Optimum passive differentiators

Abstract

A general, nth order, transfer ,!unctiOD (TF) is derived, whose time-domain response approximates optimally that of an ideal differentiator, optimality criterion chosen being the maximisation of the first n derivatives of the ramp response at t=O⁺. It is shown that transformer less, passive, unbalanced realisabillty is ensured for n<3, but for n>3, the TF is unstable. For n=3, the TF is not realisable, however, near optimum results can be obtained by perturbation of the pole locations.Optimum TFs are also derived for the additional constraint of inductorless realisability. It is.shown that TFs for n:≥ 2 are not realisable. For all n, however, near optimum results can be achieved by small perturbations of the pole locations; this is illustrated in this paper for n=2.Network realisations, for a variety of cases, are also given.