A transformer works by the alternating currents’ magnetic field of one coil of wire expanding and contracting (at 60 cycles per second, or 50 in Europe) across the coils of another winding. The heavier the flux density of this field when it crosses the other winding, the more induced voltage can occur. This density can be enhanced by proximity (winding one coil directly on top of the other) or by concentrating the magnetic field. This is where the ferrite coil comes into it. Since the iron core has a lower “reluctance” than air (in other words, the iron can concentrate the magnetic flux lines more densely), more flux lines are concentrated into crossing the coil and the transformer is more efficient.
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