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RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT ⏐⏐⏐ 585
The curves are sketched in Fig. 14.17.
I m = 12 mA
– 2 p
90°
v C
i C
V m = 30 V
0 p
2
p 3
2
p
i leads v by 90°.
FIG. 14.17
Example 14.5.
EXAMPLE 14.6 The current through a 100-mF capacitor is given.
Find the sinusoidal expression for the voltage across the capacitor.
i � 40 sin(500t � 60°)
Solution:
10
XC � � � � �20 �
Vm � ImXC � (40 A)(20 �) � 800 V
and we know that for a capacitor, v lags i by 90°. Therefore,
v � 800 sin(500t � 60° � 90°)
and v � 800 sin(500t � 30°)
2 10 �
�
5
6 1
�
��� �4 (500 rad/s)(100 � 10 5 � 10 �6 1
�
qC
F)
EXAMPLE 14.7 For the following pairs of voltages and currents,
determine whether the element involved is a capacitor, an inductor, or a
resistor, and determine the value of C, L, or R if sufficient data are provided
(Fig. 14.18):
a. v � 100 sin(qt � 40°)
i � 20 sin(qt � 40°)
b. v � 1000 sin(377t � 10°)
i � 5 sin(377t � 80°)
c. v � 500 sin(157t � 30°)
i � 1 sin(157t � 120°)
d. v � 50 cos(qt � 20°)
i � 5 sin(qt � 110°)
Solutions:
a. Since v and i are in phase, the element is a resistor, and
Vm 100 V
R ����� 5 �
Im 20 A
b. Since v leads i by 90°, the element is an inductor, and
Vm 1000 V
XL ����� 200 �
Im 5 A
so that X L � qL � 200 � or
2p
�
+
v ?
–
i
FIG. 14.18
Example 14.7.