Advance Theory

Measurement units don't matter as long as you use the same one.

A cylinder coil :

L : inductance, r : radius, N : number of turns, b : length of coil, w : length of wire, d : turns per (what ever unit you are using).

L = (r * r * N * N) / (9 * r + 10 * b)

Using this and solving for L in terms of r. and using (N = b / d and b = w / (2 * Pi * r * d)) we get

L = w / (4 * Pi * Pi * (9 * r + (10 * w) / (2 * Pi * r * d)))

Now I took the derivative in respects to r

- (d * w * w * (9 * d * Pi * r * r - 5 * w)) / (4 * Pi * (9 * d * Pi * r * r + 5 * w) ^ 2)

Set this to 0 for the maximum and solve

r = sqrt((5 * w) / d) / (3 * sqrt( Pi ))

Now replace use w = b * 2 * Pi * r * d and you get

r = b * 10 / 9

So this is the maximum inductance for a cylinder coil.

A pancake coil :

L : inductance, r : mean radius, N : number of turns, w : inner radius minus the outer radius

L = (r * r * N * N) / (8 * r + 11 * b)

Doing the same thing above we get

r = w * 11 / 8

So this is the maximum inductance for a pancake coil.

Other :

More stuff will be posted here as i learn more.  ¦¬)