Root locus

The formula

A single-loop feedback system with forward-times-feedback transfer function G(s)H(s) and an adjustable gain K has its closed-loop poles wherever the characteristic equation is satisfied:

1 + K·G(s)H(s) = 0

Read it as: the closed-loop poles are the values of s that make the loop gain equal to -1. As K changes, those roots move, and the path each one traces is a branch of the root locus.

Split the complex condition K·G(s)H(s) = -1 into two real conditions. First the angle condition, which decides whether a point s is even on the locus:

∠G(s)H(s) = ±180°, ±540°, ...   (odd multiples of 180°)

Read it as: sum the angles from every zero to the test point, subtract the angles from every pole, and if the total lands on an odd multiple of 180°, that point is a closed-loop pole for some positive K. The angle condition draws the shape; K is not in it.

Then the magnitude condition, which tells you which gain puts a pole at that point:

K = 1 / |G(s)H(s)|  =  (product of pole distances) / (product of zero distances)

Read it as: once you know a point is on the locus, the gain that parks a pole there is the product of distances from that point to all the poles, divided by the product of distances to all the zeros.

Write G(s)H(s) = N(s)/D(s) with n poles and m zeros. A few construction facts fall straight out:

branches            = n            (one per open-loop pole)
asymptotes          = n − m
centroid  σ_a       = (Σ poles − Σ zeros) / (n − m)
asymptote angles    = (2k+1)·180° / (n − m),   k = 0,1,...,(n−m−1)

The branches start on the open-loop poles at K = 0 and end on the open-loop zeros (or run off to infinity along the asymptotes) as K → ∞. The centroid is the single real-axis point all the asymptotes fan out from.

Where you meet it

Compensator tuning at a design review. You are setting the loop gain on a servo — a gimbal, a valve positioner, a rate table — and someone asks how much margin you have before it goes unstable. The root locus shows the exact K where a branch crosses the imaginary axis, so you can quote the gain limit instead of guessing.
Sizing a PID on a test stand. A thrust-vector actuator or a thermal loop is ringing. Sketching the locus with a candidate zero added (that is what the derivative term does) shows whether the extra zero pulls the dominant poles left into more damping, or whether you have just moved the problem.
Reverse-engineering a marginal loop in the field. A flight control or antenna-pointing loop that was fine on the bench oscillates once payload mass or cable stiffness shifts a pole. The locus tells you which direction the poles migrate and whether nudging the gain down buys you back your damping.
Explaining a stability call to a review board. The plot is the one control-theory picture non-specialists can read: poles in the left half-plane, good; poles crossing into the right half-plane, the loop is now an oscillator you did not order.

How it works

The whole method rests on one idea: closed-loop poles are the roots of 1 + K·G(s)H(s) = 0, and those roots are continuous functions of K. Start at K = 0 and the roots sit exactly on the open-loop poles. Push K up and they slide along smooth curves. The genius of Evans' rules is that you can draw those curves by hand — from just the open-loop pole and zero locations — without factoring a polynomial at every gain.

Work the textbook loop G(s)H(s) = 1/[s(s+2)]. Two poles, at 0 and -2, no zeros. The characteristic equation is s² + 2s + K = 0. The real-axis rule says the segment between the two poles is on the locus, so at low gain both poles are real and creeping toward each other. At K = 1 they collide at s = -1 — the breakaway point — and for any K > 1 they split off vertically into a complex conjugate pair with real part fixed at -1. Run the numbers: K = 2 gives poles at -1 ± j1, K = 5 gives -1 ± j2, K = 10 gives -1 ± j3. The two asymptotes are the vertical lines at 90° and 270° off the centroid σ_a = (0 + (−2))/2 = -1, which is exactly where the branches went. This loop never goes unstable — the real part stays at -1 no matter how hard you push. Higher gain just makes it ring faster with the same damping envelope.

Now add a third pole: G(s)H(s) = 1/[s(s+1)(s+2)]. Three poles, no zeros, so three branches and n − m = 3 asymptotes at 60°, 180°, and 300° off a centroid at -1. Two of the branches bend right and cross the imaginary axis. Solving the characteristic polynomial s³ + 3s² + 2s + K = 0, the crossing happens at K = 6, where the poles sit at -3 and ±j1.414. Below K = 6 the loop is stable; above it, two poles are in the right half-plane and the loop oscillates and grows. That single number — the gain at the axis crossing — is what the root locus exists to hand you, and it agrees exactly with a Routh-Hurwitz check on the same polynomial.

The breakaway and break-in points — where branches leave or rejoin the real axis — are the roots of dK/ds = 0. Solve K = -(s² + 2s) from the first example, differentiate, and you get s = -1, the collision point, with K = 1. A quick sanity check: plug the candidate back in and the gain must come out positive, or the point is spurious.

Gotchas that bite people:

The plain locus is for K ≥ 0 with negative feedback. Positive feedback, a negative gain constant, or a plant written in (1 − τs) form flips the sign of the loop gain and needs the complementary (0°) locus, where the angle condition becomes even multiples of 180°. Note that a right-half-plane zero by itself does not change the rule — write G(s)H(s) in factored pole-zero form and check the sign of the leading gain constant. Draw the wrong locus and your stable design is actually a bomb.
It moves one parameter. Root locus answers "what does one gain do." If you are trading two coupled gains, or the plant itself is uncertain, the single-knob picture undersells the problem. Sketch a family of loci, not one.
Dominant-pole thinking has limits. Engineers read damping and settling time off the pair of poles closest to the imaginary axis and ignore the rest. That works when the other poles are far to the left, but a zero sitting near a dominant pole, or a fast pole that is not actually fast enough, will make the real step response miss your prediction.
It assumes a linear, time-invariant plant. Saturation, backlash, and rate limits are invisible to the locus. A loop the plot swears is stable can still limit-cycle on hardware because the actuator clips. The locus is a starting point, not a flight certificate.
Poles and zeros that nearly cancel are fragile. A compensator zero placed right on top of an unstable plant pole looks clean on paper, but the tiniest model error leaves a whisker of unstable pole behind. Never cancel a right-half-plane pole with a zero and call it stable.

History

Walter Richard Evans (1920–1999) came out of Washington University in St. Louis with an electrical engineering degree in 1941 and went to work in industry — General Electric, then the aircraft and aeronautics side at companies that became Rockwell ^[1]^[2]. In the late 1940s the young field of servomechanism design had the frequency-response tools of Nyquist and Bode, but reading closed-loop pole motion straight off a gain change was still a slog of hand-factoring polynomials.

Evans' move was to work in the s-plane directly and treat the closed-loop poles as a curve parameterized by gain. He worked out the angle and magnitude conditions and, in 1948, invented both the method and a cardboard-and-transparent-arm slide-rule device called the Spirule for adding up the pole and zero angles by hand ^[1]^[2]. His first paper, "Graphical Analysis of Control Systems," appeared in the AIEE Transactions in 1948; the follow-up, "Control System Synthesis by Root Locus Method," landed in the same journal in 1950 ^[6]. Getting there was not smooth — practicing engineers liked it immediately, but theory reviewers found the manuscript hard to follow, and publication slipped ^[6]. He gathered the whole approach into the textbook Control-System Dynamics (McGraw-Hill, 1954) ^[1]. The method has been standard in every controls curriculum since, and "root locus" is still drawn by every autopilot, servo, and process-loop engineer who needs to see where the poles are headed before turning a knob.

Related tools

/tools/rc-filter
/tools/lc-resonance
/tools/vibration-natural-freq
/tools/opamp-gain
/tools/q-factor-bandwidth

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →