Question

An electrical system and its signal-flow graph representations are shown in the figure (a) and (b) respectively. The values of $G_2$ and $H_1$ respectively, are

• A. $\frac{Z_3\left(s\right)}{Z_2\left(s\right)+Z_3\left(s\right)+Z_4\left(s\right)},\frac{-Z_3\left(s\right)}{Z_1\left(s\right)+Z_3\left(s\right)}$
• B. $\frac{-Z_3\left(s\right)}{Z_2\left(s\right)-Z_3\left(s\right)+Z_4\left(s\right)},\frac{-Z_3\left(s\right)}{Z_1\left(s\right)+Z_3\left(s\right)}$
• C. $\frac{Z_3\left(s\right)}{Z_2\left(s\right)+Z_3\left(s\right)+Z_4\left(s\right)},\frac{Z_3\left(s\right)}{Z_1\left(s\right)+Z_3\left(s\right)}$
• D. $\frac{-Z_3\left(s\right)}{Z_2\left(s\right)-Z_3\left(s\right)+Z_4\left(s\right)},\frac{Z_3\left(s\right)}{Z_1\left(s\right)+Z_3\left(s\right)}$

Answer 1

The correct option is C $\frac{Z_3\left(s\right)}{Z_2\left(s\right)+Z_3\left(s\right)+Z_4\left(s\right)},\frac{Z_3\left(s\right)}{Z_1\left(s\right)+Z_3\left(s\right)}$

From KVL in both loops
In first loop,
$V_i\left(s\right)=I_1\left(s\right)Z_1\left(s\right)+\left[I_1\right(s)-I_2(s\left)\right]Z_3\left(s\right)$

$V_i\left(s\right)=I_1\left(s\right)\left[Z_1\right(s)+Z_3(s\left)\right]-I_2\left(s\right)Z_3\left(s\right)$

$\frac{V_i\left(s\right)}{Z_1\left(s\right)+Z_3\left(s\right)}=I_1\left(s\right)-\frac{I_2\left(s\right).Z_3\left(s\right)}{Z_1\left(s\right)+Z_3\left(s\right)}....\left(i\right)$

In second loop,
$\left[I_2\right(s)-I_1(s\left)\right]Z_3\left(s\right)+I_2\left(s\right)Z_2\left(s\right)+I_2\left(s\right)Z_4\left(s\right)=0$

$I_2\left(s\right)\left(Z_2\right(s)+Z_3(s)+Z_4(s\left)\right)=I_1\left(s\right).Z_3\left(s\right)$

$G_2=\frac{I_2\left(s\right)}{I_1\left(s\right)}=\frac{Z_3\left(s\right)}{Z_2\left(s\right)+Z_3\left(s\right)+Z_4\left(s\right)}$

From SFG,
$I_1\left(s\right)=V_i{G}_1\left(s\right)+I_2\left(s\right)H\left(s\right)$

$I_1\left(s\right)=V_i\left(s\right)\frac{1}{Z_1\left(s\right)+Z_3\left(s\right)}+I_2\left(s\right)\frac{Z_3\left(s\right)}{Z_1\left(s\right)+Z_3\left(s\right)}$ ..... from (i)

$\therefore H\left(s\right)=\frac{Z_3\left(s\right)}{Z_1\left(s\right)+Z_3\left(s\right)}$
(on comparing above two equations)