Square Helmholtz-to-DTC Coil Pair
A wound square Helmholtz transmitter drives a flat square planar-spiral (DTC) receiver of nested squares. The square outline fills rectangular docks and modules that a round coil would leave half-empty, and the receiver can be a PCB trace.
Worked example
Inputs. Transmitter: 5 co-located square turns at 50 mm half-side. Receiver: 5-turn square planar spiral (DTC), 2 mm pitch, turns at 50, 48, 46, 44, 42 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 | 2.925 | 8.878 | 5.929 | 0.403 |
| 30 mm | 1.333 | 8.878 | 5.929 | 0.184 |
Going from 10 mm to 30 mm lowers k from 0.403 to 0.184, a 54% drop. Of the four Helmholtz-transmitter cases, this one keeps the most coupling at 30 mm (k = 0.184).
How it was computed. The transmitter is five co-located square turns at a 50 mm half-side, evaluated side by side with the finite-wire Biot–Savart result; the receiver is five nested squares at 50, 48, 46, 44 and 42 mm half-side, each integrated over its own area. k = M / √(L_tx · L_rx).
When it's the right choice
- Rectangular charging bays, docks and industrial fixtures
- Receivers printed on a rectangular PCB or flex circuit
- Cases where transmitter depth is free but receiver thickness is not
Spiral receiver vs wound square receiver
Replacing the wound square receiver of the square Helmholtz pair with a flat square spiral costs a little mutual inductance but improves the coupling coefficient, and AirInduct Sim puts numbers on both: at a 10 mm gap M drops from 3.28 µH to 2.93 µH while k rises from 0.370 to 0.403. The spiral’s inner turns at 48, 46, 44 and 42 mm half-side enclose less area than five co-located turns at 50 mm, which lowers receiver self-inductance from 8.9 µH to 5.9 µH; since k normalises M by both self-inductances, the ratio improves. At 30 mm the same trend holds, 0.184 against 0.165. Choose this Helmholtz-to-DTC pair when the receiver must be a PCB trace or flex circuit inside a rectangular module, and the transmitter can be a wound coil in a dock or charging bay. Keep the full square Helmholtz pair when receiver thickness is not a constraint and you want the most induced voltage per ampere.
Watch out for
- Field is strongest near the conductors, so a small receiver near the centre sees less than the nominal average
- Corner field concentration can push local field above exposure limits before average power looks high
Neighbouring topologies
Square DTC-to-DTC Planar Coil Pair
Tablets, flat modules, PCB-to-PCB links. At the worked-example geometry: k = 0.465 at 10 mm.
Swap the receiver to HelmholtzSquare Helmholtz Coil Pair
Rectangular test rigs and fixtures. At the worked-example geometry: k = 0.369 at 10 mm.
Switch to a circular outlineCircular Helmholtz-to-DTC Coil Pair
Thin tags, implants, smart cards. At the worked-example geometry: k = 0.389 at 10 mm.
All eight pairs side by side, with coupling at both gaps: coil topology comparison.
Check the numbers
Run the Square Helmholtz-to-DTC Coil Pair at your geometry
Frequently asked questions
What does DTC mean for a square coil?
Descending-turn coil: the turns are nested squares in one plane, each half-side smaller than the last by the pitch. In the worked example that is 50, 48, 46, 44 and 42 mm — a square planar spiral that can be etched on a PCB.
Why does k rise when the square receiver becomes a spiral?
The spiral lowers receiver self-inductance from 8.88 µH to 5.93 µH while M only falls from 3.28 µH to 2.93 µH at a 10 mm gap, so k = M / √(L_tx·L_rx) rises from 0.370 to 0.403.
Can the square spiral receiver be a PCB trace?
Yes — that is the usual reason to pick it. AirInduct Sim reports the inductances and coupling for the trace layout; pair them with your own trace-resistance estimate to get the quality factor and efficiency limit.