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> { | : <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}{ | : <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}{ | : <math> {\eta}\% = 100-\left( \frac{11.3}{1162.556} \right) \times 100 \, </math> | ||
: <math> {\eta}\% = 99. | : <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}{ | : <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
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