# Inductance of single-layer coils on cylindrical winding forms

Input the mean radius, length and number of turns of a coil then press
'compute' button.

If the number of turns is set to zero, only the Nagaoka coefficient will be
computed.

Probably no other type of coil is so widely used as a simple helical winding,
such as is obtained by winding a single layer of wire on a cylindrical form.
Single-layer coils have the advantage, not only of simple and inexpensive
construction, but the effective (distributed) capacitance of the winding
is small. For this reason, they are especially useful in high frequency
application, except where higher inductances are required.
The calculation of the inductance of single-layer coil is based on formulas
for a cylindrical current sheet, that is, a winding where the current flows
around the axis of a cylinder of infinitesimal radial thickness on the
surface of the cylinder. Except in the case of very open helical windings,
the inductance of a single-layer coil is closely equal to that of a cylindrical
current sheet having the same number of turns N as the coil, the same mean
radius a, and length l equal to the number of turns in the coil times the
distance between centers of adjucent wires.

An exact formula for the inductance of a cylindrical current sheet was first
found in 1879 by Lorenz(1), who integrated the expression for the mutual
inductance of two equal coaxial circular filaments twice over the length
of the current sheet. Lorenz's formula is in elliptic integrals and involves
both positive and negative terms.

In early days when man have no electronic computers, many forms of series
aproximations and tables are developed. Among of them, Nagaoka's formula (2)
is widely used till now.

L = K*u*PI*a*a*N*N/l
where
L = inductance of single-layer coil (H)
K = Nagaoka coeeficient (0 < K <= 1)
= = 4/(3*¦Ð*sqrt(1-k^2))*((1-k^2)/k^2*K(k) - (1-2*k^2)/k^2*E(k) - k)
k = 1 / sqrt(1 + (l/(2*a)^2)
u = permeability (4e-7*3.1415.. for bacuum, air or nonferrous material)
a = mean radius of coil (m)
l = length of coil (m)
N = number of turns of windings
PI = 3.14159265..

If K equal to 1, this foumula expresses the inductance per unit length
of infinite single-layer coil.
K(k) is the complete elliptic integral of first kind,
E(k) is the complete elliptic integral of the second kind.

## References

(1) Lorenz, Wied. Ann. 7, 161 (1879); B. of S.Sci. Paper 169, 117.
(2) Nagaoka, Jour. Coil. Sci. Tokyo 27, 18-33, art. 6 (1909); B. of S. Sci. Paper 169, 64.