A matrix method of calculating the distribution of transient voltages in transformer windings

The paper describes a matrix method of calculating the distribution of transient voltages in a uniform transformer winding hit by an impulse wave. By subdividing the winding into n elementary coils which are identical among themselves, and neglecting the resistances of all coils and the series capacitances of non-adjacent coils, an equivalent circuit is obtained in which the voltages and currents are governed by an ordinary linear matrix differential equation of the first order with a square coefficient matrix of order 2(n¿1). Use is made of the fact that the coefficient matrix of the governing equation can be represented as a hypermatrix whose submatrices are rational algebraic functions of the so-called uniform continuant matrix of order n¿1. By means of the canonical representation of the uniform continuant matrix the governing equation is split up into n¿1 independent matrix equations whose coefficient matrices are second-order square matrices. Formulae are derived and illustrated with examples for the solution of these equations. Expressions for the transient voltages are obtained in terms of the eigenvalues and eigenvectors of the uniform continuant matrix.