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1. – 4.) Sketch the positive root locus of the system shown below with the following choices of G(s). State the asymptote angles and their centroid. Determine the arrival and departure angles at any complex pole, complex zero, and repeated pole/zero. The frequencies of any imaginary axis crossings. The locations of any repeated roots (e.g. break-in or break-away points).

1. – 4.) Sketch the positive root locus of the system shown below with the following choices of G(s). State the asymptote angles and their centroid. Determine the arrival and departure angles at any complex pole, complex zero, and repeated pole/zero. The frequencies of any imaginary axis crossings. The locations of any repeated roots (e.g. break-in or break-away points).

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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### Root Locus and System Analysis 1. **Objective**: Sketch the positive root locus for the given system with various choices of G(s). Analyze the asymptote angles and centroid. Determine the arrival and departure angles at complex poles or zeros, and identify repeated root locations, such as break-in or break-away points. Verify results using MATLAB. 2. **Feedback Control System Diagram**: - **Components**: - **Σ (Summing Junction)**: Combines the input signals, indicating a feedback system where the input is summed with the negative feedback. - **K**: Represents the system gain. - **G(s)**: The transfer function, which varies as per different versions of the problem statement. - **Flow**: - The system receives an input signal which enters the summing junction. - The difference is sent through the gain element \( K \). - The output is processed by the transfer function \( G(s) \), leading to the system output. - Feedback is taken from the output and fed back into the summing junction. 3. **Steps for Analysis**: - **Root Locus Plot**: Using various forms of \( G(s) \), plot how the roots of the characteristic equation change with varying gain \( K \). - **Asymptotes**: Calculate the angles and centroid of the asymptotes, which indicate the behavior of the roots at extreme values of \( K \). - **Complex Poles and Zeros**: Find angles of arrival and departure, contributing to system stability and transient response. - **Repeats and Crossings**: Identify repeated roots and any crossings over the imaginary axis indicating stability limits. - **MATLAB Verification**: Implement the root locus plots using MATLAB to validate the manual calculations. This detailed analysis aids in understanding system stability, response characteristics, and control system design.
Transcribed Image Text:### Root Locus and System Analysis 1. **Objective**: Sketch the positive root locus for the given system with various choices of G(s). Analyze the asymptote angles and centroid. Determine the arrival and departure angles at complex poles or zeros, and identify repeated root locations, such as break-in or break-away points. Verify results using MATLAB. 2. **Feedback Control System Diagram**: - **Components**: - **Σ (Summing Junction)**: Combines the input signals, indicating a feedback system where the input is summed with the negative feedback. - **K**: Represents the system gain. - **G(s)**: The transfer function, which varies as per different versions of the problem statement. - **Flow**: - The system receives an input signal which enters the summing junction. - The difference is sent through the gain element \( K \). - The output is processed by the transfer function \( G(s) \), leading to the system output. - Feedback is taken from the output and fed back into the summing junction. 3. **Steps for Analysis**: - **Root Locus Plot**: Using various forms of \( G(s) \), plot how the roots of the characteristic equation change with varying gain \( K \). - **Asymptotes**: Calculate the angles and centroid of the asymptotes, which indicate the behavior of the roots at extreme values of \( K \). - **Complex Poles and Zeros**: Find angles of arrival and departure, contributing to system stability and transient response. - **Repeats and Crossings**: Identify repeated roots and any crossings over the imaginary axis indicating stability limits. - **MATLAB Verification**: Implement the root locus plots using MATLAB to validate the manual calculations. This detailed analysis aids in understanding system stability, response characteristics, and control system design.
**Problem 2:** Given the transfer function \( G(s) \), we need to analyze the following equation: \[ G(s) = \frac{s^2 + 2s + 145}{(s + 8)(s^2 + 4s + 5)} \] **Problem 3:** Next, consider the transfer function \( G(s) \) described by the equation: \[ G(s) = \frac{s^2 - 4s + 20}{(s + 4)(s + 6)(s^2 + 4s + 20)} \] ### Explanation: For both problems, \( G(s) \) represents a standard form of transfer function used in control systems. The numerator in each case consists of a polynomial in \( s \), while the denominator is a product of linear and quadratic factors in \( s \). These equations are typically analyzed to understand the behavior of dynamic systems, including stability and response characteristics.
Transcribed Image Text:**Problem 2:** Given the transfer function \( G(s) \), we need to analyze the following equation: \[ G(s) = \frac{s^2 + 2s + 145}{(s + 8)(s^2 + 4s + 5)} \] **Problem 3:** Next, consider the transfer function \( G(s) \) described by the equation: \[ G(s) = \frac{s^2 - 4s + 20}{(s + 4)(s + 6)(s^2 + 4s + 20)} \] ### Explanation: For both problems, \( G(s) \) represents a standard form of transfer function used in control systems. The numerator in each case consists of a polynomial in \( s \), while the denominator is a product of linear and quadratic factors in \( s \). These equations are typically analyzed to understand the behavior of dynamic systems, including stability and response characteristics.
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