The Transfer Function and Sensitivity of a Network with n Variable Elements

Sensitivity: Old Questions, Some New Answers

“…A considerable number of people [2]- [5] were involved in the early 1960's, and a group of essentially identical computer programs emerged, all using a transformation of one sort or another, of the independent frequency variable. The most frequently used transformation s2 + co; z2 = ~ s2 + 0; of the filter.…”

“…Sorensen [4] has extended these results to include those cases where x is associated with the unilateral immittance of any type of controlled source. Also, Troop and Peskin [5] have generalized this formulation to include circuits with 12 rather than a single variable element.…”

“…If the nominal value of the characteristic equation is denoted by Q(s, x0) then Q(s, x0) = B(s) + x,A(s) = 0. (5) If Q(s, x0) has a root of multiplicity v, then it is possible to show that dp, = [-r,-,c?x]"~ (6) where, in the neighborhood of s =p,,, the Laurent expansion of AbYQb, x0) is A(s) Q(s, E (s "-p) (7) The latter result, first presented by Papoulis [8], has the computational advantage of expressing root sensitivities in terms of simple residue type calculations. The relationship between zero-pole sensitivities and the system sensitivity is established by expressing T(s, x) in the following form fi rs -zi(x)] T(s, x) = K(x) i;l PI Ls -Pi(x)l (8) where K(x) is a constant and zi(x) and p,(x) are simple zeros and poles which depend upon x.…”