90 degrees phase shifted magnetic fields

Question

If we have a closed magnetic circuit with two primary coils and one secondary coil wound on it. The primary coils are wound such that unlike poles are adjacent. Both primary coils are using rectified AC and the current in one coil is phase shifted by 90 degrees to the other coil.

This would result in an interplay of the magnetic fields where one is increasing to its peak value while at the same time the other is decreasing to zero. What would be the effect of the magnetic fields on a secondary coil? I'm not sure, however I'll take a stab at it.

If the south pole on primary1 is losing strength lenz law says the secondary will establish a north pole to arrest the waning south pole. At the same time the north pole on primary2 is gaining strength lenz law says the secondary will establish a north pole in opposition to the growing magnetic north pole. Obviously two north poles is not possible, so what is the effect on the secondary?

Answer 1

The flux that is generated by a coil is

$$\Phi(t) = L\cdot i(t)$$

With L the self-inductance. If you now state that the transformer is ideal, then you also say that the flux through all coils is the same. You find that

$$\Phi(t) = L_{p1}\cdot i_{p1}(t) = L_{p2}\cdot i_{p2}(t)$$

This equation also implies that you cannot have a 90 degrees phase shift between the currents through the primary coils. You need

$$\frac{i_{p1}(t)}{i_{p2}(t)} = \frac{L_{p2}}{L_{p1}} = constant$$

You can basically treat it like someone saying:
I have two current sources in series, what's the total current?

To which the obvious answer is: It's impossible... Unless perhaps the currents are equal.