Air-Core Coil Inductance: Wheeler Formula and Design

Sooner or later every builder needs a coil that nobody sells: an inductor of exactly 4.7 microhenries for an antenna tuner, a loading coil for a short vertical, or a tank circuit for a low-pass filter. Toroids are convenient, but they saturate and lose power at high levels, and the cores are not always available. An air core coil, wound on a plastic former or standing free in the air, has no core to saturate, adds almost no loss and can handle full power. It also has a pleasant property for a designer: its inductance can be calculated from a simple formula with an accuracy of about one percent. This guide explains the formula named after Harold Wheeler, works through examples in both directions, and gives practical advice on the diameter, the length, the wire and the Q of single-layer coils.

What Decides the Inductance

Inductance measures how much magnetic flux a coil links for a given current. For a coil without a magnetic core, the flux depends only on the geometry: the number of turns, their diameter and the length of the winding. The inductance grows with the square of the number of turns, because doubling the turns doubles the flux through each turn and also doubles the number of turns that link it. It increases with the diameter, since a larger loop encloses more area, and it decreases with length, since a long coil spreads the same turns over a larger distance and the turns couple less closely.

The material of the former and the wire does not matter for the inductance of an air coil, provided the former is not magnetic. A few turns of copper on a plastic tube have the same inductance as a few turns of silver-plated wire in the air, if the dimensions are identical. What changes is the loss, and so the Q, which is discussed below.

The Wheeler Formula

For a single-layer round coil, Wheeler’s approximation, published in 1928, gives the inductance in microhenries as L = n² d² / (18 d + 40 l), where n is the number of turns, d is the mean diameter of the coil in inches and l is the length of the winding in inches. With the radius r, which is half of d, the same formula reads L = n² r² / (9 r + 10 l). Both forms give the same numbers, and the second one is the version commonly printed in handbooks. The formula is accurate to about one percent for coils whose length is greater than 0.8 of their radius, which covers nearly every practical design. For very short and flat coils, a different expression is needed. The inductance of other shapes is tabulated in electrical engineering handbooks.

Notice that the formula requires the diameter measured between the centres of the wire on opposite sides, not the outside diameter of the former. For thick wire this makes a noticeable difference. A 25 millimetre former wound with 2 millimetre wire has a mean diameter of 27 millimetres. The length is also measured from the centre of the first turn to the centre of the last turn plus one turn pitch, that is, the number of turns times the pitch. The more carefully you measure, the better the result.

A Worked Example: Inductance From the Dimensions

Suppose you have wound 10 turns on a former with a mean diameter of 1 inch (25.4 millimetres), spread over a length of 1 inch. The radius is 0.5 inch. Then L = 100 × 0.25 / (4.5 + 10) = 25 / 14.5 = 1.72 microhenries. If you check the coil with a NanoVNA or an LC meter, you will probably find a value between 1.70 and 1.76. The guide to the NanoVNA explains how to measure an inductor with a calibrated reflection test.

Solving for the Number of Turns

More often you know the inductance you need and want to find the turns. Rearranging the formula gives n = √( L (18 d + 40 l) ) / d. Suppose you need 5 microhenries on a former of 1 inch mean diameter, with a winding length of 1.5 inches. Then n = √( 5 × (18 + 60) ) / 1 = √390 = 19.7 turns. So roughly 20 turns, spaced to cover 1.5 inches. The check is straightforward: L = 19.7² × 1 / 78 = 388 / 78 = 4.97 microhenries. In practice, wind a little more than you need, measure the coil and remove or spread a fraction of a turn to reach the target. Spreading the turns slightly reduces the inductance, and squeezing them together increases it, which is a useful way to adjust the final value.

The Best Shape for a High Q

The Q of a coil is the ratio of its reactance to its resistance at the working frequency. A high Q means low loss, and for a tuned circuit or a filter it means sharp response and little insertion loss. For a single-layer air coil, the highest Q for a given length of wire occurs when the length of the winding is about equal to its diameter, or a bit less, with the turns spaced by roughly one wire diameter. A very long, thin coil has more wire for the same inductance and therefore more resistance, while a short, fat coil has turns that interact strongly and lose some of the benefit.

The wire size should be as large as is practical. The resistance at radio frequencies is dominated by the skin effect, which pushes the current into the surface layer of the conductor, so a thick wire or a tube has a lower resistance than a thin one at the same frequency. Silver plating helps a little, and bare copper tube is the choice for high-power loading coils. Spacing the turns reduces the proximity effect between neighbouring turns, which raises the Q further. A good coil for the HF bands wound with 1.5 mm wire in this way reaches a Q of 200 to 400, while a coil wound on a toroid usually has a Q of 100 to 200.

Forms, Supports and Shielding

The former should be a good insulator with low dielectric loss. PVC pipe is cheap and gives acceptable results, but polystyrene, polyethylene, Teflon and ceramic are better, especially at high power or high frequencies where the former sits in the strong electric field of the coil. A fibreglass tube with ribs is also popular. Wind the coil under tension and fix it with a few spots of glue, because a loose turn changes the inductance and may create noise. Do not coat the whole coil with varnish; the dielectric of a coating adds loss and capacitance. A sparing use of glue at a few points is better.

A coil placed close to a metal surface loses inductance and Q, because the eddy currents in the metal oppose the field of the coil. Keep the coil away from the chassis by at least the diameter of the coil, and place coils in a filter at right angles to each other so that they do not couple. A shield can is used when the coupling must be eliminated, with the shield diameter at least twice the diameter of the coil. Remember that the shield lowers the inductance by a few percent, so wind the coil slightly large and tune it in place.

Self-Resonance and Stray Capacitance

Every coil has some capacitance between its turns, and together the inductance and the capacitance form a parallel resonance. Above that self-resonant frequency the coil behaves as a capacitor, so it is useless as an inductor. For a typical air coil of a few microhenries the self-resonance is well above the HF bands, perhaps 50 to 100 MHz, but a coil of 50 microhenries may self-resonate on 15 metres. Keep the self-resonant frequency at least a factor of two above the working frequency, and use fewer turns of a larger diameter or a spaced winding to reduce the capacitance. The relation between inductance, capacitance and resonance is covered in the article on the L network, which uses the same quantities in a matching circuit.

Where Air Coils Are Used

Air coils appear in many places in a ham station. In antenna tuners, a roller inductor is just a large air coil with a sliding contact, and the L network inside is made with air coils for power handling. In the low-pass and band-pass filters used after an amplifier, the coils see high currents and are usually air coils, as in the guides to the low-pass filter and band-pass filters. In antennas, loading coils for shortened verticals and the coils of traps, as described in the article on the trap dipole and on the loading coil and capacity hat, are air coils whose Q determines how much power is lost as heat.

Winding Tips From the Workbench

A neat coil is easier to calculate and performs better. Mark the positions of the turns on the former with a pencil or cut a shallow groove, or wind the coil together with a thread or a thin spacer wire that you remove afterwards, which gives an even pitch. Drill two small holes at the ends of the winding and pass the wire through them to anchor the ends without glue. Keep the leads short, because a lead of two centimetres adds about 20 nanohenries, which is no problem at 5 microhenries but matters for a 0.2 microhenry coil on VHF, where even the way the leads are bent changes the tuning.

When you need an adjustable coil, there are three easy ways. Tapping the winding at several turns gives steps of inductance, selected with a switch or a clip. A slug of brass or ferrite moved inside the former gives a continuous change, though a ferrite slug lowers the Q. A pair of coils, one rotating inside the other, forms a variometer. For a one-off tuner, the tapped coil with a crocodile clip is the cheapest and most reliable arrangement, as long as the unused turns do not form a resonance of their own at a frequency in use. Shorting the unused part of the coil prevents this.

Finally, think about heating at high power. The current in a coil of an antenna tuner at 100 watts may be several amperes, and the wire resistance turns part of it into heat. A coil that feels warm after a minute of transmission is a coil that wastes power and that will drift in value as it heats. Use thicker wire, or a tube, and the problem goes away.

Checking Your Result

After winding, measure the coil. An LC meter or a NanoVNA will give the inductance and the Q at a chosen frequency. Compare it with the calculation and adjust. Many online calculators implement the Wheeler formula, but it is worth doing the sum by hand once, because the numbers give you a feeling for the effect of each dimension: ten percent more diameter gives ten percent more inductance, but ten percent more length reduces it by about eight percent. Keep a note of the result for the next project.

The Bottom Line

The inductance of a single-layer air core coil in microhenries is L = n² d² / (18 d + 40 l), with the mean diameter d and the winding length l in inches, and it is accurate to about one percent for ordinary shapes. To find the turns for a wanted value use n = √( L (18 d + 40 l) ) / d, wind slightly more than you need and trim by spreading the turns. For a high Q choose a length near the diameter, turns spaced about one wire diameter apart, thick wire, a low-loss former and enough distance from metal. Stay well below the self-resonant frequency.

Frequently Asked Questions

What is the Wheeler formula?

It is an approximation for the inductance of a single-layer air coil: L in microhenries equals the turns squared times the diameter squared, divided by 18 times the diameter plus 40 times the length, with dimensions in inches.

How accurate is it?

About one percent for coils longer than 0.8 of their radius. Real coils differ slightly because of the wire thickness and the leads, so measure and adjust.

What shape gives the highest Q?

A winding length about equal to the diameter, with the turns spaced about one wire diameter apart and thick wire.

Why does a coil lose inductance near a metal box?

Eddy currents in the metal create a field that opposes the coil’s field. Keep the coil at least one coil diameter away from large metal surfaces.

Can I use a plastic pipe as a coil form?

Yes. PVC works for general use. For high power or high frequencies, polystyrene, polyethylene or ceramic forms have lower dielectric loss.

Why does my coil act as a capacitor at high frequency?

Above its self-resonant frequency the stray capacitance between the turns dominates. Use fewer turns or spaced turns to raise the self-resonance.