IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 11, NOVEMBER 2013 5533 Optimized Design of a Novel Modular Tubular Transverse Flux Reluctance Machine Dan-Cristian Popa , Dan D. Micu , Olivia-Ramona Miron , and Loránd Szabó Department of Electrical Machines and Drives, Technical University of Cluj-Napoca, Cluj-Napoca 400027, Romania Department of Electrical Engineering, Numerical Methods Research Center, Technical University of Cluj-Napoca, Cluj-Napoca 400027, Romania This paper presents a new type of tubular electrical machine with a modular construction. The structure of the machine, concerning its construction, is discussed in the first part of the paper. A semi-analytical method based on the magnetic equivalent circuit calculation is used in order to obtain the flux densities in different parts of the iron core of the machine. A Gauss elimination procedure is applied to the system of linear equations resulted from the magnetic equivalent circuit, in order to express the flux in the air gap. The problem of optimization of the traction force is analyzed. The maximization of the function is handled with the Nonlinear Conjugate Gradient method and verified with a Gauss Newton algorithm. An application of the presented theory shows the usefulness of this approach. The results provided by the optimization method applied on a designed tubular machine illustrate its advantages. A numerical analysis performed on both a designed and then optimized structure confirmed the results obtained in the optimization process. Index Terms—Electromagnetic force, gradient method, magnetic circuit, numerical analysis, tubular linear machine. I. INTRODUCTION T HE demand for linear servo-controlled high-speed actu- ation, with high precision and a high bandwidth, has in- creased considerably in the past years [1]. One of the main ad- vantages of the linear electromagnetic machines over their ro- tary-to-linear counterparts is the absence of mechanical gears and transmission systems, which offers higher performance and improved reliability [1]. Hence, electrical machines with linear movement have become more important in common applica- tions such as healthcare [2], [3], transportation [4], [5], and var- ious electrical drives [6]. Linear structures present two topologies: flat-type or tubular. The most important shortcoming of the linear machines with a single-sided flat-type structure is the existence of a significant thrust force between the two armatures [7]. This disadvantage is avoided due to the radial symmetry of the tubular machines or the double-sided flat-type structures which determines the compensation of all the thrust forces acting around the circum- ference of the air gap. The double sided structures require so- phisticated double linear guidance systems, which have to pre- cisely assure the two constant air gaps and also to guide later- ally the moving armature together with the stators. On the other hand tubular linear motors need only much simpler linear ball or sleeve bearings. Extended researches have been conducted on flat-type or tubular linear structures with permanent mag- nets (PM) [8]–[10]. Despite their very good performances (high power and force densities, servo characteristics, efficiency) the drawbacks of such machines, such as the complex manufac- turing and high costs must also be considered [7]. Hence, dif- ferent types of linear machines without permanent magnets (es- pecially those with variable reluctance) can be seen as alterna- tive solutions for different applications. In a former study on Manuscript received November 11, 2012; revised January 25, 2013 and April 19, 2013; accepted June 04, 2013. Date of publication June 18, 2013; date of current version October 21, 2013. Corresponding author: D.-C. Popa (e-mail: Dan.Cristian.Popa@emd.utcluj.ro). 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/TMAG.2013.2269537 tubular linear reluctance motors, several advantages of tubular variable reluctance machines have been demonstrated [9]. An interesting application of such a machine driving a pump for the circulatory heart assistance is reported in [2]. In this paper, a tubular machine with a modular construction is proposed. The machine under study originated from the linear transverse flux reluctance machine with a modular construction. In [12] a flat-type linear transverse flux motor with PM was pre- sented. A simpler structure, without PM, but with similar per- formances, was analyzed in [13]. In [14] the general structure of the tubular linear transverse flux reluctance machine has been studied. An extended numerical analysis of the tubular motor mentioned above was given in [15]. A comparison between the flat type and tubular transverse flux reluctance machines, showing their fault tolerant capacity and the similarities of the two structures, was presented in [16]. In [17], the technologies used for the construction of this tubular motor and the basic concepts concerning the control of the motor were presented. A prototype of the tubular machine was analyzed and experi- mental tests were performed in [18]. This paper is focused on the development of an optimized design of the studied tubular machine. The optimization process is based on the maximiza- tion of the traction force under the assumption that the volume of the machine remains unchanged. The modular tubular transverse flux reluctance machine (MTTFRM) operates on the variable reluctance principle and belongs to the transverse flux machine class. The machine is without PM, only with electromagnetic excitation on the stator and passive mover. Its modular construction is presented in the following sections. A particular feature of both the stator and the mover of the machine is the use of magnetic pieces alternating with non-magnetic spacers. A semi-analytical analysis, based on the magnetic equiva- lent circuit, is the foundation of the optimization procedure ap- plied for this machine. The force is computed similarly as is in the case of other devices operating on the variable reluctance principle [7]. The optimization procedure is performed in order to obtain the maximum traction force. An optimal solution of the objective function can be found using the direction of its gradient. In order to distinguish between the minimum or the maximum of a function, the matrix of second derivatives has to 0018-9464 © 2013 IEEE