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solve parts d and e and f  d in the picture, e and f are: e) The value of K when the root loci intersect the imaginary axis   f) Ples the root loci for the closed loop system

solve parts d and e and f  d in the picture, e and f are: e) The value of K when the root loci intersect the imaginary axis   f) Ples the root loci for the closed loop system

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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solve parts d and e and f 

d in the picture, e and f are:

e) The value of K when the root loci intersect the imaginary axis

 

f) Ples the root loci for the closed loop system

 

**Q.3 For closed loop control system** Given: \[ G(s) = \frac{K(s+9)}{s(s^2 + 4s + 11)} \] \[ H(s) = 1 \] Tasks: a) Determine the zeros and poles of the system b) Determine the root loci on the real axis c) Determine the asymptotes of the root loci d) Find the value of \( K \) to keep the system stable Instructions: 1. **Determine Zeros and Poles:** - Zeros are the values of \( s \) that make \( G(s) = 0 \). - Poles are the values of \( s \) that make the denominator of \( G(s) = 0 \). 2. **Determine Root Loci on the Real Axis:** - Root loci are the paths that the roots of the characteristic equation follow in the s-plane as a single system parameter (typically \( K \)) varies from 0 to ∞. 3. **Determine Asymptotes of the Root Loci:** - Asymptotes give the direction in which the root loci move as they tend to ∞. - The formula for asymptotes is given by the centroid \( \sigma_a \) and angle \( \theta_a \): \[ \sigma_a = \frac{\sum \text{poles} - \sum \text{zeros}}{n-m} \] where \( n \) is the number of poles and \( m \) is the number of zeros. 4. **Find the Value of \( K \) for System Stability:** - The value of \( K \) should be such that all poles of the closed-loop system have negative real parts. These tasks will help understand the stability and response characteristics of the given closed-loop control system.
Transcribed Image Text:**Q.3 For closed loop control system** Given: \[ G(s) = \frac{K(s+9)}{s(s^2 + 4s + 11)} \] \[ H(s) = 1 \] Tasks: a) Determine the zeros and poles of the system b) Determine the root loci on the real axis c) Determine the asymptotes of the root loci d) Find the value of \( K \) to keep the system stable Instructions: 1. **Determine Zeros and Poles:** - Zeros are the values of \( s \) that make \( G(s) = 0 \). - Poles are the values of \( s \) that make the denominator of \( G(s) = 0 \). 2. **Determine Root Loci on the Real Axis:** - Root loci are the paths that the roots of the characteristic equation follow in the s-plane as a single system parameter (typically \( K \)) varies from 0 to ∞. 3. **Determine Asymptotes of the Root Loci:** - Asymptotes give the direction in which the root loci move as they tend to ∞. - The formula for asymptotes is given by the centroid \( \sigma_a \) and angle \( \theta_a \): \[ \sigma_a = \frac{\sum \text{poles} - \sum \text{zeros}}{n-m} \] where \( n \) is the number of poles and \( m \) is the number of zeros. 4. **Find the Value of \( K \) for System Stability:** - The value of \( K \) should be such that all poles of the closed-loop system have negative real parts. These tasks will help understand the stability and response characteristics of the given closed-loop control system.
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