A graphical procedure for determining the gain of a servomechanism (or a specified maximum modulus less than unity

A PROBLEM of frequent occurrence in servomechanism analysis and design is that of determining the gain K of the forward-frequency transfer function $[G(s)]_{s} = j omega = left [ {K(1+a_{1}s+ ldots + a_{m}s^m)} over {s^k (1+b_{1}s + ldots + b_{n}s^n)}right ]s = j omega eqno{hbox{(1)}}$ of a single-loop unity feedback system, Fig. 1, such that the over-all frequency transfer function ${C(j omega) over R(j omega)} = {G(j omega) over 1+G(j omega)} = {vert G(j omega)vert e^{j alpha 1} over vert 1+G(j omega) vert epsilon ^{j alpha 2}} = M epsilon ^{(j alpha 1-alpha 2)} = M epsilon ^{(j alpha)} eqno{hbox{(2)}}$ has a specified maximum modulus M = Mm. Accordingly, those textbooks1¿4 which present a comprehensive integrated accountof basic servo mechanism theory advance the outline of a simple graphical procedure for determining the required gain K, providing Mm > 1.