Abstract
The best analytical formulae for the self-inductance of rectangular coils of circular cross section available in the literature were derived from formulae for the partial inductance of straight wires, which, in turn, are based on the well-known formula for the mutual inductance of parallel current filaments, and on the exact value of the geometric mean distance (GMD) for integrating the mutual inductance formula over the cross section of the wire. But in this way, only one term of the mutual inductance formula is integrated, whereas it contains also other terms. In the formulae found in the literature, these other terms are either completely neglected, or their integral is only coarsely approximated. We prove that these other terms can be accurately integrated by using the arithmetic mean distance (AMD) and the arithmetic mean square distance (AMSD) of the wire cross section. We present general formulae for the partial and mutual inductance of straight wires of any cross section and for any frequency based on the use of the GMD, AMD, and AMSD. Since partial inductance of single wires cannot be measured, the errors of the analytical approximations are computed with the help of exact computations of the six-dimensional integral defining induction. These are obtained by means of a coordinate transformation that reduces the six-dimensional integral to a three-dimensional one, which is then solved numerically. We give examples of an application of our analytical formulae to the calculation of the inductance of short-circuited two-wire lines. The new formulae show a substantial improvement in accuracy for short wires.
Highlights
The self-inductance of a straight wire may only be defined as so-called partial inductance [1, 2]
We have shown that the known formulae for calculating the partial inductance of straight wires of circular cross section lead to large approximation errors for short wires
We have presented two methods for deriving better analytic formulae for the partial inductance of straight wires of any cross section and for any frequency, and we have used them to derive formulae for the case of wires of circular cross section in the low- and high-frequency limits
Summary
The self-inductance of a straight wire may only be defined as so-called partial inductance [1, 2]. It was originally proposed by Rosa [1], he did not carry it out In this method, in addition to using the GMD as described above, the arithmetic mean distance (AMD) and the arithmetic mean square distance (AMSD) are used for calculating the normalized double area integrals of the remaining summands of equation (2), see sections 3 and 4. It consists of forming the normalized double area integrals by applying the GMD and AMD as in the mean distances method, but expanding the summands
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