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
- Nodes, which we already discussed.
- Directed edges, as you can see the above image with the directed arrows.
- Forward paths, which are started and ended on different nodes.
- 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.
- Non-touching loops: If there are two or more loops in a single SFG, then they do not touch each other.
Mason Gain Formula: [Tex] \frac{C(s)}{R(s)} = \frac{\sum_{i=1}^{N}P_{i}\Delta_{i}}{\Delta} [/Tex]
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.
Signal Flow Graph
Here, two paths are available. The transfer function will be:
[Tex]T=\frac{C(s)}{R(s)} = \frac{P_{1}\Delta_{1}+P_{2}\Delta_{2}}{\Delta} [/Tex]
[Tex]\frac{C}{R}= \frac{G1G2G4 + G1G3G4}{1-G1G4H1+G1G2G4H2+G1G3G4H2} [/Tex]
Output: [Tex]\frac{C(s)}{R(s)} = \frac{G1G4(G2+G3)}{1-G1G4H1+G1G2G4H2+G1G3G4H2} [/Tex]
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.
Table of Content
- What is a Signal Flow Graph?
- Basic Elements of Signal Flow Graph
- How to build Signal Flow Graph?
- Mason Gain Formula with Example
- Signal Flow Graph from Block Diagram
- Applications of SFG
- Advantages and Disadvantages of SFG
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