boylistad

586 ⏐⏐⏐ THE BASIC ELEMENTS AND PHASORS
200 � 200 �
L ����� 0.531 H
q 377 rad/s
c. Since i leads v by 90°, the element is a capacitor, and
Vm 500 V
XC ����� 500 �
Im 1 A
1
so that XC ��� 500 � or
qC
1
1
C ����� � 12.74 mF
q500 � (157 rad/s)(500 �)
d. v � 50 cos(qt � 20°) � 50 sin(qt � 20° � 90°)
� 50 sin(qt � 110°)
Since v and i are in phase, the element is a resistor, and
V m
50 V
R ����� 10 �
Im 5 A
dc, High-, and Low-Frequency Effects on L and C
For dc circuits, the frequency is zero, and the reactance of a coil is
X L � 2pfL � 2p(0)L � 0 �
The use of the short-circuit equivalence for the inductor in dc circuits
(Chapter 12) is now validated. At very high frequencies, X L � 2pf L
is very large, and for some practical applications the inductor can be
replaced by an open circuit. In equation form,
X L � 0 �
dc, f � 0 Hz
(14.10)
and XL ⇒ � � as f ⇒ � Hz
(14.11)
The capacitor can be replaced by an open-circuit equivalence in dc
circuits since f � 0, and
1 1
XC ����⇒�� 2pfC 2p(0)C
once again substantiating our previous action (Chapter 10). At very
high frequencies, for finite capacitances,
1
XC ��
2pf ↑ C
is very small, and for some practical applications the capacitor can be
replaced by a short circuit. In equation form
X C ⇒ � � as f ⇒ 0Hz
(14.12)
and XC � 0 � f � very high frequencies (14.13)
�