Power Equation Of A D.C. Motor

The voltage equation of a d.c. motor is given by,

V = E_b + I_a R_a

Multiplying both sides of the above equation by I_a we get,

This equation is called power equation of a d.c. motor.

VI_a = Net electrical power input to the armature measured in watts.
I_a²R_a = Power loss due the resistance of the armature called armature copper loss.
So difference between VI_a and I_a²R_a i.e. input – losses gives the output of the armature.
So E_b Ia is called electrical equivalent of gross mechanical power developed by the armature. This is denoted as P_m.

.^..       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,

P_m = Gross mechanical power developed = E_b Ia
                = VIa –  I_a²Ra
For maximum P_m, dP_m/dI_a = 0
.^..                         0 = V – 2I_aRa
.^..                         I_a = V/2Ra     i.e.        I_aRa = V/2

Substituting in voltage equation,

V = E_b + I_aRa = E_b + (V/2)
.^..                        E_b  = 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 %.