Calculation of phasor fields around A.C. transmission lines using temporal and spatial complex variables

The field variables (potentials and induced voltages, electric and magnetic fields) in the neighborhood of a.c. transmission lines are periodic functions of time and can be represented by phasors or temporal complex variables. At the same time, the point in space can be represented by a position vector, a spatial complex variable, and the field quantity as a function of that spatial complex variable. Thus, phasor fields appear as complex variables both in relation to time and space. The use of temporal and spatial complex variables has the advantages of regular complex variables: simple analytical formulations and derivations, and generalizations of real variable procedures (e.g. Taylor series expansions and closed form integration). As an application, the paper presents the calculation of inductive and capacitive effects of a three phase overhead line on a parallel telecommunication line.