Air-Core Inductor Calculator (Single-Layer Coil)
Air-Core Inductor Calculator (Single-Layer Coil)
Inductance of a single-layer air-cored coil from its diameter, length and turns by Wheeler’s 1928 formula — or the turns for an inductance you want — with Nagaoka’s exact coefficient beside it so you can see what the rule of thumb costs, plus wire length, DC resistance and Q.
Single-layer air-core coil
a coil 20 mm across and 30 mm long, 40 turns of 24 AWG, Q measured at 1,000 kHz
Wheeler’s formula, and the exact answer behind it
- a
- coil radius; b is the coil length — Wheeler wrote them in inches
- N
- number of turns, in one layer
- K
- Nagaoka’s coefficient, a function of diameter ÷ length alone; 1 for an infinitely long coil, falling towards 0 for a flat one
- μ₀
- 4π × 10⁻⁷ H/m, the magnetic constant
Worked example
a coil 20 mm across and 30 mm long, 40 turns of 24 AWG, Q measured at 1,000 kHz
In Wheeler's own units the coil is a = 0.39370 in and b = 1.18110 in, so L = a²N² ÷ (9a + 10b) = 16.15 µH
ℓ/d = 1.50, comfortably inside Wheeler's stated range (ℓ at least 0.4 d)
Nagaoka's coefficient at d/ℓ = 0.6667 is K = 0.76989, giving the exact current-sheet value L = 16.21 µH — Wheeler reads 0.36% below it
40 turns round a 20 mm mean diameter is 2.513 m of wire; 24 AWG is 0.511 mm bare (0.2047 mm²), so R = 211.7 mΩ at 20 °C
At 1,000 kHz the reactance is 101.5 Ω, so Q = 479.5 on the DC resistance alone — an upper bound, because the wire is 3.86 times two skin depths thick here
How far Wheeler’s formula is from the exact answer
| Coil length ÷ diameter | Nagaoka’s K | Wheeler against exact |
|---|---|---|
| 0.10 | 0.20332 | 256.71% |
| 0.20 | 0.31983 | 283.77% |
| 0.30 | 0.40527 | 293.72% |
| 0.40 | 0.47186 | 297.82% |
| 0.50 | 0.52551 | 299.51% |
| 0.80 | 0.63809 | 300.09% |
| 1.00 | 0.68842 | 299.62% |
| 2.00 | 0.81814 | 298.02% |
| 5.00 | 0.92009 | 297.75% |
| 10.00 | 0.95881 | 298.12% |
| 20.00 | 0.97909 | 298.46% |
Winding a single-layer coil
A single-layer solenoid is the one air-cored coil you can still calculate on paper. Harold Wheeler published the arithmetic in 1928: with the radius a and the length b both in inches, L in microhenries is a²N² divided by (9a + 10b). This page works in millimetres and converts, so the answer is the same number Wheeler’s own expression gives. The inductance rises with the square of the turns and with the area the turns enclose, and falls as the winding is stretched out, because a longer coil returns less of its own flux through itself.
What the approximation costs. Wheeler’s formula is a fit to the exact answer, which Nagaoka worked out in 1909 and which needs elliptic integrals. This page computes both: the exact current-sheet inductance is μ₀N²(πd²/4)K/ℓ, where K is Nagaoka’s coefficient and depends on nothing but the ratio of diameter to length. The error is printed beside the answer and drawn against coil shape in the chart. It stays inside 1% for any coil at least 0.4 diameters long — the condition Wheeler stated — and falls apart below that: a pancake one tenth of a diameter long reads nearly 11% low. If your coil is that short, take the Nagaoka figure and ignore Wheeler.
What neither formula includes. Both treat the winding as a smooth sheet of current, so both ignore the gaps between turns and the thickness of the wire; a real spaced winding measures a little lower, and a correction for that (Rosa’s round-wire terms) is a few per cent on a coil of a few tens of turns. Neither includes the self-capacitance between turns, which resonates with the inductance and gives every coil a self-resonant frequency above which it behaves as a capacitor — use the coil well below it, and see the LC resonant frequency calculator for what a stray capacitance does to a tuned circuit. Nothing here knows about a core: a ferrite or iron slug multiplies the inductance by an effective permeability that is a property of the core, not of the coil, and a ring core is sized from its Aₗ instead — see the toroid inductor turns calculator.
Wire, resistance and Q. The wire length is the turns times the mean turn circumference, and the DC resistance follows from the AWG size using the same constants as the AWG wire size calculator: 0.127 × 92^((36−n)/39) mm for the diameter and 17.241 Ω·mm²/km for annealed copper at 20 °C. The Q shown is the reactance divided by that DC resistance, which is an upper bound and usually an optimistic one: above a few hundred kilohertz the current crowds into the outside of the wire, and the page prints the skin depth so you can see when that has started. Proximity effect between adjacent turns makes it worse again. For the reactance on its own, the reactance calculator covers inductors and capacitors at any frequency.
Frequently asked questions
How do I calculate the inductance of an air-core coil?
Wheeler’s 1928 formula: L in microhenries = a²N² ÷ (9a + 10b), with the coil radius a and length b in inches and N the number of turns. A 20 mm coil 30 mm long with 40 turns comes to 16.15 µH.
How accurate is Wheeler’s formula?
About 1% as long as the coil is at least 0.4 diameters long, which is what Wheeler claimed and what this page checks against Nagaoka’s exact coefficient. Shorter than that it reads low — nearly 11% low at one tenth of a diameter.
How many turns do I need for a given inductance?
Inductance goes with the square of the turns, so N = √(L ÷ k) where k is the inductance of one turn on that former. Switch this page to the second mode and it does the rounding to whole turns and tells you what you actually get.
Does the wire gauge change the inductance?
Barely. It changes the mean turn diameter slightly and it decides how many turns fit in the length you have, but the inductance is set by the geometry. The gauge matters for resistance, for Q and for whether the winding fits.
Why is my measured inductance lower than the calculation?
Three usual reasons: the turns are spaced rather than touching, so the real winding is not the solid current sheet both formulas assume; the mean turn diameter is smaller than you measured; or you are near the coil’s self-resonant frequency, where the winding’s own capacitance starts to dominate.
Related calculators
References
- Wheeler HA. Simple Inductance Formulas for Radio Coils. Proceedings of the Institute of Radio Engineers, vol. 16 no. 10, October 1928, pp. 1398–1400. L (µH) = a²N²/(9a + 10b) with a and b in inches, stated as accurate to about 1% for a coil at least 0.4 diameters long.
- Nagaoka H. The Inductance Coefficients of Solenoids. Journal of the College of Science, Imperial University, Tokyo, vol. 27 art. 6, 1909. The exact current-sheet coefficient tabulated against diameter ÷ length; K = 0.68842 at d = ℓ.
- Rosa EB, Grover FW. Formulas and Tables for the Calculation of Mutual and Self-Inductance. NBS Scientific Paper 169, 1916. Maxwell’s mutual-inductance formula for coaxial circular filaments, which this page’s exact figure was checked against by numerical integration.
- ASTM B258-02. Standard Specification for Standard Nominal Diameters and Cross-Sectional Areas of AWG Sizes of Solid Round Wires Used as Electrical Conductors. ASTM International. The 0.127 × 92^((36−n)/39) mm diameter law.
- IEC 60028. International standard of resistance for copper: annealed copper at 1/58 Ω·mm²/m, 20 °C — the 17.241 Ω·mm²/km used here and on the AWG page.