Single layer coil
A single layer coil has two advantages. Firstly, like all air core coils, it is free from 'iron losses' and the non-linearity mentioned above. Secondly, single layer coils have the additional advantage of low self-capacitance and thus high self-resonant frequency. These coils are mostly used above about 3 Mhz.
In the simple case of a single layer solenoidal coil the inductance may be estimated as follows (Wheeler)
L = 0.001 N2r2 / (228r + 254l)
where L is the inductance in henrys, r is the coil radius in metres, l is the coil length in metres (>0.8r) and N is the number of turns.
This formula applies at 'low' frequencies. At frequencies high enough for skin effect to occur a correction of up to about -2% is made.
To construct a self-supporting air cored coil take a length of plain 1 millimetre diameter copper wire and hold one end in a bench vice. Take the other end in a pair of pliers and pull until the wire has stretched slightly - this will straighten it. Using a 5 millimetre diameter drill bit wrap the wire around it until enough turns have been applied. Using 'long nosed' pliers bend the ends of the coil to get them into a radial position.
Small 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.
This property also leads to a disadvantage of the air cored coil: microphony. If you need good frequency stability in the presence of vibration then wind the coil on a support made from a suitable plastic or ceramic former.