Approximating alternator output

Question

I've got this graph of the ampere output of an alternator:

or as a table read off the graph:

RPM	Ampere output
1450	42
2000	97
2500	120
3000	133
3500	143
4000	149
4500	152
5000	156
5500	159
6000	162
7000	164
8000	166

which I need to approximate using a formula, $x$ being the alternator speed in RPM (rotations per minute) and $y$ being it's output in amperes.

I've tried using the various logarithmic, square root and cubic root functions, but none of the graphs produced by those came close to the graph above.

Answer 1

Sometimes to get a good fit you need to apply a transformation to the data. I tried shifting the data horizontally to the left by 1350 amps. This is accomplished by subtracting 1350 from each of your RPM values. I was able get a pretty good logarithmic fit with a R^2 of 0.9899. The 1350 was found just by trial and error, you might find there is a different value which is more optimal.

My fit equation is,

$$Y = 30.811 \ln(X-1350) - 98.15, $$

where $Y$ is the number of amps of current and $X$ is the number of rpm's from the alternator.

Answer 2

Flipped upside down, the graph strongly resembles one of an exponentially decaying function. I therefore tried fitting to an exponential function, optimizing the fit according to the minimax criterion. This results in

$$y \approx \frac{7457}{16} \left(1 - \exp \left(-\frac{63}{65536} x\right)\right) -\frac{2435}{8}$$

which has a maximum error of about $\pm3.525$. Because of similarity in shape, I also tried minimax fits to the sigmoid function and $\tanh$, but the resulting maximum error was slightly larger, a bit larger than $\pm4$.