Electronic – Internal resistance of batteries

You're not really looking for the C rating (maximum discharge current in multiples of nominal capacity), you're looking for the adjusted capacity at your nominated discharge current.

A 1.8Ah lithium battery can theoretically give 1.8A for 1hour, or 3A for 1.8/3h = 36 minutes. HOWEVER the capacity for a battery is traditionally quoted for a 20 hour discharge. That is, a capacity rating of 1.8Ah means the battery delivered 90mA for 20 hours in testing.

The relationship between continuous current and time-to-full-discharge is NOT linear. You have discovered that when discharged at 3A, your 1.8Ah battery is delivering much less capacity (only about a quarter!) than a linear interpolation of the amp-hour rating would suggest. This is not unusual.

The better batteries will give rated capacity at several discharge time samples, or even a graph of current vs capacity. A battery intended for remote control uses will probably give better performance, as quick discharge is the intended application of these batteries.

(I use a 1500mAh 3-cell (11.1v) 25C lithium pack to power a 3A LED bike light, and I get around an hour, which is close to rated capacity given a reasonably efficient buck regulator).

Here is an excerpt from an article I found:

"To determine the available charge in a battery, simple monitoring methods are preferred. They should consume little energy and should (ideally) allow one to deduce the charge level from battery voltage. Such a voltage-only method can produce unreliable outcomes, however, because no clear correlation exists between voltage and the available charge (Figure 5). Battery voltage also depends on temperature, and dynamic relaxation effects can cause a slow increase in the terminal voltage after a reduction in load current. Thus, purely voltage-based monitoring is unlikely to provide charge-level accuracies better than 25%.

The relative charge level, often called the state of charge (SOC), is defined as the ratio of residual charge to the battery's charge capacity. Hence charge flow must be measured and monitored through a procedure called "coulomb counting." In practice, coulomb counting is accomplished by integrating the currents flowing into and out of the cell. To measure these currents with a high-resolution ADC, one typically connects a small resistor in series with the anode."

Here is a datasheet I found for an Atmel AVR352 that you could use for coulomb counting:

Even better would be to refer here.

And there are devices.

Though as far as I can see there is plenty of info on the web; i don't see how you could have missed it.