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I have a solar setup for some computing equipment that is working pretty well for me (most of the time), however it doesn't have quite enough juice to make it through the night on very overcast winter days.

While I could throw more panels and batteries at the problem it would not be economical as I would not be able to utilise that energy >90% of the time.

I have a large unused UPS transformer from a UPS that used to power this equipment and I was wondering if there is anything preventing the possibility of stepping down the 240V AC wall power to around 40V-60V DC using it, and then feeding the charger controller from this. I would of course connect this in such a way that it cannot operate at the same time as the solar input or back feed into it.

I was also wondering then if a rectification step would still be required for the DC conversion, or would the charge controller be able to handle that step? I would think a charge controller anyway rectifies so probably it is unnecessary but perhaps I am missing something?

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In principle you can do this.
In practice there may be some "issues".

You will probably need to rectify the AC and certainly should do so even if the controller handles it as it will be a non optimum input. An MPPT controller MAY have input rectification but this is a non essential part of the basic design and if present would probably be for protection against reversed polarity input. Input should be reasonably well smoother DC - not just rectified AC. High ripple AC may well interact with the MPPT algorithm which would be quite likely to "chase its tail" trying to keep up with the AC variations - which are MUCH faster than the usual input variations. Even if the sensing input is filtered enough to not track the AC ripple it could still cause problems as it may reach different conclusions about the RMS value of the input from what the inverter proper sees.

Best would be to smooth the rectified AC - the UPS may have some capacitors that you could use for this.

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