Copyright 20 13 Dorling Kindersley (India) Pvt. Ltd

3.6 
serIes connectIon oF cAPAcItors
Practical capacitors come in standard sizes with standard voltage, current and capacitance ratings. 
Series or parallel connection of such capacitors is almost always required in practical applications 
in order to meet the application requirements on voltage rating or current rating or capacitance 
rating. Such series or parallel connections of capacitors can often be represented as a single 
equivalent capacitor for analysis purposes provided we pay careful attention to initial conditions 
of the capacitors involved in the connection. We look at series connection of capacitors in this
section.
3.6.1 
series connection of capacitors with Zero Initial energy
Let n capacitor with capacitance values C
1
, C
2
, …, C
n
be with zero initial voltage and no mutual 
electrostatic coupling among them be connected in series as shown in Fig. 3.6-1.
v
1
(
t
)
v
2
(
t
)
C
1
C
n
i
(
t
)
v
(
t
)
v
(
t
)
i
(
t
)
C
eq
+
–
–
+
+
+
+
v
n
(
t
)
+
–
C
2
–
–
–
Fig. 3.6-1 
Series connection of capacitors and equivalent capacitor
The change in the voltage across a capacitor is equal to the charge dumped into it divided by 
capacitance value. Capacitors that have been connected in series have to have a common current 
and hence all of them will get only the same charge dumped on them subsequently. Therefore, the 
change in capacitor voltages across the capacitors in a series combination due to i(t) will be in 
proportion to reciprocal of their capacitance values. If all the capacitors had zero initial voltage 

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3.42
Single Element Circuits
at the instant of connection, then capacitor voltages themselves will be in proportion to reciprocal 
of capacitance values. Thus, v(t) gets distributed among the capacitors as per the following 
relation. 
Ratio of distribution of ( )
/
v t
C
C
C
C
v t
C
n
j
j
=
∴
=
1
1
1
1
1
1
2
3
:
:
:
:
( )
11
1
1
1
1
1
1
2
3
1
C
C
C
C
v t
C
C
v t
n
j
k
k
n
+
+
+ +
=
=
∑
( )
( )
/
/
If there are only ttwo capacitors ( )
and
( )
(
v t
C
C
C
 v(t)
v t
C
C
C
 v t
1
2
1
2
2
1
1
2
=
+
=
+
))
Note that smaller capacitors will take larger share of voltage in a series connection of capacitors.
Assume that this series connection is being driven by a current source i(t) across its terminals from 
t 
=
0 onwards. The voltage v(t) developed across the current source terminals will be
v t
C
i t dt
C
i t dt
C
i t dt
C
i t dt
t
t
n
t
( )
( )
( )
( )
( )
=
+
+ +
=
−∞
−∞
−∞
∫
∫
∫
1
1
1
1
1
2
1
−−∞
−∞
−∞
=
−
−
−
∫
∫
∫
∑
+
+ +








+



0
2
0
0
1
1
1
1
C
i t dt
C
i t dt
C
n
k
k
n
( )
( )



=
+






=
−
−
∫
∑
∑
∫
−
=
=
−
i t dt
V
C
i t dt
V
t
k
k
n
k
k
n
t
( )
( )
( )
(
0
1
1
0
0
1
0
eq
))
( )
( )
(
+








=
=
−
∫
∑
=
−
1
1
1
0
0
0
1
C
i t dt
C
C
V
V
t
k
k
n
k
eq
eq
eq
where
and
−−
=
∑
)
k
n
1
(3.6-1)
Hence, we can replace the series connection by a single capacitor of value C
eq
with an initial 
voltage of V
eq
(0
-
) defined as in Eqn. 3.6-1. When all the capacitors are initially relaxed, the effective 
capacitor will also be initially relaxed. Otherwise, the algebraic sum of initial voltages across the 
capacitors will give the initial voltage across the effective capacitor.
In summary, when n capacitors with zero initial condition are connected in series, one capacitor 
with its value as defined in Eqn. 3.6-1 can be used to replace the series combination as far as v
-
i 
relationship, stored charge and stored energy are concerned. All capacitors as well as effective capacitor 
will have equal charge at all t. Individual capacitor voltage will be as per the ratio of reciprocal of 
capacitance values. Total stored energy will be distributed in various capacitors as per the ratio of 
reciprocal of capacitance values.