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662 ⏐⏐⏐ SERIES AND PARALLEL ac CIRCUITS
or, finally,
PT � PR � PL � EIR cos vR � EIL cos vL � (20 V)(6 A) cos 0° � (20 V)(8 A) cos 90° � 120 W � 0
� 120 W
Power factor: The power factor of the circuit is
Fp � cos vT � cos 53.13° � 0.6 lagging
or, through an analysis similar to that employed for a series ac circuit,
E EG G G
cos vT � � ������ I I/V YT
2 P /R
� �
EI EI
G
and Fp � cos vT � �� (15.32)
Y
where G and YT are the magnitudes of the total conductance and admittance
of the parallel network. For this case,
0.3 S
Fp � cos vT ���0.6 lagging
0.5 S
Impedance approach: The current I can also be found by first finding
the total impedance of the network:
Z
ZT �
RZL (3.33 � �0°)(2.5 � �90°)
� ����
ZR � ZL 3.33 � �0° � 2.5 � �90°
8.325 �90°
��� �2 � �53.13°
4.164 �36.87°
And then, using Ohm’s law, we obtain
E 20 V �53.13°
I ����� �10 A �0°
ZT 2 � �53.13°
R-C
Refer to Fig. 15.66.
i = 14.14 sin qt
+
e
–
iR
T
R 1.67 �
FIG. 15.66
Parallel R-C network.
Phasor Notation As shown in Fig. 15.67.
Y T and Z T
a
X C
i C
1.25 �
1
1
YT � YR � YC � G �0° � BC �90° �� �0° �� �90°
1.67 �
1.25 �
� 0.6 S �0° � 0.8 S �90° � 0.6 S � j 0.8 S � 1.0 S �53.13°
1 1
ZT ����� �1 � ��53.13°
YT 1.0 S�53.13°
a c