I'm looking for a calculation which can take relative magnetic permeability, toroidal core dimensions and if relevant windings + frequency to give the VA expected.
The purpose is to either build a custom toroidal core or to use a premade core to build a transformer of a desired VA rating with a custom turn ratio.
The only relevant quantities are size of the toroid, the saturation field, the frequency, and permitted temperature of operation. Windings, permaeability, don't figure.
The cross section of the iron, the max field, and the operating frequency gives you turns per volt. The volume of copper, and the temperature you are happy to run it at, give you the current loading. Volts x current = VA.
The easiest way to get to 80% of the answer real fast is to weigh the toroid, double it to allow for the weight of copper, then interpolate between weight/VA entries in several online transformer catalogues for transformers of your type around your target VA.
You will find that 'instrument' transformers tend to have lower VA per weight than 'power' transformers, and microwave oven transformers have an exceptionally high VA to weight ratio. This is because the latter are built down to a price, up to a power, and run with fan cooling, so they push the dissipation, magnetising current and field up to levels that could not be tolerated for 'ordinary' transformers.
This choice you have to make on field and temperature strongly affects the max VA answer you get. So while it's possible to do accurate electrical calculations on the back of an envelope, thermal calculations are rather more difficult. That's why I recommend the catalogue interpolation approach!
There are more caveats of course. The permeability does matter if you are trying to design down to magnetising current specification as well as power, which is a function of max field and permeability. The permeability gives you constant mag_current*VA product, the max field you choose sets their ratio.
Power handling capacity of magnetic cores is affected by the required regulation, copper space factor and temperature rise limit. The permissible temperature rise depends upon current density,flux density, frequency, class of insulation, duty cycle and the heat dissipation capability of the transformer. That is why power rating cannot be specified precisely.The power handling capacity of the magnetic(Toroidal) cores may be determined
where:
VA = Power handling capacity (VA)
J = Current density (A/ mm2)
Bm = Maximum induction ( Tesla)
f = Frequency ( Hz)
ID = Inside diameter (mm)
Ac min = Effective cross-sectional area (mm2)
