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:**

Fig.1 Linear model of 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:

Fig. 2 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**

Fig.3 Sample the residue on the extra capacitor (discrete-time domain is used to indicate the current sample and the previous one)

**Step2: apply the sampled residue to the opposite input of the comparator during the next conversion**

Fig.4 Apply the residue to the opposite input of the comparator when the next sample is converted

**Now it comes to the discussion about choosing the value of C_R.**

**Assume **

**Then** ,

and

What will the linear model look like?

Fig. 5 Linear model and transfer functions

If ( ), and . The memory of the previous residues is ignored and only the current residue is recorded. The linear model can be simplified to:

Fig. 6 Linear model and transfer functions when k1=1 and k2=0

Take a look at the magnitude responses of the NTFs under different k:

Fig. 5 Magnitude response of NTFs under different k and compared to 1st-order noise shaping

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 [1]).

However, compared to the 1st-order modulator, this way of noise shaping is much less efficient. We could do better! How? The next post ;-).

References:

[1] 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.

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