Formulas for the inductance of rectangular tubular conductors

THE formula for the inductance per unit length of two parallel conductors of rectangular cross section, obtained by using the formal expressions for the geometric mean distance of a rectangle to itself and the geometric mean distance between two rectangles with sides parallel, is exceedingly cumbersome. More than a score of logarithms and arc tangents must be evaluated when computing the inductance for a given conductor spacing and cross section. To circumvent such lengthy calculations when computing the reactance of like parallel rectangular strap conductors, Dwight^1,2 has published curves, each plotted for a certain conductor spacing and cross section, from which the reactance can be obtained. Unless, however, the conductor spacing and the ratio of conductor thickness to breadth coincides with those values for which the curves are plotted, interpolation is necessary. Recently Roth³ has expressed the inductance of such lines in the form of a rapidly converging series, the parameters of which are the spacing and dimensions of the conductors. Usually, a few terms of this series suffice to yield a value of inductance sufficiently accurate for all design purposes. As regards rectangular tubular conductors, the only analytical literature is a paper by Dwight and Wang⁴ giving formulas for thin square tubular conductors. These are derived on the assumption that the conductor walls are so thin that the sides can be considered as but line segments. The geometric mean distances for such segments are then used to calculate the inductance. Such a procedure is equivalent to asserting that the current flows solely on the surface of the conductors. It follows, then, that the values of inductance computed on this basis will be less than the actual values. Consequently, while these formulas suffice for many calculations on square tubular conductors, if accuracy is a desideratum of design this minimization of thickness restricts the range of application of these formulas to very thin square tubular conductors. Recognizing this restriction and noting, further, that commercial conductors have rounded corners, Dwight and Wang give formulas, empirically deduced, from which corrections for thickness and for rounded corners can be calculated.