I would like to design a circuit that charges 3 Li-Ion batteries in parallel with a 1S charger, then switches the configuration to put them in series to discharge as one 11.1V battery pack. This is quite similar to the design in this answer to a previous post: https://electronics.stackexchange.com/a/219619/152168 (except I'm using 3 cells to power a Jetson TK1, and my switches are a series of relays).
This will be used as a backup battery for a 12V main supply that may occasionally cut out. This is one of my design constraints, I cannot use a different source as my main power. Aside from that, if you have any suggestions for design changes, I'd be happy to hear them.
My first question is whether or not the batteries will fall out of balance when they are discharged is series. The aforementioned old post made no mention to whether or not balance was an issue when switching between configurations. If they do become unbalanced, then I'll have to figure out a separate balance circuit (which I'd prefer to avoid). The cells are all from the same shipment, which I've been told will help them stay balanced. Also, the batteries will never be relied on for longer than 30 minutes at a time, so I will not likely be running them close to low charge.
On another note, I've found that when switching power sources with relays, the Jetson momentarily loses power and shuts down. I plan to avoid that by using some sort of capacitor to temporarily power the Jetson in that moment while the relays are switching contacts. I have very little practical experience with capacitors, so I'm not sure how I would integrate this into the circuit. Could you explain (or show) where the capacitor would go in relation to the batteries, relays, and load? You can refer to the schematic below for an idea of what I want to do. Sorry that it's so messy, I'm new to drawing circuits.
I also don't know what capacitance I will need to keep it running during the switch. I understand the C=Idt/V formula, but I'm not sure how to tell how long is needed. If you wanted to do the calculations yourself, the accepted voltage drop is 2V and the current is about 1A, so C=0.5*dt. How do I determine dt given a particular relay?
If there is a better/easier alternative to relays or any other part of my design, please let me know.