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3.6.2 
series connection of capacitors with non-zero Initial energy
We identify two cases in this situation. In the first case we consider a situation in which all the 
capacitors had same initial charge in polarity and magnitude at the instant they were put in series. 

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Series Connection of Capacitors 
3.43
If they had same initial charge to begin with, they will continue to have equal charge even after 
they have been connected in series. This is so because current is a common variable in a series 
connection. Therefore, the individual capacitor voltages will be in inverse proportion to capacitance 
values initially and subsequently. Therefore, the series connection can be replaced by an effective 
capacitor of values as per Eqn. 3.6-1 and with an initial voltage as per Eqn. 3.6-1. The equivalent 
capacitor will be equivalent in v–i relationship, in charge and in total stored energy. Total voltage and 
total stored energy will get distributed in individual capacitors in inverse proportion to capacitance 
values.
In the second case, we consider a set of capacitors with arbitrary initial charges. If they had unequal 
initial charges they will continue to have unequal charges subsequently too. Only change in charge for 
various capacitors will be equal in this case.
Therefore, in this case the effective capacitor will describe only the change in capacitor voltages 
correctly. The equivalent capacitor is equivalent with respect to 
D
v(t)–i(t) relationship, change in 
charge and change in total stored energy. Change in charge will be equal in all capacitors. Change 
in total voltage is distributed as changes in individual capacitor voltages in inverse proportion to 
capacitance values. Change in total stored energy calculated from equivalent circuit will be correct. 
This change in total stored energy is equal to sum of changes in stored energy of individual capacitors. 
But the distribution is not in inverse proportion to capacitance values.
The initial voltage across the equivalent capacitor will be given by the algebraic sum of initial 
voltages of individual capacitors in this case too. But the initial stored energy that is calculated from 
equivalent capacitor will be less than the actual initial stored energy in the system. The difference 
will represent the portion of initial energy that gets trapped and hidden in the system. This portion is 
trapped forever in the series connection and can not be taken out by other elements.
A simple example will show that there can be trapped energy in a series connection. Consider two 
capacitors with initial voltages that are equal in magnitude and opposite in polarity. The initial voltage 
observed at terminals of series combination will be zero when we connect them in series. We can not 
take out the initial energy though we know that it is there. In this case the entire initial energy becomes 
trapped energy.
The amount of trapped energy in a series connection is given by Trapped Energy 
=

 −






=
−
−
=
∑
∑
1
2
0
1
2
0
1
2
1
2
k
n
k
k
k
k
n
C V
C
V
( )
( )
eq
J.
A single capacitor 
C
eq
can replace a set of n capacitors connected in series as far as 
changes in charge, changes in voltage and changes in total stored energy are concerned.
1
1
1
1
1
2
C
C
C
C
n
eq
=
+
+
+






However, the total charge in each capacitor and total charge in 
C
eq
will not be the same 
unless all capacitors had the same charge at 
t
=
0
-
.
Similarly, the total stored energy in the system will not be the same as the total stored 
energy in 
C
eq
unless all capacitors had same charge at 
t
=
0
-
.
If the capacitors started out with different charges at 
t
=
0
-
, a portion of total initial 
energy is trapped in the circuit and gets hidden from rest of the circuit forever.

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3.44
Single Element Circuits