An analytical approach to root loci

The general algebraic equations of root loci for real are found in polar and Cartesian coordinates. A synthesis method is then suggested which leads to linear equations in the coefficients of the open-loop transfer function when closed-loop poles and their corresponding gains are specified. Equations are also found for the gain corresponding to a given point on the root locus. A superposition theorem is presented which shows how the root loci for two open-loop functions place constraints on the locus for their product. With a knowledge of the simple lower-order loci, this theorem can be used in sketching and constructing root loci.