How Do Core Construction and Amp-Turns Affect Transformer Performance?

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Core construction and amp-turns significantly influence transformer performance. Using toroidal cores and ensuring the core is laminated can enhance efficiency by preventing induced currents from circulating. A rule of thumb regarding amp-turns is essential, particularly in relation to the core's permeability. Interestingly, an early attempt with a straight rod yielded the best results, highlighting the importance of experimentation. Safety precautions are crucial when working with mains electricity.
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Off and on, I have been messing with making a little transformer. My first attempt was last winter.

They never seem to really work very well.

Is there any sort of rule regarding the amp-turns needed around a core with a particular permeability?

My earliest attempts used a straight rod. More recently, I have been winding around toroidial cores. Typically, the greater of my windings has a hundred or more turns.

Strangely, one of my early "straight rod" attempts has actually provided to have best performance, thus far.
 
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I would suggest you to break-apart a real transformer and work your way backward.
For Best performance, you need to use striped iron forming a closed loop.
But always be careful when playing with the mains.
 
More recently, I have been winding around toroidial cores
The construction of the core is critical. For decent performance it has to be laminated such that no induced current is able to circulate in the plane perpendicular to the direction of the magnetic flux.
 
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