Root locus sketching shows how the closed-loop poles of a feedback control system move as a gain parameter changes. This cheat sheet helps engineering students organize the standard sketching rules into a reliable sequence. It is useful for exams, design problems, and quick checks of stability and transient response trends.
A clear root locus sketch connects algebraic characteristic equations to controller design choices.
The core idea is to apply the open-loop transfer function G(s)H(s) and study points s that satisfy the angle and magnitude conditions. Important rules include locating open-loop poles and zeros, finding real-axis segments, computing asymptotes, and identifying breakaway or break-in points. Imaginary-axis crossings are commonly found using the Routh-Hurwitz criterion.
Once the locus is sketched, gain values and pole locations can be interpreted in terms of damping, natural frequency, settling time, and stability.
Key Facts
- For negative unity feedback, the closed-loop characteristic equation is 1 + K G(s)H(s) = 0.
- A point s is on the root locus if angle G(s)H(s) = (2q + 1)180 degrees, where q is any integer.
- The gain at a point on the locus is K = 1 / |G(s)H(s)| when the characteristic equation is 1 + K G(s)H(s) = 0.
- The number of root locus branches equals the number of open-loop poles of G(s)H(s).
- Root locus branches start at open-loop poles when K = 0 and end at open-loop zeros or at infinity as K approaches infinity.
- A point on the real axis is on the root locus if the number of real open-loop poles and zeros to its right is odd.
- The asymptote centroid is sigma_a = (sum of open-loop poles - sum of open-loop zeros) / (n - m), where n is the number of poles and m is the number of zeros.
- The asymptote angles are theta_k = (2k + 1)180 degrees / (n - m), for k = 0, 1, 2, ..., n - m - 1.
Vocabulary
- Root locus
- A plot of closed-loop pole locations in the s-plane as the gain K varies from 0 to infinity.
- Open-loop pole
- A value of s that makes the denominator of G(s)H(s) equal to zero.
- Open-loop zero
- A value of s that makes the numerator of G(s)H(s) equal to zero.
- Asymptote
- A straight-line direction followed by root locus branches that go to infinity.
- Breakaway point
- A point on the real axis where two or more root locus branches leave the real axis.
- Routh-Hurwitz criterion
- A tabular test used to determine stability and find gain values where roots cross the imaginary axis.
Common Mistakes to Avoid
- Counting real-axis segments incorrectly is wrong because only poles and zeros to the right of the test point determine whether the count is odd.
- Using the number of zeros instead of the number of poles for the branch count is wrong because every branch begins at an open-loop pole.
- Forgetting zeros at infinity is wrong because when n > m, exactly n - m branches must go to infinity along asymptotes.
- Choosing every solution of dK/ds = 0 as a breakaway point is wrong because the candidate must lie on a valid real-axis root locus segment.
- Assuming a sketch proves stability for all gains is wrong because imaginary-axis crossings and gain ranges should be checked with Routh-Hurwitz or direct substitution.
Practice Questions
- 1 For G(s)H(s) = K / [s(s + 2)(s + 5)], how many root locus branches are there, and how many asymptotes go to infinity?
- 2 For G(s)H(s) = K(s + 4) / [s(s + 1)(s + 3)], find the asymptote centroid sigma_a.
- 3 For open-loop poles at 0, -2, and -6 with no finite zeros, list the real-axis intervals that belong to the root locus.
- 4 Explain why adding a zero near the desired closed-loop pole location can reshape the root locus and affect transient response.
Understanding Root Locus Sketching Rules Reference
A reliable sketch begins with a map of the complex plane, not with a formula. Mark each pole with a cross and each zero with a circle. The locus is symmetric about the real axis because systems with real coefficients have complex roots in conjugate pairs.
This symmetry is a useful error check. If a branch leaves into the upper half plane, a matching branch must appear below it. Count how many branches must finish at finite zeros.
The remaining branches travel outward along asymptotic directions. Their paths near the origin may look curved, but far from the poles and zeros they settle toward straight lines.
The angle condition explains the shape of curved branches. At a trial point, draw vectors from every pole and zero to that point. Each vector contributes an angle.
The total angle from zeros minus the total angle from poles must be an odd multiple of one hundred eighty degrees. This is especially useful for finding the departure angle from a complex pole or the arrival angle at a complex zero. Near a pole, the angles from all other poles and zeros are nearly fixed.
Their combined effect sets the direction in which the branch initially leaves. Students often make sign errors here. Write every vector angle using one consistent reference direction before adding or subtracting them.
Breakaway points occur when two real branches meet and then leave the real axis as a complex pair. Break in points are the reverse event. A candidate can be found by expressing gain as a function of the real variable s, then setting the rate of change of gain with respect to s equal to zero.
Not every calculated candidate is valid. It must lie on an allowed real axis segment and it must give a positive gain for the feedback arrangement being studied. Testing the gain is important because algebra can produce stationary points that do not belong to the physical locus.
A quick hand sketch should show branches moving continuously. They cannot jump across the plane or stop halfway without ending at a zero.
Stability is determined by where the moving closed loop poles lie, not by the poles of the open loop system alone. For a continuous time system, every closed loop pole must remain in the left half of the s plane for stable behavior. When a branch crosses the imaginary axis, the system reaches a boundary of stability.
A Routh table gives the gain at this crossing without needing an exact plot. The corresponding auxiliary polynomial can give the oscillation frequency.
This matters in motor speed control, aircraft control, temperature regulation, and electronic amplifiers. Increasing gain can improve response at first, yet too much gain can reduce damping and create sustained oscillation.
When studying root locus, separate geometry from arithmetic. First identify poles, zeros, real axis sections, asymptotes, and likely break points. Then calculate only the values needed to confirm the sketch, such as departure angles, crossing gains, or a desired pole location.
Check the final drawing against physical expectations. Poles farther left usually mean faster decaying motion. Poles close to the imaginary axis usually mean slow decay or noticeable ringing.
Complex poles with a larger vertical distance from the real axis correspond to faster oscillation. These links turn a root locus from a drawing exercise into a design tool.