The long, full blown, Spanish version of the mathematics behind the TC calculations.....
Sorry, I didn't have enough time for a translation, and Spanish is my first language, but I can cut to the chase, in short, directly translated:
If you got enough data about resistance or resistivity variation with temperature, you can plot those points or you can try to be close to the actual data with some mathematical approximation. The simplest one is a linear regression of the data. Even simpler, and often enough to get the work done, what mathematicians called a local aproximation based on first derivative, as linear function. Enough chit-chat, let's put it plain. You get close to the actual data with this:
Where R0 is the base resistance (the fixed one with the + and - buttons on YiHi boards, or recently read after confirming the 'new coil' question on Evolv boards). That one is measured at T0 temperature, which is postulated as 20 ºC (68 ºF). Of course, R is the actual resistance as it get higher because the temperature rises. B is the local slope of the representation in the rearranged formula:
which renders (or it should) a straight line which passes through the origin (0,0). That number, which has units of inverted temperature, is the famous TCR.
I read the former message about the process and I didn't get convinced about it. Something was off, but I could get mesmerized by the explanation. If you can work with this (relatively simple) equations, is not some big piece of cake. Some examples from my own numbers:
* With actual resistivity values found in the Net, my linear correlation of the above renders B=0,00575 1/ºC instead the 0,0060 or 0,0062 values often found, but the purity of nickel is of paramount importance to fine tuning this.
* This is how it appears in a plot, using just the four points relevant to our application (from 20 through 300 ºC):
* Once you get the gist of the above formulae, you might ask for the final useful one:
, which is just a new reordering of terms..... and with actual values:
with a base resistance of 0,081 Ω and a maximum temperature set (by user, in the mod) at 210 ºC, using that nickel TCR obtained as B=0,00575 1/ºC, the maximum resistance that the mod should show is about 0,17 Ω.
Bear in mind that we are getting values with not so many significant figures (just two, in fact, with the former numbers), that is, the truncation in calculations and error bars are pretty high, and that each TC controlled PCB has its own algorithms to push the brakes (that is, lowering the voltages applied) as the temperature approaches near the maximum value, so you do not get exact figures and nail the expected numbers. When a reviewer has got actual temperature values (with a thermocouple probe or another methods), like the ones that Phil Busardo and Dirk Oberhaus have recently done, usually you see how the mod approaches the target, but the final 20 - 30 degrees are not immediately obtained, to avoid to get beyond the target.
oW! I know it, I should look for atonement after this one........imagine the original, full blown one!....
