8.6 Mutual inductance

Mutual inductance

Mutual inductance describes how a changing current in one circuit can induce an EMF in a separate, nearby circuit through their shared magnetic flux. This concept sits at the heart of transformers, wireless charging, and any system where energy transfers between magnetically coupled loops. It's measured in henries (H) and represented by $M$.

Coupled inductors

When two or more inductors sit close enough for their magnetic fields to overlap, they become coupled inductors. A changing current in one produces a time-varying magnetic flux, and some fraction of that flux threads through the second inductor, inducing an EMF there via Faraday's law.

Coupled inductors show up in transformers, inductive power transfer systems, and wireless charging devices.

Coupling coefficient

The coupling coefficient $k$ is a dimensionless number between 0 and 1 that tells you how much of one inductor's flux actually links with the other.

• $k = 0$: no shared flux at all (inductors are magnetically isolated).
• $k = 1$: every field line from one inductor passes through the other (perfect coupling).

It's calculated as:

$k = \frac{M}{\sqrt{L_1 L_2}}$

where $L_1$ and $L_2$ are the self-inductances of the two inductors. This formula also gives you a useful upper bound: $M$ can never exceed $\sqrt{L_1 L_2}$.

Leakage inductance

Not all the magnetic flux produced by one inductor links with the other. The portion that "leaks" into the surrounding space without coupling is called leakage inductance. It stores energy but doesn't contribute to mutual coupling, so it acts like an extra series inductance in each winding.

Leakage inductance can be reduced by winding the coils closer together, using a shared high-permeability core, or interleaving the windings.

Mutual inductance calculation

Calculating $M$ depends on the geometry and arrangement of the conductors. The general approach is to find how much magnetic flux from one circuit links with the other, then relate that to the current producing it.

Neumann formula

For two arbitrary current-carrying loops, the Neumann formula gives the mutual inductance directly from the geometry:

$M = \frac{\mu_0}{4\pi} \oint_{C_1} \oint_{C_2} \frac{d\vec{l_1} \cdot d\vec{l_2}}{r}$

Here $C_1$ and $C_2$ trace the two loops, $d\vec{l_1}$ and $d\vec{l_2}$ are infinitesimal directed length elements along each loop, and $r$ is the distance between those elements. The double line integral runs over every pair of elements, one from each loop.

This formula is completely general but often leads to integrals that require numerical evaluation for anything beyond simple geometries. Notice that the expression is symmetric in the two loops, which confirms the reciprocity of mutual inductance: $M_{12} = M_{21}$.

Coaxial coils

Coaxial coils share a common axis, with one coil nested inside the other (or placed end-to-end along the axis). For two tightly wound solenoids of the same length $l$ sharing a common core area, the mutual inductance is:

$M = \frac{\mu_0 N_1 N_2 \pi r^2}{l}$

where $N_1$ and $N_2$ are the turn counts and $r$ is the radius of the inner coil (the cross-sectional area that captures the shared flux). This is the same geometry used in most power transformers.

Coplanar coils

When two circular coils lie in the same plane (side by side), the mutual inductance between single-turn coils separated along their common plane can be approximated as:

$M = \frac{\mu_0 N_1 N_2 \pi r_1^2 r_2^2}{2(r_1^2 + r_2^2)^{3/2}}$

where $r_1$ and $r_2$ are the coil radii. This approximation holds when the coils are small relative to their separation. Coplanar arrangements appear in wireless power transfer pads and RFID tag readers.

Energy in coupled inductors

Mutual inductance changes how energy is stored and exchanged in a system of inductors, so you can't just add up the self-energies and call it a day.

Stored energy

The total magnetic energy in a pair of coupled inductors carrying currents $I_1$ and $I_2$ is:

$W = \frac{1}{2} L_1 I_1^2 + \frac{1}{2} L_2 I_2^2 + M I_1 I_2$

The first two terms are the familiar self-energy contributions. The third term is the interaction energy due to mutual inductance.

The sign of $M I_1 I_2$ depends on whether the currents produce fluxes that aid or oppose each other. If the fluxes reinforce, the interaction term is positive and the system stores more energy; if they oppose, it's negative.

Because stored energy must be non-negative for any combination of currents, you can show that $M^2 \leq L_1 L_2$, which is exactly the constraint captured by $k \leq 1$.

Energy transfer

When the current in one inductor changes, the induced EMF in the coupled inductor drives a current there, transferring energy between the two circuits. The rate of energy transfer scales with both $M$ and the rate of change of current $dI/dt$.

This mechanism is the operating principle behind transformers and wireless power transfer: AC current in the primary continuously pushes energy into the secondary through the shared magnetic field.

Mutual inductance applications

Transformers

A transformer consists of two or more coils wound on a shared magnetic core. The primary coil connects to the source, and the secondary coil connects to the load. By exploiting mutual inductance, a transformer can step voltages up or down, match impedances, and provide galvanic isolation between circuits.

Ideal vs. real transformers

An ideal transformer assumes $k = 1$ (perfect coupling), zero winding resistance, infinite core permeability, and no core losses. Under these assumptions:

• The voltage ratio equals the turns ratio: $V_2 / V_1 = N_2 / N_1$.
• Power is perfectly conserved: $V_1 I_1 = V_2 I_2$.

Real transformers deviate from this in several ways:

• Core losses (hysteresis and eddy currents) dissipate energy as heat.
• Copper losses ($I^2 R$) in the windings add resistive voltage drops.
• Leakage inductance means $k < 1$, so not all flux couples to the secondary.
• Finite core permeability requires a magnetizing current to establish the core flux.

Good transformer design aims to minimize all of these to approach ideal behavior.

Inductive power transfer

Inductive power transfer (IPT) wirelessly transmits power using mutual inductance. A primary coil driven by AC current creates a time-varying magnetic field. A secondary coil nearby captures part of that field and converts the induced EMF into usable power for a load or battery.

Because the coils aren't wound on a shared core, $k$ is typically much lower than in a transformer (often 0.1 to 0.5). Resonant tuning of both coils at the operating frequency compensates for weak coupling and improves efficiency. Applications include electric vehicle charging, implantable medical devices, and consumer electronics.

Wireless charging

Wireless charging is a consumer-facing application of IPT. A charging pad contains a primary coil, and the device holds a secondary coil. Standards like Qi (used by most smartphones) define coil dimensions, operating frequencies (typically around 100-200 kHz), and communication protocols so that different manufacturers' devices and pads remain compatible.

Mutual inductance vs. self-inductance

Both quantities are measured in henries and both arise from the interaction of currents and magnetic fields, but they describe different things.

• Self-inductance $L$ is a property of a single inductor. It relates the EMF induced in the same loop to the rate of change of its own current: $\mathcal{E} = -L \, dI/dt$. Self-inductance is always positive.
• Mutual inductance $M$ is a property of a pair of inductors. It relates the EMF induced in one loop to the rate of change of current in the other: $\mathcal{E}_2 = -M \, dI_1/dt$. Mutual inductance can be positive, negative, or zero depending on the relative orientation and winding sense.

For stored energy, the self-inductance contribution is $\frac{1}{2}LI^2$ (always positive), while the mutual term $MI_1I_2$ can add or subtract energy depending on current directions.

Mutual inductance in AC circuits

In AC circuits, mutual inductance introduces coupling between mesh equations, so you can't analyze each inductor in isolation.

Impedance matrix

For a pair of coupled inductors driven at angular frequency $\omega$, the voltage-current relationship can be written in matrix form:

$\begin{pmatrix} V_1 \\ V_2 \end{pmatrix} = \begin{pmatrix} R_1 + j\omega L_1 & j\omega M \\ j\omega M & R_2 + j\omega L_2 \end{pmatrix} \begin{pmatrix} I_1 \\ I_2 \end{pmatrix}$

The diagonal entries are the self-impedances, and the off-diagonal entries $j\omega M$ are the mutual impedances. This matrix formulation lets you solve for voltages and currents using standard linear algebra.

Equivalent circuits

To use familiar circuit analysis tools (Kirchhoff's laws, node/mesh analysis), you can replace coupled inductors with an equivalent network of uncoupled elements.

• T-network: Three inductors in a T configuration with values $(L_1 - M)$, $(L_2 - M)$, and $M$. This works when the dot convention gives a positive $M$.
• Pi-network: The dual of the T-network, sometimes more convenient depending on the circuit topology.
• Ideal transformer model: A leakage inductance in series with an ideal transformer, useful for analyzing real transformers where you want to separate the coupling from the imperfections.

Each representation is mathematically equivalent; you pick whichever makes the analysis simplest.

Mutual inductance measurement

Direct measurement methods

Direct methods apply a known excitation to one inductor and measure the response in the other.

1. Voltage method: Drive a known AC current through the primary coil and measure the open-circuit voltage across the secondary. Then $M = V_2 / (j\omega I_1)$, so $|M| = V_2 / (\omega I_1)$.
2. Current method: Drive the primary and measure the short-circuit current in the secondary, then extract $M$ from the circuit equations.

These methods give accurate results but require careful shielding and calibration to avoid stray coupling and measurement error.

Indirect calculation techniques

Indirect methods measure related quantities and compute $M$ from them.

1. Series-aiding/opposing method: Measure the total inductance of the two coils connected in series with fluxes aiding ($L_{\text{aid}} = L_1 + L_2 + 2M$) and opposing ($L_{\text{opp}} = L_1 + L_2 - 2M$). Then $M = (L_{\text{aid}} - L_{\text{opp}}) / 4$. This only requires a standard LCR meter.

2. Resonance method: Place the coupled inductors in a resonant circuit and measure the resonant frequency, from which $M$ can be extracted using the known capacitance and self-inductances.

Indirect techniques are often more practical in a lab setting, though their accuracy depends on how well you know the other parameters.