Question

For the network of figure shown below,
Thevenin equivalent resistance as a function of $\beta$ looking into terminal AB is

• A. $\left(\frac{5+2\beta }{3}\right)\Omega$
• B. $\left(\frac{5-2\beta }{3}\right)\Omega$
• C. $\left(\frac{2\beta }{3}\right)\Omega$
• D. $(5+2\beta )\Omega$

Answer 1

The correct option is A $\left(\frac{5+2\beta }{3}\right)\Omega$

To find out $R_{th}$

By applying KCL at node 1,
$\frac{V-1}{1}+\frac{V}{2}+\frac{V-\beta i_x}{1}=\mathrm{0...}\left(i\right)$
and $\frac{V-1}{1}=-i_x$
$V=1-i_x$
Put equaltion (ii) in equation (i)
$\frac{1-i_x-1}{1}+\frac{1-i_x}{2}+\frac{1-i_x-\beta i_x}{1}=0$
$\frac{2-2i_x-2+1-i_x+2-2i_x-2\beta i_x=0}{2}$
$3-5i_x-2\beta i_x=0$
$3=(5+2\beta )i_x$
$i_x=\frac{3}{5+2\beta }$
$R_{th}=\frac{V}{i_x}=\frac{5+2\beta }{3}\Omega$