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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.
For a long single-layer solenoid the inductance is
L = μ0μrN2A / l
with μ0 = 4π×10−7 H/m, N the number of turns, A the cross-sectional area in square metres and l the coil length in metres.
The dominant term is N2. Doubling the turns quadruples the inductance, which makes turn count by far the most powerful adjustment available — and also the easiest place to make a large error. Everything else enters linearly.
Winding the same coil on a ferrite core with μr = 1,000 multiplies the inductance by a thousand. That is why cored inductors exist: achieving a useful inductance in air demands an impractical number of turns, and every one of those turns adds resistance and loss.
The catch is that μr is not a constant. It falls as the core approaches saturation, so a cored inductor’s value depends on the current flowing through it. A part measured at 100 µH on an LCR meter at milliamp test current may be considerably less than that at its rated operating current — which is precisely when the circuit depends on it.
Introducing a small air gap in the magnetic path lowers the effective permeability, so inductance drops — but the energy the core can store before saturating rises substantially. For an inductor that must carry significant DC bias, such as a buck converter’s output choke, a gapped core is the standard answer: it trades inductance per turn for the ability to keep that inductance at full load.
Note also that the formula above assumes a long, single-layer coil. Short coils, multi-layer windings and toroids all depart from it, which is why practical designs use the core manufacturer’s AL value — inductance per turn squared for that specific core — rather than a general-purpose equation.
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.
L = μ0μrN²A/l, with μ0 = 4π×10−7 H/m, area in square metres and length in metres. It assumes a long, single-layer coil.
Inductance quadruples, because it depends on the square of turns. This makes turn count the most effective way to change inductance, and the easiest place to make a large mistake.
By roughly its relative permeability. A core with μr = 1,000 gives about a thousand times the inductance of the same winding in air, which is why practical inductors almost always use a core.
For a cored inductor, yes. Permeability falls as the core approaches saturation, so measured inductance at low test current can be much higher than the value at rated operating current.
A gap lowers effective permeability and therefore inductance, but substantially increases the energy the core can store before saturating. It is standard practice for inductors carrying significant DC bias, such as buck converter output chokes.
Inductance per turn squared for that specific core, usually in nH/turn². Multiply AL by N² to get inductance directly. It is more reliable than a general formula because it accounts for the core's actual geometry.
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