MASON'S GAIN FORMULA

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 K^th 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 K^th 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

P_{1 = G1G2G3      ∆1 = 1}

P₂ = -G₁G₄       ∆₂ = 1

Individual loops

L₁ = - G₁G₂H₁

L₂ = -G₂G₃H₂

L₃ = -G₁G₂G₃

L₄ = G₁G₄

L₅ = G₄H₂

Non touching Loops = 0