + gee) . (b) 4.69 x 10-3 5 .

13-18

Elements of the h matrix : (a) 0.4 1Z, 0 .8, (0.4K - 0.8), (2 .9 + 0.8K) S .(b) 26(K - 1)l(4K - 7), 4(1 - K)/(7 - 4K), (4 - 6K)l(4K - 7), (2 .4 - 0.8K)/(7 - 4K) .

13-20

Elements of the h matrix : 2.22 S2, 1 .44, -5 .00, 1 .5 S . (b) (1 2/1,) = - 1 .25 .

13-22

(a) - 25 . (b) 25 .

13-24 h � = 1/y,, . h,2 = -Y12/Yii . h2i = Y2Aii , h22 = (Y11Y22 - Y12Y2i)IYii .13-26

(a) A = (L2Cs 3 + 2RLCs2 + Ls + R)IL2Cs' .B = [2RL2Cs3 + (L2 + 2R2LC)s2 + 2RLs + RZIL2Cs3 . C = (2LCs2 + 1)IL2Cs3 .D = (L 2Cs3 + 2RLCs2 + Ls + R)/L 2Cs3 . (b) A = 1 .63 B = 6.30 fl, C = 1 .08 S, D = 4 .79 .

13-28

(a) h � = BID . h,2 = (AD - BC)ID . h2, _ - 11D . h22 = CID .(b) A

=

-(hiihz2 - h12h2i)/h2, . B =

-hiilh2, . C =

-h22/h2, . D =

- l/h2, .

13-30

(a) With 1, = 0, E = (V2/V,), and G = (1 2 /V,) . With V, = 0, F = -(V2/1,),and H = -(I 21I 1 ) . (b) E = DI(AD - BC), F = B/(AD - BC), G = C/(AD - BC) .H = AI(AD - BC) . (d)-EFGH parameters are : 0 .694, 1 .54 S2, 1 .04 S, 7 .31 .

13-32

h parameters . Overall h matrix = sum of the individual h matrices .

13-34

(a) A = 0, B = -h�/h2 , . C = 0 . D = - 1/h 2 , . (b) V 2/V., = RLh2,,,h2,nlR, + h �_

Chapter 1414-2

a, = 0.632 . a�, = 1 .264/(1 + 4 ar2m2 ) . a, = 0.0312 ; a2 = 7 .95 x 10-3 ;a3 = 3.55 x 10-3 ; a4 = 2 x 10-3 ; a5 = 1 .28 x 10-3 . b, � = 2.53 ,rrm/(I + 4 rr2m 2 ) .b, = 0.196 ; b2 = 0.100 ; b3 = 0.0669 ; b4 = 0.0503 ; 65 = 0.0402 .

14-4

(a) T = 10 ms . (b) Amplitude = (20/k7r)

(1 + k2 -rr2) + 1 . Phase = (ar - arc tank7r) . (c) Third harmonic : 2 .13 cos (600at + 96°) . Fifth harmonic: 1 .28 cos (1000irt + 93 .6°) .


823

14-6

(a) vI(t) = 0.15 + 0.271 cos (wt - 72 .6°) + 0.199 cos (2wt - 148°)+ 0.117 cos (3wt + 124°) + 0.0699 cos (4wt + 12°) + . . .(b) 0.15 + 0.271 cos (wt + 72.6°) + 0.199 cos (2wt + 148°) + 0.117 cos (3wt - 124°)+ 0.0699 cos (4wt - 12°) + . . . (c) v3(t) = 0 .517 sin (wt + 180°)+ 0.211 sin (2wt + 180°) + 0.194 sin (3wt) + 0 .0291 sin (4wt) + . . . .

14-10

v�(t) = 15 + 31 .2 cos (5001 - 29.4°) + 10.6 cos (1500t + 89 .6°)+ 0.0343 cos (25001 - 126°) V .

14-12

L = 0.75 H. C = 0.333 l.LF . i(t) = 13 .0 cos (1000t + 77 .5°) + 30.0 cos (2000t)+ 7.43 cos (3000t - 68.2°) nzA.

14-14 131 mW.

14-16

(a) 0.612 V . (b) 2.83 V. (c) 8.66 V .

14-18

c� = 0.75 . c,� = (1/2mIa2)[(I + jm7r)(- I)' - 1] - (1/j2arm)[1 - (- 1)']14-20

(a) T = 31 .4 ms . (b) do component = 0.75 V. (c) 0.433 cos (2001 - 22.5°) V,0.138 cos (4001 - 45°) V, 0.0646 cos (6001 - 67.5°) V.

14-22

v(t) = 4.64e -i(5000t+ 21 .8°) +

12 .5e-A3000t+ 153 .7° ) + 8 .9e -(1000t+93 .4') + 8 .9ej( 1000t+93 .4°)+ 12 .5e'(3000t+153 .7°) + 4.64e(5000'+21 .8) .

14-24 vt,(t) = 0.0596e-'(1000`-232') + 0.130e-i(500t-146') + 0.432 + 0.130ej(500t-146')+ 0 .0596eit1000-2320) .

14-28

(b) 1 .43 W. (c) f(t) = 0 .75 + 1 .13 cos (wt + 57 .5 °) + 0.478 cos (2wt - 180°)+ 0 .324 cos (3wt + 78°) + 0.238 cos (4wt - 180°) + 0.193 cos (5wt + 83°)+ 0 .154 cos (6wt - 180°) .

314-30

(a) vo(t) _

k= -3 (50/k3r)sln (kar/l0)e7('000k.t-0 .1k~) (c) v�(t) = 5+ 9.84 cos (1000irt - 18 °) + 9.36 cos (2000,irt - 36°) + 8.58 cos (3000,rrt - 54°) V.

14-32

F(jw) = (A/w2to)[(1 - cos wt, - wt. sin wt,) + j(sin wt, - wt,, cos wt,,)] .IF(jw)l = (A/w 2tn) V[2 + wltP2 - 2 cos (wt,) - 2wta sin (wt.) .14-34

(a) (4V�/jw) sine (wtP) . (b) (V)jw)[2 cos (wt�) - 2 cos (2wtp)] .

14-36

(a) 25/27r(-2 + 100) . (b) 10 r/s . (c) 1 .25 J.

14-38

(a) IF(jw)l = lOe"I- 1 . (c) Arg F(jw) = -arc tan (0 .01-).14-40

(K/Trt)[sin (wbt) - sin (wat)] .14-42

(a) V,.(jw) = (1 - e-"")/['w(1 + '0 .5w)] . (b) VR(jw) = 0 .5(1 - eJw)/(1 + j0.5w) .(c) IV,.(jw)) = [(2/w) sin (w/2)]/

1 + 0.25-2. IVR(jw)l = sin (w/2)/ N/"-1_+0.25( 2 .14-46

(a) 0.333 . (b) 3 .58 . (c) 9.50.

14-48

(a) H(jw) =

1/(jw +

I) . (b) h(t) = e-'u(t) . y(t) = 0 (t < 0) ; (l

- e-') (0 < t < 1) ;e- ' (e - I) (t > 1) .

14-50

H(jw) = (20 cos w)e -;4- .

Chapter 15

15-2

(a) (5/s)(e4` + e-45 - 2) . (b) (A/S2tp)(e-S'P - 1 + st�). (c) 2aKl(a 2 - s'-) .15-4

(a) (s2 - 16)/(s2 + 16)2 . (b) (s2 + 6s - 7)/(s2 + 6s + 25)2 . (c) se -5 '/(s2 + b2) .(d) s(s2 -

16)/(s 2 +

16)2 . (e) [(A cos D)s - (A sin D)B]l( s2 + 82) .

15-6

11 (s) _ (10/R)[sl(s2 + 100)(s + 1/RC)] . 12(s) = (1/L)[s/(s + 10) (s2 + 1/LC)] .R=20.L=0.2H.

15-8

1,(s) = 200(s + 5)/[s(s2 + 50s + 300)] . 12(s) = 50(s + 20)/[s (s2 + 50s + 300)].


15-10

(a) H(s) = 1/[R,L,C,C2S3 + L,C2S2 + R, (C, + C2)s + 1 ] .(b) H(s) = L,C,s2/[L,(C, + C2)s2 + 1] . (c) H (s) = (s2 +

1)/(s2 + 5.5) .

15-12

(a) f(t) = (1 + 5e - ` - 6e-2`)u(t) . (b) f(t) = 6(t) - 0.111u(t) - 0.444e+ 3 .56e3 ' u(t) . (c) f(t) = (e -' + 2e-2 ' + 3e-3')u(t) .

15.14

(a) f(t) = (t + 3e -' - 2)u(t) . (b) f(t) = (e - ' - te -2 t - e-2t)u(t).(c) f(t) = (-0.5t2 -

1

+ 0.5 e-' + 0.5e')u(t) . (d) 0.5(t cos t + sin t) .

15-16 i(t) = 5e-60'u(t) A.15-18

v,.(t) = [loRl(1 - aRC)](e -°' - e- "RC)u(t) .15-20

vL(t) =

-12.5e-31.51 u (t) V .

15-22

v,(t) = (0.667e-5 ' + 1 .33)u(t) V .15-24 H(s) = Ae-s'o .

15-26

(a) y(t) = [u(t) - u(t - T) + u(t - T/2) - u(t - 3T/2) + u(t - 2T)+ u(t - 3T)] . (b) y(t) _ [u(t) + (t - T)u(t - T) + (t - 2T)u(t - 2T)] .(c) y(t) _ [(T/2-rr)sin (2-rrt1T)][u(t) - 2u(t - T) + u(t - 2T)] .

15-28 Z(s) = (354 + 65 3 + 2752 + 24s)/(s4 + 1352 + 36) .

15-32

(a) H(s) = sl(s + RIL) . (b) h(t) = [8(t) - (RIL)e-R"' u(t)] .(c) v2(t) = [e-RUL u(t) - e(I-RVL) u(t - LIR)].

15-34

R = 10 12 . C = 1 .25 wF . 1, = 0.25 A.v(t) = [-2.49e-Zoo` + 50 cos (4000t - 87 .1°)]V .15-36

(a) L, = 1 .53 H. L2 = 1 .08 H . C, = 1 .58 F. C2 = 0.383 F. (b) h(t) _[-e-() .3831 cos (0.924t - 22.5°) + 2.41e-° .924, cos (0 .383t - 67.5°)] .

-3t u(t)


825

answers to even-numbered problems

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