Can Charge Controller Work Without Solar Panel?
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To get a little more specific as to why your 4.5ah battery is too small - most small ups-style agm batteries will accept no more than about .25C to .3C of inrush current. You can see this limitation on most of the case printing, but sometimes it is not there.
So, 4.5 * .25 = 1.125 amps inrush. That's about a 20 watt panel.
If you haven't received your new battery yet, you could run your two 4.5ah batteries in parallel to garner a 9ah capacity, and this would be ok with the 40 watt panel, and be just about maxed out. Before paralleling them together, try to charge each one individually first to make sure each is fully charged when you put them together as a pack so they are balanced. This is assuming they are both relatively the same age as each other and have had the same amount of usage.Comment
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