Square Helmholtz Coil Pair
A square Helmholtz coil pair is two matched multi-turn square coils on a common axis — each side winds all of its turns at the same half-side, so the full ampere-turns sit at the outer edge. In AirInduct Sim it is the square reference case and the best fit for rectangular fixtures, bays and test rigs.
Worked example
Inputs. Transmitter: 5 co-located square turns at 50 mm half-side. Receiver: 5 co-located square turns at 50 mm half-side. Coaxial, no lateral offset, 1 A at 6.78 MHz, air gaps of 10 mm and 30 mm.
| Air gap | M (µH) | L_tx (µH) | L_rx (µH) | k |
|---|---|---|---|---|
| 10 mm | 3.280 | 8.878 | 8.878 | 0.369 |
| 30 mm | 1.465 | 8.878 | 8.878 | 0.165 |
Going from 10 mm to 30 mm lowers k from 0.369 to 0.165, a 55% drop. Tripling the gap from 10 mm to 30 mm cuts mutual inductance from 3.28 µH to 1.47 µH. Both coils are identical, so L_tx = L_rx and k is simply M / L.
How it was computed. Each square turn is integrated side by side with the closed-form Biot–Savart result for a finite straight conductor; the axial field is then integrated over the receiver area on a 1 mm grid. k = M / √(L_tx · L_rx).
When it's the right choice
- Rectangular test fixtures, coil bays and calibration rigs
- Sanity-checking a square-coil design before adding spiral turns
- Teaching how a straight-sided winding differs from a round one
Square vs circular Helmholtz pair
Choose the square Helmholtz pair over the circular one when your fixture, bay or test rig is rectangular, because at the same 50 mm half-dimension the square coil encloses about 27% more area and AirInduct Sim reports a higher mutual inductance. In the worked example above, the square pair gives M = 3.28 µH at a 10 mm gap against 2.69 µH for the circular pair of radius 50 mm, yet the coupling coefficient barely moves (0.370 versus 0.366) because the self-inductance of the square coils rises in step. Swap to the circular pair when you want the textbook reference: its mutual inductance comes from a closed-form elliptic-integral expression, so it is the case to use for checking hand calculations or teaching. Keep the square pair when corners matter — square turns put more conductor near the edges of a rectangular receiver, and the solver integrates every straight side exactly, so corner field concentration shows up in the field map instead of being averaged away.
Watch out for
- Classical Helmholtz uniformity assumes a coil spacing equal to the coil radius — at other gaps the central field flattens differently
- Field concentrates near the corners, so the local peak is higher than a circular pair of the same area
Neighbouring topologies
Square DTC-to-Helmholtz Coil Pair
Worktop panels, tool and robot docks. At the worked-example geometry: k = 0.396 at 10 mm.
Swap the receiver to DTCSquare Helmholtz-to-DTC Coil Pair
Rectangular docks with PCB receivers. At the worked-example geometry: k = 0.403 at 10 mm.
Switch to a circular outlineCircular Helmholtz Coil Pair
Calibration, teaching, benchmarking. At the worked-example geometry: k = 0.366 at 10 mm.
All eight pairs side by side, with coupling at both gaps: coil topology comparison.
Check the numbers
Run the Square Helmholtz Coil Pair at your geometry
Frequently asked questions
Is /helmholtz the square or the circular Helmholtz pair?
It is the square pair. The URL predates the circular variants and is kept so existing links keep working. The classic circular Helmholtz coil pair lives at /circular_helmholtz; both use five co-radial turns per side in the worked examples, so you can compare them number for number.
How does AirInduct Sim compute mutual inductance for square Helmholtz coils?
It evaluates the field of each straight side with the exact Biot–Savart expression for a finite wire, sums the four sides and all turns, and integrates the axial component over the receiver area on a 1 mm grid. For two 100 mm, 5-turn square coils at a 10 mm gap that gives M = 3.28 µH.
What coupling coefficient does a 100 mm square Helmholtz pair reach?
With 5 turns per side at 6.78 MHz, k = 0.370 at a 10 mm gap and 0.165 at 30 mm — a 55% drop for a threefold increase in gap. Larger coils hold k further out, so sweep the gap at your real dimensions in the simulator.