Procedure for calculation of transformer model parameters

This chapter is devoted to the description of procedure for calculating parameters of the equivalent circuits of various transformers. It consists of two parts describing the order of calculation in relative units and the order of determining parameters of the model based on catalog data.

1.  Equivalent circuit parameters based on relative units

1.1.  Single-phase transformers

Basis resistances and inductances (1):

\[R_{б1}=\frac{U_{1ном}^2}{S_{ном}}, L_{б1}=\frac {R_{б1}}{2\cdot п\cdot f_{ном}} \]

\[R_{б2}=\frac{U_{2ном}^2}{S_{ном}}, L_{б2}=\frac {R_{б2}}{2\cdot п\cdot f_{ном}} \]

(1)\[R_{б3}=\frac{U_{3ном}^2}{S_{ном}}, L_{б3}=\frac {R_{б3}}{2\cdot п\cdot f_{ном}} \]

Thus, the winding parameters will be equal to (2):

\[R_1=R_{1,о.е.}\cdot R_{б1}, L_1=L_{1,о.е.}\cdot L_{б1}\]

\[R_2=R_{2,о.е.}\cdot R_{б2}, L_2=L_{2,о.е.}\cdot L_{б2}\]

(2)\[R_3=R_{3,о.е.}\cdot R_{б3}, L_3=L_{3,о.е.}\cdot L_{б3}\]

Magnetization branch parameters (3):

(3)\[R_m=R_{m,о.е.}\cdot R_{б1}, L_m=L_{m,о.е.}\cdot L_{б1}\]

1.2.  Three-phase transformers

A three-phase transformer is a connection of three single-phase transformers, the transformation ratio of each is determined as ratio of the rated secondary phase voltage of the transformer to the primary phase (4):

(4)\[K=\frac{U_{2номф}}{U_{1номф}}\]

But, if the transformer windings are connected differently, then the rated voltage of one of them can become linear, and the other phase. In order to account for such cases, adjustments shall be made to the transformer ratio calculation. They are presented in the table below. Rated voltages in the case of three-phase transformers are linear, not phase.

Winding connection method	Transformation ratio
Star-Star	K=\(\frac{U_{2ном Ф}}{U_{1ном Ф}}\)=\(\frac{U_{2ном}}{\sqrt {3}}\cdot \frac{\sqrt {3}}{U_{1ном}}=\frac{U_{2ном}}{U_{1ном}}\)
Star-Triangle I	K=\(\frac{U_{2ном Ф}}{U_{1ном Ф}}\)=\(\frac{U_{2ном}}{\sqrt {3}}\cdot \frac{1}{U_{1ном}}=\frac{U_{2ном}}{\sqrt {3} \cdot U_{1ном}}\)
Triangle I-Star	K=\(\frac{U_{2ном Ф}}{U_{1ном Ф}}\)=\(\frac{U_{2ном}}{1}\cdot \frac{\sqrt {3}}{U_{1ном}}=\frac{\sqrt {3}\cdot U_{2ном}}{U_{1ном}};\)
Triangle I-Triangle II	K=\(\frac{U_{2ном Ф}}{U_{1ном Ф}}\)=\(\frac{U_{2ном}}{U_{1ном}}\)

Basic resistance and inductance of the windings of each phase (1 - high voltage winding, 2 - medium voltage winding, 3 - low voltage winding) (5):

\[R_{б1}=\frac{U_{1номф}^2}{S_{номф}}=\frac{U_{1ном}^2}{(\sqrt {3})^2}\cdot \frac{3}{S_{ном}}=\frac{U_{1ном}^2}{S_{ном}}, L_{б1}=\frac {R_{б1}}{2\cdot п\cdot f_{ном}} \]

\[R_{б2}=\frac{U_{2номф}^2}{S_{номф}}=\frac{U_{2ном}^2}{(\sqrt {3})^2}\cdot \frac{3}{S_{ном}}=\frac{U_{2ном}^2}{S_{ном}}, L_{б2}=\frac {R_{б2}}{2\cdot п\cdot f_{ном}} \]

(5)\[R_{б3}=\frac{U_{3номф}^2}{S_{номф}}=\frac{U_{3ном}^2}{(\sqrt {3})^2}\cdot \frac{3}{S_{ном}}=\frac{U_{3ном}^2}{S_{ном}}, L_{б3}=\frac {R_{б3}}{2\cdot п\cdot f_{ном}} \]

Thus, the parameters of the windings will be equal (6):

\[R_1=R_{1,о.е.}\cdot R_{б1}, L_1=L_{1,о.е.}\cdot L_{б1}\]

\[R_2=R_{2,о.е.}\cdot R_{б2}, L_2=L_{2,о.е.}\cdot L_{б2}\]

(6)\[R_3=R_{3,о.е.}\cdot R_{б3}, L_3=L_{3,о.е.}\cdot L_{б3}\]

Magnetization branch parameters (7):

(7)\[R_m=R_{m,о.е.}\cdot R_{б1}, L_m=L_{m,о.е.}\cdot L_{б1}\]

2.  Procedure for calculation of transformer model parameters

This article is devoted to the description of procedure for calculating parameters of the equivalent circuits of various transformers. It consists of two parts describing the order of calculation in relative units and the order of determining parameters of the model based on catalog data.[1]

The rated power of transformer S\(_{rat}\), VA is taken as the basic power, their rated phase (in the case of a single-phase transformer) or linear (in the case of a three-phase transformer) voltages U\(_{1rat}\), U\(_{2rat}\), U\(_{3rat}\), V are taken as basic voltages of the windings.

The designations of parameters corresponding to the catalog data of the transformers and their units of measurement used in the calculation are given in the table below:

Catalog parameter	Unit of measurement	Designation
Power losses during idle operation	W	\(\Delta\)P\(_X\)
Current in idle test	%	I\(_0\)
Voltage in short circuit test	%	U\(_K\)
Voltage in short-circuit test with short-circuited windings of high and medium voltages	%	U\(_{K,B-C}\)
Voltage in short-circuit test with short-circuited windings of high and low voltages	%	U\(_{K,B-H}\)
Voltage in short-circuit test with short-circuited windings of medium and low voltages	%	U\(_{K,C-H}\)
Power losses in short-circuit test	W	\(\Delta\)P\(_K\)
Power losses in short-circuit test at short-circuited windings of high and medium voltage	W	\(\Delta\)P\(_{K,B-C}\)
Power losses in short-circuit test at short-circuited windings of high and low voltage	W	\(\Delta\)P\(_{K,B-H}\)
Power losses in short-circuit test at short-circuited windings of medium and low voltage	W	\(\Delta\)P\(_{K,C-H}\)

2.1.  Single-phase transformers

Transformation ratio (8):

(8)\[K=\frac{U_{2ном}}{U_{1ном}}\]

Basis resistances and inductances (9):

\[R_{б1}=\frac{U_{1ном}^2}{S_{ном}}, L_{б1}=\frac {R_{б1}}{2\cdot п\cdot f_{ном}} \]

\[R_{б2}=\frac{U_{2ном}^2}{S_{ном}}, L_{б2}=\frac {R_{б2}}{2\cdot п\cdot f_{ном}} \]

(9)\[R_{б3}=\frac{U_{3ном}^2}{S_{ном}}, L_{б3}=\frac {R_{б3}}{2\cdot п\cdot f_{ном}} \]

Formulas to calculate parameters of double-winding and three-winding transformer models are given in the table below. They are based on[1][2]

Double winding transformer

Three-winding transformer

Resistances: R\(_{1,о.е.}\)=R\(_{2,о.е.}\)=0.5\( \cdot\frac{\Delta P_{K}}{S_{ном}}\) R\(_1\)=R\(_{1,о.е.}\cdot R_{б1}\), R\(_2\)=R\(_{2,о.е.}\cdot R_{б2} \) Inductances: L\(_{1,о.е.}\)=L\(_{2,о.е.}=0.5 \cdot U_K/100\)% L\(_1\)=L\(_{1,о.е.}\cdot L_{б1}\), L\(_2\)=L\(_{2,о.е.}\cdot L_{б2}\) Active resistance and inductance of the magnetization branch: R\(_{m,о.е.}\)=\(\frac{S_{ном}}{\Delta P_{X}}\), R\(_m\)=R\(_{m,о.е.}\cdot R_{б1}\) L\(_{m,о.е.}\)=100%/I\(_0\), L\(_m\)=L\(_{m,о.е.}\cdot L_{б1}\)”

Power losses in short-circuit experience reduced to windings of three-winding transformer: \(\Delta\)P\(_{K,B}\)=0.5\(\cdot\) (\(\Delta\)P\(_{K,B-Н}+\Delta\)P\(_{K,B-С}-\Delta\)P\(_{K,C-Н}\)) \(\Delta\)P\(_{K,С}\)=0.5\(\cdot\) (-\(\Delta\)P\(_{K,B-Н}+\Delta\)P\(_{K,B-С}+\Delta\)P\(_{K,C-Н}\)) \(\Delta\)P\(_{K,Н}\)=0.5\(\cdot\) (\(\Delta\)P\(_{K,B-Н}-\Delta\)P\(_{K,B-С}+\Delta\)P\(_{K,C-Н}\)) * Voltage in short circuit test reduced to windings of three-winding transformer: U\(_{K,B}\)=0.5\(\cdot\) (U\(_{K,B-Н}+U_{K,B-С}-U_{K,C-Н}\)) U\(_{K,C}\)=0.5\(\cdot\) (-U\(_{K,B-Н}+U_{K,B-С}+U_{K,C-Н}\)) U\(_{K,Н}\)=0.5\(\cdot\) (U\(_{K,B-Н}-U_{K,B-С}+U_{K,C-Н}\)) In all models of three-winding transformers, when setting parameters through catalog parameters, the primary winding corresponds to the high voltage winding, the secondary to the medium voltage winding, and the tertiary to the low voltage winding. Winding resistances: R\(_{1,о.е.}=\frac{\Delta P_{К,В}}{S_{ном}}\) R\(_1\)=R\(_{1,о.е.}\cdot\)R\(_{б1}\) R\(_{2,о.е.}=\frac{\Delta P_{К,C}}{S_{ном}}\) R\(_2\)=R\(_{2,о.е.}\cdot\)R\(_{б2}\) R\(_{3,о.е.}=\frac{\Delta P_{К,Н}}{S_{ном}}\) R\(_3\)=R\(_{3,о.е.}\cdot\)R\(_{б3}\) Inductanceof windings: L\(_{1,о.е.}=U_{К,В}\)/100%, L\(_1\)=L\(_{1,о.е.}\cdot\) L\(_{б1}\) L\(_{2,о.е.}=U_{К,C}\)/100%, L\(_2\)=L\(_{2,о.е.}\cdot\) L\(_{б2}\) L\(_{3,о.е.}=U_{К,Н}\)/100%, L\(_3\)=L\(_{3,о.е.}\cdot\) L\(_{б3}\) Active resistance and inductance of the magnetization branch: R\(_{m,о.е.}=\frac{S_{ном}}{\Delta P_X}\), R\(_m\)=R\(_{m,о.е.}\cdot\) R\(_{б1}\) L\(_{m,о.е.}\)=100%/I\(_0\), L\(_m\)=L\(_{m,о.е.}\cdot\) L\(_{б1}\)”

2.2.  Three-phase transformers

A three-phase transformer is a connection of three single-phase transformers, the transformation ratio of each is determined as ratio of the rated secondary phase voltage of the transformer to the primary phase (10):

(10)\[K=\frac{U_{2номф}}{U_{1номф}}\]

But, if the transformer windings are connected differently, then the rated voltage of one of them can become linear, and the other phase. In order to account for such cases, adjustments shall be made to the transformer ratio calculation. They are presented in the table below. Rated voltages in the case of three-phase transformers are linear, not phase.

Winding connection method (primary-secondary)	Transformation ratio
Star-Star	K=\(\frac{U_{2ном Ф}}{U_{1ном Ф}}\)=\(\frac{U_{2ном}}{\sqrt {3}}\cdot \frac{\sqrt {3}}{U_{1ном}}=\frac{U_{2ном}}{U_{1ном}}\)
Star-Triangle I	K=\(\frac{U_{2ном Ф}}{U_{1ном Ф}}\)=\(\frac{U_{2ном}}{\sqrt {3}}\cdot \frac{1}{U_{1ном}}=\frac{U_{2ном}}{\sqrt {3} \cdot U_{1ном}}\)
Star-Triangle I	K=\(\frac{U_{2ном Ф}}{U_{1ном Ф}}\)=\(\frac{U_{2ном}}{\sqrt {3}}\cdot \frac{1}{U_{1ном}}=\frac{U_{2ном}}{\sqrt {3} \cdot U_{1ном}}\)
Triangle I-Triangle I	K=\(\frac{U_{2ном Ф}}{U_{1ном Ф}}\)=\(\frac{U_{2ном}}{U_{1ном}}\)
Triangle I-Star	K=\(\frac{U_{2ном Ф}}{U_{1ном Ф}}\)=\(\frac{U_{2ном}}{1}\cdot \frac{\sqrt {3}}{U_{1ном}}=\frac{\sqrt {3}\cdot U_{2ном}}{U_{1ном}};\)
Triangle I-Triangle II	K=\(\frac{U_{2ном Ф}}{U_{1ном Ф}}\)=\(\frac{U_{2ном}}{U_{1ном}}\)
Triangle II-Triangle II	K=\(\frac{U_{2ном Ф}}{U_{1ном Ф}}\)=\(\frac{U_{2ном}}{U_{1ном}}\)
Triangle II-Star	K=\(\frac{U_{2ном Ф}}{U_{1ном Ф}}\)=\(\frac{U_{2ном}}{1}\cdot \frac{\sqrt {3}}{U_{1ном}}=\frac{\sqrt {3}\cdot U_{2ном}}{U_{1ном}};\)
Triangle II-Triangle I	K=\(\frac{U_{2ном Ф}}{U_{1ном Ф}}\)=\(\frac{U_{2ном}}{U_{1ном}}\)

Basic resistance and inductance of windings of each phase (1 - high voltage winding, 2 - medium voltage winding, 3 - low voltage winding) (11):

\[R_{б1}=\frac{U_{1номф}^2}{S_{номф}}=\frac{U_{1ном}^2}{(\sqrt {3})^2}\cdot \frac{3}{S_{ном}}=\frac{U_{1ном}^2}{S_{ном}}, L_{б1}=\frac {R_{б1}}{2\cdot п\cdot f_{ном}} \]

\[R_{б2}=\frac{U_{2номф}^2}{S_{номф}}=\frac{U_{2ном}^2}{(\sqrt {3})^2}\cdot \frac{3}{S_{ном}}=\frac{U_{2ном}^2}{S_{ном}}, L_{б2}=\frac {R_{б2}}{2\cdot п\cdot f_{ном}} \]

(11)\[R_{б3}=\frac{U_{3номф}^2}{S_{номф}}=\frac{U_{3ном}^2}{(\sqrt {3})^2}\cdot \frac{3}{S_{ном}}=\frac{U_{3ном}^2}{S_{ном}}, L_{б3}=\frac {R_{б3}}{2\cdot п\cdot f_{ном}} \]

The formulas to calculate parameters of double-winding and three-winding transformer models are similar to those given above for single-phase transformers due to the fact that voltages in the short-circuit experiments are phase values and all powers are summarized for three phases, due to which the coefficients of all powers (both losses and total) are the same and reduce each other. Calculation formulas are given in the table below.

Double winding transformer

Three-winding transformer

Resistances: R\(_{1,о.е.}\)=R\(_{2,о.е.}\)=0.5\( \cdot\frac{\Delta P_{K}}{S_{ном}}\) R\(_1\)=R\(_{1,о.е.}\cdot R_{б1}\), R\(_2\)=R\(_{2,о.е.}\cdot R_{б2} \) Inductances: L\(_{1,о.е.}\)=L\(_{2,о.е.}=0.5 \cdot U_K/100\)% L\(_1\)=L\(_{1,о.е.}\cdot L_{б1}\), L\(_2\)=L\(_{2,о.е.}\cdot L_{б2}\) Active resistance and inductance of the magnetization branch: R\(_{m,о.е.}\)=\(\frac{S_{ном}}{\Delta P_{X}}\), R\(_m\)=R\(_{m,о.е.}\cdot R_{б1}\) L\(_{m,о.е.}\)=100%/I\(_0\), L\(_m\)=L\(_{m,о.е.}\cdot L_{б1}\)”

Power losses in short-circuit experience reduced to windings of three-winding transformer: \(\Delta\)P\(_{K,B}\)=0.5\(\cdot\) (\(\Delta\)P\(_{K,B-Н}+\Delta\)P\(_{K,B-С}-\Delta\)P\(_{K,C-Н}\)) \(\Delta\)P\(_{K,С}\)=0.5\(\cdot\) (-\(\Delta\)P\(_{K,B-Н}+\Delta\)P\(_{K,B-С}+\Delta\)P\(_{K,C-Н}\)) \(\Delta\)P\(_{K,Н}\)=0.5\(\cdot\) (\(\Delta\)P\(_{K,B-Н}-\Delta\)P\(_{K,B-С}+\Delta\)P\(_{K,C-Н}\)) * Voltage in short circuit test reduced to windings of three-winding transformer: U\(_{K,B}\)=0.5\(\cdot\) (U\(_{K,B-Н}+U_{K,B-С}-U_{K,C-Н}\)) U\(_{K,C}\)=0.5\(\cdot\) (-U\(_{K,B-Н}+U_{K,B-С}+U_{K,C-Н}\)) U\(_{K,Н}\)=0.5\(\cdot\) (U\(_{K,B-Н}-U_{K,B-С}+U_{K,C-Н}\)) In all models of three-winding transformers, when setting parameters through catalog parameters, the primary winding corresponds to the high voltage winding, the secondary to the medium voltage winding, and the tertiary to the low voltage winding. Winding resistances: R\(_{1,о.е.}=\frac{\Delta P_{К,В}}{S_{ном}}\) R\(_1\)=R\(_{1,о.е.}\cdot\)R\(_{б1}\) R\(_{2,о.е.}=\frac{\Delta P_{К,C}}{S_{ном}}\) R\(_2\)=R\(_{2,о.е.}\cdot\)R\(_{б2}\) R\(_{3,о.е.}=\frac{\Delta P_{К,Н}}{S_{ном}}\) R\(_3\)=R\(_{3,о.е.}\cdot\)R\(_{б3}\) Inductanceof windings: L\(_{1,о.е.}=U_{К,В}\)/100%, L\(_1\)=L\(_{1,о.е.}\cdot\) L\(_{б1}\) L\(_{2,о.е.}=U_{К,C}\)/100%, L\(_2\)=L\(_{2,о.е.}\cdot\) L\(_{б2}\) L\(_{3,о.е.}=U_{К,Н}\)/100%, L\(_3\)=L\(_{3,о.е.}\cdot\) L\(_{б3}\) Active resistance and inductance of the magnetization branch: R\(_{m,о.е.}=\frac{S_{ном}}{\Delta P_X}\), R\(_m\)=R\(_{m,о.е.}\cdot\) R\(_{б1}\) L\(_{m,о.е.}\)=100%/I\(_0\), L\(_m\)=L\(_{m,о.е.}\cdot\) L\(_{б1}\)”

3.  References

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[1] (1,2)

Novash I. V., Rumyantsev Yu.V. Calculation of parameters of a three-phase transformer model from the MatLab-Simulink library, taking into account the saturation of the magnetic core//Energy. News of higher educational institutions and energy associations of the CIS. 2015. №1. URL.

[2]

Idelchik V.I. Electrical systems and networks: Textbook for universities. - Moscow: Energoatomizdat, 1989. - 592 p.: ill.