As I understand
You need to have an inverter to convert 12v DC to 110v AC in order to
charge HyperJuice battery
is that the battery does NOT work with a 12V adapter. The battery comes with a proper charger. That text says you need an extra module ('inverter') which converts 12V back to 110V... Which can then feed the actual charger included with the original Hyperjuice battery.
That proper charger works from 90V to 240V or so, and is compatible with 'the world', as long as you can plug it in :) The 'world' has many different sockets.
I managed to obtain 6 x 18650 Batteries from an old laptop.
This is your first problem. Those old batteries are probably tired and will struggle to supply the required current. Individual cells may have different internal resistances and capacities, so balancing is advised.
Solution A - Use only a 1S3P (or more in parallel) Pack instead and
use a TP4056-based USB 5V Charger.
Bad idea. The battery will charge very slowly, and the booster will waste power. The pack and wiring will have to handle 14A+ discharge current.
Solution B (BMS and '12.6V' charger)
If the BMS includes balancing then it should work, provided the '12.6V' charger is designed for 3.7V Lithium cells. Without balancing, some cells could reach peak voltage before others and then the BMS would terminate the charge early, resulting in a partially charged, out of balance battery.
The BMS won't cut on discharge until at least one cell has dropped to a dangerously low voltage. After a few cycles the cells will start dying. To protect the battery you should install an alarm or cutoff that doesn't let any cell go below 3.2V.
Solution C - Individually Protect Each CELL with a 1S BMS, AND use a
3S BMS
Overkill, but perhaps (depending on the balancers) not enough! Many balancers work on the principle of bypassing charging current when the cell reaches peak voltage (4.2V). The problem with this method is that if the balancer can't bypass all the current then the cell will continue to be overcharged (until the protection circuit kicks in).
Solution D - The Proper Balanced Method , which would need a use a of
bulky balance charger
Again, how well this will work depends on the particular charger. Some contain 3 isolated circuits that charge each cell individually. This is the most reliable method of balance charging, but the control panel has to communicate with all 3 chargers while maintaining isolation, so it is mostly used in simple low-end chargers that may be unreliable.
More sophisticated balancing chargers have an LCD screen and are fully programmable. Their balancers usually work throughout the charge cycle so the cells start to become balanced before reaching peak voltage, but most of them have relatively weak balancers. The main advantage is that the LCD screen shows you the cell voltages, so you can cut the charge rate down to help balance the pack if necessary. The display also shows how much charge is put in, so you can gauge the health of the pack.
A good balance charger may be bulkier, but will be more powerful and gives you much more control and flexibility. Many can also do Nicad/NiMH, LiFPO4 and Lead acid batteries. One charger may be all you need to charge many different devices.
Best Answer
I am not sure the question belongs to the forum as its more towards the consumer electronics side. Anyways TA means Terminal-Adapter. And yes there is a big difference between charging through a wall adapter and through the USB.
In order to charge fast, we need to provide the charge at a faster rate, aka more current to be provided at the source.
A USB does not have a large current output and thus charges the phone slowly, whereas a wall adapter can provide large amounts of current which charges the batteries faster.