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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:

  1. Does this really happen?

  2. If so, has it been put to any use? This could be used as an AC filter

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If you connect the two windings in phase, then you get an inductor. It has the same value as the inductance of either winding by itself, but it is effectively wound with copper having twice the area. As it's based on a transformer, it's a pretty lousy inductor, with a very low saturation current, and very low energy storage for the amount of iron used.

If you connect the two windings in anti-phase, then you get (for an ideal transformer) a short circuit. The inductance is zero, as there is no magnetic field for any current. No magnetic field means no change of magnetic field, means no back emf, means zero impedance. You have a short circuit, to both DC and AC.

In a real transformer, a DC signal experiences the winding resistance. An AC signal experiences the winding resistance plus the residual leakage inductance due to less than 100% coupling of the two windings.

This anti-phase connection of a transformer or coupled inductor is used in two principal places. As a transformer for detecting unbalanced currents in ground fault interrupters. And as a coupled inductor for putting inductance in the common mode of a signal or power feed pair, without filtering the differential signal present.

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The inductances cancel out and a large current will be drawn, limited by the DC resistance and the voltage applied (ideally). Not what you would typically want.

Maybe you are thinking of a common mode choke which has the coils on a common core connected so that current flowing in one coil and out the other faces little impedance, but common mode voltage applied across the pair faces a large impedance. enter image description here

Most properly made off-line switching supplies have a filter using a common mode choke on the input to reduce current from the switching noise. The inductance across the coils (in phase parallel) is usually in the millihenry range so a high impedance at hundreds of kHz switching frequencies but the impedance for normal mode current is very low.

In this case the low impedance to AC normal mode current is not necessarily desirable, but a pair of inductors that would have that common mode inductance in parallel, and not saturate with the large DC normal mode current would be prohibitively large and expensive. Check out the size of, say, two 20mH 10A inductors, for example, vs. a 10mH 10A common mode choke (kg vs grams).

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schematic

simulate this circuit – Schematic created using CircuitLab

As an example of the 2nd case, an Ayrton-Perry winding is essentially a 1:1 transformed with the windings connected such that their magnetic fields cancel. This reduces the inductance theoretically to zero. It is used in applications like wire-wound resistors where a winding is required, but inductance is undesirable. In this case it was never intended to be used as a transformer, but the principle is the same.

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