Analytical calculation of the magnetic field produced by electric power lines

The magnetic field produced by electric power lines is usually calculated numerically with the use of a computer. However, the analytical calculation of the magnetic field is preferable because it results in a mathematical expression for showing its dependences on the various parameters of the line arrangement. A method to derive the analytical formula of the magnetic field vector produced by any power line is developed in this paper. The specific formulas for the magnetic field produced by any single circuit line in flat, vertical, or delta arrangement, as well as for hexagonal lines considered as double circuit lines in super bundle or low reactance phase arrangements or as six-phase lines, are given. The derived formulas are valid at any point with practical importance, close to or far from the line. The development of the method is made possible with the use of new kinds of numbers, named double complex numbers, to represent the magnetic field vector in the vicinity of power lines. Double complex numbers remarkably simplify the mathematical expressions for the magnetic field vector. Using these numbers, it is observed that the infinite terms of the magnetic field multipole expansion, for flat single circuit lines and for lines exhibiting polygonal symmetry are contracted, resulting in simple formulas for the magnetic field vector, which is used to derive the formulas for the resultant value of the magnetic field. The general formula of the magnetic field vector produced by an arbitrary power line is a rational function of the distance from it. Through the given expressions for the coefficients of this function numerator and denominator, the formula for the magnetic field vector produced by any power line can be derived.