Determination of the maximum modulus, or of the specified gain, of a servomechanism by complex-variable differentiation

A problem of frequent occurrence in servomechanism analysis and design is that of determining the maximum modulus Mm, and the angular frequency at which it occurs, of the over-all frequency-transfer function M(j¿) = C(j¿)/R(j¿). Those textbooks1¿4 which present a comprehensive, integrated account of basic servomechanism theory advance, in considerable detail, two procedures for determining the maximum modulus Mm of M(j¿) for a unity feedback system, namely, by plot of the transfer function G(j¿) on a chart of circles of constant values of modulus |M(j¿)|, or on a Nichol´s chart of circles of constant values of modulus |M(j¿)|. Further, if the feedback transfer function H(s) although nonunity is yet a pure numeric H(s) = Kn, then, according to well-known theory $eqalignno{{C(j , omega) over R (j , omega)} = M (j omega) cr = {1 over K_{h}} , {G(j omega) K_h over 1 +G(j omega) K_{h}} cr = {1 over K_{j}} times M_{1}(j omega) hbox{(1)}}$ and Mm can yet be determined by obtaining M1m through plot of G1(j¿) = G(j¿)Kh as just mentioned and thence calculating Mm from Mm = M1m/Kh.