Development of an Analytical Method to Predict the Behaviour of the Magnetic Field in PM Linear Motors with Halbach Array

This paper describes the application of the method of separation of variables and the use of Fourier series for solving the Laplace ́s and Poisson ́s equations on the study and analysis of the magnetic field produced in a linear motor with Halbach array. Equations for predicting the 2D magnetic flux density distribution produced in the air gap were developed. The model was validated by means of finite element analysis and by measurements carried on a prototype of the linear motor. Theoretical results helped understand the behaviour of the magnetic flux density in the air gap and to obtain the values of the static propulsion force and normal force in such machine.


I. INTRODUCTION
Fourier series can be applied on the solution of many problems in Electrical Engineering, especially on the study of electrical circuits and on the electric and magnetic field analysis. In the magnetic field analysis, it can be employed for predicting the magnetic flux density distribution in electromagnetic devices and for calculating the involved forces. The Fourier series was presented by Jean Baptiste Joseph Fourier (1768-1830) in his Théorie Analytique de la Chaleur, in 1822 [1]. It allows the representation of a periodic nonsinusoidal function in terms of an infinite sum of sines and cosines.
Maxwell´s equations form a set of partial differential equations that can describe the behaviour of magnetic and electric fields and the relations between them. In order to obtain the equations of fields in the electric motors, the solution of Laplace´s and Poisson´s equations can be involved. These equations are subject of study in Electrical Engineering courses, principally due to their application on electromagnetic problems.
On the study of the magnetic field in electromagnetic linear motors, the equation of the magnetic flux density demands the representation of the device under study in a model by using a suited system of co-ordinates, where boundary conditions are imposed. The model is based on the magnetic circuit of the device and must be divided into regions, and the electric and magnetic properties of those regions must be defined. The conditions of the magnetic field between the regions are specified, too.
can develop movement through a straight line directly by means of a linear propulsion force produced by electromagnetic interaction between the translator and the stator. This type of motor does not require mechanical systems that convert rotation into linear movement. It is suitable for applications that require development of propulsion force in a specific direction.
In graduate and undergraduate courses, the study and development of electromagnetic linear devices is one of the lines of work in Electrical Engineering. In such undergraduate courses, the study of the Laplace´s and Poisson´s equations for the solution of the electric or magnetic field is present on Electromagnetism syllabus. Commonly, the bibliography suggests exercises where the application of those equations is used for the solution of the electric potential in problems related to the electric field content.
A proposed case of study about the use of Fourier series, Laplace´s and Poisson´s equations applied to an electromagnetic device is a simple electromagnetic linear motor that can rely on a mover with permanent magnets and a stator with armature windings mounted around a slotless ferromagnetic armature core. In such devices, the mover can develop movement through a straight line by means of a propulsion force. The focus of an analytical study is to obtain the equations of the magnetic field and the involved forces.
In this paper, the equations of the magnetic field and the propulsion and normal forces produced by a device with permanent magnets arranged in a Halbach array are deduced. In the device, a nonferromagnetic plate holds the permanent magnets, because in that kind of array the magnetic field is established through a path that involves only one polar surface of the permanent magnets. As part of the study, a prototype of the device was built in the laboratory and was employed for validation purposes of the analytical model.

II. STUDY OF A DC LINEAR MOTOR WITH PERMANENT MAGNETS IN A QUASI-HALBACH ARRAY
The analytical model presented in this paper was deduced for a DC linear motor as shown by Fig. 1. It has a mover or translator with permanent magnets in a quasi-Halbach array topology and a stator with armature winding mounted around a slotless ferromagnetic armature core [9]. Table I presents the principal characteristics of the linear motor constructed. It has nine NdFeB permanent magnets arranged in a quasi-Halbach array assembled on an aluminium plate. The armature core is made of devices, the mover can develop movement through a straight line by means of a linear propulsion force. The array was mounted in order to have the magnetic flux established through a path that involves only one polar surface of the permanent magnets, the air gap and the ferromagnetic armature core. This means that the magnetic flux that passes through the upper polar surface of array core is very weak, and a nonmagnetic core can be used only to support the permanent magnets mechanically.
The mass reduction due to the employment of a nonmagnetic material in the PM assembly is desirable when high acceleration is required. The magnetic flux density distribution in the air gap presents a sinusoidal shape in a Halbach array instead a flattened sinusoidal shape produced in the PM linear motors without this kind of array. In the case of a flattened sine shape, the wave becomes squarer, with many odd number harmonics. That results in electromagnetic force ripple and variable force.  corresponding to the side of the square polar area of each permanent magnet.
The distribution of the magnetic flux density established in the air gap depends on the magnetic field produced by the permanent magnets and on the magnetic field produced by the currents in the phases of the armature windings. The magnetic field due to the permanent magnets was analysed separately from the magnetic field produced by the armature windings. So, the magnetic field due to the current in the windings is not considered by the present analysis and the region between the boundaries O and B has the same magnetic properties as the air. In the free space of air, the magnetic flux density vector, B  , and the magnetic field vector, H  , are related by , where 0  is the magnetic permeability of the vacuum. The analytical model was developed in terms of the magnetic scalar potential, ψ [5]. So, the magnetic field, H  , is equal to the negative gradient of ψ , according to . Applying this equation to the second Maxwell´s , and taking into account the relation between B  and H  , one can obtain the Laplace´s equation in terms of the magnetic scalar potential for the air gap, g ψ , given by 0 In 2D rectangular coordinates, it can result in: (1) A solution method to (1) involves the determination of a field function, which satisfies the Laplace´s equation, the imposed boundaries and the field conditions in the permanent magnets region [2]- [4]. The method of separation of variables allows solving (1). Taking into account that g ψ can be expressed as the product of two functions, X(x), and Y(y), (1) is modified to X(x)Y(y) = y) (x, ψ g [4] [5], where X(x) is a function that depends only on x, and Y(y) depends only on y. After some operations, one can get (2).
The sum of the terms in the left side of (2) is equal to zero, and the variables are independent, so, each term is equal to a constant and the variables can be separated according the set of two equations presented in (3) The magnetic scalar potential was considered equal to zero on the planes   Fig. 3 shows the graphs of the components of the magnetization vector. The Fourier series that represents the behaviour of the x-component of the magnetization vector is presented in (6). The y-component is represented by (7). The homogeneous term of pm ψ is calculated following the same steps employed to obtain (5) and it is equal to (8).
The particular term, p ψ , is solved by Poisson´s equation , presented in (9).
The equation of the magnetic scalar potential in the PM region, pm ψ , is defined by (11): Constants 1 k , 2 k , 3 k and 4 k are obtained by means of the boundary conditions. The boundary conditions are the following: , is equal to zero and this results in , the y-component of the magnetic field in the air gap, .
Those conditions allow one to obtain the equations of the potentials. Eq. (12) presents the expressions of the magnetic scalar potential in the air gap: The x-component of the magnetic flux density in the air-gap, where ' y M and ' p ψ are given respectively by (14) and (15).
The y-component of the magnetic flux density in the air-gap, By using the expression of    The calculation of the magnetic field in the region between boundaries O and B , Fig. 2 . This is the Laplace's equation for the magnetic vector potential in the region of the armature phases and it can be expressed by (18): The solution of (15) is equal to the sum of two terms, or  [7], or: In the solution of (19) is employed the same steps of the solution of (1) using the method of separation of variables. The magnetic vector potential is consider equal to zero on the planes The particular solution is equal to: where   A results in: In the ferromagnetic armature core, under the plane y = 0, the magnetic vector potential also The region between the boundaries B and G, Fig. 2 The region between the boundaries G and P, Fig. 2, is also free from currents. The equation of the magnetic vector potential also assumes the form of the Laplace's equation and is given by (26  B and G , Fig. 2. The conditions of the magnetic field are the following [6]: . This condition implies that 6 k must be equal to zero; is the magnetic permeability of the permanent magnets,  Fig. 2; . This condition implies that 11 k must be equal to zero.
The resulting equations of the magnetic vector potential in the winding region and in the free space in air are given, respectively, by: )(e e (e a g l m n(l g nl g nl

D. The Normal Force
A normal force is also present, and it is the result of the magnetic attraction between the permanent magnets and ferromagnetic core of the armature. The propulsion force was obtained by Laplace´s force. In the analysis of the normal force, Maxwell Tensor was employed for obtaining its equation.  [10]. The integration of y dF over a closed surface that involves entirely the permanent magnets produces the normal force the acts between the permanent magnets and the ferromagnetic material of the armature. The lower surface of the closed surface is located on the boundary G. The upper surface is located on   y where the magnetic field is considered equal to zero, so, the integration over this surface is zero. On the laterals sides, the integration over the terms yx T and yz T cancel one to each other [2]. By this way, the normal force results of the integration of yy T through the surface located on boundary G ( g l y  ), where are located lower polar surfaces of the permanent magnets, according (40).   ,.. where: using force sensing resistors. It provides an inverse change in resistance in response to an increase/decrease in applied force. Fig. 5(a) presents the test rig for measurement of propulsion force with the actuator suspended by means of a structure that keeps it static. During the tests, only four phases located under permanent magnets with normal magnetization were fed by current. Fig. 5(b) shows the force sensing resistor and its position in the structure used for measurements. The measured values were obtained by means of a gaussmeter. The behaviour of the magnetic flux density in the plane y = 6 mm presents a sinusoidal shape as expected instead a flattened shape.        Fig.9. Graph of the normal force vs. current in the coils of the armature winding.