Inductance in series

L₁ and L₂ are two coils of self-inductance L₁ and L₂ connected in series, Let M be the mutual inductance between the two coils. The coils are placed such that the direction of current is the same in both coils. Then the fluxed in L₁ and L₂ will also be the same direction and the coupling is called positive coupling. Hence whenever the current in the coils changes, the emf induced due to self-induction and mutual induction will be in same direction.
    For the first coil, the total induced e.m.f. is
        e₁ = L₁ dI/dt + M dI/dt                                … (1)
    For the second coil, we have
        e₂ = L₂ dI/dt + MdI/dt                                … (2)

Total e.m.f., e = e₁ + e₂ = [L₁ + L₂ + 2M] dI/dt                            … (3)

. : Equivalent Inductances L = L₁ + L₂ + 2M                            … (4)
In the case of negative coupling, one of the coils will be reversed so that the current in one coil is opposite to that in the other coil. Then,
        L = L₁ + L₂ – 2M



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