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.