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3.4 
PArALLeL connectIon oF Inductors
We develop the equivalent of n inductors in parallel in this section. We consider only magnetically 
uncoupled inductors in this section. 
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Parallel Connection of Inductors 
3.29
i
n
(
t
)
i
2
(
t
)
i
1
(
t
)
i
(
t
)
i
(
t
)
v
(
t
)
v
(
t
)
L
1
L
2
L
n
L
eq
+
–
+
–
Fig. 3.4-1 
Parallel connection of ‘
n
’ inductors
The change in inductor current over a time interval is given by the V-sec applied to it during that 
time interval divided by value of inductance. We assume zero initial currents at t 
=
0 and hence 
i t
L
v t dt
k
to n
k
k
t
( )
( )
= +
=
−
∫
0
1
1
0
for 
The V-sec product applied to all the inductors will be equal since they all share a common voltage 
from t 
=
0
-
onwards. Hence, the currents in inductors will be in inverse proportion to the inductance 
values. i.e.,
i t i t
i t
L
L
L
n
n
1
2
1
2
1
1
1
( ) : ( ) :
: ( )
:
:
=
 
Further, applying KCL at the 
+ 
ve node of voltage source in the circuit in Fig. 3.4-1, we get,
i t
i t
i t
i t
L
L
L
v t dt
L
n
n
t
( )
( )
( )
( )
( )
=
+
+ +
=
+
+ +






=
−
∫
1
1
1
2
0
1
1
1
1
eqq
v t dt
t
( )
0
−
∫
Therefore, we see that a single inductor L
eq
can represent the n inductors in parallel as far as the 
v–i relationship is concerned. This implies that the source will not be able to distinguish between the 
parallel combination of n inductors and a single inductor that is the parallel equivalent.
All the inductors had zero initial flux linkage and zero initial energy. The change in flux linkage in 
an inductance over a time interval is same as the V-sec applied to it during that time interval. Since all 
of them started at zero flux linkage as per our assumption, it follows that all of them will have equal 
flux linkage at all t. 
Now, we look at the stored energy picture. We know that the stored energy function E(t) of an 
inductor 
=
0.5Li
2
, where i is the instantaneous current. However, we need another formulation for the 
same function now. We proceed as below shown in the following:
y
y
( )
( )
( )
t
v t dt
t
=
+
=
+
−
−
∫
0
0
initial flux linkage V-s added
Also,
volt sec if 
y
y
( )
( )
( )
( )
( ) ( )
(
) (
t
Li t
E t
L i t
t i t
L
=
∴
=
[ ]
=
=
−
1
2
1
2
1
2
2
2
iinitial flux linkage is zero)
Hence the stored energy at any instant in an inductor is proportional to square of V-s product 
dumped into it till that instant provided the inductor was initially relaxed. Therefore, in the parallel 
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3.30
Single Element Circuits
connection of inductors under consideration, the stored energy in the inductors will be in inverse 
proportion to their inductance values. They all share the same V-sec product at the same instant.
E
t
E t
E t
E t
L
L
L
n
n
total
volt-se
( )
( )
( )
( )
=
+
+ +
=
+
+ +






1
2
1
2
1
2
1
1
1
cc
volt-sec
Energy stored in an initially relax
eq
[
]
=
[
]
=
2
2
1
2
L
eed inductor 
eq
L
Therefore, the equivalent inductance L
eq 
gives the correct value for stored energy. Note that we have 
shown this only for a parallel connection of initially relaxed inductors.
A single inductor 
L
L
i
i
n
eq
=
1
1
1
/
=
∑
can represent a set of n inductors in parallel. Voltage, 
V-sec and flux linkage are common variables in parallel connection of initially relaxed 
inductors. Total current and total stored energy will be shared by the inductors in 
inverse proportion to their inductance value.
It can be shown that the conclusions arrived above are valid for inductors with non-zero initial current 
connected in parallel provided they have equal initial flux linkages. In the case of parallel connection of 
inductors with unequal initial flux linkages, circulating currents in local loops formed by the inductors 
can appear under certain conditions, thereby trapping a portion of total initial energy and hiding that 
portion of initial energy from rest of the circuit forever. Example 3.4.2 illustrates this situation.