MOSFET Amplifier Review

Cheatsheet Content

### MOSFET Amplifier Configurations This section summarizes common MOSFET amplifier configurations and their general characteristics. #### Common Source (CS) - **Characteristics:** Negative gain. - **Typical Application:** Voltage amplification. #### Common Source with Source Degeneration - **Characteristics:** Improved linearity, reduced gain. - **Output Conductance:** $G_m R_{out}$ #### Common Gate (CG) - **Characteristics:** Positive gain, good high-frequency response, current buffer. #### Common Drain (CD) / Source Follower - **Configuration:** Gate driven, output taken from source. - **Voltage Gain ($V_o/V_{in}$):** $$ \frac{V_o}{V_{in}} = \frac{R_S || r_o}{R_S || r_o + \frac{1}{g_m}} \approx \frac{R_S}{R_S + \frac{1}{g_m}} $$ - Where $r_o$ is the output resistance of the MOS transistor. - **Characteristics:** Voltage buffer, gain close to 1. #### Transconductance ($G_m$) - **Common Source:** $G_m = g_m$ - **Common Drain:** $G_m = -g_m$ - **Common Source with Source Degeneration:** $$ G_m = \frac{g_m}{1 + g_m R_S} $$ - **Common Gate:** $G_m = g_m$ ### Output Resistance ($R_{out}$) This section details how to calculate the output resistance for various MOSFET configurations. 1. **Ideal Current Source:** $R_{out} = \infty$ 2. **Common Source (without $R_S$):** $R_{out} = r_o$ 3. **Common Gate (looking into source):** $R_{out} = \frac{1}{g_m}$ 4. **Common Source with Source Degeneration (looking into drain):** $R_{out} = r_o (1 + g_m R_S)$ 5. **Common Drain (looking into source):** $R_{out} = \frac{1}{g_m}$ ### Multi-Stage Amplifier Analysis Example analysis of a two-stage amplifier. #### DC Gain Calculation For the given two-stage amplifier: 1. Find DC gain $V_o/V_{in}$. 2. Find transfer function $V_o/V_{in}(s)$ (CS). **Step-by-step for DC Gain:** - $V_x = V_o \frac{R_{o2}}{R_{o2} + 1/g_{m2}}$ - $V_x \approx V_o$ (assuming $R_{o2} \gg 1/g_{m2}$) - $V_o / V_{in} = -g_{m1} R_D$ (assuming $R_G$ has no current flow) - Therefore, the DC gain is: $$ \frac{V_o}{V_{in}} \approx -g_{m1} R_D $$ ### Frequency Response with Miller Capacitance Analyzing the frequency response of a two-stage amplifier using Miller approximation for input and output capacitances. #### Capacitances - **Input Capacitance ($C_{in}$):** $$ C_{in} = C_f (1 + |A_{ol}|) = C_f (1 + g_m R_D) $$ - **Output Capacitance ($C_o$):** $$ C_o = C_f (1 + \frac{1}{|A_{ol}|}) = C_f (1 + \frac{1}{g_m R_D}) $$ #### Poles - **Input Pole ($P_{in}$):** $$ P_{in} = \frac{1}{C_{in} R_{in}} $$ - Where $R_{in} = R_G$ - **Output Pole ($P_o$):** $$ P_o = \frac{1}{C_o R_o} $$ - Where $R_o = \frac{1}{g_{m2}}$ #### Transfer Function The overall transfer function $A(s)$ is given by: $$ A(s) = \frac{V_o}{V_{in}}(s) = \frac{-g_m R_D}{(1 + \frac{s}{P_{in}})(1 + \frac{s}{P_o})} $$ ### Op-Amp Analysis This section covers ideal and non-ideal op-amp analysis. #### Ideal Op-Amp Analysis For an ideal op-amp, find $V_o/V_{in}$. - **Assumption:** $V_1 = V_2$ (Virtual Short). - **Equation 1 (Voltage Divider at non-inverting input):** $$ V_1 = V_o \frac{R_3}{R_3 + R_4} $$ - **Equation 2 (KCL at inverting input):** $$ \frac{V_{in} - V_2}{R_2} = \frac{V_2 - V_o}{R_1} $$ - **Substitute $V_1$ into $V_2$ in Equation 2:** $$ \frac{V_{in} - V_o \frac{R_3}{R_3 + R_4}}{R_2} = \frac{V_o \frac{R_3}{R_3 + R_4} - V_o}{R_1} $$ - **Rearranging to solve for $V_o/V_{in}$:** $$ V_{in} \frac{R_1}{R_2} = V_o \left( \frac{R_1 R_3}{R_2(R_3+R_4)} - \frac{R_3}{R_3+R_4} + 1 \right) $$ #### Non-Ideal Op-Amp Analysis (Finite $A_o$ and $V_{os}$) Consider an op-amp with finite open-loop gain $A_o$ and input offset voltage $V_{os}$. - **Equation 1 (Op-amp output definition):** $$ V_o = A_o (V_{in} - V_2) $$ - **Equation 2 (Voltage at inverting input with $V_{os}$):** $$ V_2 = \frac{R_2}{R_1 + R_2} V_o - V_{os} $$ - **Substitute Equation 2 into Equation 1:** $$ V_o = A_o \left( V_{in} - \left( \frac{R_2}{R_1 + R_2} V_o - V_{os} \right) \right) $$ - **Solve for $V_o$:** $$ V_o \left( 1 + A_o \frac{R_2}{R_1 + R_2} \right) = A_o V_{in} + A_o V_{os} $$ - **Final Expression for $V_o$:** $$ V_o = \frac{A_o}{1 + A_o \frac{R_2}{R_1 + R_2}} V_{in} + \frac{A_o V_{os}}{1 + A_o \frac{R_2}{R_1 + R_2}} $$ - This shows the gain for $V_{in}$ and the effect of $V_{os}$ on the output.