# Signal Flow Graph (SFG)

• 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:

x2 = A x1

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., X1
• An output node or sink contain only the incoming branches. i.e., X3
• A chain node contain both incoming and outgoing branches. i.e., X2
• A path is a continuous, unidirectional succession of branches along which no node is passed more than ones. i.e., X1 to X2 to X3 ### 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., X1 to X2 to X3 to X4 , and X1 to X2 to X4 , are forward paths.
• A feedback path or feedback loop is a path which originates and terminates on the same node. i.e.; X2 to X3 and back to X2 is a feedback path. ### Self Loop

• A self-loop is a feedback loop consisting of a single branch. i.e.; A33 is a self loop. ### 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 X1 to X2 to X3 to X4 is A21A32A43
• The loop gain is the product of the branch gains of the loop. i.e., the loop gain of the feedback loop from X2 to X3 and back to X2 is A32A23.

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:
• [G2(s) H1(s)] [G4(s) H2(s)]
• [G2(s) H1(s)] [G4(s) G5(s) H3(s)]
• [G2(s) H1(s)] [G4(s) G6(s) H3(s)]

• 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.

1. Input node.
2. Output node.
3. Forward paths.
4. Feedback paths (loops).
5. Determine the loop gains of the feedback loops.
6. Determine the path gains of the forward paths.
7. Non-touching loops To find the solution of above problem and to understand above concepts clearly, watch the following video.