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580 ⏐⏐⏐ THE BASIC ELEMENTS AND PHASORS
i C = ?
C
+
vC = Vm sin qt
–
FIG. 14.11
Investigating the sinusoidal response of a
capacitive element.
and current will be known, and the opposing voltage (velement) can be
determined for any sinusoidal current i.
Our investigation of the inductor revealed that the inductive voltage
across a coil opposes the instantaneous change in current through the
coil. For capacitive networks, the voltage across the capacitor is limited
by the rate at which charge can be deposited on, or released by, the
plates of the capacitor during the charging and discharging phases,
respectively. In other words, an instantaneous change in voltage across
a capacitor is opposed by the fact that there is an element of time
required to deposit charge on (or release charge from) the plates of a
capacitor, and V � Q/C.
Since capacitance is a measure of the rate at which a capacitor will
store charge on its plates,
for a particular change in voltage across the capacitor, the greater the
value of capacitance, the greater will be the resulting capacitive
current.
In addition, the fundamental equation relating the voltage across a
capacitor to the current of a capacitor [i � C(dv/dt)] indicates that
for a particular capacitance, the greater the rate of change of voltage
across the capacitor, the greater the capacitive current.
Certainly, an increase in frequency corresponds to an increase in the
rate of change of voltage across the capacitor and to an increase in the
current of the capacitor.
The current of a capacitor is therefore directly related to the frequency
(or, again more specifically, the angular velocity) and the capacitance
of the capacitor. An increase in either quantity will result in an
increase in the current of the capacitor. For the basic configuration of
Fig. 14.10, however, we are interested in determining the opposition of
the capacitor as related to the resistance of a resistor and qL for the
inductor. Since an increase in current corresponds to a decrease in
opposition, and iC is proportional to q and C, the opposition of a capacitor
is inversely related to q (� 2pf ) and C.
–
e
+
i C
+ –
v C
C
Opposition inversely
related to f and C
FIG. 14.10
Defining the parameters that determine the opposition of a capacitive element
to the flow of the charge.
We will now verify, as we did for the inductor, some of the above
conclusions using a more mathematical approach.
For the capacitor of Fig. 14.11, we recall from Chapter 10 that
and, applying differentiation,
dv C
� dt
i C � C
dv C
� dt
d
� �(Vm sin qt) � qVm cos qt
dt
�