Journal of ELECTRICAL ENGINEERING, VOL. 62, NO. 4, 2011, 220{226 CALCULATION AND MEASUREMENT OF COIL INDUCTANCE PROFILE IN TUBULAR LINEAR RELUCTANCE MOTOR AND ITS VALIDATION BY THREE DIMENSIONAL FEM ∗ Ali MOSALLANEJAD | Abbas SHOULAIE This paper reports a study of coil inductance profile in all positions of plunger in tubular linear reluctance motors (TLRMs) with open type magnetic circuits. In this paper, maximum inductance calculation methods in winding of tubular linear reluctance motors are described based on energy method. Furthermore, in order to calculate the maximum inductance, equivalent permeability is measured. Electromagnetic finite-element analysis for simulation and calculation of coil inductance in this motor is used. 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. K e y w o r d s: tubular linear reluctance motors, maximum inductance, inductance profile, FEM analyzes 1 INTRODUCTION The tubular linear reluctance motor is a series of coils activated sequentially to pull the plunger along the bore. Linear Motors have been widely studied using the elec- Note that the plunger is only pulled, and is never pushed. tric circuit theory [1{5]. The linear reluctance motors This is a disadvantage of the TLRM when it is compared were considered under both ac and dc supply. Although to other synchronous accelerators that can be pushed and the ac motors have lower efficiency than the dc ones, they pulled by choosing the relative polarity of the armature are employed in many applications [6]. The linear motors and stator windings. differ in both construction and type. One of the known Tomczuk and Sobol [6] investigated tubular linear re- motors in the group of linear motors is tubular linear re- luctance motors (TLRMs) in various types of magnetic luctance motors (TLRMs), which can operate in different circuits. They analyzed magnetic field and calculated in- modes such as self- oscillating, switched-oscillating and tegral parameters of the field and also determined static accelerator. A tubular linear reluctance motor consists characteristics and electromagnetic parameters of the mo- of two main parts: a moving part, and stator. The stator tor. Also, in [7], the performance of the linear reluctance consists of an exciting coil that excites the magnetic field. oscillating motor operating under ac and dc supply is in- Also, the moving part is a ferromagnetic material which vestigated. Design of a reluctance accelerator and dis- is called plunger. The tubular linear reluctance motor in cusses the methods used in its design is described in [8]; its simplest form is presented in Fig. 1 [6, 7]. it also discusses the methods of control of the accelerator A tubular linear reluctance motor is an electric mo- and its predicted performance. In [9] Mendrela demon- tor in which velocity and magnetic force are produced by strated the design and principle of operation and perfor- the tendency of its movable part (plunger) to move to a mance of a linear reluctance self-oscillating motor. This position where the inductance of the excited winding is paper examined a mathematical model of a motor, per- maximized or the reluctance to the flow of magnetic flux mitting analysis of the dynamics and transients occurring is decreased. The resistance to the creation of magnetic in this circuit. flux in the material around the coil of the motor is called reluctance. Ferromagnetic material in the plunger reduces Many researches have been investigating TLRMs be- the reluctance and therefore magnetic force is developed havior, mostly focused on evaluation of TLRMs opera- due to the change in the reluctance of the material sur- tion modes and structure. While, to our knowledge, no rounding the coil as the plunger moves. The ferromagnetic research has concentrated on calculating maximum in- plunger has a greater magnetic permeability than the air ductance or profile inductance theoretically and in all it replaces. As a consequence, the magnetic flux can be the mentioned references, the minimum and maximum formed more easily when the plunger is centered on the inductance is obtained by measuring method. Even it is coil. At this point, the reluctance is minimum for a given mentioned in [10] that The dynamic inductance of the flux level; it is also the position of least energy. When motors was not calculated on the basis of the magnetic displaced from the centered position, magnetic forces will field analysis. They often were obtained with using mea- always act to restore the plunger to its centered position. sured inductance only. ∗ Electrical Engineering Department, Iran University of Science and Technology (IUST), Iran University of Science and Technology, Narmak, Tehran, Iran, PO Box 1684613114, [email protected] DOI: 10.2478/v10187-011-0035-x, ISSN 1335-3632 ⃝c 2011 FEI STU Journal of ELECTRICAL ENGINEERING 62, NO. 4, 2011 221 as follows h i (π ) L = L 1 + cos x + L ; (1) m l min L − L L = max min ; (2) m 2 where Lmin is the inductance of the coil without the plunger and Lmax is the inductance of the coil with the plunger placed in the centre of the coil, so that (x = 0). In other words, inductance should be determined in Fig. 1. Tubular linear reluctance motor with open magnetic sys- tem [4] the presence (Lmax) and in the absence of the plunger (Lmin). According to (1), inductance profile calculation is done for two different basic positions of plunger in this motor. The two positions take place when the coil is without the plunger or the plunger is fully inside the coil. The two aforementioned positions yield the minimum and maxi- mum inductance, respectively. In other words, Eq. (1) re- ceives the measured maximum and minimum inductances and estimates the inductance profile for the other posi- tions. But our proposed method is able to calculate the coil inductance profile proportional to each plunger posi- Fig. 2. Tubular linear reluctance motor with open magnetic system tion. when the plunger placed in the centre of the coil There are many methods for calculation of coil mini- mum inductance. Most of them are experimental [12{17]. While, no methods are presented in literature to calcu- late the maximum inductance. But in this paper a new approach in maximum inductance calculation is proposed based on energy method. 2.1 Maximum Inductance Calculation Applying electrical energy to the Tubular Linear Re- Fig. 3. Schematic of the iron core luctance Motor, the plunger is pulled into the coil due to a strong magnetic field generated by a coil current. When plunger enters to coil, the inductance increases and In this paper, a new approach in maximum inductance reaches to its maximum value. calculation is proposed based on energy method. In en- The coil inductance with the plunger placed in the ergy method, it is necessary to calculate the field inten- centre of the coil, so that x = 0, is maximum that is sity inside the iron core. Furthermore, in order to calcu- shown in Fig. 2. late the maximum inductance, equivalent permeability is In this paper energy method is utilized in order to measured. Therefore, in this paper, field intensity inside calculate the maximum inductance. The stored energy in the iron core is calculated with regard to eddy current. an inductor is actually stored in its surrounding magnetic Finally, calculation, simulation, and experimental results field. Hereby, an explicit formula for the stored energy in of coil inductance are compared. a magnetic field is obtained. The stored energy in the coil when a current I flows through the winding is as follows 1 2 METHODS OF THE INDUCTANCE W = LI2 (3) PROFILE CALCULATION 2 where L is the self-inductance. Also the stored energy in Operation of the tubular linear reluctance motor de- the solenoid can be rewritten as follows: pends on the inductance profile of the machine. The ma- chine inductance is related to machine dimensions such B2 as the coil and plunger, excitation currents, and plunger W = V; (4) µ0 position. Therefore, calculation of motor inductance with NI a good accuracy is very important. The winding induc- B = µ0 ; (5) tance, L, is determined according to plunger position. l Considering that the inductance of the coil is dependent where V is the volume of the solenoid. The above equa- on the position of the plunger, the inductance is estimated tions are simply used when coil is without the plunger. 222 A. Mosallanejad | A. Shoulaie: CALCULATION AND MEASUREMENT OF COIL INDUCTANCE PROFILE IN TUBULAR ::: If plunger enters the coil, it is necessary that the field If the inner diameter of winding is equal to iron core intensity inside the iron core to be calculated. diameter, then the above equation has a very high accu- Usually in electrical motors the iron core is laminated, racy, otherwise its accuracy will be decreased; In TLRMs, while in TLRMs, it is integrated. Therefore, the high eddy (12) is not usable since there is air gap between plunger current is produced inside the core. and coil; therefore in this paper, equivalent magnetic per- Figure 3 shows a cylindrical core of the linear reluc- meability (µe ) is measured by the experimental results. tance motors without coil. In Fig. 3, R is the cores radius Substituting (11) and (10) into (9), Equation (9) can and lc is the length of the iron core. The current i(t) in be rewritten in the form the winding flows in the ϕ direction of the cylindrical Z µ H (r; t)l R coordinate system shown in Fig. 3. c c = Ni(t) + l J (r; t) dr ; 0 < r < R : µ c θ With flowing current in the coil, in one dimensional e r (14) analysis, the flux line and the magnetic field intensity From (21), we can write vector in the core has only the component along the z axis Z which depends only on the r coordinate along the cores h R i @Hc(r; t)l lc µe @ lc µe radius and time, t, and the eddy current density vector = Jθ(r; t)dr = − Jθ(r; t) : @r l µ @r l µ has the ϕ-directed component J (r; t) only.