Basic Elements of Signal Flow Graph

Signal Flow Graphs are a crucial component of control systems. Furthermore, the control system is one of the most significant subjects in Electronics. It is primarily covered in the sixth semester of the B.Tech syllabus, though individual universities may alter it based on their syllabus hierarchy. To understand Signal Flow Graph let's understand the Control System first, then we will dive into the main topic. When we put some inputs into a particular electronic device, it computes the signal or data and results in an output, this is what a Control System does.

A similar process is used with the Signal Flow Graph. Engineers can quickly compute and comprehend how the system operates by inserting the signal at one end and adding electronic devices in between to compute data using algebraic equations.

What is a 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.

Before we begin, let's understand the basic elements of a Signal Flow Graph, we will see an actual Signal Flow Graph.

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:

• Input Node
• Output Node
• Mixed Node.

To identify a node, in Signal Flow Graph it mainly represents with circles, dots, you can find it in below images.

• Input Node: It is a node where we provide only inputs and it has only outgoing branches. To identify the input node, always follow the arrow, if it enters the node then it is input node.

• Output Node: It is a node that has only incoming branches. To identify the output node, always follow the arrow, if it goes out from node then it is output node.

• Mixed Node: It is a node that has both incoming and outgoing branches. To identify the mixed node, always follow the arrow, if it has both incoming and outgoing arrow then it is mixed node.

Branch

It is a line segment which joins two nodes. Since we use this in electronics equipment, which likewise has gain in both positive and negative, also it has directions, we must provide them the branch in order to finish our signal flow graph. Let's use a little diagram to ensure we grasp this properly.

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

We shall go into more information about this in the article's lower section.

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.

• Step1: Create the first loop, and required branches with the y₂ and y₄ equation.

• Step2: Create the second loop, and don't forget the directions with the y₃ equation, we have to be careful with the directions. Give directions according to the equations.

• Step3: Create the main branch with directions, and connect all the nodes with branches.

• Step4: Now we have to combine the whole structure and we will get the whole Signal Flow Graph, with proper notations and directions.
  

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.

Basic Elements of Signal Flow Graph related to Mason Gain Formula

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

P_i : gain of the i^th forward path

∆: determinant of the graph

∆_i : path-factor for the i^th path

The determinant of the graph (∆) and the path-factor for the i^th 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.

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.

• R(s) is the input point, C(s) is the output point.

• The circle with a cross is called summing point(S), and the branches meet at the dot point is called take-off point.

• The "G" inside a box called the gain, there can be many gain blocks in a single block diagram.

• Since electronics components also provide feedback, "H" inside the box is referred to as the feedback.

In the image below we can see the 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

• Control System analysis is very much dependent on Signal Flow Graph, because it really helps to calculate and create transfer functions.

• In communication systems, SFGs are employed to model and analyze signal paths, helping engineers to understand the flow of information through a system.

• Signal Flow Graphs are applied to analyze electrical networks, representing voltages and currents.

Advantages and Disadvantages of SFG

There are some list of Advantages and Disadvantages of SFG given below :

Advantages

• It provides a clear view of systems flow of information and system dynamics, which helps engineers to easily understands the complex circuits.

• Signal Flow Graphs are useful for examining a system's feedback loops. The influence of feedback on the performance and stability of a system can be recognized and assessed by engineers.

• The transfer function of a system can be systematically derived using the Mason's Gain Formula related to SFGs. This is especially helpful for designing control systems.

Disadvantages

• It provides clear picture of complex systems, but sometimes if there are large systems and to many interactions, then handling large numbers of nodes and branches become really hard.

• Signal Flow Graphs are most effective for linear time-invariant (LTI) systems. They may not be as suitable for modeling nonlinear or time-varying systems.

• Signal flow graphs, particularly for systems with delays, may not explicitly reflect causalities in the time domain, despite their ability to depict cause-and-effect linkages.

Conclusion

Signal Flow Graphs offers valuable features with it, as engineers can easily understand the complex circuits, with proper calculations of gains and loops with it. With Mason Gain Formula it becomes more effective as it helps engineers to get the transfer function from the Signal Flow Graph. Also, it helps to breakdown a complex block diagrams and calculate its components values easily. But as systems get more complex, SFGs become less effective, and they might not be the best option for nonlinear or time-varying systems. Engineers should use them sparingly depending on the particular needs of their study, as the usefulness of each modelling tool depends on the features of the system being studied.