Square Helmholtz → Helmholtz
A square Helmholtz pair puts two matched multi-turn square coils on a common axis, producing the most uniform field region of any topology in the simulator. It is the reference case: if you are checking the solver or teaching the physics, start here.
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
- Laboratory and teaching setups where field uniformity matters more than packaging
- Sanity-checking a result against the textbook Helmholtz condition
- Applications that tolerate a transmitter surrounding the receiver volume
Watch out for
- True Helmholtz uniformity assumes the separation equals the coil radius — deviate and the central field flattens differently
- Square geometry concentrates field at the corners, so peak field is higher than a circular pair of the same area
Run your geometry.
How is mutual inductance calculated for a square Helmholtz to Helmholtz pair?
AirInduct evaluates the Neumann double integral between every transmitter turn and every receiver turn, then sums the contributions. Square turns are integrated segment by segment along each of the four straight sides, using the closed-form Biot–Savart result for a finite straight conductor. 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.