The efficiency of a Transformer

In our other article All-day Efficiency of Transformer, we have learned what is all-day efficiency and how to calculate it. In this article, we want to know the efficiency of a transformer and the calculation formula.

What is the Transformer Efficiency?

A transformer's ordinary or commercial efficiency is defined as the ratio of output power to input power.

Efficiency = η = Output / Input

Efficiency = η= Output / (Output + Losses) 

As Input = Output + Losses

If we go into more detail about the transformer losses, just remind the previous articles about the transformer loss calculation.

Transformer Copper Loss the ,

WC = I12 x R1   or I22 x R2

Transformer Iron Loss,

WI = Hysteresis Loss + Eddy Current Loss = WI = WH + WE

Suppose to the primary side of transformer…

Primary Input = P1 = V1 x I1 Cosθ1

Efficiency = η = Output / Input

Efficiency = η = (Input – Losses) / input 

(As Output = Input – Losses)

Efficiency = η = (Input – Copper losses – Iron Losses)/Input

Efficiency = η = (P1 – WC – WI) / P1

Efficiency = η = (V1 x I1 Cosθ1 – I12  x R1 – WI) / V1 x I1 Cosθ1

Efficiency = η = 1- (I12 x R1 / V1I1 Cosθ1) – (WI / V1 x I1 Cosθ1)

Or

Efficiency = η = 1- (I1 x R1 / V1 Cosθ1) – (WI / V1 x I1 Cosθ1)

Differentiating both sides w.r.t I1.

Dη / dI1 = 0 – ( R1 / V1 Cosθ1) + (WI /V1 x I12 Cosθ1)

Dη / dI1= – ( R1 / V1 Cosθ1) + (WI / V1 x I12 Cosθ1)

For maximum efficiency, the value of (Dη / dI1) should be minimum i.e.

Dη / dI1 = 0

Thus, the above equation can be rewritten as:

R1 / (V1 Cosθ1) = (WI /V1 x I12 Cosθ1)

Or, WI = I12 x R1or       I22 x R2

Iron Loss = Copper Loss

Thus, the transformer will give the maximum efficiency when its copper loss is equal to iron loss:

I2 = √ (WI / R2)

The value of output current (I2) is the factor that makes it possible to equal the value of copper loss and iron loss (i.e. copper loss = iron loss).

The range of transformer efficiency

The transformer's efficiency is generally in the range of 95 – 99 %. The efficiency can be as high as 99.7% for great power transformers with very low waste. Ideally, the efficiency of the Transformer should be hand 100%, but practically it is not possible due to various types of losses such as core losses or ohmic loss.

Case Study of Transformer Efficiency

A 500 KVA transformer has 2500 watts iron loss, and 7500 watts copper loss at full load. The power factor is 0.8 lagging. Calculate transformer efficiency at full load,

maximum efficiency of the transformer,

output KVA corresponding to maximum efficiency,

transformer efficiency at half load.

Solution: Transformer rating = 500 KVA

Transformer output power = 500,000 x 0.8 = 400,000 watts

Iron losses (Pi) = 2500 W

Full load copper loss (Pcu) = 7500 W

 

Transformer Efficiency at Full Load

= [(output power)/(output power + Pi +Pcu)] x 100

 

= [(400,000)/(400,000 + 2500 + 7500)] x 100

 

= 97.56% (Ans)

 

Maximum Efficiency of Transformer

 

For maximum efficiency, Copper loss (Pc) = Iron losses (Pi) = 2500 W

 

= [(output power)/(output power + Pi +Pc)] x 100

 

Therefore, maximum efficiency = [(400,000)/(400,000 + 2500 + 2500)] x 100

 

= 98.76% (Ans),

Output KVA Corresponding to Maximum Efficiency
 

= full load KVA x √(Pi/Pc)

 

= 500 x √(2500/7500)

= 500 x √0.333 = 166.5 KVA (Ans)

You may know the details about the electrical transformer from the following articles:

 

1. Working Principle of Transformer;
2. Transformer Construction;
3. Core-type Transformers;
4. Shell-type Transformers;
5. Elementary Theory of an Ideal Transformer;
6. E.M.F. Equation of Transformer;
7. Voltage Transformation Ratio;
8. Transformer with losses but no Magnetic Leakage;
9. Transformer on No-load;
10. Transformer on Load;
11. Transformer with Winding Resistance but no Magnetic Leakage;
12. Equivalent Resistance;
13. Magnetic Leakage;
14. Transformer with Resistance and Leakage Reactance;
15. Simplified Diagram;
16. Total Approximate Voltage Drop in Transformer;
17. Exact Voltage Drop;
18. Equivalent Circuit Transformer Tests;
19. Open-circuit or No-load Test;
20. Separation of Core Losses;
21. Short-Circuit or Impedance Test;
22. Why Transformer Rating in KVA?;
23. Regulation of a Transformer;
24. Percentage Resistance, Reactance, and Impedance;
25. Kapp Regulation Diagram;
26. Sumpner or Back-to-back-Test;
    
27. The efficiency of a Transformer;
28. Condition for Maximum Efficiency;
29. Variation of Efficiency with Power Factor;
30. All-day Efficiency;
31. Auto-transformer;
32. Conversion of 2-Winding Transformer into Auto-transformer;
33. Parallel Operation of Single-phase Transformers;
34. Questions and Answers on Transformers;
35. Three-phase Transformers;
36. Three-phase Transformer Connections;
37. Star/Star or Y/Y Connection;
38. Delta-Delta or ∆/∆ Connection;
39. Wye/Delta or Y/ Connection;
40. Delta/Wye or ∆/Y Connection;
41. Open-Delta or V-V Connection;
42. Power Supplied by V-V Bank;
43. Scott Connection or T-T Connection;
44. Three-phase to Two-Phase Conversion and vice-versa;
45. Parallel Operation of 3-phase Transformers;
46. Instrument Transformers;
47. Current Transformers;
48. Potential or Voltage Transformers.