Basic Elements of Signal Flow Graph (original) (raw)

Last Updated : 9 Mar, 2026

Signal Flow Graphs are a crucial component of control systems. Furthermore, the control system is one of the most significant subjects in Electronics. To understand the Signal Flow Graph, let's understand the Control System first, and then we will dive into the main topic.

Signal Flow Graph

A **Signal Flow Graph (SFG) is a directed edge and node-based visual depiction of the dynamics of a system. Variables or signals are represented by nodes, while the signal flow between them is shown by edges. SFGs are widely used in control theory because they provide an easy-to-understand representation of intricate system relationships, which helps with analysis and optimization.

Basic Elements of Signal Flow Graph

Nodes and branches are the two basic elements of a Signal Flow Graph.

Node

It is a point that represents a variable or a signal. There are three types of nodes:

To identify a node in a Signal Flow Graph, it is mainly represented with circles or dots; you can find it in the images.

Branch

A branch represents a line segment connecting two nodes in a signal flow graph. Electronic equipment may produce positive or negative gain, and signal movement follows a specific direction. Branch representation therefore includes both gain and direction to complete a signal flow graph. A simple diagram helps in understanding branch connection and signal direction clearly.

y221

Demo Signal Flow Graph

Here we can see the black dots (y1, y2, y3, y4) they are the nodes of the Signal flow graph, and the lines are the branches. As we see that these branches has some directions that is defined by the arrow symbols. So, we can say a,b,c are the positive gain branches and the d is negative gain branch that is why it is in negative sign.

Characteristics of SFG

  1. Nodes
  2. Branches or Edges
  3. Forward Paths
  4. Single Loops
  5. Non-touching Loops
  6. Mason Gain Formula

How to build Signal Flow Graph?

Let's start building of an new signal flow graph with an algebraic equation, it will help us to understand the complex graphs and how electronics engineers use this in Control System.

**Things to remember: While calculating the result we use the node value (like: y1, y2, y3...) and multiply that with the gain (a, b, c). The value of the node is the resultant value that is obtained by adding both values of a branch to another node.

Here "y1, y2, y3, y4, y5, y6" are the nodes, and "a, b, c, d, e, f, g" are the gains in the Signal Flow Graph respectively.

**Equations:

y2 = ay1, y3 = y2b + y5g, y4 = y3c + y2f, y5 = y4d and y6 = y5e

Here we will create the Signal Flow Graph step by step, you can also check the given image below.

step_by_step_signal_flow_graph

Signal Flow Graph from Equations

Mason Gain Formula with Example

The Mason's Gain Formula is a mathematical tool used in control system engineering to calculate the overall transfer function of a signal flow graph.

  1. Nodes, which we already discussed.
  2. Directed edges, as you can see the above image with the directed arrows.
  3. Forward paths, which are started and ended on different nodes.
  4. Loops, which are the close paths in SFG, stared and ended in same node, but passed throw other nodes as well. A SFG can contain many loops.
  5. Non-touching loops: If there are two or more loops in a single SFG, then they do not touch each other.

**Mason Gain Formula: \frac{C(s)}{R(s)} = \frac{\sum_{i=1}^{N}P_{i}\Delta_{i}}{\Delta}

where,

N: total number of forward paths

Pi : gain of the ith forward path

∆: determinant of the graph

∆i : path-factor for the ith path

The determinant of the graph (∆) and the path-factor for the ith path (∆i) are defined as follows:

∆i : 1 - (loop gain which does not touch the forward path)

∆: 1 - Σ(all individual loop gains) + Σ(gain product of all possible combinations of two non-touching loops) - Σ(gain product of all possible combinations of three non-touching loops) + ....

In this formula the loops of the Signal Flow Graph is very important. In the next example we will see how can we get a transfer function from this formula.

Transfer function T, R is input, C is output, G are the gains and H are the feedbacks of a transfer function.

y11

Signal Flow Graph

Here, two paths are available. The transfer function will be:

T=\frac{C(s)}{R(s)} = \frac{P_{1}\Delta_{1}+P_{2}\Delta_{2}}{\Delta}

\frac{C}{R}= \frac{G1G2G4 + G1G3G4}{1-G1G4H1+G1G2G4H2+G1G3G4H2}

**Output: \frac{C(s)}{R(s)} = \frac{G1G4(G2+G3)}{1-G1G4H1+G1G2G4H2+G1G3G4H2}

Signal Flow Graph from Block Diagram

In the field of electronics engineering, block diagrams are used to simplify intricate circuits. The block diagram shows numerous electronic components as well as input and output. As a result, we must carefully comprehend this before drawing the Signal Flow Graph. Prior to that, we must comprehend the jargon.

In the image below we can see the block diagram to Signal Flow Graph:

y1

Block Diagram to Signal Flow Graph

R(s) and C(s) is the input and output respectively.

As you can see that in the block diagram there are two summing point so we have mentioned them with S1,S2 in the Signal Flow Graph, and with t1,t2,t3 we mentioned the take-off points.

As, G1 and G2 are in a loop, so we do the same for the Signal Flow Graph also. And feedbacks are in negative so we mentioned it with -H1 and -H2.

So, this is how we made the Signal Flow Graph from Block diagram.

Applications of SFG

Advantages and Disadvantages of SFG

Advantages

Disadvantages