Table of Contents

Toggle- Signal Flow Graph (SFG) is an Alternative method to block diagram representation.
- Like block diagrams it does not consist of blocks, summing points and take-off points.
- It only of a network in which nodes are connected by directed branches.
**Definition:**A signal flow graph is graphical representation of the relationship between variables of set of linear algebraic equations.**Advantage:**The availability of a flow graph gain formula, also called Mason’s gain formula.

## Fundamentals of Signal Flow Graphs

Consider a simple equation below and draw its signal flow graph:

**x _{2} = A x_{1}**

The signal flow graph of the equation is shown below;

- ‘A’ is a constant gain.
- Every variable in a signal flow graph is called a
**Node**. - Every transmission function in a signal flow graph is called a
**Branch**. - Branches are always
**unidirectional**. - The arrow in the branch denotes the
**direction**of the signal flow.

## Important Signal Flow Graph Terms

### Node and Path

- An
**input node**or source contain only the outgoing branches. i.e.,*X*_{1 } - An
**output node**or sink contain only the incoming branches. i.e.,*X*_{3} - A
**chain node**contain both incoming and outgoing branches. i.e.,**X**_{2} - A
**path**is a continuous, unidirectional succession of branches along which no node is passed more than ones. i.e.,**X**_{1}to X_{2}to X_{3}

### Forward Path and Feedback Loop

**Now, have a look at next figure.**

- A forward path is a path from the input node to the output node. i.e., X
_{1}to X_{2}to X_{3}to X_{4 }, and X_{1}to X_{2}to X_{4 }, are forward paths. - A feedback path or feedback loop is a path which originates and terminates on the same node. i.e.; X
_{2}to X_{3}and back to X_{2}is a feedback path.

### Self Loop

- A
**self-loop**is a feedback loop consisting of a single branch. i.e.;is a self loop.*A*_{33}

### Path Gain and Loop gain

- The
**gain**of a branch is the transmission function of that branch. - The
**path gain**is the product of branch gains encountered in traversing a path. i.e. the gain of forwards path*X*_{1}to X_{2}to X_{3}is*to X*_{4}*A*_{21}A_{32}A_{43} - The
**loop gain**is the product of the branch gains of the loop. i.e., the loop gain of the feedback loop fromto*X*_{2}and back to*X*_{3}is*X*_{2}*A*_{32}A_{23}_{.}

**Now, have a look at next figure.**

### Touching and Non-touching Loop

- Two loops, paths, or loop and a path are said to be
**non-touching**if they have no nodes in common. - Non-touching loop gains:
- [G
_{2}(s) H_{1}(s)] [G_{4}(s) H_{2}(s)] - [G
_{2}(s) H_{1}(s)] [G_{4}(s) G_{5}(s) H_{3}(s)] - [G
_{2}(s) H_{1}(s)] [G_{4}(s) G_{6}(s) H_{3}(s)]

- [G

- A branch having gain 1 can be added at the input as well as output node such node is called
**dummy node.** - Dummy node will not affect transfer function of the system.
- Note that, dummy nodes can be added only at the input and output node.

## Problem on Signal Flow Graph

Consider the signal flow graph below and identify the following.

- Input node.
- Output node.
- Forward paths.
- Feedback paths (loops).
- Determine the loop gains of the feedback loops.
- Determine the path gains of the forward paths.
- Non-touching loops

To find the solution of above problem and to understand above concepts clearly, watch the following video.