Calculation of Feedback Control Systems by Log-Root-Locus : 2 nd Report, Response Characteristics of Closed-Loop Transfer Function

Abstract

Analytical properties of closed-loop transfer functions can be investigated by Log-Root-Locus method and expressed by phase-loci and gain-loci (Fig. 12). For calculating indicial response, residues can be calculated by (7·5) and (7·6) on log (1+s) chart. Tangent of Log-Root-Locus and differential of gain along Log-Root-Locus give directly the phase and magnitude of the residues as (7·12) and (7·13) and a corrected protractor (Fig. 13) is conveniently used for measuring those values. Frequency response of closed-loop transfer function can be found either setting brake points at characteristic roots on Log-Root-Locus (Fig. 15) or directly calculating the gain and the phase for s=jw again using log (1+s) chart as Fig. 16. As an example, a simple servomechanism (Fig. 17) is synthesized by Log-Root-Locus method (Figs. 18 & 19), and its response characteristics are calculated (Figs. 20 & 21). Multi-loopsy stem is treated using successively Log-Root-Locus for each loop as in the case of root-locus (Fig. 22), of which a servomechanism with tachometric feedback is calculated as an example.