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Why is it that in full wave rectifiers do you divide the secondary coil by half as stated in this book:

Page from book

(Source)

What concept am I missing?

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what concept am I missing?

If you use a rectifier bridge (four rectifiers), you don't need a center tap. However, if you have only two rectifiers, you need a center tap otherwise there is no path for current.

Consider the following diagram

enter image description here

If there were no center tap, and the load were in series with the diodes, there would be (essentially) no current since one or the other diode is reverse biased during the entire cycle.

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