Plot a Root locus of the system 12 005 K²009 10? G(s)H(s) = thot S ( 5 + 2 + 2 j ) ( S + 2 − 2jj 1212 191
Plot a Root locus of the system 12 005 K²009 10? G(s)H(s) = thot S ( 5 + 2 + 2 j ) ( S + 2 − 2jj 1212 191
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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![### Plotting the Root Locus of a System
To plot the root locus of the system described by the transfer function \( G(s)H(s) \), we use the following expression:
\[ G(s)H(s) = \frac{K}{s(s + 2 + 2j)(s + 2 - 2j)} \]
Where:
- \( s \) is a complex frequency variable,
- \( K \) is a gain.
#### Steps to Plot the Root Locus:
1. **Identify Poles and Zeros**:
- Poles are the values of \( s \) that make the denominator zero.
- For this system, the poles are at \( s = 0 \), \( s = -2 - 2j \), and \( s = -2 + 2j \).
- There are no zeros for this system since there are no values of \( s \) that make the numerator zero.
2. **Plot Poles and Zeros on the Complex Plane**:
- Poles are represented by '×' marks.
- Zeros are represented by 'o' marks (none in this case).
3. **Draw Root Locus Paths**:
- Start from the positions of the poles.
- As the gain \( K \) increases from 0 to infinity, the paths show where the closed-loop poles move on the complex plane.
4. **Determine Breakaway and Break-in Points**:
- Identify points where multiple root locus branches either diverge or converge.
5. **Asymptotes of Root Locus**:
- Determine the directions that branches go to as \( s \) approaches infinity.
- The number of asymptotes is the difference between the number of poles and zeros.
6. **Calculate Centroid and Asymptote Angles**:
- Centroid: The average of the pole locations subtracted by the average of the zero locations.
- Angle: Given by \( \frac{(2k+1)180°}{n-m} \) where \( k \) is an integer, \( n \) is the number of poles, and \( m \) the number of zeros.
#### Detailed Explanation of the System's Function:
- The transfer function provided involves complex poles, which indicate oscillatory behavior.
- The positioning of poles along with the imaginary axis indicates the system's potential damped or](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F320f74c5-b563-40ea-a836-e50a07174a66%2F874cd204-c489-4ad9-8a99-ad58bc02efaa%2Fhvltd8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Plotting the Root Locus of a System
To plot the root locus of the system described by the transfer function \( G(s)H(s) \), we use the following expression:
\[ G(s)H(s) = \frac{K}{s(s + 2 + 2j)(s + 2 - 2j)} \]
Where:
- \( s \) is a complex frequency variable,
- \( K \) is a gain.
#### Steps to Plot the Root Locus:
1. **Identify Poles and Zeros**:
- Poles are the values of \( s \) that make the denominator zero.
- For this system, the poles are at \( s = 0 \), \( s = -2 - 2j \), and \( s = -2 + 2j \).
- There are no zeros for this system since there are no values of \( s \) that make the numerator zero.
2. **Plot Poles and Zeros on the Complex Plane**:
- Poles are represented by '×' marks.
- Zeros are represented by 'o' marks (none in this case).
3. **Draw Root Locus Paths**:
- Start from the positions of the poles.
- As the gain \( K \) increases from 0 to infinity, the paths show where the closed-loop poles move on the complex plane.
4. **Determine Breakaway and Break-in Points**:
- Identify points where multiple root locus branches either diverge or converge.
5. **Asymptotes of Root Locus**:
- Determine the directions that branches go to as \( s \) approaches infinity.
- The number of asymptotes is the difference between the number of poles and zeros.
6. **Calculate Centroid and Asymptote Angles**:
- Centroid: The average of the pole locations subtracted by the average of the zero locations.
- Angle: Given by \( \frac{(2k+1)180°}{n-m} \) where \( k \) is an integer, \( n \) is the number of poles, and \( m \) the number of zeros.
#### Detailed Explanation of the System's Function:
- The transfer function provided involves complex poles, which indicate oscillatory behavior.
- The positioning of poles along with the imaginary axis indicates the system's potential damped or
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