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772 ⏐⏐⏐ METHODS OF ANALYSIS AND SELECTED TOPICS (ac)<br />

Z T<br />

2<br />

4 � 4 �<br />

3 � 4 �<br />

1 3<br />

2 � 3 �<br />

In the study of dc networks, we found that if all of the resistors of<br />

the � or Y were the same, the conversion from one to the other could<br />

be accomplished using the equation<br />

For ac networks,<br />

R � � 3R Y or R Y �<br />

Z� � 3ZY or ZY � � Z�<br />

�<br />

3<br />

(17.19)<br />

Be careful when using this simplified form. It is not sufficient for all the<br />

impedances of the � or Y to be of the same magnitude: The angle associated<br />

with each must also be the same.<br />

EXAMPLE 17.20 Find the total impedance Z T of the network of Fig.<br />

17.48.<br />

Z B<br />

2<br />

Z A<br />

Z C<br />

1 3<br />

FIG. 17.48<br />

Converting the upper � of a bridge configuration to a Y.<br />

R �<br />

� 3<br />

2<br />

Z 3<br />

N A<br />

Z1 Z2 1 3<br />

Solution:<br />

ZB � �j 4 ZA � �j 4 ZC � 3 � j 4<br />

Z<br />

Z1 �<br />

BZC (�j 4 �)(3 ��j 4 �)<br />

�� � ����<br />

ZA � ZB � ZC (�j 4 �) � (�j 4 �) � (3 ��j 4 �)<br />

(4 ��90°)(5 �53.13°) 20 ��36.87°<br />

� ��� � ��<br />

3 � j 4<br />

5 ��53.13°<br />

� 4 � �16.13° � 3.84 ��j 1.11 �<br />

Z (�j 4 �)(3 ��j 4 �)<br />

Z2 �<br />

AZC �� � ���<br />

ZA � ZB � ZC 5 � ��53.13°<br />

� 4 � �16.13° � 3.84 ��j 1.11 �<br />

Recall from the study of dc circuits that if two branches of the Y or �<br />

were the same, the corresponding � or Y, respectively, would also have

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