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Problem 3.24PP
UseMason’s rule to determine the transfer function between R(s) and Y(s) in Fig.
Figure Block diagram
Step-by-step solution 
Step-by-step solution
step 1 of 10
(b )
Construct signal flow graph from Figure 3.52 in textbook.
-H j
- H a
Figure 1
Step 2 of 10
From Figure 1, the fon/vard path gains are 2 and loop path gain is € 
Consider Mason’s rule for Figure 1.
Determine the first Fonvard path gains from Figure 1.
G 2 _________________G 4
G i G e
Figure 2
Step 3 of 10 ^
Refer Figure 2 and write the equation for first fonvard path gain.
Pi = ...... (1)
Thus, the first fonvard path gain is G f i2G^G^
Determine the Second Fonvard path gains from Figure 1.
G i G e
G i Gs
Figures
Step 4 of 10
From Figure 3, write the second fonvard path gain.
P2 = G iG jG jG j...... (2)
Thus, the second fonvard path gain is G,G)G5G^. 
Determine the first loop path gain from Figure 1.
- H i
Figure 4
Step 5 of 10
From Figure 4, write the first loop path gain.
/ ,= -G ,G jG ,G * //, (3)
Thus, the first loop path gain is
Determine the second loop path gain from Figure 1.
- H a
Figure 5
Step 6 of 10
From Figure 5, write the second loop path gain.
I2 =-G,G,G^G,i¥, (4)
Thus, the second loop path gain is -G jG jG^Gj/f^ 
Determine the third loop path gain from Figure 1.
-H i
Gi Gs
Figure 6
Step 7 of 10
From Figure 6, write the third loop path gain.
/ , = -G ,G ,G ,G */f, (5)
Thus, the third loop path gain is -GjGjGjG^Z/j 
Determine the fourth loop path gain from Figure 1.
G\
Step 8 of 10
From Figure 7, write the third loop path gain.
U =-G ,G ,G jG */f^ (6)
Thus, the fourth loop path gain is -G fijG^GJH^ 
Determine fifth loop path gain from Figure 1.
Step 9 of 10
From Figure 8, write the fifth loop path gain.
....... (7)
Thus, the fifth loop path gain is —G4H 2 
Determine the sixth loop path gain from Figure 1.
- H 2
Step 10 Of 10 ^
From Figure 9, write the fifth loop path gain.
G , /7 j...... (8)
Thus, the sixth loop path gain is —GjJVj 
Consider mason’s gain formula.
P ,*P l (9)
Where,
p^,P2 is the fonvard path gains.
is the loop path gains.
Substitute Equation {1),(2),(3),(4),{5),(6),(7) and (8) in Equation (9).
y(»)_____________________ g,G,G.G.+G,G,G.G.
I+ G 1G2G4G4/ / } ̂ -G iG j^tG^/f^+G|G)GjG4/f) + G|G}G5G4/ f 4 -\-G^H2 -¥G^H2 
_______________G |G ,(G ,G , + G ,G ,)_____________
1 + / / , ( G , + G ,)+ G ,G ,(G ,G , + G ,G ,) ( / / , + f f .
Thus, the transfer function of the system is
G |G ,(G ,G , + G ,G ,)