IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 42, NO. 6, NOVEMBER/DECEMBER 2006 1437 Space Mapping Optimization of a Cylindrical Voice Coil Actuator Laurentiu Encica, Student Member, IEEE, Juraj Makarovic, Elena A. Lomonova, Member, IEEE, and André J. A. Vandenput, Senior Member, IEEE Abstract—An optimal design problem of an electromagnetic actuator is formulated by defining the set of design variables, the constraints, and the optimality criterion. Solving such a problem is a difficult and time-expensive task when many variables, con- straints, and conflicting objectives are involved, and when high accuracy is required. In order to determine the solution in an efficient manner, the space mapping technique is investigated. A cylindrical voice coil actuator is chosen as a proof-of-concept example. The numerical results show that the approach is viable, and an obtained design is verified by measurements. Index Terms—Design optimization, electromagnetic actuators, equivalent circuits, finite elements, space mapping. I. I NTRODUCTION T HE EFFICIENCY of electromagnetic actuators or their manufacturing and exploitation costs can be ameliorated by defining in the design and simulation stage an optimal design problem [6], which involves a rather high number of design variables and nonlinear constraints and, possibly conflicting, objective functions. This paper deals with the case of a single- objective constrained optimization design problem that can be defined by min x∈X  F (x, d)|g (i) inq (x, d) ≤ 0,g (j) eq (x, d) = 0; A · x  ≤ b; x ∈ X ⊂ R n ; d ∈ R m ; i =1,...,r; j =1,...,t  (1) where F (x, d) is the objective function; x =[x 1 ,x 2 ,...,x n ] is the vector of design variables defined on the feasible set X; d =[d 1 ,d 2 ,...,d m ] defines a vector of design parameters, i.e., geometrical sizes that have given values and do not vary during optimization; and a set of linear and nonlinear and equality and inequality constraints is given by A · x  ≤ b and g (j) eq (x, d)=0, g (i) inq (x, d) ≤ 0, respectively. Paper IPCSD-06-072, presented at the 2005 IEEE International Electric Machines and Drives Conference, San Antonio, TX, May 15–18, and approved for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Electric Machines Committee of the IEEE Industry Applications So- ciety. Manuscript submitted for review September 11, 2005 and released for publication July 12, 2006. This work was supported by the Dutch Ministry of Economic Affairs under Project IOP-EMVT 02201. L. Encica, E. A. Lomonova, and A. J. A. Vandenput are with the Electromechanics and Power Electronics Group, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands (e-mail: l.encica@tue.nl; j.makarovic@tue.nl; e.lomonova@tue.nl; a.j.a.vandenput@tue.nl). J. Makarovic was with the Electronics and Power Electronics Group, Eind- hoven University of Technology, 5600 MB Eindhoven, The Netherlands. He is now with the ASML, 5504 DR Veldhoven, The Netherlands. Color versions of all figures are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIA.2006.882672 In order to achieve high solution accuracy, the electromag- netic quantities are evaluated by means of complex numer- ical models, e.g., finite-element (FE) models. However, the optimization process becomes time-expensive. Alternatively, the use of less complex models, which can be analytical, or equivalent circuit formulations, is common in the research field of engineering optimization. This second approach has the advantage of a fast solution but lacks the required precision. A method to solve (1) in an accurate and time-inexpensive manner is desirable. The space mapping (SM) technique [2], [3] was introduced as an efficient tool for the optimization of microwave systems, and in [4], it was applied successfully for the design of two electromechanical systems. The underlying idea of SM is to solve (1) by exploiting two models of the studied physical system, namely: 1) a “fine” model (accurate but difficult to evaluate) and 2) a “coarse” model (easy to evaluate but inaccu- rate), and approximating a link (mapping) between the design variables of the two models. A review of the state of the art of the SM technique is given in [1]. Its implementation in the field of electromagnetic actuator design in the context of constraint optimization is a rather unexplored topic. From the existing variants of the technique, the aggressive SM (ASM) is chosen to solve the optimization problem of a cylindrical voice coil actuator (CVCA). The principles of the SM technique and the ASM algorithm are presented in Section II. In Section III, the electromagnetic design specifications of the problems are introduced. They are further translated into the objective function and the constraints of the optimization problems in a mathematical formulation. Two numerical examples are detailed. Initially, an existing ac- tuator is considered. Its response characteristics are determined by measurements and FE simulation and used to formulate an optimal design problem in an attempt to improve the existing geometry and to check the performance of the SM technique. Next, a different set of design specifications is introduced, and a new geometry of the CVCA is obtained. The solution is verified with the help of a standard optimization routine. Moreover, a test setup is realized, and the design is verified by measurements. The results are given and conclusions are discussed in Sections IV and V, respectively. II. SM A. SM Concept Consider an accurate model f (x f ): R n → R m denoted as a “fine” model. The set of variables x f that minimizes 0093-9994/$20.00 © 2006 IEEE