## 6 November 2017

### Transformer Efficiency and Condition for Maximum Efficiency

• The efficiency of the transformer at any load power factor is defined as the ratio of the output power and input power.

Efficiency
= ( Output power / Input power ) × 100%
= ( Output power / Output power + Losses ) × 100%

= ( Output power / Output power + Iron losses + Copper losses ) × 100%
= kVA Cos F / ( kVA Cos F + Wi + Wcu ) × 100%
Where

Cos F = Load power factor
Wi = Iron losses
Wcu = Full load copper losses
kVA = Rating of the transformer
Why the efficiency of the transformer is much higher than any other rotating machines?
• There are no moving or rotating parts in the transformer i.e. the mechanical losses is zero so the transformer efficiency is much more higher than any other rotating devices.

How the efficiency of transformer varies with load current?

Condition For Maximum Efficiency
• The transformer iron losses are practically constant at constant voltage, but copper losses varies as square of load current therefore the efficiency is affected by the factor load current.
• Let us find out the condition for maximum efficiency.

Efficiency
= ( Output power / Input power ) × 100 %
h   = ( Input power – Total losses / Input power ) × 100 %
h  =  [ ( V1I1 Cos F – Wcu – Wi ) / V1I1 Cos F ] × 100 %
h  =  [ ( V1I1 Cos F – I12R01 – Wi ) / V1I1 Cos F ] × 100 %
= 1 – ( I12R01 / V1I1 Cos F ) –  ( Wi / V1I1 Cos F )
=  1 – ( I1R01 / V1 Cos F ) –  ( Wi / V1I1 Cos F )
As the load current is variable parameter, the transformer efficiency becomes maximum when
dh / dI1 = 0
= 0 – ( R01 / V1 Cos F ) + ( Wi / V1I12 Cos F )
( R01 / V1 Cos F ) = ( Wi / V1I12 Cos F )
I12 R01 = Wi
Wcu  = Wi
The transformer efficiency becomes maximum when total copper losses are equal to iron losses.
At what load current efficiency becomes maximum?
For maximum efficiency Wcu  = Wi
I12 R01 = Wi
I22 R02 = Wi
I2 = √ ( Wi / R02 )
I2 = [ I2full / I2full ] √ ( Wi / R02 )
= I2full √ ( Wi / I2full2 R02 )
= I2full √ ( Wi / Wcu )
= I2full √ ( Iron losses / Full load copper losses )
At what kVA rating the efficiency of transformer becomes maximum?
I2 = I2full √ ( Wi / Wcu )
V2I2   = V2 I2full √ ( Wi / Wcu )
V2I2 / 1000   = [ V2 I2full / 1000 ] √ ( Wi / Wcu )

kVA = kVAfull √ ( Wi / Wcu )

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