Selecting analytical method of leakage inductance calculation for optimizing high-voltage test transformer design

AN ACTUAL PROBLEM in transformer design is to optimize its design, since it allows you to create a competitive transformer. One of the objective functions in optimization algorithms for high-voltage test transformers is the ratio of the leakage inductive reactance of the transformer, reduced to the secondary side, to the load capacitance. Since the objective functions in the optimization procedure must be found many times, it is advisable to use approximate analytical calculation methods to calculate the leakage inductance. THE PURPOSE of the article is to analyze the error of the traditional analytical method for calculating the leakage inductance of power transformers with a rectangular axial cross-sections of the windings, and to develop a refined analytical method that takes into account the main design feature of high-voltage test transformers, which consists in the trapezoidal shape of the secondary winding cross-section. The refined method takes into account the change in the number of turns in the radial direction while maintaining the basic assumptions of the traditional method. METHODS. The error is studied by comparing the calculation results using these methods with the results of a numerical calculation of leakage inductance based on a 3D magnetostatic field, which are accepted as accurate. RESULTS. It is shown that the calculation error using the method proposed in the article is significantly reduced compared to the error of the method developed for power transformers with rectangular winding cross-sections. The ranges of changes in the relative geometric parameters of the base transformer have been determined, in which the error of this calculation method does not exceed 10%. The TGI 50/100 transformer, whose rated power is 5 kVA, was chosen as the base transformer. The developed analytical method is recommended for high-voltage test transformers optimization algorithms.

1. K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, "A fast and elitist multiobjective genetic algorithm: NSGA-II," in IEEE Transactions on Evolutionary Computation, vol. 6, no. 2, pp. 182-197, April 2002, doi: 10.1109/4235.996017

2. G.N. Petrov. Elektricheskie mashiny. Chast' pervaya. Vvedenie. Transformatory. M: Energiya. 1974, 240 s.

3. Butyrin P.A., Alpatov M.E. Analiticheskaya teoriya transformatorov M: Nacional'nyj issledovatel'skij universitet MEI. 2019. C. 112

4. Ouyang, W. G. Hurley and M. A. E. Andersen, "Improved Analysis and Modeling of Leakage Inductance for Planar Transformers," in IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 7, no. 4, pp. 2225-2231, Dec. 2019, doi: 10.1109/JESTPE.2018.2871968.

5. L. M. R. Oliveira and A. J. M. Cardoso, "Leakage Inductances Calculation for Power Transformers Interturn Fault Studies," in IEEE Transactions on Power Delivery, vol. 30, no. 3, pp. 1213-1220, June 2015, doi: 10.1109/TPWRD.2014.2371877.

6. Dawood K., Isik F., Kömürgöz Kırış G. Comparison of Analytical Method and Different Finite Element Models for the Calculation of Leakage Inductance in Zigzag Transformers, ELEKTRONIKA IR ELEKTROTECHNIKA, cilt.28, sa.1, pp.16-22, 2022.

7. A. Sharma and J. W. Kimball, " Evaluation of Transformer Leakage Inductance Using Magnetic Image Method,” in IEEE Transactions on Magnetics, vol. 57, no. 11, pp. 1-12, Nov. 2021, Art no. 8401912, doi: 10.1109/TMAG.2021.3111479.

8. M. Lambert, F. Sirois, M. Martinez-Duro and J. Mahseredjian, "Analytical Calculation of Leakage Inductance for Low-Frequency Transformer Modeling," in IEEE Transactions on Power Delivery, vol. 28, no. 1, pp. 507-515, Jan. 2013, doi: 10.1109/TPWRD.2012.2225451.

9. Hernandez, F. de Leon and P. Gomez, "Design Formulas for the Leakage Inductance of Toroidal Distribution Transformers," in IEEE Transactions on Power Delivery, vol. 26, no. 4, pp. 2197-2204, Oct. 2011, doi: 10.1109/TPWRD.2011.2157536.

10. M. Lambert, F. Sirois, M. Martinez-Duro and J. Mahseredjian, "Analytical Calculation of Leakage Inductance for Low-Frequency Transformer Modeling," in IEEE Transactions on Power Delivery, vol. 28, no. 1, pp. 507-515, Jan. 2013, doi: 10.1109/TPWRD.2012.2225451.

11. R. Torchio, "A Volume PEEC Formulation Based on the Cell Method for Electromagnetic Problems From Low to High Frequency," in IEEE Transactions on Antennas and Propagation, vol. 67, no. 12, pp. 7452-7465, Dec. 2019, doi: 10.1109/TAP.2019.2927789

12. Zhujkov A.V., Kubatkin M.A., Larin V.S., Matveev D.A., Nikulov I.I., Hrenov S.I. K opredeleniyu induktivnostej rasseyaniya obmotok transformatorov // Elektrotekhnika. 2019. №8. c. 46-53.

13. M. D’Antonio, S. Chakraborty and A. Khaligh, "Planar Transformer With Asymmetric Integrated Leakage Inductance Using Horizontal Air Gap," in IEEE Transactions on Power Electronics, vol. 36, no. 12, pp. 14014-14028, Dec. 2021, doi: 10.1109/TPEL.2021.3089606.

14. Butyrin, Pavel A., Semen D. Dubickij, i Nikolaj V. Korovkin. \"Chislennoe modelirovanie elektromagnitnyh polej: mul'tifizicheskie zadachi, instrumentarij i obuchenie.\" Elektrichestvo 6 (2019): s.51- 58.

15. COMSOL Multiphysics User’s Guide, p.563.

16. E. D. Paramonov, I. Sakhno Liudmila, I. Sakhno Olga and E. Y. Kochetkova, "Analysis of Methods for Calculating Leakage Inductance of a High-Voltage Test Transformer," 2024 Conference of Young Researchers in Electrical and Electronic Engineering (ElCon), Saint Petersburg, Russian Federation, 2024, pp. 680-684, doi: 10.1109/ElCon61730.2024.10468519.