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658 ⏐⏐⏐ SERIES AND PARALLEL ac CIRCUITS
Y L – Y C
j
Y C
Y L
0.2136 S
20.56°
Y R
Y T
FIG. 15.59
Admittance diagram for the network of
Fig. 15.58.
+
1
1
YL � BL ��90° � � ��90° � � ��90°
XL
8 �
� 0.125 S ��90° � 0 � j 0.125 S
1
1
YC � BC �90° � � �90° � � �90°
XC 20 �
� 0.050 S ��90° � 0 � j 0.050 S
b. YT � YR � YL � YC � (0.2 S � j 0) � (0 � j 0.125 S) � (0 � j 0.050 S)
� 0.2 S � j 0.075 S � 0.2136 S ��20.56°
1
c. ZT ���� � 4.68 � �20.56°
0.2136 S ��20.56°
or
ZRZLZC ZT ����
ZRZL � ZLZC � ZRZC (5 � �0°)(8 � �90°)(20 � ��90°)
�������
(5 � �0°)(8 � �90°) � (8 � �90°)(20 � ��90°)
� (5 � �0°)(20 � ��90°)
800 � �0°
�����
40 �90° � 160 �0° � 100 ��90°
800 �
800 �
��� ���
160 � j 40 � j 100 160 � j 60
800 �
���
170.88 ��20.56°
� 4.68 � �20.56°
d. The admittance diagram appears in Fig. 15.59.
On many occasions, the inverse relationship YT � 1/ZT or ZT �
1/YT will require that we divide the number 1 by a complex number
having a real and an imaginary part. This division, if not performed in
the polar form, requires that we multiply the numerator and denominator
by the conjugate of the denominator, as follows:
1 1
1 (4 ��j 6 �) 4 � j 6
YT����� � �� ��
� �� ���22 ZT 4 ��j 6 � 4 ��j 6 � (4 ��j 6 �) 4 � 6
4 6
and YT � � S � j � S
52 52
To avoid this laborious task each time we want to find the reciprocal
of a complex number in rectangular form, a format can be developed
using the following complex number, which is symbolic of any impedance
or admittance in the first or fourth quadrant:
1
�
a 1 � jb 1
1
�
a1 � jb1 a1 � j b1 �
a1 � j b1 � � �� � �
a 1 � j b 1
�a 2 1 � b 2 1
1 a1
or �� � �2 (15.27)
a1 � jb1 a1 � b 2 b1
� � j �2 1 a1 � b 2 �
1
Note that the denominator is simply the sum of the squares of each
term. The sign is inverted between the real and imaginary parts. A few
examples will develop some familiarity with the use of this equation.
a c