Wheeler's 1982 formulas verified

Discussion in 'Electronic Design' started by The Phantom, Jun 15, 2005.

1. The PhantomGuest

About April 7 I posted some formulas due to Harold Wheeler for calculating the inductance
of solenoids wound on circular and square forms. I tweaked his formulas to get .02%
accuracy, and included a correction for the circular form. I've now added a correction
for the square form.

My main further work on these formulas has been to verify their accuracy. I have found
that their only substantial errors occur for small diameter solenoids, and I have now
verified that these formulas will provide stated accuracy for solenoids with few turns,
all the way down to the single turn case.

Let D = the diameter of a circular solenoid, d = the diameter of the wire used to wind the
solenoid, l = the length of the winding and p = the pitch of the winding (center to center
distance between two adjacent turns). I found that if D/d > 5, the formulas have an error
less than .2% and if D/d > 50, the formulas have an error less than .02%. These errors
are not influenced by the length of the winding, so for all ratios of D/l the error will
be as just described. Also, the correction for pitch (Grover's table 38) goes all the way
to a ratio of 100 for p/d, which is a winding with *very* widely spaced turns. Grover
doesn't specifically discuss the error for such widely spaced windings, but in the preface
to the book, he says that the tables are intended to provide an error of 1 part in a
thousand.

Just to give an idea of what this means, a D/d of 5 is about like winding 4 gauge wire on
a pencil; D/d would be like winding 24 gauge wire on a pencil. It's apparent that the
inductance of a typical solenoid can easily be calculated with an accuracy of better than
..1%, provided its physical dimensions can be measured with that accuracy.

To verify the accuracy of the formulas I needed to accurately determine the inductance of
a single ring of round wire, the ring having radius D/2 and wire diameter d. Grover gives
formula (119a) on page 143 of his book. For D=1 cm and d=1 cm, formula (119a) gives an
inductance of .0020699 uH. The GMD method (Grover, pp 17-25) gives an inductance of
..003455 uH using the exact formula for the mutual inductance of two circular filaments and
the GMD of a wire of diameter 1 cm. These two values are very different; which one to
believe?

On page 9, Grover says that a method of calculating the inductance of "Actual Circuits and
Coils" is to integrate a basic formula such as the one for the mutual inductance of
circular filaments over the cross section of a winding. As he says, "Such direct
integration is, in general, too difficult". But as he says elsewhere, it can be done
numerically. I was able to do this for the case of a circular ring of wire, and for a
ring with D=1 cm and d=1 cm, as above, the numerical integration gave an inductance of
..003966 uH, a result much closer to the GMD method result than to the result of formula
(119a). Formula (119a) is not very accurate for small diameter rings. It can be improved
by adding more terms from series formula (119).

However, I wanted to see if I could verify the result of the numerical integration
somehow, and what I did was this: In chapter 13, page 94, et. seq., are tables for
calculating the inductance of circular coils of rectangular cross section. I considered a
coil of 1 turn of wire of mean radius .5 cm and diameter 1 cm. This is a c/2a ratio
(Grover's nomenclature) of 1, and table 21 gives a value of Po' of 7.112. This would give
an inductance of .003556 uh, but we must apply a correction for the fact that the
rectangular cross section of the coil is not completely filled with copper. Grover gives
the correction on page 99, formula (96). Multiplying .003556 by 1+(.739 * .155), we get
..003963 uH, a value very close to that obtained with the numerical integration described
in the paragraph above.

Grover says on page 9, last paragraph, that the case of the inductance of a circular coil
of rectangular cross section was solved by direct integration. Table 21 was no doubt
generated by this method and the values are exact to the number of figures shown. I
believe this because when I do the integration numerically, I get exactly his values.

So, by this method I can get exact values for the inductance of a small diameter circular
ring of round wire. This is how I verified that the Wheeler formulas (for solenoids with
circular cross section) with correction have the accuracy stated above. Because they are
based on the Nagaoka function, they actually give a better result for the inductance of a
ring (single turn) of wire than Grover's (approximate) formula (119a) for small diameter
rings. I didn't check the formulas for square cross section coils as thoroughly, but
several spot checks gave similar accuracy.

The formulas with corrections are posted over on ABSE.

2. Reg EdwardsGuest

About April 7 I posted some formulas due to Harold Wheeler for
calculating the inductance
================================

What did you use for your standards of inductance?

3. The PhantomGuest

I didn't use any *physical* standards of inductance, if that's what you mean.

There a number of physical geometries for inductors that have known *exact* closed form
solutions. For example, Maxwell gives an exact formula for the mutual inductance between
two circular filaments. See his "A Treatise on Electricity and Magnetism", section 701.
The exact, closed form expression for the inductance of a cylindrical current sheet was
found in 1879 by Lorenz. This is how the NBS standards of Grover's day were produced. A
geometry was selected such that the inductance could be known just by precise measurements
of the dimensions of the inductor.

From these formulas, further expressions may be derived for more practical inductors such
as a solenoid wound with round wire. Grover, in his book "Inductance Calculations:
Working Formulas and Tables", shows how this is done. Chapter 16 is devoted to
single-layer coils on cylindrical winding forms, for example. One can calculate the value
of such inductors to much greater accuracy than the formulas I have given, but that
requires using the tables in the Grover book; six figure accuracy is attainable that way,
assuming you have measurements of the inductor dimensions to six figure accuracy.
Wheeler published an earlier, well known formula in 1928 which has errors up to 10 percent
or more for certain extreme ratios of diameter/length. In 1982 he published an updated
pair of formulas that were .1% accurate approximations to the inductance of a cylindrical
current sheet for all ratios of diameter/length. Add to those the corrections for round
wire windings found in Grover, and you have quite accurate formulas.

The purpose of these formulas is to provide inductance calculations accurate to better
than .1 percent for all (well, nearly all) diameters and lengths and for the formulas to
be simple so they can just live in your programmable calculator.