In the previous post, I’ve shared some basics of sigma-delta ADC. In this post, before we look at the noise-shaping SAR ADC, let’s again do a warm-up.
The z-domain linear model for a 1st-order sigma-delta modulator:
The linear model has the same transfer functions as the one in Fig.6 of the previous post, where a delaying integrator is used as the loop filter.
Before the quantizer, the modulator is doing two tasks:
1. Δ: generate the conversion residue R (=U-V)
2. ∑: add all the previous residues
Keep this in mind. Now let’s try to make the SAR do the noise shaping.
A conventional charge-redistribution SAR ADC:
When the conversion is complete for an N-bit SAR, the magnitude of the voltage generated at the top plate of the DAC represents the difference between the sampled input and a representation constructed from decisions of the high-weighted N-1 bits:
If we do one extra switching of the DAC array based on the final decision of LSB, we recalculate the voltage generated at the DAC top plate:
Yes! We catch the conversion residue! It is further simplified as follows:
According to Fig.1, the simplified equation can be rewritten as .
Then we need to sample this residue and store it somewhere else. How about this method?
Step 1: sample the residue on the extra capacitor
Step2: apply the sampled residue to the opposite input of the comparator during the next conversion
Now it comes to the discussion about choosing the value of C_R.
What will the linear model look like?
If ( ), and . The memory of the previous residues is ignored and only the current residue is recorded. The linear model can be simplified to:
Take a look at the magnitude responses of the NTFs under different k:
Noise does be shaped! In addition, it seems using a small residue sampling capacitor is fairly good compared to larger ones (Note that the kT/C noise during residue sampling presents itself to the comparator input and can also be shaped together with the quantization noise and the input-referred comparator noise ).
However, compared to the 1st-order modulator, this way of noise shaping is much less efficient. We could do better! How? The next post ;-).
 J. A. Fredenburg and M. P. Flynn, “A 90-MS/s 11-MHz-Bandwidth 62-dB SNDR Noise-shaping SAR ADC”, JSSC, vol.47, 2012.