Efficiency of a D.C. Machine

For a d.c. machine, its overall efficiency is given by,

Let P_out = total output of a machine

P_in = total input of a machine

P_cu = variable losses

P_i = constant losses

then P_in = P_out + P_cu+ P_i

1.1 Condition for Maximum Efficiency

       In case of a d.c. generator the output is given by,
                     P_out = VI
                    P_cu= variable losses = I² R_a= I² R_a
                    I_a= I                              .................. neglecting shunt field current

The efficiency is maximum, when the denominator is minimum. According to maxima-minima theorem,

Thus for the maximum efficiency, the condition is,
Variable losses = Constant losses

1.2 Current at Maximum Efficiency

From the condition of maximum efficiency, the current through the d.c. machine at the time of of maximum efficiency can be obtained.

For shunt machines : The I_sh is constant and the loss VI_sh is treated to be the part of constant losses. The variable losses are I_a² R_a.

At maximum efficiency,
I_a² R_a= P_i = (Stray + shunt field losses)

This is the armature current at maximum efficiency. Neglecting I_sh, I_a = I_L is the line current of the machine.

For series machines : The current through series field is same as armature current which is same as line current. Hence the constant losses are only mechanical losses while the variable losses are the copper losses in armature as well as series field winding due to the armature current.

At maximum efficiency,
I_a² ( R_a+ R_se) = P_i = Mechanical losses