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Root loci are usually plotted for variations in the gain. Sometimes we are interested in the variations of the closed-loop poles as other parameters changed. For a unity feedback system with the forward transfer function G(s) = 1/[s(s+a)], sketch the root locus as a is varied from 0 to infinity.

Root loci are usually plotted for variations in the gain. Sometimes we are interested in the variations of the closed-loop poles as other parameters changed. For a unity feedback system with the forward transfer function G(s) = 1/[s(s+a)], sketch the root locus as a is varied from 0 to infinity.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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Root loci are usually plotted for variations in the gain. Sometimes we are interested in the variations of the closed-loop poles as other parameters change. For a unity feedback system with the forward transfer function \( G(s) = \frac{1}{s(s+\alpha)} \), sketch the root locus as \( \alpha \) is varied from 0 to infinity. **Explanation:** This task involves creating a root locus plot, which is a graphical representation used in control systems to show how the roots of a system change with variation in a certain parameter, typically gain. In this case, the parameter of interest is \( \alpha \), which varies from 0 to infinity. **Graph/Diagram:** - **Axes:** The x-axis represents the real part, and the y-axis represents the imaginary part of the complex plane. - **Poles and zeros:** Identify the poles and zeros of the system. Here, the poles are at \( s = 0 \) and \( s = -\alpha \), and there are no zeros. - **Root locus path:** The paths of these poles as \( \alpha \) increases from 0 to infinity should be sketched on the complex plane. **Steps to Sketch:** 1. **Starting Points:** At \( \alpha = 0 \), the poles are at \( s = 0 \) and \( s = 0 \) (superimposed). 2. **Movement:** As \( \alpha \) increases, the pole at \( s = -\alpha \) moves leftwards along the real axis. 3. **Asymptotes:** Since there's a difference in the number of poles and zeros, asymptotes assist in visualizing the movement direction at infinity. 4. **Behavior at Infinity:** The poles continue to diverge along the real axis. This exercise portrays how the closed-loop poles move in response to changes in parameter \( \alpha \), providing valuable insights into system stability and dynamics.
Transcribed Image Text:Root loci are usually plotted for variations in the gain. Sometimes we are interested in the variations of the closed-loop poles as other parameters change. For a unity feedback system with the forward transfer function \( G(s) = \frac{1}{s(s+\alpha)} \), sketch the root locus as \( \alpha \) is varied from 0 to infinity. **Explanation:** This task involves creating a root locus plot, which is a graphical representation used in control systems to show how the roots of a system change with variation in a certain parameter, typically gain. In this case, the parameter of interest is \( \alpha \), which varies from 0 to infinity. **Graph/Diagram:** - **Axes:** The x-axis represents the real part, and the y-axis represents the imaginary part of the complex plane. - **Poles and zeros:** Identify the poles and zeros of the system. Here, the poles are at \( s = 0 \) and \( s = -\alpha \), and there are no zeros. - **Root locus path:** The paths of these poles as \( \alpha \) increases from 0 to infinity should be sketched on the complex plane. **Steps to Sketch:** 1. **Starting Points:** At \( \alpha = 0 \), the poles are at \( s = 0 \) and \( s = 0 \) (superimposed). 2. **Movement:** As \( \alpha \) increases, the pole at \( s = -\alpha \) moves leftwards along the real axis. 3. **Asymptotes:** Since there's a difference in the number of poles and zeros, asymptotes assist in visualizing the movement direction at infinity. 4. **Behavior at Infinity:** The poles continue to diverge along the real axis. This exercise portrays how the closed-loop poles move in response to changes in parameter \( \alpha \), providing valuable insights into system stability and dynamics.
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