A root locus plot computes the locus of closed-loop poles for a transfer function while itâ€™s under gain feedback. The gain is swept over a wide range of values, typically from a small value near 0 to a larger one.

For brevity, the transfer function can be denoted as
*GH*(*s*), which represents a ratio of polynomials in *s*, and
can be written as:

where *N*(*s*) is the numerator polynomial and
*D*(*s*) is the denominator polynomial. The root locus gain is
represented as *K.*

The following diagram presents the structure Embed uses to perform Root Locus calculations:

The closed-loop poles are the roots of the characteristic equation (or denominator of the closed-loop transfer function) and can be computed as the roots of:

For small values of *K*, the closed-loop poles are
approximately the roots of *D*(*s*) = 0 (the open-loop poles); and for
large values of *K* (positive or negative), the closed-loop poles are
approximately the roots of *N*(*s*) = 0 (the open-loop 0s). The root
locus plot presents the paths of the closed-loop poles as the value of *K*
varies.