Circular Helmholtz-to-DTC Coil Pair
A circular Helmholtz transmitter drives a flat circular planar-spiral (DTC) receiver whose turns step inward in one plane. It suits designs where the transmitter can be a compact wound coil but the receiver — a tag, implant or card — has to stay thin.
Worked example
Inputs. Transmitter: 5 co-located circular turns at 50 mm radius. Receiver: 5-turn circular planar spiral (DTC), 2 mm pitch, turns at 50, 48, 46, 44, 42 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 | 2.380 | 7.615 | 4.912 | 0.389 |
| 30 mm | 1.051 | 7.615 | 4.912 | 0.172 |
Going from 10 mm to 30 mm lowers k from 0.389 to 0.172, a 56% drop. The spiral receiver has 35% less self-inductance than the 7.6 µH transmitter, which is why k (0.389) ends up higher than for the matched circular Helmholtz pair even though M is lower.
How it was computed. The transmitter field comes from elliptic integrals for a circular loop; the receiver’s five spiral turns are each integrated over their own disc, so inner turns contribute less flux than outer ones. k = M / √(L_tx · L_rx).
When it's the right choice
- Powering a thin sensor, tag or card from a wound reader coil
- Implant and on-skin work where receiver thickness is the binding constraint
- Fixtures where the transmitter can sit around or under the working volume
Helmholtz-to-DTC vs DTC-to-Helmholtz
Reversing the pair — putting the planar spiral on the transmitter side instead of the receiver — leaves the mutual inductance essentially unchanged, which AirInduct Sim confirms numerically: 2.380 µH for this Helmholtz-to-DTC pair against 2.384 µH for the DTC-to-Helmholtz pair at a 10 mm gap. That is reciprocity at work; M is a property of the two windings, not of which one is driven. What does change is where the inductance sits. Here the transmitter carries the larger 7.6 µH winding and the receiver the smaller 4.9 µH spiral, so the receiver presents a lower impedance to its load and needs a larger resonating capacitor at the same frequency. Choose this direction when the device you are powering must stay thin — a sensor tag, implant or smart card — while the transmitter is free to be a compact multi-turn coil around or beneath the working volume. Choose the reverse when the pad must be flat and the device has room for a wound coil.
Watch out for
- A receiver smaller than the transmitter intercepts only part of the flux, so k falls well below the matched-size example
- Receiver tilt costs more coupling than lateral offset for a flat spiral
Neighbouring topologies
Circular DTC-to-DTC Planar Coil Pair
Qi-style phone and watch charging. At the worked-example geometry: k = 0.461 at 10 mm.
Swap the receiver to HelmholtzCircular Helmholtz Coil Pair
Calibration, teaching, benchmarking. At the worked-example geometry: k = 0.366 at 10 mm.
Switch to a square outlineSquare Helmholtz-to-DTC Coil Pair
Rectangular docks with PCB receivers. At the worked-example geometry: k = 0.403 at 10 mm.
All eight pairs side by side, with coupling at both gaps: coil topology comparison.
Check the numbers
Run the Circular Helmholtz-to-DTC Coil Pair at your geometry
Frequently asked questions
Does it matter which coil is the transmitter in a Helmholtz/DTC pair?
Not for mutual inductance, which is reciprocal: AirInduct Sim gives 2.380 µH for Helmholtz-to-DTC and 2.384 µH for DTC-to-Helmholtz at a 10 mm gap. It matters for packaging and impedance, because the side carrying the wound coil also carries the larger self-inductance.
How does AirInduct Sim model a circular planar-spiral receiver?
As concentric circular turns of descending radius — 50, 48, 46, 44 and 42 mm for a 2 mm pitch — each integrated over its own disc and summed. That captures the fact that inner turns link less transmitter flux than outer ones, which a single equivalent loop would overstate.
How fast does coupling fall with gap for a Helmholtz-to-DTC pair?
For 50 mm coils, k drops from 0.389 at 10 mm to 0.172 at 30 mm, a 56% loss. Through a real enclosure the gap is rarely under a few millimetres, so simulate the actual stack-up rather than contact.