Question

The gain $\frac{Y}{R}$ for the system with the following signal flow graph using Mason's gain formula is _________.

• A. 1.913

Answer 1

The correct option is A 1.913
2 forward paths:
$P_1=1\times 5\times 1=5;$
$P_2=1\times 2\times 3\times 4\times 1=24$

4 loops in the diagram,

$L_1=2\times -1=-2$
$L_2=3\times -1=-3$
$L_3=4\times -1=-4$
$L_4=5\times -1\times -1\times -1=-5$

Using Mason's gain formula,

$\frac{Y}{R}=\frac{P_1{\Delta }_1+P_2{\Delta }_2}{\Delta }$

$\Delta =1-L_1-L_2-L_3-L_4+L_1{L}_3$
$\Delta =1+2+3+4+5+8=23$
${\Delta }_1=1-L_2=1+3=4$
${\Delta }_2=1$

$\therefore \frac{Y}{R}=\frac{5\times 4+24\times 1}{23}=\frac{44}{23}=1.913$