MICRO-461: Exercise 10

Dear Prof. Enz, dear TAs,

I have some questions about the exercise 10.

1) In the problem 1, for eq.5, why do we only keep the term in $\omega_{RF} - \omega_{LO}$ and do not use the term in $\omega_{RF} + \omega_{LO}$?

2) In the problem 2, for eq. 11a & 11b, why does a differential drive for the input pair imply a time offset of $t-\frac{T_{LO}}{2}$ for the m term?

3) In the problem 2, for eq. 14, why do we get this expressions for the IM2? how do we find it?

4) In the problem 2, for eq. 16, why do we set V_m = V_{ov} to get eq. 17? How do we get the right-sided expression of eq. 16?

Thank you,

Alexandre

Re: Exercise 10

par Hung-Chi Han, mercredi, 22 juin 2022, 20:33

Hi Alexandre,

Thanks for your questions.

1) Nice question. The derivation is a bit tricky. It doesn't matter you have cos(\omega_LO) or sin(\omega_LO) since they just simply describe a square wave. If you use sin wave, you can easily explain (4) to (5). However, the amplitude is the most important thing as question asked.

2) V_LO on the differential pair input can be regarded that you active M2 and M3 alternatively. So they have the time shift by half of the period.

3) it is also similar to (7) to (8) in problem 1. The square-law equation describes the long-channel device in strong inversion and in saturation.You can easily find it in text book for MOSFET, given as ID = (1/2) \mu C_ox (W/L) (V_G - V_T)^2

4) Actually V_m is always V_ov, it might be confused with previous problem. Because (14) calculates the IM2 product of M1 transistor, where V_m is nothing else but V_G - V_T = V_ov, according to square-law.

Cheers,
Hung-Chi

Re: Exercise 10

par Alexandre Nicolas Lechartier, samedi, 25 juin 2022, 14:17

Dear Dr. Hung-Chi,

Thank you for all your answers!

4) I think I didn't understand what the notation V_m is used for. Is the notation V_m used for the overdrive voltage of transistors or is it just a notation of a voltage in the circuit?

Best,
Alexandre