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Power Equation of a D.C. Motor

The voltage equation of a d.c. motor is given by,
                V = Eb + Ia Ra
       Multiplying both sides of the above equation by Ia we get,
       This equation is called power equation of a d.c. motor.
       VIa = Net electrical power input to the armature measured in watts.
       Ia2Ra = Power loss due the resistance of the armature called armature copper loss.
       So difference between VIa and Ia2Ra i.e. input - losses gives the output of the armature.
       So Eb Ia is called electrical equivalent of gross mechanical power developed by the armature. This is denoted as Pm.
...       Power input to the armature - Armature copper loss = Gross mechanical power
                                                                                            developed in the armature.
1.1 Condition for Maximum Power
       For a motor from power equation it is known that,
       Pm = Gross mechanical power developed = Eb Ia
                = VIa -  Ia2Ra
       For maximum Pm, dPm/dIa = 0
...                         0 = V - 2IaRa
...                         Ia = V/2Ra     i.e.        IaRa = V/2
       Substituting in voltage equation,
                           V = Eb + IaRa = Eb + (V/2)
...                        Eb  = V/2             .................Condition for maximum power
Key Point : This is practically impossible to achieve as for this, current required is much more than its normal rated value. Large heat will be produced and efficiency of motor will be less than 50 %.
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