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?
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?
