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:

[Tex] \frac{C}{R}=\frac{P_{1}∆_{1}}{∆} [/Tex]

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

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:

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

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:

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

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.