The calculation is based on Harold A. 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]

$$L(H)=\frac{10-6 a^2 N^2}{[ b (10 + 9a/b)]}[Henrys]$$

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.

Here are some more information (TBD):

The single layer air core coil formula is most accurate when the coil length is greater than 0.67r and the frequency is less than 10 MHz. As the frequency goes above 10MHz, the formula becomes less accurate, because parasitics dominate the circuit. In all cases, the length is 4 times the radius.

Formula in Inch Units: L = r2N2 / (9r + 10A); N = (L(9r + 10A) / r2)1/2

Formula in Metric Units: L = 0.394r2N2 / (9r + 10A); N = (L(9r + 10A) / 0.394r2)1/2 Where:

- L = inductance (in microhenries).
- r = radius of coil (in inches or cm)
- N = number of turns.
- A = length of winding (in inches or cm)

Formula: Q = XL / RS, Where:

- XL = 2πƒL. where ƒ = Frequency (Hz); L = Inductance in Henries.
- RS is determined by multiplying the length of the wire used to wind the coil by the D.C. resistance per unit length for the wire gage used.
- Q changes dramatically as a function of frequency.
- At lower frequencies, Q is very good because only the D.C. resistance of the windings has an effect which is very low.
- As frequency goes up, Q will increase up to about the point where the skin effect and the combined distributed capacitances begin to dominate.
- From then on, Q falls rapidly and becomes 0 at the self resonance frequency of the coil.

• Spread the windings. Air gaps between the windings decrease the distributed capacitances.

• Use a ferrite core or powdered iron to wind the coil on. This will increase the permeability of the space around the core.

• Decrease the series resistance of the windings by increasing the wire gage used. Larger wire has a lower resistance per unit length.

The presence of conducting material in the vicinity of the coil disposed in such a way as to form a shorted-turn coupled to the coil, and especially of any metal screening-can or box, will cause the effective inductance to be less than that predicted for an isolated coil.

The presence of any non-conductive ferromagnetic material in the vicinity of the coil will cause the inductance to be greater than that predicted by formulae which assume that μ=μ0.

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