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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

Inductance in series

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