Electronic – Safe temperature range for 18650 battery – powerful flashlight

I think what they're trying to do here is 'trick' the phone's battery charging intelligences. Li-On batteries are very touchy and have somewhat complex charging strategies. It all boils down to determining something called State-Of-Charge (SOC). SOC is just a percentage in the end, but arriving at the SOC number relies on a large number of factors that are not always easy to read and sometimes must be indirectly inferred. For instance, let's assume that you have a cell phone with a Li-On battery that is 3.7V and 1000mAH. We'll start with it being fully charged, so we know SOC is 100%. As you use your device you're drawing current out of the battery and the battery's voltage will drop - eventually. By measuring the current and monitoring the voltage you can guess what the SOC is. One problem is that the voltage isn't terribly useful in determining SOC because it doesn't change very much until the battery is nearly empty - that is NOT something you want to do to a Li-On battery. So you're mainly relying on the current.

So your SOC is being estimated throughout the usage. It gets low - 50% maybe - so you plug it in to charge. While it's charging, it monitors the charge current and battery voltage to determine when SOC is 100% again. Only, due to errors in measurement it says that the charge is complete when SOC is actually only 95%. Now your phone thinks 95% is fully charged - and it remembers this for future reference because it doesn't want to over-charge the batteries (this is also very bad). So essentially it's trying to read when the battery is full by measuring what goes in and guessing where that puts the SOC based on past results.

The errors aren't large so during normal charge/discharge usage you won't notice a problem. But sometimes the errors can stack up and your phone thinks its fully charged when it has little or no charge - it goes straight from full to empty and due to the incorrect SOC calculation the phone won't try to charge the battery more because it doesn't want to damage it.

In these cases you have to reset the SOC state. I have a Droid Incredible 2 and I've done it by removing the battery and holding the power button for 30 seconds, then putting the battery in and charging the phone while its off. This always fixes the issue where the battery thinks its full but drops down to something like 10% very quickly and the issue where it thinks its at 10% but has much more charge left.

The strategy outlined in your post is obviously trying to recalibrate the SOC or trick the algorithm somehow. Having never developed a charger that relied on SOC I can't say whether it will work but it seems like a lot of effort for a questionable amount of benefit. If your battery is acting really funny try what I suggested first.

Here's an alternative way to resolve your problem or figure out if your problem is physical or mathematical. Lets look at the problem from another angle and see if your measurements give the same result or a different one.

Your physical model is, you have a single heat source and a fixed path from that source to the environment, with a fixed thermal mass. Throw away all the details of the properties of aluminum, your preliminary measurement of the heat sink thermal resistance etc. With your simple (e.g. lumped-element) model, the response to turning on the heat source will be a curve like

\$T(t) = T_\infty - (T_\infty-T_0) \exp(-t/\tau)\$.

First, this shows you will need three measurements to work out the curve because you have three unknowns: \$\tau\$, \$T_\infty\$, and \$T_0\$. Of course one of these measurements can be done before the experiment starts to give you \$T_0\$ directly.

If you know \$T_0\$ and you take two measurements, you'll have

\$T_1 = T_\infty - (T_\infty-T_0) \exp(-t_1/\tau)\$

\$T_2 = T_\infty - (T_\infty-T_0) \exp(-t_2/\tau)\$

and in principle you can solve for your two remaining unknowns. Unfortunately I don't believe these equations can be solved algebraicly, so you'll have to plug them in to a nonlinear solver of some kind. Probably there's a way to do that directly in Excel, although for me it would be easier to do in SciLab, Matlab, Mathematica, or something like that.

So my point is, if you solve the problem this way, and you still get the same answer as you've already gotten, you know there is something wrong with your physical model --- an alternate thermal path, a nonlinear behavior, etc.

If you solve it this way and you get an answer that matches the physical behavior, then you know you made some algebraic or calculation error in your previous analysis. You can either track it down or just use this simplified model and move on.

Additional comment: If you do decide to just use this phenomenological model to solve your problem, consider taking more than two measurements before trying to predict the equilibrium temperature. If you have just two measurements, measurement noise is likely to cause some noticeable prediction errors. With additional measurements, you can find a least-squares solution that'll be less affected by measurement noise.

Edit Using your data, I tried two different fits:

The red curve was for a single exponential response, fitted as

\$T(t) = 33.4 - 38.6\exp(-t/81.96)\$

The green curve was for a sum of two exponentials, fitted as

\$T(t) = 36.86 - 35.82\exp(-t/81.83) - 5.42\exp(-t/383.6)\$.

You can see that both forms fit the data nearly equally for the first 100 s or so, but after about 200 s the green curve is clearly a better fit. The red curve is very nearly flattened out at the end, whereas the green curve still shows a slight upward slope, which is also apparent in the data.

I think this implies

1. You need a slightly more complex model to get a good match for your data, particularly in the tail, which is exactly what you're trying to characterize. The extra term in the model probably comes from a second thermal path out of your device.

2. It will be very difficult for a fitter to distinguish the part of the response due to the main path from the part due to the secondary path, using only, say, the first 100 s of data.