Question

In the circuit shown in the figure, the transistors M1 and M2 are operating in saturation. The channel length modulation coefficients of both the transistors are non-zero. The transconductance of the MOSFETs M1 and M2 are \( g_{m1} \) and \( g_{m2} \), respectively, and the internal resistance of the MOSFETs M1 and M2 are \( r_{o1} \) and \( r_{o2} \), respectively. Ignoring the body effect, the ac small signal voltage gain \( \frac{\partial V_{\text{out}}}{\partial V_{\text{in}}} \) of the circuit is 

• A. \( -g_{m2} (r_{o1} || r_{o2}) \)
• B. \( -g_{m2} \left( \frac{1}{g_{m1}} || r_{o2} \right) \)
• C. \( -g_{m1} \left( \frac{1}{g_{m2}} || r_{o1} r_{o2} \right) \)
• D. \( -g_{m2} \left( \frac{1}{g_{m1}} || r_{o1} || r_{o2} \right) \)

Hint:

For small signal analysis of MOSFET circuits, the voltage gain is determined by the transconductances \( g_m \) and the resistances in the circuit. Use the parallel combination of resistances when calculating the effect of multiple resistors in the path.

Answer 1

The Correct Option is D

In the given circuit, the two transistors M1 and M2 are operating in saturation. The circuit configuration is such that M1 acts as the input transistor, while M2 is the output transistor. We are tasked with determining the small signal voltage gain of the circuit, ignoring the body effect. Step 1: Understand the basic small signal model for MOSFETs.
In a small signal analysis, the voltage gain \( \frac{\partial V_{\text{out}}}{\partial V_{\text{in}}} \) is affected by the transconductance of the MOSFETs and the resistances in the circuit. We can compute the voltage gain using the following relationships: - The transconductance \( g_{m1} \) of transistor M1 controls the current change with respect to input voltage. - The resistance \( r_{o1} \) and \( r_{o2} \) represent the output resistance of the MOSFETs, which affects the voltage drop across them. Step 2: Apply the small signal voltage gain formula.
The small signal voltage gain is influenced by the interaction of \( g_{m1} \), \( g_{m2} \), and the resistances \( r_{o1} \), \( r_{o2} \). Using the small signal analysis approach for a two-stage transistor circuit, the voltage gain formula can be derived as: \[ A_v = -g_{m2} \left( \frac{1}{g_{m1}} || r_{o1} || r_{o2} \right) \] This formula accounts for the effect of both transistors M1 and M2 on the overall voltage gain, considering their respective transconductances and internal resistances. Step 3: Conclusion.
Thus, the ac small signal voltage gain of the circuit is \( -g_{m2} \left( \frac{1}{g_{m1}} || r_{o1} || r_{o2} \right) \), which corresponds to option (D). Final Answer: \( -g_{m2} \left( \frac{1}{g_{m1}} || r_{o1} || r_{o2} \right) \)