Extensive Multilinear Algebraic Transformer Model 79 Extensive Multilinear Algebraic Transformer Model Rawid Banchuin 1 and Roungsan Chaisricharoen 2 ABSTRACT An extensive s-domain multilinear algebraic model of the transformer has been proposed. This model is alternatively a tensor algebraic model because the multilinear algebra is alternatively the tensor algebra. Unlike the traditional matrix-vector approach, which relies on conventional linear algebra, this model, which in turn uses the multilinear algebra that is of higher dimension and is thus more generic, is ap- plicable to those recently often cited transformers which often have unconventional characteristics such as frequency variant parameters, time variant param- eters, and fractional impedance. Examples of such transformers are on-chip monolithic transformers, dy- namic transformers, and transformers with fractional impedances. The imperfect coupling has been con- sidered, and a multiple winding transformer has also been assumed. Applications of the proposed model to the chosen recent transformers with unconventional characteristics is presented. The effects of failure of Kron’s postulate on power invariant and valid- ity of duality invariant, which pertain to the men- tioned issues, are also discussed. The proposed ex- tensive model is more inclusive and up to date than the matrix-vector based model and previous algebraic models. However, it is more complicated. Keywords: Dynamic Transformer, Fractional Mu- tual Inductance, Multilinear Algebra, On-chip Mono- lithic Transformer, Tensor Algebra 1. INTRODUCTION Transformers have been used in various electri- cal engineering applications for decades. Accord- ing to the simplicity of the algebraic based analysis in the complex frequency domain (s-domain), trans- formers have been traditionally modelled in the s- domain by using a classical linear algebra (matrix- vector algebra) approach where the impedances have been modelled by using linear functions. Moreover, those parameters which comprise the coefficients of Manuscript received on November 6, 2018 ; revised on May 28, 2019. Final manuscript received on March 6, 2020. 1 The author is with Graduated school of IT and Faculty of Engineering, Siam University, Bangkok, Thailand., E-mail: rawid b@yahoo.com 2 The author is with School of IT, Mae Fah Luang University, Chiangrai, Thailand., E-mail: roungsan.cha@mfu.ac.th DOI: 10.37936/ecti-cit.2020141.153727 the impedance functions have been assumed to be constant. Therefore this traditional approach works well with a conventional transformer which employs linear impedance functions and constant circuit pa- rameters. Unfortunately this is not applicable to many recently developed transformers including on- chip monolithic transformers of both passive and ac- tive types [1]-[13], dynamic transformers [14]-[16] and those with fractional impedances which is termed the fractional-order mutual inductance [17]. These re- cently developed transformers have been applied in many electronic circuits [2]-[13], [17]. Severe error may occur in calculations using the traditional model [6]. This is because these recent transformers em- ploy unconventional characteristics such as frequency variant circuit parameters, time variant circuit pa- rameters, and fractional impedances, all of which are far beyond the scope of the traditional approach. In the last few decades, a very powerful mathe- matical tool entitled multilinear algebra, or tensor algebra, which is the generalization of the matrix- vector algebra, has been applied to electrical en- gineering [18]-[24]. Tensor algebraic modelling at- tempts of the transformer have been proposed [19], [20], [24]. Unfortunately, the results of [19] and [20] are also inapplicable to recent transformers because these works used order 2 tensors as the impedance and current/voltage transformation arrays and used order 1 tensors for current/voltage arrays. They also used linear impedance functions and assumed that all parameters which comprise the coefficients of the current, voltage and impedance functions are neither time nor frequency dependent. Therefore these pre- vious modelling attempts are effectively identical to the aforementioned traditional matrix-vector algebra based modelling approach. They also fail to model many modern transformers. Moreover, these previ- ous attempts were performed assuming that the cou- pling is perfect, thus the coupling factor is fixed at 1. Unfortunately, the coupling can be imperfect in many recent transformers. This occurs, for example, in passive on-chip transformers because the magnetic flux linkage is weak [1]-[5], and in active on-chip trans- formers since lossy active couplings have been utilized [7]–[13]. Their coupling factors can be lower than 1. In practice, such coupling factors have arbitrary val- ues between 0 and 1. Despite the failures of [19] and [20], tensor algebra is still very promising due to its greater generality compared to matrix-vector algebra. Therefore we de-