$$(a)$$ Draw the equivalent circuit and evaluate the gain $$\left( {{V_0}\,\,vs\,\,{V_i}} \right)$$ of the circuit for
![GATE EE 1991 Analog Electronics - Operational Amplifier Question 11 English](https://imagex.cdn.examgoal.net/yFis4jd9EEMsfBIWQ/4hCo6y4B9sKUImtxhayrJT3IcqJdV/dmNoXthjd4qqUuOQrHFY6K/image.png)
$$(i)$$ $${V_i} \le 0$$ $$\,\,\,\,\,\left( {ii} \right)\,\,\,0 < {V_i} < 5V\,\,\,\,\,\,$$ $$\,\,\,\,\,\left( {iii} \right)\,\,\,{V_i} > 5V\,\,\,\,\,\,$$
$$(b)$$ Sketch the gain $$\left( {{V_0}\,\,\,{V_s}\,\,\,{V_i}} \right)$$ characteristics of the above circuit and label the salient features.
![GATE EE 1991 Analog Electronics - Small Signal Modeling Question 3 English 1](https://imagex.cdn.examgoal.net/mdKliPejpZ9g4nbwk/53eN42zdWPuywpyn16A68bKVpKdso/yb4WGRBbwyzX2DQrprN1uQ/image.png)
![GATE EE 1991 Analog Electronics - Small Signal Modeling Question 3 English 2](https://imagex.cdn.examgoal.net/tBw2Oy75vu35YrkRP/RmvA3E1Y2awZxhWKnK9Yku9KHK2qy/t68ArjoI6FXFjTRb1e4bcd/image.png)
$$(a)$$ Simplify the circuit by applying thevenin's theorem to biasing network $${R_1},{R_2}$$ at
$$\,\,\,\,\,\,\,$$ the base of the transistor.
$$(b)$$ Assuming $${C_s}$$ to be a short for frequency range considered. Draw the small signal
$$\,\,\,\,\,\,\,$$ $$a.c.$$ model of the circuit obtained in $$(a)$$ by using the simple model for the
$$\,\,\,\,\,\,\,$$ transistor shown in figure.
$$(c)$$ Evaluate the small signal gain $$\left( {{{{V_0}} \over {{V_i}}}} \right)$$ of the amplifier.
$$a=$$ ____________
$$K=$$ ___________