CHAPTER 3 SOLUTIONS
2/20/10
3-1) a) I0
=
b) I rms
V0
=
V mp
=
R R Vrms V m
=
=
R
2R
170/ p
=
15 170 2(15)
=
3.60 A.
=
5.66 A.
c) P = I 2 R = 5.66 2 (15) = 480 W . d) S
= Vrms rms I rms rms =
e) pf =
P
=
S
170 � � (5.66) = 679 VA. � � �2�
480 W 679 VA
=
0.707 = 70.7%
3-2) a) I 0 Vo
=
Vrms N 1 N 2
= 12
A.; I 0
V m
; Vm p 754
=
=
2 240 533
b) I o � = Io
=
V 0 R
= Vop =
� V0
=
I 0 R = (12)(20) = 240 V.
240 p = 754 V .
= 533 V .
=
0.45
N 2 N 1
=
12 0.45
=
26.7 A.
3-3) a ) pf
=
P S
=
P V s , rms I rms
; I rms
=
V s ,rms R
; VR , rms
V m � � �2 �/ R V /R 2 � � pf = = = V m � V s , rms I rms �Vm � 2 � �2 �/ R � � � � �2� 2 R , rms
=
=
Vm 2
; Vs , rms =
V m 2
1 2
b) Displa Displacem cement ent pf = cos(q1 - f ) = cos(0) = 1 I1
=
V 1 R
=
1 V m R 2
�0; pf
=
cos(q1 - f 1 ) DF ;
\
DF =
1 2
3-4) Using Eq. 3-15, a ) i(wt ) = Z
Vm Z
sin(wt - q ) +
R 2 + (w L) 2
=
=
V m Z
(sin q ) e -w t /wt
12 2 + (377(0.012)) 2
= 12.8 W
�w L � �377(0.012) � q = tan -1 � �= tan -1 � �= 0.361 rad R 12 � � � � wt =
w L R
=
377(0.012) 12
=
0.377
i (wt ) = 13.2 sin(wt - 0.361) + 4.67 e -w t /0.377 : b = 3.50 rad = 20 1 � b) I avg = 4.36 A. ( numerical integration) c ) I rms
=
d ) pf =
6.70 A. ( numerical integration) P = I r2ms R = (6.70) 2 (12) = 538 W. P S
=
538 (120)(6.70)
=
0.67
3-3) a ) pf
=
P S
=
P V s , rms I rms
; I rms
=
V s ,rms R
; VR , rms
V m � � �2 �/ R V /R 2 � � pf = = = V m � V s , rms I rms �Vm � 2 � �2 �/ R � � � � �2� 2 R , rms
=
=
Vm 2
; Vs , rms =
V m 2
1 2
b) Displa Displacem cement ent pf = cos(q1 - f ) = cos(0) = 1 I1
=
V 1 R
=
1 V m R 2
�0; pf
=
cos(q1 - f 1 ) DF ;
\
DF =
1 2
3-4) Using Eq. 3-15, a ) i(wt ) = Z
Vm Z
sin(wt - q ) +
R 2 + (w L) 2
=
=
V m Z
(sin q ) e -w t /wt
12 2 + (377(0.012)) 2
= 12.8 W
�w L � �377(0.012) � q = tan -1 � �= tan -1 � �= 0.361 rad R 12 � � � � wt =
w L R
=
377(0.012) 12
=
0.377
i (wt ) = 13.2 sin(wt - 0.361) + 4.67 e -w t /0.377 : b = 3.50 rad = 20 1 � b) I avg = 4.36 A. ( numerical integration) c ) I rms
=
d ) pf =
6.70 A. ( numerical integration) P = I r2ms R = (6.70) 2 (12) = 538 W. P S
=
538 (120)(6.70)
=
0.67
3-5) Using Eq. 3-15, a ) i(wt ) = Z
Vm Z
sin(wt - q ) +
R 2 + (w L) 2
=
=
V m Z
(sin q ) e -w t /wt
10 2 + (377(0.015)) 2
= 11.5 W
w L � � �377(0.015) � q = tan -1 � �= tan -1 � �= 0.515 rad � R � � 10 � wt =
w L R
=
377(0.015) 10
=
0.565
i (wt ) = 14.8 sin(wt - 0.515) + 7.27 e -w t /0.565 : b = 3.657 rad =20 9.5 � b) I avg = 5.05 A. ( numerical integration) c ) I rms
=
d ) pf =
7.65 A. ( numerical integration) P = Ir2ms R = (7.65) 2 (10) P S
=
584 (120)(7.65)
=
= 584
W.
0.637 = 63.7%
3-6) Using Eq. 3-15,
a ) i(wt ) = Z
Vm Z
sin(wt - q ) +
R 2 + (w L) 2
=
=
V m Z
(sin q ) e -w t /wt
152 + (377(0.08)) 2
= 33.7 W
w L � � �377(0.08) � q = tan -1 � �= tan -1 � �= 1.11 rad � R � � 15 � wt =
w L R
=
377(0.08) 15
=
2.01
i (wt ) = 10.1sin(wt - 1.11) + 9.02 e -w t /2.01 : b = 4.35 rad = 25 0 � b) I avg = 4.87 A. ( numerical integration) c ) I rms
=
d ) pf =
6.84 A. ( numerical integration) P = Ir2ms R = (6.84) 2 (15) = 701 W. P S
=
701 (240)(6.84)
=
0.427 = 42.7%
3-7) Using an i!a" io! #o!", $ 48 & 'o an a!ag! c*!nt o' 2 +.
8.0A
Current Iavg = 2 A fr R = 48 !ms 4.0A (1".#00m$2.00%0)
Average Current
0A 0s
5ms I(R1)
10ms AVG(I(L1)) Time
15ms
20ms
3-8) Using Eqs. 3-22 an 3-23, a ) i(wt ) =
Vm Z
sin(wt - q ) -
V dc R
+
Ae -wt /wt
V �a /wt � Vm sin(a - q ) + dc � e R � � Z
A = � Z
R 2 + (w L)2
=
=
10 2 + (377(.075) 2
= 30.0 W
�w L � �377(.075) � q = tan -1 � �= tan -1 � �= 1.23 rad R 10 � � � � wt =
w L R
a = sin -1
=
377(0.075)
V dc V m
=
10 100 240 2
=
2.83
= 0.299
rad = 17. 1�
i (wt ) = 11.3sin(wt - 1.23) -10 + 21.2 e -w t /2.83; b = 3.94 rad I avg
= 3.13
b) I rms
=
c ) pf =
A. (numerical integration) Pdc
= Vdc I avg = (100)(3.13) = 313
4.81 A. ( numerical integration) PR
P S
=
313 + 231 (240)(4.81)
=
0.472 = 47.2%
� = 226
=
W .
2 I rms R = (4.81) 2 (10) = 231 W.
3-9) Using Eqs. 3-22 an 3-23, a ) i(wt ) =
Vm
sin(wt - q ) -
Z
V dc R
+
Ae -wt /wt
V �a /wt � Vm sin(a - q ) + dc � e R � � Z
A = �
Z
R 2 + (w L) 2
=
=
12 2 + (377(0.12) 2
= 46.8 W
w L � � �377(0.12) � q = tan -1 � �= tan -1 � �= 1.31 rad � R � � 12 � wt =
w L R
a = sin -1
=
377(0.12)
V dc
12 48
=
V m
120 2
=
3.77
= 0.287
rad = 16.4 �
i (wt ) = 3.63sin(wt - 1.31) - 4.0 + 7.66 e -w t /3.77; b = 4.06 rad I avg
= 1.124
b) I rms
A. ( numerical integration) Pdc
= 1.70
c ) pf =
P S
=
= Vdc Iavg = (48)(1.124) = 54.0
A. ( numerical integration) PR
54.0 + 34.5 (120)(1.70)
=
� = 233
=
2 I rms R = (1.70) 2 (12) = 34.5 W.
0.435 = 43.5%
3-10) Using Eq. 3-33, i (wt ) =
Vm w L
(cos a
- cos wt ) +
V dc w L
(a - wt)
� V � � 48 � a = sin -1 � dc �= sin -1 � �= 0.287 rad . V 120 2 � � �m � i (wt ) = 4.68 - 4.50 cos(wt) -1.23wt A.; b I o
=
1 2p
b
�i(wt )d (w t ) = 2.00 A.; P a
dc
=
W .
= 4.483
I oVdc
=
rad
� = 257
2.00(48) = 96 W .
3-11)
%00&
200& L = 0.25
100&
0& 0s
5ms AVG(&(V'))
10ms
Time
15ms
20ms
3-12) 0.14 'o 50 (51 ).
100&
(1"."#0m$51.15") 50&
L = 0.14
0& 0s
5ms AVG(&(V'))
10ms
Time
15ms
20ms
3-13) Using Eq. 3-34, a) V0
=
Vm p
=
120 2 p
=
54.0 V .; I 0
=
V 0 R
=
54 12
=
4.50 A.
b)
n n n n 0 54.02 12.00 4.50 1 84.85
25.6
3.31
2 36.01
46.8
0.77
4
91.3
0.08
7.20
! t!#s !on n 1 a! insigni'icant.
3-14)
$*n a tansi!nt !sons! "ong !no*g to aci!! st!a-stat! !s*"ts (!.g., 1000#s). ! !a-to !a "oa c*!nt is aoi#at!" 1.48 +, so#!at "ag! tan t! 1.35 + otain! *sing on" t! 'ist a#onic. (! in*ctanc! so*" ! s"igt" "ag!, ao*t 0.7 , to co#!nsat! 'o t! aoi#ation o' t! ca"c*"ation.)
3-15) a) I 0
=
I1
=
V m p R V 1 Z 1
=
=
50
= 3.98 A. 4p V m / 2
R
R 2 + (w L) 2 L =
125 2p 60
=
2
+ (w L)
=
2
25
=
9 + (w L) 2
R =
2
+ (w L)
25 0.199
2
=
0.05 I 0
= 125 W
=
0.199 A.
�w L
0.33 !
) + <ic! si#*"ation *sing an i!a" io! #o!" gi!s 0.443 + - in t! st!a stat!. is co#a!s it 2(1)2(0.199)0.398 + -.
3-16) a ) V0 I 0
=
V m
p V0 - V dc
=
170
=
= 54.1 V p 54.1 - 24
=
R 10 Dio � 1 A. �2 I1 � I1 V m
V1
=
Z1
=
2 V 1 I 1
=
=
170 2 85 0.5
=
3.01 A.
= 0.5
A.
= 85 V
R 2 + (w L) 2 �w L
= 170 W =
170
= 450 m! . 377 b) Pdc = IavgVdc = (3.01)(24) = 72.2 W .
L =
c ) P R P R
=
=
2 I rms R; I rms
�I
=
2 n , rms
� (3.01) 2 + (0.5 / 2) 2
=
3.12 A.
(3.12)2 (10) = 97.4 W .
3-17) a) = $> 10310-31 s; =/ 60. it = ?? , t! !on!ntia" !ca is ! s#a"" an t! o*t*t o"tag! as "itt"! aiation. ) Eact !q*ations: q
= - tan
-1
(w R" ) + p
= - tan
-1
(377) + p
= 1.5573
rad = 90.15
V m sin q = 200sin(90.15 � ) = 199.9993 sin a
- sin q e
- (2p +a +q )/ w R"
=
0 � a = 1.391 rad
= 79.72 �
DVo = Vm (1 - sin a ) = 3.21 V .
c) +oi#ation o' Eq. 3-51: DVo
�
V m
fR"
=
200 (60)(103 )(10 -3 )
= 3.33 V .
3-18) a) $ 100 &: = $> (100)10-3 0.1 s; =/ 6. = - tan
q
-1
(w R" ) + p
= - tan
-1
(37.7) + p
= 1.5973
rad = 91.52
V m sin q = 200sin(91.52 � ) = 199.93 sin a
- sin q e
) - ( 2p +a + q )/w R"
DVo = Vm (1 - sin a ) = DVo
�
V m
fR"
=
=0
� � a = 1.0338 rad = 59.23
28.16 V . ( e#act)
200 (60)(100)(10 -3 )
= 33.3 V .
( appro#imation)
) $ 10 &: = $> (10)10-3 0.01 s; =/ .6. q
= - tan
-1
(w R" ) + p
= - tan
-1
(3.77) + p
= 1.830
rad = 104.9
V m sinq = 200sin(104.9� ) = 193.3 ) sin a - sin q e - (2p +a + q )/w R"
=0
DVo = Vm (1 - sin a ) = 143.2 DVo
�
V m
fR"
=
� � a = 0.2883 rad = 16.5
V . ( e#act)
200 (60)(10)(10 -3 )
= 333 V .
( appro#imation)
n (a) it =/6, t! aoi#ation is #*c #o! !asona"! tan () !! =/0.6.
3-19) a) it > 4000 @A, $> 4 s., an t! aoi#ation o' Eq. 3-51 so*" ! !asona"!. DVo
�
V m
fR"
=
120 2 (60)(4)
=
0.707 V .
) it > 20 @A, $> 0.02, ic is on t! o! o' on! so*c! !io. !!'o!, t! aoi#ation i"" not ! !asona"! an !act !q*ations #*st ! *s!. q
= - tan
-1
(w R" ) + p
a = 0.5324 rad
=
= - tan
-1
((377)(1000)(20(10) -6) + p =1.703 rad = 97.6 )
30.5 �( numericall$ from %&. 3 - 43)
DVo = Vm - Vm sin a = 83.6
V.
3-20) a) it > 4000 @A, $> 2 s., an t! aoi#ation o' Eq. 3-51 so*" ! !asona"!. DVo
�
V m
fR"
=
120 2 (60)(2.0)
= 1.41 V .
) it > 20 @A, $> 0.01, ic is on t! o! o' on! so*c! !io. !!'o!, t! aoi#ation i"" not ! !asona"! an !act !q*ations #*st ! *s!. q
= - tan
-1
(w R" ) + p
= - tan
-1
((377)(500)(20(10) -6) + p = 1.83 rad =104.9 )
( numericall$ = 16.5 �
a = 0.2883 rad
DVo = Vm - Vm sin a = 121 V .
from %&. 3 - 43)
3-21) Ao# Eq. 3-51 "
=
V m fRDV o
=
�
60(750)(2)
= 1,886
m F
� 2 � 1� 1�= sin � �= 1.417 rad = 81.2 V 120 2 � � m � � sin a � � w" cos a + = Vm � �= 18.7 A. R � �
a �sin -1 � 1 I D , pea'
120 2
V I D , avg � m R
=
DV o
-
0.226 A.
3-22) +ss*#ing o is constant an !q*a" to #, Vo2
Vm2
V m2
P � � � R = R R P
=
(120 2) 2 50
=
576 W
Ao# Eq. 3-51 "
=
V m fRDV o
=
�
60(576)(1.5)
=
3,270 m F
� 1.5 � 1� 1�= sin � �= 1.438 rad = 82.4 V � 120 2 � m � � sin a � � w" cos a + = Vm � �= 28.1 A. R � �
a �sin -1 � 1 I D, pea'
120 2
I D, avg =
V m R
=
DV o
0.295 A.
-
3-23) Using t! !'inition o' o! 'acto an #s 'o# Eq. 3-53, pf =
P S
V m
1-
2
=
=
2 Vrms /R
(V s , rms )( I s , rms ) a
2 Vrms /R
=
(Vs, rms )(Vrms / R)
=
Vrms Vs, rms
sin2a
+
p 2p V m / 2
1
=
1-
2
a p
sin 2a
+
1
=
2p
2
-
a 2p
+
sin 2a 4p
3-24) a ) Vo
=
b) P = =
V m 2p
2 Vrms
2 80.9 2 100
c) S
; V rms
R
120 2
P =
(1 + cos a ) =
1=
=
V m
0.785 p
2 +
120 2 2p 1-
a p
(1 + cos 45 � ) = 46.1 V . sin 2a
+
2p
sin(2(0.785)) 2p
= 80.9
V .
65.5 W .
= V s , rms I rms =
�80.9 � (120) � �= 97.1 VA; pf 100 � �
P
=
S
=
65.5 97.1
=
0.674 = 67.4%
3-25) a ) vo
=
I o R = (2.5)(30) = 75 V
�2p V a = cos -1 � o �V m b) P = Vo ,rms P
=
=
2p
(1 + cos a )
�
�2p (75) � cos -1 � or 1.143 rad - 1� = 65.5 � 240 2 � � �
- 1� =
V o2,rms R V m 2
147.62
c) S
V m
=
30
1=
a p
+
sin 2a 2p
=
240 2 2
1-
1.143 p
+
sin(2(1.143)) 2p
= 147.6
V .
726 W .
= = V s , rms I rms
147.6 � P 726 � (240) � 1181 VA ; pf = = = = 0.615 = 61.5% � S 1181 � 30 �
3-26) a ) i(wt ) = 5.42sin(w t - 0.646) +1.33 e -w t /0.754 A. a
=
25�= 0.524 rad , b = 3.79 rad
b) I o
=
c ) I rms
1
= 217 � ( numericall$)
b
i (wt ) d (w t ) = 1.80 A. �
2p a =
1
b
i (wt )d (w t ) = 2.80 A.; 2p � 2
Po
=
PR
=
2 I rms R = (2.80) 2 25 = 193 W.
a
3-27) a ) i(wt ) = 3.46sin(w t - 0.615) - 6.38 e -w t /0.707 A. a
=
60�= 1.047 rad , b = 3.748 rad
b) I o
=
c) I rms
1
=
215 �( numericall$)
b
i (wt ) d (w t ) = 0.893 A. �
2p a =
1
b
i (wt ) d (w t ) = 1.50 A.; 2p � 2
Po
=
PR
=
2 I rms R = (1.50) 2 40 = 90.3 W.
a
3-28) B 46C. Do a aa#!tic s!! 'o a"a. Us! t! !'a*"t (D!a) io!, an *s! $on 0.01 'o t! sitc. +"a o' 46 !g!!s !s*"ts in aoi#at!" 2 + in t! "oa.
3-29) B 60.5C. Do a aa#!tic s!! 'o a"a. Us! t! !'a*"t (D!a) io!, an *s! $on 0.01 'o t! sitc. +"a o' 60.5 !g!!s !s*"ts in aoi#at!" 1.8 + in t! "oa.
3-30) Ao# Eq. 3-61, a ) i (wt ) = 4.29sin(wt - 1.263) - 4.0 + 7.43 e -w t /3.142 A., 0.873 �w t �3.95 rad I o
=
b
1
i (wt )d (w t ) = 1.04 A., �
2p a
b) I rms
1
Pdc
=
I oVdc
b
i (wt )d (w t ) = 1.67 A.; 2p �
=
(1.04)(48) = 50.1 W .
=
2
PR
=
2 I rms R = (1.67) 212 = 33.5 W .
a
c) pf =
P S
=
50.1 + 33.5 (120)(1. 67)
=
0.417 = 41.7%
3-31) Ao# Eq. 3-61, a ) i(wt ) = 2.95sin(wt - 0.515) - 0.96 + 3.44e I o
=
w t /0.565
A., 1.047 �w t �3.32 rad
b
1
i (wt )d (w t ) �
2p a
b) I rms
-
1
=
0.454 A., Pdc
I oVdc
2
=
0.830 A.; PR
a
c) pf =
P S
=
=
(0.454)(96) = 43.6 W .
b
i (wt )d (w t ) 2p �
=
=
43.6 + 69.0 (240)(0.830)
=
0.565 = 56.5%
=
2 I rms R = (0.830)2 100 = 69.0 W .
3-32) B 75C. +"a 75 !g!!s gi!s 35 in t! c o"tag! so*c!. +n $on 0.01 'o t! sitc an n 0.001 'o t! io! (i!a" #o!").
3-33) Ao# Eq. 3-61, a ) i(wt) I o
=
1
= 5.99sin(wt -1.50) - 24.0 + 29.3 e
A., 0.873 �w t �4.24 rad
b
i (wt )d (w t ) = 1.91 A., �
2p a
b) I rms
-w t /14.1
1
Pdc
=
I oVdc
(1.91)(48) = 91.6 W .
=
2 I rms R = (2.93) 2 2 = 17.1 W.
b
i (wt ) d (w t ) = 2.93 A.; 2p �
=
=
2
PR
a
3-34) B 81C 3-35) L
di(t )
= Vm
dt di (t ) 1
sin w t - V dc
= EVm sin w t - Vdc F or dt L di (w t ) 1 = EVm sin w t - V dc F d (w t ) w L
i (wt ) = Vm
1 w L
w t
(V �
m
sin wt - Vdc ) d (w t )
a
I o
=
(cos a - cos wt ) +
V dc
(a - wt ) w L w L i (wt) = 4.34 - 7.58cos w t - 1.82wt A., 1.309 �wt �4.249 =
1
b
i (wt )d (w t ) = 1.91 A. �
2p a
3-36) 0 s !n <1 on, 00 !n D2 on I o
=
Vo
, Vo
R
\ I o =
V m 2p R
=
1 2p
p
�
Vm sin(wt ) d (wt ) =
a
V m 2p
(1 + cos a )
(1 + cos a )
3-37)
� I L ( s � ; ( s = w Ls = 377(1.5)(10) 3 = 0.566 � � V m � � 5(0.452) � u = cos 1 � 1�= 10.47� � 120 2 � V � ( L ( s � 120 2 � 5(.566) � Vo = m � 11� �= 53.57 V . �= p � 2V m � p � 2 2(120) � u = cos -1 � 1-
-
-
(compared to
V m p
= 54.0
V .)
<ic!: Us! a c*!nt so*c! 'o t! constant "oa c*!nt:
W
D1 to D2
D2 to D1
3-38) * 20C. $*n t! si#*"ation "ong !no*g 'o st!a-stat! !s*"ts. Ao# t! o! o*t*t, t! co##*tation ang"! 'o# D1 to D2 is ao*t 20 !g!!s, an 'o# D2 to D1 is ao*t 18 !g!!s. Got! tat t! ti#! ais is cang! to ang"! in !g!!s !!.
3-39) $*n t! si#*"ation "ong !no*g 'o st!a-stat! !s*"ts. Ao# t! o! o*t*t, t! co##*tation ang"! 'o# D1 to D2 is ao*t 16.5 !g!!s, an 'o# D2 to D1 is ao*t 14.7 !g!!s. Got! tat t! ti#! ais is cang! to ang"! in !g!!s !!.
3-40) +t Ht I, D2 t*ns on, D1 is on !ca*s! o' t! c*!nt in < (s!! Aig. 3-17). VL; v LS di D1 d ( wt )
=
at wt = p
= Vm sin w t =
V m w L s + u,
Ls
di D1
did 1 d (w t )
w t
sin(wt ) d (wt) + i �
D1
(p )
p
i D1
=
0=
V m w L s
cos(p + u) = - cos u � 0 = \u =
dt
= w LS
� I L ( s � � � V m �
cos -1 � 1-
E- 1 - cos(p + u)F + I L Vm w L s
(-1 + cos u) + I L = -
V m w Ls
cos u + I L
