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410 ⏐⏐⏐ CAPACITORS
+
E
–
Q T
V 1
Q 1
V 2
Q 2
+ – + – + –
FIG. 10.61
Series capacitors.
V 3
Q 3
time between 0 and t 1. The same can be said about any portion of the
voltage waveform that has a constant slope.
An important point to be gained from this discussion is that it is not
the magnitude of the voltage across a capacitor that determines the current
but rather how quickly the voltage changes across the capacitor. An
applied steady dc voltage of 10,000 V would (ideally) not create any
flow of charge (current), but a change in voltage of 1 V in a very brief
period of time could create a significant current.
The method described above is only for waveforms with straight-line
(linear) segments. For nonlinear (curved) waveforms, a method of calculus
(differentiation) must be employed.
10.13 CAPACITORS IN SERIES AND PARALLEL
Capacitors, like resistors, can be placed in series and in parallel.
Increasing levels of capacitance can be obtained by placing capacitors
in parallel, while decreasing levels can be obtained by placing capacitors
in series.
For capacitors in series, the charge is the same on each capacitor
(Fig. 10.61):
Applying Kirchhoff’s voltage law around the closed loop gives
E � V1 � V2 � V3 However, V � � Q
C �
so that
QT �
CT
Q1 Q2 Q3 ������ C1 C2 C3
Using Eq. (10.29) and dividing both sides by Q yields
(10.29)
(10.30)
which is similar to the manner in which we found the total resistance of
a parallel resistive circuit. The total capacitance of two capacitors in
series is
(10.31)
The voltage across each capacitor of Fig. 10.61 can be found by first
recognizing that
QT � Q1 or C T E � C 1V 1
Solving for V 1:
Q T � Q 1 � Q 2 � Q 3
1 1 1 1
�� � �� � �� � �� C C C C
T
1
2
C
CT � �
1C2 �
C1 �
V1 � CTE �C1
C 2
3