# How to find the optimal section of a magnetic core of a transformer and of the windings?

I'm designing, as a hobby project, a simple step down transformer with a power rating of 300VA. These are the requirements of the design:

• Rated nominal power: S_n = 300 VA
• Nominal frequency: 50Hz
• Nominal voltages: primary E=230V, secondary 50V.
• Iron core made with E-I iron sheets.

I have no requirements on the size and I assume a conservative 1 T peak induction value. Please note that voltages are in RMS while induction is a peak value.

The following is the process I am following for the design:

An empirical formula for determining the core area: $$A = 1.37 \sqrt{S_n} = 23.72 cm^2$$

From first principles, the number of primary turns should be roughly equal to:

$$N_1 =\frac{\sqrt{2}E}{2 \pi f B_{peak} A} = 436$$

and of course the number of secondary turns: $$N_2 = N_1 \frac{50}{220}=95$$

As for the wire, I am designing the wire section by allowing a maximum current density at the nominal current. The primary nominal current is given by: $$I_1 = \frac{300}{230} = 1.30 A$$ by allowing a maximum current density of $$\delta = \frac{I_1}{S_w} = 2.5\frac{A}{mm^2}$$ which leads to a section for the primari winding of about 0.52 mm^2. Then I do the same for the secondary.

Now my questions. I'm pretty confident the design process is fine for a small air cooled transformer, however I am not that satisfied with the empirical formulas used for the core section and the winding section. I took these formulas from a manual on how to build transformers, however it is not explained how the formulas are derived and what assumptions have been made.

1) Could you explain me what process is behind the formula, if you are familiar with it, for determining the core section and what assumptions have been made? If you are not, but you know a better approach, could you please explain it to me "from first principles"?

2) The same goes for the process behind finding the section of the wires for the winding. It seems that a current density of 2.5A/mm^2 is a common maximum limit for small transformers but why is this? Where does this rule of thumb come from? Same as above could you please derive a "closed" formula for finding the winding wires'section? I know how to find the section for distribution wires, but this process is not applicable to transformers.

I could just plug in the formulas and let them do their job, but I'd rather understand how they work, what limitations they imply and how to play with them. If you can suggest other approaches please do. My main problem with these formulas is just that I see no reasoning behind them but just "blind trust" and while that is fine sometimes, I would still get a better grasp of the problem they are trying to solve.

PS: for the windings, I am using common enamelled copper wire.