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Does the current double along with the voltage?

For example, if a center tapped secondary is listed as 100-0-100 V @100 mA, does it have equal current or twice the current of one expressed as 100 V 50 mA -0- 100 V 50 mA.

This second example is how the values are shown on an engineering drawing for the transformer in a popular brand of oscilloscope. It shows a center tapped secondary with 383 V on each leg and .180 listed with each side.

Would this be a 180 mA 760 V center tapped winding or 360 mA?

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They are speaking of each pole separately. The L1 pole is 100mA. The L2 pole is 100mA.

So you can either use it as 200V@100mA (ignoring center tap) or get 100mA from each pole.

You cannot get 200mA from one pole and not use the other pole.

They do make transformers that are designed to work that way, where there are actually two independent 100V secondary windings, and you can either parallel them for 200mA@100V, or connect them nose to tail for split-phase 200V@100mA with a center tap. Here is one.

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It is important to remember when thermal loading a wire or winding, 200% load 50% of the time is not the same as 100% load... because of how Ohm's Law interacts with Watt's law, 200% overload is actually 400% thermal load! Thus 200% overload half the time is actually 200% thermal load.

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