This sum looks bad on the surface, but it is actually convergent for all polynomial transfer functions. It is pretty easy to convince yourself of why this would be. For a polynomial transfer function, the corrected sum error function has coefficients of 0 for the constant, linear, and quadratic terms. Because the kth term of the sum is (2^k) * E_sc(x / (2^k)), so if E_sc = x^3, the term reduces to x/(2^2k). If, however, there is a linear term, then the kth term just reduces to x, and the sum obviously diverges. This gives a simple test for whether the sum diverges. If you can approximate the corrected sum error function as a Maclaurin series for an arbitrarily small domain centered around zero, then the coefficient of the linear term of the Maclaurin expansion is the the first derivative at zero. So, if the first derivative of the corrected sum error function at zero is zero, the sum will converge.

The thing is, if over some domain centered on zero you can approximate the transfer function error as a Maclaurin series, the corrected sum error will have a first derivative of zero at zero because it always has coefficients of zero for the constant, linear, and quadratic terms. Also, as you approach zero, the cubic term in the corrected sum error is dominant, and for a E_sc(x) = x^3, this sum equals (4/3)*E_sc(x). In fact, for any polynomial fit term Ax^n in E_sc(x), this sum will be equal to A*(2^(n-1))/(2^(n-1) - 1)*x^n. So you can do a polynomial fit over part of the domain of the sum error function and use that to avoid infinite recursion.

I am looking at using a bipolar sum error to marginally reduce the sensitivity to error near zero. The error term is (V(DAC,GND) - V(GND,DAC)) - (V(DAC,CT) - V(CT,DAC) + V(CT,GND) - V(GND,CT)). I'll also take a closer look at the topology near zero. I should include that even with my spline fitted sum error function without the correction factor, the shape of the transfer function converged around five terms in the sum, and I could not differentiate the plots between six and 20 terms by eye, which is good enough when the limits of the vertical axes are less than 200 ppb fs.

E(v) = E_t(v) + sum(i = 0 to inf)( (2^i) * Esc(v / (2^i))

Sorry for my late response. I came to the same equation seems the begin of exploring the idea of R-R divider idea.

The equation itself speak for fundamental limitation of algorithm, even if if is quite easy to implement in recursive software algorithm.

I suspended any HW development until I'm satisfied with sim results.

I was playing with idea of combining of multiple Sum=0 divider based on 1*R/1*R n*R/m*R.

I have tried applying Kalman filter to mitigate disadvantages between different resistor divider configurations. I'm still not satisfieed with results. There many parameters. Just to mention few of them error sensitivity, calibration points coverage, measurement time. sensitivity for temperature and time drifts.

I'm almost at he point to give up the idea of multiple divider ratios.

What is next - I'm considering to explore the initial idea from Echo88 and use DAC + resistive string or not DAC at all

With regards to the ideal topology for a source like this, I haven't compared the math between this and the string DAC, but I can say for noise, the limiting factor is definitely the DUT with the design I am using. Rod White at New Zealand's NMI has published some work about using a similar principle but with various series and parallel combinations of four resistors to measure linearity errors of bridges for resistance thermometry down to the 100 ppb level, and this might be worth checking out for ideas.

Developing the HW for this was really not that much work, I think the total time I spent was around a week. I believe that having a prototype, potentially with the ability to implement different methods of testing linearity, is probably going to yield more productive results at a certain point. Given the simplicity of the actual schemes, having tested this board, I would say the best way of approaching the design is to implement all of them on one board. Actually, if you just feed the string DAC with an IC DAC, that gives you all everything you need, and you can test all three of those possibilities. Just add a handful of muxes, an MCU with isolated UART to USB, and a reference, and that's it. If I were going to make this again, I would probably also include something where the DAC voltage bootstraps a, say, 2V048 or 2V5 reference, which feeds a divider between itself and the DAC voltage. Then you could calculate the sum error by measuring the bias voltage and the other components of the sum, so you would be getting something like 8, 9, 1, and 10 V for bias, the two centertap readings, and the total, respectively. This would be useful to be able to probe the average concavity around zero without needing to use points in the sum that are spaced very close together. It would give similar information to that available with the string DAC but with better resolution. Imperfect CMRR for the bootstrapped reference would just give horizontal scale compression. Whatever your design, I would recommend being able to bias with bipolar references to cancel out residual thermocouple errors. Also, the DAC11001 is overkill, but that's not really news.