How does the number of branches in a root locus relate to the number of poles and zeros?

• A. Branches equal the number of zeros
• B. Branches equal the number of poles
• C. Branches equal the sum of poles and zeros
• D. Branches equal the difference of poles and zeros

Answer: (C)

The number of branches in a root locus is directly related to the concept of poles and zeros in a control system. Specifically, the branches of the root locus correspond to the locations of the closed-loop poles as the gain of the system is varied.

The correct assertion here is that the number of branches equals the sum of the number of poles and zeros present in the open-loop transfer function. This is because every pole contributes a branch to the root locus as it moves from the complex plane to the zeroes, and every zero can also attract branches to itself.

In practical terms, this means that if you have a system with a certain number of poles and a certain number of zeros, the root locus will illustrate the possible positions of all the closed-loop poles as system parameters change. For example, if there are three poles and two zeros, there would be five total branches in the root locus representation.

Understanding this relationship helps in analyzing system stability and performance through the root locus method, as adjusting the gain modifies how these branches behave and consequently affects the system's response.