The text presents a control systems question from an examination or academic exercise, focusing on the topic of root locus in control theory.---**Question 12:**Consider the transfer function \( G[s] = \frac{(s+3)^2 + (1)^2}{s[(s+1)^2 + (1)^2]} \) and let \( G_c[s] = K \). How many root locus branches are there?Options:- (a) 0- (b) 1- (c) 2- (d) 3---### Explanation:In this problem, you are given a transfer function \( G[s] \) and are asked to determine the number of root locus branches when connected to a controller \( G_c[s] = K \).- **Numerator Analysis:** The numerator \((s+3)^2 + 1^2\) indicates the presence of poles that can affect the root locus plot.- **Denominator Analysis:** The denominator \(s[(s+1)^2 + 1^2]\) contains both a term at the origin (s = 0) and a quadratic term, which will affect the system's stability and dynamics.- **Root Locus Branches:** The total number of root locus branches corresponds to the total number of poles in the open-loop transfer function \( G[s]H[s] \) where \( H[s] = G_c[s] = K \). Count the poles in the denominator and any additional poles at infinity due to unmatched zeros. This analysis helps in determining the number of branches for the root locus plot and is crucial for understanding system behavior in control systems. **Block Diagram Analysis**The diagram represents a signal processing system with the following components:1. **Input Signal (R[\(\omega\)])**: - This is the initial input to the system.2. **Summation Block (\(\Sigma\))**: - The input signal R[\(\omega\)] is fed into a summation block where it is added to a feedback signal. - The output of the summation block is E[\(\omega\)], which represents the error or difference after feedback.3. **Controller Block (Gc[\(\omega\)])**: - E[\(\omega\)] is processed through this block. This represents a controller that modifies the signal, typically to achieve desired system behavior. 4. **Plant Block (G[\(\omega\)])**: - The output of the controller (Gc[\(\omega\)]) is then processed by the plant block. - This block represents the actual system being controlled.5. **Output Signal (Y[\(\omega\)])**: - The final output of the system after processing through both the controller and plant.6. **Feedback Loop**: - The output Y[\(\omega\)] is looped back as a feedback into the summation block. - This feedback is crucial for adjusting the input to the controller, aiming to minimize the error over time and achieve stable system behavior.This block diagram is typical in control systems, illustrating the interaction between forward and feedback paths to influence system dynamics.

The given problem is from control system. The number of root locus branch is Number of finite open…

Answered: 12. , Let G[s]: = there? (a) 0 (b) 1…

12. , Let G[s]: = there? (a) 0 (b) 1 (c) 2 (d) 3 (s+3)²+(1)² s[(s+1)²+(1)²] and Ge[s] = K. How many root locus branches are

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