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Modelling of transmission lines using idempotent decomposition Marcano, Fernando José


The modelling of wave propagation in multiconductor transmission line involves full matrices for the wave propagation and characteristic impedances functions. Modal decomposition, as in the fdLine model in the EMTP, leads to an elegant and numerically efficient solution, even in the presence of frequency dependent parameters. The advantages of modal decomposition are lost, however, when the transformation matrix relating modal and phase quantities cannot be assumed constant and real but is complex and changes with frequency. This is the case, for instance, when there is strong conductor asymmetry in multicircuit transmission lines and cable systems. A number of alternatives have been proposed to solve the problem of frequency dependent transformation matrices: from frequency synthesis of the transformation matrices to working directly in the phase domain. Both of these approaches, however, have drawbacks. Direct synthesis of the transformation matrices with stable rational functions is difficult because the eigenvectors that make up the columns of these matrices are not uniquely defined at each frequency point. Direct phase-domain modelling is also difficult because an N-phase transmission line has N propagation modes and N time delays and the N2 elements of [Aphase] are a combinations of these basic travelling times and modes. The idempotent Line Model (idLine) expresses the line propagation function as a matrix directly in phase coordinates [Aphase] (thus avoiding modal transformation matrices), but the expression is in terms of the N natural propagation modes (thus avoiding mixed-up travelling times). With idempotent decomposition, the line propagation matrix can be written as a combination of the modal propagation functions with the idempotent matrices as weighting factors. As opposed to the eigenvectors, which are defined only up to an arbitrary complex constant, the idempotent coefficient matrices are uniquely defined at each frequency point. In the idempotent line model, each scalar modal propagation function is synthesised in the frequency domain using a rational function approximation for the wave shaping and the mode's travelling time for the wave delay. The elements of the idempotent matrices are relatively simple functions of frequency that can also be synthesised using rational function approximations. The proposed model is very accurate and numerically stable. A number of simulations are presented and comparisons are made between the new model, the traditional fdLine model, and the "exact" solution obtained with the frequency domain program FDTP.

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