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In the original development of z-transform electromagnetic transient analysis in power systems, transmission-line impulse responses and surge-impedance functions formed initially in the frequency domain were mapped into the z-plane by bilinear transformation, although it is known that this transformation is one that gives rise to a form of distortion error. In now providing means by which this error may be avoided, methods are investigated in the paper by which transmission-line response functions can be synthesised directly in the z-plane. Given the high accuracy in function definition which this can provide, an examination is made of the relationships that are to be satisfied between the step length in the time-domain and transmission-line aerial-and earth-mode transit times. In particular, it is shown that residual errors arising from the frequency dependence of the earth-mode transit time are amenable to analysis and then, based on this, to control by interpolation. Drawing on trial analyses of a 400 kV single-circuit untransposed line, the paper then evaluates the further advances in z-transform electromagnetic transient analysis that these further developments can achieve.