Flow Graphs of Laplace Transform

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    Here’s the next part of Nasir’s tutorial on Laplace Transform.

    Introduction

    We know that Laplace transformation is a way of transforming integral and differential equations into algebraic equation of s-domain. The expression of Laplace transformation is given as:

    Flow Graphs of Laplace Transform 1

    In the preceding article we discussed the simulation diagrams. A simulation diagram is basically the pictorial representation of the systems. It shows the functioning of each component and expresses the flow of signals.

    A simple block diagram notation is given as:

    Flow Graphs of Laplace Transform 2

    For a better understanding of the working of a system we may convert the transform simulation diagrams into flow graphs. A flow graph represents continuous working of a system.

    Definition of Flow Graphs

    A flow graph is a simpler representation of a block or simulation diagram. It is the graphical embodiment of the association of dependent and independent variables. Below is the example of a simple flow graph:

    Flow Graphs of Laplace Transform 3

    Elementary Building Blocks of a Flow Graph

    Following are the basic elements of a flow graph:

    Node:

    A node is basically the junction at which the variable is present. It signifies a variable. There are two types of nodes:

    1. Input Node: a node that has only outgoing branches is called Input node.
    2. Output Node: a node that has only incoming branches is called an output node.

    Path:

    The assortment of all the uninterrupted branches, having the same direction is known as a path.

    A forwarded Path is a path that initiates from an input node and dismisses at the output node is called a forwarded path. A forwarded path circulates once and only once from each node.

    Branch:

    The path that joins two nodes is called a branch. A branch mainly consists of a direction and a gain. The Direction shows whether the connection is coming to the node or going out of it.

    Loop:

    If a path ceases at the node it initiated from, then such a path is called a Loop. Just like a forwarded path a loop doesn’t pass any node more than once.

    Loop Gain:

    The product of all the branch gains present in a traversing path is called loop gain. To figure loop gain out we use Mason’s gain formula, which states:

    Flow Graphs of Laplace Transform 4

    Where,

    • N is the total number of forwarded paths between Yin and Yout
    • Mk is the gain of the Kth forwarded path
    • Transient Response Analysis of Control Systems_symbolis the determinant of the graph

    Example 1

    Flow Graphs of Laplace Transform 5

    Going from left (R) to the right (Y), we find one forward path M1=G(s).

    We are having only one loop which touches the frontward track.

    Flow Graphs of Laplace Transform 6

    Example 2

    Flow Graphs of Laplace Transform 7

    Where,

    R3R4/R1R2 (two non touching twists along with an increase)

    Transient Response Analysis of Control Systems_symbol1=1 (is a forward path touched by each twists)

    M1=R3R4/R1R2 (only forward route)

    -R3/R1, -R3/R1 and -R4/R2 (three feedback twists having a three increases)

    Flow Graphs of Laplace Transform 8

    Example 3

    Flow Graphs of Laplace Transform 9
    M1= G1G2G3G4G5 and M2= G6G4G5 are forward pathways.

    Where,

    Transient Response Analysis of Control Systems_symbol1 = 1 (each twist/loop comes in contact with first forward pathway).

    Transient Response Analysis of Control Systems_symbol2 = 1+G2H1 (unlike former pathway, first twist doesn’t touch the second pathway).

    Flow Graphs of Laplace Transform 10

    Example 4

    To make things more interesting, we are taking a complex example which is not easily solved by clock diagram procedure.

    Flow Graphs of Laplace Transform 11

    Where,

    Flow Graphs of Laplace Transform 12

    are three forward pathways.

    Flow Graphs of Laplace Transform 13

    are the increases of feedback twists.

    Flow Graphs of Laplace Transform 14

    These are the cofactors involved.

    All the twists will touch except for the twists L4 and L7 which will not be touched by L5 and loop L4 which will not be touched by loop L3.

    The determinant Δ is given by:

    Flow Graphs of Laplace Transform 15

    Block Diagram of Signal Flow Graph

    Flow Graphs of Laplace Transform 16

    By utilizing Mason’s Gain Formula, we will find out the transfer function.

    Taking in account the equation written below:

    Flow Graphs of Laplace Transform 17

    Supposing the initial conditions to be zero, taking the differential equation written below:

    Flow Graphs of Laplace Transform 18

    After applying Laplace transform, the result will be:

    Flow Graphs of Laplace Transform 19

    Again we will utilize the Mason’s Gain formula to obtain the transfer function:

    Flow Graphs of Laplace Transform 20

    Conclusion

    So in a nut shell the signal flow graph is a modified form of simulation diagrams. The working of a system can be analyzed with ease if we construct a flow graph with help of the function and simulation diagram.

    With flow graphs, the learning part of our Laplace transform comes to an end. We have discussed from the very basics of the Laplace transform and its orientation to its working and simulation diagrams along with the graphs.

    It the upcoming at last part of tutorial we will demonstrate maximum number of example we can. This will help us in comprehending the implementation of the Laplace transform. We shall start with some basic functions and then move further to demonstrate the working of each part of this tutorial.

    As usual I thank you all for reading my tutorials!

    Nasir.

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