Root Locus

The characteristic equation 1 + KGH = 0 contains K, so its roots depend on it. Each gain gives a different set of closed-loop poles, and sweeping gain traces a path through the s-plane.

At K = 0 the equation reduces to the open-loop denominator, so every branch begins at an open-loop pole. Branches end at open-loop zeros or run off to infinity along asymptotes.

1. 1Low gain — poles real and far apart. Overdamped, slow, no overshoot.
2. 2Rising gain — they slide toward each other along the real axis. Faster, still overdamped.
3. 3Breakaway — they meet and leave the axis as a complex pair. This point is exactly ζ = 1, so the breakaway gain is the largest gain with no overshoot.
4. 4Higher still — the poles move toward the imaginary axis, so overshoot and ringing grow.
5. 5Axis crossing — the stability limit, at the gain and frequency Routh-Hurwitz gives from a row of zeros.

The specification regions from the second-order topic can be drawn on the same plane. Design becomes: where does the locus enter the region I want?

And if it never enters — a common outcome — no gain will do, and a compensator must reshape the locus itself. That is the entire justification for lead and lag networks: adding a pole or zero bends the path.

• Branches start at open-loop poles and end at zeros or infinity.
• Real-axis segments lie to the left of an odd count of poles and zeros.
• Asymptote angles are (2k+1)·180°/(n − m).
• The locus is symmetric about the real axis, since complex roots of a real polynomial come in conjugate pairs.

Every rule follows from the angle condition: a point is on the locus when ∠GH = 180°, since only then can a positive K satisfy 1 + KGH = 0. Software draws these now; the rules survive because they build intuition for which way the poles will move.