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In Transformers two coils are used for mutual induction (check transformers theory) and so that the magnetic flux generated in primary coil gets transferred to secondary coil we need inductance which is only generated in case of an AC current flowing through a coil (check faraday's law).

DC current has no frequency so no inductance(2*pi*f*L) so no induced emf !

Comment

Inductance is not 'generated'; it is a natural quantity just like resistance or capacitance and is quite independent of whether the supply is d.c. or a.c. (although it can vary with the quantity of current passing through the circuit). And the equation 2 pi f L is that for inductive reactance, not inductance!

Transformers are a.c. machines, because an varying current is required in order to create a varying magnetic flux around the core of the transformer. When a varying flux links the secondary winding, it induces a voltage into that winding through the process of mutual induction (not 'inductance'!). If d.c. is applied, the resulting flux will be constant and mutual induction will not occur. Furthermore, by applying a d.c. supply to the transformer, you risk the chance of burning out the primary winding, as its resistance is very low compared with its inductive reactance (opposition to a.c.), and the resulting current can be much larger than it's designed to carry.

Transformers CAN work with a.c. providing the current is continously interrupted, which is the principle of operation behind a car's ignition coil.

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