DC generator losses and efficiency

The power lost inside the machine while converting mechanical power into electrical power is called loss. The ratio of electrical output to mechanical input is the efficiency.

Types of losses

Losses
 |-- Copper losses (variable)
 |     armature I_a^2 R_a | field I_f^2 R_f | brush contact loss
 |-- Iron (core) losses (constant)
 |     hysteresis | eddy current
 |-- Mechanical losses (constant)
 |     bearing friction | brush friction | windage
 +-- Stray load loss (small)

1. Copper losses: armature winding (I_a²R_a), shunt field (I_sh²R_sh or V·I_sh) and series field (I_se²R_se); brush contact loss is brush drop × I_a. Armature copper loss is about 30 to 40 % of full-load losses.
2. Iron losses: hysteresis loss (K_h B^1.6 f V) and eddy-current loss (K_e B² f² t²V) in the armature core. Constant for constant flux and speed.
3. Mechanical losses: friction at bearings and brushes, windage loss (air friction).
4. Stray load losses: due to flux distortion and other effects.

Constant losses W_c = iron + mechanical + shunt field loss. Variable losses = I_a²R_a.

Efficiency

• Mechanical efficiency: η_m = E_g I_a / (mechanical input)
• Electrical efficiency: η_e = V I_L / (E_g I_a)
• Overall efficiency: η = output / input = output / (output + losses)

Derivation (shunt generator)

Let output = V I_L, armature current I_a = I_L + I_sh ≈ I_L (I_sh is small).

Armature copper loss  = I_a^2 R_a ≈ I_L^2 R_a
Constant loss         = W_c
η = V I_L / (V I_L + I_L^2 R_a + W_c)

Dividing numerator and denominator by I_L:

η = V / (V + I_L R_a + W_c / I_L)

Condition for maximum efficiency

η is maximum when the denominator is minimum: d/dI_L (V + I_L R_a + W_c/I_L) = 0

R_a − W_c / I_L² = 0, so I_L² R_a = W_c

That is, the efficiency is maximum when the variable loss equals the constant loss. The load current at maximum efficiency is I_L = √(W_c / R_a).