You are correct that in each winding, the magnetic field varies in phase with the current in the windings. What you're having a problem with is the concept of flux being 'annihilated' at where the cores are joined.
It's helpful here to think about 'magnetic circuits'. Think about a single phase transformer for a moment; the core completes a loop that passes through the windings, so the field from the windings has a closed path. Now think about a three phase transformer. Look at the phase A winding. It has a certain amount of field that needs to be returned from one end of the winding to the other. You could just close it on itself, and do the same with phases B and C, and have three separate single-phase transformers, and it would get the job done, but it would be wasteful of material. Consider that the phase relationship of the currents means that, at any given moment, the fields from phases B and C added together are equal and opposite to that of phase A. It doesn't matter which phase you look at, the fields from the other two add to cancel. You see, where you were surmising that the fields annihilated eachother, what in fact happens is that they complement one another, and provide the right amount of magnetic return path. This lets you use less core material, and so economics dictates that's the way to go.
It's a bit like what happens to the currents in a Y-connected three phase load; the currents sum to zero, but it's not that they annihilate one another, it's that they form balanced return paths for one another.
A typical distribution network in Australia will look something like the below.
The "MV" section is a delta-connected "three-wire" system, so you are correct in asserting that there is no neutral wire. However, there is a path for neutral or "zero-sequence" currents to flow to ground, via the earthing 'zig-zag' transformer that is installed for this purpose. (The reasons for installing a earthing transformer deserve a separate question and answer.)
There are a few phenomena that may give rise to neutral current on a MV transmission line, but unbalanced LV loads, which cause a current to flow in the LV star-point/neutral, don't cause MV neutral current.
Why is that?
The picture above shows a delta HV, grounded-star LV system. There is a single-phase load which draws 1 unit (1 p.u.) of current from LV winding 1, with the current returning via the LV neutral.
What happens on the HV?
Each of the transformer's HV and LV windings are magnetically coupled by iron cores, so that the law of "amp-turns balance" must apply. I.e. conservation of energy applies between the pairs of HV and LV windings, HV1-LV1, HV2-LV2, and HV3-LV3.
That means that a 1 p.u. current on winding LV 1 must be balanced out by a 1 p.u. current on winding HV1. And since no current flows in LV2 or LV3, no current may flow in HV2 or HV 3 either.
By Kirchoff's Current Law, the 1 p.u. current in Winding HV1 must be sourced from HV line L1 and HV line L2. That is:
For a delta-HV, grounded-star-LV system, single-phase LV loads appear as phase-to-phase loads on the HV system.
This answers your original question: no matter how unbalanced the load on the LV side, no neutral current will flow on the HV side, so no neutral wire is needed.
This leads to the question of: "If no neutral wire is needed on the delta-connected system, why do we bother putting an earthing transformer on it?"
A couple of reasons I can think of - though I am uncertain on these, so don't quote me here...
- Without a connection to earth, the delta network would float relative to ground and might be at any arbitrary potential relative to ground. I.e. the MV system could rise up to 132,000V above ground voltage. The earthing transformer is needed to tie the MV system to ground and keep it from floating to dangerous voltages.
- 'Neutral' zero-sequence currents do flow on the MV network, i.e. from
capacitive line charging current. (Edit 2015-09-22: The charging current is balanced under normal conditions.) The earthing transformer gives these zero-sequence currents a place to go.
- The earthing transformer will be the most attractive return path for any short-circuit fault current resulting from a line-ground fault. So it's an attractive place to put a earth-fault detection relay.
Best Answer
3 phase to single phase is not possible simply, you would need a lot of extra stuff, like phase shifters (load sensitive), or motor-generator (big'n'heavy), or rectifier-inverter (the preferred route these days).
If you want to stick to wiring, then there are two options.
You can use one of the output phases with respect to the star neutral. This will load only a single phase.
Or, you can use two phase outputs with respect to each other, without using the neutral line. This will get you sqrt(3) more voltage, but no more current. This will load two of the phases equally.