Eight coil topologies, one solver

Outline. Circular coils are evaluated with complete elliptic integrals, and their coupling does not change when the device rotates about the shared axis. Square coils are built from straight segments using the exact Biot–Savart result for a finite wire; at the same half-dimension a square encloses about 27% more area than a circle, which raises mutual inductance.

Transmitter type and receiver type. A Helmholtz coil puts every turn at the full radius, so it links the most flux per turn but also carries the most self-inductance. A DTC, or descending-turn coil, is a planar spiral whose turns step inward by a fixed pitch; it is flat enough to print on a PCB, and its lower self-inductance often gives it a higher coupling coefficient. The direction matters for packaging, not for mutual inductance, which is reciprocal.

Start from the mechanics. If one side must be flat — a pad under a surface, a PCB inside a phone or tablet — that side is a DTC spiral. If both must be flat, you are choosing between the circular and square DTC-to-DTC pairs, and the outline of the product usually decides it.

If you need a reference you can check by hand, use the circular Helmholtz coil pair, which AirInduct Sim solves in closed form. For rectangular rigs and bays, the square Helmholtz coil pair gives the highest mutual inductance of the eight (3.28 µH at 10 mm) even though its k is among the lowest.

Then check the gap. Across all eight pairs, k falls by 54–57% when the gap triples from 10 mm to 30 mm at this size, so the enclosure stack-up matters more than the choice between neighbouring topologies. Square outlines hold coupling slightly better with distance because they enclose more area.

Finally, remember that a higher k is not the whole story: link efficiency depends on k together with the quality factor of each coil, and printed spirals usually have more resistance than wound copper.

• Mutual inductance calculator
• Coupling coefficient calculator
• Helmholtz coil calculator