Mutual Inductance and the Coupling Coefficient
How much flux do they share?
Current flows in the gold coil on the left, and the coil on the right receives its flux. The faint loops that wrap only the left coil are flux leaking away to the side, and the large gold loops that wrap both coils are flux the two share. Raise the coupling coefficient k from 0 to 1 and the faint loops turn into gold loops one by one.
From self to mutual
A single coil links the flux produced by its own current, and that self-linkage is the self-inductance L. Place a second coil beside it, and part of that flux threads the second coil as well. The quantity that measures how much threads the second coil is the mutual inductance M.
k is the linking fraction
Only part of the flux one coil produces passes through the other; the rest leaks off to the side as leakage flux. In the plot, the faint loops around the left coil alone are exactly that leakage flux. Define the linking fraction as the coupling coefficient k, and 0 ≤ k ≤ 1. At k = 0 the two coils have no magnetic connection at all, and at k = 1 every bit of one coil's flux passes through the other, which is the ideal transformer.
Why the ceiling is √(L₁L₂)
The flux reaching the second coil cannot exceed the flux the first coil has, and the same holds in the other direction. Taking both directions of coupling together, M stops at the geometric mean of the two inductances, √(L₁L₂). So we write M = k√(L₁L₂), and k states, as a number between 0 and 1, how far M has climbed toward that ceiling. The two coils in this plot have L₁ = 40 mH and L₂ = 10 mH, so the ceiling is 20 mH.
Coupling seen in the voltage
In the sinusoidal steady state, the voltages of the two coils are tied together as V₁ = jωL₁I₁ + jωMI₂, V₂ = jωMI₁ + jωL₂I₂. The M term that appears in both equations acts as the bridge between the two circuits. The larger M is, the more strongly the current on one side shows up in the voltage on the other.
Back to the first screen
As you raised k from 0 to 1, the faint leakage lines turned one by one into gold linking lines wrapping both coils. That fraction is exactly k, and M climbed only by k times the ceiling √(L₁L₂). At k = 1 every flux line passed through both coils and M touched the ceiling. Coupling, in the end, is a question of how much flux is shared.