22. Optimum Passive Differentiators

A general, nth order, the transfer function (TF) is derived, whose time-domain response approximates optimally that of an ideal differentiator, optimality criterion chosen being the maximization of the first n derivatives of the ramp response at t = 0⁺. It is shown that transformerless, passive, unbalanced realizability is ensured for n < 3, but for n > 3, the TF is unstable. For n = 3, the TF is not realizable, however, near optimum results can be obtained by perturbation of the pole locations. Optimum TFs are also derived for the additional constraint of inductorless realizability. It is shown that TFs for n ≥ 2 are not realizable. For all n, however, near optimum results can be achieved by small perturbations of the pole locations; this is illustrated in this chapter for n = 2. Network realizations, for a variety of cases, are also given.