What is mutual inductance?
When current i₁ flows in coil 1 it produces a magnetic field, and part of that field threads coil 2. The total flux linkage of coil 2 (flux through each turn, summed over its N₂ turns) is proportional to i₁. The constant of proportionality is the mutual inductance:
M is measured in henries, like self-inductance. It is reciprocal (M₁₂ = M₂₁), so it does not matter which coil you drive. For coils in air, with no magnetic or conducting material nearby, M depends only on geometry: the shapes and sizes of the two windings, their turn counts, the axial gap between them and any lateral or angular misalignment.
In a wireless power link M is the quantity that actually transfers energy. The induced voltage jωM·I₁ drives the receiver, and the impedance the receiver reflects back into the transmitter is (ωM)²/Z₂. Its normalised form, the coupling coefficient k = M/√(L₁L₂), is what designers usually quote, but every circuit equation ultimately needs M itself.
What is the Neumann formula?
For two closed thin-wire loops C₁ and C₂, the mutual inductance is a double line integral around both loops, known as the Neumann formula (Jackson, ch. 5; Grover, ch. 1):
It follows from writing the flux through loop 2 as the circulation of the vector potential of loop 1, Φ = ∮ A·dl₂, and substituting the Biot–Savart vector potential A = (μ₀I/4π) ∮ dl₁/|r₁ − r₂|. Two properties are worth noting. The integrand contains only the dot product of the two line elements, so perpendicular segments contribute nothing. And the formula is symmetric in the two loops, which is why M is reciprocal. For compact windings of N₁ and N₂ turns, M scales as N₁N₂.
The Neumann formula is exact for thin filaments in a magnetostatic field, which makes it the reference method for air-core coils. It needs no mesh and no boundary at infinity: the only inputs are the two current paths. The practical difficulty is the 1/|r₁ − r₂| singularity when the loops come close, which makes naive numerical quadrature slow to converge. Production solvers therefore do as much of the integral analytically as possible. AirInduct Sim evaluates the Neumann integral in closed form wherever one exists, using complete elliptic integrals for circular loops and exact straight-segment formulas for square loops, and combines it with the Biot–Savart field for its 3D, heatmap and contour plots. Because the method is deterministic, the same geometry always returns the same M, which is useful when a result has to be reproduced in a report or a paper. Its limits are the magnetostatic, air-core assumptions, discussed below.
What is Maxwell's formula for two coaxial circular loops?
For two coaxial circular filaments of radii R₁ and R₂ separated axially by d, the Neumann integral can be done exactly. The result, due to Maxwell (Treatise, vol. 2, §701) and tabulated by Rosa and Grover and by Grover, is:
K and E are the complete elliptic integrals of the first and second kind. Here k is the elliptic modulus, not the coupling coefficient. Both integrals are evaluated to machine precision in a handful of steps with the arithmetic–geometric mean. When the loops are far apart (d ≫ R) the modulus is small and the bracket suffers numerical cancellation; it is better evaluated in the rearranged form (2/k)·[(1 − k²/2)K − E] with a series that avoids subtracting nearly equal numbers. In that limit the result tends to the magnetic dipole approximation:
As a check, for two single-turn 50 mm-radius loops 0.5 m apart, Maxwell’s formula gives 95.82 pH and the dipole approximation 98.7 pH, a difference of 3 %. Closer in, the dipole formula overestimates M badly, which is why the full expression matters for real wireless power gaps.
Babic and Akyel extended the closed-form approach to circular coils with lateral and angular misalignment; their papers are a common benchmark for numerical solvers.
How do you calculate M for square or misaligned coils?
Square loops. Each side is a straight segment, and only parallel sides contribute. The mutual inductance of two parallel straight filaments has an exact closed form (Grover), so the Neumann integral for two axis-aligned square loops reduces to a sum of eight exact terms, valid for any gap and lateral offset.
Laterally offset circles, or a circle paired with a square. No simple closed form exists, but the inner integral is still exact: it is the vector potential of the circular loop, itself expressed with K and E. Only the outer integral, around the second loop, has to be done numerically. Because the integrand is smooth and periodic, the trapezoidal rule converges very quickly, and the number of points can be doubled until successive answers agree to one part in a million.
Misalignment can reverse the sign. As the receiver slides sideways, part of it moves outside the transmitter and catches return flux. For two equal 50 mm-radius coils at a 20 mm gap, M falls to zero at a lateral offset of 82.2 mm (1.64·R) and becomes negative beyond it. That null is a real design hazard for charging pads, and one reason pads are designed and tested over a range of lateral offsets rather than at perfect alignment only.
Worked example: two 50 mm coils at a 20 mm gap
Take two identical circular coils, radius 50 mm, 10 turns each, coaxial and 20 mm apart. These are the default inputs of the free mutual inductance calculator, and every number below is computed on this page with the same code.
| Case | Result |
|---|---|
| Coaxial, Maxwell formula (N₁N₂ = 100) | 6.754 µH |
| Single-turn M (one filament pair) | 67.54 nH |
| Brute-force Neumann sum, 1,000 × 1,000 points | 67.54 nH (relative difference 3.6e-13) |
| Same coils, 25 mm lateral offset | 4.959 µH (−26.6 %) |
| Two 100 mm square coils, 10 turns, 20 mm gap | 8.268 µH |
| 50 mm-radius circle (10 turns) over 80 mm square (5 turns) | 2.922 µH |
With 1 mm diameter wire, each circular coil has a self-inductance of about 24.04 µH (thin-wire formula L = N²μ₀R[ln(8R/a) − 2]), so the coupling coefficient is k = M/L = 0.281. Driving the transmitter with 1 A RMS at 100 kHz induces an open-circuit voltage of V₂ = ωM·I₁ = 4.24 V RMS in the receiver.
The brute-force sum agrees with Maxwell’s formula to the precision shown, which is a useful sanity check when you implement either method yourself.
What limits the accuracy of these formulas?
The filament formulas above are exact for their model. Real coils depart from that model in four ways:
- Winding cross-section. Lumping N turns into one filament is accurate when the winding is small compared with the gap. A planar spiral, where turns step inward, should be modelled turn by turn: the mutual inductance is then the sum of M over every pair of turns.
- Magnetic and conducting material. Ferrite raises M and k; aluminium or steel nearby lowers them through eddy currents. Neither is in the air-core formula.
- Frequency. M is magnetostatic, which holds while the coil is much smaller than the wavelength and well below its self-resonant frequency.
- Current distribution. Skin and proximity effects move the current within the conductor, which matters mainly for very close, wide conductors.
Biot–Savart vs FEM explains when each limit bites and what to use instead. The validation page lists the closed-form identities AirInduct Sim is checked against.
References
- J. C. Maxwell, A Treatise on Electricity and Magnetism, vol. 2, Clarendon Press, Oxford (1873), §701. archive.org/details/electricandmag02maxwrich
- E. B. Rosa and F. W. Grover, “Formulas and tables for the calculation of mutual and self-inductance,” Bulletin of the Bureau of Standards 8(1), 1–237 (1912). doi.org/10.6028/bulletin.185
- F. W. Grover, Inductance Calculations: Working Formulas and Tables, Van Nostrand (1946); Dover reprint (2004), ISBN 978-0-486-49577-4. openlibrary.org/isbn/9780486495774
- J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley (1999), ISBN 978-0-471-30932-1. www.wiley.com/en-us/Classical+Electrodynamics%2C+3rd+Edition-p-9780471309321
- S. I. Babic and C. Akyel, “Calculating mutual inductance between circular coils with inclined axes in air,” IEEE Trans. Magnetics 44(7), 1743–1750 (2008). doi.org/10.1109/TMAG.2008.920251