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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)
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
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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