Root Locus Concept

In control systems, the behavior of a closed-loop system depends directly on the location of its poles in the s-plane. The root locus is a graphical technique that shows how the closed-loop poles move in the s-plane as the open-loop gain K is varied from zero to infinity. It gives a complete picture of system stability and transient response for all possible gain values without solving the characteristic equation repeatedly.

Understanding root locus is essential for GATE and university exams because it connects open-loop pole-zero data directly to closed-loop stability. Rather than computing roots for every K, the root locus lets you predict dominant pole behavior, damping ratio, and stability margins from a single diagram.

Core Concept Explanation

Consider a standard unity feedback closed-loop system with open-loop transfer function G(s)H(s) = KN(s)/D(s). The closed-loop characteristic equation is 1 + KG(s)H(s) = 0, which means the closed-loop poles are the values of s satisfying this equation for a given K. As K changes, these pole locations change, tracing out paths called the root locus.

At K = 0, the characteristic equation reduces to D(s) = 0, meaning the closed-loop poles coincide exactly with the open-loop poles. At K approaching infinity, the dominant behavior comes from N(s) = 0, so branches terminate at the open-loop zeros. If there are more poles than zeros, the extra branches go to infinity along asymptotes. This explains why root locus always starts at open-loop poles and ends at open-loop zeros or infinity.

The angle condition for a point s0 to lie on the root locus is that the net angle contribution from all open-loop poles and zeros must equal an odd multiple of 180 degrees. Mathematically: angle of G(s0)H(s0) = (2q+1)180 degrees for any integer q. This condition is used to test whether a given point belongs to the locus.

The magnitude condition states that for any point already confirmed to be on the locus, the gain K is found as K = 1 / |G(s0)H(s0)|. This means once you locate a desired closed-loop pole position on the locus, you can directly calculate the gain needed to place the pole there.

Mathematical Expression

The closed-loop characteristic equation forms the mathematical foundation. For a unity feedback system with open-loop gain K and transfer function G(s), the closed-loop poles satisfy 1 + KG(s) = 0, or equivalently KG(s) = -1. This complex equation has two parts: the angle condition and the magnitude condition, both derived from writing -1 in polar form as having magnitude 1 and phase 180 degrees (or any odd multiple).

If G(s) has m zeros z1, z2, ..., zm and n poles p1, p2, ..., pn, then the angle condition becomes: sum of angles from zeros minus sum of angles from poles = (2q+1) x 180 degrees. The magnitude condition gives: K = product of distances from s to all poles divided by product of distances from s to all zeros.

Practical Understanding

The root locus directly reveals stability: any portion of the locus that lies in the right half of the s-plane (RHP) corresponds to unstable closed-loop operation. As K increases, if branches cross into the RHP, the system becomes unstable beyond that gain. The crossing point on the imaginary axis gives the critical gain at which the system is marginally stable.

In practice, designers use root locus to select K for desired transient specifications. For example, a desired damping ratio of 0.707 corresponds to poles at 45 degrees from the negative real axis. By finding the intersection of this 45-degree line with the root locus, the required K can be read directly from the magnitude condition. This makes root locus a powerful design tool beyond just a stability check.

Solved Numerical Example

Consider an open-loop transfer function G(s) = K / (s(s+2)). The closed-loop characteristic equation is s^2 + 2s + K = 0. Using the magnitude condition, find K when the closed-loop poles are at s = -1 + j1. First verify the point lies on the locus using the angle condition, then apply the magnitude formula.

Given:
G(s) = K / [s(s+2)], test point s0 = -1 + j1

Why this formula applies:
Magnitude condition: K = |s0| x |s0+2| / 1 (no finite zeros)

Formula:
K = product of pole distances from s0 / product of zero distances from s0

Substitution:
|s0| = |(-1+j1)| = sqrt(1+1) = sqrt(2)
|s0+2| = |(1+j1)| = sqrt(1+1) = sqrt(2)

Calculation:
K = sqrt(2) x sqrt(2) = 2

Final Answer:
K = 2 (closed-loop poles at -1+j1 and -1-j1 for K=2)

Exam Tip: Root locus always has exactly n branches (n = number of OL poles). Branches start at OL poles for K=0 and end at OL zeros or infinity. On GATE, if asked how many branches go to infinity, the answer is always n - m where n = poles and m = finite zeros.

Quick Revision

• Root locus is the plot of closed-loop pole locations in the s-plane as gain K varies from 0 to infinity.
• Number of branches equals number of open-loop poles (n). Each branch starts at an OL pole (K=0) and ends at an OL zero or infinity (K to inf).
• Angle condition: angle of G(s)H(s) = (2q+1) x 180 degrees for integer q. Used to check if a point is on the locus.
• Magnitude condition: K = 1/|G(s0)H(s0)|. Used to find gain at any point on the confirmed locus.
• Locus crossing the imaginary axis indicates marginal stability. Locus in RHP means unstable closed-loop system.
• Root locus is symmetric about the real axis because complex poles always appear in conjugate pairs.
• Exam trap: The angle condition applies to standard negative feedback. For positive feedback, the condition changes to even multiples of 180 degrees.