I am working out a circuit for a micro controller device. I am reading two voltages thru an analog to digital converter, performing a conversion calculation on the results and using the result to adjust some other parameter.

The conversion is of the form:

\(T = K (\frac {X}{Y}-1)\)

where:

K is a constant

X is a measured voltage dependent upon an external resistor

Y is a measured voltage reference

We can also say the following about X and Y:

\(Y \leq X\leq 5\)

\(0 \leq Y\leq 5\)

I chose this scheme for a number of reasons, not the least of which is the exact value of the driving 5 volt supply voltage cancels out of the equation.

Now my goal is to select a Y to minimize errors when computing T. To predict that I am equating the magnitude of the change due to X to the magnitude of the change due to Y:

Compute derivatives:

\(\frac {dT}{dX} = \frac {K}{Y}\)

\(\frac {dT}{dY} = -K \frac {X}{Y^2}\)

Equate the magnitudes:

\(\frac {K}{Y}=K \frac {X}{Y^2}\)

\(Y=X\)

At this point I start waving my hands trying to explain why setting the constant term Y to 2.5 will result in the least errors in the system, but all I can say is it "feels" right.

Does someone catch my drift here and can offer a firm mathematical basis for selecting Y?