Transfer Function of a FET Differential Amplifier

[CoolDude420]

Homework Statement

[Taken from Razavi's Design of Analog CMOS Integrated Circuits 2nd edition]

Homework Equations

The Attempt at a Solution

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I'm not too sure what to do with this question. Here's what I think of the circuit

This doesn't look like a differential amplifier to me since there's only 1 input. From what, I see, M1 has a constant DC bias current through it of I1. I see the following relationship
$$I_2=I_{SS}-I_1$$
Leading to,
$$V_{out2}=V_{DD}-(I_{SS}-I_1)R_1$$

Note that I2 is the current through R1 and M2

As Vin small signal changes, this I1 is perturbed, which in turn perturbs I_2 which in turns adjusts voltage drop across R1 which leads to Vout2 having a small-signal change. Now, I'm not sure how to get Vout1/Vin or Vout2/Vin. I was thinking of maybe adjusting the equation up there and sub in the Id equation for I1 but that would leave me with a quadratic and squared Vin term.

Also if there's any experienced analog people out there. Is my thought process correct for understanding the circuit or along the right lines?

 

[phyzguy]

As Vin small signal changes, this I1 is perturbed, which in turn perturbs I_2 ...

But you are given that I1 is an ideal current source. That means it is a constant current regardless of Vin and Vout1, so changes in Vin cannot "perturb" I1.

 

[CoolDude420]

But you are given that I1 is an ideal current source. That means it is a constant current regardless of Vin and Vout1, so changes in Vin cannot "perturb" I1.

Hmm good point. In that case then, I have another question. What about like a standard CS source stage with a current source like this

How do these circuits operate then?

 

[NascentOxygen]

If M1 between D and S appears as a variable resistance controlled by V_GS then the voltage across that varying resistance, V_DS, can vary even while I_D is held fixed.

 

[phyzguy]

Hmm good point. In that case then, I have another question. What about like a standard CS source stage with a current source like this()
How do these circuits operate then?

I1 doesn't change, but Vout does change as a function of Vin.

In your original circuit, I didn't say Vout1 doesn't change. It does change as you change Vin.