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a c
or E � V R � V L � V C
which can also be verified through vector algebra.
Phasor diagram: The phasor diagram of Fig. 15.38 indicates that
the current I is in phase with the voltage across the resistor, lags the
voltage across the inductor by 90°, and leads the voltage across the
capacitor by 90°.
Time domain:
i � �2�(10) sin(qt � 53.13°) � 14.14 sin(qt � 53.13°)
vR � �2�(30) sin(qt �53.13°) � 42.42 sin(qt � 53.13°)
vL � �2�(70) sin(qt � 36.87°) � 98.98 sin(qt � 36.87°)
vC � �2�(30) sin(qt � 143.13°) � 42.42 sin(qt � 143.13°)
A plot of all the voltages and the current of the circuit appears in Fig.
15.39.
–
�
2
98.98 V
70.70 V
42.42 V
36.87°
0
53.13°
90°
v L
�
FIG. 15.39
Waveforms for the series R-L circuit of Fig. 15.35.
Power: The total power in watts delivered to the circuit is
PT � EI cos vT � (50 V)(10 A) cos 53.13° � (500)(0.6) � 300 W
or PT � I 2 R � (10 A) 2 (3 �) � (100)(3) � 300 W
or
PT � PR � PL � PC � VRI cos vR � VLI cos vL � VC I cos vC � (30 V)(10 A) cos 0° � (70 V)(10 A) cos 90° � (30 V)(10 A) cos 90°
� (30 V)(10 A) � 0 � 0 � 300 W
Power factor: The power factor of the circuit is
Fp � cos vT � cos 53.13° � 0.6 lagging
Using Eq. (15.9), we obtain
R 3 �
Fp � cos v �����0.6 lagging
ZT
5 �
�
2
e
v R
i
3
�
2
v C
V C
SERIES CONFIGURATION ⏐⏐⏐ 643
j
I
V L –
V C
36.87°
53.13°
V R
E
V L
FIG. 15.38
Phasor diagram for the series R-L-C circuit of
Fig. 15.35.
2�
5
�
2
3�
�t
+