Mason's Gain Formula
The relationship between an input variable and an output variable of a signal flow graph is
given by the net gain between the input and output nodes and was the overall gain of
the system. Mason's gain formula for the determination of the overall system gain is given by

1285
where Pe path gan of the forward path: a determinant of the graph = 1 - Gum of loop
gains of all individual loops) sum of gain products of all possible combinations of two non-
touching loops) um of gain products of all possible combinations of three non-touching!
loop) e
4:1- P- P- P . .

where gain product m-th possible combination of non-touching the value of
system.
A for the part of the graph not touching the K-th forward path: and T overall gain of the
Let us illustrate the use of Mason's formula by finding the overall gain of the signal flow
graph shown in Fig. 2.35. The following conclusions are drawn by inspection of this signal flow
graph
1. There are two forward paths with path gains

P
Fig. 2.36 (a)
P PE , 36
Fig. 2.36 (
2. There are five individual loops with loop gains
P a
Fig. 2.36 (c

"Non touching implies that no node in common between the two.
CONTROL SYSTEMS ENGINEERING

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66

Fig. 2.36 (d)
Fig. 2.36 (e)
Fig. 2.36 )
P41 a 4552
Fig. 2.36 g
P6 aa2
3. There are two possible combinations of two non-touching loops with loop gain products

P Pia aag2 44
Fig. 2.36 h
Fig. 2.36 (i)
4. There are no combinations of three non-touching loops, four non-touching loops, etc.

MAT

P=P
m4
Hence from equation. (2.85)

Therefore

4 1-(agga2+a 2+4t a 2+aa552) +(agaa44+a23a35aga
5. First forward path is in touch with all the loops. Therefore, 4, 1. The second forward
path is not in touch with one loop (Fig. 2.36 (). Therefore, A2 1-a

T = PA+PAR

From equation. (2.85), the

1 - Agas - 023434 42 - 4442334455223432044 + 0230 35 52044