Circular Helmholtz → Helmholtz
Two matched circular multi-turn coils on a shared axis. This is the classical Helmholtz arrangement and the most analytically tractable case in the solver, evaluated directly with complete elliptic integrals.
The simulator computes the field from the Biot–Savart law and the mutual inductance from the Neumann double integral, then reports the coupling coefficient k for the geometry you enter. Nothing is meshed and nothing is fitted, so the result is reproducible to the digit.
Use it for
- Benchmarking, calibration and coursework where the analytic answer is known
- Designs needing a large, uniform central field volume
- Any case where you want the fastest and most numerically exact path through the solver
Watch out for
- Uniformity degrades quickly once separation departs from the coil radius
- A surrounding transmitter is often impractical in a real product enclosure
Run your geometry.
How is mutual inductance calculated for a circular Helmholtz to Helmholtz pair?
AirInduct evaluates the Neumann double integral between every transmitter turn and every receiver turn, then sums the contributions. Circular turns are evaluated with complete elliptic integrals (K and E), which is why the circular cases are the fastest and the most numerically exact in the solver. Because this is a closed-form evaluation rather than a meshed field solve, the same geometry always returns the same number and there is no discretisation error to tune away.
What coupling coefficient should I expect from this topology?
There is no single figure — k depends on the radii, turn counts, separation and alignment you choose, and it falls off steeply with distance. That is precisely why this page links to the simulator rather than quoting a number: enter your geometry and read k directly. As a rough orientation, closely spaced pairs of similar size can reach k above 0.5, while a gap comparable to the coil radius typically drops k below 0.1.
Do I need to install anything?
No. The solver runs from the browser and the free tier covers individual research and coursework use. Desktop and mobile builds run the identical kernels if you prefer a local install.
Is this accurate enough to replace a full-wave FEM tool?
For inductive WPT in the magnetostatic regime — coils small relative to a wavelength, no dominant ferrite or lossy material in the field volume — the closed-form result is exact, not approximate. If your design depends on shielding, ferrite loading, strong proximity effects in litz bundles, or radiative behaviour, a full-wave or FEM tool is the right instrument and we will tell you so.