Calculations:Transformer Efficiency and Regulation

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.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = \frac{100P_o}{P_i}=\frac{P_i-P_L}{P_i}=\left(1-\frac{P_L}{P_o+P_L}\right) \times 100 \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta} = 100 \times \left(1-\frac{P_L}{P_o+P_L}\right) \, }

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = \, } efficiency (%)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {P_o} = \, } output

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {P_i} = \, } input

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {P_L} = \, } losses

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = 100 \times \left(1-\frac{W_i+x^2 W_c}{x S P_o+W_1+x^2 W_c}\right) \, }

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = \, } efficiency (%)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {W_i} = \, } no-load loss (W)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {W_c} = \, } load loss at full load at reference temperature (W)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {x} = \, } per unit load

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {S} = \, } 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.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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 \, }

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = \, } efficiency (%)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {P_o} = \, } no-load loss (kW)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {n} = \frac{Load}{Rated Power} \, } (kVA)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {P_k} = \, } load loss (kW)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {cos \phi} = \, } power factor

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {U_{\phi n}} = \, } voltage drop (or rise) %

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {U_{ \phi n}} = n \times \left(r cos \phi + x sin \phi \right) + \frac{\left[ n \times \left(r sin \phi - x cos \phi \right) \right]^2}{200} \, }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z=0.6+j5.97=6.0 \Omega \, }

Efficiency at 100% load unity power factor

At unity power factor (full load)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r cos \phi + x sin \phi = 0.6 \times 1 + 5.97 \times 0 = 0.6 \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r sin \phi - x sin \phi = 0.6 \times 0 - 5.97 \times 1 = -5.97 \, }

Voltage DropFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 0.6 + \frac{- 5.97^2}{200} = 0.778\% \, }

As per IEC Standard

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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 \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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 \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = 100-\left( \frac{11.3}{1502.3} \right) \times 100 \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = 99.248\% \, }

As per ANSI Standard

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = 100 \times \left(1-\frac{W_i+x^2 W_c}{x S P_o+W_1+x^2 W_c}\right) \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = 100 \times \left(1-\frac{2.3+9.0}{1500+2.3+9.0}\right) \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = 99.252\% \, }

Efficiency at 100% load 0.8 power factor

At 0.8 power factor (full load)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r cos \phi + x sin \phi = 0.6 \times 0.8 + 5.97 \times 0.6 = 4.062 \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r sin \phi - x sin \phi = 0.6 \times 0.6 - 5.97 \times 0.8 = -4.416 \, }

Voltage DropFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 4.062 + \frac{- 4.416^2}{200} = 4.159\% \, }

As per IEC Standard

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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 \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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 \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = 100-\left( \frac{11.3}{1162.556} \right) \times 100 \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = 99.028\% \, }

As per ANSI Standard

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = 100 \times \left(1-\frac{W_i+x^2 W_c}{x S P_o+W_1+x^2 W_c}\right) \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = 100 \times \left(1-\frac{2.3+9.0}{1500 \times 0.8 + 2.3 + 9.0}\right) \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = 99.067\% \, }

Efficiency at 50% load unity power factor

At unity power factor (50% load)

Voltage DropFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 0.5 \times 0.6 + \frac{(- 5.97 \times 0.5)^2}{200} = 0.3445\% \, }

As per IEC Standard

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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 \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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 \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = 100-\left( \frac{11.3}{750.05} \right) \times 100 \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = 99.493\% \, }

As per ANSI Standard

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = 100 \times \left(1-\frac{W_i+x^2 W_c}{x S P_o+W_1+x^2 W_c}\right) \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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) \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = 99.502\% \, }

Efficiency at 50% load 0.8 power factor

At 0.8 power factor (50% load)

Voltage DropFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 0.5 \times 4.062 + \frac{(- 4.416 \times 0.5)^2}{200} = 2.055\% \, }

As per IEC Standard

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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 \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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 \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = 100-\left( \frac{11.3}{580.178} \right) \times 100 \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = 98.052\% \, }

As per ANSI Standard

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = 100 \times \left(1-\frac{W_i+x^2 W_c}{x S P_o+W_1+x^2 W_c}\right) \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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) \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\eta}\% = 98.131\% \, }

Transformer Regulations

The exact formula for calculation of regulation is as follows:

When the load is lagging:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Reg = \sqrt{(R+F_P)^2+(x+q)^2}-1 \, }

When the load is leading:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Reg = \sqrt{(R+F_P)^2+(x-q)^2}-1 \, }

where:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_P = \, } power factor of load

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q = \sqrt{1-{F_P}^2} \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R = \, } resistance factor of transformer

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R = \frac{\text{Load Loss in kW}}{\text{Rated kVA}} \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = \, } resistance factor of transformer

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = \sqrt{Z^2-R^2} \, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z = \, } impedance factor

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z = \frac{\text{Impedance kVA}}{\text{Rated kVA}} \, }

References

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