Condition for Maximum Efficiency of DC Machine

We have discussed about construction & working of DC machine. In this article  condition for maximum efficiency in DC machine will be derived. 

Condition for Maximum Efficiency in DC Machine

In this DC generator is taken as reference to find out maximum efficiency .The DC generator efficiency is perpetual but varies with load. Think through a shunt generator supplying a load current IL at a terminal voltage V.

Then Generator output  =  VIL
Generator input  =  Output + Losses

=  VI+ Variable losses + Constant losses
=  VI+ I2R+ Wc
=  VIL + (IL + Ish2)Ra + Wc  ( ∵  Ia = IL + Ish)

The shunt field current Ish is generally small as compared to Iand, therefore, can be neglected.
Generator input   =  VIL + I2a Ra + Wc

Now                Efficiency   η  =  output / Input

=  VI/ (VIL + I2a Ra + Wc
=  1 / {1+[(ILRa/V)+(Wc/VIL)]}

The efficiency is always maximum when the denominator of above equation  is minimum i.e.,
                                              d/dI{( ILRa/V) + (Wc+VI2L)} =0
                                             (Ra/V) – (Wc/ VI2L) =0
                                              Ra/V = Wc/VI2L
                                               I2LRa = Wc

I.e. Variable loss = Constant loss (IL ≈ Ia)
The load current corresponding to maximum efficiency is given by;
I= √ Wc/Ra

Therefore, the efficiency of a DC generator will be maximum when the variable loss is equal to the constant loss.