PROBLEM 038 · Hard · Common-mode rejection
Tail Risk
A pair with a real tail: reject the common mode
The problem
Build a differential pair with load resistors to VDD and an NMOS transistor as the tail current source, with its gate on an Input source named “vbt”. The inputs are Input sources named “inp” and “inn” at the same common-mode level. The outputs are “outp” and “outn”.
A common-mode signal moves the tail node and changes the current through each side; the stage’s single-ended common-mode gain is about R_D/(2·r_tail) where r_tail is the tail transistor’s output resistance. The ratio of the single-ended differential gain to that common-mode gain is the CMRR, about gm·r_tail. A long tail channel gives a high r_tail.
The test bench
- Supply: VDD = 1.8 V.
- The test bench hangs a 200 fF load capacitor from each of “outp” and “outn” to ground.
- It first applies the stimulus to “inp” only and measures the differential gain. It then applies the same stimulus to both “inp” and “inn” and measures the common-mode gain at “outp”. CMRR is their ratio in dB.
Numbers to hit
- Differential gain at least 12 dB
- Common-mode rejection ratio at least 35 dB
- Output common-mode level between 0.8 V and 1.5 V
- Power at most 250 µW
- Every transistor in saturation
Design it yourself
Draw the circuit on a schematic canvas, size every transistor, and see it simulated in your browser with ngspice and the SkyWater SKY130 models. Every transistor lights up green, gold or red by operating region as you move the sliders.
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- Two's CompanyTwo matched transistors share a tail current and turn a voltage difference into a differential output.
- Pair, FlippedFlip the pair upside down: PMOS inputs, a tail current from VDD, loads to ground. It works with inputs near ground.
- Slow DownAdd a resistor in each source: the gain drops, but the pair stays linear for a much larger input.
- LimboA pair only works while its tail transistor stays saturated. Find how much common-mode range your design really has.
- Load BearingUse PMOS diodes as the loads: the gain is no longer a resistor ratio but a transistor ratio.
- Mirror TailReal tails come from mirrors. Take a 25 µA reference, scale it to 100 µA, and use it to bias the pair.