I've been looking deeper into the idea of performing feedback analysis for MOSFET differential pair configurations. This particular amplifier circuit has been giving me some trouble, where I am looking to find the closed-loop gain \$(v_{o}/v_{s}) \$:
The circuit states that the transconductances of the MOSFETs in the pair are equal but the transconductance of the 3rd MOSFET is not \$(g_{M1}=g_{M2}\ne\ g_{M3}) \$. Output resistances of the transistors are also taken to be infinity.
If we continue under the assumption that \$R_1=R_2\$ and perform our standard \$A\beta\$ feedback analysis, I arrive at the following equations:
$$A=-g_{M3}R_{eq}g_{M1}R_{1},\space\ \beta=\frac{R_5}{R_5+R_4}$$
where \$R_{eq}=(R_{22}||R_3)=((R_4+R_5)||R_3)\$ with the final closed-loop gain expression being of the form:
$$\left( \frac{v_o}{v_s} \right)=\frac{A}{1+A\beta}$$
Here is my issue. I do not see how the gain can be solved for unless \$R_1\$ is taken to be equal to \$R_2\$. This assumption allows us to convert the differential pair into a half circuit and makes solving the \$A\$ circuit relatively straightforward.
Is this a legitimate assumption though, or can the circuit be solved assuming \$R_1\ne R_2\$?