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This feels like a simple question, but my google game is letting me down:

The below flyback transformer has two secondary coils that are meant to be connected in parallel, as per the below image, and a turns ratio of 1:1:1:0.5 (pri:sec:sec:aux). If I link the two coils in parallel, is the turns ratio still 1:1? Has it changed to 1:0.5 as though adding inductors in parallel? Is it 1:2 as though you doubled the output of a single coil? What would happen if I connected them in series instead? I'm new to designing any kind of power supply and am struggling with some of these concepts, although I'm familiar with the general idea of how a flyback converter works.

enter image description here

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What you said are two ways to reduce the leakage inductance in a flyback converter. I often use this technique.

Connected in parallel - It will be 1:1, but the current will be double.

Connected in series - It will be 1:2, you obtain double turns.

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Think of two parallel wires as a single wire of twice the area. The turns ratio is now clearly still 1:1.

Connected in series, the turns ratio is 1:2.

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Has it changed to 1:0.5 as though adding inductors in parallel?

Two identical inductors (\$L\$) that are not sharing a common magnetic field have a net inductance of \$L/2\$ when connected in parallel. Inside a transformer, the two windings do share the same magnetic field hence they produce (near) identical terminal voltages and can be paralleled to increase current drive capability just like you can with paralleling identical batteries.

What would happen if I connected them in series instead?

Just like with batteries connected in series, the voltage doubles.

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