In d.c machine the armature  rotates, a voltage is generated in the coils. this generated voltage is called d.c. machine emf. In case of a generator, the emf of rotation is called the generated e.m.f. and Er = Eg .  In case of a motor, the emf of rotation is known as  back e.m.f. (or counter e.m.f.) and Er = Eb . 
the expression is the same of both condition of operation.

Let  Φ  =  useful flux per pole in webers
        P  =  total number of poles
        z  = total number of conductors in the armature 
        n  = Speed of rotation of armature in revolution per second 
        A  = number of parallel path through the armature between the brushes of                       opposite polarity
      Z/A = number of armature conductor in series for each parallel path

Since the flux per pole is Φ , each  conductor cuts a flux PΦ in one revolution.

Generated voltage per conductor
           
Since n revolution are made in one second, one revolution will be made in 1/n second. therefore the time for one revolution of the armature is 1/n second.

The average value of generated voltage per conductor
                                               
Generated voltage voltage E is determined by the number of armature conductors in series in any one path between the brushes. therefore , total voltage generated 

 E = (average voltage per conductor)  x (number of conductors in series per path)

                                          E = nPΦ x Z/A

Equation is called the e.m.f. equation of d.c. machine.


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