Let's break down the problem and derive the expression for the **transit frequency** \( f_T \) for the given MOS device. --- ### **Step 1: Understanding the Circuit** - **Given:** - The source and drain **AC voltages** are held at ground. - Only the **gate** is driven by an input AC current \( i_{in} \). - Capacitances: \( C_{GS} \) (gate-source), \( C_{GD} \) (gate-drain). #### **Small-Signal Equivalent** - Since source and drain are at AC ground: - The gate sees **two capacitors to ground**: \( C_{GS} \) and \( C_{GD} \). - The input current \( i_{in} \) splits between these two capacitors. --- ### **Step 2: Current Gain Definition** - The frequency where the **current gain** drops to **1** is the **transit frequency \( f_T \)**: \[ \left| \frac{i_{out}}{i_{in}} \right| = 1 \quad \text{at} \quad f = f_T \] --- ### **Step 3: Small-Signal Analysis** - **Apply KCL at the gate node:** \[ i_{in} = i_{C_{GS}} + i_{C_{GD}} \] Since both **source** and **drain** are at AC ground: \[ i_{C_{GS}} = C_{GS} \frac{dV_{gs}}{dt} \] \[ i_{C_{GD}} = C_{GD} \frac{dV_{gd}}{dt} \] But \( V_{gs} = V_{gd} \) (since source and drain AC voltages are zero), so: \[ i_{in} = (C_{GS} + C_{GD}) \frac{dV_g}{dt} \] - **Output Current:** - The current through \( C_{GD} \) (to ground) is **output current**: \[ i_{out} = i_{C_{GD}} = C_{GD} \frac{dV_g}{dt} \] --- ### **Step 4: Current Gain Expression** \[ \frac{i_{out}}{i_{in}} = \frac{C_{GD}}{C_{GS} + C_{GD}} \] But, **at high frequencies**, the gate current is also affected by the transistor's transconductance \( g_m \): - If the MOSFET has a transconductance \( g_m \), then at the gate: - \( i_{out} = g_m V_{gs} \) - \( i_{in} = j \omega (C_{GS} + C_{GD}) V_{gs} \) So, the **current gain** becomes: \[ \left| \frac{i_{out}}{i_{in}} \right| = \left| \frac{g_m V_{gs}}{j\omega (C_{GS}+C_{GD}) V_{gs}} \right| = \frac{g_m}{\omega (C_{GS} + C_{GD})} \] --- ### **Step 5: Find \( f_T \) ("Transit Frequency")** Set the current gain to 1: \[ \frac{g_m}{2\pi f_T (C_{GS} + C_{GD})} = 1 \] \[ f_T = \frac{g_m}{2\pi (C_{GS} + C_{GD})} \] --- ## **Final Expression** \[ \boxed{ f_T = \frac{g_m}{2\pi (C_{GS} + C_{GD})} } \] --- ### **Summary** - **Transit frequency \( f_T \)** is the frequency where the current gain drops to 1. - **Expression:** \( f_T = \frac{g_m}{2\pi (C_{GS} + C_{GD})} \) - **Assumption:** Both source and drain are at AC ground; only gate is driven. If you need the derivation considering only the capacitive divider (ignoring \( g_m \)), let me know!