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This paper presents a model for analyzing a five-phase fractional-slot permanent magnet tubular linear motor (FSPMTLM) with the modified winding function approach (MWFA). MWFA is a fast modeling method and it gives deep insight into the calculations of the following parameters: air-gap magnetic field, inductances, flux linkages, and detent force, which are essential in modeling the motor. First, using a magnetic circuit model, the air-gap magnetic density is computed from stator magnetomotive force (MMF), flux barrier, and mover geometry. Second, the inductances, flux linkages, and detent force are analytically calculated using modified winding function and the air-gap magnetic density. Finally, a model has been established with the five-phase Park transformation and simulated. The calculations of detent force reveal that the end-effect force is the main component of the detent force. This is also proven by finite element analysis on the motor. The accuracy of the model is validated by comparing with the results obtained using semianalytical method (SAM) and measurements to analyze the motor’s transient characteristics. In addition, the proposed method requires less computation time.

Permanent magnet linear synchronous motors (PMLSM) have been developed for many years [

An accurate model is important in analyzing and controlling PMTLMs. In this regard, the calculation of air-gap magnetic field is critical in modelling the motor. This is because the air gap flux density distribution has a deep influence on the thrust ripples [

Modified winding function approach is another analytical method which gives insight into the calculations of parameters without considering the complicated boundary conditions. It is a simple, fast modelling method for motors. This paper shows the detailed calculations of the parameters for a five-phase FSPMTLM, such as the detailed calculation of the detent force, which are not expressed distinctly in [

Figure

Specifications of the PMTLM.

Symbol | Quantity | Value |
---|---|---|

| Axial width of module | 18 mm |

| Axial width of slot open | 12 mm |

| Axial width of armature | 10 mm |

| Axial thickness of module core (leg) | 3 mm |

| Axial space of slot pitch | 21 mm |

| Pole pitch | 15 mm |

| Height of armature coil | 30 mm |

| Height of module | 35 mm |

| Air gap (mechanical clearance) | 1 mm |

| Axial magnetic separation distance | 3 mm |

| Axial ferromagnetic core width between PMs | 7 mm |

| Axial PM width | 8 mm |

| Outer radius of tubular reaction rail | 15 mm |

| Inner radius of tubular reaction rail | 9 mm |

| Axial width of stator | 102 mm |

| Number of turns | 280/slot |

Physical structure of PMTLM and the schematic outline of the halved motor cross section: (a) motor physical; (b) cross section of the motor.

MWFA was first proposed for solving air-gap eccentricity in rotary machines [

Different from the rotary machines, the analysis of the modified winding function requires creating an appropriate coordinate system on the stator. As shown in Figure

Due to the influence of the fractional slot on the winding function distribution, the modified winding function of the five-phase FSPMTLM is defined as the product of winding function and the winding factor

Winding function of phase “

In Figure

The generation of air gap flux density is due to the current flowing in each phase. Hence, the flux density in phase “

Calculation of the inverse air gap function

Flux paths due to the stator slot and mover saliency.

The detailed computations of the inverse air-gap function

In terms of the modified winding function theory, the inductances and the permanent magnet flux linkages are derived from the air-gap function and the modified winding functions. Both are computed for the volume of the air-gap section of the motor with the effective air-gap radius

The schematic diagram of computational volume in the air gap.

In Figure

The self and mutual inductances are computed in the computational volume, as shown by [

The permanent magnet flux linkages are derived from the air-gap flux density produced by permanent magnets and the modified winding functions, as computed by

The detailed calculations of the effective air-gap computational radius

The stator voltage equations and mechanical thrust equations compose the mathematical model of PMLMs [

The detent force is the interaction force between the mover magnets and stator slots without the stator currents flowing [

Using the Virtual Work Method (VWM),

The detailed calculations of the detent force obtained by using the MWFA are shown in Appendix

Figure

Calculations of slot force, end force and detent force distribution versus mover position.

Five-phase FSPMTLM model using finite element method.

Seen from Figures

As seen from Section

Using (

In papers [

Establishment of the studied motor model on Matlab/Simulink platform.

In Figure

Subsystem of the proposed model: (a) current balance subsystem, (b) mechanical balance subsystem.

In Figure

The simulations using the proposed model to analyze the motor transients are carried out. This is done for the three cases given in [

The velocity of the mover is set to 18 mm/s and there is no load. The simulation results of the mover position (versus time) and the velocity of the mover (versus time) are shown in Figures

Figure

Transient characteristic of the mover position under no-load for

Simulation results of the mover velocity under no-load for

The velocity of the mover is set to 1 m/s and no load is installed on the mover. The simulation results of the mover position (versus time) and the velocity (versus time) are shown in Figures

Figure

Transient characteristics of the mover position under no-load for

Simulation results of the mover velocity under no-load for

An additional mass

Transient characteristic of the mover position under an additional inertial load with the mass

Simulation results of the mover velocity under an additional inertial load with the mass

The additional mass brings out more ripples in the mover position than in Cases

In this paper, a method based on modified winding function theory was applied to model a five-phase fractional-slot permanent magnet tubular linear motor. The proposed method has provided the detailed computational expressions of the air-gap flux density, the inductance, flux linkages, and the detent force. Due to the fact that it has a property of fast modeling no matter whether the motor is healthy or not, it can make up for the time-consuming remodeling of the motor using finite element analysis and semianalytical method when the faults arise or supply power changes. The analytical results showed that the slot-effect force had been greatly reduced by this structural design, and the main component of the detent force is the end-effect force. This is as had been demonstrated by using the finite element model. In this proposed approach, a model with five-phase Park transformation had been established and simulated. The simulation results for transients on the motor were analyzed and were close to those of semianalytical method and measurements. In this way, the accuracy of the model using MWFA was validated.

As shown in Figure

Plugging (

Then, the expression (

All-phase modified winding function

The air gap magnetic field is the product of the PM field with unslotted stator and the relative air gap permeance [

By using Fourier series expansion theory, the inverse air gap function

Combining (

In light of the paper [

Because air gap

As shown in Figure

Seen from the five-phase Park transformation matrix

Applying

When the harmonics of the inductances are ignored, the direct-axis and quadrature-axis inductances are as follows:

The permanent magnet flux linkages using the five-phase Park transformation matrix

The amplitude of the permanent magnet flux linkages is given by

The authors declare that there is no conflict of interests regarding the publication of this manuscript.