Figure illustrates two coils coupled in series opposition as their fluxes are in opposite direction as per dot convention.
Figure: Inductive Coupling in Series (Flux Opposing)
Let us consider coil 1 and coil 2 individually.
Self-induced e.m.f e =- L1 (di /dt)
Mutually induced e.m.f e1′ = M (di /dt) [due to change of current in coil 2]
Self-induced e.m.f e2 =- L2 (di /dt)
Mutually induced e.m.f e2′ = M(di/dt) [due to change of current in coil 1]
Therefore the total induced e.m.f of the combination can be written as
e = e1 + e2 + e1′ + e2′
e =- L1 (di /dt) - L2( di/ dt) + 2 M (di/dt) =- (L1 + L2 - 2M ) (di/dt)
If L is the equivalent inductance of the combination, then we may write
e =- L( di/ dt)
From Eqs (39) and (40), we get
- L (di / dt )= - (L1 + L 2 - 2M ) (di /dt)
or, L = L1 + L2 - 2M