I've never seen it analyzed that way - but I suspect it's the wrong way to look at it given your result with a 10 bit "accurate" capacitor. The noise from a mismatch will be a tone, not white like thermal noise.

The capacitor mismatch to check out is the major transition - usually when the MSB turns on. e.g. 0111 1111 => 1000 0000 but you can relax the matching if the first few MSBs are thermometer coded.

I reran the numbers - disclaimer, it has been a long while since I've done this... easier to run in Matlab if someone was so inclined...

Let's say we have a 16 bit dac with LSB=100 fF (σ=0.1%) and an MSB = 2^15*100fF = 3.2768 nF (σ=5.5e-6). The σ of the MSB (8000) will be 18 fF. Also, the σ of the 7FFF code will be essentially 18 fF since the cap for that code is only 100 fF less. Thus the σ of the difference will be sqrt(2)*18 fF = 25 fF, or a quarter LSB. This will give you the DNL σ so if you are only interested in DNL (or monotonicity) it is better than 16 bits (one σ). I think this is how your formula analyzes it (but perhaps it should be 2^N-1*1u for looking at the quantization noise at the major transition).

In hindsight that isn't so surprising since 100 fF is a rather large LSB cap resulting in a total cap of 6.4 nF.

But INL is what causes the big tones/intermodulation. INL σ isn't straight forward - I forgot the formula (which is only an approximation anyway) but I think the worst case is on the order of sqrt(2^16)*σ_LSB which is 256 LSBs. INL will give you tones depending on the shape so figuring the tone σ isn't straightforward, but it is going to be closer to 8-10 bits than 16 bits. However, I believe that formula overestimated the INL so you really have to run a Monte Carlo or find someone with a better memory than me .

from Rob:  "Let's say we have a 16 bit dac with LSB=100 fF (σ=0.1%) and an MSB = 2^15*100fF = 3.2768 nF (σ=5.5e-6). The σ of the MSB (8000) will be 18 fF. Also, the σ of the 7FFF code will be essentially 18 fF since the cap for that code is only 100 fF less."

Where do you get:  (σ=5.5e-6) ?
Where do you get: 18 fF ?

To me, you gate the X sigma of the MSB transition banks (ex, code 1000..., code 0111... cap values).  
(Where X >= 3, since 1σ is not going to get you great yield either.)

Hi RobG and wave,


actually my formula is simply measuring the worst case deviation from the ideal code. So it isn't INL or DNL. As you you seem to be saying, if you want to be scientific, you could check the DNL by checking the variance of the 0111 code and the 1000 code, and then adding them together (multiply by X for X-sigma). But I don't think the answer is going to go from 16-bit to 10-bit. I think the real issue with the formula is that I assumed the 0.1% variation was in the LSB. Since the LSB cap is quite small, I don't think it will have such a small variation.


regards,
Aaron