Mason's Gain Formula (With Examples) | Control Systems - Electrical Engineering (EE) PDF Download

The relation between an input variable and an output variable of a signal flow graph is given by Mason's Gain Formula.

For determination of the overall system, the gain is given by:
Where,

P_k = forward path gain of the Kth forward path.

∆ = 1 - [Sum of the loop gain of all individual loops] + [Sum of gain products of all possible of two non-touching loops] + [Sum of gain products of all possible three non-touching loops] + .......

∆k = The value of ∆ for the path of the graph is the part of the graph that is not touching the Kth forward path.

Forward Path

From the above SFG, there are two forward paths with their path gain as -

Loop

There are 5 individual loops in the above SFG with their loop gain as -

Non-Touching Loops

There are two 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,

Example

Draw the Signal Flow Diagram and determine C/R for the block diagram shown in the figure.
The signal flow graph of the above diagram is drawn below

The gain of the forward paths

P₁ = G₁G₂G₃      ∆1 = 1

P₂ = -G₁G₄       ∆2 = 1

Individual loops

L₁ = - G₁G₂H₁

L₂ = -G₂G₃H₂

L₃ = -G₁G₂G₃

L₄ = G₁G₄

L₅ = G₄H₂

Non touching Loops = 0