Question

In the circuit, \( I_{DC} \) is an ideal current source. The transistors \( M_1 \) and \( M_2 \) are assumed to be biased in saturation, wherein \( V_{in} \) is the input signal and \( V_{DC} \) is fixed DC voltage. Both transistors have a small signal resistance of \( r_{ds} \) and transconductance of \( g_m \). The small signal output impedance of this circuit is: 

• A. \( 2r_{ds} \)
• B. \( \frac{1}{g_m} + r_{ds} \)
• C. \( g_m r_{ds}^2 + 2r_{ds} \)
• D. infinity

Hint:

In a cascode amplifier, the small-signal output resistance is significantly increased due to the multiplication effect from the transconductance and output resistance of the upper transistor: \( R_{out} \approx g_m r_{ds}^2 + 2r_{ds} \).

Answer 1

The Correct Option is C

This is a cascode amplifier structure. The small signal output impedance of a cascode circuit can be approximated using the following expression: \[ R_{out} \approx r_{ds2} + \left(1 + g_{m2}r_{ds2} \right)r_{ds1} \] Assuming \( r_{ds1} = r_{ds2} = r_{ds} \) and \( g_{m1} = g_{m2} = g_m \), we substitute: \[ R_{out} \approx r_{ds} + (1 + g_m r_{ds})r_{ds} = r_{ds} + r_{ds} + g_m r_{ds}^2 = 2r_{ds} + g_m r_{ds}^2 \] So the total small signal output impedance is: \[ R_{out} = g_m r_{ds}^2 + 2r_{ds} \]