The root locus is a graphical method for showing how the closed-loop poles of a feedback control system move in the s-plane as the gain K changes. It matters because pole locations determine whether a system is stable and how it responds to inputs or disturbances. Engineers use the plot to connect algebraic transfer functions with visible design choices.
A well drawn root locus quickly shows which gains give fast, slow, oscillatory, or unstable behavior.
For a unity feedback system with open-loop transfer function G(s), the closed-loop poles satisfy 1 + K G(s) = 0. As K increases from 0 to infinity, each branch of the root locus starts at an open-loop pole and moves toward an open-loop zero or toward infinity along an asymptote. Points in the left half of the s-plane usually represent stable decaying motion, while points in the right half represent growing unstable motion.
By adding poles, zeros, or choosing K, a controller designer can reshape the locus to meet stability, damping, and settling time goals.
Understanding Engineering: The Root Locus
A root locus comes from two conditions that every possible pole location must satisfy. First is the angle condition. The combined angles from all open-loop zeros and poles to a trial point must add up to an odd multiple of one hundred eighty degrees.
This condition tells engineers where branches are allowed to lie. Second is the magnitude condition. It gives the gain required for that allowed point.
In practice, the angle condition is used to sketch the shape, then the magnitude condition labels useful locations with gain values. This separation is helpful because it turns a difficult polynomial problem into geometry on a plane.
Several drawing rules reveal the overall pattern before any detailed calculation. On the real axis, a point belongs to the locus when the number of real poles and zeros to its right is odd. When branches must travel far from the origin, they follow straight asymptotes.
Their directions depend on how many more poles than zeros the system has. Their common meeting region is found from the average position of the poles minus the average position of the zeros, with the required counting included. These rules matter because a rough sketch can expose a dangerous path toward instability early in a design.
Branches do not always remain separate. Two branches can meet on the real axis, then leave it as a complex conjugate pair. This is called a breakaway point.
The reverse event is a break-in point. Both can be found by expressing gain as a function of position and finding where that gain has a turning point. Branches can leave a complex pole at particular departure angles or arrive at a complex zero at particular arrival angles.
Students often make errors here by measuring angles in inconsistent directions. A clear diagram, with every angle measured from the positive real direction, prevents many sign mistakes.
The most important design check is where the locus crosses the imaginary axis. At that gain, the system has sustained oscillation in the ideal mathematical model. A slightly larger gain may place poles in the right half plane, where the response grows instead of dying away.
Engineers can calculate this boundary using a Routh stability table, then confirm it on the plot. They often choose a gain safely away from the boundary because real components have tolerances, loads change, and models are never exact. A motor speed controller, drone attitude loop, room temperature controller, and audio feedback circuit all face this same issue.
Pole position gives more detail than a simple stable or unstable label. Poles far to the left usually produce quicker decay, but pushing them too far can require excessive control effort or make the system sensitive to noise. Complex poles near the imaginary axis tend to create ringing and overshoot.
A designer may draw lines of constant damping ratio and constant natural frequency on the same plane to identify a region with acceptable settling time and overshoot. Adding a controller zero can pull branches toward a preferred region. Adding a pole can slow the response or create extra phase lag.
The plot is therefore a guide, not a final answer. Engineers still test the chosen gain with time responses, frequency response, actuator limits, and uncertainty.
Key Facts
- Closed-loop characteristic equation for unity feedback: 1 + K G(s) = 0.
- Root locus branches show the locations of closed-loop poles as K varies from 0 to infinity.
- Each branch starts at an open-loop pole when K = 0.
- Each branch ends at an open-loop zero or goes to infinity if there are more poles than zeros.
- A continuous-time system is stable if all closed-loop poles have negative real parts.
- For a complex pole s = σ + jω, the decay rate is set by σ and the oscillation frequency is set by ω.
Vocabulary
- Root locus
- A plot of the paths followed by closed-loop poles in the complex s-plane as a system gain changes.
- Closed-loop pole
- A root of the closed-loop characteristic equation that determines a mode of the feedback system response.
- s-plane
- The complex plane used in control engineering, with real part σ on the horizontal axis and imaginary part jω on the vertical axis.
- Gain K
- A multiplier in the loop transfer function that changes pole locations and therefore changes the system response.
- Stability boundary
- The imaginary axis in the continuous-time s-plane, separating stable left-half-plane poles from unstable right-half-plane poles.
Common Mistakes to Avoid
- Thinking the root locus is a time graph, which is wrong because it plots pole locations in the complex s-plane rather than output versus time.
- Ignoring the imaginary axis stability boundary, which is wrong because crossing into the right half-plane means the continuous-time closed-loop system becomes unstable.
- Assuming larger K always improves performance, which is wrong because increasing gain can move poles toward low damping or instability.
- Confusing open-loop poles with closed-loop poles, which is wrong because the root locus starts at open-loop poles but represents closed-loop pole locations for different gains.
Practice Questions
- 1 A unity feedback system has G(s) = 1/(s(s + 4)). Write the closed-loop characteristic equation for gain K and find the range of K for stable closed-loop poles.
- 2 For a closed-loop pole at s = -3 + j4, find the damping ratio ζ and natural frequency ωn. Use ωn = sqrt(σ^2 + ω^2) and ζ = -σ/ωn.
- 3 A root locus branch moves closer to the imaginary axis as K increases. Explain what this suggests about settling time, oscillation, and stability margin.