### Efficiency of a D.C. Machine

For a d.c. machine, its overall efficiency is given by,
Let             Pout = total output of a machine
Pin  = total input of a machine
Pcu  = variable losses
P = constant losses
then           Pin = Pout + Pcu  + Pi

1.1 Condition for Maximum Efficiency
In case of a d.c. generator the output is given by,
Pout = VI
Pcu  = variable losses = I2 Ra = I2 Ra
I= 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 Ish is constant and the loss VIsh is treated to be the part of constant losses. The variable losses are Ia2 Ra.
At maximum efficiency,
Ia2 Ra = P = (Stray + shunt field losses)
This is the armature current at maximum efficiency. Neglecting Ish, Ia = IL 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,
Ia2 ( Ra + Rse) = P = Mechanical losses

See solved examples on eff and power loss of dc motor

Related Articles :
Torque Equation of a D.c. Motor
Characteristics of D.C. Shunt Motor
Applied Voltage Control
Four Point Starter