Circular Helmholtz Coil Pair
A circular Helmholtz coil pair is two matched multi-turn circular coils on a shared axis — the classical arrangement found in physics textbooks. AirInduct Sim solves it with the closed-form Neumann expression in complete elliptic integrals, so it is the fastest and most hand-checkable case in the simulator.
Worked example
Inputs. Transmitter: 5 co-located circular turns at 50 mm radius. Receiver: 5 co-located circular turns at 50 mm radius. 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 | 2.692 | 7.359 | 7.359 | 0.366 |
| 30 mm | 1.162 | 7.359 | 7.359 | 0.158 |
Going from 10 mm to 30 mm lowers k from 0.366 to 0.158, a 57% drop. Because no grid is involved, these figures are reproducible to the last digit and can be checked against a hand calculation in a few lines.
How it was computed. This pair takes the closed-form path: M from the Neumann integral in complete elliptic integrals K and E, and each self-inductance from Maxwell’s thin-loop formula with a 0.5 mm wire radius. No grid is involved. k = M / √(L_tx · L_rx).
When it's the right choice
- Benchmarking, calibration and coursework where the analytic answer is known
- Round coils where placement rotation about the axis must not matter
- Fast first estimates before moving to spiral or square variants
Circular vs square Helmholtz pair
The circular Helmholtz pair is the right reference case whenever you need a number you can verify by hand, and AirInduct Sim evaluates it with the closed-form Neumann expression in complete elliptic integrals rather than a grid. That makes it the fastest topology in the simulator and the natural starting point for coursework, calibration and benchmarking. Its square neighbour, which lives at /helmholtz, trades that analytic purity for footprint: with the same 50 mm half-dimension and five turns per side, the square pair reaches M = 3.28 µH at a 10 mm gap versus 2.69 µH here, because a square of side 100 mm encloses roughly 27% more area than a circle of radius 50 mm. The coupling coefficient ends up almost identical (0.366 circular, 0.370 square), so the choice is rarely about k. Pick the circular pair for round coils, rotational symmetry and hand-checkable answers; pick the square pair when the enclosure, PCB outline or test bay is rectangular.
Watch out for
- Uniformity degrades quickly once coil spacing departs from the coil radius
- Self-inductance here uses Maxwell’s thin-loop formula, so it does not capture the winding cross-section of a thick multi-layer coil
Neighbouring topologies
Circular DTC-to-Helmholtz Coil Pair
Earbud, stylus and wearable pads. At the worked-example geometry: k = 0.390 at 10 mm.
Swap the receiver to DTCCircular Helmholtz-to-DTC Coil Pair
Thin tags, implants, smart cards. At the worked-example geometry: k = 0.389 at 10 mm.
Switch to a square outlineSquare Helmholtz Coil Pair
Rectangular test rigs and fixtures. At the worked-example geometry: k = 0.369 at 10 mm.
All eight pairs side by side, with coupling at both gaps: coil topology comparison.
Check the numbers
Run the Circular Helmholtz Coil Pair at your geometry
Frequently asked questions
Why does the circular Helmholtz pair solve faster than the other topologies?
Two coaxial circular loops have an exact mutual-inductance formula in complete elliptic integrals, so AirInduct Sim skips the field grid and receiver-area integration that the other seven topologies need. It is typically 50–100× faster than the grid path at moderate grid sizes.
How can I check the circular Helmholtz result by hand?
Use M = Nt·Nr·μ0·√(R1·R2)·[(2/m − m)·K(m²) − (2/m)·E(m²)] with m² = 4·R1·R2 / ((R1 + R2)² + h²). For R1 = R2 = 50 mm, five turns each and h = 10 mm, that gives 2.69 µH, matching the simulator. The Helmholtz coil calculator and mutual inductance calculator do the same arithmetic.
What coupling coefficient does a 50 mm-radius circular Helmholtz pair reach?
With five turns per side, k = 0.366 at a 10 mm gap and 0.158 at 30 mm. Self-inductance is 7.36 µH per coil from Maxwell’s thin-loop formula with a 0.5 mm wire radius; a thicker wire lowers L slightly and nudges k up.