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differential gain vs regular

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firewind

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we all know that differential ckt puts out 2x the output voltage swing but how come it's small signal voltage gain is the same as the equivalent regular non-differential pair. Let's say a CS with diode connect would have Av=gm1/gm2, the differential gain uses half ckt so it would have the same gain? Sorry, i don't understand that... it doesn't follow the logic. Shouldn't differential gain be twice as large?

Also, if you add a source degeneration. Your differential gain will decrease by half as much. Let's said for the same setup you add a Rss for the tail. You will get Acm=Rd/2 /[1/2gm + Rss] but the non-differential setup will put out Av= Rd/2/ [ 1/gm + Rss].... you see that 1/2gm? that would reduces its gain by 1/2... again, shouldn't it be 2x bigger?

Added after 1 minutes:

I got these results from Razavi book btw. So they should be correct. Please help me understand thanks

Added after 2 hours 20 minutes:

That's alright if you guys can't answer that question, could you please help me understand CMRR and PSSR then? I what's it about the equation derivation from Razavi book is kinda hard to understand. thanks
 

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