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Wheeler's 1982 formulas verified

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

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  1. The Phantom

    The Phantom Guest

    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 Edwards

    Reg Edwards Guest

    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 Phantom

    The Phantom Guest

    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.
     
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