## Single-layer coil inductance calculator

Most of the formulas for the inductance of a coil are valid for the current sheet approximation, where the current flows in an indefinitely thin surface around the coil diameter. This is the same as assuming the coil wound with an indefinitely thin tape with negligible separation between turns. If the separation between turns is not small, a correction factor should be applied. Moreover, at high frequencies the current crowds towards the inside of the coil so the effective radius where the current flows become smaller. Sometimes it is suggested to use the internal radius of the coil instead of the wire mean radius in the calculations, in order to compensate for this effect. However the difference between the low- and high-frequency inductances is usually not large [1]. To compute accurately the inductance of any kind of coil (or also of more complicated conductiong structures) one can use an electromagnetic simulator, as explained here. Regarding current sheet inductance formulas for single-layer coils, one of the most widely known is the one by Wheeler [2], which states (after converting to metric units):

where d is the coil diameter in meters, n the number of turns and l the coil length in meters. The above formula is accurate within 1 % for l>0.4d ; for shorter coils one can use the well-known Nagaoka formula [3] (which has the inconvenience of requiring a list of tabulated values for different diameter/length ratios) or other asymptotic approximations [4]. Some useful formulas, applicable for any diameter to length ratio, are presented in [4] and [5]; the following form implements the latter, for which the maximum relative error is stated to be less than 3 ppm. The Q value is computed here using the formula of [6]; take the resulting value with care, since the limits of validity are not clear to me.