An analytical method for obtaining the root locus with posititve and negative gain

An analytical procedure for obtaining the root locus with positive and negative gain is described. Solutions of various systems consisting of rational polynomial and/or transcendental functions of
s
are obtained. The root locus for a varying time constant is investigated. Accurate root locus curves can be plotted from the analytical solution. The gain or time constant is calculated for each point on the root locus. Application of the method is particularly suitable for obtaining the root locus of a characteristic equation containing a transcendental function or with a varying time constant for which the graphical method is difficult to apply.