How it works
Root locus construction rules: locus starts at open-loop poles (K=0) and ends at open-loop zeros (K=∞). Number of branches equals number of open-loop poles n; (n−m) branches go to infinity along asymptotes, where m is the number of finite zeros. Asymptote angles = (2q+1)×180°/(n−m) for q = 0, 1, 2, ... Centroid of asymptotes σ_a = (Σpoles − Σzeros)/(n−m). Points on the real axis are on the locus if the total number of open-loop poles and zeros to the right is odd. Breakaway point: solve dK/ds = 0 where K = −1/G(s)H(s).
Key points to remember
For G(s) = K/[s(s+2)(s+4)]: n=3, m=0, three asymptotes at angles 60°, 180°, 300°, centroid = (0+−2+−4−0)/3 = −2. Gain at a specific point s₀ on the locus: K = 1/|G(s₀)|. The jω-axis crossing (marginally stable gain) is found by substituting s = jω into the characteristic equation and solving for ω and K — or directly from the Routh array's auxiliary polynomial. Adding a zero to G(s) pulls the locus toward the left half-plane (improving stability); adding a pole pushes it rightward. PD control effectively adds a zero, while PI control adds a pole at the origin.
Exam tip
The examiner always asks you to sketch the root locus of a given G(s)H(s) showing asymptotes, centroid, real-axis segments, and the jω crossing — compute the centroid numerically and find the breakaway point using dK/ds = 0.