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- Mason's Gain Formula
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# Control Systems - Signal Flow Graphs

Signal flow graph is a graphical representation of algebraic equations. In this chapter, let us discuss the basic concepts related signal flow graph and also learn how to draw signal flow graphs.

## Basic Elements of Signal Flow Graph

Nodes and branches are 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 us consider the following signal flow graph to identify these nodes.

The

**nodes**present in this signal flow graph are**y**and_{1}, y_{2}, y_{3}**y**._{4}**y**and_{1}**y**are the_{4}**input node**and**output node**respectively.**y**and_{2}**y**are_{3}**mixed nodes**.

### 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**.

## Construction of Signal Flow Graph

Let us construct a signal flow graph by considering the following algebraic equations −

$$y_2=a_{12}y_1+a_{42}y_4$$

$$y_3=a_{23}y_2+a_{53}y_5$$

$$y_4=a_{34}y_3$$

$$y_5=a_{45}y_4+a_{35}y_3$$

$$y_6=a_{56}y_5$$

There will be six **nodes** (y_{1}, y_{2}, y_{3}, y_{4}, y_{5} and y_{6}) and eight **branches** in this signal flow graph. The gains of the branches are a_{12}, a_{23}, a_{34}, a_{45}, a_{56}, a_{42}, a_{53} and a_{35}.

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 −

**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

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.

### Example

Let us convert the following block diagram into its equivalent signal flow graph.

Represent the input signal $R(s)$ and output signal $C(s)$ of block diagram as input node $R(s)$ and output node $C(s)$ of signal flow graph.

Just for reference, the remaining nodes (y_{1} to y_{9}) are labelled in the block diagram. There are nine nodes other than input and output nodes. That is four nodes for four summing points, four nodes for four take-off points and one node for the variable between blocks $G_1$ and $G_2$.

The following figure shows the equivalent signal flow graph.

With the help of Mason’s gain formula (discussed in the next chapter), 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.