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Therefore,
RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT ⏐⏐⏐ 581
dv C
iC � C � C(qVm cos qt) � qCVm cos qt
or iC � Im sin(qt � 90°)
where Im � qCVm Note that the peak value of iC is directly related to q (� 2pf ) and C,
as predicted in the discussion above.
A plot of vC and iC in Fig. 14.12 reveals that
for a capacitor, iC leads vC by 90°, or vC lags iC by 90°. *
�
dt
If a phase angle is included in the sinusoidal expression for vC, such
as
vC � Vm sin(qt � v)
then iC � qCVm sin(qt � v � 90°)
Applying
cause
Opposition ��
effect
and substituting values, we obtain
Vm Vm 1
Opposition ������ Im qCVm qC
which agrees with the results obtained above.
The quantity 1/qC, called the reactance of a capacitor, is symbolically
represented by XC and is measured in ohms; that is,
1
XC � � (ohms, �) (14.6)
qC
In an Ohm’s law format, its magnitude can be determined from
Vm �
Im
X C � (ohms, �) (14.7)
Capacitive reactance is the opposition to the flow of charge, which
results in the continual interchange of energy between the source and
the electric field of the capacitor. Like the inductor, the capacitor does
not dissipate energy in any form (ignoring the effects of the leakage
resistance).
In the circuits just considered, the current was given in the inductive
circuit, and the voltage in the capacitive circuit. This was done to avoid
the use of integration in finding the unknown quantities. In the inductive
circuit,
v L � L di L
�dt
*A mnemonic phrase sometimes used to remember the phase relationship between the
voltage and current of a coil and capacitor is “ELI the ICE man.” Note that the L (inductor)
has the E before the I (e leads i by 90°), and the C (capacitor) has the I before the E
(i leads e by 90°).
I m
–
p
2
C: i C leads v C by 90°.
90°
i C
V m
v C
0 p p 3 2p
2
2 p
FIG. 14.12
The current of a purely capacitive element
leads the voltage across the element by 90°.
qt