Question: Given a simple circuit of a draw around 28mA (when supplied 3.3v). I'm trying to maximize run time on a battery, but working with what I've got on hand... I have the following batteries:

* 5000mAh 3S LiPO (12.6V fully charged)
* 1700mAh 2S LiFE (7.2V)
* 600mAh 1S LiPO (4.2V)

I have the following regulators:

* MCP1700 3.3V LDO
* LM2931 5.0V LDO

The MCP1700 Can't handle the larger batteries. So I see my options are:

* Use the tiny 1S and the MCP1700
* Use the 3S and the LM2931 output into the MCP1700
* As above but with the 2S

Which should last longer?

Obviously the 5000mAh battery is by far the largest, but that's a pretty inefficient setup. The 2S is somewhat more efficient and still about 3x the size of the 1S. The 1S should be by far the most efficient, but the battery is tiny.

I can, of course, just try each and measure. Thought I'd ask here for insights.

Answer: 
LDO regulators don't "transform" energy, they are "linear" regulators. Whichever goes into the load, nearly the same current will be drawn from the battery.

So, within conditions of your task, the largest battery has 5 A-h capacity, the medium one has 1.7 A-h, and the smallest one has 0.6 A-h.

Since the same current (0.028 A) will be consumed from a battery in all cases, the run time will be 178 hours (5/0.028), 60.7 hours, and 21.4 hours correspondingly. I assume that regulators have negligible control losses (quiescent current) in their feedback/control circuitry. If not, this "control current" needs to be added to the 0.028 A load. 

ADDITION: MCP1700 uses MOSFET as regulating element, with control circuitry consuming a negligible 1.6 uA (max 4 uA) additional current from input. For LM2931 the "ON" current is 2.5 mA typical, up to 30 mA, so LM2931 might last half of MCP1700 time. True, the MCP1700 can't handle input voltages above 6V, but there must be other low-quiescent-current regulators. And, as Tony Stewart hints, DC-DC switching converters would do a much better job.
```
                       +/-15%   Ah  Wh    TPS82150 Buck Wh-out  hrs @ 92 mW
5000mAh 3S LiPO (12.6V) 10.8  5000  54    86%            46     505 h
1700mAh 2S LiFE (7.2V)  7     1700  11.9  87%            10     113 h
 600mAh 1S LiPO (4.2V)  3.6    600  2.16  92% LDO         2     22 h

 Meanwhile a 3.3V LDO with 7V in avg is 60% efficient
                      with 10.8V avg is 30% efficient TPS71533

But chances the 5000mAh is only 2100mAh are very high unless proven. (Fake Chinese specs)

```

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> * A Lithium-ion cell starts at 4.2 but drops to 3.6 avg and drop out at 3V typ. So 3.3V out of a MOSFET LDO with 3.6 average is 92% efficient. Or more... since if Vin-Vdrop (50mV type) goes below 3.3 V , the output is now tracking battery. Ie 99% efficient best case
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