Efficiency Maximization of the Air Core Double-Sided Permanent Magnet Linear Synchronous Motor

Air core permanent magnet linear synchronous motor (ACPMLSM) recommended for applications in which the accurate control of speed and position is required. Omission of the core in the primary of the motor reduces the detent force and increases the controllability of the motor. In this paper, the flux density of different parts of an ACPMLSM is calculated using both the Maxwell’s equations and finite element method (FEM). A precise flux density model is presented and the design of the air core double sided permanent magnet linear synchronous motor is optimized for the efficiency maximization using the genetic algorithm method.


I. INTRODUCTION
The permanent magnet linear synchronous motors (PMLSM) have taken more attention than linear induction motors because of high force density and efficiency, low losses, satisfactory dynamic performance and easy control [1]- [6].However, the detent force generated by the slotted structure and the end effect caused by the limited length of the moving part are the main disadvantages of slotted linear motors.Some techniques such as skewing the slots and optimizing the permanent magnet (PM) and the width of the primary winding have been already suggested to reduce the detent force [5]- [7].Position and speed control of a PMLSM can be improved by mitigation or alleviation of the detent force.Since using the air-core instead of the iron core in a permanent magnet linear synchronous motor considerably reduces the detent force; the air core machine is usually recommended for high precise applications.
Some investigations are carried out on the optimal design of the PMLSM.It has been shown that some increases in the thrust can be achieved by modifying the shape or dimension of the permanent magnet and width of the winding [7], [8].The ripple reduction has been the other concern of earlier studies [7].Authors in [9], have been optimized the motor design in order to reduce the electrical time constant.In [10] increasing the motor thrust and reducing the magnet Manuscript received June 15, 2012; accepted February 16, 2013.consumption have been the goals for the optimization.In [11], the output power maximization or loss minimization has been done for a tubular permanent magnet linear synchronous motor.
In general, the efficiency of a small PMLSM employed in a control system, is not considered as the main objective, and so the most efforts as mentioned earlier have been conducted to improve the transient responses of the machine.However, the efficiency is one of the most important criteria when employing a motor for the energy conversion purposes.Due to the increasingly development of the linear motors in the industry, the efficiency optimization needs more attention.
In this paper, efficiency optimization of a four pole ACPMLSM is investigated.The paper is organized as the following: in section 2, the basic structure of the ACPMLSM is presented and the procedures to evaluate the air gap flux density distribution are discussed.The flux density distributions are obtained by the use of both Maxwell equations as an analytical method and the finite element method as a numerical method.Finally, a comparison between the results is given in section 2. Section 3 involves the design formulations and modelling tasks of the ACPMLSM defining the existing relationships among some performance characteristics and the design parameters.The optimization algorithm and the results are presented in section 4. Finally, the conclusion and discussion are presented in section 5.

A. Motor structure
The structure of an ACPMLSM is shown in Fig. 1 in which the moving part of motor is a short primary and consists of a three phase air core winding.Efficiency Maximization of the Air Core Double-Sided Permanent Magnet Linear Synchronous Motor The secondary part involves the N and S permanent magnets located on the surfaces of the back irons.The primary and the secondary parts are separated by an air gap g.The parameters and the dimensions of the motor are presented in Table I.
Supposing that the primary winding is not excited, an ACPMLSM, as shown in Fig. 2, contains two layers of iron extended along x axis, two layers of PMs and a layer of air gap.
The following assumptions are made for the magnetic field calculation [3], [4]: 1) All the regions (shown in Fig. 2) are extended along x ± ; 2) The end effects along z axis are neglected; 3) The permeability of the PM is considered equal to vacuum permeability µ 0 ; 4) The permeability of the secondary yoke is infinite.
where A is the magnetic vector potential.Flux density B is obtained by using the vector potential as The subscripts determine the regions and 2

J
is the equivalent current density defining the whole permanent magnet effects that can be expressed as the Fourier series [4] 2 1,3,...

sin( )
with coefficients where is the spatial frequency.Considering Maxwell's equations ( ), the boundary condition that must be satisfied at the border surface of two materials is as below [4] 1 2 where n ˆ is a unit vector normal to the boundary surface directed from region 2 to region 1 and H is the magnetic field intensity.Equation ( 5) can be expressed as below [4]: Given the boundary conditions ( 6) and the excitation 2 J the X-Y components of the flux density can be expressed as below [4]: In the rest of this section, the flux density of the primary winding is calculated using the method in [1].In this method, at the first step the magnetic potential of the primary winding s A is calculated based on the magneto motive force (MMF) of the traveling magnetic field.At the next step, the distribution of the flux density s B is calculated using the magnetic potential.According to the method in [1], the amplitude of the ν th harmonic of MMF produced by the primary winding is equal to where a I is the RMS value of the phase current, p is the number of pole pairs, ν w k is the winding factor for the th ν harmonic and c K is the Carter's coefficient.According to Laplace equation and the boundary conditions the components of the flux density generated by the primary winding in the middle of the air gap can be expressed as follows: )

C. Analytical and FEM results of flux density
In this section, the results of the flux density obtained by the analytic method i.e.Maxwell's equation, is compared to those evaluated by the FEM.X and Y components of the air gap flux density distribution are presented as a function of displacement in Fig. 3 and Fig. 4 when the motor is open circuited.Y component of the flux density in the middle of the magnetic air gap is illustrated in Fig. 5.  Nonlinearity and saturation of the iron core are considered by FEM analysis using the corresponding B-H curves of the ferromagnetic materials.In addition, structural complexity such as slotting can be simply simulated by FEM.As depicted in Figs. 3 and 4, the results obtained by the analytic method and FEM are nearly similar.
In the ACPMLSM, the effective length of the magnetic air gap is large.Therefore, as shown in Figs. 4 and 5  Fig. 7 to Fig. 9 illustrate a few samples of the first harmonic of the vertical and horizontal air gap flux densities generated individually by the PMs and primary currents which are evaluated using Maxwell's equations and FEM.As seen in the figures, the air gap flux density is mainly produced by the PM.Also a comparison between the first harmonics of the flux densities calculated by the analytical model and evaluated by FEM reveals the validity of the analytical modeling approach of the flux densities.

III. BASIC EQUATIONS OF SYNCHRONOUS MOTOR
The phasor diagram of a synchronous motor is shown in Fig. 10.The following equations are deduced from the vector diagram of Fig. 10 [1]: where δ is the load angle which is the angle between the terminal phase voltage 1 V and the no-load voltage f E .
Parameters sd X and sq X are the d and q axis reactance respectively, and 1 R is the resistance per phase of the primary winding.10.The vector diagram of the synchronous motor.
According to ( 13) and ( 14), the d and q currents are as below [1]: The rms value of the primary current can be evaluated versus 1 V , f E , sd X , sq X , δ and 1 R using 2 2 .
A. Thrust calculation Enduring d-q model of the machine in a synchronously rotating reference frame is used for the optimal design purposes.In this model, the iron saturation is neglected and the flux density distribution along the air gap assumed to be sinusoidal.Therefore, the motor thrust can be calculated by [12] ( ) ( ) in which PM λ is the linkage flux per phase caused by the PMs and d L and q L are the d axis and q axis inductances, respectively.For a PM motor made by the rare earth materials, the relative permeability is somewhat equal to 1 and thus the material is far to be saturated so that and q L yields to a small component of the reluctance force which is neglected here.Therefore, the mean value of the thrust is simply written as

B. Losses calculation
For an ACPMLSM, the resistance of the primary winding is considerable compared with the leakage reactance.The copper loss of an ACPMLSM is obtained by 2 1
Iron loss consists of the hysteresis loss and eddy-current loss.It can be estimated using [13] 2 2 3 , [ ], t h e h m e m where m B is the maximum flux density, h k , e k and β are constants.Constant parameter β has a value from 1.8 to 2.2 depending on the laminated materials.Variations of the flux density in the yoke can be approximated by Fig. 11.At the first stage of designing, an initial estimated value is chosen for the maximum flux density of the yoke.The eddycurrent loss density of the yoke is expressed as [14] 2 3 , max

8
, [ ], where s v is the linear speed and max , y B is the maximum value of the yoke flux density as shown in Fig. 11.In (22), the longitudinal component of the eddy-current loss has been ignored.This component is simply taken into account by a constant c k with the value chosen within 1.1 to 1.2 [14].The eddy-current loss of the motor is then written as [14]  is the output power and add P is the additional losses consisting the mechanical and stray losses.

IV. OPTIMIZATION
The optimization problem with n number of variables, m number of constraints and the objective function ( ) , is defined as [15] ( ), where the parameter K is defined as In the previous equation ( ) determines the limits of the design variables.The design parameters are optimized considering the objective functions defined such as thrust per volume or mass maximization, volume of the PM minimization and power per volume or mass maximization.In this paper, the design parameters are optimized for the efficiency maximization using the genetic algorithm.
For the purpose of maximizing the efficiency of the motor, four design variables are optimized.These four variables are height of the PM, air-gap, height of the primary winding and height of the secondary yoke.The width of the motor, current density and pole pitch are assumed constants specified properly.

A. Constraints
Thermal stress and demagnetizing boundary of the PMs are considered as constraints.In addition, to prevent saturation of the yoke, the minimum values of its dimensions are considered.
The thermal stress which depends on the total losses of the motor is given as [16] To avoid demagnetizing the PMs, primary currents must be limited so that [16] sin , where D B is the critical or minimum allowable magnetic flux density of the PMs.The value of D B is about -0.2 tesla for the Nd-Fe-B magnet.
A minimum value for the height of the secondary yoke can be obtained by using the following equation

B. Simulation results of optimization
As mentioned earlier, efficiency maximization is the objective of the optimization problem of this paper.The rated power out P and linear speed s ν of the motor are assumed 1000 W and 4.65 m/s at 50 Hz supply frequency respectively.The additional loss is assumed 4% of the output power.The design variables and their variations ranges are listed in Table II.The temperature limit of the motor is assumed 100 °C and the heat exchange coefficient of the surface is assumed h=24.6 (W/m 2 °C).The maximum flux density of the yoke is Optimized design variables obtained by the genetic algorithm are given in Table III.The results show that the efficiency of the optimized motor has been increased by 4.23% in comparison with the initial design.Also, despite the copper loss has reduced by 59.7%, the iron loss has increased by 140%.However, the influence of the copper loss on the efficiency is more than that of the iron loss, so the efficiency of the optimized machine is higher.
Harmonic contents of the flux density produced by the PMs are evaluated using FEM.These Harmonics for the initial design and optimized machine are shown in Fig. 12.
The amplitude of the first harmonic of the flux density before and after the optimization is 0.62 and 0.78 Tesla, respectively showing an increase of the flux density for the optimized machine.However, as expected an increase of the

Fig. 3 .
Fig. 3. X component of the flux density in the middle of the air gap.

Fig. 4 .
Fig. 4. Y component of the flux density in the middle of the air gap.

Fig. 5 .Fig. 6 .
Fig. 5. Y component of the flux density in the middle of the magnetic air gap.

Fig. 7 .Fig. 8 .
Fig. 7.The first harmonic of vertical component of the air gap flux density produced by the PM using analytical method and FEM.

Fig. 9 .
Fig.9.The vertical and the horizontal components of the air gap flux density generated by the primary currents at ωt=0 using analytical method and FEM.

TABLE I .
PARAMETERS OF THE STUDIED MOTOR.