Circular DTC-to-Helmholtz Coil Pair
A flat circular planar-spiral (DTC) transmitter drives a wound multi-turn circular receiver. This is the familiar consumer layout: a flush pad underneath, a wound coil inside the earbud case, stylus or wearable.
Worked example
Inputs. Transmitter: 5-turn circular planar spiral (DTC), 2 mm pitch, turns at 50, 48, 46, 44, 42 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 | 2.384 | 4.912 | 7.615 | 0.390 |
| 30 mm | 1.052 | 4.912 | 7.615 | 0.172 |
Going from 10 mm to 30 mm lowers k from 0.390 to 0.172, a 56% drop. The 4.9 µH pad is the lower-inductance side here, which keeps transmitter voltage modest for a given drive current — useful when the pad runs from a low-voltage USB supply.
How it was computed. The pad field is the sum of five circular loops at 50, 48, 46, 44 and 42 mm radius, each evaluated with elliptic integrals; the wound receiver collects flux over its full 50 mm disc, five times. k = M / √(L_tx · L_rx).
When it's the right choice
- Earbud, stylus and wearable charging pads
- Transmitters that must sit flush under a desk, tray or console surface
- Retrofits onto an existing flat charging surface
Wound receiver vs spiral receiver
Keeping the flat spiral transmitter but swapping the wound receiver for a second spiral raises the coupling coefficient, and AirInduct Sim quantifies the gain: k climbs from 0.390 to 0.461 at a 10 mm gap and from 0.172 to 0.197 at 30 mm. The reason is not more shared flux — mutual inductance actually falls slightly, from 2.38 µH to 2.26 µH — but the lower self-inductance of the spiral receiver, whose inner turns enclose less area. Because k divides M by the square root of both self-inductances, trimming L_rx from 7.6 µH to 4.9 µH outweighs the small loss in M. So why keep a wound receiver at all? Its co-radial turns all sit at the full radius where linkage is strongest, so it delivers slightly more induced voltage per ampere of pad current and suits a device with some depth. Choose this DTC-to-Helmholtz pair for earbud cases, styluses and wearables with a wound coil; choose DTC-to-DTC when both sides must be flat PCB traces.
Watch out for
- Coupling falls off sharply with vertical gap — check k at your real enclosure thickness, not at contact
- Lateral misalignment is the dominant real-world failure mode; sweep it before committing
Neighbouring topologies
Circular Helmholtz Coil Pair
Calibration, teaching, benchmarking. At the worked-example geometry: k = 0.366 at 10 mm.
Swap the receiver to DTCCircular DTC-to-DTC Planar Coil Pair
Qi-style phone and watch charging. At the worked-example geometry: k = 0.461 at 10 mm.
Switch to a square outlineSquare DTC-to-Helmholtz Coil Pair
Worktop panels, tool and robot docks. At the worked-example geometry: k = 0.396 at 10 mm.
All eight pairs side by side, with coupling at both gaps: coil topology comparison.
Check the numbers
Run the Circular DTC-to-Helmholtz Coil Pair at your geometry
Frequently asked questions
Why is a spiral pad with a wound receiver so common in consumer charging?
The pad has to be flat to sit under a surface, while small devices such as earbud cases often have a few millimetres of depth for a wound coil. At 50 mm and a 10 mm gap this pair reaches k = 0.390 in AirInduct Sim, comparable to a matched wound pair, without a bulky transmitter.
Why does a five-turn spiral have less inductance than five wound turns?
In a DTC spiral each turn steps inward by the pitch, so the inner turns enclose less area and link less of each other’s flux. With a 2 mm pitch the five-turn spiral gives 4.91 µH against 7.62 µH for five co-radial turns at 50 mm.
What gap should I simulate for a charging pad?
The real distance between the copper planes: pad cover, device back and any air or adhesive layers. For this pair, k is 0.390 at 10 mm but 0.172 at 30 mm, so a few millimetres of enclosure changes the design noticeably.