WEEK 04: SIGNAL FLOW GRAPHS

Sections: Configuration | BD to SFG | Algebra | Mason's Gain Rule | Example | Student's Tasks

Mason's Gain Rule

Mason's Gain Rule The transfer function of a graph is equal to the quotient of the summation of the product of the path gain and its respective co-factor; and the graph determinant.

Path Gain, P_n The transmittance of each forward path between a source and a sink node.

Loop Gain, L_n The transmittance of each closed succession of branches not passing through any node twice. Non-touching Loops are loops which do not share a common node. Touching Loops are loops in which one or more nodes are commonly shared by each loop.

Graph Determinant, D The sum of all the loop transmittances, from the formula given below:

D = 1 - SL₁ + ZL₂ - SL₃ + . . . + SL_n

Where:
L₁ = transmittance of each closed loop.
SL₁ = sum of all L₁'s
L₂ = the product of transmittances of two (2) non-touching loops.
SL₂ = sum of all L₂'s
L₃ = the product of transmittances of non-touching loops taken three (3) at a time.
SL₃ = sum of all L₃'s
L_n = the product of transmittances of non-touching loops taken (n) at a time.
SL_n = sum of all L_n's

Co-Factor of Pn, D_n The determinant of the remaining subgraph when the path which produces Pn is removed, or the determinant of the graph formed after deleting all the loops [Ln] touching the particular path.

Sections: Configuration | BD to SFG | Algebra | Mason's Gain Rule | Example | Student's Tasks

Example

Given:

Solution: Mark all close loops, that is:

A. Loops:
For L₁'s:
La = -4 / ( s²+s )
Lb = -s
Lc = -56 / ( s+8 )
Ld = -6 / s
Le = (10) x( 1/s )x( 1/s )x[ 3/(s+3)]x[ s/(s+2) ] = 30/(s)( s+2 )( s+3 )

For L₂'s:
(LaLb)=[ -4 / ( s²+s ) ] x (-s) = 4 / (s+1)
(LaLc)=[ -4 / ( s²+s ) ] x [-56 / ( s+8 )] = 224 / [ ( s²+s ) x (s+8)]
(LaLd)=[ -4 / ( s²+s ) ] x [-6 / s] = 25 / [s x ( s+8 )]
(LbLc)=(-s) x [-56 / ( s+8 )] = 56s / (s+8)
(LbLd)=(-s) x [-6 / s] = 6

For L₃'s:
(LaLbLc)=[ -4 / ( s²+s ) ] x (-s) x [-56 / ( s+8 )] = - 224 / [(s+1) x (s+8)]
(LaLbLd)=[ -4 / ( s²+s ) ] x (-s) x [-6 / s] = - 24 / (s²+s)

B. Paths:

P₁ = (1)x[1/( s+3 )]x[ 1 /( s²+s ) ] x(10)x( 1/s )x( 1/s )x(1)

P₂ = (1)x[1/( s²+4 )]X[ 8 /( s+8 )]x( 1/s )x( 1/s )x(1)

C. Co-Factor:
From
D = 1 - [La+Lb+Lc+Ld+Le] + [(LaLb) + (LaLc) + (LaLd) + (LbLc) + (LbLd) ] - [(LaLbLc) + (LaLbLd)]
Removing the loops that touched P₁:
D₁ = 1 - Lb = 1 - (-s) = 1+s
Removing the loops that touched P₂:
D₂ = 1 - [La + Lb] + (LaLb)
D₂ = 1 - [ -4 / ( s²+s )] - [ -s ] + [ 4 / (s+1)]

D. Graph Determinant:
D = 1 - [La+Lb+Lc+Ld+Le] + [(LaLb) + (LaLc) + (LaLd) + (LbLc) + (LbLd)] - [(LaLbLc) + (LaLbLd)]
D = s² + 29s +6

Sections: Configuration | BD to SFG | Algebra | Mason's Gain Rule | Example | Student's Tasks

Student's Corner

Seatwork No. 1
Convert the following Block Diagram into a Signal Flow Graph and solve to the Transfer Function using the Mason's Gain Rule, if G = 10 / (s+10).

Assignment No. 1
Solve for the six transfer functions using the Mason's Gain Rule.

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