The electrical performance of an unknown device was measured by applying a voltage V across the device and then measuring the current through it. The results are shown in the table:
- Table: Voltage-Current Properties of Unknown Device
Input V (V) 1.0 2.0 3.0 3.5 5.0 6.0 7.5 9.5 10. 13. 20.
Output I (A) 0.74 0.58 0.49 0.48 0.42 0.40 0.36 0.34 0.32 0.30 0.26
You feel confident that the functional relationship between the input voltage and the output current is empirical.
State the type of empirical equation (i.e., linear, power, or exponential) that best fits the measured data. Prove your conclusion by submitting graphs for each of the possible cases.
Note: You do not have to extract the empirical equation’s parameters.
This question revolves around your ability to visualize the trend that data exhibits once plotted. Start by plotting the actual points:
We can tell immediately that this is not a linear relationship. If it were a polynomial relationship, you might expect a local minimum or maximum, but we don't see any. It could be a power relationship, as we see some possible asymptotes, mainly the vertical one at x = 0. Finally, it could be an exponential relationship without a vertical asymptote. To me, it's a competition between the last two to see which fits best.
Here, we'll overlay a linear best fit (red), polynomial best fit of degree 2 (green), power function best fit (blue), and an exponential function best fit (purple):
Now, these best fits are calculated by minimizing the distance from each point to the line of best fit given the prototype equations given. Here, it is plain to see that the power function best fit wins out. This aligns with the asymptotes that could be noted from the data.
Now, you might object to my using a function of only degree 2, wondering why I didn't use degree 3 or 11. First, you'll want a reason to add those degrees, which I am hard pressed to do with a smooth-looking curve monotonically decreasing in y value and monotonically increasing in slope, but neither ever reaching zero. Second, the higher-degree of polynomial you fit, the fewer degrees of freedom you have, and the less-valid your equation is as a generalized fit to describe the relationship between voltage and current. In other words, you'll get an better "fit" than the parabolic curve with a higher degree polynomial--and an absolutely perfect fit if you use a polynomial of degree eleven--but it won't be anywhere near as valid as the power function fit.