But where does the quartic formula for the spiderweb coil come from?
The equation for a spiderweb coil is most often given in the form below:
L = inductance
r =mean radius
n = number of turns
b = coil depth
which means that given the dimensions of the coil and number of turns, one can determine the inductance of the coil.
This is OK... If you're shooting for a specific inductance to have an LC tank resonate at a certain frequency, you can use a few iterations of this equation to eventually find the number of turns you'll need while building the coil.
But let's rewrite the equation so that we can find the number of turns in one shot, given the desired inductance. First, a couple of definitions are in order:
b (coil depth/spread) = w*n (wire diameter times number of turns)
r (mean radius) = (coil inner diameter + coil spread) / 2
if we use the innermost radius of the coil instead of inner diameter or mean radius, we get
rm = ri + w*n/2 (where rm is mean radius and ri is inner radius)
So substituting all that in, and collecting terms gives us the equation (with x instead of n, and r being coil innermost radius instead of mean radius):