Correct replacement transformer for commercial flyback converter

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I have an MPPT charge controller that has recently started behaving strangely and upon investigation I found a very hot mosfet and a KY286-TX1089NLT transformer (datasheet:http://www.kynix.com/uploadfiles/pdf9675/TX1089NLT.pdf ) that has clearly seen more heat than it should have.

The manufacturer isn't particularly helpful in providing information about this. I've been able to figure out what's going on, but not why this happened in the first place nor what is required to repair it.
From what I could determine, the particular circuit is a step-up converter that generates 15V from the original 5V. I have no idea what values are used, it is hard to decipher. The final 15V regulator is a 78l15, so I know the voltage on that end is below 35V and the current will be below 0.1A.

The input signal is a 5V 40khz square wave generated by a PIC microcontroller.
I suspect, though I'm not 100% sure yet, that this 15V is used to drive the main mosfet. This particular controller switches in the positive line (as is the proper way to do it), so a higher-than-input voltage is required. I think this is it. That's why it needs to be isolated.
My question: What would be the simplest way to replace either the whole step-up converter with another -- with the proviso that it MUST be isolated -- or to somehow replace that transformer with something off the shelf?
I do not presently have the equipment to measure inductors. I do own an oscilloscope but lack a function generator. This makes it hard to really get into it further, which is why I would also strongly consider another off-the-shelf solution to just replace the whole converter, possibly with something that's even more efficient.

If you are 100% sure about the schematic you provide I would estimate the transformer to be a certain type of 1:4 or 1:5 job, depending on the smoothing capacitor and current drain.
If then, again you are 100% sure about the schematic not doing anything fiddly with the generated 40kHz on any other point (another winding? The incoming "+5V"?) this is an exceptional candidate for replacing with an isolated DC/DC module.
A very good reason to build this like they did is that in mass production the components may be cheap compared to a 1W5 avg / 3W peak DC/DC module.
Another very good reason is needing the resulting 40kHz wave at another point, or secretly tapping off another voltage somewhere. So you do need to make absolutely sure you don't interrupt anything but the 5V to 15V conversion.
The safest thing to do is leave the 15V regulator in place and replace the remaining circuit by a DC/DC unit that makes (one way or another) at least 18V, since we do not know the ripple requirements on the output. If it's just a gate voltage (and you can absolutely verify that it is) 100mVpp on 15V won't be much of an issue and you can go for 15V DC/DC blocks. In the latter case you may need to add an extra capacitance on the output (with or without Schottky and/or small resistor) to make the module's regulation less susceptible to the MOSFET's peak gate current.
From my friend’s advice, I then try it with that at a conservative current limit shortly to see if that works okay, since I can know and control the parameters with those and what happens upon a mistake with a DC/DC module is always a bit guess work.
Another thing to be very weary of is that the PIC might turn the auxiliary DC/DC off by stopping the signal and I may have to account for that in any of the possible fixes, since then it may assume its "off-ness" at other points in control loops. So certainly consider that in any further research.
As to the "laboured" square, that's possibly explained with the parallel capacitor, though I'd still want some kind of a protection zener to prevent the MOSFET from getting into dangerous waters. If there is none of that at all, that is my first clue they might have not spent enough time on the DC/DC design.

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Hi kingblert,

As this is not a PI-specific issue, perhaps others here on the Forum can feel free to comment.

-PI-ODO