Mutual Inductance & Coupling

Intuition

How much of the beam lands on the target

Shine a torch at a small object across a room and only part of the beam actually falls on it. Bring the object closer, or narrow the beam, and the fraction that lands goes up. Move it aside and the fraction drops away, even though the torch is putting out exactly as much light as before.

Two coils behave that way with magnetic field. A coil carrying a current produces a field around itself, and if a second coil sits in that field, some of the field passes through it. How much depends on how close the two are, how they are oriented, and whether anything guides the field from one to the other. When the current in the first coil changes, the field changes, and a changing field through the second coil produces a voltage in it — the same effect that gives a single coil its inductance, except that the cause is now in a different piece of wire.

That is mutual inductance, written M and measured in henries. It says how many volts appear across the second winding for a given rate of current change in the first. No wire runs between them. Nothing conducts. The two are joined only by a magnetic field, and one can be at a completely different voltage from the other.

Everything that depends on this idea depends on that last sentence. A transformer moves power across a barrier that carries no current. A wireless charger moves power across an air gap. And, unhelpfully, a switching supply moves noise into an adjacent signal loop by exactly the same mechanism, which is why coupling is something to design out as often as in.

Practitioner

The coupling coefficient, and the voltage it delivers

Mutual inductance depends on both windings and on how well they share their field. That last part is captured by a single number between zero and one, the coupling coefficient k:

At k = 1 every line of flux from one winding passes through the other, which is the ideal case and is never quite reached. At k = 0 the two ignore each other entirely. Loosely coupled air-cored coils sit at a few tenths; windings sharing a closed ferrite core reach well above 0.99.

The working relation is the same shape as the one for a single coil, with M in place of L:

Mutual inductance is symmetric, which is not obvious and is genuinely useful: the M that couples winding one into winding two is the same M that couples two into one. A single number describes the pair, whichever way the energy happens to be flowing.

Orientation matters as much as proximity. Two coils on the same axis couple strongly; the same two turned at right angles couple hardly at all, because the field from one runs parallel to the turns of the other and passes through almost none of them. That is a layout tool rather than a curiosity, and it is why the inductors in adjacent switching converters on one board are often deliberately mounted at ninety degrees to each other.

Engineer

Measuring M without knowing k

Series-connecting two coupled windings gives a total inductance that depends on which way round they are wired, because the mutual term either adds to or subtracts from the self terms. Wired so their fluxes reinforce:

Reverse either winding and the flux from one now opposes the flux from the other, so the mutual term changes sign and the same pair measures a good deal less. That gives a measurement route that needs no knowledge of the geometry at all — take both readings and the difference is four times M.

That is the standard bench method, and an LCR meter with a soldering iron beside it is all the apparatus it needs. It also gives k, since k is M divided by the geometric mean of the two self-inductances, both of which the same meter reads directly.

The dot convention exists to record which way round is which. Two windings on a schematic carry a dot at one end of each, and the convention is that current entering both dotted ends produces flux that aids. Without those dots a coupled pair is ambiguous, and the ambiguity is not academic — the sign of M decides whether a flyback converter's output is positive or negative, and whether a feedback winding stabilises an oscillator or stops it dead.

The energy stored in a coupled pair is not simply the sum of what each winding stores alone. A cross term appears, proportional to M and to the product of the two currents, and it is positive or negative depending on the winding sense. It is that term that lets a transformer take energy in at one winding and deliver it out of another, and its presence is why the two windings cannot be treated as independent components that happen to be nearby.

The model has honest limits. A single k treats the coupling as one number, whereas real windings couple differently at different frequencies as skin effect and capacitance change the current distribution. It also assumes a linear core: once the core saturates, both self-inductances collapse and M with them. And it says nothing about the capacitance between the windings, which provides a second, entirely electrostatic coupling path — the one that carries common-mode noise straight through a transformer that was fitted to block it.

Professional

Leakage, and what the missing flux costs

The flux that fails to link both windings is called leakage, and in a power design it is usually the dominant nuisance.

Leakage inductance behaves as an ordinary inductor in series with each winding, and it does everything an ordinary inductor does. It stores energy that has nowhere to go when a switch opens, producing the voltage spike that destroys switching transistors in flyback converters. It limits how fast current can transfer between windings, which shows up as a droop on the output of a pulse transformer. And it forms a resonant circuit with the winding capacitance, producing the ringing seen on every real transformer waveform.

Improving coupling is mostly a matter of construction. Interleaving primary and secondary layers, using a closed magnetic path rather than an open rod, filling the winding window and keeping the windings physically concentric all raise k. Every one of those also raises the capacitance between the windings, and a design that needs isolation rather than efficiency deliberately goes the other way — a safety-isolating transformer has separated bobbin sections precisely so the two windings do not couple electrostatically, and accepts the leakage that comes with it.

Unwanted coupling is the same physics without the intent. A current loop on a board radiates a field, and any nearby loop is a secondary winding whether or not anyone drew it that way. The cure is geometric: shrink the loop areas, keep the return path directly under its outgoing conductor, orient any magnetic components so their fields do not link, and put distance or a shield between the aggressor and the victim. Every one of those reduces M, which is the only quantity that matters.

Coupling can also be exploited deliberately against noise. A common-mode choke is two windings wound so that the wanted differential current produces opposing fluxes that cancel, leaving the part almost invisible to the signal, while common-mode current produces aiding fluxes and meets the full series-aiding inductance. The same two numbers from Layer 3 describe both behaviours; only the winding sense differs.

Coupled inductors in power conversion sit between the two worlds. A multiple-output flyback uses one core for several outputs, so the outputs track each other well but cross-regulation is limited by leakage between them. A coupled-inductor buck converter uses the mutual term to shape ripple current in ways two separate inductors could not manage. In both, M is a design parameter rather than a parasitic, and the winding sense is part of the schematic.