Battery Ampere-hour rating vs Battery Amps (not an experienced user)
Your hunch that batteries have a current limitation is correct. In general, it's hard to tell the current rating [A] from capacity [A·h]. You have to look it up in the datasheet. A lot depends on the design of the battery.
For example: coin cells with 500mAh capacity may have only 3mA max current.
Another (opposite) example: automotive starter battery with 40Ah capacity may have 500A max current.
Lead-acid batteries are interesting in this respect, because there are two distinct types.
- Starter lead-acid batteries are designed specifically to deliver high peak current for a short period of time. Deep discharge, however, dramatically shortens the life of a starter battery. So, it's not suited for routine operation at high depths of discharge. Your typical starter battery in the automobile works at very shallow depth of discharge.
- Deep cycle lead-acid batteries are designed (as name suggests) to discharge further. But they can not provide as much instantaneous current.
Here's an example datasheet for a deep cycle battery. Have a look at the nominal capacity on p.1. Notice that capacity depends on discharge current (i.e. the rate of discharge).
-
Depth of Discharge Starter Battery Deep-cycle Battery
100% 12–15 cycles 150–200 cycles
50% 100–120 cycles 400–500 cycles
30% 130–150 cycles 1,000 and more cycles
(Source.)
p.s. If you want to read-up, here's an excellent web site on batteries - Battery University.
No, not necessarily.
Amp-hours tells you how much energy is in the battery, the amps tells you how quickly that energy can be delivered (For the most part, for a fixed voltage).
Your suspicion is well founded, because the Amp-hour doesn't tell the story on the batteries wattage capability.
Car batteries post their peak amperage because that is important when starting a vehicle. For your purposes, you'll want to find a deep cycle battery. Significantly discharging a car battery pretty much ruins it. Deep cycle batteries are designed to be discharged to a much greater degree, down to as low as 20%, and those are rated such that they can deliver the rated current continuously.
You may be able to find a suitable battery, depending on how long you are going to run your microwave. A decent 20+ amp rated deep cycle should still be able to unload 150 amps for a minute or two. Don't run the battery full blast non-stop though, it will overheat. Shop around for batteries for boats, RVs, etc to find good batteries.