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Published: 25 January 2015

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Most often in different soft and online calculators the multilayer inductor is calculated by a simple empirical formula H.А. Wheeler:
 L  inductance (µH)
 N  number of turns
 R  average radius of winding (mm)
 l  length of winding (mm)
 c winding thickness (mm)
In previous versions of Coil32 this formula was also used.
But ... Any empirical formula not reflect the real physical phenomena and has limited accuracy of calculations. In this case the limit is quite serious. The Wheeler formula with sufficient accuracy (± 1%) suitable for calculating only the coil with limited geometry of winding. What does this mean? This means that the cross section of the coil winding as possible must have a square shape. In other words, the winding length "l" should be roughly equal to the thickness of winding "c". But we see that the formula does not take into account the thickness of the insulation of the wire, therefore 1% deviation increases significantly even with "friendly" to the formula winding geometry. The introduction of the socalled "packing factor" does not help the situation. In practice often we have to make quite long coil, winding length of which is significantly greater than the thickness and with a small number of layers. Such coils the formula calculates absolutly wrong. The limiting case where the multilayer coil get down into a single layer. The calculation by the above formula gives absolutely wrong result!
Why, then, do enjoy this limited formula? Since, the multilayer coil with a square cross section of the winding is optimal. Inner winding radius also needs to be equal to its length. At this geometry of winding, the maximum inductance is obtained with a minimum wire length. As a consequence  loss in the coil is also minimal, which is important, for example in the design of crossover inductors.
Where is the solution of the problem? You can use the more accurate empirical formulas, such as published in the book "Radio Engineer's Handbook  1943 F.M. Terman pp 60..63". He gives a formula for short and long coils with the calculated coefficients. You can also use other empirical formulas. But ... First  we are again dealing with the empirical formula, which, because of its limitations somewhere again may be wrong, and secondly  such calculation method was acceptable in the middle of the XX century, when it was necessary to use a pencil and paper , tabular data, and column addition. In XXI century, when according to the Moore's law the computing power doubles every two years, you can use a more perfect algorithm.
This numerical method bases on the formulas that the physicist J.C. Maxwell has proposed in the late XIX century in his famous work  "A Treatise on Electricity and Magnetism." (Maxwell, James Clerk; "A Treatise on Electricity and Magnetism", Vol. 2, Third Edition, Dover 1954. Art. 701, "To find M by Elliptic Integrals", pp. 338340.) Maxwell gives a formula for calculating the mutual inductance of two coaxial circular loops:
 M  mutual inductance,
 r1, r2  radii of loops,
 k  coefficient depending on the distance between loops.
 K and E  elliptic integrals of the first and second kind.
This is the basic formula that displays real physics.
A multilayer coil can be represented as a large number of coaxial circular filaments. Calculating the selfinductance of each turn, and mutual inductance of each possible pair of turns and summing it all, we get a selfinductance of the multilayer coil. Such a calculation under the force of modern computers. It is also necessary to consider an amendment to the fact that the real wire is not infinitely thin. Work on the creation of such a programming algorithm has been done by the Canadian radioamateur Bob Weaver in 2012, as well as Brazilian fan of Tesla coils Dr. Antônio Carlos M. de Queiroz in 2005.
Calculation of the multilayer air core coil in Coil32 (version v8.0 or later ) is based on these decisions. It is assumed that the winding has the form as shown below.
 d is the diameter of the wire on copper;
 k  the diameter of the wire with insulation, or the distance between the centers of adjacent turns
In Coil32 the turns of multilayer coil are adding one by other. The selfinductance is calculated for each new turn and also the mutual inductance is calculated between this turn and all previos turns. The result is summed to total inductance. If the winding form is not the same as on figure, the calculation not will be correct. In this case it is easier to measure than to calculate and we can only estimate required number of turns for multilayer coil.
Coil is "virtually wound" until the required inductance will be reached, the latter layer may be incomplete. Along the way, calculated the total length of the wire and its DC resistance. The same calculation algorithm is implemented in the multilayer coil online calculator. The application have possibility to calculate multilayer air core inductors with insulating layers. You can enable the corresponding checkbock to calculate this coil. On figure you can see the schema of winding of multilayer coil with insulating pads. You can select number of the winding layers between insulating pads, for example two as on the figure.
Audiophiles should not forget that the length of the winding should be equal to the thickness. To achieve this, we have to repeat the calculation several times.
References:
 Numerical Methods for Inductance Calculation  Robert Weaver 2012 (Part 1 –> Elliptic Integrals > MultiLayer Coils)
 Classical Calculation for Mutual Inductance of Two Coaxial Loops in MKS Units  Kurt Nalty 2011
 Mutual Inductance and Inductance Calculations by Maxwell’s Method  Dr. Antônio Carlos Moreirão de Queiroz 2005
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Comments
1) The calculation of the thickness of the coil (c). I believe it should be k*number of layers.
2) The necessary length of the wire. There are different results produced between the older version (9.8) and the newer version (10.0
Thank you. I'll be looking...
The errors have corrected.
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