Coil Inductance Calculator calculates inductance from wire radius, loop diameter, coil length, turns, and relative permeability while displaying AL value, Q factor, flux, and energy.
The Long-Solenoid Formula Behind the Inductance Value
Self-inductance here comes from one equation: $$L = \mu_r \mu_0 \frac{N^2 A}{l}$$ μ₀ is the permeability of free space, a fixed constant equal to $4\pi \times 10^{-7}$ H/m. μᵣ is the relative permeability of whatever’s inside the coil — 1 for air or a plastic form, tens to low thousands for a ferrite or iron core.
N is the turn count, squared, because both the flux each turn generates and the flux each turn links scale with N. A is the coil’s cross-sectional area, found from the loop diameter as $A = \pi (d/2)^2$, and l is the coil’s axial length — the distance the windings actually span, not the wire length.
Wire radius never enters this equation. It sets the wire’s resistance and how many turns fit into one layer, covered further down, but it has zero effect on the inductance value itself — the same turns, diameter, length, and core wound with thicker or thinner wire report identical L.
The same geometry produces two more figures the calculator reports alongside L: permeance, $P = \mu_0 \mu_r A / l$, and its reciprocal, reluctance. Divide inductance by turns squared and you get the A_L value — nanohenries per turn squared — the number ferrite core manufacturers publish so a winding can be sized without redoing the field geometry from scratch. All three describe the same magnetic circuit; they just answer slightly different questions.
A 100-Turn Air-Core Coil, Worked Step by Step
Take a 100-turn coil wound on a 20 mm diameter form, 50 mm long, with 0.4 mm diameter wire (0.2 mm radius) and no magnetic core, so μᵣ = 1. Cross-sectional area comes first: $A = \pi (0.01\text{ m})^2 = 3.1416 \times 10^{-4}\text{ m}^2$.
Plug that into the solenoid formula: $L = (1)(1.2566\times10^{-6})(100^2)(3.1416\times10^{-4}) / 0.05 = 7.896\times10^{-5}$ H, or 78.96 µH. Every term in that line traces straight back to an input field — no scaling factors, no assumed defaults beyond what was actually typed in.
The same inputs put total wound wire length at $100 \times \pi \times 0.02\text{ m} = 6.283$ m, which works out to 0.84 Ω of DC resistance through 0.4 mm copper wire — small next to the coil’s 49.61 Ω of reactance at 100 kHz, covered next.
What A_L Value, Q Factor, and the 100 kHz Test Point Actually Mean
For the coil above, A_L works out to 7.90 nH per turn², with a matching permeance of 7.90 nH and a reluctance of roughly 126.65 million ampere-turns per weber. A higher A_L means fewer turns are needed to hit a target inductance — drop in a ferrite core and A_L climbs by whatever μᵣ that core carries, with no change to turns or geometry.
Reactance and Q factor are both computed at a fixed 100 kHz test point, not whatever frequency the coil will actually run at. Reactance came out to 49.61 Ω against 0.84 Ω of DC resistance, giving a Q factor near 59 — that just means resistive loss is close to a rounding error next to reactive impedance at that frequency. The same coil’s Q drops sharply at lower frequencies, since reactance falls with frequency while DC resistance stays roughly fixed.
The formula itself assumes an ideal long solenoid, and the calculator flags that assumption the moment coil length drops below coil diameter — an aspect ratio under 1. Below that point the field no longer stays uniform along the axis, flux leaks out the ends more than the formula accounts for, and the reported inductance reads higher than what a real coil would measure; the tool still returns a number, it just stops being one you can trust without a correction factor.
Turns Squared and Core Material Move the Number — Wire Size Doesn’t
Turn count matters more than any other input, because it’s squared: doubling turns from 100 to 200 quadruples inductance, all else equal, while doubling coil length only cuts it in half. Area follows the same squared relationship with diameter, since $A \propto d^2$ — a coil twice as wide, wound the same way, reports four times the inductance.
Core material is the single biggest lever of all. Swap μᵣ from 1 (air) to 100 (a mid-range ferrite) and inductance rises a hundredfold with the same wire, same turns, same geometry — nothing else in the formula scales that aggressively.
Wire radius shows up in a different place entirely: $\text{max single-layer turns} = l / (2r_w)$. For the worked example that’s $0.05 / 0.0004 = 125$ turns — go past that and the requested turns won’t physically fit in one layer along a 50 mm form, so the calculator flags a multi-layer warning without changing the L value shown, since the single-layer formula has no way to account for a second layer’s different field geometry.
Common Questions About Coil Inductance
How do you calculate the inductance of an air-core coil?
Use $L = \mu_0 N^2 A / l$ with μᵣ set to 1, since air and most non-magnetic coil forms don’t concentrate flux the way a ferrite or iron core does. Turns, cross-sectional area, and coil length are the only three inputs that matter once the core term drops out.
Does wire gauge affect a coil’s inductance?
Not directly — the solenoid formula has no wire-diameter term in it. Wire gauge only changes DC resistance and how many turns physically fit in one layer along the coil’s length.
What’s the difference between inductance and A_L value?
Inductance (L) is the total property of the finished coil; A_L is inductance per turn squared, $A_L = L / N^2$. Core manufacturers publish A_L so a target inductance can be solved backward into a turn count without redoing the field geometry.
Why is my measured inductance lower than the calculated value?
The formula assumes an ideal long solenoid with uniform flux along its full length; real coils leak flux at both ends, and that leakage grows as the coil gets short and fat relative to its diameter. A coil with length shorter than its diameter sits furthest from the ideal-solenoid assumption this formula is built on.
What does Q factor mean for an inductor?
Q factor is reactance divided by resistance at a given frequency — it measures how much of the coil’s impedance is genuinely reactive versus wasted as resistive loss. A Q near 60 means resistance contributes almost nothing to total impedance at that test frequency; the same coil’s Q drops at lower frequencies because reactance falls while resistance holds roughly steady.
How much does core material change inductance?
By a factor equal to the core’s relative permeability, μᵣ, since it’s a direct multiplier in the formula. Air and most plastics sit at μᵣ ≈ 1; ferrites commonly run from the tens into the low thousands, which is why a ferrite-core coil needs far fewer turns to reach the same inductance as an air-core one.
How many turns fit on a coil in a single layer?
Divide the coil’s length by twice the wire radius: $l / (2r_w)$. Go past that number and the winding has to stack into a second layer, which the formula doesn’t model — the reported inductance stops matching what a multi-layer winding will actually measure.
This model doesn’t account for skin effect or proximity effect at high frequency — the reactance and impedance figures use the same DC resistance value regardless of frequency, so real AC resistance at 100 kHz and above runs higher than what’s shown here. It also assumes constant permeability with no core saturation or hysteresis loss, and treats winding turns as bare wire in direct contact with no insulation gap between layers.