Integrated Circuit Amplifiers

NMOS Current Mirror

\begin{matrix} \frac{I_{O}}{I_{R E F}} = \frac{(W / L)_{2}}{(W / L)_{1}} (1 + \frac{V_{O} - V_{G S}}{V_{A 2}}) \\ r_{o 2} = \frac{V_{A}}{I_{O}} \end{matrix}

\begin{matrix} \frac{I_{O}}{I_{R E F}} = \frac{(W / L)_{2}}{(W / L)_{1}} (1 + \frac{V_{O} - V_{G S}}{V_{A 2}}) \\ r_{o 2} = \frac{V_{A}}{I_{O}} \end{matrix}

\begin{matrix} \frac{I_{O}}{I_{R E F}} = \frac{m}{1 + \frac{m + 1}{β}} (1 + \frac{V_{O} - V_{B E}}{V_{A 2}}) \\ m = \frac{I_{S 2}}{I_{S 1}} = \frac{Area of EBJ of Q_{2}}{Area of EBJ of Q_{1}} \\ r_{o 2} = \frac{V_{A}}{I_{O}} \end{matrix}

A_{i s} = \frac{g_{m 2}}{g_{m 1}} = \frac{(W / L)_{2}}{(W / L)_{1}}

\begin{matrix} r_{o 1} = \frac{1}{λ_{2} I_{D}}, r_{02} = \frac{1}{λ_{2} I_{D}} \\ A_{v} = - g_{m} (r_{01} | | r_{02}) = - \frac{1}{2} g_{m} r_{o} \\ R_{i n} = \infty, R_{o} = r_{o} = (r_{01} | | r_{02}) \end{matrix}

\begin{matrix} A_{v} = - g_{m} r_{o} \\ R_{i n} = r_{π}, R_{o} = r_{o} \end{matrix}

\begin{matrix} Assuming v_{G 1} = v_{G 2}, \\ V_{D 1} = V_{D 2} = V_{D D} - \frac{I}{2} R_{D} \\ V_{G S 1} = V_{G S 2} = \sqrt{\frac{I}{k_{n}}} + V_{t} \\ V_{G_{m a x}} = V_{D D} - \frac{I}{2} R_{D} + V_{t} \end{matrix}

\begin{matrix} i_{D 1} + i_{D 2} = I \\ Currents i_{D 1} and i_{D 2} can be changed by varying v_{i d} : \\ - \sqrt{\frac{2 I}{k_{n}}} < v_{i d} < \sqrt{\frac{2 I}{k_{n}}} \end{matrix}

\begin{matrix} i_{D 1} = \frac{I}{2} + \sqrt{k_{n} I} (\frac{v_{i d}}{2}) \sqrt{1 - \frac{(v_{i d} / 2)^{2}}{I / k_{n}}} \\ i_{D 2} = \frac{I}{2} - \sqrt{k_{n} I} (\frac{v_{i d}}{2}) \sqrt{1 - \frac{(v_{i d} / 2)^{2}}{I / k_{n}}} \end{matrix}

\begin{matrix} v_{D 1} = - g_{m} R_{D} \frac{v_{i d}}{2} \\ v_{D 2} = g_{m} R_{D} \frac{v_{i d}}{2} \\ v_{o d} = v_{D 2} - v_{D 1} = g_{m} R_{D} v_{i d} \\ \frac{v_{o d}}{v_{i d}} = g_{m} R_{D} \end{matrix}

A_{d} = \frac{v_{o d}}{v_{i d}} = g_{m} (R_{D} | | r_{o})

\begin{matrix} v_{i d} = v_{B 1} - v_{B 2} \\ I_{E 1} = \frac{I}{1 + e^{- v_{i d} / V_{T}}} \\ I_{E 2} = \frac{I}{1 + e^{v_{i d} / V_{T}}} \end{matrix}

A_{d} = \frac{v_{o d}}{v_{i d}} = g_{m} R_{c}