I think the really accurate systems measure the current from the battery and integrate it over time. What you want to do is model what is going on inside the battery. If you know the amp-hours rating of the battery, and if you are also monitoring the charging situation, then you can start with an assumption of a fully-charged battery and go from there. If you are installing disposables, this is harder to do because you don't know the history of the battery being installed. The voltage curves are poor indicators of remaining battery charge, except near the end when it is almost empty. But proper modeling of charge and discharge current can more accurately predict capacity in the middle of the range as well. Googe for "gas gauge chips" or something like that.
This is not too difficult a problem. Manufacturers will publish discharge curves for their cells at several different current draws. I would find an example of one from an Alkaline cell and one from NiMH and use them. Let me find an example so you can see what I'm talking about if you're not familiar with the idea.
Look at the graph labeled "Discharge Characteristics". This one doesn't show the curve at different currents, but some do.
Can the user set which type of battery is installed? If not, you will need to take a guess at which of the types you have in order to calculate the remaining capacity, which is more difficult and less certain; 1.4v could be either a low alkaline or a fully charged NiMH. There are also some temperature effects, but you may not need to go to that level of detail depending on your application.
First, all of the discharge curves that I have seen are either for the constant R or for the constant I. There are no curves for the constant power consumption.
Second, figuring out a curve is not a big problem however there is no point in reinventing the wheel. I am almost certain there should be a formula already available since the task looks so typical. Do you now of such formula?
The terminal impedance of some batteries are fairly linear for *much* of their range, but some are not (certainly some NiCd constructions are not
- they are logarithmic), but the final 10% is very non-linear.
Assuming a constant load, a typical NiMH has a fairly linear discharge until about 20% remaining capacity, and then the output voltage starts to increase it's slope. I could be convinced that the terminal impedance of NiMH is logarithmic. Note I have not done those experiments (I am not using NiMH in current products), but the curves look very close to the typical ones I worked with for other chemistries (NiCd being among them).
My findings from experimentation with NiCd (remembering that this is 15 years ago) was that the terminal impedance was proportional to ln(Q).
If you were to prefix your above formula with (X ln) where X is some constant of proportionality, you may well be close. Interesting problem.
I assume the impedance ~ 1/remaining charge. This seems logical since the impedance should be reverse proportional to the amount of the active stuff inside the battery.
However the problem is NOT of what percent of the initial charge is in the battery. What we really need to know is how much time the battery will last under a given power consumption! Unfortunately, this leads to the nonlinear differential equation of the
2nd order, and I don't yet know how to approach it...
You will end up with a vague guess. The terminal voltage is dependent on the Temperature, State of charge, Number of cycles on the cells. And for Nickel Batteries The number of partial discharges. Of course NiMH and Alkaline are totally different. You may get acceptable results with Alkaline cells. Otherwise get a gas guage IC (TI or Maxium).
NiMH and NiCd batteries have a very flat V vs. charge curve for much of their capacity, then drop off suddenly. If all you do is monitor voltage then you can't really say much until the battery is close to dead.
Dry cell batteries have a much more linear drop off, so you could probably do a better job of it -- I suspect you could just interpolate between 1.5V and 0.9V and do pretty well.
Ultimately what I see done is charge monitoring by current integration
-- this is a PITA, so if there were a better way some IC vendor would have it patented and encapsulated in epoxy, ready to sell to you.
For the constant power, steady state operation, normal conditions, and at the discharge currents about 0.06C the remaining operating time for NiMH is fairly linear with the voltage. The 100% corresponds to 1.25V, the 0% is at 1.08V per cell.
This actually is in the good agreement with the theory that the impedance of the battery ~ 1/charge.
First, the gas gauge has to know if the battery was replaced, and what is the initial state of the battery which was placed in. And it costs money whereas the voltmeter does it for free.
The research I did showed that that dynamic impedance of a battery is always a good indicator of the remaining charge. That may sound unremarkable, but the battery manufacturers did not want it known (they make money, after all, by selling new batteries).
The only thing to determine is the actual impedance v. remaining charge. That's going to vary for every chemistry and construction.
I filed patent disclosure on this technique in 1991.