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Chapter 9, Problem 69.

Find the equivalent admittance Y eq of the circuit in Fig. 9.76.

Figure 9.76

For Prob. 9.69.

Chapter 9, Solution 69.

1

 

 

 

 

1

 

 

 

 

 

1

1

 

 

 

 

 

 

 

 

 

=

 

 

 

 

 

+

 

 

=

 

 

(1+ j2)

 

 

 

 

Y

 

 

4

- j2

4

 

 

 

 

o

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Y =

 

4

 

 

=

 

(4)(1j2)

= 0.8 j1.6

 

 

1+ j2

5

 

 

 

o

 

 

 

 

 

 

 

 

 

 

Yo + j = 0.8 j0.6

 

 

 

 

1

 

 

 

 

1

 

 

 

1

1

 

 

 

 

 

 

 

 

=

 

 

 

 

+

 

 

 

+

 

 

 

= (1) +( j0.333) +(0.8 + j0.6)

Yo

1

- j3

 

0.8 j0.6

1

 

 

=1.8 + j0.933 = 2.028 27.41°

 

 

Y

 

 

 

o

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Y

 

= 0.4932 - 27.41°= 0.4378 j0.2271

o

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Y

 

+ j5 = 0.4378 + j4.773

 

 

 

 

o

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

=

1

+

 

 

 

1

 

 

 

= 0.5 +

0.4378 j4.773

Y

 

 

 

2

 

0.4378 + j4.773

 

22.97

 

eq

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

= 0.5191j0.2078

 

 

 

 

Y

 

 

 

 

 

eq

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Y

 

=

0.5191j0.2078

= 1.661 + j0.6647 S

 

 

eq

 

 

 

 

 

 

 

 

 

 

 

 

0.3126

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PROPRIETARY MATERIAL. © 2007 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission.

Chapter 9, Problem 70.

Find the equivalent impedance of the circuit in Fig. 9.77.

Figure 9.77

For Prob. 9.70.

PROPRIETARY MATERIAL. © 2007 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission.

Chapter 9, Solution 70.

Make a delta-to-wye transformation as shown in the figure below. a

Zan

Zbn

Zeq

n Zcn

b

c

8

2

-j5

Zan =

(-j10)(10 + j15)

=

(10)(15 j10)

= 7 j9

 

5 j10 +10 + j15

15 + j5

 

Zbn =

(5)(10 + j15)

= 4.5 + j3.5

 

 

 

 

 

 

 

15 + j5

 

 

 

Zcn =

(5)(-j10)

= -1j3

 

 

 

 

 

 

 

 

 

15 + j5

 

 

 

Zeq = Zan +(Zbn + 2) || (Zcn

+8 j5)

 

Zeq = 7

j9

+(6.5 + j3.5) || (7 j8)

 

Zeq = 7

j9

+

(6.5 + j3.5)(7 j8)

 

 

 

 

 

 

 

13.5 j4.5

 

Zeq = 7

j9

+5.511j0.2

 

 

 

Zeq =12.51j9.2 = 15.53 -36.33°

PROPRIETARY MATERIAL. © 2007 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission.

Chapter 9, Problem 71.

Obtain the equivalent impedance of the circuit in Fig. 9.78.

Figure 9.78

For Prob. 9.71.

PROPRIETARY MATERIAL. © 2007 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission.

Chapter 9, Solution 71.

We apply a wye-to-delta transformation. j4

Zab

a b

Zac Zbc

Zeq

1

-j2

c

Zab =

2 j2 + j4

=

2 + j2

=1j

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

j2

 

 

 

j2

 

Zac =

 

2 + j2

=1+ j

 

 

 

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Zbc =

2 + j2

 

= -2 + j2

 

 

 

 

 

 

 

 

- j

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

j4 || Zab = j4 || (1j) =

 

 

( j4)(1j)

=1.6 j0.8

 

 

1+ j3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1|| Zac

 

=1|| (1+ j) =

(1)(1+ j)

= 0.6 + j0.2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 + j

 

 

j4 || Zab +1|| Zac = 2.2 j0.6

 

 

1

=

 

 

1

+

 

1

 

+

 

1

 

 

 

 

 

 

 

- 2 + j2

 

2.2 j0.6

 

 

Zeq

 

- j2

 

 

 

=j0.5 0.25 j0.25 +0.4231+ j0.1154

=0.173 + j0.3654 = 0.4043 64.66°

Zeq = 2.473 -64.66°= 1.058 – j2.235

PROPRIETARY MATERIAL. © 2007 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission.

Chapter 9, Problem 72.

Calculate the value of Z ab in the network of Fig. 9.79.

Figure 9.79

For Prob. 9.72.

PROPRIETARY MATERIAL. © 2007 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission.

Chapter 9, Solution 72.

Transform the delta connections to wye connections as shown below.

 

 

a

 

 

 

 

 

 

 

 

 

j2

j2

 

 

 

 

 

 

 

-j9

-j18

 

 

 

 

j2

 

 

 

 

 

 

 

 

 

 

 

 

 

R1

 

R2

 

 

 

 

 

R3

 

 

 

 

b

 

 

 

 

 

- j9 || -j18 = -j6 ,

 

 

 

 

 

R1

=

(20)(20)

 

= 8 ,

R 2 =

(20)(10)

= 4 ,

 

 

20 + 20 +10

 

 

50

 

 

 

R 3 = (20)(10) = 4

 

 

 

 

 

 

50

 

 

 

Zab

= j2 +( j2 +8) || (j2 j6 + 4) + 4

 

 

 

Zab

= 4 + j2 +(8 + j2) || (4 j4)

 

 

 

Zab

= 4 + j2 + (8 + j2)(4 j4)

 

 

 

 

 

 

 

12 - j2

 

 

 

Zab

= 4 + j2 +3.567 j1.4054

 

 

 

Zab

= 7.567 + j0.5946

 

 

 

PROPRIETARY MATERIAL. © 2007 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission.

Chapter 9, Problem 73.

Determine the equivalent impedance of the circuit in Fig. 9.80.

Figure 9.80

For Prob. 9.73.

Chapter 9, Solution 73.

Transform the delta connection to a wye connection as in Fig. (a) and then transform the wye connection to a delta connection as in Fig. (b).

a

j2

j2

 

 

-j9

-j18

j2

 

 

 

R1

 

R2

 

R3

 

b

PROPRIETARY MATERIAL. © 2007 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission.

Z1 =

 

 

( j8)(-j6)

=

48

= -j4.8

 

 

 

j8 + j8 j6

 

 

 

 

 

 

 

 

 

j10

 

Z2 = Z1 = -j4.8

 

 

 

 

 

 

 

 

Z3

=

 

 

( j8)( j8)

=

- 64

= j6.4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

j10

j10

 

(2 + Z1 )(4 + Z2 ) +(4 + Z2 )(Z3 ) +(2 + Z1 )(Z3 ) =

 

 

 

 

 

 

 

 

 

(2 j4.8)(4 j4.8) +(4 j4.8)( j6.4) +(2 j4.8)( j6.4) = 46.4 + j9.6

Za

=

 

 

46.4 + j9.6

 

=1.5 j7.25

 

 

 

 

j6.4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Zb

=

 

 

46.4 + j9.6

 

= 3.574 + j6.688

 

 

 

4 j4.8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Zc

=

 

 

46.4 + j9.6

 

=1.727 + j8.945

 

 

 

2 j4.8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

j6 || Zb =

(6 90°)(7.583 61.88°)

 

= 07407 + j3.3716

 

 

 

 

 

 

 

 

 

 

 

 

3.574 + j12.688

 

- j4 ||

Za

=

 

(-j4)(1.5 j7.25)

= 0.186 j2.602

 

 

 

 

 

 

 

 

 

 

 

1.5 j11.25

 

j12 ||

Zc

=

(12 90°)(9.11 79.07°)

= 0.5634 + j5.1693

 

 

1.727 + j20.945

 

 

 

 

 

 

 

 

 

 

Zeq

= ( j6 || Zb ) || (-j4 || Za + j12 || Zc )

Zeq

= (0.7407 + j3.3716) || (0.7494 + j2.5673)

Zeq = 1.508 75.42° Ω = 0.3796 + j1.46

PROPRIETARY MATERIAL. © 2007 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission.

Chapter 9, Problem 74.

Design an RL circuit to provide a 90 o leading phase shift.

Chapter 9, Solution 74.

One such RL circuit is shown below.

 

20 V

20

+

 

+

Vi = 1 0°

j20

j20 Vo

 

 

Z

We now want to show that this circuit will produce a 90° phase shift.

Z = j20 || (20 + j20) =

 

( j20)(20 + j20)

=

- 20 + j20

= 4(1

+ j3)

 

 

 

20 + j40

 

 

 

1+ j2

V =

 

Z

 

V =

4 + j12

(1 0°) =

1+ j3

=

1

(1+ j)

 

Z + 20

 

 

 

6 + j3

 

 

 

i

 

24 + j12

 

3

 

 

 

 

 

j20

 

 

 

j 1

 

 

 

j

 

 

 

 

 

 

 

V =

 

 

 

V

=

 

 

 

 

 

(1+ j)

=

 

 

= 0.3333 90°

 

 

 

 

 

 

 

 

 

 

 

o

 

20 + j20

 

1+ j 3

 

 

 

3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This shows that the output leads the input by 90°.

PROPRIETARY MATERIAL. © 2007 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission.

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