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666 ⏐⏐⏐ SERIES AND PARALLEL ac CIRCUITS
I C
j
I R
I L – I C
53.13°
36.87°
FIG. 15.74
Phasor diagram for the parallel R-L-C
network of Fig. 15.71.
– �
i C
E
–
�
2
90°
I
I L
+
90°
53.13°
0
or I � I R � I L � I C
Phasor diagram: The phasor diagram of Fig. 15.74 indicates that
the impressed voltage E is in phase with the current IR through the
resistor, leads the current IL through the inductor by 90°, and lags the
current IC of the capacitor by 90°.
Time domain:
i � �2�(50) sin qt � 70.70 sin qt
iR � �2�(30) sin(qt � 53.13°) � 42.42 sin(qt � 53.13°)
iL � �2�(70) sin(qt � 36.87°) � 98.98 sin(qt � 36.87°)
iC � �2�(30) sin(qt � 143.13°) � 42.42 sin(qt � 143.13°)
A plot of all of the currents and the impressed voltage appears in
Fig. 15.75.
e
i R
�
2
36.87°
i
i L
�
3
�
2
FIG. 15.75
Waveforms for the parallel R-L-C network of Fig. 15.71.
Power: The total power in watts delivered to the circuit is
PT � EI cos v � (100 V)(50 A) cos 53.13° � (5000)(0.6)
� 3000 W
or PT � E 2 G � (100 V) 2 (0.3 S) � 3000 W
or, finally,
PT � PR � PL � PC � EIR cos vR � EIL cos vL � ELC cos vC � (100 V)(30 A) cos 0° � (100 V)(70 A) cos 90°
� (100 V)(30 A) cos 90°
� 3000 W � 0 � 0
� 3000 W
Power factor: The power factor of the circuit is
Fp � cos vT � cos 53.13° � 0.6 lagging
2�
�t
a c