# Circuit Optimization

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#### ErnieM

Joined Apr 24, 2011
8,040
Circuit Optimization

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?

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