The voltage waveforms aren't out of phase when measured across each half of the load individually based upon a single, in-phase current through each resistance.
They are when you reference them to their common point.
If you select three points at random along any conductor in which AC is flowing, then choose the middle one of those three as your reference, the voltages at the other two will appear out of phase.
Go back to the circuit I described above to the point before we added the neutral, so it's just R1 & R2 in series across the full secondary winding. It's a simple 2 wire circuit with only one current path so it can't be anything but single phase, agreed?
Now select the point between R1 & R2 as your reference and compare the voltages appearing at the top of R1 and the bottom of R2. They'll be 180 out of phase, because you're using a point between them as your reference. But it isn't a 2-phase arrangement, is it?
Because if presented with something described as 2-phase 3-wire, there has got to be a common point of reference as you can't get 2 phases out of 3 wires unless they share one. So given that we would expect any polyphase system to have phases with non-zero phase angles which would be more logical for something described as 2-phase 3-wire - sharing a point which got 2 phases with 0° and two different voltages, or sharing a point which got 2 phases with a non-zero angle and the same voltages?
You're hung up on whether you can see
voltages which are out of phase with each other compared to a common reference point. You need to look at it from the point of view of how many
currents of a different phase (relative to each other) you can get by connecting across different combinations of supply wires.
Now add your neutral between the junction of R1/R2 and the c.t. of the xfmr. If R1=R2 then no current flows in the neutral. Neither I1 nor I2 change in magnitude nor in their relationship to each other, nor does the polarity of the p.d. across each resistance change. Agreed?
No, absolutely and fundamentally disagree because of the meaning of the centre point when you call it a 2-phase 3-wire system, as per above.
So you think when connecting a third wire between the common point of R1/R2 and the xfmr c.t. in which no current flows because R1=R2 that the voltages across the loads and/or the currents through them somehow change in direction or relationship to each other? How do you think that happens when the third wire has merely connected points together which are already at the same potential so that no current actually flows in that third wire?
As soon as you introduce that common point, and designate it 2-phase, you stop measuring everything from one end point to the other and start measuring from the centre to each end point.
Why? What do you think is so special about the centre point of the transformer, taking the system as a whole? Because it's the supply wire designated as neutral? Because it's the point which is earthed? What exactly?
If the c.t. is earthed but not extended as a neutral, as with 110V CTE site tool supplies, do you still think that should be called 2 phase because there are two voltages at the ends of the winding which are 180 out of phase relative to earth?
But the bottom is not + wrt the centre it is -
In any secondary winding one end will be +ve and the other -ve at any given instant relative to some point mid way. It has to be because the current is flowing in one given direction through the whole length of the winding at any given moment. The simple 2-wire 240V secondary on that rural pole xfmr feeding one house is no exception - Select a mid point and on any given half cycle one end will be +ve with respect to it and the other -ve (with one end of the winding earthed then obviously the midpoint will also be changing its polarity with respect to earth on each half cycle, but it doesn't alter the fact that on any half cycle one end of the winding will still be +ve and the other -ve with respect to the mid point).
So with conventional current flow, there is current from the top to the centre and from the centre to the bottom.
With the top end of the winding +ve and bottom -ve, conventional current flow within the winding will actually be from bottom to top. It's the source.
So in one half the current is flowing from L1 to N and in the other it is flowing from N to L2.
If we're taking L1 as being the top and L2 the bottom, then within the xfmr secondary current will be flowing L2 to N and N to L1. However -
Different directions. Out of phase by 180°.
That isn't different directions. The current is flowing in
one direction through the whole winding from L2 to L1, passing N on the way. Yes, at the precise point of the winding at which N is connected the current must obviously be arriving from the L2 side and departing toward L1. But if you select
any point in a single conductor, whether a xfmr winding or just a piece of wire, the current must be arriving from one side and leaving via the other.
No current has changed direction or changed its phase relationship to another current. No p.d. across a load, or a part of a load has changed in voltage, polarity, or phase relationship either. Yes?
No.
You agree that with R1=R2 no current flows through the neutral wire which has been added? So how can any of the voltages and currents which existed before have changed?
From the perspective of N, one L is +, the other is -, they are not both + or both -.
From the perspective of N one current is flowing one way and the other is flowing the other way.
From the perspective of the xfmr c.t. point that was happening before we added the neutral wire.