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The inductance of a coil depends on its geometry and the core material. For a single-layer air-core coil, Wheeler's formula gives an accurate estimate from the diameter, length, and number of turns. For a toroid, the closed magnetic path and the core's permeability set the inductance, which is why ferrite and iron-powder toroids give large inductance in a small size.
| Type | Formula |
|---|---|
| Air-core (Wheeler) | L(µH) = d²N² / (18d + 40ℓ), inches |
| Toroid | L = μ0μrN²h·ln(OD/ID)/(2π) |
| Turns dependence | L ∝ N² (both types) |
| Permeability | μ0 = 4π×10−7 H/m |
Inductance scales with the square of the number of turns, so doubling the turns roughly quadruples the inductance. Air-core coils have µr = 1; magnetic cores multiply the inductance by their relative permeability.
It depends on the geometry and core. For a single-layer air-core coil, Wheeler's formula uses the diameter, length, and turns; for a toroid, the formula uses the core dimensions and permeability.
An accurate approximation for a single-layer air-core coil: L(µH) = d²N²/(18d + 40ℓ), with the coil diameter d and length ℓ in inches. This calculator converts your mm inputs automatically.
Each turn links the flux produced by all the turns, so both the flux and the linkage scale with N — making the inductance proportional to N².
μr is how much more magnetic flux a core carries compared with air. Air is 1; ferrites can be hundreds to thousands, multiplying the inductance by that factor.
The magnetic path is closed inside the ring, so almost all the flux stays in the high-permeability core, giving large inductance and low stray field in a compact size.
Wheeler's formula is within a few percent for typical single-layer coils where the length is comparable to the diameter. Very short or very long coils are less accurate.
Add turns (L ∝ N²), use a magnetic core (multiply by μr), or increase the coil diameter. Adding a core is by far the most effective.
Ferrite has very high permeability (great for filters and small inductors) but saturates easily; iron-powder has lower permeability but handles more current and higher frequencies with less loss.
Only slightly through the coil dimensions. Thicker wire changes the length and diameter a little, but inductance is mainly set by turns and geometry, not wire size.
The inductance per turn-squared (nH/N²) given for a specific core. If you know AL, inductance is simply L = AL × N² — a quick alternative to the full formula.
Wheeler's single-layer formula is for one layer. Multi-layer coils need a different formula, as the extra layers add mutual coupling and change the geometry.
Real coils vary with winding tightness, core tolerance, frequency, and nearby metal. Treat the calculation as a close estimate and trim turns to fine-tune.
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