# R-2R Ladder Resistors Mismatch

I am still trying to understand what kind of errors are caused in the R2R ladder depending on the amount of resistor mismatch relative to the others. For example if you have a mismatch of 12% in the MSB of an 8 bit ADC I think you would have a gain error since the total resistance would not be equal to R and and thus the inverting amplifier would not be balanced. Also I think there would be non-monotonic behavoir and DNL errors. On the other hand I think that you would have DNL and gain error even with the slightest mismatch(such as 0.2%) in the MSB resistor but you would not have non-monotonic behavior.

On the other hand if you have an error in the lower resistors such as for example LSB(+3) in an 8 bit DAC you would have both DNL and INL errors with the slightest mismatch such as for example (0.2% mismatch) and you would only have non monothonic behavoir if the mismatch is larger than a certain amount (for example 12%).

Can somebody please verify if my assumptions are correct?

## 2 Answers

First thing to do is writing a function of the setup, like

$$F(R_1, R_2, R_3,...R_N)=....$$

Then you have to calculate he partial derivatives for each element and multiply with resistance tolerance value:

$$\dfrac{\partial F}{\partial R_n}\cdot R_{nTolerance}$$

Then a reliable error limit would be:

$$E = \bigg[ \lvert \dfrac{\partial F}{\partial R_1}\cdot R_{1Tolerance} \rvert + \lvert \dfrac{\partial F}{\partial R_2}\cdot R_{2Tolerance} \rvert+\lvert \dfrac{\partial F}{\partial R_3}\cdot R_{3Tolerance} \rvert +....+ \lvert \dfrac{\partial F}{\partial R_N}\cdot R_{NTolerance} \rvert\bigg]$$

• But how you would know what kind of error you have – suyol854 Nov 19 '17 at 16:44
• @user7075815 The same type of the function: volts, increments, ... – Marko Buršič Nov 19 '17 at 22:31

So you have a R-2R-4R-8R-16R.... DAC, and the MSB R is either 12% low or 12% high.

If 12% low in value, the current will be 12% high. Now continue your reasoning about INL and DNL. I like how you are approaching this. Now take the next step.

Perhaps draw a sketch of the output current, and draw the resultant code-errors as the Successive Approximation algorithm proceeds.

• It is an R-2R ladder not a weighted resistor DAC, all the resistors are R and 2R – suyol854 Nov 20 '17 at 8:12