Corelated currents DNL in thermometer DAC

[yefj]

Hello, I understand that DNL is measured as the standart deviation of subtraction of two voltages where each one is normal distributed.

for example:
if our step is from 6I to 7I then then its I4+I2 to I4+I2+I1 to 111 in binary weighted where as in thermometer it 6I1 to 7I1.
DNL of binary: stadrt diviation of subtraction of random number is sqrt of their power 2 sum)
sqrt(4+2 +0+4+2+1)= sqrt(13)

in thermometer our current distributions are corelated so we subrtaract and getting DNL of 1.
But the formula shown bellow is much more comlicated.how come we just subtract?
Thanks.

 

[sutapanaki]

DNL is not a measure of the statistical deviation of the difference of two voltages. DNL i s the deviation of the step from one voltage to the nex, compared with the average step of the DAC. Then, on top of that you can ask about the statistical variation of the DNL or the error in the step.
In that respect, for thermometer DACs, you always step from one code to the next by adding one unit element - current in your example. And you compare it with the average value of that unit element, which the average LSB step. DNL=(I-Iavg)/Iavg. To find the variance you simply have
var[(I-Iavg)/Iavg] = var(I/Iavg)=var(I)/(Iavg^2) and thus sigma=sigma(I)/Iavg