The output waveform deduced from the dot convention in a transformer [closed]

Question

In the figure above, shouldn't the output waveform of the first transformer be the only one in phase? Why I am saying that? I am saying that because the dots are up in both sides and the red terminal of the meter is connected to the up side of the secondary. Following the same reasoning, the output in the last one should be reversed with respect to the input, because even though the dots are both down, the red terminal of the meter is connected to the up part of the secondary. The solution in the book says that the output of the third transformer is the only one with 180° phase difference with the input, whereas the rest are in phase, so my question is how?

The figure is taken form the textbook titled "Circuit Analysis Theory and Practise" for its author Miller.

Answer 1

The way you have to look at it is "current into the dot on the primary is current out of the dot on the secondary" when the dots are on the same side. Think of the dot as a "current portal" that bridges the magnetic interface. So for the following cases:

• (1) Dots same side. Current into the dot primary, current out of dot secondary. Same polarity. In phase.

• (2) Dots opposite side. Current out of the dot primary, current into the dot secondary. Same polarity. In phase.

• (3) Dots opposite side. Current into the dot primary, current out of the dot secondary. Opposite polarity. Out of phase.

• (4) Dots same side. Current out of the dot primary, current into the dot secondary. Same polarity, In phase.

Note: that cases 1 and 4 are identical. Flipping both dots at the same time is like switching both the primary conductors on the transformer and switching the secondary conductors on the transformer. The effect is 180 degree phase swap on the primary and 180 degree phase swap on the secondary yielding a 360 degree phase change which degenerates into no phase change at all.

The same can be said for case 2 and 3. But the salient point is that the polarity is reversed in this comparison.