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I never knew it was harmful to the battery to do rapid charging.
Well,
everything is harmful to a battery, eventually.
It's a matter of tradeoffs.
The last numbers I saw on the fast vs trickle thing were 100 charge cycles if done fast, 500 if done slow. Now ignore for a sec all the other details, like the exact battery design, how fast "fast" is, etc. Consider simply that charging fast might save you hundreds of hours a year and the cost is, say, $20 over that time for the extra cells you have to buy. It'd be worth fast charging in that case.
I don't know what the numbers on 9Vs are. Maybe it's something horrific like 10 charge cycles if done "fast" (again, not bothering to put an exact value on that term) instead of hundreds. In that case, you might be content to let them charge overnight.
Just one example of another thing that will hurt NiMHs is trickle charging too long. This is why a smart charge controller can still be useful even when trickle charging.
The last numbers I saw on this, though, was that trickle charging for up to 30 days was OK. Probably that's optimistic and probably it doesn't apply to all NiMH battery designs, but what if you decide you'll never let a battery cook for more than, say, two days when it really only needs half a day to charge? Well then, you might decide we don't care about intelligent charge termination, as I did for the
PIMETA v2.
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Which trickle charge design do you recommend then?
The PIMETA v2's trickle charger is documented
here.
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what battery holds the most charge?
The mAh numbers you find are fairly reliable. You might find some that stretch the truth by maybe 10%, but that's nothing in a field where, as you've found, values already vary by over 100%. Your 350 mAh number for a LiIon "9V" is low, actually: I have one here that claims 500 mAh, and testing backs that up.
There are lots of other factors to consider when choosing a battery.
In the case of LiIon, they're dangerous if charged improperly, and charging properly isn't as easy as with nickel chemistries, so many DIYers -- including myself -- don't DIY chargers for them. I only have one headphone amp that runs on a 9V LiIon, and it charges outside the amp on a professionally designed charger. That way, even if the battery explodes, it's less likely to cause damage than if it's inside the amp.
Another big factor is terminal voltage: no "9V" battery is actually 9.0000 volts, not even alkalines. And, all of them drop in terminal voltage nonlinearly over time as they're depleted, they don't stay at one particular voltage the entire time. That can affect useful life just as much as mAh, especially at low voltages like these. I have an article on that topic
here.
Another factor to consider is reliability. Some cell designs will put up with a lot more abuse than others. I have little to say on this topic because I don't go through nearly enough NiMHs in a year to gain any kind of long-term sense of reliability. In general, lower mAh numbers are likely to correlate better with longevity, because it means they're not pushing the design limits as hard. That's not to say that high-mAh long-life batteries don't exist, but I'd expect them to be pricey.
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Is it possible that I can run the amp and recharge using the same plug or will I need a separate one?
Yes. Again I refer you to the PIMETA v2's power supply design. There's an entire section of the docs on it,
here, much of which goes into battery issues.