3d finite element analysis of a hybrid stepper motor

IN DEGREE PROJECT ELECTRICAL ENGINEERING,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2019

3D Finite Element Analysis of a Hybrid Stepper Motor

YUANYI FAN

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE

3D Finite Element Analysis of aHybrid Stepper Motor

YUANYI FAN

Degree Project in Electrical Energy ConversionDate: September 25, 2019Supervisor: Dr. Bin LiuExaminer: Dr. Oskar WallmarkSchool of Electrical Engineering and Computer ScienceHost company: ABBSwedish title: 3D FEM-analys av en hybrid stegmotor

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Abstract

Hybrid stepper motors are being applied to more and more industrial regionsdue to their low cost compared with servo motors and prominent performance.Many industrial applications require accurate and effective methods for pre-dicting a motor’s performance at the design stage. The geometry of the motorsis complicated and the magnetic saturation effect is also serious, giving riseto the difficulty of understanding the transient behavior of the motors. Fur-thermore, the drive circuit and control algorithm are more sophisticated thanthose of traditional AC or DC motors. Lastly, the losses of the motors createthe rising of temperature, while the thermal effect and dynamic performanceaffect each other.

All these factors can be solved by simulating a hybrid stepper motor witha model combining the effect of electromagnetic field, control algorithm, andmotor loss together. In this thesis, a three-dimension (3D) finite elementmodel is developed in the software Maxwell for studying motor character-istics. The electromagnetic field is analyzed in a static state. The simulatedback electromagnetic force is verified by experiments. The feasibility of full-step control algorithm is analyzed. The vector control algorithm is applied tothe model through co-simulation of Simulink and Maxwell in Simplorer. The3D model is proved to be unrealistic for co-simulation. In the end, this the-sis summarizes the modeling experience and gives recommendations on thetransient simulation of the motor.

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Sammanfattning

Hybridstegsmotorer appliceras i fler ochfler industriapplikationer tack varederas låga kostnad och förbättrad prestanda jämfört med servomotorer. Mångabranschapplikationer kräver exakta och effektiva metoder för att förutsägamotorns prestanda redan i konstruktionsstadiet. Motorns geometri är kompli-cerad och den magnetiska mättnadseffekten är också betydande, vilket försvå-rar modelleringen. Dessutom är drivkretsen och styralgoritmen mer sofistike-rad än den för traditionella växel- eller likströmsmotorer. Vidare så resulterarmotorns förluster i temperaturökningar vilka påverkar dynamiska.

Alla dessa faktorer kan studeras genom att simulera hybrida stegmotorermed en modell som kombinerar effekten av elektromagnetiskt fält, kontrol-lalgoritm och motorförluster tillsammans. I detta examensarbete utvecklas entredimensionell finit elementmodell i programvaran Maxwell för att studeramotorns elektromagnetiska egenskaper. Det elektromagnetiska fältet analy-seras i ett statiskt tillstånd. Den beräknade mot-EMK:n har verifieras genomexperiment. Vektorkontrollalgoritmen tillämpas på modellen genom samsi-mulering i Simulink och Maxwell i Simplorer. Den tredimensionella model-len visade sig vara orealistisk för samsimulering. Till sist summeras uppnådaerfarenheter och rekommendationer för fortsatt arbete ges.

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Acknowledgement

I want to express my sincere gratitude to my knowledgeable and experiencedsupervisor, Dr. Bin Liu. He leads me in the right direction and helps me a lotin modifying my experiments and thesis. His positive working attitude andoptimistic attitude are what I should learn in my life. Also, I want to thankmy examiner associate professor, Oskar Wallmark, for his precious recom-mendation and continuous guidance.

Then, I want to thank my colleagues for helping me a lot in my daily lifeduring my stay in ABB Sweden Research Center. Having Fika and lunchtogether with them is really happy.

In the end, I would like to offer my thanks to my family, giving me a warmback up; To my beloved girlfriend, Mengyao Jiang, giving me support everyday; And to my friends, Helin Zhou, Feifan Liu, etc, always encouraging meand providing a lot of help.

Contents

1 Introduction 11.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Scope and Challenge . . . . . . . . . . . . . . . . . . . . . . 11.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Previous Work/Literature Review . . . . . . . . . . . . . . . . 31.5 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Hybrid Stepper Motor 92.1 Stepper Motor . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Basic Structure and Working Principle . . . . . . . . . . . . . 102.3 Detent Torque and Holding Torque . . . . . . . . . . . . . . . 122.4 Motor Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.1 Copper Loss . . . . . . . . . . . . . . . . . . . . . . 142.4.2 Iron Loss . . . . . . . . . . . . . . . . . . . . . . . . 14

3 FEM Modeling of Motors 173.1 Theory of Electromagnetic Field . . . . . . . . . . . . . . . . 173.2 Introduction to Maxwell . . . . . . . . . . . . . . . . . . . . 193.3 Co-simulation of Maxwell and Simulink . . . . . . . . . . . . 24

4 Simulation 294.1 3D Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Static State . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3 Back EMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4 Full-Step Control . . . . . . . . . . . . . . . . . . . . . . . . 354.5 Vector Control . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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5 Conclusions and Future Work 455.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2.1 2D Model . . . . . . . . . . . . . . . . . . . . . . . . 465.2.2 Verification of FEM Model . . . . . . . . . . . . . . . 465.2.3 Thermal Simulation . . . . . . . . . . . . . . . . . . 47

Bibliography 49

Chapter 1

Introduction

1.1 Purpose

This thesis work aims to investigate the feasibility of simulating the transientbehavior of hybrid stepper motors by a universal three-dimension (3D) FiniteElement Method (FEM) commercial software. The method proposed in thisthesis is beneficial for predicting characteristics of this kind of motor in prac-tical industrial applications. The research provides motor designers and userswith reference to apply the motor in multiple working circumstances.

1.2 Scope and Challenge

The work is a continuation of previous master thesis projects and some re-sults are directly used in this thesis. In order to analyze the complex tran-sient behavior of hybrid stepper motors, a precise model should be built andverified using existing instruments. The thesis intends to find an effectivemodeling method for this kind of motor and operate corresponding simula-tions properly for representative working conditions. Restricted by time andinstruments, this work only does a limited number of simulations. The elec-tromagnetic field is analyzed in a static state. The back electromotive force issimulated and the result is verified by experimental results. The simple open-loop control algorithm is applied to the numerical model. Also, the relativelycomplicated vector control algorithm is tried to apply on the model throughco-simulation of Simulink and Maxwell. Although there are many kinds ofhybrid stepper motors on the market, this thesis work only studies a specificmotor. That is to say, the effect of motor dimension is not studied in thisthesis.

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2 CHAPTER 1. INTRODUCTION

The challenge of this project is mainly on establishing an FEM model.The model needs to be verified with the motor by trial and error: a modelis built and simulated, and then the model is changed to get a better simula-tion result which is closer to the experimental result. However, the simulationof 3D FEM is very time-consuming and heavily relies on experience. Thus,another challenge lies in shortening the simulation time, which is realizedthrough model reduction. As previous studies simulated the motor only instatic solvers, the biggest challenge of this project is thinking out how to sim-ulate the motor in a transient solver and how to verify the effectiveness of thesimulation. That will be detailedly discussed in Section 1.4.

1.3 Background

A stepper motor is an electromechanical component that produces a corre-sponding angular displacement or line displacement when it is applied withan electrical pulse signal. It is widely used as an actuator in various modernindustrial products, such as reversing radar, printers, and lighting system ofautomobile [1]. In comparison with servo motors, the most outstanding meritof stepper motors is being cheap. Therefore, researchers and engineers havetried to replace servo motors in some applications with stepper motors, whichcould yield huge economic benefits. The use of stepper motors is growingrapidly that the market share of stepper motors currently accounts for about17% of the global drive motor market [2].

In the case of low control accuracy requirements, stepper motors can beopen-loop controlled by digital signals, making them easy to construct a sim-ple, inexpensive but reliable control system. However, under open-loop con-trol, stepper motors have poor control accuracy, which limits their range ofapplication. Therefore, in the past three years, three projects have been con-ducted to explore the control strategy of stepper motors with higher posi-tional accuracy on the premise of reducing hardware cost. In these projects,microstepping and vector control strategy were applied to an inexpensive po-sition sensor and a hybrid stepper motor. The control strategy satisfied thedemand for position accuracy and the achievement was published in threemaster thesis [3, 4, 5].

However, when the hybrid stepper motor is attempted to provide power toan automation machine, the power density is too small to meet industrial de-mand. That is to say, under normal load, the operation speed of that machineis too slow, thereby reducing the production efficiency. It is obliged to enlargethe power of the hybrid stepper motor. Greater power means a larger current

CHAPTER 1. INTRODUCTION 3

is supplied to the motor. However, for one thing, a large current may causethe motor to heat up too much and the temperature is too high, which maydamage the motor. It is well known that the temperature and dynamic charac-teristics influence each other, especially in over-heated condition. For anotherthing, the heating energy comes from the losses of motor, meaning that thelarge excitation current would alter the motor efficiency, i.e. the economicvalue.

Therefore, a relative precise model combined the dynamic and thermalfeatures of a hybrid stepper motor should be developed so as to help re-searchers investigating the performance limit and understanding the transientbehavior.

1.4 Previous Work/Literature Review

Numerical methods, mainly FEM methods, and analytical methods are thetwo most extensively employed methods to explore an electrical motor’s per-formance. The foundation of analytical methods is the electric and magneticcircuit, while the foundation of numerical methods is geometry treatment [6].The excellent merits of analytical methods are that they deliver outcomesswiftly and could trace physical context underlying calculation, and couldwell define cause and effect, but their limitation is achieved when saturationeffect and complicated geometries are considered [6]. In contrast with ana-lytical methods, although numerical methods could overcome the limitationof complex geometry and saturation effect, they need too much computationresource and calculation time. In hybrid stepper motors, geometry complexityis greatly increased by small teeth and the saturation effect in airgap is verysevere, which arise difficulties of achieving a precise model. Based on thesetwo methods, a lot of studies has been performed by previous researchers tomodel and simulate hybrid stepper motors.

Only a few studies purely based on analytical methods have been con-ducted. For driving hybrid stepper motors by sinusoidal currents, which drivesynchronous motors, the torque characteristics were studied by Mizutami,Hayashi, and Matsui [7] and Matsui, Nakamura, and Kosaka [8], using equiv-alent magnetic circuits based on permeance. A voltage equation was deducedfrom that magnetic circuits with respect to motor dimension. By this analyti-cal method, the instantaneous torque and detent torque were validly predictedand also analyzed by Fourier transforms. This method facilitates motor de-signers preliminarily predicting operation characteristics of the motors underspecific driving current in the design stage. Bêkir, El Amraoui, and Gillon [9]


pointed out an analytical method computing the permeance curves in tooth/airregion and it was verified by a 2D FEM simulation. Referring to the resultedpermeance curves, a dynamic model of a hybrid stepper motor is built. How-ever, the response of position and torque is lack of experimental verification.

More work has been done by combining the analytical model togetherwith 2D models.

A magnetic circuit of a hybrid stepper motor was analyzed by Rao andPrasad [10] for various design topologies using 2D finite element analysisusing PDE toolbox in Matlab. The results provide methods for improvingmotor performance, such as cogging torque and steady-state torque.

Stuebig and Ponick [6] worked out an analytical model combined withthe 2D finite element model. Studying the geometry of the motor, the authorsdeveloped the analytical model described by an equivalent magnetic circuitcontaining the corresponding magnetic permeances of hybrid stepper motorparts. Newton–Rathson method is adopted to simulate this model. The mag-netic permeance of the airgap in this analytical model is calculated by a 2DFEM model as the analytical method is unable to deal with the saturation ef-fects. This combined model is verified by 3D FEM models and experiment.In comparison with 3D FEM models, this model is 18 000 times faster.

Any analytical method is incapable of getting reliable results for the satu-ration effect in the area of tooth/airgap. In cope with this issue in hybrid step-per motors, Jenkins, Howe, and Birch [11] used a 2D finite element model togenerate the permeance curves. The authors applied these curves to a non-linear model represented as a lumped parameter network. The measurementin the experiment validates the effectiveness of this model in predicting torquecharacteristics, winding inductance and back electromotive force (EMF) con-stants.

Kang and Lieu [12] proposed a lumped parameter model combined with2D FEM model for torque analysis of hybrid stepper motors. The 2D FEMmodel is used to compute the permeance in airgap under given magnetomotiveforce (MMF) and rotor position. This method provides a continuous functionfor exploring static characteristics.

Hybrid stepper motors can be analyzed by 3D or 2D FEM models. 3DFEM analysis is much more precise than 2D FEM analysis. However, mostof the hybrid stepper motor FEM analysis attempts to create 2D models otherthan 3D models because 3D models require significantly more computingresources than 2D models [13]. With the development of computer perfor-mance, 3D analysis can be directly conducted for the motors [14]. However,a 3D analysis still takes a very long time in the development period of the mo-


tors [6]. Effectively reducing the computational complexity of FEM analysisis especially critical in engineering. Thus, some work has been done in orderto make a trade-off between simulation time and accuracy by developing anequivalent 2D finite element model.

Jang et al. [15] proposed a virtual magnetic thin layer barrier in an equiv-alent 2D FEM model. The assumption in this paper is that the permeabilityof the barrier is very small so that the flux can hardly traverse it and the axialflux is changed into radial flux. The results are verified by the static torqueresulted from experiments and 3D analysis. In contrast to 3D finite elementanalysis, this method could reduce the simulation time to 1/30. Li, Lu, andShen [16] transformed a 3D finite element model of a hybrid stepper mo-tor into an equivalent 2D finite element model. The ring-shaped permanentmagnet with axial magnetization direction is transformed into two equivalentmagnets with radial magnetization direction. This model excellently predictsthe static torque characteristics of the motor by post-processing. The 2D mod-els in these two articles are based on geometry features of the motors.

Ionica et al. [14] introduced a 3D numerical model of a hybrid stepper mo-tor in the design phase. The authors studied the flux distribution in the motorand the detent torque characters. They found that factors, such as airgap andmaterials, have a deep influence on motor performance. The result shows thatthe magnetic saturation effect is severe in the airgap. The authors claimed that2D finite element models of hybrid stepper motors are unacceptable due tothe considerable discrepancies between simulation and experiment. Kosaka,Pollock, and Matsui [17] explored the effect of isolation layers between lam-inations by 3D finite element analysis of hybrid stepper motors. Isolationlayers are analyzed by a cross lamination model based on reluctance in orderto provide an equivalent magnetic effect in the perpendicular direction of thelamination staking direction. The result is verified by torque characteristics inexperiments for two kinds of hybrid stepper motors with different permanentmagnetic materials. These two articles directly adopt 3D models for analyz-ing the motors.

Oswald and Herzog [18] studied the traits of both 3D reduced models and2D models in predicting static torque characteristics of a hybrid stepper mo-tor. They argued that only limited certain features can be analyzed by 2Dmodels, such as static torque variation. The reduced 3D models are capa-ble of overcoming the drawbacks of 2D models and taking the merits of 3Dfull models without increasing simulation time. Rajagopal, Singh, and Singh[19] compared flux density and torque characteristics of hybrid stepper motorswith different tooth-geometry using 2D finite element method and 3D finite


element method respectively in order to find the optimal tooth design for spe-cific motor performance requirement. Later, Praveen et al. [13] developed theprevious method and studied the tooth-geometry effect only using a 2D finiteelement model. The structure of trapezoidal teeth with unequal tooth pitch isproved to be the best design verified by simulation and experiment. Both 2Dmodels and 3D models are analyzed in these three articles.

However, all of the literature mentioned above just verified the effective-ness of FEM models in a static state without verifying the effectiveness ofthe models in a transient state. They also did not take the thermal effect intoconsideration and not researched motor efficiency. Through experiments, De-rammelaere et al. [20] researched the efficiency of a hybrid stepper motor indifferent control algorithm and the result showed that the efficiency can beimproved from 20% to 50% by current reduction. A modeling method shouldbe developed for studying the efficiency of the motor.

To apply motors in industrial applications, it is necessary to study thecontrol algorithm of motors, where motors are generally modeled in Simulink.Morar [21] built a model of a bipolar hybrid stepper motor using some powerelectronics models and motor models in the existing Power Blocks modulein Simulink and simulated the motion control of the motor under differentloads. This method is run with too many assumptions. For example, the detenttorque is assumed to be perfectly sinusoidal changed. Also, this method cannot explore the effect of motor geometry and electromagnetic field. However,simulation in Simulink can be used as a basic evaluation of control algorithms.

ANSYS Maxwell is a versatile electromagnetic finite element analysissoftware that could perform both transient and static state simulations of mo-tors for optimizing a motor design. Therefore, many finite element simula-tions of motors are conducted by Maxwell. Apart from general simulation, itis also accessible to conduct co-simulation of Simulink and Maxwell by Sim-plorer, creating a model that is much close to the real situation. Makolo [22]used the Simplorer in ANSYS to combine a PI controller in Simulink with atwo-dimensional transient model of a permanent magnet synchronous motor(PMSM) in Maxwell for simulating a wind turbine. This method could simul-taneously study control algorithms, geometry and magnetic fields of motors.

To sum up, for motor design, only the modeling of hybrid stepper motorsin a static state has been studied by analytical methods and FEM methods.The simulation results in static simulation, however, can only qualitativelycompare the dynamic performance of different motors. Although the motionbehavior of motors could be simulated by Power Blocks in Simulink, it ne-glects too many factors, such as thermal effect caused by motor loss. The


existing literature also does not provide a model that is capable of calculatingthe motors’ loss. Therefore, a precise model of hybrid stepper motors undertransient conditions considering all these effects should be developed.

1.5 Contribution

In this thesis, the control algorithm of hybrid stepper motors designed inSimulink is connected with Maxwell to do co-simulation, where transientsimulation is conducted. Therefore, this thesis opens the way to the transientsimulation of this kind of motor so that more problems are directly settledand the results are more easily observed and more conformed to reality. Al-though not so many simulations are completed due to time limitation, thisthesis summarizes the experience of transient simulation and provides rec-ommendations to commercial numerical simulation software developers andusers. As Maxwell is capable of calculating the losses of motors simulta-neously with simulating dynamic performance, subsequent researchers couldconduct co-simulation taking thermal effect into consideration, based on thisthesis.

1.6 Thesis Outline

Chapter 1 points out the research background and significance. Chapter 2introduces the basic theory of hybrid stepper motors. Chapter 3 exhibits themodeling methods and process. Chapter 4 describes the experiment and sim-ulation, and discusses the results. Chapter 5 gives the conclusions and futurework.

Chapter 2

Hybrid Stepper Motor

2.1 Stepper Motor

A stepper motor is an electromagnetic actuator that converts an electrical im-pulse into a corresponding discrete angular rotation. It has an extensive ap-plication and plays a critical role in the modern applications of life, office,and industry [15]. Beyond that, it generates a stepwise rotation when a se-quential impulse is fed in. Its rotation direction is determined by the impulsesequence. While the motor is under rated load, the same impulse results in thesame rotation step regardless of the load variation. Besides, the position errorof each step is under 10% and the error will not be accumulated from onestep to the next [14]. Therefore, the motor operation can be easily controlledby just controlling the impulse rate of the input signal. This basic featureallows it to be used directly in the open-loop control. Compared with closed-loop control, open-loop control effectively reduces the cost of control. Thus,stepper motors have a wide range of applications where the control accuracyrequirements are not high.

Stepper motors can be mainly divided into three types [23] :(1) Variable Reluctance(VR) stepper motor: Copper wires are wound

around the stator to form windings. Its rotor is composed of soft iron. Inthe operation, the rotor always rotates to the position with the smallest reluc-tance. The operating principle of this motor is the same as that of a switchedreluctance motor. This motor has several merits, such as simple structure, lowcost, small step size, and good high-speed performance. However, its appli-cation range is limited by the defect of low torque, inefficiency, large heatgeneration, loud noise, and bad reliability.

(2) Permanent Magnet(PM) stepper motor: The rotor is not tooth-

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10 CHAPTER 2. HYBRID STEPPER MOTOR

shaped. There are permanent magnets on the rotor. At the same time, therotor and the stator have the same number of poles. Compared with VR step-per motors, the permanent magnets in PM stepper motors provide additionalmagnetomotive force, resulting in larger output torque. This motor has twooutstanding advantages: high efficiency and large output torque. Neverthe-less, its drawbacks also include poor accuracy and large step size (typically7.5◦ or 15◦).

(3) Hybrid stepper motor: Multi-phase windings are on the stator. Therotor is made from the permanent magnet and silicon steel. In addition, theimprovement of the position precision can be realized by the small teeth onstator and rotor. As the name suggests, a hybrid stepper motor is a combina-tion of a VR stepper motor and a PM stepper motor. Meanwhile, it owns themerits of the two motors, such as small step size, high efficiency, large outputtorque, and good dynamic performance. However, the complex structure andhigh cost are the main drawbacks. From the point of performance, hybridstepper motors are the best stepper motors.

Although stepper motors have been widely used, they cannot simply bedriven by DC power or AC power as conventional motors. Indeed, it canonly be used when it is combined with pulse signals, power drive circuits,etc. into a control system. Control algorithm also significantly influences theperformance of stepper motors. Therefore, it is not easy to use a stepper motorand many efforts have been done for its more application.

2.2 Basic Structure and Working Principle

Due to the outstanding performance, a hybrid stepper motor is selected as theresearch object. It was disassembled as shown in Figure 2.1.

Figure 2.1: Disassembled Stepper Motor [5].

CHAPTER 2. HYBRID STEPPER MOTOR 11

Figure 2.2: Side view and cross-sections of the hybrid stepping motor [24].

The simplified half part of a hybrid stepper motor is shown in Figure 2.2.The stator has eight poles distributed along the circumference. There are sev-eral teeth on the face of the pole. Copper wire wraps around poles so asto forming windings, which would be used to control the motion of motorthrough controlling the current in windings. A rotating shaft, two iron cores,and a ring-shaped permanent magnet constitute the rotor. The two iron coresare respectively installed at the two ends of the permanent magnet. They ex-hibit the opposite magnetic polarities respectively due to the permanent mag-net. The rotor core is uniformly distributed with small teethwhich are of thesame shape and size as that of the teeth on the stator, and the iron cores at bothends miss each other by a half distance between two neighbor teeth. Theseteeth greatly improve the position resolution of the stepper motor.

The two-phase hybrid stepping motor shown in Figure 2.1 has 50 smallteeth uniformly distributed on the stator, eight magnetic poles evenly dis-tributed on the stator. The position resolution of this two-phase stepper motoris 1.8◦.There are two windings wound on the magnetic pole, with six smallteeth on each poles. The wiring of windings on the stator pole is shown in Fig-ure 2.1. The wires on poles 1, 3, 5, and 7 are connected to form the A-phasewinding. The wires on poles 2, 4, 6, and 8 are connected to form the B-phasewinding. If the teeth of poles 1 and 5 are in a tooth-to-tooth state with respectto the rotor teeth, the teeth of poles 3 and 7 must be in a tooth-to-slot statewith respect to rotor teeth. Thus, when phase A is energized and the magneticfield direction generated by poles 1 and 5 is toward to the circle center, the


magnetic field direction generated by poles 3 and 7 is toward to the oppositedirection.

The magnetomotive force of the motor is provided by the permanent mag-net and winding coil. The magnetization direction of the permanent magnetis perpendicular to the direction of the magnetic field generated by the coil.Thus, the magnetic field of the motor is distributed in three dimension space.When the motor is in operation, the rotor always tries to find the magneticpath with the minimum reluctance. Also, every pole drags the teeth whichhave the opposite magnetic field direction and pushes the teeth which havethe same magnetic field direction, resulting in a tangential force along theaxial direction of the rotor, resulting in a tangential force along the axial di-rection of the rotor. The reluctance mainly generated by the airgap betweenthe rotor and the stator. In a tooth-to-tooth state, the airgap is pretty smallthat generally less than 0.1 mm, thereby increasing the production cost of thehybrid stepping motor.

2.3 Detent Torque and Holding Torque

Hybrid stepper motors are characterized by detent torque and holding torque.Nowadays, improving these two torques is the main task of electrical motordesigners at the design stage.

When a hybrid stepper motor is in a state of rest and not energized, asmall torque must be applied to the rotor in order to break through the staticequilibrium state [25]. This torque is called detent torque, also known asresidual torque [26]. This effect is caused by the permanent magnet in therotor, drawn to the poles of the stator [26]. In conventional permanent magnetmotors, this effect is also called the cogging effect. In general, it is a universalfeature distinguishing whether a motor has a permanent magnet.

As the detent torque must be overcome to rotate the motor, the outputtorque and output power are reduced. Also, the magnetic field is distributedunevenly in the motor and the detent torque is variated with rotor position,causing the variation of output torque. Furthermore, it would definitely bringsome bad impact to motor controllers. Zhou [5] pointed out that both thespeed controller and the position controller are limited and affected by detenttorque because it enlarges the peak-to-peak error in the PID controller. On an-other hand, detent torque is pretty rewarding for deceleration and maintainingthe position of the motor.

When the windings are powered and the rotor is at the balanced position,an external torque should be applied to a hybrid stepper motor so as to rotate


the rotor one full step [26]. This torque is called holding torque. In contrastto servo motors, holding torque enables hybrid stepper motors keeping rotorson equilibrium position disregarding external load. The larger the windingcurrent is, the larger the holding torque is. However, the saturation effect inthe motor is pretty severe. Thus, the value of holding torque is limited bythe current amplitude and the magnetic saturation effect. Typically, the outputtorque at low speed approaches the holding torque. The increase of speedis accompanied by continuous attenuation of output torque and variation ofoutput power. Thus, holding torque is a crucial parameter for evaluating theperformance of a hybrid stepper motor. Holding torque is averagely 5 to 20times of the detent torque [25].

2.4 Motor Loss

The loss of a hybrid stepper motor is mainly composed of mechanical lossPmech, copper loss PCu and iron loss PFe. The energy utilization of a hybridstepper motor is shown in Figure 2.3. The efficiency can be expressed as :

η =Pout

Pin=

Pout

Pout + Pmech + PCu + +PFe(2.1)

where Pout and Pin are the output power and input power respectively.

Figure 2.3: Energy utilization of the hybrid stepper motors.

Mechanical losses are mainly caused by friction, which accounts for asmall part of total losses. A hybrid stepper motor would be heated up by thethermal energy converted from these losses. However, the increase of tem-perature lowers the magnetomotive force of the permanent magnet, therebyreducing the output torque. Moreover, some materials inside motors, such asplastic, would deform or even melt at high temperatures. Due to the different


thermal expansion coefficients of various parts of the hybrid stepper motor,structural stress and air gap changes in cold shrinkage and thermal expansion,which will affect the dynamic response of the motor, resulting in losing stepsin the case of high-speed operation. Therefore, overheating would cause themotor to malfunction or even be damaged and it is the main cause of motorfailure. In general, a hybrid stepper motor operating temperature does notexceed 80 ◦C. Therefore, operating the hybrid stepper motor in larger outputpower in the premise of no overheat is of great importance.

2.4.1 Copper Loss

Copper loss is caused by the resistance of the winding when currents passthrough motor. It is proportional to the square of the current I and can beexpressed as:

Pcu = I2R (2.2)

where R is the resistance in the winding. Through theoretical calculationand experimental measurement, Derammelaere et al. [20] verified that copperloss is the main loss source of hybrid stepper motors and the copper lossgenerally takes up more than 50% percent of input energy. The authors alsopointed out that the efficiency of the motor is pretty low because of the highcurrent amplitude, the comparatively winding resistance of the stator, and thepoor scale of torque/current. Therefore, the copper loss could be reducedby selecting a motor with small winding resistance and reducing the currentduring operation. A proper control algorithm would improve the efficiency ofthe hybrid stepper motor by current reduction.

2.4.2 Iron Loss

Iron loss PFe is the sum of hysteresis loss Ph and eddy current loss Pe:

PFe = Ph + Pe (2.3)

Magnetic hysteresis of ferromagnetic material is the phenomenon that thevariation of the magnetic flux density B lags behind the variation of the mag-netic field strength H, which is induced by winding current. When the currentin winding varies through a cycle, energy streams from the power source tothe iron core in a certain period, and energy comes back the power sourcein another certain period. Nevertheless, the energy streaming into the core isgreater than the energy streaming back to the core due to the hysteresis effect.


Thus, net energy flows from the power source to the iron core during a cycleof current variation. When the hybrid stepper motor is operating, the wind-ings are powered by approximate alternating current. Hence, the iron core isin an alternating magnetic field and hysteresis loss is generated [27]. The lossin such process is called hysteresis loss Ph. According to experimental andtheoretical analysis, the hysteresis loss is [27]:

Ph = KhBnmax f (2.4)

where Kh is the hysteresis loss factor, determined by the material propertiyand volume of iron core, Bmax is the maximum magnetic flux density. Thevalue of n is generally between 1.5 and 2.5 [27]. Therefore, Selecting propermaterial for iron core could effectively reduce the hysteresis loss.

When an alternating current is passed through the windings of a hybridstepper motor, the magnetic flux generated by the current is also alternating.Therefore, not only the induced electromotive force is generated in the coil,but also the induced electromotive force and the induced current are generatedin the iron core. This current is called eddy current and it circulates in a planeperpendicular to the direction of the magnetic flux. The loss generated by theeddy current in the core is called the eddy current loss Pe. It can be seen asa pure resistance circuit with a certain electromotive force when an iron coreis subjected to an eddy current. The loss can, therefore, be decreased by in-creasing the resistance. One option is to adopt material with a big strengthcoefficient; the other is to use a laminated iron core due to the greater resis-tance of the longer electrical circuit. For iron cores stacked from silicon steelsheets, eddy current loss can be written as [27]:

Pe = KeB2max f 2 (2.5)

where Ke is the eddy loss factor, determined by the material property and thelamination thickness of iron core.

Experiments can readily measure the iron loss of motors, whereas it ispretty time-consuming. For motor design, a technique of predicting the lossshould be developed. Analytical methods and FEM methods are widely usedfor conventional motors. However, because of the complicated distributionof magnetic fields in hybrid stepper motors, no loss simulation for them hasbeen performed so far.

Chapter 3

FEM Modeling of Motors

3.1 Theory of Electromagnetic Field

In the 19th century, Maxwell’s equations were proposed by the British physi-cist James Clark Maxwell, based on previous researcher’s studies. They area collection of partial differential equations that describe the connection be-tween electric field, magnetic field, the density of charge, and the density ofcurrent [28]:

∇ · D = ρ

∇ · B = 0

∇ × E = −∂B∂t

∇ ×H = J +∂D∂t

(3.1)

where D is the electric flux density, ρ is the electric volume charge density,B is the magnetic flux density, E is the electric field, H is the magnetic field,J is the current density. The first equation, Gauss’s law, defines how electriccharges produce electric fields. The second equation, Gauss’s law for mag-netism, demonstrates that there are no magnetic monopoles, in analogy withelectric charges. The third equation, Faraday’s law, explains how variant amagnetic field induces an electric field. The last equation, Ampere’s law withMaxwell’s addition, presents that magnetic fields can be produced by twomeans: electric current and displacement current [28, 29].

The property and mutual relationship of electric fields and magnetic fieldsare described by Maxwell’s equations. These equations are valid in macroscale,whereas the quantum effect must be considered in microscale. Maxwell’sequations are the theoretical basis of electromagnetic devices and components

17

18 CHAPTER 3. FEM MODELING OF MOTORS

such as electric power, electric motor, electromagnetic wave, and etc.Because the motor running frequency is lower than the radio frequency,

the displacement current can be neglected in the process of calculating themagnetic fields of the motor, i.e. ρ = 0 and ∂D

∂t = 0 [29]. And Equation 3.1can be refined as:

∇ · D = 0

∇ · B = 0

∇ × E = −∂B∂t

∇ ×H = J

(3.2)

In electromagnetic media, field vectors have the following linear relation-ship:

D = εE = ε0εrEJ = σEB = µH = µ0µrH

(3.3)

where ε is the permittivity, ε0 is the vacuum permittivity, εr is the relativepermittivity, σ is the electrical conductivity, µ is the permeability, µ0 is thevacuum permeability, and µr is the relative permeability.

In order to form an independent partial differential equation for electricfield or magnetic field, the magnetic vector potential A is introduced:

B = ∇ × A (3.4)

Thus, the Faraday’s law can be rewritten as [29]:

∇ × E = −∇ ×∂A∂t

(3.5)

If the reduced electric scalar potential Vpot is introduced, Equation 3.5 canbe rewritten as [29]:

E = −∂A∂t− ∇Vpot (3.6)

Taking Equation 3.6 into Equation 3.2 and Equation 3.3, the result is [29]:

∇2Aµ− σ

∂A∂t

= −J + σ∇Vpot (3.7)

The field distribution of magnetic potential and electric potential can be ac-quired by the numerical solution of Equation 3.7, deducing various physicalquantities in electromagnetic field.

CHAPTER 3. FEM MODELING OF MOTORS 19

3.2 Introduction to Maxwell

Numerical methods are applied to solve the partial differential equations inSection 4.1. Finite element method (FEM) and finite difference method (FDM)are the two main numerical methods for this issue. According to Cavka et al.[30], in comparison with other numerical methods, FEM owns a favorabletrait that neither the formulation nor the computer code needs to be changedwhen dealing with complex geometry and inhomogeneity material. It alsousually creates symmetric and sparse matrix systems reducing the require-ment of computer memory. These are the reasons why it is widely used in en-gineering. The solving process of FEM is divided into four steps in sequence[31]: 1) Region discretization: The solution region is discretized into subre-gions or elements with finite number, 2) Element analysis: A typical elementis analyzed to acquire governing equation, 3) Element assemblage: All of thesubregions or elements in the solution region are assembled, 4)Solution: Thegenerated system of equations is solved.

Based on the theories mentioned above, many types of commercial FEMsoftware have been developed for analyzing electromagnetic problems. Amongthem, Maxwell is one of the most outstanding software in the industry. It isextensively used in the analysis of electromagnetic components in industrialapplications such as sensors, regulators, motors, transformers, and other in-dustrial control systems. It was originally produced by ANSOFT in 2003.Later, ANSOFT was acquired by ANSYS in 2008. Nowadays, Maxwell isa functioning module in ANSYS EM Suite and ANSYS provides some in-terface for Maxwell with other modules of ANSYS, such as Static ThermalAnalysis, to do some co-simulation. The user-friendly interface of Maxwellis intuitive and easy to use. This software has the following advantages overother finite element analysis software:

(1) It processes data very efficiently.(2) With easy-to-follow drawing function and model input port, it can

easily import geometric models generated by other CAD software, such asSolidWorks.

(3) It has powerful splitting mesh function. Manual splitting meshes andautomatic splitting meshes are provided to users. The shape and density of themesh are flexible and the energy error can be reduced to any specified value.

(4) It has the ability of performing linear and nonlinear analysis.Therefore, Maxwell is selected to model the hybrid stepper motor in this

thesis.The steps of doing simulation in Maxwell are:


1. Select the modeling modeThere are two types of modeling mode in Maxwell, including Maxwell

2D, Maxwell 3D. In most cases of motor analysis, as the magnetization direc-tion of the permanent magnet and the direction of the magnetic field generatedby windings are in the same plane, modeling motors in Maxwell 2D could geta pretty precise result. The number of nodes generated in Maxwell 2D is alsomuch smaller than that of Maxwell 3D, resulting in much faster calculationspeed. However, Maxwell 2D cannot analyze the electromagnetic field dis-tribution of motors in axial direction. Simulation in Maxwell 3D could get amore precise result and do more analysis with higher cost.

2. Select the solverMaxwell provides three solvers for electric fields and magnetic fields. For

this thesis work, the motor only needs magnetic field analysis. The Magneto-static solver and the Transient solver of magnetic fields are for the static anal-ysis and transient analysis respectively. The Eddy Current solver is mainlyfor the analysis of eddy effect and thermal effect.

3. Draw the motorAlthough the way of drawing in Maxwell is similar to that of professional

CAD software, Maxwell lacks some functions, such as crop. Thus, drawinga motor and importing it into Maxwell would be a great choice. Some motorspare parts with complicated geometry are difficult to be drawn by hand, suchas windings. Maxwell also provides some User Defined Primitive models forthese spare parts, as shown in Figure 3.1.


Figure 3.1: Spare parts provided by User Defined Primitive.

As drawing a motor by hand is time-consuming and skilled, RMxprt inMaxwell could provide twelve kinds of motors automatically, as shown inFigure 3.2. Users just need to input the parameters of the researched motorwithout trivial drawing to get a 2D or 3D model. The generated model couldbe directly imported into Maxwell solvers. This function greatly facilitatesthe modeling of motors.


Figure 3.2: Motors provided by RMxprt.

After drawing all objects, some virtual objects, whose material is vacuum,should be added. For instance, Region and Band define the border of thesolution space and the motion space respectively.

4. Assign materialsA rich material library is provided by Maxwell. Also, new materials can

be added to the material library by inputting parameters. It should be espe-cially noted that the magnetization direction of permanent magnet materialneeds to be specified. The B-H curve of new silicon steel sheet material canbe formed through inputting discrete points in that curve.

5. Assign boundaryIn essence, Maxwell aims to solve a series of differential equations in

space, on which boundary has a large influence. Therefore, setting up a cor-rect boundary condition is of great importance for an accurate result. TheHelp in this software shows that: The default boundary condition in Maxwellis Nature and Neumann. The magnetic field H continues across the boundaryin Nature boundary condition, such as an interface between two objects. His tangential to the boundary and flux cannot cross the boundary in Neumannboundary condition, such as the boundary of Region. Normally, the defaultboundary condition could meet the solving requirements.

The geometry of many electromagnetic products, such as transformersand motors, is symmetrical or periodically variable. Therefore, the modelingand the calculation for these products can be simplified by analyzing a partof them. In these analyses, it is necessary to establish not only the general


boundary conditions representing the behavior of the electromagnetic fieldbut also the Symmetry or Matching (Master and Slave) boundary conditions.

6. Assign excitationBoth current and permanent magnets can be an excitation which is the

source of electromagnetic fields. In an analysis of motors, the excitation cur-rent should be defined. There are three modes to define excitation in windings,including current mode, voltage mode, and external mode. It is convenient toset up a dataset describing the waveform of a current or a voltage. In voltagemode, not only the voltage amplitude but also the resistance, the reluctance,the initial current amplitude should be assigned, whereas only the currentamplitude needs to be assigned in current mode. External circuits or con-trol algorithms could provide excitations for MAXWELL models in externalmode. Thus, the co-simulation of Simulink and Maxwell could be realized inthe external mode.

7. Assign solution parametersSolution parameters are the outputs of simulation, including torque and

force. In a motor, the rotor is compromised by several parts and all of themshould be selected together while assigning solution parameters.

8. Split meshIn Magnetostatic solver, an initial mesh is automatically generated before

calculation. With the iterative calculations of solvers, the mesh is continuallyrefined and become thicker until the solution accuracy is met. Such mesh iscalled adaptive mesh. Thus, it is not a must to split the mesh manually in thestatic solver. Also, adding manually split mesh would speed up meshing inMagnetostatic solver.

However, there is no adaptive mesh in the transient solver. It is obligedto split the mesh manually, which relies on experience. Fortunately, the influ-ence of mesh on results is small. For motor analysis, after adding inner layersin the gap between Band and rotor, the influence of mesh even can be ignored.

9. Setup of analysisThe analysis setup of Magnetostatic solver is to set up the maximum num-

ber of passes and the error percent between two passes. When one of the se-tups is met by the result, the calculation would stop. The analysis setup ofTransient is to set up the simulation step size and duration.

10. Lookup resultWhen a simulation is stopped, Maxwell could generate the results in the

form of a figure or a data table. The data tables can be imported to othersoftware, such as Matlab, for further post-processing.


3.3 Co-simulation of Maxwell and Simulink

As a hybrid stepper motor is an actuator in digital products, it would alwayscombine with a microprocessor and drive circuit so as to form an automa-tion system. Therefore, the co-simulation can be very rewarding for industry.There are two methods for combining Maxwell with Simulink.

The first method directly uses the external excitation in Maxwell. Firstly,the excitation for all windings should be set as External. Then, Set up co-simulation with Simulink should be right-clicked in excitation option. Then,the edit window will appear, as shown in Figure 3.3. The Edit Circuit optionwill automatically open the Simulink dialog for Maxwell link assignment, asshown in Figure 3.4. Now, the Simulink model can be edited with a preparedcontrol algorithm. After editing the Simulink model, this model should besaved and closed. Lastly, the Simulink model can be imported to Maxwellby clicking Import Circuit in Figure 3.3. The following step is the same asnormal Maxwell simulation process.

Figure 3.3: The window of Setup Co-Simulation with Simulink.


Figure 3.4: The Simulink dialog for Maxwell link assignment.

This method owns several advantages, including simpleness, requiringless memory, and high reliability. However, the input port only can receivecurrent signals, and the output port can only export voltage signals. In gen-eral, the application scope of this method is very narrow, as it cannot do someco-simulations whose control algorithms have position feedback.

The second method is based on the Simplorer module in ANSYS EMSuite. Simplorer is a powerful mechatronics simulation platform, as it pro-vides plenty of physical models. A Simploerer model could integrate a Simulinkmodel, a Maxwell model, and a mechatronics model together as a whole sys-tem, which is much closer to practical industrial electromechanical systems.

The settings of this method are very similar to that of the first method. De-sign Settings can be opened. Right-clicking the Model in Project Managerand then left-clicking the Set Symmetry Multiplier would open the windowof Design Settings where the option Advanced Product Coupling should beenabled. The winding excitation is set as External.

Then, in Simplorer, the Maxwell Transient-Transient Coupling windowand Simulink Interface window can be opened in SubCircuit under TwinBuilder, shown in Figure 3.5 and Figure 3.6. A Maxwell transient modelcan be imported to Simplorer through the Maxwell Transient-Transient Cou-pling window. The Simulink Interface window could specify the input andoutput ports of the Simulink model. After that, there is a Simulink block inSimplorer.


Figure 3.5: The Maxwell Coupling window in Simplorer.

Figure 3.6: The Simulink interface in Simplorer.

The last step is coupling Simulink model with Simplorer model. It shouldbe noted that this method needs to configure path settings in MATLAB. In the


installation path of ANSYS EM Suite, there are corresponding path folderswhich should be set as a path of MATLAB/Simulink. After changing thename of the S-function block in the Simulink model as ’AnsoftSFunction’, acoupling window would automatically appear, shown in Figure 3.7. Then, theSimplorer model can be imported to Simulink through this window. Now, theCo-Simulation could run after clicking the ’Start’ button in Simulink.

Figure 3.7: The Simplorer interface in Simulink.

Chapter 4

Simulation

4.1 3D Model

The selected motor is shown in Figure 2.1.Due to the periodic structure of hybrid stepper motors, half or one-quarter

model can be used for analysis and calculation under periodic boundary con-dition. This method reduces the time of splitting mesh and computation, andthe amount of stored data. However, in the process of applying periodicboundary conditions in this project, the software always reported an errorand could not run. Therefore, the whole motor magnetic field is taken as thesolution area.

As there is no hybrid stepper motor model provided by RMxprt, the 3Dmodel of the hybrid stepper motor in this thesis was drawn manually. Figure4.1 shows the 3D model of the motor. There are two identical permanentmagnets in the rotor. Thus, this motor can be seen as a combination of twonormal hybrid stepper motors. To simplify the analysis of the rotor, this modeldoes not take the end winding into account.

29

30 CHAPTER 4. SIMULATION

(a) Front view (B-B’) (b) Top view (A-A’)

Figure 4.1: The sectional view of the 3D model.

There are four linking screws in this motor. As they are made from aferromagnetic material, they have some effect on motor performance.

4.2 Static State

When the windings are not energized, the rotor is in a balanced position, asshown in Figure 4.1. The static analysis is conducted for this state. After 10adaptive passes, the energy error is less than 2%. The mesh plot is shown inFigure 4.2 and this simulation took four hours. The calculated result of gen-erated torque is 9.75 × 10−4 Nm, which can be neglected as it is approximateto the real value 0 Nm. According to Figure 4.3 and Figure 4.4, the polesin a teeth-to-teeth state would have larger magnetic field density. Therefore,if there is small torque applied to the rotor, the motor would generate de-tent torque which takes it back to the initial position because of the unevenmagnetic field density distribution. The area of stator near screws have largermagnetic field density, meaning that linking screws have an impact on thegenerated torque.

CHAPTER 4. SIMULATION 31

Figure 4.2: The mesh plot when windings are not energized and the rotor isin balanced position.

(a) Top view (b) Bottom view

Figure 4.3: The magnetic field density plots when windings are not energizedand rotor is in balanced position.



Figure 4.4: The magnetic field density steamline plots when windings are notenergized and rotor is in balanced position.

When phase A is energized by 1A and the rotor is in a balanced position,the magnetic field density plot and the streamline plot are shown in Figure4.5 and Figure 4.6 respectively. The contrast between the energized state andno energized state demonstrate that the windings enlarge the magnetic fielddensity a lot. The comparison between the two streamline plots is not soevident because of the saturation effect.


Figure 4.5: The magnetic field density plots when the excitation current inwinding is 1 A and rotor is in balanced position.



Figure 4.6: The magnetic field density steamline plots when the excitationcurrent in winding is 1 A and rotor is in balanced position.

4.3 Back EMF

In operation of motors, a back electromotive force (EMF) is generated, greatlyreducing the amplitude of the current in windings. The simulation exploringthe back EMF is the first step to verify the effectiveness of the FEM model inthe dynamic simulation of motors.

The scheme of the test rig is shown in Figure 4.7. The tested speed is 60rpm, 120 rpm, and 180 rpm and the amplitude of the back EMF generatedby the hybrid stepper motor is 4 V, 8 V and 12 V respectively. Accordingto e = Ktv the maximum speed of the hybrid stepper motor cannot exceed360rpm when the drive voltage is 24 V.

Figure 4.7: The scheme of the test rig.


As Section 4.2 mentioned, the mesh of the transient simulation in Maxwellshould be split by hand. The mesh of this transient simulation is shown inFigure 4.8. The results are displayed in Figure 4.9, Figure 4.10 and Figure4.11 respectively. The results present that this model is valid. However, as themesh is generated automatically after inputting mesh parameters, the shapeand the size of the mesh cannot be defined by users, causing the mesh ofsome region is not split well. Maxwell cannot import mesh from professionalmesh splitting software. This should be modified by Maxwell developers.

Figure 4.8: The mesh in transient simulation.

Figure 4.9: The waveforms of back EMF in 60rpm.




4.4 Full-Step Control

The most convenient control algorithm for stepper motors is the open-loopcontrol algorithm, divided into full-step control, half-step control, and mi-crostepping control. After simplification and equivalence, the excitation schemesand the equilibrium positions for hybrid stepper motors are shown in Figure4.12.

Full-step control is the most basic way to drive motors. In this method,there are four equilibrium positions. Since the two phases are excited by turnsand only one phase at a moment is excited, this method demands minimal in-put energy in contrast with other methods. Half-step drive is an upgradedversion of full-step control since an extra equilibrium position is added in the


middle of the two equilibrium positions in full-step control. This provides thetwo phases with equal currents. As a result, there are eight equilibrium posi-tions in half-step control and the motor resolution is being doubled. However,it also requires more energy.

(a) Full-step

(b) Half-step

(c) Microstepping

Figure 4.12: Open-loop Control [20].

Inspired by half-step control, microstepping control is proposed by previ-ous researchers. When the two phases are excited by currents with differentamplitude, the equilibrium position will be biased toward the stator pole withlarger current amplitude. Based on this phenomenon, a series of equilibriumpositions could be added between the two equilibrium positions in full-stepcontrol. The current in the two windings is stepwise changed according to thecontours of the sine and cosine, respectively:

i∗a = I0cos(Nrθre f )

i∗b = I0sin(Nrθre f )(4.1)

where Nr is the number of teeth on one side of the rotor, I0 is the currentamplitude, and θre f is the referent mechanical degree. Using microstepping


control, the motor resolution is greatly increased and the output torque is alsomuch stable than that of full-step control.

Though the open-loop strategy only provides discrete positions, it reducesthe cost of using hybrid stepper motors as there are no position sensors. Also,this open-loop strategy is much simpler than other motor control algorithms.Thus, the first step of simulating hybrid stepper motors in industry applicationis to explore the full-step control.

Figure 4.13 shows an instance of full-step control starting from the staticstate with the excitation current amplitude of 1A. There is no load applied tothe motor. In each step, the rotor would firstly attain a large angular accelera-tion as well as a pretty high angular speed. Then, the speed gradually sloweddown until zero due to the effect of the damping torque. The step angle ofeach step is about 1.8◦, which means there is no missing step. Therefore, themotor can keep running forever. The average speed is 50 rpm. The speedvaries pretty sharply, which would definitely cause large noise.

If a load torque and excitation are applied to the motor, there will be amissing step, as the rotor cannot get enough angular acceleration. Therefore,the amplitude of the excitation current should be enlarged in order to keep themotor running in steady-state. Figure 4.14 shows an instance applied with theload torque of 0.13 Nm with the excitation current amplitude of 3A. In orderto avoid missing step, the excitation period also changed. The average speedis 85.7 rpm. Therefore, the speed cannot be freely controlled under a specificload in the full-step control algorithm. Though the generated motor torque inFigure 4.14 is about twice of the generated torque in Figure 4.13, the appliedload torque is small due to the effect of the large damping torque resultingfrom the high speed. Thus, the efficiency of the motor would be reduced.

In summary, the full-step control cannot meet the requirement of industrialapplication where the load and the speed should be independently controlled.Also, sharply speed variation, large noise, and low efficiency also limits itsapplication. As half-step control and microstepping control are based on full-step control, they have the same limitations. Thus, a feedback control algo-rithm could be the solution for avoiding missing step in real-time.


(a)

(b)

(c)

(d)

Figure 4.13: Waveforms of full-step control with no load and current ampli-tude of 1A: (a) Winding current (b) Position (c) Speed (d) Torque.


(a)

(b)

(c)

(d)

Figure 4.14: Waveforms of full-step control with load and current amplitudeof 3A: (a) Winding current (b) Position (c) Speed (d) Torque.


4.5 Vector Control

As Figure 4.12(c) displays, if the step size of microstepping is very tiny, thecurrent of the windings can be identified as AC current. Therefore, controlalgorithms for AC motors can also be implemented in hybrid stepper motors.Vector control, also known as Field Oriented Control (FOC), is one of themost commonly used control algorithms for AC motors. The vector controlof hybrid stepper motors uses a transformation of the coordinate, called Parktransformation, which transforms the currents in two windings under αβ co-ordinate system into two corresponding current components under dq system.The vector analysis for the hybrid stepper motor in this thesis is shown inFigure 4.15.

Figure 4.15: Vector analysis for the hybrid stepper motor [5].

The stator windings A and B are on the stationary α-axis and β-axis re-spectively, and the d-axis and q-axis are rotary and synchronized with therotation of the rotor. The angle between these two coordinate systems is θe. Itcan be concluded from Figure 4.15 that the currents in winding A and B areconverted from the stationary αβ coordinate system to the rotating dq coordi-nate system. The Park transformation for this hybrid stepper motor is :

id = iAcosθe + iBsinθe

iq = −iAsinθe + iBcosθe(4.2)


And the inverse Park transformation is:

iA = idcosθe − iqsinθe

iB = id sinθe + iqcosθe(4.3)

Zhou [5] and Fitzgerald et al. [32] point out that the current on d-axle shouldbe set as zero and the current on the q-axle could control the output torque soas to get the maximum output power as well as improve the working efficiencyof the motor. The output torque can be expressed as :

T = Kmiq (4.4)

where Km is the coefficient for motor torque.According to Equation 4.3, vector control has a high requirement on the

precision of position estimation. However, there is always some noise in theprocess of acquiring the position of a rotor. To solve this problem, the vectorcontrol is always utilized together with an Extended Kalman Filter.

In order to completely study the transient dynamic characteristics of hy-brid stepper motors, it is obliged to simulate the vector control together withFEM simulation, which could be realized through co-simulation. The vec-tor control algorithm for speed control has been developed by Ronquist andWinroth [3], Wallin [4], and Zhou [5]. Based on their work, the Simulinkmodel and the Simplorer model in this simulation are shown in Figure 4.16and Figure 4.17 respectively. The referent speed can be transformed into ref-erent position by integration. Each referent position has a corresponding ref-erent d and q value. In dq system, PI controllers are adopted for controllingthe current. With Park transformation, the currents in dq system are trans-formed into corresponding currents in αβ system. Then, the PWM moduleand H-bridge transform the referent currents into corresponding voltage sig-nal, which drives the motor.


Figure 4.16: The scheme of the Simulink model.

Figure 4.17: The scheme of the Simplorer model.

In this co-simulation model, the reference speed is set as 20 rpm to studythe motor in steady-state. As the time step in this model is 0.05 ms and thesimulation time is set as 200 ms, 4000 steps should be calculated. However,when the simulation has run for about 10 ms, the software reports an errorand the simulation is automatically stopped. The operation of the motor isnot in steady-state. The waveforms of the current in two windings are shownin Figure 4.18. In this simulation, the current waveform profiles are not veryclose to a sinusoidal waveform, whereas the current waveform profiles in areal hybrid stepper motor are a sinusoidal waveform. Thus, this model shouldbe modified in the future.


Figure 4.18: The current waveforms of co-simulation.

Normally, a time step in Simulink is calculated very quickly if the Simulinkis not coupled with other software. However, a time step in this 3D Simulinkmodel would take several minutes. If the transient simulation could be fin-ished, it would take several days. The calibration with the real system wouldtake even longer. As a time step in 2D transient simulation takes only about0.1 s, this co-simulation could be finished in several hours. Therefore, a 2Dmodel should be developed in the future to meet the requirement of the co-simulation.

Chapter 5

Conclusions and Future Work

5.1 Conclusion

In this thesis, a 3D FEM model is proposed for the transient simulation of ahybrid stepper motor. When the motor is energized, the current in windingsenlarges the magnetic field density a lot. The transient model is verified by aback EMF experiment. The full-step control strategy has limitation in termsof control speed, and control torque. Therefore, in order to avoid the problemsof full-step control, vector control is proposed as the solution. however, theco-simulation of Maxwell together with Simulink for vector control is notaccomplished. But, the results from this thesis do conclude that it is indeedpossible to perform co-simulation, which is very evident from the currentwaveforms as described in Section 4.5. In order to fully accomplish potential,one must overcome the limitations imposed by the computation time and thenumber of steps involved in the computation.

In conclusion, the transient simulation of hybrid stepper motors should becombined with a control algorithm to overcome the problem of missing step.However, 3D FEM models need too much computation time and computationresources. Therefore, a 2D FEM model should be developed to meet therequirement of co-simulation.

Some problems severely affect the process of this project. In the begin-ning, the accuracy of the dimension measurement is very bad. For example,the teeth in this motor are very small and very hard to be measured. To geta reliable dimension, the model is calibrated with experimental data by con-ducting numerous simulation. As the simulation is very time consuming, thecalibration generally takes a long time.

The Maxwell software should also be improved. As Maxwell is one of the

45

46 CHAPTER 5. CONCLUSIONS AND FUTURE WORK

most handy electromagnetic FEM software, it has no scripting support. To be-gin with, the drawing function is very weak in comparison with professionalCAD software, such as SolidWorks. The work of drawing could be sped up byimporting model created in professional software. As every modification ofdimensions needs drawing the model, it is also time-consuming. As modelingmotors in RMxprt only needs inputting parameters without hand drawing, themodeling efficiency will be greatly enhanced if the model of hybrid steppermotors could be provided by RMxprt. Although the mesh splitting functionof Maxwell in static simulation is very powerful, the mesh split in transientsimulation is not so efficient as there are always existing some mesh elementswhose ratio of maximum side length to minimum side length is too large. Thatwould distort results. Therefore, Maxwell should provide interface importingmesh split in professional mesh splitting software. Nowadays, a CPU usuallyhas several cores as distributed computing could greatly reduce computingtime. The difference in computing speed between using the different numberof cores in this project is very small, which is a waste of computing resource.It also means that even a supercomputer cannot reduce the calculation time.Thus, more effort should be done by software developers to enhance the dis-tributed computing of Maxwell.

5.2 Future Work

5.2.1 2D Model

The simulation of 3D models consumes a very long time and has a high re-quirement on computer hardware. The mesh elements in 3D models and 2Dmodels are polyhedrons and polygons respectively. The number of mesh el-ements in a 2D model is also much less than that in a 3D model. Thus, thecalculation time of a 2D model is much less. A hybrid stepper motor shouldbe firstly analyzed in an analytical model, which provides a theoretical basisfor model transformation. Then, the parameters of a 3D model can be trans-formed into the corresponding parameters in a 2D model. In this way, thesimulation speed would be greatly improved, and the FEM model could becoupled with a control algorithm in Simulink.

5.2.2 Verification of FEM Model

Due to time and instrument limitation, this thesis work only verified the reli-ability of the 3D model in the back EMF experiment. In order to calibrate the

CHAPTER 5. CONCLUSIONS AND FUTURE WORK 47

rotor position in the simulation, a full-step controller should be developed toconduct real-time experiments. For co-simulation, the current and the voltageshould also be calibrated.

5.2.3 Thermal Simulation

In the future, after solving the issue of electromagnetic dynamics, the lossesin hybrid stepper motors also need to be studied. The effect of hysteresisand eddy current can be simulated in Maxwell, where the heating power canbe obtained. As hybrid stepper motors are simply cooled by natural cooling,the Steady-state Thermal module in ANSYS could simulate the temperaturerise and there is no need for computational fluid dynamics (CFD) simulation.In this way, the motor working limitation under safety temperature can beobtained.

Bibliography

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