Question

The transconductance of MOSFET in saturation is

• A. \(k(V_{GS} - V_T)\)
• B. \(2k(V_{GS} - V_T)\)
• C. \(kV_{DS}\)
• D. Constant

Hint:

Remember that since the drain current in saturation depends on the square of the voltage term (\(I_D \propto V_{ov}^2\)), its derivative (transconductance \(g_m\)) will always be linearly proportional to that overdrive voltage term: \(2k(V_{GS} - V_T)\).

Answer 1

The Correct Option is B

Concept: The drain current \(I_D\) of an n-channel enhancement MOSFET operating in the saturation region is given by the square-law expression: \[ I_D = k(V_{GS} - V_T)^2 \] where:
• \(V_{GS}\) is the gate-to-source voltage.
• \(V_T\) is the threshold voltage of the device.
• \(k\) is the conduction parameter, defined as \(k = \frac{1}{2}\mu_n C_{ox}\left(\frac{W}{L}\right)\). The transconductance (\(g_m\)) represents the change in drain current output caused by a change in the gate-to-source input voltage. It is calculated by taking the partial derivative of the drain current expression with respect to \(V_{GS}\): \[ g_m = \frac{\partial I_D}{\partial V_{GS}} \]

Step 1: Differentiate the saturation drain current equation with respect to \(V_{GS}\). Using the chain rule for differentiation on the drain current expression: \[ g_m = \frac{\partial}{\partial V_{GS}} \left[ k(V_{GS} - V_T)^2 \right] \] Bring the exponent 2 to the front and multiply by the derivative of the inner term: \[ g_m = k \cdot 2(V_{GS} - V_T)^{2-1} \cdot \frac{\partial}{\partial V_{GS}}(V_{GS} - V_T) \] Since \(\frac{\partial}{\partial V_{GS}}(V_{GS} - V_T) = 1 - 0 = 1\), the expression simplifies to: \[ g_m = 2k(V_{GS} - V_T) \] This derived expression matches Option (B).