Calculations:Transformer Efficiency and Regulation: Difference between revisions

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= Transformer Efficiency and Regulation =
= Transformer Efficiency and Regulation =
 
[[File:Power-Transformer.jpg|center|500px]]
== Introduction ==
== Introduction ==


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: <math> {P_o} = \, </math> output
: <math> {P_o} = \, </math> output
: <math> {P_i} = \, </math> input
: <math> {P_i} = \, </math> input
: <math> {P_LL} = \, </math> losses
: <math> {P_L} = \, </math> losses


For power factor of "S" and load of "x" per unit, the formula for efficiency is
For power factor of "S" and load of "x" per unit, the formula for efficiency is
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: <math> {\eta}\% = 100-\left(\frac{P_o+n^2 P_k}{n S_n cos \phi \left(1-\frac{U_{ \phi n}}{100} \right) + P_o + n^2 P_k}\right) \times 100 \, </math>
: <math> {\eta}\% = 100-\left(\frac{P_o+n^2 P_k}{n S_n cos \phi \left(1-\frac{U_{ \phi n}}{100} \right) + P_o + n^2 P_k}\right) \times 100 \, </math>
: <math> {\eta}\% = 100-\left(\frac{2.3+9.0}{1500 \times 1 \left(1-\frac{0.778}{100} \right) + 2.3 + 9.0}\right) \times 100 \, </math>
: <math> {\eta}\% = 100-\left(\frac{2.3+9.0}{1500 \times 1 \left(1-\frac{0.778}{100} \right) + 2.3 + 9.0}\right) \times 100 \, </math>
: <math> {\eta}\% = 100-\left( \frac{11.3}{1499.63} \right) \times 100 \, </math>
: <math> {\eta}\% = 100-\left( \frac{11.3}{1502.3} \right) \times 100 \, </math>
: <math> {\eta}\% = 99.248\% \, </math>
: <math> {\eta}\% = 99.248\% \, </math>


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: <math> {\eta}\% = 100-\left(\frac{P_o+n^2 P_k}{n S_n cos \phi \left(1-\frac{U_{ \phi n}}{100} \right) + P_o + n^2 P_k}\right) \times 100 \, </math>
: <math> {\eta}\% = 100-\left(\frac{P_o+n^2 P_k}{n S_n cos \phi \left(1-\frac{U_{ \phi n}}{100} \right) + P_o + n^2 P_k}\right) \times 100 \, </math>
: <math> {\eta}\% = 100-\left(\frac{2.3+9.0}{1500 \times 0.8 \left(1-\frac{4.159}{100} \right) + 2.3 + 9.0}\right) \times 100 \, </math>
: <math> {\eta}\% = 100-\left(\frac{2.3+9.0}{1500 \times 0.8 \left(1-\frac{4.159}{100} \right) + 2.3 + 9.0}\right) \times 100 \, </math>
: <math> {\eta}\% = 100-\left( \frac{11.3}{1161.39} \right) \times 100 \, </math>
: <math> {\eta}\% = 100-\left( \frac{11.3}{1162.556} \right) \times 100 \, </math>
: <math> {\eta}\% = 99.027\% \, </math>
: <math> {\eta}\% = 99.028\% \, </math>


'''As per ANSI Standard'''
'''As per ANSI Standard'''
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: <math> {\eta}\% = 100-\left(\frac{P_o+n^2 P_k}{n S_n cos \phi \left(1-\frac{U_{ \phi n}}{100} \right) + P_o + n^2 P_k}\right) \times 100 \, </math>
: <math> {\eta}\% = 100-\left(\frac{P_o+n^2 P_k}{n S_n cos \phi \left(1-\frac{U_{ \phi n}}{100} \right) + P_o + n^2 P_k}\right) \times 100 \, </math>
: <math> {\eta}\% = 100-\left(\frac{2.3+9.0}{0.5 \times 1500 \times 1 \left(1-\frac{0.3445}{100} \right) + 2.3 + 0.5^2 \times 9.0}\right) \times 100 \, </math>
: <math> {\eta}\% = 100-\left(\frac{2.3+0.5^2 \times 9.0}{0.5 \times 1500 \times 1 \left(1-\frac{0.3445}{100} \right) + 2.3 + 0.5^2 \times 9.0}\right) \times 100 \, </math>
: <math> {\eta}\% = 100-\left( \frac{11.3}{758.72} \right) \times 100 \, </math>
: <math> {\eta}\% = 100-\left( \frac{11.3}{750.05} \right) \times 100 \, </math>
: <math> {\eta}\% = 99.493\% \, </math>
: <math> {\eta}\% = 99.493\% \, </math>


'''As per ANSI Standard'''
'''As per ANSI Standard'''
: <math> {\eta}\% = 100 \times \left(1-\frac{W_i+x^2 W_c}{x S P_o+W_1+x^2 W_c}\right) \, </math>
: <math> {\eta}\% = 100 \times \left(1-\frac{W_i+x^2 W_c}{x S P_o+W_1+x^2 W_c}\right) \, </math>
: <math> {\eta}\% = 100 \times \left(1-\frac{2.3+9.0}{0.5 \times 1500+2.3+0.5^2 \times 9.0}\right) \, </math>
: <math> {\eta}\% = 100 \times \left(1-\frac{2.3+0.5^2 \times 9.0}{0.5 \times 1500+2.3+0.5^2 \times 9.0}\right) \, </math>
: <math> {\eta}\% = 99.502\% \, </math>
: <math> {\eta}\% = 99.502\% \, </math>


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At 0.8 power factor (50% load)
At 0.8 power factor (50% load)
:Voltage Drop<math> = 0.5 \times 4.062 + \frac{(- 4.416 \times 0.5)^2}{200} = 2.055\% \, </math>
:Voltage Drop<math> = 0.5 \times 4.062 + \frac{(- 4.416 \times 0.5)^2}{200} = 2.055\% \, </math>
'''As per IEC Standard'''
: <math> {\eta}\% = 100-\left(\frac{P_o+n^2 P_k}{n S_n cos \phi \left(1-\frac{U_{ \phi n}}{100} \right) + P_o + n^2 P_k}\right) \times 100 \, </math>
: <math> {\eta}\% = 100-\left(\frac{2.3+0.5^2 \times 9.0}{1500 \times 0.8 \left(1-\frac{2.055}{100} \right) + 2.3 + 0.5^2 \times 9.0}\right) \times 100 \, </math>
: <math> {\eta}\% = 100-\left( \frac{11.3}{580.178} \right) \times 100 \, </math>
: <math> {\eta}\% = 98.052\% \, </math>
'''As per ANSI Standard'''
: <math> {\eta}\% = 100 \times \left(1-\frac{W_i+x^2 W_c}{x S P_o+W_1+x^2 W_c}\right) \, </math>
: <math> {\eta}\% = 100 \times \left(1-\frac{2.3+ 0.5^2 \times 9.0}{0.5 \times 1500 \times 0.8 + 2.3 + 0.5^2 \times 9.0}\right) \, </math>
: <math> {\eta}\% = 98.131\% \, </math>
== Transformer Regulations ==
The exact formula for calculation of regulation is as follows:
When the load is lagging:
: <math> Reg = \sqrt{(R+F_P)^2+(x+q)^2}-1 \, </math>
When the load is leading:
: <math> Reg = \sqrt{(R+F_P)^2+(x-q)^2}-1 \, </math>
where:
: <math> F_P = \, </math> power factor of load
: <math> q = \sqrt{1-{F_P}^2} \, </math>
: <math> R = \, </math> resistance factor of transformer
: <math> R = \frac{\text{Load Loss in kW}}{\text{Rated kVA}} \, </math>
: <math> x = \, </math> resistance factor of transformer
: <math> x = \sqrt{Z^2-R^2} \, </math>
: <math> Z = \, </math> impedance factor
: <math> Z = \frac{\text{Impedance kVA}}{\text{Rated kVA}} \, </math>
== References ==
'''Power and Distribution Transformers - Practical Design Guide<br>'''
© 2021 K.R.M. Nair<br>
CRC Press

Latest revision as of 02:40, 20 September 2023

Transformer Efficiency and Regulation

Power-Transformer.jpg

Introduction

The calculation of efficiency differs from IEC 60076 and ANSI C57.12 Standards because of the difference in defining rated kVA by these standards.

IEC 60076 Standard and ANSI C57.12 Standard

The rated power is defined by IEC 60076 as “When the transformer has rated voltage applied to a primary winding, and rated current flows through the terminals of a secondary winding the trans- former receives the relevant rated power for that pair of windings”.

This implies that it is a value of apparent power input to the transformer, including its own absorption of active and reactive power. The apparent power that the transformer delivers to the circuit connected to the terminals of the secondary winding under rated loading differs from the rated power. The voltage across the secondary terminals differs from the rated voltage by the voltage drop (or rise) in the transformer.

This is different from the definition as per ANSI C57.12.00 standard, where the rated kVA is defined as

The rated kVA of a transformer shall be the output that can be delivered for the time specified at rated secondary voltage and rated frequency without exceeding the specified temperature rise limitations under prescribed conditions of test, and within the limitations of established standards.

Calculation as per ANSI Standard

The efficiency of a transformer is the ratio of its useful power output to its total power input.

where

efficiency (%)
output
input
losses

For power factor of "S" and load of "x" per unit, the formula for efficiency is

where

efficiency (%)
no-load loss (W)
load loss at full load at reference temperature (W)
per unit load
power factor per unit

The calculation of efficiency differs from IEC 60076 and ANSI C57.12 Standards because of the difference in defining rated kVA by these standards.

Calculation as per IEC Standard

The voltage drop (or rise) of the transformer secondary will have to be considered, while the efficiency is calculated, as per IEC standard.

where

efficiency (%)
no-load loss (kW)
(kVA)
load loss (kW)
power factor
voltage drop (or rise) %

Example Calculations

An example for a typical power transformer.

Transformer parameters

kVA – 1500
Voltage ratio – 11/0.433
% Impedance – 6.0
% Resistance – 0.6
% Reactance – 5.97
No-load loss – 2.3 kW
Load loss – 9.0 kW

Transformer Impedance

Efficiency at 100% load unity power factor

At unity power factor (full load)

Voltage Drop

As per IEC Standard

As per ANSI Standard

Efficiency at 100% load 0.8 power factor

At 0.8 power factor (full load)

Voltage Drop

As per IEC Standard

As per ANSI Standard

Efficiency at 50% load unity power factor

At unity power factor (50% load)

Voltage Drop

As per IEC Standard

As per ANSI Standard

Efficiency at 50% load 0.8 power factor

At 0.8 power factor (50% load)

Voltage Drop

As per IEC Standard

As per ANSI Standard

Transformer Regulations

The exact formula for calculation of regulation is as follows:

When the load is lagging:

When the load is leading:

where:

power factor of load
resistance factor of transformer
resistance factor of transformer
impedance factor

References

Power and Distribution Transformers - Practical Design Guide
© 2021 K.R.M. Nair
CRC Press