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 I_L at a terminal voltage V.

Read Here : Alternator EMF Eqaution

Then Generator output  =  VI_L

Generator input  =  Output + Losses

=  VI_L + Variable losses + Constant losses

=  VI_L + I2_a R_a + W_c

=  VI_L + (I_L + I_sh²)R_a + W_c  ( ∵  I_a = I_L + I_sh)

The shunt field current I_sh is generally small as compared to I_L and, therefore, can be neglected.

Generator input   =  VI_L + I²_a R_a + W_c

Now                Efficiency   η  =  output / Input

=  VI_L / (VI_L + I²_a R_a + W_c

=  1 / {1+[(I_LR_a/V)+(W_c/VI_L)]}

The efficiency is always maximum when the denominator of above equation  is minimum i.e.,

                                              d/dI_L {( I_LR_a/V) + (W_c+VI²_L)} =0

                                                        Or

                                             (R_a/V) – (W_c/ VI²_L) =0

                                                        Or

                                              Ra/V = Wc/VI²_L

                                                       Or

                                               I²_LR_a = W_c

I.e. Variable loss = Constant loss (I_L ≈ I_a)

The load current corresponding to maximum efficiency is given by;

                                             

I_L = √ W_c/R_a

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