# Control System Mason Gain Formula

Control System Mason Gain Formula with tutorial, introduction, classification, mathematical modelling and representation of physical system, transfer function, signal flow graphs, p, pi and pid controller etc.

# 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,

Pk = 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