Analytical Approximation for the Inductance of Circularly Cylindrical Two-Wire Transmission Lines with Proximity Effect

H A Aebischer,H Friedli

doi:10.7716/aem.v7i1.516

Abstract

The paper describes a simple analytical approximation for the inductance of two-wire transmission lines of circularly cylindrical wires with proximity effect. It yields precise results up to very high frequencies, and also at all interaxial distances between the wires above some limit. Its accuracy is established by comparison to numerical computations and to measurements. It is shown that the proximity effect cannot be neglected unless the interaxial distance between the wires amounts to at least four wire diameters. Further, images of the current distribution in various situations are discussed.

Highlights

At high frequencies inductance decreases compared to its value at DC due to skin effect, which has been well documented for wires of circular, annular, and rectangular cross section [1] – [6]
We describe a simple yet precise analytical approximation for the inductance of a cylindrical two-wire transmission line that takes the proximity effect into account
Equations (11) – (14) for L and equation (18) for C allow to compute the precise wave impedance of twowire transmission lines, even in function of the frequency, via the equation Zwave = √L/C. It seems that not much information is available on the inductance of cylindrical two-wire transmission lines in the literature, and that its calculation requires extensive numerical computations

Summary

Introduction

At high frequencies inductance decreases compared to its value at DC (direct current) due to skin effect, which has been well documented for wires of circular, annular, and rectangular cross section [1] – [6]. Et al presented results of precise inductance and resistance measurements on rectangular loop coils composed of circularly cylindrical copper wires at various wire distances and frequencies [5]. Some of these measurements entailed the proximity effect. We describe a simple yet precise analytical approximation for the inductance of a cylindrical two-wire transmission line that takes the proximity effect into account. We verify it by means of systematic numerical calculations for a set of interaxial wire distances and frequencies.

Numerical solution method

Results and discussion

Analytical approximation

Comparison with measurements

Conclusions