### Feedback Control Systems - Root Locus Analysis#### Problem Statement:For the following feedback control systems:a. **Sketch its Root Locus**b. **Show clearly, with arrows, the direction the closed-loop poles are moving as K goes from 0 to ∞.**c. **Find the Asymptotes (intersection and angles), and show them on the Root Locus.**d. **Indicate on the Root Locus the Break-in and Break-away points.**#### System Block Diagram:``` + +-----------+ R(s) -----> O -----> E(s) -----> K -----> (s^2 + 17s + 70) -----> C(s) | | | | +---------------------------------------------+```The given transfer function in the system is: \[ \frac{1}{s + 5} \]### Explanation:1. **Sketch the Root Locus**: - Root Locus is a graphical representation of the closed-loop poles of a control system as a function of the loop gain K. - For the given system, you would plot the poles and zeros of the transfer function \( \frac{s^2 + 17s + 70}{s^2 - 6s + 8} \).2. **Show Closed-loop Pole Directions with Arrows**: - Indicate the points on the graph where the poles move as K varies from 0 to ∞. The direction of movement is shown with arrows.3. **Find and Show Asymptotes**: - Calculate the asymptotes for the Root Locus by using the formula: - Asymptote angles: \[ \theta_A = \frac{(2q+1)180^\circ}{n - m} \] Where n is the number of poles, and m is the number of zeros. - The point where the asymptotes intersect the real axis (σ), given by: \[ \sigma = \frac{\sum \text{Poles} - \sum \text{Zeros}}{n - m} \] - Show these asymptotes and their angles on the Root Locus graph.4. **Indicate Break-in and Break-away Points**: - Break-in and break-away points occur where the poles move toward or away from each other on the real

The loop transfer function is given below,

For the following feedback control systems: a. Sketch its Root Locus b. Show clearly, with arrows, the direction the closed-loop poles are moving as K goes from 0 to ∞ e. Find the Asymptotes (intersection and angles), and show them on the Root Locus d. Indicate on the Root Locus the Break-in and Break-away points C(s) R(s) + E(s) K s2+17s + 70 s2-6s+8 s+5

For the following feedback control systems: a. Sketch its Root Locus b. Show clearly, with arrows, the direction the closed-loop poles are moving as K goes from 0 to ∞ e. Find the Asymptotes (intersection and angles), and show them on the Root Locus d. Indicate on the Root Locus the Break-in and Break-away points C(s) R(s) + E(s) K s2+17s + 70 s2-6s+8 s+5

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

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Question

### Feedback Control Systems - Root Locus Analysis

#### Problem Statement:

For the following feedback control systems:
a. **Sketch its Root Locus**
b. **Show clearly, with arrows, the direction the closed-loop poles are moving as K goes from 0 to ∞.**
c. **Find the Asymptotes (intersection and angles), and show them on the Root Locus.**
d. **Indicate on the Root Locus the Break-in and Break-away points.**

#### System Block Diagram:
```
+ +-----------+
R(s) -----> O -----> E(s) -----> K -----> (s^2 + 17s + 70) -----> C(s)
| |
| |
+---------------------------------------------+
```
The given transfer function in the system is:

\[ \frac{1}{s + 5} \]

### Explanation:

1. **Sketch the Root Locus**:
- Root Locus is a graphical representation of the closed-loop poles of a control system as a function of the loop gain K.
- For the given system, you would plot the poles and zeros of the transfer function \( \frac{s^2 + 17s + 70}{s^2 - 6s + 8} \).

2. **Show Closed-loop Pole Directions with Arrows**:
- Indicate the points on the graph where the poles move as K varies from 0 to ∞. The direction of movement is shown with arrows.

3. **Find and Show Asymptotes**:
- Calculate the asymptotes for the Root Locus by using the formula:
- Asymptote angles: \[ \theta_A = \frac{(2q+1)180^\circ}{n - m} \]
Where n is the number of poles, and m is the number of zeros.
- The point where the asymptotes intersect the real axis (σ), given by:
\[ \sigma = \frac{\sum \text{Poles} - \sum \text{Zeros}}{n - m} \]
- Show these asymptotes and their angles on the Root Locus graph.

4. **Indicate Break-in and Break-away Points**:
- Break-in and break-away points occur where the poles move toward or away from each other on the real

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