Cable System Transients

Chapter 3Theory of Wave Propagation in Cables

Akihiro Ametani

Graduate School of Electrical Engineering, Doshisha University, Kyoto, Japan

3.1 Modal Theory

A single-phase cable is, in general, composed of two conductors, a core and a metallic sheath (shield) as explained in Chapter 2. Thus, it is necessary to deal with a multi-conductor system in an analysis of voltages and currents of the cable. A basic theory of the multi-conductors involves matrix algebra for the impedance and the admittance which are given in a matrix form. A theory of eigenvalues and vectors or diagonalization of a matrix is learnt as one of basic mathematics in any university engineering department. In a distributed-parameter circuit, such as a cable, the theory of eigenvalues and vectors relates to the wave propagation along a conductor. This is called “Modal theory” or “Natural mode theory” [1–3], because an eigenvalue of the product of the impedance and the admittance gives wave propagation related parameters, that is, the attenuation constant and the propagation velocity on the conductor in a modal domain, but not in a phase domain.

The modal theory is used to transform voltages, currents, impedances and admittances in a phase domain to those in a modal domain, and is used to evaluate propagation constants, characteristic impedances and transformation matrices in the modal domain. By transforming those back to the phase domain, we can obtain voltages, currents, characteristic impedances, and so on ...