4th stage gives you equalization
This statement/ claim in the literature, is the work of marketers, not engineers. It is false, and a disservice to the consumer.
An Equalization is an intentional overcharge, requiring voltages in the mid 15s to 16v range required to max out the specific gravity of each cell, to attempt to equalize them. After many discharge/recharge cycles the SG of each individual cell can vary quite dramatically and affect battery performance negatively, and possibly very negatively.
No RV converter I know of supplies voltages over 14.8v without intentional opening of the case and manipulation of internal potentiometers or the addition of resistors or potentiometers, and thusly CANNOT to a proper Equalization.
Apparently the overpaid marketers decide that 'equalization' is a spiffier word and will sell more products, or are just completely and overwhelmingly ignorant of battery charging.
The well marketed "4th stage" is actually destratification. Every 18 hours or so, when the battery has been held at float voltage and is not bubbling, the stronger acid sinks to the bottom of the cells and is more corrosive to the bottom of the plates. Applying voltages even briefly in the 14's causes gassing and thus mixes up the electrolyte, destratifying it. Perhaps the lawyers got in there and decided they can get away using the Spiffier word because the electrolyte is equalized in strength in each individual cell after 14.xx volts mixes it up, but 14.x will not make every cell in the battery equal compared to each other.
Always beware of the marketing departments and their claims within both the advertising and the literature. They are just a bunch of people sitting around figuring out how to increase sales with little or no Knowledge of what they are selling or the technology behind it.
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You can use the parallax as a power supply, but if the battery is hooked up and fully charged when you hook it up, the parallax will continue to charge it and cause more water loss and positive plate shedding and a quicker battery demise
If you hook up a regular automatic battery charger, it might see the voltage supplied by the parallax and assume the battery is already full and not even start.
There is all sorts of scenarios where this might work and scenarios where it will not work and scenarios where you need to go through a lot of effort just to make it work, and still have issues with over or under charging.
Are you willing to alter the sequence of hooking up various charging sources, or waiting for surface charge to dissipate, or placing loads on the system to manipulate battery voltage just to get the regular battery charger do decide to kick into gear?
If you do all this, and do get the battery charger to start working properly, it might get confused and shut off if any loads are placed on the 12 volts system. It might get confused if you decide to plug in the parallax, and shut off.
There is a reason that the parallax was cheap. Go with something more modern and save yourself a bunch of headaches and effort. and possibly a set of fried batteries.
This is a perfect example where being 'frugal' can cost a lot more in the long run. If you already bought it, then it already has, it will just take a while to realize it.