The given equation represents a transfer function commonly used in control systems and signal processing.**Equation:**\[ G(s) = \frac{1}{s^4 + 8s^3 + 44s^2 + 112s + 160} \]**Description:**The transfer function $ G(s) $ is expressed as the reciprocal of a polynomial in the complex frequency variable $ s $. The denominator is a fourth-degree polynomial, $ s^4 + 8s^3 + 44s^2 + 112s + 160 $. This type of function is essential for analyzing the dynamics of systems in the Laplace domain, allowing for the study of system behavior in response to various inputs. 1. – 4.) Sketch the positive root locus of the system shown below with the following choices of $ G(s) $. State the asymptote angles and their centroid. Determine the arrival and departure angles at any complex pole, complex zero, and repeated pole/zero. The frequencies of any imaginary axis crossings. The locations of any repeated roots (e.g., break-in or break-away points). Verify your results using Matlab to obtain numerical data. Your sketches and the Matlab results should be displayed on the same scale.**Block Diagram Explanation:**The block diagram consists of a standard feedback control system with the following components:1. **Summation Block ($\Sigma$):** - This block sums the input signals. It has two inputs: a positive input (from the left) and a negative feedback input (from the bottom).2. **Gain Block ($K$):** - A gain block that multiplies the input signal by a constant $K$.3. **Transfer Function Block ($G(s)$):** - This block represents the system transfer function $G(s)$, which transforms the input to output based on system dynamics.4. **Feedback Loop:** - The output from $G(s)$ is fed back to the summation block, completing the control loop. The goal is to analyze the root locus of this feedback system for various choices of $G(s)$, determining critical properties related to system stability and dynamics.

Given Transfer function G(s)·H(s)=N(s)D(s)=1s4+8·s3+44·s2+112·s+160 The characteristic equation…

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)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

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Question

The given equation represents a transfer function commonly used in control systems and signal processing.

**Equation:**

\[ G(s) = \frac{1}{s^4 + 8s^3 + 44s^2 + 112s + 160} \]

**Description:**

The transfer function $ G(s) $ is expressed as the reciprocal of a polynomial in the complex frequency variable $ s $. The denominator is a fourth-degree polynomial, $ s^4 + 8s^3 + 44s^2 + 112s + 160 $. This type of function is essential for analyzing the dynamics of systems in the Laplace domain, allowing for the study of system behavior in response to various inputs.

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). Verify your results using Matlab to obtain numerical data. Your sketches and the Matlab results should be displayed on the same scale.

**Block Diagram Explanation:**

The block diagram consists of a standard feedback control system with the following components:

1. **Summation Block ($\Sigma$):**
- This block sums the input signals. It has two inputs: a positive input (from the left) and a negative feedback input (from the bottom).

2. **Gain Block ($K$):**
- A gain block that multiplies the input signal by a constant $K$.

3. **Transfer Function Block ($G(s)$):**
- This block represents the system transfer function $G(s)$, which transforms the input to output based on system dynamics.

4. **Feedback Loop:**
- The output from $G(s)$ is fed back to the summation block, completing the control loop.

The goal is to analyze the root locus of this feedback system for various choices of $G(s)$, determining critical properties related to system stability and dynamics.

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