Mutual Inductance of Coupled Coils

Two coils that share flux have a mutual inductance: change the current in one and a voltage appears in the other. How much depends on how much of the first coil's flux the second one actually catches, and that fraction is the coupling coefficient $k$. The relation $M = k\sqrt{L_1 L_2}$ separates the two things that matter — how good each coil is on its own, and how well they see each other. The geometric mean appears because $M$ is a reciprocal quantity: the voltage coil 2 induces in coil 1 uses the same $M$ as the other way round, so the expression has to be symmetric in $L_1$ and $L_2$.

Numbers set the ranges. A 100 mH and a 400 mH coil at $k = 0.5$ give $M = 0.5\sqrt{0.1 \times 0.4} = 0.1\ \text{H}$. Two air-cored coils a few centimetres apart rarely exceed $k = 0.3$. Windings on a common ferrite core, interleaved, reach 0.95 to 0.99. A deliberately gapped flyback transformer might sit near 0.9, and the leakage the missing tenth represents is not a defect but the design's energy store.

$k$ cannot exceed 1, and the reason is conservation rather than convention: coil 2 cannot intercept more flux than coil 1 produced. That ceiling is what makes $k = 1$ the ideal transformer, where the turns ratio alone determines everything and $M = \sqrt{L_1 L_2}$ exactly. Everything a real transformer does differently from the ideal — leakage reactance, regulation under load, the ringing at a switching edge — is the gap between its $k$ and 1.

Coupling is not always wanted, and this page is as useful for avoiding it as for achieving it. Two inductors on the same board with a stray $k$ of 0.05 are enough to ruin a filter's stopband, which is why toroids are preferred over drum cores and why adjacent inductors are mounted at right angles. Beyond that: $k$ is not a constant of the pair but of the geometry, so it changes if a core saturates, a winding shifts, or a shield is added. It is frequency-dependent in practice, because at high frequency the flux path changes and core permeability falls. And the sign of $M$ is a winding-direction convention — the dots on a schematic — not a property of the coils; this page works in magnitudes and leaves the polarity to the dots.