Root Locus Interview Questions

Q1. What is the root locus and what does it show?

The root locus is a plot of the closed-loop pole locations in the s-plane as the open-loop gain K varies from zero to infinity, showing every possible closed-loop pole for every possible gain value. For a DC motor position control system, the root locus shows exactly how the two poles associated with mechanical and electrical dynamics move as the proportional gain is increased from zero toward instability. The root locus is the most direct way to choose a gain that places the closed-loop poles at a desired damping ratio and natural frequency.

Follow-up: What do the closed-loop poles approach at very high gain?

Q2. State the rules for sketching a root locus.

The key rules are: the number of branches equals the number of open-loop poles; branches start at open-loop poles (K=0) and end at open-loop zeros or infinity; the real-axis segments to the left of an odd number of real poles plus zeros are on the locus; asymptote angles are (2k+1)·180°/n-m for k=0,1,... and the asymptote centroid is at (Σpoles - Σzeros)/(n-m). For G(s) = K/[s(s+2)(s+4)] with n=3 and m=0, three asymptotes go at 60°, 180°, and 300° from the centroid at (-6/3) = -2. The angle condition 180° test confirms whether any arbitrary s-plane point lies on the root locus.

Follow-up: How do you find the breakaway point on the real axis?

Q3. How do you determine the gain K at a specific point on the root locus?

The gain K at any point s0 on the root locus is calculated using the magnitude condition: K = |product of distances from s0 to all poles| / |product of distances from s0 to all zeros|. For a system with poles at 0, -3, and -5, the gain at the breakaway point s = -1.5 is the product of distances 1.5 × 1.5 × 3.5 divided by 1 (no zeros), giving K ≈ 7.875. This graphical measurement, originally done with a ruler, is now computed automatically in MATLAB's rlocfind function.

Follow-up: What is the significance of the gain value at the point where the root locus crosses the imaginary axis?

Q4. What happens to the root locus when you add a zero to the open-loop transfer function?

Adding a left-half-plane zero attracts the root locus branches toward it, pulling the locus to the left and away from the right-half plane, which generally improves stability and allows higher gain. In a poorly damped motor control loop, adding a PD controller zero at s = -5 pulls the root locus away from the imaginary axis, allowing 10× more gain before instability compared to proportional control alone. The closer the zero is to the dominant poles, the stronger the attraction and the greater the stabilizing effect.

Follow-up: What happens if the added zero is in the right-half s-plane?

Q5. What happens when you add a pole to the open-loop transfer function?

Adding a pole pushes the root locus toward the right-half plane, tending to destabilize the system and reducing the maximum gain before instability. Adding a lag compensator pole at s = -0.1 to a third-order system shifts the asymptote centroid to the right and causes one locus branch to approach the new pole slowly, reducing the gain margin. The general rule is that poles push the locus right (destabilize) and zeros pull it left (stabilize), which is why adding integral action (a pole at origin) always makes stability harder to maintain.

Follow-up: Why does adding a pole at the origin always reduce stability?

Q6. How do you find the breakaway point on the real axis?

The breakaway point is found by differentiating the characteristic equation 1 + K·G(s)H(s) = 0 with respect to s, setting dK/ds = 0, and solving — the real solutions that lie on the root locus are the breakaway (and break-in) points. For G(s) = K/[s(s+4)], setting d/ds[−s(s+4)] = 0 gives −2s − 4 = 0, so the breakaway point is at s = −2. At the breakaway point, two closed-loop poles that travel along the real axis meet and depart into the complex plane as gain increases, marking the onset of oscillatory behavior.

Follow-up: What is the difference between a breakaway point and a break-in point?

Q7. How does the root locus relate to the closed-loop step response?

The location of the dominant closed-loop poles on the root locus directly determines the damping ratio ζ and natural frequency ωn of the step response: poles at angle θ from the negative real axis have ζ = cos(θ), and their distance from the origin equals ωn. For a servomotor positioning system, selecting K to place the dominant poles at s = -4 ± j4 gives ζ = 0.707 and ωn = 5.66 rad/s, predicting 4.3% overshoot and a settling time of about 1 second. The root locus thus provides a direct graphical link between gain choice and time-domain specifications.

Follow-up: What is the relationship between pole angle and damping ratio?

Q8. What does it mean when the root locus crosses the imaginary axis?

When the root locus crosses the imaginary axis, the closed-loop poles are purely imaginary at that gain value, meaning the system is marginally stable and will exhibit sustained oscillations at the imaginary-axis crossing frequency. For a third-order system, the imaginary-axis crossing gain matches the ultimate gain Ku found by the Routh-Hurwitz criterion, confirming that both methods give the same stability limit. Above this gain, the closed-loop poles move into the right-half plane and the system becomes unstable.

Follow-up: How would you find the exact gain at the imaginary axis crossing using the Routh array?

Q9. What is a root locus compensator design and how is it done?

A root locus compensator design adds poles and zeros to the open-loop transfer function to reshape the root locus so that it passes through the desired closed-loop pole location at a reasonable gain. For a DC motor position loop that needs dominant poles at s = -3 ± j3√3 (ζ = 0.5, ωn = 6), a lead compensator zero is placed near the desired poles at s = -2 and the pole is placed at s = -20 to attract the locus to that region. The angle condition is then verified to confirm the desired point lies on the locus after compensation, before calculating the required gain.

Follow-up: What is the angle condition check in root locus compensator design?

Q10. What is the angle of departure from a complex pole?

The angle of departure is the direction in which the root locus leaves a complex open-loop pole, calculated as 180° minus the sum of angles subtended by all other poles plus the sum of angles from all zeros to that pole. For a complex pole at s = -1 + j2 in a system with another pole at s = -3, the angle contributions from the other poles and zeros are summed and subtracted from 180° to give the departure angle, which determines whether the locus initially moves toward or away from the imaginary axis. Knowing the departure angle is critical when designing compensators for systems with lightly damped resonant modes.

Follow-up: How does the angle of departure change if you add a zero near the complex pole?

Q11. How many branches does a root locus have and what determines this?

The number of root locus branches equals the number of open-loop poles (the order of the denominator of the open-loop transfer function), because each branch represents one closed-loop pole trajectory as gain varies. A system with transfer function K·(s+1)/[s(s+2)(s+5)(s+10)] has four poles and therefore four root locus branches. Each branch starts at one of the four open-loop poles at K = 0 and terminates at either the single finite zero at s = -1 or at one of three asymptotes going to infinity.

Follow-up: Where do the branches that go to infinity travel, and how do you calculate the asymptote directions?

Q12. What is the significance of the centroid of asymptotes in root locus?

The centroid of asymptotes σa = (Σpoles - Σzeros)/(n - m) is the point on the real axis from which all asymptotes radiate, and a centroid far to the left indicates that increasing gain is relatively safe for stability, while a centroid near zero or positive signals potential instability at moderate gains. For G(s) = K/[s(s+1)(s+3)] the centroid is at (0-1-3-0)/3 = -4/3 ≈ -1.33, indicating the asymptotes radiate from s = -1.33. Moving the centroid left — for example by adding a left-half-plane zero — is one way to improve stability across all gain ranges.

Follow-up: Can the centroid of asymptotes ever be in the right-half plane?

Q13. What is a positive or 0° root locus and when does it arise?

A positive (0°) root locus applies when the open-loop gain is negative, and the angle condition changes to require even multiples of 180° — the locus now lies on real-axis segments to the left of an even number of poles and zeros. Negative feedback systems with an inverting sensor or a physical sign reversal in the plant effectively have negative gain, requiring analysis with the 0° root locus to correctly predict closed-loop behavior. In some op-amp feedback circuits where the feedback path inverts the signal, the designer must use the complementary root locus to find the correct closed-loop poles.

Follow-up: Give a practical circuit example where negative gain makes the 0° root locus necessary.

Q14. How does the root locus change when a pole-zero pair is added close together?

A pole-zero pair added close together (a dipole) has minimal effect on the overall root locus shape because their angle contributions nearly cancel, but it can significantly affect the gain at a specific point and may introduce a slow residual mode if not carefully placed. Adding a lag compensator dipole — zero at s = -0.1 and pole at s = -0.01 — to a speed controller barely changes the root locus shape but increases the low-frequency gain by a factor of 10, reducing steady-state error without meaningfully affecting transient performance. The dipole's slow pole also introduces a lightly excited but very slow mode that takes many seconds to decay, which must be acceptable for the application.

Follow-up: Why does a lag compensator dipole placed far to the left of the dominant poles have negligible effect on transient response?

Q15. What is the closed-form condition for a point to lie on the root locus?

A point s in the s-plane lies on the root locus if and only if the phase of G(s)H(s) equals an odd multiple of 180°: ∠G(s)H(s) = (2k+1)·180° for integer k. For the point s = -1 + j1 in the system G(s) = K/[s(s+2)], the angles from each pole are computed as arctan(1/1) = 45° from pole at 0 and arctan(1/1) = 45° from pole at -2, summing to 90° — which is not 180°, so this point does not lie on the root locus. This angle condition is the theoretical foundation for all the graphical rules used to sketch root loci.

Follow-up: How would you use MATLAB to verify whether a specific point lies on the root locus?