What is the coupling coefficient k?
Two coils with self-inductances L₁ and L₂ and mutual inductance M have a coupling coefficient
Physically, k is the geometric mean of the fractions of each coil’s flux that links the other. A power transformer with a shared ferrite core has k close to 1. A wireless power link has a large air gap, so most of each coil’s flux returns without reaching the other coil: k is well below 1, and the unshared flux appears in the circuit as leakage inductance, L₁(1 − k²) seen from the primary.
Because M scales with N₁N₂ and each self-inductance with N², the turn counts cancel to first order. That makes k the cleanest single number for comparing coil geometries, and the first thing to check in any design. The mutual inductance article covers M in depth.
How do you calculate the coupling coefficient?
There are three routes, depending on what you already have:
- From L₁, L₂ and M (from a datasheet, a measurement or a field solver): divide, as in the formula above.
- From geometry. Compute M with the Neumann integral (Maxwell’s elliptic-integral formula for coaxial circular loops) and each L with a self-inductance formula, for example L = N²μ₀R[ln(8R/a) − 2] for a compact circular coil of wire radius a.
- By measurement. Measure coil 1 with coil 2 open (L_oc) and shorted (L_sc):
Most online coupling calculators only rearrange k = M/√(L₁L₂), which assumes you already know M. In practice M is the hard part, because it depends on the full three-dimensional geometry: coil shapes and sizes, the gap, and lateral or angular misalignment. For air-core coils it can be computed exactly from the Neumann integral, with no meshing, which is how both the free coupling coefficient calculator on this site and AirInduct Sim work. AirInduct Sim reports M, L_tx, L_rx and k together for eight coil topologies, including circular and square Helmholtz pairs, planar descending-turn spirals (DTC) and the cross pairs between them, and can sweep the gap or offset to show how k changes. When the coils sit on ferrite or near metal, the air-core value is a starting point only, and a finite-element model or a measurement should confirm the final number.
What are typical coupling coefficient values?
Rather than quote numbers without a geometry, the table below computes k for representative air-core geometries with the same exact method used by the calculators (filament coils, thin-wire self-inductance). Ferrite backing, which most commercial pads use, raises k above these values.
| Scenario | Geometry | k |
|---|---|---|
| Phone-size pad | 2 × circular, R = 20 mm, 10 turns, 0.5 mm wire, 5 mm gap | 0.417 |
| Same pad, 15 mm gap | as above, 15 mm gap | 0.16 |
| Vehicle-pad scale (air core) | 2 × square, 500 mm side, 10 turns, 2 mm wire, 150 mm gap | 0.158 |
| Implant-scale link | R = 20 mm transmitter, R = 5 mm receiver, 5 turns each, 0.2 mm wire, 20 mm gap | 0.0165 |
| Mid-range resonant | 2 × circular, R = 300 mm, 5 turns, 3 mm wire, 1 m gap | 0.00684 |
Two patterns stand out. First, k depends mainly on the ratio of gap to coil size: the phone-size pad at a 15 mm gap and the vehicle-scale pad have almost the same k because the gap is a similar fraction of the coil size in both. Second, a small receiver facing a larger transmitter, as in implants and small sensors, has a low k even at a short gap, because most of the transmitter’s flux misses the receiver.
How does k change with distance and misalignment?
The chart shows k for two identical circular coils (radius 50 mm, 10 turns, 1 mm wire; L = 24.04 µH each), computed on this page from the exact Neumann mutual inductance. The blue curve varies the axial gap with the coils coaxial; the copper curve keeps a 20 mm gap and slides the receiver sideways.
Distance. While the gap is small compared with the coil radius, k falls gently. Around one radius it falls steeply, and far away the coils behave like two magnetic dipoles, so M, and therefore k, decays as the cube of distance. Between 100 mm and 150 mm in this example k drops by a factor of 2.58, heading towards the far-field dipole ratio (150/100)³ = 3.375. A useful rule follows directly: in this example k is already 0.1 when the gap equals the coil radius, so a link that needs k above about 0.1 needs coils at least as large as the transfer distance.
Lateral misalignment. Sliding the receiver sideways reduces the shared flux, then makes the receiver catch return flux of the opposite sign. k passes through zero and turns negative. The null position depends on gap and coil shapes, so check it for your own geometry with the mutual inductance calculator. Angular misalignment (tilt) also reduces k; Babic and Akyel give closed-form results for inclined circular coils.
Larger transmitters than receivers, or a transmitter made of several overlapping coils, widen the region where k stays near its peak at the cost of lower peak k.
Is a higher coupling coefficient always better?
Usually, but not without limit. Efficiency depends on the figure of merit U = k√(Q₁Q₂), so higher k lets you reach a given efficiency with lower-Q coils (see WPT efficiency and kQ). But two resonant circuits that are coupled more tightly than their loaded Q allows become over-coupled: the transfer response splits into two peaks either side of the design frequency, and the input phase can have several zero crossings (bifurcation). Sample, Meyer and Smith describe this frequency splitting experimentally; Wang, Covic and Stielau derive the bifurcation conditions for the basic compensation networks. The compensation topologies article covers how to design around it.
In practice, a link that must work over a range of gaps sees k vary by several times, so the design target is usually a k range, not a single value. Sweeping the gap and offset across the expected range, then designing the compensation and control for the worst case, is the safer workflow.
References
- G. A. Covic and J. T. Boys, “Inductive power transfer,” Proceedings of the IEEE 101(6), 1276–1289 (2013). doi.org/10.1109/JPROC.2013.2244536
- A. P. Sample, D. A. Meyer and J. R. Smith, “Analysis, experimental results, and range adaptation of magnetically coupled resonators for wireless power transfer,” IEEE Trans. Industrial Electronics 58(2), 544–554 (2011). doi.org/10.1109/TIE.2010.2046002
- C.-S. Wang, G. A. Covic and O. H. Stielau, “Power transfer capability and bifurcation phenomena of loosely coupled inductive power transfer systems,” IEEE Trans. Industrial Electronics 51(1), 148–157 (2004). doi.org/10.1109/TIE.2003.822038
- 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
- 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