Mason's Gain Formula in Control System (original) (raw)

Last Updated : 9 Mar, 2026

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.

**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

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) + ....

Important Terminologies of Mason's Gain Formula

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

Signal Flow Graph Showing Different Elements

Signal Flow Graph Showing Different Elements

**Forward Path

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

**Loop

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

**Non-Touching Loops

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

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

Where,

**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

Signal FLow Graph

Signal Flow Graph

**Solution:

No. of forward path(N) = 1

Gain of Forward Paths (P1) = 1*G1G2G3G4G5

No. of individual loops:

Non-Touching Loops (Combination of two):

Non-Touching Loops (Combination of three):

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

ex2

Signal Flow Graph

**Solution:

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

**Transfer Function:

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

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

SIgnal FLow Graph

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:

Non-Touching Loops (Combination of two)

∆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

**Advantages

Disadvantages

Applications