Is there another method than the Area product method to size the core of a transformer
Is there another method than the Area product method to size the core
  of a transformer


These days, for ferrites, 50 kHz is quite low and (if designing a ferrite core transformer), you would use the various manufacturer's guides for choosing the best ferrite material for the application. For instance, 3F3 from Ferroxcube is good for 500 kHz. So, choose your ferrite supplier (there are many) and focus in on the operating frequency you are likely to use in order to pick the material.

But is that (3F3) material available in the core set that most appeals to you?


This comes with experience. If you are just designing a power transformer for 50/60 Hz you don't really get the option of what to use i.e. the laminates will likely be silicon steel and be available in a few handy sizes. The path is defined but, for ferrites there are many more options and 3F3 is just one option. Regards the core sets, there are plenty and, I would encourage anyone interested, to check out the various manufacturer's websites for the core set that most likely covers the options.

That doesn't really get you very far in sizing the core; you might have chosen a ferrite material that is good for the operating frequency but you still have little idea about how much power throughput it might support. You know that the saturation level shouldn't be too bad and, for ferrites, 400 mT is usually too much and so you might set a limit of 200 mT. 

200 mT peak flux density has nothing to do with peak power handling capability in most regular transformer applications such as forward converters (mentioned in the question text); the magnetic field is only produced by the primary winding and the magnetization current. That current is the current that flows when the secondaries are unloaded. That current continues to flow (barely altered) when the secondaries carry full load current. The magnetic flux in the core remains (largely) that same whether on no load or full load.

This fact allows you to design the primary winding. At the chosen operating frequency, you need to have sufficient primary inductance so that the peak ampere-turns (that produce magnetization flux) are not so great so as to cause too much saturation: -

$$B_{MAX} = \mu_e\cdot H_{MAX}$$

Where \$H_{MAX}\$ is the peak ampere turns divided by the core length (called \$\ell_e\$ in most data sheets for cores). So now you are juggling a few open things and my advice is to step in and take a punt on a particular core set that fits within your physical space requirements. By taking that punt you get \$\ell_e\$ and \$\mu_e\$ delivered to you and you can "trial" a primary inductance value and modify that by altering the number of turns. The core set data will also provide you with another parameter, \$A_L\$ and that allows you to calculate inductance i.e. \$L = N^2\cdot A_L\$ (or inductance is proportional to the square of the turns, N).

From this, and the operating frequency and the voltage supply, you can calculate ampere turns and divide that by \$\ell_e\$ to get the maximum H-field value. Multiply that by \$\mu_e\$ to get \$B_{MAX}\$. If the B-field is too much then more turns might be the answer.

More turns might seem counter-intuitive given that ampere turns (or magneto motive force) will appear to increase but consider that inductance is proportional to turns squared and you will see that if you double the turns, you will quadruple the inductance and quarter the amps hence doubling the turns reduces the H-field by 2.

You may find that the core set you took a punt on just isn't big enough or you may find that it's easily big enough and you can probably choose a smaller core set. You may also, at this point, find that adding a small gap between the cores can be seriously beneficial in some applications.

By now, you should realize that for ferrites, the formulaic route is not recommended by me; I always go down the same path of going through the calculations (as above) and trialing a ferrite core set that might fit the bill. I do it every time because there are more things to consider than are covered in the formula at the top of the question.

Power throughput - so you've chosen a core set that appears to fit the bill in terms of saturation requirements and you've got the number of turns for the primary based on avoiding that saturation limit. But, is there enough "space" to wind those turns with a thick enough copper wire so that \$I^2\cdot R\$ losses on full load are avoided? This is the next consideration.

In the original question it assumed a current density of 420 amps per square cm and you might use a similar figure or, you just work out the resistance of the copper wire and try to keep power losses low enough not to reduce the efficiency of the transformer below (say) 90%. You may find, as stated earlier that you can't fit thick enough copper wire in the available space to get the number of turns that give you the right inductance. You clearly need to leave enough space for the secondary winding so don't forget that; you could assume that 50% of the available space is used for primary and 50% for the secondary but, again, this is down to the application.

For instance, if designing a step up transformer to produce a few kV from a DC supply of 24 volts (forward converter recommended or H bridge), the amount of insulation needed between layers of the secondary winding will mean that virtually 90% of the free space to put windings needs to be allocated to the secondary. Horses for courses.

I could go on (but won't) other than to say that the formulaic route doesn't impart much in the way of knowledge and doesn't cover all the options when designing a modern ferrite based forward converter. I've purposely not mentioned flyback transformers because that opens up the reasons for gapping a core set and is pretty much beyond the scope of the question.