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(b) Draw the signal flow graph of a system represented by the following block diagram and find C/R using Mason's gain formula. C R a b. e

(b) Draw the signal flow graph of a system represented by the following block diagram and find C/R using Mason's gain formula. C R a b. e

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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The task is to draw a signal flow graph for a control system based on a given block diagram and calculate the transfer function \( C/R \) using Mason's Gain Formula. ### Block Diagram Description: - **Inputs and Outputs:** - The system begins with an input \( R \) which leads to the output \( C \). - **Blocks:** - The blocks represent system components with specific transfer functions labeled as \( a, b, c, d, e, f, g, h, i, \) and \( j \). - **Connections:** - The input \( R \) passes through block \( a \) and then encounters a summing junction which receives additional feedbacks. - The signal then proceeds through blocks \( c, b, i, j \), and \( f \) sequentially with intermediate feedback and feedforward paths intersecting. - From block \( i \), a feedback path feeds back into the summing junction before block \( c \). - After block \( j \), the output branches into two paths: one proceeds to block \( f \) and then to \( g \), and the other feeds into block \( h \). - Outputs of blocks \( g \) and \( d \) go into a summing junction leading to the final system output \( C \). - Feedback is provided from the output \( C \) back to the input summing point via block \( e \). ### Instructions for Constructing the Signal Flow Graph: 1. **Nodes:** - Create nodes for each junction/interconnection in the diagram. 2. **Branches:** - Draw directed branches for each signal path between nodes. - Label each branch with the corresponding transfer function block (i.e., \( a, b, c, \) etc.). 3. **Loops and Paths:** - Identify and label forward paths from \( R \) to \( C \). - Identify any loops and their respective gains for applying Mason's Gain Formula. ### Mason’s Gain Formula: To determine the transfer function \( \frac{C}{R} \), use Mason's Gain Formula: \[ T = \frac{1}{\Delta} \sum_{k=1}^{N} P_k \Delta_k \] Where: - \( T \) is the overall system gain. - \( P_k \) is the gain of the kth forward path. - \( \
Transcribed Image Text:The task is to draw a signal flow graph for a control system based on a given block diagram and calculate the transfer function \( C/R \) using Mason's Gain Formula. ### Block Diagram Description: - **Inputs and Outputs:** - The system begins with an input \( R \) which leads to the output \( C \). - **Blocks:** - The blocks represent system components with specific transfer functions labeled as \( a, b, c, d, e, f, g, h, i, \) and \( j \). - **Connections:** - The input \( R \) passes through block \( a \) and then encounters a summing junction which receives additional feedbacks. - The signal then proceeds through blocks \( c, b, i, j \), and \( f \) sequentially with intermediate feedback and feedforward paths intersecting. - From block \( i \), a feedback path feeds back into the summing junction before block \( c \). - After block \( j \), the output branches into two paths: one proceeds to block \( f \) and then to \( g \), and the other feeds into block \( h \). - Outputs of blocks \( g \) and \( d \) go into a summing junction leading to the final system output \( C \). - Feedback is provided from the output \( C \) back to the input summing point via block \( e \). ### Instructions for Constructing the Signal Flow Graph: 1. **Nodes:** - Create nodes for each junction/interconnection in the diagram. 2. **Branches:** - Draw directed branches for each signal path between nodes. - Label each branch with the corresponding transfer function block (i.e., \( a, b, c, \) etc.). 3. **Loops and Paths:** - Identify and label forward paths from \( R \) to \( C \). - Identify any loops and their respective gains for applying Mason's Gain Formula. ### Mason’s Gain Formula: To determine the transfer function \( \frac{C}{R} \), use Mason's Gain Formula: \[ T = \frac{1}{\Delta} \sum_{k=1}^{N} P_k \Delta_k \] Where: - \( T \) is the overall system gain. - \( P_k \) is the gain of the kth forward path. - \( \
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