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Draw the root loci of the following pole-zero configurations, as the gain K goes from 0 to infinity. a. Calculate the number of asymptotes in each case and (Approximately) mark the center of asymptotes in the rlocus graph. b. Point the breakaway and break-in points on each rlocus graph with an arrow, if any. jeo s-plane splane s-plane splane jeo X X s-plane s-plane

Draw the root loci of the following pole-zero configurations, as the gain K goes from 0 to infinity. a. Calculate the number of asymptotes in each case and (Approximately) mark the center of asymptotes in the rlocus graph. b. Point the breakaway and break-in points on each rlocus graph with an arrow, if any. jeo s-plane splane s-plane splane jeo X X s-plane s-plane

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Root Locus Analysis for Various Pole-Zero Configurations** In this module, we will explore how to draw the root loci for different pole-zero configurations as the gain \( K \) varies from 0 to infinity. We will also calculate the number of asymptotes and approximate the center of these asymptotes. Additionally, we will identify the breakaway and break-in points, if any, for each configuration. ### Pole-Zero Configurations and Their Root Loci: #### 1. First Configuration Graph Description: - One zero and one pole on the real axis. - The zero is located at \( s = -1 \). - The pole is located at \( s = -2 \). Analysis Steps: a. Number of asymptotes: 0 b. No breakaway or break-in points. #### 2. Second Configuration Graph Description: - Two poles on the real axis. - The poles are located at \( s = -2 \) and \( s = -3 \). Analysis Steps: a. Number of asymptotes: 1 (at infinity; the asymptote is simply the real axis). b. Possible breakaway point at \( s = -2.5 \). #### 3. Third Configuration Graph Description: - Two poles and one zero on the real axis. - Zeros at \( s = -1 \) and \( s = -2 \). - A pole at \( s = -3 \). Analysis Steps: a. Number of asymptotes: 2 b. No breakaway or break-in points. #### 4. Fourth Configuration Graph Description: - Two zeros and three poles on the real axis. - Poles at \( s = -1 \), \( s = -2 \), and \( s = -3 \). - A zero at \( s = 0 \). Analysis Steps: a. Number of asymptotes: 1 (asymptotes intersect at s = -1) b. Possible breakaway point between \( s = -1 \) and \( s = -2 \). #### 5. Fifth Configuration Graph Description: - One zero and two poles on the real axis. - Poles at \( s = -1 \) and \( s = -3 \). - A zero at \( s = 2 \). Analysis Steps: a. Number of asymptotes: 1 (asymptotes intersect
Transcribed Image Text:**Root Locus Analysis for Various Pole-Zero Configurations** In this module, we will explore how to draw the root loci for different pole-zero configurations as the gain \( K \) varies from 0 to infinity. We will also calculate the number of asymptotes and approximate the center of these asymptotes. Additionally, we will identify the breakaway and break-in points, if any, for each configuration. ### Pole-Zero Configurations and Their Root Loci: #### 1. First Configuration Graph Description: - One zero and one pole on the real axis. - The zero is located at \( s = -1 \). - The pole is located at \( s = -2 \). Analysis Steps: a. Number of asymptotes: 0 b. No breakaway or break-in points. #### 2. Second Configuration Graph Description: - Two poles on the real axis. - The poles are located at \( s = -2 \) and \( s = -3 \). Analysis Steps: a. Number of asymptotes: 1 (at infinity; the asymptote is simply the real axis). b. Possible breakaway point at \( s = -2.5 \). #### 3. Third Configuration Graph Description: - Two poles and one zero on the real axis. - Zeros at \( s = -1 \) and \( s = -2 \). - A pole at \( s = -3 \). Analysis Steps: a. Number of asymptotes: 2 b. No breakaway or break-in points. #### 4. Fourth Configuration Graph Description: - Two zeros and three poles on the real axis. - Poles at \( s = -1 \), \( s = -2 \), and \( s = -3 \). - A zero at \( s = 0 \). Analysis Steps: a. Number of asymptotes: 1 (asymptotes intersect at s = -1) b. Possible breakaway point between \( s = -1 \) and \( s = -2 \). #### 5. Fifth Configuration Graph Description: - One zero and two poles on the real axis. - Poles at \( s = -1 \) and \( s = -3 \). - A zero at \( s = 2 \). Analysis Steps: a. Number of asymptotes: 1 (asymptotes intersect
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