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Ferrite Transformer

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Wire size is based on how many wire turns will fit the core aperture, and on (I^2)R losses.
 
It's determined by the power the transformer must handle. From that you pick a core size and then wire size based upon the number of turns needed and resistive losses as Mike stated. Sometimes it's somewhat of a iterative process.
 
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If it's a high frequency switching transformer, you will also need to consider skin and proximity affects in your choice.
 
Try this excellent book
TRANSFORMER AND INDUCTOR DESIGN HANDBOOK
COLONEL WM. T. MCLYMAN
 
You can reduce skin effect by wither using litz wire or by using 3, or 4 or more thinner wires instead of one thick one.
 
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