Analytical Solution of the Voltage Distribution Problem Along Insulator Strings of Nonuniform Disks

This paper deals with deriving the analytical solution to the problem of determining the voltage distribution along the suspension insulator strings composed of a mixture of clean and polluted nonidentical disks having nonuniform ground and line stray capacitances. A system of linear equations describing the voltage distribution was formed based on the insulator string equivalent lumped circuit and transformed into a tridiagonal matrix equation. Explicit recursive formulae for calculating the tridiagonal system matrix determinant, adjugate, and inverse were derived. Closed-form non-recursive formulae in terms of special second-order matrices for computing the entries of the adjugate or adjoint matrix were also obtained. Moreover, an alternative method for forming the tridiagonal system matrix adjugate and inverse matrix in terms of diagonal and triangular matrices and their transpose was developed. The said formulae enable one to calculate the string efficiency and the voltage distribution along the insulator string for the general case where nonuniformity of the disks and stray capacitances are imposed and different contamination levels are prescribed. The formulation approach to the problem and the analytical solution obtained admits the inclusion of tower design and environmental effects such as due to wind velocity and the use of corona rings and grading rings.