The tensor — A new engineering tool

An Extensive Tensor Algebraic Model of Transformer

“…As a result, I P ik , I Sik , V P ik and V Sjk are functions of complex frequency only, and the obtained current/voltage tensors are able to model the time dependencies of current/voltage functions caused by those of the circuit parameters despite the fact that these tensors are purely in the s-domain. This cannot be achieved if an order 1 tensor has been assumed similarly to [19], [20] and [24] because a tensor of order 1 is mathematically equivalent a vector. Before we proceed further, it should be mentioned here that the dimensions of I P ik and V P ik are N ×K, but those of I Sjk and V Sjk are M × K.…”

“…In the last few decades, a very powerful mathematical tool entitled multilinear algebra, or tensor algebra, which is the generalization of the matrixvector algebra, has been applied to electrical engineering [18]- [24]. Tensor algebraic modelling attempts 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.…”

“…Tensor algebraic modelling attempts 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.…”

“…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-rived a tensor algebraic model of the on-chip monolithic transformer in [24] where the effects of imperfect coupling and frequency dependent circuit parameters, which are common to the on-chip transformer, have been taken into account.…”

“…Therefore we de-rived a tensor algebraic model of the on-chip monolithic transformer in [24] where the effects of imperfect coupling and frequency dependent circuit parameters, which are common to the on-chip transformer, have been taken into account. However, this previous work used order 2 tensors for the impedance and current/voltage transformation arrays and used order 1 tensors for current/voltage arrays similarly to [19] and [20]. Thus that model also fails to model some recent transformers, particularly those with time variant circuit parameters.…”