7.7 Mutual inductance

Definition of mutual inductance

Mutual inductance describes how a changing current in one circuit induces a voltage in a nearby circuit through electromagnetic coupling. It quantifies the strength of that interaction and is central to how transformers, wireless chargers, and many other coupled systems work.

Two coils don't need to touch for one to influence the other. As long as the magnetic field from one coil passes through the other, changing the current in the first coil will induce an EMF in the second. Mutual inductance, denoted $M$, tells you exactly how much EMF you get per unit rate of change of current.

Magnetic flux linkage

Magnetic flux linkage measures the total magnetic flux threading through all the turns of a coil. For a single turn, you multiply the magnetic flux density $B$ by the area $A$ of the loop (accounting for the angle between the field and the loop's normal). For a coil with $N$ turns, the total flux linkage is:

$\Lambda = N\Phi$

• Flux linkage depends on the relative orientation and proximity of the coils
• Measured in weber-turns (Wb·turns), which is equivalent to volt-seconds (V·s)
• The greater the flux linkage between two coils, the stronger their mutual inductance

Faraday's law of induction

Faraday's law is the reason mutual inductance exists. It states that the induced EMF in a circuit equals the negative rate of change of magnetic flux linkage through that circuit:

$\varepsilon = -N\frac{d\Phi}{dt}$

Here, $\varepsilon$ is the induced EMF, $N$ is the number of turns, and $\frac{d\Phi}{dt}$ is the rate of change of magnetic flux through each turn. The negative sign reflects Lenz's law: the induced EMF opposes the change that caused it.

For mutual inductance specifically, if the current $i_1$ in coil 1 is changing, the EMF induced in coil 2 is:

$\varepsilon_2 = -M\frac{di_1}{dt}$

This is the defining relationship. $M$ connects the rate of current change in one coil to the voltage that appears across the other.

Units of mutual inductance

Mutual inductance is measured in henries (H), named after Joseph Henry. One henry means that a current changing at 1 ampere per second in one coil induces 1 volt in the other coil.

$1\text{ H} = 1\frac{\text{V·s}}{\text{A}} = 1\frac{\text{Wb}}{\text{A}}$

In practice, most mutual inductances you'll encounter are much smaller than 1 H. Common subunits include millihenries (mH, $10^{-3}$ H) and microhenries (μH, $10^{-6}$ H).

Factors affecting mutual inductance

Several physical and geometric factors determine how strongly two coils are coupled. Knowing these helps you predict and control mutual inductance in real designs.

Coil geometry

• The shape and size of each coil matter. Larger coil areas capture more flux, increasing $M$
• A coaxial arrangement (coils sharing the same axis) maximizes mutual inductance because the field from one coil passes most directly through the other
• The aspect ratio (length-to-diameter ratio) of a coil affects how its magnetic field spreads out, which in turn affects coupling

Number of turns

More turns in either coil means more flux linkage, which increases mutual inductance. The relationship is proportional to the product of the turn counts:

$M \propto N_1 N_2$

So doubling the turns on coil 1 doubles $M$, and doubling turns on both coils quadruples it. There are practical limits, though: more turns add resistance and parasitic capacitance, which can hurt performance at higher frequencies.

Core material

The material inside or around the coils has a huge effect on $M$.

• Ferromagnetic cores (iron, ferrite) concentrate the magnetic field and dramatically increase mutual inductance. The relative permeability $\mu_r$ of the core material directly multiplies the field strength.
• Air-core designs have much lower mutual inductance but avoid problems like core saturation and hysteresis losses, making them better for high-frequency applications.
• At high field strengths, ferromagnetic cores saturate, and their permeability drops. This is a real design constraint for transformers.

Distance between coils

Mutual inductance drops off quickly as you move coils apart. The magnetic field weakens with distance, so less flux from coil 1 threads through coil 2. This is a critical factor in wireless power transfer and RFID, where the gap between transmitter and receiver directly limits performance.

Calculation of mutual inductance

Self-inductance vs. mutual inductance

Self-inductance $L$ describes how a coil's own changing current induces an EMF in itself. Mutual inductance $M$ describes how a changing current in one coil induces an EMF in a different coil.

They're connected by the coupling coefficient:

$M = k\sqrt{L_1 L_2}$

This equation is worth memorizing. It tells you that $M$ can never exceed $\sqrt{L_1 L_2}$, since $k$ maxes out at 1.

Coupling coefficient

The coupling coefficient $k$ is a dimensionless number between 0 and 1 that describes how effectively two coils share magnetic flux.

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

• $k = 1$: Perfect coupling. All flux from one coil links the other. This is the ideal (never quite achieved in practice).
• $k = 0$: No coupling at all. The coils are magnetically independent.
• Tightly wound transformers with ferromagnetic cores can reach $k \approx 0.95$ to $0.99$
• Air-core coils at a distance might have $k < 0.1$

The value of $k$ depends on coil geometry, orientation, separation, and core material.

Neumann's formula

For a more fundamental calculation, Neumann's formula gives the mutual inductance between two arbitrary current loops:

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

Here, $\mu_0$ is the permeability of free space, the integrals run over both loop paths, and $r$ is the distance between each pair of line elements. This double line integral is rarely solved by hand for complex geometries, but it forms the theoretical basis for numerical methods and simpler approximations.

Applications of mutual inductance

Transformers

Transformers are the most common application of mutual inductance. A changing current in the primary winding creates a changing magnetic flux that induces an EMF in the secondary winding.

• Step-up transformers increase voltage (more turns on the secondary)
• Step-down transformers decrease voltage (fewer turns on the secondary)
• Isolation transformers provide electrical isolation between circuits while transferring power
• Transformer efficiency depends directly on the coupling coefficient. Higher $k$ means less energy lost to leakage flux.

Wireless power transfer

Wireless chargers for phones and electric vehicles use mutual inductance to transfer energy across an air gap. Since there's no shared core, the coupling coefficient is lower than in a transformer, so these systems often use resonant inductive coupling: both coils are tuned to the same resonant frequency, which greatly improves efficiency even at modest $k$ values.

RFID technology

RFID systems use mutual inductance for near-field communication. The reader generates an alternating magnetic field, and a passive RFID tag nearby picks up energy through mutual inductance. The tag then modulates the coupling to send data back to the reader. Reading distance is limited by how quickly mutual inductance falls off with distance.

Mutual inductance in circuits

Series vs. parallel connections

When two mutually coupled inductors are connected in series, the total inductance depends on whether their fields aid or oppose each other:

• Aiding (fields in same direction): $L_{\text{total}} = L_1 + L_2 + 2M$
• Opposing (fields in opposite directions): $L_{\text{total}} = L_1 + L_2 - 2M$

The sign depends on the dot convention, which indicates the relative winding direction of the coils. For parallel connections, the analysis is more involved because you must account for coupling polarity, and the total inductance can either increase or decrease depending on the configuration.

Equivalent circuit models

To simplify analysis of mutually coupled inductors, two standard equivalent circuits are used:

• T-equivalent circuit: Replaces the coupled pair with three inductors arranged in a T-shape. The series elements are $(L_1 - M)$ and $(L_2 - M)$, and the shunt element is $M$.
• π-equivalent circuit: Uses two inductors and an ideal transformer representation.

Both models give identical results and are chosen based on which makes a particular analysis easier.

Energy stored in coupled inductors

The total energy stored in a system of two coupled inductors is:

$E = \frac{1}{2}L_1 i_1^2 + \frac{1}{2}L_2 i_2^2 + M i_1 i_2$

The first two terms are the self-energy of each inductor. The third term, $M i_1 i_2$, is the mutual energy. It can be positive or negative depending on the relative directions of the currents (coupling polarity). For the total energy to be physically meaningful (non-negative), this places a constraint: $M \leq \sqrt{L_1 L_2}$, which is consistent with $k \leq 1$.

Measurement techniques

Bridge methods

Classical bridge circuits like the Maxwell-Wien bridge and Carey Foster bridge measure mutual inductance by balancing it against known standards. These methods offer high accuracy but require careful setup and are best suited for laboratory environments.

Resonance methods

These methods place the coupled inductors in an LC circuit and measure how mutual coupling shifts the resonant frequency. A Q-meter can determine mutual inductance with good accuracy, especially for high-Q (low-loss) inductors. This approach works well when you need measurements at a specific operating frequency.

Digital measurement systems

Modern LCR meters and impedance analyzers automate the measurement process. They use digital signal processing to measure mutual inductance across a wide frequency range and often report additional parameters like the coupling coefficient and quality factor in a single sweep.

Mutual inductance in electromagnetic theory

Maxwell's equations

Mutual inductance is rooted in Maxwell's equations. Faraday's law directly gives you the induced EMF from a changing magnetic flux, which is the physical basis for mutual inductance. Ampère's law (with Maxwell's displacement current correction) describes how currents and changing electric fields produce magnetic fields. Together, these equations fully describe the electromagnetic coupling between circuits.

Magnetic vector potential

The magnetic vector potential $\mathbf{A}$ is defined by:

$\mathbf{B} = \nabla \times \mathbf{A}$

Using $\mathbf{A}$ can simplify mutual inductance calculations, especially in complex geometries. Neumann's formula, for instance, can be derived from the vector potential. In multi-conductor systems, expressing fields in terms of $\mathbf{A}$ often makes the math more tractable.

Reciprocity theorem

A fundamental result: the mutual inductance between two circuits is the same regardless of which one carries the source current.

$M_{12} = M_{21}$

This means it doesn't matter whether you drive current through coil 1 and measure the effect on coil 2, or vice versa. You get the same $M$. This symmetry simplifies the analysis of any coupled system and follows from the symmetry properties of the underlying electromagnetic Green's functions.

Challenges and limitations

Stray capacitance effects

Parasitic capacitance between windings creates unintended resonances at high frequencies, altering the effective mutual inductance. This is more pronounced in tightly coupled systems and must be accounted for in high-frequency transformer design.

Frequency dependence

Mutual inductance can vary with frequency due to several effects:

• Skin effect pushes current to the surface of conductors, changing the effective inductance
• Proximity effect redistributes current in nearby conductors
• Eddy currents in core materials cause additional losses
• The complex permeability of magnetic materials changes with frequency

Accurate high-frequency models need to incorporate these effects.

Magnetic saturation

Ferromagnetic cores behave nonlinearly at high field strengths. Once the core saturates, its permeability drops sharply, reducing mutual inductance and distorting waveforms. Designers must check the core's B-H curve to ensure the operating point stays in the linear region under expected conditions.

Advanced concepts

Mutual inductance matrices

When a system has $N$ coupled inductors, the coupling is described by an $N \times N$ inductance matrix. Diagonal elements are the self-inductances $L_i$, and off-diagonal elements are the mutual inductances $M_{ij}$. This matrix is symmetric (because $M_{ij} = M_{ji}$) and is essential for analyzing multi-winding transformers and coupled inductor networks.

Leakage inductance

Not all magnetic flux from one winding links the other. The flux that "leaks" corresponds to leakage inductance, which acts as a series inductance that doesn't participate in energy transfer between windings. Leakage inductance affects voltage regulation and transient response in transformers. It can be reduced through interleaved winding techniques and careful core geometry.

Mutual inductance in multi-winding systems

Systems with three or more coupled windings (like three-phase transformers) require tracking multiple mutual inductance terms. Each pair of windings has its own $M$, and the full behavior is captured by the inductance matrix. Analyzing these systems involves solving coupled differential equations for the currents and flux distributions in each winding.