Question 1

What is the root locus method?

Accepted Answer

The root locus is the set of paths traced by the closed-loop poles (roots of 1 + KG(s)H(s) = 0) as the gain K varies from 0 to infinity. At K=0 the locus starts at open-loop poles; as K→∞ it ends at open-loop zeros or along asymptotes to infinity. The plot gives a visual picture of stability and transient behavior as a function of gain.

Question 2

How are root locus asymptotes calculated?

Accepted Answer

There are n−m asymptotes (n poles, m zeros). The asymptote angles are (2k+1)×180°/(n−m) for k=0,1,…,n−m−1. The centroid (real-axis intersection) is σ_a = (Σpole real parts − Σzero real parts)/(n−m). For G(s) = 1/[s(s+2)(s+4)], n=3, m=0, asymptote angles are 60°/180°/300° and centroid σ_a = −2.

Question 3

How do you read damping ratio ζ and natural frequency ωn from the root locus?

Accepted Answer

For a complex closed-loop pole σ ± jωd: natural frequency ωn = √(σ²+ωd²) is the distance from the origin; damping ratio ζ = −σ/ωn; the angle from the negative real axis is θ = arccos(ζ). Design goals such as ζ ≥ 0.5 and ωn ≥ 5 rad/s define a region in the s-plane; choose K to place the dominant poles inside that region.

Question 4

What gain K makes the system marginally stable?

Accepted Answer

Substitute s = jω into the characteristic equation, separate real and imaginary parts, and solve simultaneously (Routh-Hurwitz criterion). The resulting gain K_c is the critical gain at marginal stability (sustained oscillation). In the Ziegler-Nichols method, K_c and the period of oscillation T_c are used directly to set P/I/D gains.

Root Locus Control System Designer

Root Locus Construction Rules