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Network analysis, in terms of S-parameters, is invariably carried out in modern textbooks by employing the power-wave concept. It is shown that this approach is not relevant to practical measurement; furthermore, the complication of square roots, real parts, conjugates, moduli and conjugate transpose matrices is carried throughout the mathematics. These difficulties are overcome by employing the voltage-wave concept with complex normalisation, whereby the analysis is carried out in terms of port-voltage and port-current relationships, from which, power flow relationships are readily obtained in a straightforward manner. The new method of network analysis presented is not only completely general and rigorous, but simple to apply and faster than any other known technique. It is based on a once-for-all solution of the most general problem of all in 3-port analysis, namely, a 3 Ã 3 renormalising matrix, each element of which embodies three renormalising factors. Examples are given of the derivation of some 2-port and 3-port network equations, but in a more general form in which the port normalising impedances are not only different but complex. It is concluded that the voltage-wave S-parameter approach, with complex normalisation, should not only be revived and taught in academic institutions, but that it should take precedence over the power-wave approach which has a very restricted application. In a companion paper, a reappraisal is made of computer-corrected network analyser design and calibration.