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3.3 
serIes connectIon oF Inductors
A single equivalent inductor can replace many inductors connected in series for specific analysis 
purposes. We consider series connection of n inductors that have no mutual magnetic coupling among 
them. Let the inductance values be L
1
, L
2
, …, L
n
. We assume that they have the same initial current in 
the same direction at t 
=
0
-
. Let the applied voltage be v(t) and the current in the series combination 
be i(t). See Fig. 3.3-1.
i
(
t
)
i
(
t
)
i
(
t
)
v
1
(
t
)
v
2
(
t
)
v
n
(
t
)
L
1
L
2
L
n
L
eq
v
(
t
)
v
(
t
)
v
(
t
)
+
– +
+
–
–
+
–
+
+
–
–
Fig. 3.3-1 
Series connection of ‘
n
’ inductors
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3.28
Single Element Circuits
Applying KVL along with the element equation of inductor, we get 
v t
v t
v t
v t
L
di t
dt
L
di t
dt
L
di t
dt
n
n
( )
( )
( )
( )
( )
( )
( )
(
=
+
+ +
=
+
+ +
=
1
2
1
2
L
L
L
L
di t
dt
L
di t
dt
L
L
L
L
n
n
1
2
1
2
+
+ +
=
= +
+ +
)
( )
( )
eq
eq
where
Thus, a series connection of n inductors may be replaced by an equivalent inductor with an 
inductance value equal to sum of the inductance values of n inductors as far as the v–i relationship 
is concerned. The total applied voltage across the combination is shared by the various inductors in 
direct proportion to inductance value. i.e., 
v t
v t
v t
v t
v t v t
v t
L L
L
v t
n
n
n
j
( )
( )
( )
( )
( ) : ( ) :
: ( )
:
:
( )
=
+
+ +
=
1
2
1
2
1
2
==
=
L
L
v t
j
n
j
eq
for
( )
1
The sum of flux linkages in the individual inductors is same as the flux linkage of the equivalent 
inductor. Flux linkage is shared in proportion to the inductance value. Similarly the total energy stored 
in all inductors put together is the same as the energy storage calculated using equivalent inductance 
value.
Thus, series equivalent of n inductors with no mutual coupling and with same initial 
current is ‘equivalent’ with respect to 
v
–
i
relation, flux linkage and stored energy. The 
voltage across the series combination, total flux linkage and total stored energy are 
shared among the inductors in direct proportion to their inductance value.
Connecting two or more inductors with unequal initial currents is a situation in which the idealised 
model of an inductance does not work for a physical inductor. A physical inductor has series resistance 
representing its winding resistance, a parallel resistance representing losses in its core (if an iron core 
is used) and a parallel capacitance representing the aggregate effect of inter-winding capacitance. 
Connecting two inductors with different initial currents in series will result in high frequency 
oscillating currents and voltages everywhere in the circuit due to the winding capacitance of the
inductors.