(Solved): An NMOS transistor has parameters \( \mathrm{W}=10 \mu \mathrm{m}, L=1 \mu \mathrm{m}, k^{\prime}= ...

An NMOS transistor has parameters \( \mathrm{W}=10 \mu \mathrm{m}, L=1 \mu \mathrm{m}, k^{\prime}=190 \mu \mathrm{A} / \mathrm{V}^{2}, \lambda=0.02 \mathrm{~V}^{-1}, \mathrm{t}_{\mathrm{ox}}=80 \AA \AA_{\mathrm{A}}, \phi_{\mathrm{F}}=0.3 \mathrm{~V} \), \( \mathrm{V}_{\mathrm{TO}}=0.6 \mathrm{~V} \), and \( \mathrm{N}_{\mathrm{A}}=5 \times 10^{15} \mathrm{~cm}^{-3} \). (1) Sketch the \( I_{D S} \) verse \( V_{D S} \) characteristics for \( V_{D S} \) from 0 to \( 2 V \) and \( V_{G S}=1 V \). (2) Sketch the \( I_{D S} \) verse \( V_{G S} \) characteristics for \( V_{D S}=2 V \) and \( V_{G S} \) varies from 0 to \( 2 V \) with \( V_{S B}-0 V \) and \( 1 V \). (3) Derive and sketch the small-signal equivalent circuit for the device with \( V_{G S}=1 \mathrm{~V}, V_{D S}=2 \mathrm{~V} \), and \( V_{S B}=1 \mathrm{~V} \). Please also draw the capacitances in the small-signal model though you do not need to do the calculation. Please note that \( \varepsilon_{\mathrm{sio2}}=0.34 \times 10^{-12} \mathrm{Fcm}^{-1} \) and \( \varepsilon_{\mathrm{si}}=1.06 \times 10^{-12} \mathrm{Fcm}^{-1} \) \[ \begin{array}{c} V_{T H}=V_{T H .0}+\gamma\left(\sqrt{2 \phi_{F}+V_{S B}}-\sqrt{2 \phi_{F}}\right) \\ \gamma=\frac{\sqrt{2 q \epsilon_{S S} N_{A}}}{C_{o x}} \\ C_{o x}=\frac{\epsilon_{S i O 2}}{t_{o x}} \end{array} \] \( K=k^{\prime} \frac{W}{L} \), where \( k^{\prime}=\frac{\mu_{n} c_{o x}}{2} \)