Posynomial Models of Inductors for Optimization of Power Electronic Systems by Geometric Programming Andrija Stupar*, Josh A. Taylor**, Aleksandar Prodic* *- Laboratory for Integrated SMPS; ** - Institute for Sustainable Energy University of Toronto Toronto, Canada a.stupar@utoronto.ca Abstract- When optimizing power electronic converters for multiple objectives, such as power density and efficiency, the optimization of the magnetic components is often the most challenging and time-consuming task. In order to perform optimization quickly and efficiently, it would be advantageous to formulate converter optimization as a geometric program, a proven convex optimization method. In order to optimize for losses and volume via a geometric program however, all loss and volume models of the various components must be in the form of posynomials. While some loss models, such as those of semiconductors, are naturally in posynomial form or easily transformed, this is not the case for inductors. This paper presents a derivation of posynomial loss, volume, temperature, and saturation models for families of inductive components, based both on simulation and on adapting familiar analytical models into approximate posynomial form. The terms of the derived posynomial models are the inductor design variables, such as the number of turns, the air gap, and so forth. This allows inductors to be optimized for multiple design objectives as a geometric program. I. INTRODUCTION Inductors account for a major part of the volume and losses of power electronic converters [1], [2], [3], [4], [5], and special care must be paid to their design when optimizing convert- ers for efficiency and power density. In fact, the magnetic component optimization sub-problem can be said to be the most complex and challenging part of converter optimization [2], [3], [6]. Meaningful optimization of magnetic components depends on accurate calculation of their losses under diverse operating conditions. The accurate calculation of core losses is especially challenging, and the entire problem is very complex [7] as the effects of all electric and magnetic operating conditions (including DC pre-magnetization) must be taken into account together with core and winding temperature, and the thermal cross-coupling between the core and the windings. A diverse literature on inductor optimization exists, focusing either on analytically-derived models, such as in [1], [3], [8], [4], measurement-based simulation models [7], [6], and finite- element simulation models [9], [10]. [9] and [10] are notable for their use of genetic algorithms for optimization, while the remainder utilize mostly exhaustive search. 978-1-5090-1815-4/16/$31.00 ©2016 To the authors' best knowledge, no publications have at- tempted to frame inductor optimization for power electronics as a convex optimization problem. Convex optimization is an extremely powerful framework for the solving of diverse optimization problems [11], and is a field which has been extensively studied. Powerful software solvers for convex optimization problems exist, such as [12], which allow the computation of optimal solutions efficiently and quickly. Al- though many real-world engineering problems are inherently non-convex, they can often be closely approximated by convex problems or divided into convex sub-problems [11]. Geometric programming, which has been used to to solve circuit-sizing problems [13], [14], [15] is a promising convex optimization approach for use in power electronics [16]. However, geometric programming operates on special types of polynomial functions, called posynomials. Loss models for inductors are not inherently posynomial, and therefore the purpose of this paper is to explore and demonstrate the derivation of posynomial models for inductor losses, inductor volume, and inductor temperature. In Section II, geometric programming is briefly presented. In Section III, appropriate posynomial functions that can be used to model inductors are discussed and a general fitting procedure is presented. In Section IV, posynomial models of inductors are derived from accurate and comprehensive simu- lations. An alternative approach, in which posynomial models are derived from standard well-known analytical equations, is given in Section V. Examples of posynomial models fitted to data and the accuracy of the fits are given in both sections. Section VI presents conclusions. II. GEOMETRIC PROGRAMMING A geometric program (GP) has the following form n minimize fo(x) = L (aolxl kOLlXlol2 ...X n k O ln ) (1) 1=1 n subject to fi(X) = L(allxlkill ...xnkiln)::; l,i = 1, ... ,m 1=1 hj(x) = ajXlkjlx2kj2 ...xnkjn = l,j = 1, ... ,p