Mutual inductance

Mutual inductance $M_{12}$ between two coils is the ratio of the flux linkage in coil 2 to the current that produced it in coil 1:

$M_{12}=\frac{N_2{\Phi }_{12}}{I_1}\text{(H)},$

where ${\Phi }_{12}$ is the magnetic flux through one turn of coil 2 due to current $I_1$ in coil 1, and $N_2$ is the number of turns on coil 2. Self-inductance ($L=N\Phi /I$) is “how strongly a coil links its own flux”; mutual inductance is “how strongly two coils link each other’s flux.”

Units are henries, same as self-Inductance.

Where the flux linkage comes from

Current $I_1$ produces a Magnetic field ${\mathbf{B}}_1$ via Biot-Savart law (or Ampère’s law when symmetry allows). The flux of this field through one turn of coil 2:

${\Phi }_{12}={\int }_{S_2}{\mathbf{B}}_1\cdot d{\mathbf{s}}_2.$

With $N_2$ identical turns linked by the same per-turn flux, the flux linkage of coil 2 is ${\Lambda }_{12}=N_2{\Phi }_{12}$. Divide by the source current $I_1$ to get $M_{12}$.

Reciprocity

$M_{12}=M_{21}.$

Driving coil 1 to link coil 2 produces the same coupling coefficient as driving coil 2 to link coil 1, for given geometry. This is non-obvious — the two integrals look very different geometrically — but follows from the symmetry of the Vector magnetic potential integrand ${\mu }_0/(4\pi R)$ under exchange of source and field points. Equivalently it comes from energy considerations: mutual magnetic energy depends symmetrically on the two currents.

In practice, $M_{12}=M_{21}$ means you only need to compute one direction. Drop the subscript and write simply $M$ when the geometry is fixed.

Coupling coefficient

The mutual inductance is bounded by the geometric mean of the self-inductances:

$|M|\le \sqrt{L_1{L}_2}.$

Define the coupling coefficient

$k=\frac{M}{\sqrt{L_1{L}_2}},0\le k\le 1.$

$k=1$ is ideal coupling: every flux line from coil 1 also passes through coil 2. Achieved (approximately) by tightly winding both coils on the same iron core, as in a real transformer. $k\ll 1$ corresponds to loosely coupled coils — most flux escapes.

What it lets you do

Mutual inductance is the foundation of:

• Transformers. Primary current $i_1(t)$ generates time-varying flux that links the secondary. By Faraday’s law, the induced secondary voltage is $v_2=Md{i}_1/dt$ (plus self-induction terms). With $k\approx 1$ and turns ratio $N_2/N_1$, the transformer relations $v_2/v_1=N_2/N_1$ follow.
• Wireless power transfer. Two coils across a small gap; maximizing $M$ maximizes power transfer at resonance.
• Crosstalk. Unintentional $M$ between adjacent traces on a PCB or wires in a cable couples signals from one circuit into another. Twisted-pair cabling, careful routing, and ground planes all aim to drive $M$ toward zero.
• Induction motors and generators. The rotor and stator are magnetically coupled circuits; $M$ depends on rotor position, giving a time-varying coupling that converts electrical to mechanical energy.

Two coupled-circuit equations

When two coils share mutual inductance, each circuit equation picks up a cross-term:

$v_1=L_1\frac{d{i}_1}{dt}+M\frac{d{i}_2}{dt},$ $v_2=M\frac{d{i}_1}{dt}+L_2\frac{d{i}_2}{dt}.$

The off-diagonal $M$ encodes “current in the other coil shows up as voltage on this one.” The sign of $M$ depends on coil orientation — denoted by the dot convention in circuit schematics (a dot on each coil marks one terminal; if both currents enter their dotted terminals, the cross-term is $+Mdi/dt$; if one enters and the other exits, it’s $-Mdi/dt$).

Worked example: two coaxial solenoids

A long solenoid of length $l$, $N_1$ turns, cross-section $S$, currying $I_1$, with a short secondary coil of $N_2$ turns wound around the middle. Interior field of the primary: $B_1={\mu }_0{N}_1{I}_1/l$.

Flux through one turn of the secondary: ${\Phi }_{12}=B_{1}S={\mu }_0{N}_1{I}_{1}S/l$.

Flux linkage: ${\Lambda }_{12}=N_2{\Phi }_{12}={\mu }_0{N}_1{N}_2{I}_{1}S/l$.

Mutual inductance:

$M=\frac{{\Lambda }_{12}}{I_1}=\frac{{\mu }_0{N}_1{N}_{2}S}{l}.$

Notice the product $N_1{N}_2$ instead of the $N^2$ that appears in self-inductance — one factor from the field source, one from the flux linkage on the receiving side. Reciprocity check: computing $M_{21}$ by driving the short coil (more involved, since its field isn’t uniform) gives the same answer.