small signal analysis solve

Question

I have found this small signal analysis circuit in my exercise.

^{simulate this circuit – Schematic created using CircuitLab}

I tried both \$\pi\$-model and T-model but can't go anywhere. When trying to figure out \$V_o\$ can't decide whether take only \$R_L\$ or \$R_L\$ || \$(R_1 +R_2)\$ as DC current also flow through \$R_1\$ and \$R_2\$ . For other parameters like \$V_{in} , R_{in} , R_{out} \$ can't even make any idea.

How can I solve this and please,explain in detail so that in future I can solve this kind of circuits by myself. Thanks in advance.

Answer 1

It's easy to see that the small-signal drain voltage is given by

$$v_d = -i_d R_L||(R_1 + R_2)$$

so that's all there is to that.

If this were a true common-gate circuit, the small-signal gate-source voltage would be

$$v_{gs} = v_g - v_s = 0 - v_{sig} = - v_{sig}$$

but, in this circuit, the gate is not at signal common so we have

$$v_{gs} = v_g - v_s = v_d \frac{R_1}{R_1 + R_2} - v_{sig}$$

And, recalling that

$$i_d = g_mv_{gs}$$

you should have all you need to finish the problem.