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I would like to share a few observations pertaining to the TI84+CE charging/discharging process. There does not seem to be any available info on this subject in the internet, but if I missed it, I would appreciate pointing me to the relevant source.

Follows a description of the experiments I performed.

I wrote a simple program in TI-BASIC. The program was generating a series of pseudo random numbers using randInt( and accumulating number counts in bins. The only I/O instruction used in the program was Output( used for displaying the counter.

1. When the program was running and the charging cable was plugged in the calculator, the calculator did not recognize the charger until the program was finished. The charger was recognized when the calculator was running TI-BASIC program, but was executing input command, e.g., Input(.

2. The program was run on the fully charged calculator for 5 hours. After it finished, the battery gauge was still showing "battery full" status. After subsequent running the same program for 9 hours 25 minutes, the gauge dropped to approx. 1/4 of the maximum reading. (This indicated that the gauge is nonlinear, but, in my opinion, it's acceptable since it's not a precise measurement tool.) During the time the program was running, the gauge was not refreshed, and kept showing the maximum reading. It dropped rapidly after the program finished running.

3. Connecting/disconnecting the charger when the calculator is turned off makes the calculator to turn on. When the calculator is being charged, it is turned off, and then turned on, the charging process is stopped for a second or two (i.e., the current drawn from the charger drops to almost zero, the charging LED turns off), and resumed after this time.

Closer analysis of the battery charging/discharging process required making a custom adapter for the battery to monitor its voltage and current, and, because of that, has not been performed. (Maybe when I have some free time I will give it a shot.) The battery used in TI84+CE is manufactured by Samsung, and the power adapter implements Samsung quick charge algorithm, which made the external energy measurement somewhat hard to do.
Very interesting observations - This information could be useful on WikiTI: http://wikiti.brandonw.net/index.php?title=Calculator_Documentation

Also, welcome to the forums!
Another interesting note, a new TI-84 Plus CE running at full brightness can last you about 17 hours of non-stop use! My 4 year old CE's battery lasted about 12 hours (note the battery health is drastically different between the calculators). Not too shabby for a 1,200mAh battery. Of course normal humans don't leave their calculator on for 17 hours straight and don't keep it at full brightness all the time so it's not a perfectly realistic test, but still informative! Or so I'm told.
The battery life time may also depend on what the calculator was doing during during the test.
The calculator executing program takes more current than the calculator running idle. During characterization of charging process I observed fluctuations of the charging current for different conditions, including this one, and here there definitely was a difference of several mA. Again, this was not a precise measurement.
Also my suspicion is that the current drawn by a calculator executing program from RAM differs from the current drawn during executing program from flash.
Small differences, but they can have an impact over about 20 hours period of time.
To get precise measurements I have to come up with a jig allowing to get access to battery terminals. I'd like to characterize the battery gauge vs. battery voltage. (During my experiments it became obvious that the battery status is derived from the battery voltage and --maybe--drawn current history, and not true charge monitoring; modern gas gauge chips use this method in connection with the battery model programmed in, so it's nothing out of ordinary.)
But given number of things I have on my plate at the moment, it may take some time. I'll post the results when I finally get to it.
Follow the results of the TI84+CE power test. Sorry for the long post.

A. CURRENT DRAWN BY CALCULATOR
The test was performed using a new genuine battery for TI84+CE. This one was made in China.
The currents drawn from the battery in particular operation modes are:
[1] Calculator idle, default brightness: 41.2 mA
[2] Calculator idle, minimum brightness: 22.4 mA
[3] Calculator idle, maximum brightness: 63.9 mA
[4] Calculator executing TI-BASIC program, maximum brightness: 75.2 mA
[5] Calculator executing TI-BASIC Input instruction, maximum brightness: 64.2 mA
[6] Calculator off: 9.7 uA

Notes:
1. The battery voltage during test was 3.8V
2. The difference between currents drawn while executing program from flash vs. executing program from RAM was negligible.
3. The resistance of ampmeter used in tests [1]-[5] was 2.05 Ohm, the resistance of ampmeter used in test [6] was 998 Ohm.
4. For details pertaining to the TI-BASIC program used in this test see my previous post.

B. BATTERY GAUGE TEST
This test was started with the fully charged battery. The calculator was running a TI-BASIC program, display brightness was set to maximum. The program was periodically stopped, and the voltage measurement and the state of the gauge were registered.
The battery got completely discharged (and the calculator died) after 15 hours and 40 minutes of continuous running.
During this time the battery gauge was showing full state for first 5 hours (roughly), values between full and empty for next 7 hours (roughly), and empty state for remaining 3 hours and 40 minutes. It means that the gauge shows something different than full or empty for only about 50% of the battery life time.
The battery voltage thresholds corresponding to full and empty states were approx. 3.9V and 3.7V, respectively.
It is important to mention that the gauge is not linear between the full end empty states, and the gauge behavior is often far from unambiguous (see conclusions).

Notes:
1. The total energy transferred from the battery to the calculator was 1184 mAh (value calculated from measurements.)
2. When the voltage started getting closer to 3V, the display backlight started getting visibly dimmer. Apparently the battery voltage must have dropped very closely to the Vf of the white LEDs.

C. CONCLUSIONS
The conclusions are *not* very favorable for TI designers.
The curve of voltage during discharge for LiON battery is nonlinear. Because of this, one cannot use use the battery voltage as a discharge indicator and expect accurate results. (It is possible for other chemistries though, e.g., alkaline.)
The first solution allowing to get accurate state of charge (SOC) for LiON batteries was to monitor the charge flowing in and out of the battery. This solution, far superior than voltage monitoring, still had many flaws, and it was later replaced by the intelligent voltage/current monitors. They, using battery models, can quite accurately predict the SOC. These circuits have ability of learning over time thus increasing the accuracy of measurement. There are many of chips dedicated to this function available, and they are inexpensive.
Alas, it looks like the TI84+CE gauge just shows the voltage between two voltage levels. The fact that the voltage range is relatively small (~200 mV) makes things even worse. On top of it, the measuring algorithm often times seems to get confused, e.g., it does not update the gauge after the charger is disconnected from the calculator. (When the battery is charged, its voltage is always higher than when its not charge, thus after disconnecting the charger, if the voltage remains in the active gauge area, the gauge reading should drop. The calculator refreshes the gauge only once every many seconds, but even after waiting a long time the gauge was not dropping.)
Thus, serious usefulness of the battery gauge is rather questionable.

Note: additional anomaly was discovered when the battery was charged starting from empty state. (It's not really "empty", since the protection circuit built into a battery disconnects terminals if the voltage drops below ~2.7V.) If the charger was plugged in when the almost empty battery was in the calculator, the orange LED was getting lit, but the charging current was not flowing into the battery. When the calculator was turned off, the current immediately jumped up to 0.45 A, which is the normal charging current for empty battery. (For those who thought of initial trickle charging of empty battery: if it were the case, the current would not jump up when the calculator was turned off. Unless, it is, turning the calculator off disables trickle charging, but this would be even worse bug.)
  
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