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. 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. These figures are reproducible to the last digit.
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. 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 simplicity 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 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
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 with a 0.5 mm wire radius; a thicker wire lowers L slightly and nudges k up.