Root locus rules and how to draw one
The root locus shows how the closed-loop poles move as a gain K increases. A few rules let you sketch it by hand, and an online tool can draw it exactly. The branches are the roots of D(s) + K·N(s) = 0 as K goes from 0 to infinity.
Start, end, and number of branches
There is one branch per open-loop pole. Branches begin at the open-loop poles when K = 0 and end either at the open-loop zeros or at infinity as K grows. The number of branches going to infinity equals the number of poles minus the number of finite zeros.
Real-axis segments and asymptotes
A point on the real axis is on the locus if the number of poles and zeros to its right is odd. The branches heading to infinity follow asymptotes that meet the real axis at the centroid σ = (Σpoles − Σzeros)/(n − m) and leave at angles (2q + 1)·180°/(n − m).
Breakaway points and jω crossings
Where two branches meet on the real axis and split into the complex plane is a breakaway point, found where dK/ds = 0 on the locus. Where a branch crosses the imaginary axis marks the gain K at which the closed loop becomes marginally stable — a key value for choosing K.