I am considering the case of an isolating transformer because both the coils are identical. If we connect both of these in parallel, they will act as mutually linked inductors if they are connected in phase. But what if they are connected out of phase(positive of one coil to the negative of the other), such that the magnetic field generated by one coil is cancelled out by the other?
According to the equations, if there is no magnetic field, there will be no current passing through the part. Which means to an AC signal this arrangement will offer infinite resistance, but for a DC signal it will act as a simple resistor.
Perhaps it can also be thought of as a transformer. A signal is applied to the primary, the same signal appears at the secondary. This signal is then superimposed on to the primary with a phase difference of pi, hence cancelling it out.
My questions are:
Does this really happen?
If so, has it been put to any use? This could be used as an AC filter