This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 1 On the Equivalence Between the Maxwell-Garnett Mixing Rule and the Debye Relaxation Formula Bartlomiej Salski, Member, IEEE, and Malgorzata Celuch, Member, IEEE Abstract—This paper presents a closed-form noniterative transformation of the Maxwell-Garnett mixing rule for biphased mixtures to the triple-pole Debye relaxation formula. For the first time, it is formally proven that such a transformation is complete for conductive constituent materials. In other words, the Maxwell-Garnett representation of any biphased mixture of any conductive materials always has its formal equivalent in the Debye form with three poles at most. For specific aspect ratios of ellipsoidal inclusions, the number of poles reduces to one or two, which is formally proven herein, while in previous studies, a single-pole Debye model was arbitrarily assumed. The proposed transformation provides Debye parameters as an explicit function of a mixture composition, which is competitive to alternative techniques based on laborious curve-fitting algorithms. The newly proposed approach is of particular importance to time-domain modeling of dilute mixtures, where the Maxwell-Garnett mixing rule is usually approximated with available dispersive models. Computational examples given in this paper show advantages of the presented method over previous Maxwell-Garnett to Debye conversion algorithms, in terms of accuracy, robustness, and computational cost. Index Terms—Carbon, composite materials, computational elec- tromagnetics, dispersion, electromagnetic wave absorption. I. INTRODUCTION A RTIFICIALLY composed inhomogeneous materials have recently attracted a lot of attention within the research and industrial communities. One of the main reasons is that they offer very interesting possibilities of modifying bulk electromagnetic properties of composite materials with practically no influence on their mechanical properties. Thus, classical construction materials used for a variety of applica- tions may also acquire new functions, such as screening of electromagnetic waves or their reflectionless absorption. This concerns, for example, polymer composites doped with carbon fibers or nanotubes [1]. Due to a complicated microscopic geometry of such mixtures, a quantitative knowledge about their electromagnetic properties is not that straightforward so it has become essential to develop models (either analytical or numerical) providing macroscopic electromagnetic properties Manuscript received December 20, 2011; revised April 20, 2012; accepted April 27, 2012. This work was supported in part by the Polish National Centre for Research and Development under Contract ERA-NET-MNT/14/2009. B. Salski is with QWED, Warsaw 02-078, Poland (e-mail: bsalski@qwed. com.pl). M. Celuch is with the Institute of Radioelectronics, Warsaw University of Technology, Warsaw 00-665, Poland (e-mail: mceluch@ire.pw.edu.pl). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2012.2201743 of such mixtures. Those models must be sufficiently accurate to allow virtual prototyping with as little as possible of costly and time-consuming experiments. It is frequently required to pre- dict electrical properties of a particular mixture, or, reversing the problem, adjusting the mixture composition providing the desired material properties. The only practical approach is to develop models of macroscopic homogeneous equivalents of microscopically inhomogeneous mixtures. Otherwise, direct electromagnetic analysis of an inhomogeneous material com- prising nanoscale insertions would typically lead to prohibitive requirements for computer resources. Typically, the models of effective permittivity of mixtures are specified in a frequency domain [2]. Thus, they are not directly applicable in time-domain simulations. One of the advantages of time-domain methods is the possibility of extracting wideband characteristics from a single simulation. To profit from such a possibility, one needs to apply a time-domain model, which is accurate in a wide frequency band. Typically, time-consuming and often cumbersome procedures for fitting the effective per- mittivity formula with one of the dispersive models available in the time-domain are applied. Such an approach is often used to fit the classical Maxwell-Garnett mixing rule [3], quantifying effective permittivity of dilute mixtures, with the Debye relax- ation formula, the implementation of which is known in time- domain computational routines [4], [5]. There are a few methods frequently applied to convert one dispersive model to another, such as genetic algorithms [6] or other large-scale optimization methods [7]. A common disadvantage of those methods, how- ever, is that there is no certainty that the convergence between the two models will be achieved. Moreover, if a curve-fitting algorithm fails to converge, it is not clear whether it is due to an erroneously chosen optimization algorithm, a wrong starting point, or simply because the models are nonequivalent. Recently, to overcome the aforementioned drawbacks, the concept of a noniterative transformation of the Maxwell-Gar- nett mixing rule to the single-pole Debye model has been intro- duced by the authors of this paper [8], [9]. It has been shown that, under explicitly specified conditions, the transformation is accurate and robust. A similar approach has been independently investigated in [10] and [11], where the authors focused on the conditional equivalence between the single-pole Debye model and the Maxwell-Garnett formula for the mixtures with spher- ical and cylindrical inclusions. The authors of [10] and [11] have provided a solution for several types of constituent materials, such as nondispersive, dispersive, and conductive. It can also be recalled that an alternative method, converting a biphased Frohlich model to the Lorentz-Debye dispersive model, has also been recently published [12]. 0018-9480/$31.00 © 2012 IEEE