RobG wrote on Jul 30th, 2014, 8:47am:SNIKE wrote on Jul 29th, 2014, 9:39pm:Sorry still confused.
So can I use a capacitors whose matching is just 9 Bit to generate a 12 bit output,assuming my first stage is 4Bit (3.5bit).
I'm 99% sure you need a 12 bit accurate DAC on the first stage because the errors are multiplied by the increased first stage gain, but after looking at the Kelly paper [Journal of Solid State circuits, Dec 2001] I wonder if I'm wrong.
If anyone can knows why this is true (or false) I'd like to know. I can agree with the paper that the DNL improves by sqrt(2) (contrary to the Murmann claim of 2x) but the number of elements doubles so I think the INL stays the same.
Murmann might be keeping the unit capacitor the same size and doubling the total capacitance, which would give 2x improvement in DNL and sqrt(2) improvement in INL.
The DAC linearity requirement is reduced in a pipelined ADC using a multi-bit first stage because the coarse conversion of the first stage happens BEFORE the residue is generated, it's as simple as that.
If you have a 12-bit converter, and a 3.5-bit first stage, you're converting the 3 MSBs of the your input signal before it even touches the op amp. The output residue will then be converted by the backend of the ADC, which in this case is a 9-bit converter.
This effect is one of the main reasons you use a multi-bit first stage (the other is power dissipation). The idea is if you use a multi-bit first stage you can make a 12-bit linear ADC with 10-bit capacitor matching (for instance).
BTW I made some lab measurements a few years ago that were consistent with Kelly's paper (assuming the matching data from TSMC was accurate).
Rob is probably right about the kT/C driving the cap size. That is usually the case in my designs (which is why I like the pure 1.5-bit/stage architecture so much). Be careful when calculating kT/C requirements for your converter as there are a lot of mistakes in the literature.
Murmann (he's a helpful guy!) has an excellent tutorial on calculating noise in a switched-cap amplifier the right way:
"Thermal Noise in Track-and-Hold Circuits: Analysis and Simulation Techniques" in IEEE Solid-State Circuits Magazine, June 2012