Frequency-dependent transformation matrices for untransposed transmission lines using Newton-Raphson method

The frequency-dependent aspects of transmission line transformation matrices along with their asymptotic behaviours at high and low frequencies are thoroughly investigated in this paper. The Newton-Raphson (NR) method for evaluating the transformation matrices as smooth functions of frequency is introduced. A different technique which utilizes a conventional diagonalization algorithm and a correlation technique for tracking the order of the eigenvectors and eigenvalues is used to confirm the validity of the NR method. Transformation matrices for typical line configurations are evaluated and discussed. The paper concludes that the NR method is more efficient and appropriate for use in the time domain frequency-dependent line models in the Electromagnetic Transient Program (EMTP).