Root Locus Techniques: Complete Beginner's Guide Guide.

The root locus plots how closed-loop poles migrate through the s-plane as gain K goes from 0 to infinity, for open loop K x G(s)H(s). Each point on the curves is a pole location for one K. Since poles dictate stability, speed and overshoot, watching them move tells the whole story of what gain does.

Root locus turns gain selection visual: see the exact K of instability (branches crossing the imaginary axis), the fastest response, the best damping. It is the standard tool for lead/lag compensator design and for seeing why some systems can never be stabilized by gain alone.

Key Concepts

• Poles and zeros: n branches start at open-loop poles (K=0, marked x) and end at zeros (o) or infinity.
• Real-axis rule: a real-axis point is on the locus if poles+zeros to its right total an odd number.
• Asymptotes: n - m branches to infinity at angles +-180 x (2q+1)/(n - m), from the real-axis centroid.
• Centroid: sigma_a = (sum of poles - sum of zeros)/(n - m).
• Breakaway/break-in: real-axis departure/arrival points, from dK/ds = 0.
• Angle condition: angle of K x G(s)H(s) = +-180 x (2q+1) degrees on the locus.
• Magnitude condition: |K x G(s)H(s)| = 1 gives K at any locus point.

Formulas

• Asymptote angles: theta = +-180 x (2q+1)/(n - m), q = 0, 1, 2, …
• Centroid: sigma_a = (sum poles - sum zeros)/(n - m).
• Breakaway: from 1 + K x G(s)H(s) = 0 write K = -1/(GH), set dK/ds = 0, keep roots on real-axis segments.
• Assumptions: single positive varying gain K, LTI system, negative-feedback 180-degree convention.

Worked Example

Root locus of K/(s x (s+2)), unity feedback:

1. Poles at 0 and -2 (n=2), no zeros. Real-axis segment (-2, 0) is on the locus (one pole to the right).
2. Asymptotes: +-90 degrees from centroid (0 + -2)/2 = -1.
3. Breakaway: K = -s(s+2); dK/ds = -(2s+2) = 0 → s = -1, valid on the segment. Branches meet at -1 and leave vertically.
4. j omega crossing: s^2 + 2s + K = 0 at s = j x omega forces omega = 0, K = 0: never crosses for K > 0.
5. Stable for all positive K; damping is best near the breakaway (K=1, double pole at -1) and falls as poles climb the vertical asymptotes.

Common Misconceptions

• "Zeros move with K." Open-loop zeros are fixed; only closed-loop poles travel.
• "Breakaway is always midway." True for the simple two-pole case; generally solve dK/ds = 0.
• "Higher K always destabilizes." Many systems stay stable for all K; some are conditionally stable only above a minimum gain.

Real-World Applications

• Servo tuning: read K for a target damping ratio off the plot.
• Compensators: lead/lag networks reshape the locus leftward.
• Conditional stability: spot gain bands of stability between crossings.
• Autopilots: gain schedules verified against locus families.

Root locus draws every possible closed-loop pole as gain varies, from a few rules: real-axis segments, centroid asymptotes, dK/ds = 0 breakaways, angle/magnitude conditions. The K/(s(s+2)) example met at -1 and rose vertically, stable for all K. Sketch it once by hand and gain selection stops being guesswork.