Mr. Aketchescobenz - A Tvet Trainer 19/10/2019

MR.
AKETCHESCOBENZ – A TVET TRAINER 19/10/2019
SIGNAL FLOW GRAPH

Introduction
Block diagram reduction is the excellent method for determining the
transfer function of the control system. However, in a complicated
system, it is very difficult and time-consuming process that is why
an alternate method, i.e., SFG was developed by S.J Mason (Samuel
Jefferson Mason) which relates the input and output system
variables graphically. In the signal flow graph, the transfer
function is referred to as transmittance.
It should be noted that the signal flow graph approach and the
block diagram approach yield the same information.
The advantage in signal flow graph method is that, using Mason's

gain formula the overall gain of the system can be computed
easily. This method is simpler than the tedious block diagram
reduction techniques.
Basic Elements of Signal Flow Graph

Signal flow graph is a graphical representation of algebraic
equations.
Characteristics of SFG: SFG is a graphical representation of the

relationship between the variables of a set of linear algebraic
equations. It doesn't require any reduction technique or process.
Basic Elements of Signal Flow Graph

Nodes and branches are the basic elements of signal flow graph.
MR. AKETCHESCOBENZ – A TVET TRAINER 19/10/2019
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Path: A path is a continuous set of branches traversed in the

direction indicated by the branch arrows i.e., the path that exists
from the input node to the output node is known as forward path.
Node
Node is a point which represents either a variable or a signal. There
are three types of nodes:
1. input node,
2. output node and
3. Mixed node.
Input Node (Source) − It is a node, which has only outgoing

branch.
Output Node (sink) − It is a node, which has only incoming

branch.
Note: Any non-input node can be made an output node by adding a

branch with gain= 1.
Mixed Node − It is a node, which has both incoming and outgoing

branches.
Example:
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Let us consider the following signal flow graph for
Number of forward paths, N = 2.

1. First forward path is –
P1: y1→y2→y3→y4→y5→y6.
First forward path gain, P1=abcde.
2. Second forward path is - y1→y2→y3→y5→y6.
3. Second forward path gain, P2=abge.
4. Number of individual loops, L = 5.
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Non-touching loops: Loop is said to be non-touching if they

do not have any common node. i.e. if they do not share at
least one node (Loops with no common nodes)
Examples
Non-Touching Loops: Examples:

L1 and L2 are touching loops L1 and L3 & L2 and L3 are non-
touching loops,
Open path. If no node is re-visited, the path is open.
Forward path: A path from an input node (source) to an output

node (sink) that does not re-visit any node.
Path gain: the product of the gains of all the branches in the path.
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Loop: It is a path that starts and ends at the same node. It is a closed
path (it originates and ends on the same node, and no node is
touched more than once).
Loop gain: the product of the gains of all the branches in the loop.
Non-touching loops: Non-touching loops have no common nodes.
Example
Let us consider the following signal flow graph to identify these
nodes.
 The nodes present in this signal flow graph are y1, y2,
y3 and y4.
 y1 and y4 are the input node and output node respectively.
 y2 and y3 are mixed nodes.
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Branch
Branch is a line segment which joins two nodes. It has
both gain and direction. For example, there are four branches in
the above signal flow graph. These branches have gains of a, b,
c and -d.
RULES
Rule 1: Incoming signal to a node through a branch is given by the
product of a signal at previous node and the gain of the branch.
Check below solved examples.
Rule 2: Cascaded (series)

branches can be combined to give a single branch whose
transmittance is equal to the product of individual branch
transmittance. Check below solved examples.
Note: No intermediate incoming or outgoing branches between x1

and x4.
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Rule 3: Parallel branches

Parallel branches may be represented by single branch whose
transmittance is the sum of individual branch transmittance. Check
below solved examples.
Ruled 4: A mixed node can be eliminated by multiplying the

transmittance of outgoing branch (from the mixed node) to the
transmittance of all incoming branches to the mixed node. Check
below solved examples.
Rule 5: A loop may be eliminated by writing equations at the input

and output node and rearranging the equations to find the ratio of
output to input. This ratio gives the gain of resultant branch.
Check below solved examples.
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Signal flow graph Reduction

The signal flow graph of a system can be reduced either by using the
rules of signal flow graph algebra or by using Mason's gain formula.
For signal flow graph reduction using the rules of signal flow graph,
write equations at every node and then rearrange these equations to
get the ratio of output and input (transfer function).
The signal flow graph reduction by above method will be time

consuming and tedious. S.J.Mason developed a simple procedure to
determine the transfer function of the system represented as a signal
flow graph. He developed a formula called by his name Mason's
gain formula which can be directly used to find the transfer function
of the system.
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Construction of Signal Flow Graph

Let us construct a signal flow graph by considering the following
algebraic equations −
There will be six nodes (y1, y2, y3, y4, y5 and y6) and
eight branches in this signal flow graph. The gains of the branches
are a12, a23, a34, a45, a56, a42, a53 and a35.
To get the overall signal flow graph, draw the signal flow graph for
each equation, then combine all these signal flow graphs and then
follow the steps given below −
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Simple Process of Calculating Expression of Transfer Function

for Signal Flow Graph
 First, the input signal to be calculated at each node of the
graph. The input signal to a node is summation of product of

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transmittance and the other end node variable of each of the

branches arrowed towards the former node.
 Now by calculating input signal at all nodes will get numbers
of equations which relating node variables and transmittance.
More precisely, there will be one unique equation for each of
the input variable node.
 By solving these equations we get, ultimate input and output of
the entire signal flow graph of control system.
 Lastly by dividing inspiration of ultimate output to the
expression of initial input we calculate the expiration of
transfer function of that signal flow graph
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If P is the forward path transmittance between extreme input and

output of a signal flow graph. L1, L2…………………. loop
transmittance of first, second,.….. loop of the graph. Then for first
signal flow graph of control system, the overall transmittance
between extreme input and output is
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Here in the figure above, there are two parallel forward paths.
Hence, overall transmittance of that signal flow graph of control
system will be simple arithmetic sum of forward transmittance of
these two parallel paths.
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As the each of the parallel paths having one loop associated with it,
the forward transmittances of these parallel paths are
Conversion of Block Diagrams into Signal Flow Graphs

Follow these steps for converting a block diagram into its
equivalent signal flow graph.
 Represent all the signals, variables, summing points and take-
off points of block diagram as nodes in signal flow graph.

 Represent the blocks of block diagram as branches in signal
flow graph.
 Represent the transfer functions inside the blocks of block
diagram as gains of the branches in signal flow graph.

 Connect the nodes as per the block diagram. If there is
connection between two nodes (but there is no block in

between), then represent the gain of the branch as one. For
example, between summing points, between summing point
and takeoff point, between input and summing point, between
take-off point and output.
Consider the block diagram below
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The transfer function is referred as transmittance in signal flow

graph. Let us take an example of equation y = Kx. This equation can
be represented with block diagram as below
The same equation can be represented by signal flow graph, where x

is input variable node, y is output variable node and a is the
transmittance of the branch connecting directly these two nodes.
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Again: Let us convert the following block diagram into its

equivalent signal flow graph.
The following figure shows the equivalent signal flow graph.
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With the help of Mason’s gain formula, you can calculate the
transfer function of this signal flow graph. This is the advantage of
signal flow graphs. Here, we no need to simplify (reduce) the signal
flow graphs for calculating the transfer function.
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Approach 2 – A simpler and a straighter forward method
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CONSIDER THE SIGNAL FLOW GRAPH BELOW AND

IDENTIFY THE FOLLOWING
a) Input
Input Node (source) − It is a node, which has only outgoing
branches.
b) Output Nodes
Output Node (sink) − It is a node, which has only incoming
branches.
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c) Forward Paths
Forward path: A path from an input node (source) to an output
node (sink) that does not re-visit any node
d) Feedback Paths or Loops
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Feedback Paths or Loops
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e) Self loop(s):
It starts and ends at the same node
f) Forward path gains:

Path gain: the product of the gains of all the branches in the path.
There are two forward path gains;
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g) Identify loops:
Loop: It is a path that starts and ends at the same node. It is a closed
path (it originates and ends on the same node, and no node is
touched more than once).
h) Path Gains of the Forward Paths
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g) Identify non-touching loop gains:

Non-touching loops are loops with no common nodes
Non-touching loop gains;
Loop Gains of the Feedback Loops
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Mason’s Rule (Mason, 1953)
Samuel Jefferson Mason’s provides a formula to calculate the same

overall transfer function
The transfer function, C(s)/R(s), of a system represented by a signal-
flow graph.
Let us now discuss the Mason’s Gain Formula. Suppose there are
‘N’ forward paths in a signal flow graph. The gain between the input
and the output nodes of a signal flow graph is nothing but
the transfer function of the system. It can be calculated by using
Mason’s gain formula.
Mason’s gain formula is
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Example2: we want to obtain transfer function for the following

signal flow graph
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Solution:
Step 1: Obtain Forward path
From the flow graph, we can observe that in total two forward paths
present i.e
Step 2: Total number of loops

Now, as we already found our required forward paths, we require a
total number of loops.
There is a total of 5 loops in the signal flow graph if you observe
carefully.
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MR.

AKETCHESCOBENZ – A TVET TRAINER 19/10/2019

SIGNAL FLOW GRAPH

The advantage in signal flow graph method is that, using Mason's

Basic Elements of Signal Flow Graph

Characteristics of SFG: SFG is a graphical representation of the

Basic Elements of Signal Flow Graph

Path: A path is a continuous set of branches traversed in the

Input Node (Source) − It is a node, which has only outgoing

Output Node (sink) − It is a node, which has only incoming

Note: Any non-input node can be made an output node by adding a

Mixed Node − It is a node, which has both incoming and outgoing

MR. AKETCHESCOBENZ – A TVET TRAINER 19/10/2019

Let us consider the following signal flow graph for

Number of forward paths, N = 2.

MR. AKETCHESCOBENZ – A TVET TRAINER 19/10/2019

Non-touching loops: Loop is said to be non-touching if they

Non-Touching Loops: Examples:

Forward path: A path from an input node (source) to an output

MR. AKETCHESCOBENZ – A TVET TRAINER 19/10/2019

Non-touching loops: Non-touching loops have no common nodes.

MR. AKETCHESCOBENZ – A TVET TRAINER 19/10/2019

Rule 2: Cascaded (series)

Note: No intermediate incoming or outgoing branches between x1

MR. AKETCHESCOBENZ – A TVET TRAINER 19/10/2019

Rule 3: Parallel branches

Ruled 4: A mixed node can be eliminated by multiplying the

Rule 5: A loop may be eliminated by writing equations at the input

MR. AKETCHESCOBENZ – A TVET TRAINER 19/10/2019

Signal flow graph Reduction

The signal flow graph reduction by above method will be time

MR. AKETCHESCOBENZ – A TVET TRAINER 19/10/2019

Construction of Signal Flow Graph

MR. AKETCHESCOBENZ – A TVET TRAINER 19/10/2019

MR. AKETCHESCOBENZ – A TVET TRAINER 19/10/2019

Simple Process of Calculating Expression of Transfer Function

graph. The input signal to a node is summation of product of

transmittance and the other end node variable of each of the

MR. AKETCHESCOBENZ – A TVET TRAINER 19/10/2019

MR. AKETCHESCOBENZ – A TVET TRAINER 19/10/2019

If P is the forward path transmittance between extreme input and

MR. AKETCHESCOBENZ – A TVET TRAINER 19/10/2019

MR. AKETCHESCOBENZ – A TVET TRAINER 19/10/2019

Conversion of Block Diagrams into Signal Flow Graphs

off points of block diagram as nodes in signal flow graph.

diagram as gains of the branches in signal flow graph.

connection between two nodes (but there is no block in

Consider the block diagram below

MR. AKETCHESCOBENZ – A TVET TRAINER 19/10/2019

The transfer function is referred as transmittance in signal flow

The same equation can be represented by signal flow graph, where x

MR. AKETCHESCOBENZ – A TVET TRAINER 19/10/2019

MR. AKETCHESCOBENZ – A TVET TRAINER 19/10/2019

MR. AKETCHESCOBENZ – A TVET TRAINER 19/10/2019

MR. AKETCHESCOBENZ – A TVET TRAINER 19/10/2019

Again: Let us convert the following block diagram into its

The following figure shows the equivalent signal flow graph.

MR. AKETCHESCOBENZ – A TVET TRAINER 19/10/2019