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.