Transformer losses; maximum efficiency

Types of losses

1. Core (iron) losses occur in the core and are practically constant from no-load to full load (since V1 and f are constant).

• Hysteresis loss: Pₕ = ηₕ Bm^1.6 f V, reduced by using silicon steel / CRGO.
• Eddy current loss: Pₑ = kₑ Bm² f² t² V, reduced by thin insulated laminations.

2. Copper (I²R) losses occur in the primary and secondary windings and vary as the square of the load current: Pcu = I1² R1 + I2² R2 = I1² R01.

3. Stray and dielectric losses (small): leakage-flux losses in the tank and structure, and insulation losses (insignificant at normal voltages).

Since a transformer has no moving part, there are no friction or windage losses, so efficiency is very high (95–99 %).

Condition and expression for maximum efficiency

Let x be the fraction of full-load kVA, S the rating (VA), cos φ the load power factor, Pi the iron loss and Pcu the full-load copper loss.

η = x S cos φ / ( x S cos φ + Pi + x² Pcu )

Efficiency is maximum when the denominator-to-numerator ratio is minimum. Divide numerator and denominator by x:

η = S cos φ / ( S cos φ + Pi/x + x Pcu )

η is maximum when (Pi/x + x Pcu) is minimum:

d/dx (Pi/x + x Pcu) = −Pi/x² + Pcu = 0
 =>  Pi = x² Pcu

Condition for maximum efficiency: iron loss = copper loss at that load.

Load at maximum efficiency: x = √(Pi/Pcu), i.e. output kVA = S √(Pi/Pcu).

Maximum efficiency:

η_max = x S cos φ / ( x S cos φ + 2 Pi )

since the copper loss equals the iron loss at that point.

Example: S = 50 kVA, Pi = 400 W, Pcu = 900 W, cos φ = 1: x = √(400/900) = 0.667; η_max = 33 333/(33 333 + 800) = 97.7 %.