# Control Systems Signal Flow Graphs - Control Systems

## What are single flow graphs?

Signal flow graph is known as a graphical representation of algebraic equations. This chapter discusses about the basic concepts related signal flow graph It also throws light on how to draw signal flow graphs.

## Basic Elements of Signal Flow Graph

Nodes and branches are considered as the basic elements of signal flow graph.

### Node

Node is a point which represents either a variable or a signal. There are three types of nodes — input node, output node and mixed node.
• Input Node − It is a node, which has only outgoing branches.
• Output Node − It is a node, which has only incoming branches.
• Mixed Node − It is a node, which has both incoming and outgoing branches.

## Example

Let’s see the following signal flow graph to identify these nodes.
• The nodes present in this signal flow graph arey1, y2, y3andy4.
• y1andy4are the input node and output noderespectively.
• y2andy3are mixed nodes.

## Branch

Branch is called as a line segment which connects two nodes which may be gain and direction. For example, there are four branches in the above signal flow graph. These branches have gains of a, b, cand -d.

## Construction of Signal Flow Graph

Let’s construct a signal flow graph with following algebraic equations −

${y}_{2}={a}_{12}{y}_{1}+{a}_{42}{y}_{4}$ ${y}_{3}={a}_{23}{y}_{2}+{a}_{53}{y}_{5}$ ${y}_{4}={a}_{34}{y}_{3}$ ${y}_{5}={a}_{45}{y}_{4}+{a}_{35}{y}_{3}$ ${y}_{6}={a}_{56}{y}_{5}$

If there are six nodes (y1, y2, y3, y4, y5and y6) and eight branches in this signal flow graph. The gains of the branches area12, a23, a34, a45, a56, a42, a53and a35.
You need to draw the signal flow graph for each equation to get the overall signal flow graph. After that add all these signal flow graphs. Then follow below steps: −
Step 1 − Signal flow graph for ${y}_{2}={a}_{13}{y}_{1}+{a}_{42}{y}_{4}$
is shown in the following figure.
Step 2 − Signal flow graph for ${y}_{3}={a}_{23}{y}_{2}+{a}_{53}{y}_{5}$is shown in the following figure.
Step 3 − Signal flow graph for${y}_{4}={a}_{34}{y}_{3}$ is shown in the following figure.
Step 4 − Signal flow graph for ${y}_{5}={a}_{45}{y}_{4}+{a}_{35}{y}_{3}$is shown in the following figure.
Step 5 − Signal flow graph for ${y}_{6}={a}_{56}{y}_{5}$is shown in the following figure.
Step 6 − Signal flow graph of overall system is shown in the following figure.

## Conversion of Block Diagrams into Signal Flow Graphs

Following steps are used for the conversion of 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.

### Example

Let us convert the following block diagram into its equivalent signal flow graph.
You can present the input signal $R\left(s\right)$and output signal $C\left(s\right)$of block diagram as input node $C\left(s\right)$and output node $C\left(s\right)$of signal flow graph.
The other nodes(y1to y9)are labelled in the block diagram. You can find the nine nodes other than input and output nodes. Among them four nodes are for four summing points and other four nodes for four take-off points and one node for the variable between blocksG1andG2.
Let’s see the below figure which shows the equivalent signal flow graph.
Next chapter discusses about the Mason’s gain formula which is used to calculate the transfer function of this signal flow graph. For this no need to simplify (reduce) the signal flow graphs for calculating the transfer function.