Coil Resistance Formula
It is quite straightforward
to derive an equation which will give you the coil resistance to an accuracy of
better than +/10%. This equation uses the main parameters of outer radius, RO,
inner radius, RI, length L and wire diameter as
illustrated in fig 1 below.
Fig
1 Coil parameters
The
amount of space between adjacent turns will depend on how neatly the coil is wound.
If we consider the wire to be wound neatly so that each strand effectively occupies
a total crosssectional area of
then ratio between the effective occupied area and the actual crosssectional
area of the wire is given below.
This
is a reasonable assumption since each layer of the coil lies at a slight helix
angle opposite to the preceding layer so the turns don't fall into the gaps in
the lower layer. However for the general case we will adopt a general ratio, F,
and term it the coil 'filling factor'. I would think that in most instances F
will lie around /4
(0.79).
If
we assume this ratio holds throughout the whole volume of the coil we can express
the volume of copper wire, Vc, in terms of the total volume, VT, as follows:
Now
that we have an expression for the volume of wire in the coil we can easily determine
the actual length of wire, Lw, as follows:
Now
that we have an expression for the length of the wire we need one more substitution
to express the resistance in terms of the coil parameters. Taking eqn 3 and substituting
it into the standard resistance equation
gives us the desired result:
So now we have
the coil resistance, R_{C}, expressed in terms of the wire resistivity,,
the coil dimensions, and the wire diameter, .
If we assume a
filling factor of ^{}/_{4}
then this reduces to:
Something which
should be noted is that the resistance depends on the .
This means that Rc changes by a large factor for a small change in .
Lets look at an example:
Suppose we have
two coils, A and B, with the following dimensions,
Ro = 20mm, RI =
10mm, L = 20mm, A
= 1mm, B
= 1.2mm.
We will use
= 1.7E5 mm
Working out the
resistance of each coil gives  Coil A = 0.41,
Coil B = 0.20.
So you can see
that small changes in the wire diameter will have a large effect on the resulting
resistance.
This formula gives
fairly accurate results as shown by the comparison between the measured and calculated
test coil resistances. There are several things
which can throw the result off; The winding may not be a perfect hollow cylinder
 it might be slightly oval in shape; The wire diameter may not be the exact
stated diameter. The resistivity of the wire may not be correctly known  a best
guess would be to use the value for pure copper (assuming the wire is copper of
course); Finally the windings may not form such a regular pattern and this will
alter the ratio of wire volume/total volume (filling factor).
