Which of the following statement is always 'TRUE' with respect to root locus analysis?

Explanation:

Root Locus Analysis

Definition: Root locus analysis is a graphical method used in control systems to study how the roots of a system's characteristic equation change with variations in a system parameter, typically the gain (K). It provides critical insights into the stability and behavior of the system as the gain is adjusted. The root locus plot helps determine the locations of poles and zeros of the closed-loop transfer function as a function of the gain.

Correct Option Analysis:

The correct option is:

Option 1: The root loci start at the open loop poles.

This statement is always true with respect to root locus analysis. Root locus plots are constructed based on the principle that the loci of the roots (poles) of the closed-loop transfer function start at the poles of the open-loop transfer function when the gain (K) is zero. This is a fundamental concept in root locus theory, as the open-loop poles represent the initial locations of the system's poles before any feedback is applied.

Reasoning:

To understand why this statement is correct, consider the characteristic equation of the closed-loop transfer function:

1 + KG(s) = 0

Here, G(s) is the open-loop transfer function, and K is the gain. When K = 0, the characteristic equation reduces to:

G(s) = 0

This equation indicates that the roots of the closed-loop system are the same as the poles of the open-loop transfer function. As the gain K increases, the roots (poles) move along the root locus paths, but they always start at the open-loop poles.

 Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: Number of separate root loci is the number of open-loop poles.

This statement is incorrect. The number of separate root loci is determined by the total number of poles and zeros of the open-loop transfer function, not just the number of poles. If there are more poles than zeros, the number of root loci corresponds to the total number of poles. However, if the number of poles equals the number of zeros, the paths will merge and terminate at the zeros.

Option 3: The root loci terminate at open-loop poles.

This statement is incorrect. The root loci do not terminate at the open-loop poles. Instead, the root loci terminate at the open-loop zeros or go to infinity if the system has more poles than zeros. This is a key feature of root locus analysis, as the termination points depend on the locations of the open-loop zeros and the relative number of poles and zeros.

Option 4: The root loci break away at open-loop poles.

This statement is incorrect. Root loci do not break away at the open-loop poles. Breakaway points occur at locations along the root locus paths where multiple branches of the root locus diverge or converge. These points are calculated based on the derivative of the characteristic equation and do not necessarily coincide with the open-loop poles.