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(jω) = C(jω)/R(jω). Those textbooks1¯
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which present a comprehensive, integrated account of basic servomechanism theory advance, in considerable detail, two procedures for determining the maximum modulus Mm of M(jω) for a unity feedback system, namely, by plot of the transfer function G(jω) on a chart of circles of constant values of modulus |M(jω)|, or on a Nichol's chart of circles of constant values of modulus |M(jω)|. 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(jω) = G(jω)Kh as just mentioned and thence calculating Mm from Mm = M1m/Kh.