Root Locus Construction Rules

From time to time there will be a thread on the construction of a root locus. These were methods used long before computers achieved wide availability. They are especially useful for transfer functions of third and higher orders.

For a system with a loop transfer function G(s)H(s), the first five rules are:

1. The number of branches of the root locus equals the number of closed loop poles.
2. The root locus is symmetrical about the real axis.
3. On the real axis for K > 0 the root locus exists to the left of an odd number of real-axis, finite open-loop poles and/or finite open-loop zeros
4. The root locus begins at the finite and infinite poles of G(s)H(s) and ends at the finite and infinite zeros of G(s)H(s).
5. The root locus approaches straight lines as asymptotes as the locus approaches infinity. Further the equation of the asymptotes is given by the real axis intercept, σa and angle, θa

This is not an exhaustive list, but should suffice to get started.