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In case of DC motor even if DC voltage is applied, due to commutator an AC current flows through armature ,then due to this changing current, does any voltage is induced across the field winding? Thanks in advance

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short answer

Of course, Mot. always produces BEMF with rising speed so current depends on the difference due to loading and inertial acceleration demand.

e.g. Vbemf( back EMF internally generated) for no-load speed minus V for loaded speed is what the motor coil sees.

When at max no load speed if no losses, ideally current = 0 but of course there are always some excitation current, eddy current loss and commutation losses at full speed with no-load condition.

The No-load speed is often a linear constant ratio with kV/RPM until these losses rise then reduces slightly in an efficient motor. Then speed reduces up to 20% with loading at max power if matched to load.

If torque is field current regulated then the voltage is high impedance, so usually current to ground increases magnetic coupling to increase current and thus torque and thus acceleration to some regulated speed with a tach.

If it uses a PM or permanent magnets, then there is no field winding.

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Not really. Although the current through the armature windings is alternating, really rotating relative to the rotor's own frame of reference, the purpose of the commutator is to keep the flux generated by the armature at right angles to the main field, so relative to the field coils the flux isn't changing much (there's some high frequency ripple from commutation). The terminal voltage that the motor produces is the back emf generated in the armature windings, plus the resistive losses in the windings and the brushes. If it's a series field motor you can measure the terminal voltage and the voltage at the brushes to show this, the difference will be the resistive loss across the field windings. On a shunt field winding, the field current is determined solely by the resistance of the winding.

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