Ali MOSALLANEJAD1, Abbas SHOULAIE2 Iran University of Science and Technology, Tehran, Iran (1)& (2) Calculation of Coil Inductance in Tubular Linear Reluctance Motor using Three Dimensional FEM Abstract. This paper reports a study of coil inductance in every plunger positions in tubular linear reluctance motors (TLRMs) with open type magnetic circuits. In this paper, all inductance (minimum and maximum inductance) calculation methods in winding of tubular linear reluctance motors are described. Furthermore, a new approach in maximum inductance calculation based on energy method is proposed. Electromagnetic finite-element analysis is used for motor simulation and coil inductance calculation. Moreover, inductance of a prototype TLRM was measured with experimental methods. Simulation results of coil inductance calculation using 3-D FEM with coil current excitation is compared to theoretical and experimental results. The comparison yields a good agreement. Streszczenie. Przedstawiono analizę indukcyjności cewki stosowanej w rurowym reluktancyjnym silniku liniowym z otwartym obwodem magnetycznym. Zastosowano metodę elementu skończonego do symulacji silnika i obliczania indukcyjności cewki. (Obliczenia indukcyjności cewki w rurowym liniowym silniku reluktancyjnym) Keywords: Tubular linear reluctance motors, minimum and maximum inductance, FEM analyses Słowa kluczowe: silnik reluktancyjny, indukcyjność, metoda element skończonego Introduction and pull by choosing the relative polarity of the armature Using electric circuit theory, linear motors under both ac and stator windings. and dc supplies have been greatly studied [1– 5]. It must be TLRMs have been investigated in various types of mentioned here that linear dc motors have lower efficiency magnetic circuits with analyzing magnetic field, calculating than ac ones. However, they are widely used in lots of integral parameters of the field and determining static applications [6]. Linear motors have several types which characteristics of the motor in [6]. Moreover the differ in construction. TLRM is a linear motor that can performance of the motor under both ac and dc supplies is operate in different modes of operation such as self- studied in [7]. oscillating, switched-oscillating and accelerator. Design rules of a reluctance accelerator are described in A TLRM consists of two major parts i.e. a moving part, [8]; this paper also discusses the accelerator control also known as plunger, and a stator. Stator is a winding methods and their predicted performances. TLRM different which is fed by a dc source and produces the magnetic modes of operation are discussed completely in [9]. This field. A TLRM in its simplest form is illustrated in Fig. 1 paper also discusses the design and modeling of TLRM in [6],[7]. self-oscillating mode. Equivalent circuit parameters of TLRM are obtained using experimental method in [10]. Also in [11], a permanent magnet linear oscillating accelerator is analyzed theoretically and validated via experimental results. The coupled field-circuit model of the three-stage reluctance accelerator is presented in [12]. Another work has studied the design features of variable-air gap, cylindrical and variable reluctance actuators [13]. Many other researches that have been investigating TLRMs behavior, mostly focused on evaluation of TLRM’s operation modes and structure. While, to our knowledge, no research has concentrated on calculating minimum and maximum inductance, and, in all the mentioned references, Fig 1. Tubular linear reluctance motor with open magnetic circuit the minimum and maximum inductance is obtained by measurement method. Even it is mentioned in [14] that” The A TLRM is a linear motor which its operation is based on dynamic inductance of the motors was not calculated on the the tendency of its movable part to move to a position with basis of the magnetic field analysis. They often were higher inductance or, in other words, lower reluctance. This obtained with using measured inductance only”. is exactly how force and velocity is produced in this motor. In this paper, all methods of minimum inductance As the plunger moves into the coil, its ferromagnetic calculation are described and a new approach in maximum material reduces the reluctance and therefore magnetic inductance calculation is proposed based on energy force is developed due to the change in motor reluctance. method. In the proposed method, it is necessary to The ferromagnetic plunger has a greater magnetic calculate the field intensity inside the iron core. Therefore, permeability than the air it replaces. As a consequence, the in this paper, field intensity inside the iron core is achieved magnetic flux can be formed more easily when the plunger with regard to eddy current. Finite element method is used is placed in the center of the coil. At this point, the for motor simulation and its inductance calculation in every reluctance is at its minimum for a given flux level; it is also plunger position. For evaluating simulation and calculation the position of the least energy. When displaced from the results, a prototype TLRM was built and its inductance was centered position, magnetic forces will always act to restore measured. Finally, calculation, simulation, and experimental the plunger to its previous position. TLRM is consisted of a results of coil inductance are compared. series of coils activated sequentially to pull the plunger along the coil. Note that the plunger is only pulled, and is Methods of the inductance calculation never pushed. This is a disadvantage of TLRMs when In the following sections, all the methods used for compared to other synchronous accelerators that can push TLRMs windings inductance calculation are described. Coil PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 87 NR 11/2011 271 inductance calculation must be performed for two main where: r : radius of the ith ring, : radius of the jth ring, plunger positions i.e. when the plunger is completely i rj outside the coil, and when the plunger is fully inside the coil. d ij : axial distance between ith and jth rings and m is The two mentioned positions yield to the minimum and obtained from: maximum inductances, respectively. In the first subsection, several methods for calculating the minimum inductance is 4r r presented, and in the second subsection the proposed m i j (8) 2 2 method for calculating maximum inductance is explained. ri rj di j A. Minimum inductance calculation K(m) and E(m) represent the first and second type of Operation of the tubular linear reluctance motor elliptical integrals, respectively. The self inductance of the ith depends strongly on the motor inductance profile. ring is obtained from [16]: Therefore, calculating the motor inductance with a good accuracy is very important. The motor inductance is 8r proportional to different factors such as, coil and plunger i (9) Li ri ,hi 0ri Ln 0.5 dimensions, excitation currents, and plunger position. The hi winding inductance, , is determined according to the L th plunger position. Since the coil inductance is dependent on where hi is the diameter of the i ring conductor of the motor the plunger position, the inductance profile is estimated as coil. The self inductance of the coil is given by: follows: nn (1) L Lm 1 cos x Lmin l (10) Ln1,n1 Li Li j 11 i L L i j (2) L max min m 2 In the above equation, Lij and Li are presented in (7) and where: Lmin : Coil inductance when the plunger is out of the (9), respectively. coil, L : Coil inductance when the plunger is completely Another Method for calculating coil minimum inductance max in TLRM is explained in the following. A simplified form of inside the coil, so that (x= 0). an open magnetic type of TLRM without plunger is shown in The coil minimum inductance is approximately equal to Fig. 2 [17]. air core reactor inductance. In the following, all methods of minimum inductance calculation are presented. In general, the coil inductance L is determined from the magnetic flux linkage λ as: N (3) L i i (4) m Ni Where: N is the number of coil turns, m is the permeance of the main flux . A (5) m 0 r l Fig. 2. Schematic showing the simplified coil geometry Substituting (5) into (4) and (4) into (3) yields: N 2 A The self inductance of the coil is given by: (6) (11) 2 L 0r L dN (H ) l Equation (6) is the main formula for inductance where, calculation, but using (6) for calculating TLRMs minimum 2 0.1 inductance is not accurate enough. Therefore, other methods with more satisfactory results should be extracted. (12) 2 1 0.45 One of the most common models in the electromagnetic 3 2 analysis of electrical motors and electromagnetic devices l c are current filament models. In this paper the current (13) , filament methodology is used to calculate coil minimum d d inductance in TLRM. d d (14) d 0 1 The axisymmetric configuration of the coil allows the 2 subdivision of the coil into coaxial rings, in which a uniform current density is assumed. In above equation: l length of coil, c thickness of coil, Since the assumed rings are coaxial, the following formula is used for determining the mutual inductance d i coil inner diameter, d o coil outer diameter. th th between i and j rings [15]: Experimental results indicate that the minimum inductance calculated using (10) and (11) has a satisfactory ri r j m accuracy. It should be mentioned that the response time of Li j 2 0 1 Km E m m 2 current filament method is increased if the number of coil (7) turns is increased. As turn numbers of our TLRM was considerably high, equation (11) was used for calculating the minimum inductance. 272 PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 87 NR 11/2011 An approximated formula for calculating minimum In Fig.4, R is the core’s radius and l is the length of the inductance has been presented by [18] as: c iron core.