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Op-amp with a MOSFET

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The figure above shows a MOSFET with its gate terminal, G, connected to its drain terminal, D. Assume that the MOSFET is operated in its saturation region and it is characterized by the equation:

i_D = \begin{cases} \dfrac{K}{2}(V_{GS}-V_{T})^2 & \text{for } V_T \lt V_{GS} \leq V_{DS}\\ 0 & \text{for } V_{GS} \lt V_T\end{cases}

where i_{DS} is the drain-to-source current when a voltage v_{GS} is applied across its gate-to-source terminals. Assume V_T = 0 for all parts of this problem, including part 1.

For all analytical solutions, use vIN for v_{IN}, vOUT for v_{OUT}, K for K, vDS for v_{DS} and iD for i_{D}.

###Part 1

What is the current in the MOSFET's drain terminal, i_{D}, in terms of its drain-to-source voltage v_{DS} for v_{GS} \gt 0?

i_{D}[A] =

###Part 2

Determine analytically the output voltage v_{OUT} as a function of the input voltage v_{IN} for v_{IN} > 0 for the circuit above.

v_{OUT} =

Determine numerically the output voltage when the input voltage is 2V, given K = 1 mA/V^2 and V_{T} = 0 V.

v_{OUT} [V] =

###Part 3

Determine analytically the output voltage v_{OUT} as a function of the input voltage v_{IN} for v_{IN} > 0 for the circuit above.

v_{OUT} =

Determine numerically the output voltage when the input voltage is 0.5 V, given K = 1 mA/V^2 and V_{T} = 0 V.

v_{OUT} [V] =