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Published: 08 April 2015

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Inductor with noncircular winding
Quite often, the inductor is wound on a coilformer having a square or polygonal shape. Such inductor can be represented as an equivalent cylindrical coil with the same winding length and number of turns. The easiest way is geometric. Either accept as the equivalent a cylindrical coil having an area of a cross section just the same as a polygon or with the same perimeter  the truth lies somewhere in the middle.
In 1946 F. Grover in the work "Calculation of inductance  working formulas and tables" leads tabular data of polygonal coil with matching to it cylindrical equivalent, that had been getting from experimental measurements. On the basis of these tables, R. Weaver in 20102012 derived an empirical formula that allows calculating the polygonal coil numerically.
Denote the radius of the circumcircle of the polygon as r_{0}. It is obvious that 2r_{0} = D_{0}. The crosssectional area of the polygon is:
And its perimeter:
We denote the radius of the circle with length equal to the perimeter of the polygon as r_{P} and the radius of the circle with the same area as the polygon  r_{A}, then it is obvious that r_{P} = P/2π, and r_{A} = √(A/π).
As you know, the inductance of a infinitely long solenoid is proportional to its crosssectional area. It would seem enough to determine r_{A}, but it's not right for short coils. The equivalent radius r_{E} = 2D can be found as the average value between r_{P} and r_{A} by introducing the correction factor k_{W}, which varies from one when the length of the coil tends to zero and to zero  when the length of the coil goes to infinity. As a result, the equivalent winding radius of the coil is determined by the following formulae:
where: l  winding length of the coil, and D_{0} diameter of a circle describing the polygon. The coefficient 368 selected empirically in order to the calculation had an accordance with F. Grover's tables.
The Coil32 uses this R. Weaver's numerical method. The error of calculation by this method is less than ±1.5% for the coils with a triangular crosssection of winding and becomes much smaller when you increase the number of sides of the coil.
Reference:
 Numerical Methods for Inductance Calculation (Coils on Polygonal Coil Forms)  Copyright 2010, 2014, Robert Weaver
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