Construction of Root Locus

The number of branches in a root locus equals the number of closed-loop poles in the system. Each branch represents the path of one closed-loop pole as the gain varies. The root locus is symmetrical about the real axis, meaning branches appear in mirror-image pairs for complex poles.

The root locus originates at the open-loop poles and terminates at the open-loop zeros. As gain increases, the closed-loop poles trace paths from the pole locations toward the zero locations. When the locus reaches the zeros, the system poles stabilize with rising gain.

The root locus exists on the real axis where the sum of the total number of finite open-loop poles and zeros to the left is odd. This rule applies for positive gain values and helps identify which segments of the real axis contain locus branches.

Asymptotes are straight lines that the root locus approaches as it extends toward infinity. Their equations are determined by the real-axis intercept and specific angles based on the number of poles and zeros. These asymptotes guide the locus paths from the poles to infinity as gain increases.

If a function approaches infinity as s approaches infinity, the system has a pole at infinity. If it approaches zero, it has a zero at infinity. These conditions influence the asymptotic behavior and determine how the root locus behaves as gain becomes very large.

The root locus is symmetrical about the real axis because complex poles in linear systems always occur in conjugate pairs. This symmetry reflects the fundamental property that complex roots of characteristic equations with real coefficients must be conjugates, ensuring mirror-image behavior across the real axis.

The root locus method visualizes how closed-loop poles migrate with increasing gain, enabling engineers to analyze system stability and response. By plotting and calibrating the root locus, designers can predict system behavior across operating conditions and tune control parameters to achieve desired performance and stability.