Coupling coefficient calculator, with the k-vs-gap curve.

The coupling coefficient k is the fraction of one coil’s magnetic flux that links the other, defined as k = M/√(L1·L2), and it always lies between 0 and 1. A value near 1 means almost all flux is shared, as in a transformer; wireless power links typically run from about 0.01 for loosely coupled resonant systems to around 0.5 for closely spaced charging pads. Because k sets how much power a link can transfer and how efficiently, it is the first number to check in any coil design. This calculator gives k two ways: from measured or simulated L1, L2 and M, or straight from coil geometry, computing both self-inductances with thin-wire formulas and the mutual inductance with the exact Neumann integral. The chart shows how quickly k falls as the gap grows. AirInduct Sim, the full simulator, extends this to spiral windings, misalignment sweeps and field maps.

Mutual inductance M comes from the Neumann integral, exactly as in the mutual inductance calculator: Maxwell’s elliptic-integral formula for coaxial circles, an exact side-by-side integral for squares, and converged quadrature for offset or mixed pairs.

Self-inductance of each coil uses the standard thin-wire formulas for a loop of round wire (wire radius r_w), scaled by N² for a tightly bundled winding:

• These are the high-frequency forms (current on the wire surface). At DC, replace −2 with −1.75 and −0.774 with −0.524; the difference is a few percent.
• Valid when the loop is much larger than the wire (R ≳ 10 r_w); the calculator warns otherwise.
• Multi-turn coils are modelled as a tightly packed bundle: N² scaling with the winding’s geometric-mean radius (within ~4 % of a turn-by-turn sum). A spread-out planar spiral has lower L than this.
• The chart recomputes k over a range of gaps with your sizes, offset and wire.

Two identical circular coils, radius 50 mm, 10 turns of 1 mm diameter wire (r_w = 0.5 mm), coaxial. Each coil has L = 24.04 µH. As the gap opens:

Doubling the gap from 50 mm to 100 mm (one to two radii) cuts k by a factor of about three and a half, which is why coil size has to grow with transfer distance.

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

• Spiral (DTC) and Helmholtz windings with each turn at its true radius, so L and k reflect the real layout.
• Sweeps of k over gap and misalignment in one run, with the field plots behind each point.
• Q-factor, Litz-wire AC resistance and matching-network design from the same coil model.
• An optimizer that finds the geometry for a target k within your size limits.

• Mutual inductance calculatorM between two circular or square coils from turns, size, axial gap and lateral offset — or from k, L1 and L2.
• 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.