A Tensor Algebraic Model of On-chip Monolithic Transformer

A tensor algebraic model of on-chip monolithic transformer has been proposed where the effects of imperfect coupling which is typical for on-chip monolithic transformer, have also been considered. Unlike the traditional modelling as a simple linear function of complex frequency, the impedance of each winding has been mathematically modelled in term of arbitrary order rational polynomial in s-domain for covering any possible electrical impedance characteristic of the winding including those frequency dependent. It has been found that the proposed model which is applicable to on-chip monolithic transformer with arbitrary primary and secondary windings can accurately capture the characteristics of voltages and currents of the candidate transformer for various decades of frequency.

I. Introduction

On-chip monolithic transformer has been adopted in many analog/mixed signal circuits and systems nowadays, e.g., Low Noise Amplifiers (LNA), mixers, and oscillators etc. Originally, passive on-chip monolithic transformer was constructed based on a spiral inductor [1]–[5]. Later, an active on-chip monolithic transformer has been proposed [6], [7]. Such transformer which relies on active coupling of CMOS gyrator-C based active inductors, has been adopted in various analog/mixed signal circuits and systems e.g., quadrature oscillators [8], voltage controlled oscillators [9], current-mode phase-lock loops [10], [11] and QPSK modulators [12] etc., which are obviously employed in many areas in circuits and systems engineering. For analysis design of any analogmixed signal circuit and system, the precise mathematical models of its basis components have been found to be beneficial. Obviously, a very powerful mathematical tool entitled tensor algebra has been applied to electrical engineering decades ago until nowadays [13]–[18] where electrical engineering oriented tensor algebraic analyses involving those of traditional high voltage passive transformer circuits have been proposed [15]. Unfortunately, the obtained results are inapplicable to the on-chip monolithic transformer because the previous analysis has been performed by assuming that all coupling factors are fixed at 1 and all mutual impedances are purely inductive due to the perfect couplings. This is not the case in the on-chip monolithic transformer of both types which their couplings are typically imperfect since the magnetic flux linkage is weak in the passive on-chip monolithic transformer and lossy active couplings are employed in the active type. As a result, tensor algebraic modelling of the on-chip monolithic transformer has been found to be an interesting research question.