The 40-watt solar panel can only add 16Ah to the battery bank, so if you're using a Lead-acid or AGM small 12v battery you'll need a 30Ah battery. But, I would recommend a 50Ah battery but for lithium-ion a 20Ah battery will be a best suit [pdf]
Through California's SGIP program, qualified low-income homeowners can get solar + battery installed by a licensed contractor for free. You own your power — with no monthly payments, no leases, no loans, no surprise costs, and no strings attached. See if you qualify now — it only takes 30 seconds. [pdf]
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