I don't want to argue - that seems your intent.
I know you want a magic formula - you have made that
clear. There is no single formula for you. Those who
have replied have made that clear.
We've tried to show you some of the factors that are
involved. You insist on ignoring them. So, we can't
help you, no matter how hard we try. Sorry.
"A formula yes, but no it's not magic. It simply requires considering
all the significant variables which I'd hoped others would assist
with, but obviously nobody else wants to do more than argue instead of
putting thought into what such an equation would look like. If you
say "it depends", then that should be in an equation. If you say some
other thing depends too, then that too can be put into the equation."
"There is an equation that could be made. How accurate the answer from
it would depend on how complete it was. I came here looking to make
as complete an equation as possible but it seems everyone else is
apathetic about the idea and only wants to tell me I don't "need" to
know or that it can't be done."
"It's not magic. A caveman looking at a bic lighter would think that
is magic but is it? Your great great grandfather, if he were alive
today, might think a computer is magic, but is it? Just because
someone doesn't know something it doesn't become magic. It would just
require someone bothering to do so, which I was attempting but it
seems I'm going it alone because others think it's too hard or they're
too lazy or whatever the reason. That's fine, nobody is compelled to
do anything but if they had no assistance (some replying did!) then as
always they should've just moved on to the next thread."
I find it curious that you reply only to those who either disagree with you
or do not give you a formula that automagically takes into account every
design parameter and spits out the answers you demand. And who is the lazy
one? You are asking for free advice, and you are getting a lot of bang for
your buck. Contract a design engineer at $100/hour and they will not be
The factors involved in your design are numerous:
Input voltage range
Physical size constraints
Costs (NRE, materials, labor)
Environmental conditions (temp, humidity, altitude, etc.)
Safety (UL, NEMA, Medical, etc.)
Overcurrent and short circuit protection
Reverse connection protection
Charging current/voltage/time/temperature profile
You have constrained the parameters by saying that the charger circuit has
been designed and you are not willing or able to change that. So you are
only looking for a cheap and dirty AC to DC power supply that will meet
your needs. Your charger circuit must have specifications for what it can
accept as raw DC input, and what sort of current it will draw under all
conditions. IOW, you need to model this circuit with a "formula" or SPICE
model. Then you must decide if you want to use an off-the-shelf transformer
or design one specifically to meet your needs. An OTS solution will be
initially cheaper, but you will sacrifice something, like efficiency or
physical size that will not be optimized. You will need to choose one that
meets (but probably exceeds) your specifications.
There are many simplified formulas that have been given to you, that will
give results close enough to choose from several OTS transformers that
should meet your specs. You can also model the transformer pretty well in
SPICE. But you will, sooner or later, need to build a prototype and test
the hell out of it. After a few hours of testing, you can pretty well
characterize the transformer, and then fine tune your selection. Or you can
use the results to design a custom transformer that exactly meets your
needs. But tranny design gets rather complex when you are pushing the
limits of efficiency, cost, size, regulation, and other factors.
You have all the information you need to design, build, and simulate your
DC front-end circuit. We do not have all the information we need to model
your charger load, and even if we did, it is up to you to actually design
the thing. As I said before, SPICE is essentially a mathematical formula
that you can modify with all the parameters you want, and run step and
sweep analyses to make sure your initial design is OK. Then you can tweak
it ad nauseum to get it as close to perfect as it will ever be.