What does a Biot–Savart coil solver compute?
The Biot–Savart law gives the magnetic field of a known current path in free space by summing the contribution of every current element:
Integrating the related vector potential around a second loop gives the Neumann formula for mutual inductance, and for circles and straight segments both integrals have closed forms in elliptic integrals or elementary functions (see mutual inductance). The method rests on three assumptions: the current path is prescribed, every material is non-magnetic and non-conducting apart from the coils (μ = μ₀), and the problem is quasi-static, so fields at every point respond instantly to the current.
Within those assumptions nothing is approximated. There is no mesh, no truncation boundary and no discretisation error: the remaining error is the geometric idealisation, such as treating a winding as filaments.
When is a closed-form Biot–Savart solver enough?
- Air-core and PCB coils with no ferrite and no large metal parts within a few coil diameters.
- Coupling, inductance and field questions: M, L, k, induced voltage, field maps and exposure screening of the coil’s own field.
- Electrically small coils. The total conductor length should be a small fraction of the wavelength, so the current is the same all the way round.
- Design exploration, where thousands of geometries are evaluated in sweeps or by an optimizer and speed decides how much of the design space you can see.
For a 10-turn, 40 mm-radius coil (2.51 m of wire), the electrical size at common wireless power frequencies is:
| Frequency | Free-space wavelength | Wire length / wavelength |
|---|---|---|
| 85 kHz | 3530 m | 0.00071 |
| 125 kHz | 2400 m | 0.001 |
| 6.78 MHz | 44.2 m | 0.057 |
| 13.56 MHz | 22.1 m | 0.11 |
At kilohertz frequencies the coil is electrically tiny and the quasi-static assumption is excellent. At 6.78 MHz and above, the same coil is a noticeable fraction of a wavelength, so self-resonance and current variation along the wire need checking.
AirInduct Sim is built on the closed-form approach and is honest about its scope. Its solver computes the Biot–Savart field and the Neumann mutual inductance directly for eight topologies, circular and square Helmholtz pairs, circular and square descending-turn planar spirals and the four cross pairs, and reports M, L_tx, L_rx, k, induced voltage and peak field without meshing. That makes it fast enough to drive sweeps, an optimizer and an AI Copilot whose every number is checked against solver output. The same properties mark its limits. It models coils in air, so it does not predict the effect of ferrite sheets, aluminium shields or nearby steel, the eddy-current losses those parts cause, or the distributed capacitance that sets self-resonance. For those, the honest workflow is to explore the geometry with a fast closed-form tool, then confirm the final design in a finite-element or full-wave solver and on a prototype.
When do you need FEM or a full-wave solver?
| Effect | Why closed-form misses it | Typical tool |
|---|---|---|
| Ferrite backing or cores | Permeable material reshapes the field and raises L and k | Magnetostatic or eddy-current FEM |
| Aluminium or steel shields | Induced eddy currents oppose the field and add loss | Eddy-current (time-harmonic) FEM |
| AC resistance, Q | Skin and proximity effect redistribute current in the conductor | FEM with resolved conductors, or analytic loss models |
| Self-resonance | Needs turn-to-turn capacitance and current variation along the wire | Full-wave solver or capacitance extraction |
| Radiated emissions | Quasi-static model has no radiation | Full-wave (FDTD, FEM, MoM) |
| SAR and induced fields in tissue | Needs tissue conductivity and geometry | Quasi-static or full-wave FEM/FDTD with body models |
| Temperature | Needs losses plus heat transfer | Coupled electromagnetic–thermal FEM |
FEM discretises the whole region, including materials and air, and solves Maxwell’s equations on the mesh. It handles arbitrary materials and induced currents, at the cost of mesh-dependent accuracy, longer solve times and more setup. Jin’s textbook covers the method in depth. Exposure assessment of real products follows dedicated procedures such as IEC/IEEE 63184, which combine measurement and computation; a coil-only field map is a screening step, not a compliance result.
Worked example: how fast does a Neumann sum converge?
Even inside the Biot–Savart world, how you evaluate the integrals matters. Two single-turn loops of radius 50 mm, 20 mm apart, have an exact mutual inductance from Maxwell’s formula of 67.54 nH. The table evaluates the raw Neumann double sum with n points on each loop:
| Points per loop | Neumann sum | Relative error |
|---|---|---|
| 8 | 71.54 nH | 0.059 |
| 16 | 67.65 nH | 0.0017 |
| 32 | 67.54 nH | 2.1e-6 |
| 64 | 67.54 nH | 4.4e-12 |
| 128 | 67.54 nH | 1.5e-14 |
With 8 points the answer is off by several per cent, because the 1/|r₁ − r₂| kernel is sharply peaked when the gap is small compared with the spacing between quadrature points. Once the spacing is well below the gap, convergence is extremely fast for smooth closed loops, reaching machine precision here at 64 points. The catch is that the required density scales with the smallest distance in the problem: adjacent turns of a planar spiral 1–2 mm apart need far more points than two loops 20 mm apart. A solver that does the inner integral analytically avoids this sensitivity, which is why the free mutual inductance calculator and AirInduct Sim use elliptic-integral forms wherever they exist. The same reasoning explains why a FEM model of the same air-core pair needs a fine mesh near the conductors to reach comparable accuracy.
How should you combine the two methods?
- Explore with closed-form. Choose topology, size, turns and gap; sweep gap and misalignment; find the k range the design must tolerate.
- Add the circuit. Size compensation and estimate efficiency from k and Q (see compensation topologies and WPT efficiency).
- Confirm with FEM once ferrite, shields and housings are known, and use it for losses, thermal behaviour and exposure in the real enclosure.
- Measure a prototype. Inductance, k (open/short method) and Q on an impedance analyser close the loop on both models.
The comparisons of 2D vs 3D FEM tools, general-purpose FEM suites and the best WPT simulation software cover the FEM tools in more detail.
References
- J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley (1999), ISBN 978-0-471-30932-1. www.wiley.com/en-us/Classical+Electrodynamics%2C+3rd+Edition-p-9780471309321
- J.-M. Jin, The Finite Element Method in Electromagnetics, 3rd ed., Wiley–IEEE Press (2014), ISBN 978-1-118-57136-1. www.wiley.com/en-us/The+Finite+Element+Method+in+Electromagnetics%2C+3rd+Edition-p-9781118571361
- F. W. Grover, Inductance Calculations: Working Formulas and Tables, Van Nostrand (1946); Dover reprint (2004), ISBN 978-0-486-49577-4. openlibrary.org/isbn/9780486495774
- S. I. Babic and C. Akyel, “Calculating mutual inductance between circular coils with inclined axes in air,” IEEE Trans. Magnetics 44(7), 1743–1750 (2008). doi.org/10.1109/TMAG.2008.920251
- IEC/IEEE 63184:2025, Assessment methods of the human exposure to electric and magnetic fields from wireless power transfer systems (3 kHz to 30 MHz). Supersedes IEC PAS 63184:2021. standards.ieee.org/ieee/63184/10156/
- ICNIRP, “Guidelines for limiting exposure to electromagnetic fields (100 kHz to 300 GHz),” Health Physics 118(5), 483–524 (2020). doi.org/10.1097/HP.0000000000001210