An Extensive Tensor Algebraic Model of Transformer

Main Article Content

Rawid Banchuin
Roungsan Chaisrichaoren

Abstract

An extensive s-domain tensor algebraic model of the transformer has been proposed. Unlike the traditional matrix-vector approach which relies on the conventional linear algebra, this model which in turn assumes the multilinear algebra that is of higher dimension thus more generic, is applicable to those recently often cited transformers which often employ the unconventional characteristics i.e. frequency variant parameters, time variant parameters and fractional impedance. The examples of such transformers are the on-chip monolithic transformer, the dynamic transformers and the fractional mutual inductance etc. The imperfect coupling has been considered and multiple winding transformer has also been assumed. The applications of the proposed model to the chosen recent transformers with those unconventional characteristics have been presented. The effects of failure of Kron’s postulate on power invariant and validity of duality invariant which are worthy of mentioned issues, have also been discussed. The proposed extensive model is more inclusive and up to date than the matrix-vector based model and those foregoing tensor algebraic models albeit it is more complicated.

Article Details

How to Cite
[1]
R. Banchuin and R. Chaisrichaoren, “An Extensive Tensor Algebraic Model of Transformer”, ECTI-CIT, vol. 14, no. 1, pp. 79-91, Apr. 2020.
Section
Research Article

References

[1] J.R. Long, “Monolithic Transformers for Silicon RF IC Design,” IEEE Journal of Solid-State Circuits, Vol. 35, No. 9, pp. 1368-1382, 2000.
[2] J.J. Zhou and D.J. Allstot, “Monolithic Transformers and Their Application in A differential CMOS RF Low-Noise Amplifier,” IEEE Journal of Solid-State Circuits, Vol. 33, No. 12, pp. 2020–2027, 1998.
[3] B. Jansen, K. Negus, and D. Lee, “Silicon Bipolar VCO Family for 1.1 to 2.2 GHz with Full-Integrated Tank and Tuning Circuits,” Proceeding of 43rd IEEE International Solid State Circuits Conference (ISSCC’97), pp. 392-393, 1997.
[4] M. Zannoth, B. Kolb, J. Fenk, and R. Weigel, “A Fully Integrated VCO at 2 GHz,” IEEE Journal of Solid-State Circuits, Vol. 33, No. 12 pp. 1987–1991, 1998.
[5] J.R. Long and M.A. Copeland, “A 1.9 GHz Low-Voltage Silicon Bipolar Receiver Front-End for Wireless Personal Communication Systems,” IEEE Journal of Solid-State Circuits, Vol. 30, No. 12, pp. 1438–1448, 1995.
[6] R. Akbar, I.M. Filanovsky, J.K. Järvenhaara, and N. Tchamov, “An Asymmetric VHF Self-Oscillating DC-DC Converter with Integrated Transformer,” Proceedings of the 58th IEEE International Midwest Symposium on Circuits and Systems (MWSCAS’15), pp. 1-4, 2015.
[7] F. Yuan, “CMOS Gyrator-C Active Transformers,” Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS’07), pp. 3812–3815, 2007.
[8] F. Yuan, “CMOS Gyrator-C Active Transformers,” IET Circuits Devices and Systems, Vol. 1, No. 6, pp. 494-508, 2007.
[9] A. Tang, F. Yuan and E. Law, “CMOS Active Transformers and Their Applications in Voltage-Controlled Quadrature Oscillators,” Analog Integrated Circuits and Signal Processing, Vol. 62, No. 1, pp. 83-90, 2010.
[10] A. Tang, F. Yuan and E. Law, “Low-Noise CMOS Active Transformer Voltage-Controlled Oscillators,” Proceeding of IEEE 50th International Midwest Symposium on Circuits and Systems (MWCAS’07), 2007, pp.1441-1444.
[11] D. DiClemente, F. Yuan and A.Tang, “Current-Mode Phase-Locked Loops with CMOS Active Transformers,” IEEE Transaction on Circuits and Systems—II: Express Brief, Vol. 55, No. 8, pp.1441-1444, 2008.
[12] A. Tang, G. Zhu and F. Yuan, “Current-Mode Phase-Locked Loop with Constant-Q Active Inductor CCO and Active Transformer Loop Filter,” Analog Integrated Circuits and Signal Processing, Vol. 74, No. 2, pp. 365-375, 2013.
[13] A. Tang, F. Yuan and E. Law, “A New CMOS Active Transformer QPSK Modulator with Optimal Bandwidth Control,” IEEE Transaction on Circuits and Systems—II: Express Brief, Vol. 55, No. 1, pp.1-15, 2008.
[14] B. M. Novac, I. R. Smith, and M. C. Enache “Accurate Modeling of the Proximity Effect in Helical Flux-Compression Generators”, IEEE Transaction on Plasma Science, Vol. 28, No. 5, pp. 1353-1355, 2000.
[15] S.-D. Chen, R.-L. Lin and C.-K. Cheng, “Magnetizing Inrush Model of Transformers Based on Structure Parameters”, IEEE Transaction on Power Delivery, Vol. 20, No. 3, pp. 1947-1954, 2005.
[16] A. Young, A. Neuber and M. Kristiansen “Modeling and Simulation of Simple Flux-Trapping FCGs Utilizing PSpice Software”, IEEE Transaction on Plasma Science, Vol. 38, No. 8, pp. 1794-1802, 2010.
[17] A. Soltan, A.G. Radwan, and A.M. Soliman, “Fractional‐Order Mutual Inductance: Analysis and Design,” International Journal of Circuit Theory and Applications, Vol. 44, No. 1, pp. 85-97, 2016.
[18] G. Kron, “Non-Riemannian Dynamics of Rotating Electrical Machinery,” Journal of Mathematical Physics, Vol. 13, No. 2, pp. 103–194, 1934.
[19] A. Boyajian, “The Tensor-A New Engineering Tool,” Electrical Engineering, Vol. 55, No. 8, pp. 856-862, 1936.
[20] L.V. Bewley, “Tensor Algebra in Transformer Circuits,” Electrical Engineering, Vol. 55, No. 11, pp. 1214-1219, 1936.
[21] A.A. Myl'nikov and A.I. Prangishvili, “Homological and Cohomological Invariants of Electric Circuits,” Automation and Remote Control, Vol. 63, No. 4, pp. 578-586, 2002.
[22] A. E. Petrov “Tensor Method and Dual Networks in Electrical Engineering”, Russian Electrical Engineering, Vol. 79, No. 12, pp. 645–654, 2008.
[23] A.A. Myl'nikov and A.I. Prangishvili, “Method of Solving The Generalized Problem on Eigenvalues of Multi-Dimensional Circuits, II,” Journal of Mathematical Sciences, Vol. 186, No. 5, pp. 795-797, 2012
[24] R. Banchuin and R. Chaisricharoen, “A Tensor Algebraic Model of On-Chip Monolithic Transformer,” Proceedings of 18th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON’18), pp. 1-5, 2018.
[25] T.G. Kolda and B.W. Bader. “Tensor Decompositions and Applications,” SIAM Review, Vol. 51, No. 3, pp. 455-500, 2009.
[26] Q. Liqun, S. Wenyu and W. Yiju, “Numerical Multilinear Algebra and Its Applications” Frontiers of Mathematics in China, Vol. 2, No.4, pp. 501-526, 2007.
[27] L. De Lathauwer L and B. De Moor, “From Matrix to Tensor: Multilinear Algebra and Signal Processing,” Institute of Mathematics and Its Applications Conference Series, Vol. 67, pp. 1-16, 1996.
[28] R. Banchuin and R. Chaisricharoen “A Complete Equivalent Circuit Model of CMOS Gyrator-C Active Transformer”, Proceedings the 3rd Joint International Conference on Information & Communication Technology, Electronic and Electrical Engineering (JICTEE'10), pp. 364-369, 2010.
[29] B.T. Krishna and K.V.V.S. Reddy, “Active and Passive Realization of Fractance Device of Order 1/2,” Active and Passive Electronic Components, Vol. 2008, pp. 1-5, 2008.
[30] J.K. Delson, Networks involving Ideal Transformers, Ph.D. Dissertation, California Institute of Technology, CA., U.S.A., 1953.