To design a half bridge transformer, first design a virtual buck converter with your vout. Decide what is your max duty cycle (say 0.7).
So you then design a buck with whatever vin gives your vout with your chosen D.
Then think about that “virtual” vin value.
That virtual vin is what you must get out of your transformer when it has your minimum actual vin at its primary side.
(but remember that with ½ bridge your actual vin is half of the actual vin, so to speak…..because of the divider caps)
So you then do the transformer equation vpri/vsec = Npri/Nsec
….thats pretty much it.
If you are wondering about how many turns exactly to use, and what exact core etc, and you want an exact formula which spits out everything for you in one go then this is the wrong field for you……you have to do it iteratively and you converge to the solution.
But this is good as it enables you to use your artistic side.
You can use any core or wire you like as long as………………..
Solution is not too big
Wire loss not too high
Core loss not too high
Coupling not too poor.
Skin effect is accounted for
You can physically terminate the wire to the former pins.
Etc etc etc….
Here is my smps maths course to give you some of the equations for magnetics
https://massey276.wixsite.com/maths
and here is my electronics degree advisor
https://massey276.wixsite.com/electronicsdegree