Induced emf:

Induced emf in coil 1 (because of mutual inductance M between the two coils) suppoaing permeability of mutual flux path is constant

e1 = M (di₂ /dt)

also    e₁ = N₁  (d φ₂₁ /dt)

M  di₂ / dt       = N   d φ/ dt

Or  M = N₁ (d φ₂₁ / di₂)

Assuming the zero initial conditions

M = N1 (φ₂₁ /i₂)

Multiply Eq. (1.9) and (1.12)

M²       = ( N₂  (φ₁₂/  i)( N₁ (φ₂₁/ i ₂)

Now we describe the coefficient of coupling

K = (φ₁₂ /φ₁) = φ₂₁ / φ₂

or         K = Useful flux / Total flux

                   ≤ 1

Under ideal conditions, while there is no leakage, then K = 1 (that means. 100% coupling between two coils). In any practical case, K is always less than 1.

From Eq

φ₁₂  = K φ₁      and       φ₂₁  = K φ₂

Put these values of φ 21 and φ 12 in Eq. (13)

M ² = (N₂ K φ₁  /i₁  ) (N₁  (K φ₂ / i₁)

= K ² (N₁ φ₁ / i₁) ( N₂ φ₂ / i₂)

= K² L₁ L₂

Or

So M can be positive or negative based upon the winding sense.