The calculation is based on Wheeler’s 1928 formula for a single-layer solenoid which is given in its original form as:

L = a² N² / (9a + 10b) [microHenries] , b > 0.8a

Where b is the coil length in inches, and a is the radius in inches.

To convert this formula to SI units, we will use the symbols r = radius, D = 2r = diameter, l = solenoid length.

Factoring b from the denominator gives:

L = 10-6 a² N² / [ b (10 + 9a/b)] [Henrys]
The quantity a/b is dimensionless, and so we can immediately substitute in the denominator:

L = 10-6 a² N² / [ b (10 + 9r/l)] = 10-6 a² N² / [ b (10 + 4.5 D/l)]
Factoring 10 from the denominator gives:

L = 10-7 N² ( a² / b ) / (1 + 0.45 D/l) [Henrys]

where..

- L is the inductance in Henry
- D is the coil diameter in meters
- r is the radius in meters (or D/2)
- l is the lenght of the coil in meters
- N is the number of turns

noteThis formula applies at ‘low’ frequencies (<3MHz) using enameled copper wire (magnet wire) close wound.

Tip 1Small reductions in the inductance obtained can be achieved by pulling the turns apart slightly. This will also reduce self-resonance. Other combinations of wire and coil diameter may be tried but best results are usually obtained when the length of the coil is the same as its diameter.

Tip 2 If you need good induction stability in the presence of vibration then wind the coil on a support made from a suitable non magnetic plastic or ceramic former and lock the windings using epoxy glue or other suitable adhesive.