Mutual inductance calculator, from coil geometry.

Mutual inductance M is the voltage induced in one coil per unit rate of change of current in the other, and for two air-core coils it depends only on geometry: turns, size, shape, the gap between them and any lateral misalignment. It is measured in henries and equals the magnetic flux linking the receiver per ampere flowing in the transmitter. Most online mutual inductance calculators only rearrange M = k√(L1·L2), which means you already need k. This calculator works from the coils themselves: it evaluates the Neumann integral directly, using Maxwell’s elliptic-integral formula for coaxial circular loops and an exact or adaptive integral for square and misaligned coils. The result is the magnetostatic mutual inductance of filament coils, which holds at typical wireless power frequencies. AirInduct Sim uses the same first-principles approach in its full simulator, adding spiral and multi-turn layouts, field maps and the circuit around the link.

Every result is the Neumann double integral over the two coil outlines, multiplied by the turn counts:

Coaxial circular loops (radii R₁, R₂, gap d) use Maxwell’s closed form with complete elliptic integrals evaluated by the arithmetic–geometric mean:

Square loops (side a, sides parallel): perpendicular sides contribute nothing, and each pair of parallel sides has an exact closed-form integral, so square–square M is exact for any gap and offset.

Offset circular loops and circle–square pairs: the inner integral is the exact vector potential of the circular loop (again via K and E); the outer integral runs around the other coil with spectrally convergent quadrature, doubled until successive results agree to better than 1 part in 10⁶.

• Coil 1 sits at z = 0; coil 2 is in a parallel plane at the gap, offset along x (along a side for squares).
• Each coil is N coincident filament turns: valid when the winding cross-section is small compared with the gap.
• Air core, no nearby metal or ferrite, magnetostatic (wavelength much larger than the coils).
• μ₀ = 4π × 10⁻⁷ H/m. Inputs in mm; results shown with SI prefixes to 4 significant figures.

Two identical circular coils, 10 turns each, radius 50 mm, 20 mm apart:

The same gap with other shapes:

A half-radius misalignment costs about a quarter of the mutual inductance here. Divide M by √(L₁L₂) to get k — the coupling coefficient calculator does that with self-inductances included.

This calculator is free to use and answers one question at a time. The full AirInduct Sim simulator takes the same physics further:

• Planar spiral (DTC) and Helmholtz windings modelled turn by turn, not as one lumped filament.
• Full 3D field maps — surface, heatmap and contour — so you can see where the flux goes.
• Parameter sweeps over gap, offset and size, and an optimizer that searches for the best geometry.
• Matching networks, rectifier and thermal analysis, plus PDF, KiCad, DXF and SPICE export.

• Coupling coefficient calculatork from coil geometry (with both self-inductances) or from L1, L2 and M, plus a k-versus-gap curve.
• Helmholtz coil calculatorCentre field, required current and optimal spacing for circular and square Helmholtz pairs, with axial uniformity.
• Square Helmholtz coil calculatorSquare pairs at the 0.5445·a spacing: centre field, current for a target field and the uniform region.
• WPT compensation capacitor calculatorSeries–series and series–parallel resonant capacitors from L1, L2, M and frequency, with capacitor voltage stress.
• WPT efficiency calculatorMaximum link efficiency and figure of merit U = k√(Q1Q2), the optimal load and the k needed for a target η.