Mason's Gain Formula in Control System

Mason's Gain Formula, also known as Mason's Rule or the Signal Flow Graph Method, is a technique used in control systems and electrical engineering. It provides a systematic way to analyze the transfer function of a linear time-invariant (LTI) system, especially those with multiple feedback loops and complex interconnections. Let's delve deeper into Mason's Gain Formula with a more detailed explanation. In this article, we will learn Mason's Gain Formula and problem-solving with the help of a signal flow graph by Mason's Gain Formula.

What is Mason's Gain Formula?

Mason's Gain Formula (MGF) is a method for finding the transfer function of a linear signal flow graph (SFG).

"The relationship between the input variables and the output variables of a signal flow diagram is given by the net gain of the input and output nodes, called the total gain of the system."

MGF is an alternate method to find the transfer function of the system algebraically by labeling each signal. It writes down the equation that how one signal depends on other signals, and then solving the multiple equations for the output signal in terms of the input signal.

Mason's Gain Formula

Mason's gain formula for the determination of the overall system gain is given by:

T = \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) + ....

Important Terminologies of Mason Gain Formula

The important terminologies which will be used in Mason Gain Formula are:

• Path: It is that traversal of the connected branches where no node can be encountered more than once.
  
• Forward Path: It is the traversal of the path from input to output node.
  
• Forward Path Gain: While moving through forward path the product of the gain encountered in traversing this path is called as the forward path gain.
  
• Loop: It is the traversal of the path which originates and terminates at the same node.
  
• Non Touching Loops: Loops which do not share the common node.
  
• Loop Gain: It is the gain of the path traversed along a loop.

Let us consider a signal flow graph for understanding the above elements:

Forward Path

Form the above signal flow graph (SFG) image, there are two forward paths with their path gain as:

• P1=ACEH
• P2=AGH

Loop

There are 4 individual loops in the above SFG with their loop gain as:

• L1=BC
• L2=EHD
• L3=F
• L4=GHDB

Non-Touching Loops

There are ONE possible combinations of the non-touching loop with loop gain product as -

• L1.L3=BCF

In above SFG, there are no combinations of three non-touching loops, 4 non-touching loops and so on.

Where,

• ∆₁ = 1 (since all loops are touching P₁)
• ∆₂ = 1 (since all loops are touching p2)
• ∆ = 1- [L1+L2+L3+L4] + [L1.L3]
• ∆=1 - (BC + EHD + F + GHDB) + BCF

Transfer Function:

\frac{C}{R}= \frac{P_{1}∆_{1}+P_{2}∆_{2}}{∆}

\frac{C}{R} = \frac{ACEH+AGH}{1-(BC+EHD+F+GHDB)+BCF}

Solved Examples on Mason Gain Formula

Example 1: Find the transfer function of the following signal flow graph

Solution:

No. of forward path(N) = 1

Gain of Forward Paths (P1) = 1*G1G2G3G4G5

No. of individual loops:

• L1 = -G1H1
• L2 = -G3H3
• L3 = -G4H4
• L4 = -G5H5
• L5 = -G1G2G3H2

Non-Touching Loops (Combination of two):

• L1L2 = G1G3H1H3
• L1L3 = G1G4H1H4
• L1L4 = G1G5H1H5
• L24 = G3G5H3H5
• L4L5 = G1G2G3G5H2H5

Non-Touching Loops (Combination of three):

• L1L2L4 = -G1G3G5H1H3H5
  

Here,

∆₁ = 1 (since all loops are touching p1)

∆ = 1 - (L1+L2+L3+L4+L5) + (L1L2+L1L3+L1L4+L2L4+L4L5) - (L1L2L4)

Transfer Function:

\frac{C}{R}=\frac{P_{1}∆_{1}}{∆}

\frac{C}{R} = \frac{G1G2G3G4G5}{(1+G1H1+G3H3+G4H4+G5H5+G1G2G3H2)+(G1G3H1H3+G1G4H1H4+G1G5H1H5+G3G5H3H5+G1G2G3G5H2H5)+(G1G3G5H1H3H5)}

Example 2: Find the transfer function of the following signal flow graph

Solution:

There are two forward paths and one loop. So, we have

• P₁=a
• P₂=b
• L₁=c
• ∆₁ = ∆₂ = 1 (since all loops are touching P₁ & P₂)
• ∆= 1 - c

Transfer Function:

\frac{C(s)}{R(s)}= \frac{a + b}{1 -c}

Example 3: Find the transfer function of the following signal flow graph

No. of forward path(N) = 3

The gain of Forward Paths:

1. P1=G1G2G3
2. P2=G4G3
3. P3=G5

No. of individual loops:

• L1=G1H1
• L2=G6

Non-Touching Loops (Combination of two)

• L1L2=G1G6H1

∆₁ = ∆₂ = ∆₃ = 1 (since all loops are touching P1,P2 &P3)

∆=1 - (G1H1+G6) + G1G6H1

Transfer Function:

\frac{C(s)}{R(s)}= \frac{G1G2G3 +G3G4 + G5}{1-G1H1-G6 + G1G6H1}

Advantages & Disadvantages of Mason's Gain Formula

Advantages

• Simplicity: Mason's Gain Formula provides a systematic approach for determining the overall gain of the complex control system.

• Comprehensive: It can handle system with multiple feedback loops, making it applicable to a wide range of control system designs.

• Versatility: Mason's Gain Formula can be applied to both linear and time-invariant systems, making it a versatile tool in control system analysis.

• Visualization: It helps in visualizing the different paths and loops within a system, aiding engineers in understanding the system's behavior.

Disadvantages

• Complexity for Large Systems: For systems with a large number of loops and paths, the calculation involved in Mason's Gain Formula can become complex and time consuming.

• Limited to Linear Systems: It is specifically designed for linear time-invariant systems and may not be applicable to nonlinear or time-variant systems without appropriate modifications.

• Assumption of Non Touching Loops: The formula assumes that the loops within the system do not touch or intersect each other, which may not always be the case in real-world systems.

• Limited Practical Insight: While Mason's Gain Formula calculates the over all gain, it may not provide deep insights into the dynamic behavior or stability of the system, which might be essential for certain application.

Application of Mason’s Gain Formula:

Mason’s Gain Formula finds application in various aspects of control system analysis:

• Stability Analysis: Masons Gain formula help in stabilizing the system by calculating the poles and zeroes of the overall transfer function.

• Closed-Loop Systems: Mason's gain formula helps in analysis of closed loop system, considering feedback and determining the effect of feedback on system performance.

• Transient and Steady-State Response: The formula helps in understanding how the system responds to transient and steady-state inputs.

• Filter Design: Mason’s Gain Formula assists in designing filters by allowing engineers to analyze the frequency response of the system.

Conclusion

Mason's Gain Formula is a powerful and systematic method for analyzing the overall gain of complex control systems, especially those with multiple loops and paths. Its advantages include simplicity, comprehensiveness, versatility, and the ability to visualize system components. However, it has limitations when applied to large systems, nonlinear or time-variant systems, and situations where in-depth insights into system behavior are necessary. Engineers should consider these factors and choose appropriate techniques based on the specific characteristics of the system they are analyzing.