Equivalent Inductance of Series Connected Inductor

When two or more inductors are connected in such a way that same current flows through them, then it is said that they are connected in series. Like series connection of resistors, series connected inductors can be replaced by an equivalent inductor.



Let us assume that ‘n’ inductors having inductance of L1, L2, L3,…Ln Henry are connected in series as shown in figure below. Since same current I is flowing through each of the inductors therefore voltage drop across inductors L1, L2, L3, …..Ln will be L1di/dt, L2di/dt, L3di/dt, …..Lndi/dt respectively.
Therefore from Kirchhof’s voltage law,
V = L1di/dt + L2di/dt + L3di/dt + ……..+ Lndi/dt
    = di/dt (L1 + L2 + L3 +…….+Ln)  …………….(1)
If the series connected inductors are replace by a single equivalent inductor Le then
V = Ledi/dt  …………………….(2)
Comparing equation (1) and (2) we get,
Le = L1 + L2 + L3 +…….+Ln
“It shall be kept in mind that the above equation for equivalent inductance is applicable only when there is no mutual coupling between the inductors.”

Mutually Coupled Inductors

Two inductors are said to be mutually coupled if their magnetic field link each other. Linking of magnetic field between two inductors can either be in additive mode or in subtractive mode. If the magnetic field linking an inductor is in the same direction as produced by inductor then it is called additive mode. Similarly if the magnetic field linking inductor is in opposition with the magnetic field produced then it is called subtractive mode.
Based on the kind of mutual coupling, the equivalent inductance of series connected inductor will change.

Equivalent Inductance for Additive Mutual Coupling

Let us assume that two inductors of inductance L1 and L2 are connected in series. Their mutual inductance is M. As the mutual coupling is assumed to be additive therefore the emf induced in the inductor due to mutual coupling will be in the same direction as the direction of emf due to self inductance.

V = (L1di/dt + Mdi/dt) + (L2di/dt + Mdi/dt)
But V = Ledi/dt
Therefore the equivalent inductance is given as
Le = L1 + L2 + 2M

Equivalent Inductance for Subtractive Mutual Coupling

Let us assume that two inductors of inductance L1 and L2 are connected in series. Their mutual inductance is M. As the mutual coupling is assumed to be subtractive therefore the emf induced in the inductor due to mutual coupling will be in opposite direction as the direction of emf due to self inductance.

V = (L1di/dt – Mdi/dt) + (L2di/dt – Mdi/dt)
But V = Ledi/dt
Therefore the equivalent inductance is given as
Le = L1 + L2 – 2M

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