# Spiral-shaped variable inductor

*Pieter-Tjerk de Boer, PA3FWM pa3fwm@amsat.org*

(This is an adapted version of part of an article that I wrote for the Dutch
amateur radio magazine *Electron*, December 2015.)

I have written about variable coils for antenna tuners whose inductance increases faster than linearly with the position of the control knob, namely my own exponentially switched coil and the cone-shaped roller inductor. In the meantime, I found another possibility in the book "75 Jahre Sendertechnik bei AEG-Telefunken" (by W. Burkhardtsmaier, 1978): a coil in the form of a flat spiral; see the first figure. I then found a much smaller version at the "De Lichtmis" radio fleamarket, see the second figure. The idea is that by moving the contact over the spiral, a smaller or larger part of the spiral is used.

How would the inductance of such a structure vary?
Let's call the number of turns being used *n*, counting from the center.
Then the circumference of the turns being used increases linearly in *n*.
The inductance of a single, sufficiently large turn is directly proportional to its circumference.
The inductance of *n* turns of the same size is directly proportional to *n* if the turns do not "see"
each other at all, or to *n*^{2} if they are coupled to each other perfectly.
(I wrote more about this earlier.)
In case of our spiral coil, the turns are only partially coupled: the coupling between two successive turns it relatively strong,
but the coupling between e.g. the first and the last turn is rather weak because the first turn encapsulates only a small part
of the magnetic field of the much larger outer turn.
Combining this with the increasing circumference discussed above,
we can expect that the inductance as a whole varies somewhere between *n*^{2} and *n*^{3},
which has indeed been confirmed by measurements I did on the second coil.

B.t.w., in the case of the first coil, this structure was not chosen because of the desired variation of the inductance, but because of the power: this coil was part of a 500 kW shortwave broadcast transmitter and had to endure currents up to 350 A. With a normal roller coil it was not feasible to have sufficient contact pressure and stability for this, but with this flat spiral structure it was feasible.