Principle of Mason gain formulae

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Jan 20, 2024 03:23 AM
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The Mason Gain Formula is a method used in control systems engineering to determine the transfer function of a complex system, especially when the system is represented as a signal flow graph. Developed by Samuel Jefferson Mason, it provides a systematic way to calculate the overall gain of a system with multiple paths and loops.

Key Concepts of the Mason Gain Formula:

  1. Signal Flow Graph: This is a graphical representation of a control system, showing the flow of signals through the system's components. Nodes represent system variables, and directed branches represent functional relationships between these variables.
  1. Forward Paths: A forward path is a path from the input node to the output node with no node visited more than once.
  1. Loops: A loop is any closed path in the signal flow graph, again with no node visited more than once. Loops can be self-contained or may touch other loops.
  1. Non-Touching Loops: These are loops in the graph that do not share any common nodes.

The Mason Gain Formula:

The formula to calculate the transfer function of a system is given by:
Where:
  • is the gain of the forward path.
  • is the system determinant and is calculated as
    • Here, are individual loop gains.
    • The terms are alternately subtracted and added, starting with subtraction.
  • is the determinant of the forward path and is calculated like but only considering loops that do not touch the forward path.

Steps to Apply Mason's Gain Formula:

  1. Draw the Signal Flow Graph: Represent the system with nodes and branches.
  1. Identify Forward Paths: Find all possible forward paths from input to output.
  1. Identify Loops: Find all loops in the graph.
  1. Calculate Loop Gains: Determine the gain of each loop.
  1. Identify Non-Touching Loops: Find sets of loops that don't touch each other.
  1. Calculate and : Use the formula to compute the system determinant and the determinants for each forward path.
  1. Apply Mason's Gain Formula: Calculate the transfer function using the formula.

Example:

Consider a system with:
  • Two forward paths and with gains.
  • Three loops with individual gains.
  • Some non-touching loop combinations.
The transfer function can be calculated by identifying the gains of the forward paths, and the loop gains, and using the Mason Gain Formula with the calculated values for and .

Practical Use:

Mason's Gain Formula is particularly useful in control system analysis and design, where signal flow graphs are complex, and it is difficult to intuitively understand the relationship between the input and output. The formula simplifies the process of finding the transfer function in such scenarios.
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