3D Finite Element Analysis of a Hybrid Stepper Motor

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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 TECHNOLOGY

SCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE

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3D Finite Element Analysis of a Hybrid Stepper Motor

YUANYI FAN

Degree Project in Electrical Energy Conversion Date: September 25, 2019

Supervisor: Dr. Bin Liu

Examiner: Dr. Oskar Wallmark

School of Electrical Engineering and Computer Science Host company: ABB

Swedish title: 3D FEM-analys av en hybrid stegmotor

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Abstract

Hybrid stepper motors are being applied to more and more industrial regions due to their low cost compared with servo motors and prominent performance.

Many industrial applications require accurate and effective methods for predicting a motor’s performance at the design stage. The geometry of the motors is complicated and the magnetic saturation effect is also serious, giving rise to the difficulty of understanding the transient behavior of the motors. Fur- thermore, the drive circuit and control algorithm are more sophisticated than those of traditional AC or DC motors. Lastly, the losses of the motors create the rising of temperature, while the thermal effect and dynamic performance affect each other.

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

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Sammanfattning

Hybridstegsmotorer appliceras i fler ochfler industriapplikationer tack vare deras låga kostnad och förbättrad prestanda jämfört med servomotorer. Många branschapplikationer kräver exakta och effektiva metoder för att förutsäga motorns 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 resulterar motorns förluster i temperaturökningar vilka påverkar dynamiska.

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

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Acknowledgement

I want to express my sincere gratitude to my knowledgeable and experienced supervisor, Dr. Bin Liu. He leads me in the right direction and helps me a lot in modifying my experiments and thesis. His positive working attitude and optimistic attitude are what I should learn in my life. Also, I want to thank my 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 life during my stay in ABB Sweden Research Center. Having Fika and lunch together with them is really happy.

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

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Chapter 1 Introduction

1.1 Purpose

This thesis work aims to investigate the feasibility of simulating the transient behavior of hybrid stepper motors by a universal three-dimension (3D) Finite Element Method (FEM) commercial software. The method proposed in this thesis is beneficial for predicting characteristics of this kind of motor in practical industrial applications. The research provides motor designers and users with 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 results are directly used in this thesis. In order to analyze the complex transient behavior of hybrid stepper motors, a precise model should be built and verified using existing instruments. The thesis intends to find an effective modeling method for this kind of motor and operate corresponding simulations properly for representative working conditions. Restricted by time and instruments, this work only does a limited number of simulations. The electromagnetic field is analyzed in a static state. The back electromotive force is simulated and the result is verified by experimental results. The simple open- loop control algorithm is applied to the numerical model. Also, the relatively complicated vector control algorithm is tried to apply on the model through co-simulation of Simulink and Maxwell. Although there are many kinds of hybrid stepper motors on the market, this thesis work only studies a specific motor. That is to say, the effect of motor dimension is not studied in this thesis.

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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 model is built and simulated, and then the model is changed to get a better simulation result which is closer to the experimental result. However, the simulation of 3D FEM is very time-consuming and heavily relies on experience. Thus, another challenge lies in shortening the simulation time, which is realized through model reduction. As previous studies simulated the motor only in static solvers, the biggest challenge of this project is thinking out how to simulate the motor in a transient solver and how to verify the effectiveness of the simulation. That will be detailedly discussed in Section 1.4.

1.3 Background

A stepper motor is an electromechanical component that produces a corresponding angular displacement or line displacement when it is applied with an electrical pulse signal. It is widely used as an actuator in various modern industrial products, such as reversing radar, printers, and lighting system of automobile [1]. In comparison with servo motors, the most outstanding merit of stepper motors is being cheap. Therefore, researchers and engineers have tried to replace servo motors in some applications with stepper motors, which could yield huge economic benefits. The use of stepper motors is growing rapidly that the market share of stepper motors currently accounts for about 17% of the global drive motor market [2].

In the case of low control accuracy requirements, stepper motors can be open-loop controlled by digital signals, making them easy to construct a simple, inexpensive but reliable control system. However, under open-loop control, stepper motors have poor control accuracy, which limits their range of application. Therefore, in the past three years, three projects have been conducted 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 position sensor and a hybrid stepper motor. The control strategy satisfied the demand for position accuracy and the achievement was published in three master thesis [3, 4, 5].

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

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CHAPTER 1. INTRODUCTION 3

is supplied to the motor. However, for one thing, a large current may cause the motor to heat up too much and the temperature is too high, which may damage the motor. It is well known that the temperature and dynamic characteristics influence each other, especially in over-heated condition. For another thing, the heating energy comes from the losses of motor, meaning that the large excitation current would alter the motor efficiency, i.e. the economic value.

Therefore, a relative precise model combined the dynamic and thermal features of a hybrid stepper motor should be developed so as to help researchers investigating the performance limit and understanding the transient behavior.

1.4 Previous Work/Literature Review

Numerical methods, mainly FEM methods, and analytical methods are the two most extensively employed methods to explore an electrical motor’s performance. The foundation of analytical methods is the electric and magnetic circuit, while the foundation of numerical methods is geometry treatment [6].

The excellent merits of analytical methods are that they deliver outcomes swiftly and could trace physical context underlying calculation, and could well define cause and effect, but their limitation is achieved when saturation effect and complicated geometries are considered [6]. In contrast with analytical methods, although numerical methods could overcome the limitation of complex geometry and saturation effect, they need too much computation resource and calculation time. In hybrid stepper motors, geometry complexity is greatly increased by small teeth and the saturation effect in airgap is very severe, which arise difficulties of achieving a precise model. Based on these two methods, a lot of studies has been performed by previous researchers to model and simulate hybrid stepper motors.

Only a few studies purely based on analytical methods have been conducted. For driving hybrid stepper motors by sinusoidal currents, which drive synchronous motors, the torque characteristics were studied by Mizutami, Hayashi, and Matsui [7] and Matsui, Nakamura, and Kosaka [8], using equivalent magnetic circuits based on permeance. A voltage equation was deduced from that magnetic circuits with respect to motor dimension. By this analytical method, the instantaneous torque and detent torque were validly predicted and also analyzed by Fourier transforms. This method facilitates motor designers preliminarily predicting operation characteristics of the motors under specific driving current in the design stage. Bêkir, El Amraoui, and Gillon [9]

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pointed out an analytical method computing the permeance curves in tooth/air region and it was verified by a 2D FEM simulation. Referring to the resulted permeance 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 together with 2D models.

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

Stuebig and Ponick [6] worked out an analytical model combined with the 2D finite element model. Studying the geometry of the motor, the authors developed the analytical model described by an equivalent magnetic circuit containing the corresponding magnetic permeances of hybrid stepper motor parts. Newton–Rathson method is adopted to simulate this model. The magnetic permeance of the airgap in this analytical model is calculated by a 2D FEM model as the analytical method is unable to deal with the saturation effects. 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 saturation effect in the area of tooth/airgap. In cope with this issue in hybrid stepper motors, Jenkins, Howe, and Birch [11] used a 2D finite element model to generate the permeance curves. The authors applied these curves to a nonlinear model represented as a lumped parameter network. The measurement in the experiment validates the effectiveness of this model in predicting torque characteristics, winding inductance and back electromotive force (EMF) con- stants.

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

Hybrid stepper motors can be analyzed by 3D or 2D FEM models. 3D FEM analysis is much more precise than 2D FEM analysis. However, most of the hybrid stepper motor FEM analysis attempts to create 2D models other than 3D models because 3D models require significantly more computing resources than 2D models [13]. With the development of computer performance, 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-

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

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

Ionic˘a et al. [14] introduced a 3D numerical model of a hybrid stepper motor in the design phase. The authors studied the flux distribution in the motor and the detent torque characters. They found that factors, such as airgap and materials, have a deep influence on motor performance. The result shows that the magnetic saturation effect is severe in the airgap. The authors claimed that 2D finite element models of hybrid stepper motors are unacceptable due to the 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. Isolation layers are analyzed by a cross lamination model based on reluctance in order to provide an equivalent magnetic effect in the perpendicular direction of the lamination staking direction. The result is verified by torque characteristics in experiments for two kinds of hybrid stepper motors with different permanent magnetic materials. These two articles directly adopt 3D models for analyzing the motors.

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

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

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

To apply motors in industrial applications, it is necessary to study the control algorithm of motors, where motors are generally modeled in Simulink.

Morar [21] built a model of a bipolar hybrid stepper motor using some power electronics models and motor models in the existing Power Blocks module in Simulink and simulated the motion control of the motor under different loads. This method is run with too many assumptions. For example, the detent torque is assumed to be perfectly sinusoidal changed. Also, this method can not 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 analysis software that could perform both transient and static state simulations of motors for optimizing a motor design. Therefore, many finite element simulations of motors are conducted by Maxwell. Apart from general simulation, it is 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 a two-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 motors in a static state has been studied by analytical methods and FEM methods.

The simulation results in static simulation, however, can only qualitatively compare the dynamic performance of different motors. Although the motion behavior 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

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

1.5 Contribution

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

1.6 Thesis Outline

Chapter 1 points out the research background and significance. Chapter 2 introduces the basic theory of hybrid stepper motors. Chapter 3 exhibits the modeling methods and process. Chapter 4 describes the experiment and simulation, and discusses the results. Chapter 5 gives the conclusions and future work.

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Chapter 2 Hybrid Stepper Motor

2.1 Stepper Motor

A stepper motor is an electromagnetic actuator that converts an electrical impulse into a corresponding discrete angular rotation. It has an extensive application 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 impulse sequence. While the motor is under rated load, the same impulse results in the same rotation step regardless of the load variation. Besides, the position error of each step is under 10% and the error will not be accumulated from one step to the next [14]. Therefore, the motor operation can be easily controlled by just controlling the impulse rate of the input signal. This basic feature allows 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 accuracy requirements 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. In the operation, the rotor always rotates to the position with the smallest reluctance. The operating principle of this motor is the same as that of a switched reluctance motor. This motor has several merits, such as simple structure, low cost, small step size, and good high-speed performance. However, its application range is limited by the defect of low torque, inefficiency, large heat generation, 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, the rotor and the stator have the same number of poles. Compared with VR stepper motors, the permanent magnets in PM stepper motors provide additional magnetomotive force, resulting in larger output torque. This motor has two outstanding advantages: high efficiency and large output torque. Neverthe- less, its drawbacks also include poor accuracy and large step size (typically 7.5^◦or 15^◦).

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

Although stepper motors have been widely used, they cannot simply be driven by DC power or AC power as conventional motors. Indeed, it can only be used when it is combined with pulse signals, power drive circuits, etc. into a control system. Control algorithm also significantly influences the performance of stepper motors. Therefore, it is not easy to use a stepper motor and 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 the research object. It was disassembled as shown in Figure 2.1.

Figure 2.1: Disassembled Stepper Motor [5].

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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 several teeth on the face of the pole. Copper wire wraps around poles so as to forming windings, which would be used to control the motion of motor through controlling the current in windings. A rotating shaft, two iron cores, and a ring-shaped permanent magnet constitute the rotor. The two iron cores are respectively installed at the two ends of the permanent magnet. They ex- hibit the opposite magnetic polarities respectively due to the permanent magnet. The rotor core is uniformly distributed with small teethwhich are of the same shape and size as that of the teeth on the stator, and the iron cores at both ends miss each other by a half distance between two neighbor teeth. These teeth greatly improve the position resolution of the stepper motor.

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

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

The magnetomotive force of the motor is provided by the permanent magnet and winding coil. The magnetization direction of the permanent magnet is 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 magnetic path with the minimum reluctance. Also, every pole drags the teeth which have the opposite magnetic field direction and pushes the teeth which have the same magnetic field direction, resulting in a tangential force along the axial direction of the rotor, resulting in a tangential force along the axial direction of the rotor. The reluctance mainly generated by the airgap between the rotor and the stator. In a tooth-to-tooth state, the airgap is pretty small that generally less than 0.1 mm, thereby increasing the production cost of the hybrid 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 motor designers at the design stage.

When a hybrid stepper motor is in a state of rest and not energized, a small torque must be applied to the rotor in order to break through the static equilibrium state [25]. This torque is called detent torque, also known as residual torque [26]. This effect is caused by the permanent magnet in the rotor, drawn to the poles of the stator [26]. In conventional permanent magnet motors, this effect is also called the cogging effect. In general, it is a universal feature distinguishing whether a motor has a permanent magnet.

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

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

2.4 Motor Loss

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

η = Pout

= Pout

Pout+ Pmech+ PCu+ +PFe

(2.1) where Pout and Pinare 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 a small part of total losses. A hybrid stepper motor would be heated up by the thermal energy converted from these losses. However, the increase of temperature lowers the magnetomotive force of the permanent magnet, thereby reducing the output torque. Moreover, some materials inside motors, such as plastic, would deform or even melt at high temperatures. Due to the different

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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 steps in the case of high-speed operation. Therefore, overheating would cause the motor to malfunction or even be damaged and it is the main cause of motor failure. In general, a hybrid stepper motor operating temperature does not exceed 80^◦C. Therefore, operating the hybrid stepper motor in larger output power 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 pass through motor. It is proportional to the square of the current I and can be expressed as:

Pcu = I²R (2.2)

where R is the resistance in the winding. Through theoretical calculation and experimental measurement, Derammelaere et al. [20] verified that copper loss is the main loss source of hybrid stepper motors and the copper loss generally takes up more than 50% percent of input energy. The authors also pointed out that the efficiency of the motor is pretty low because of the high current amplitude, the comparatively winding resistance of the stator, and the poor scale of torque/current. Therefore, the copper loss could be reduced by selecting a motor with small winding resistance and reducing the current during operation. A proper control algorithm would improve the efficiency of the 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 the variation of the magnetic flux density B lags behind the variation of the magnetic field strength H, which is induced by winding current. When the current in winding varies through a cycle, energy streams from the power source to the iron core in a certain period, and energy comes back the power source in another certain period. Nevertheless, the energy streaming into the core is greater than the energy streaming back to the core due to the hysteresis effect.

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

Ph = K^hBⁿ_maxf (2.4)

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

When an alternating current is passed through the windings of a hybrid stepper 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 generated in the iron core. This current is called eddy current and it circulates in a plane perpendicular to the direction of the magnetic flux. The loss generated by the eddy current in the core is called the eddy current loss Pe. It can be seen as a pure resistance circuit with a certain electromotive force when an iron core is subjected to an eddy current. The loss can, therefore, be decreased by increasing the resistance. One option is to adopt material with a big strength coefficient; the other is to use a laminated iron core due to the greater resistance of the longer electrical circuit. For iron cores stacked from silicon steel sheets, eddy current loss can be written as [27]:

Pe = KeB²_maxf² (2.5)

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

Experiments can readily measure the iron loss of motors, whereas it is pretty time-consuming. For motor design, a technique of predicting the loss should be developed. Analytical methods and FEM methods are widely used for conventional motors. However, because of the complicated distribution of magnetic fields in hybrid stepper motors, no loss simulation for them has been performed so far.

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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 are a collection of partial differential equations that describe the connection between electric field, magnetic field, the density of charge, and the density of current [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 electric charges produce electric fields. The second equation, Gauss’s law for mag- netism, demonstrates that there are no magnetic monopoles, in analogy with electric charges. The third equation, Faraday’s law, explains how variant a magnetic field induces an electric field. The last equation, Ampere’s law with Maxwell’s addition, presents that magnetic fields can be produced by two means: electric current and displacement current [28, 29].

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

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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 the magnetic fields of the motor, i.e. ρ = 0 and ^∂D_∂t = 0 [29]. And Equation 3.1 can be refined as:

∇ · D= 0

∇ · B= 0

∇ × E= −∂B

∂t

∇ × H= J

(3.2)

In electromagnetic media, field vectors have the following linear relationship:

D= E = 0rE J= σE

B= µH = µ0µrH

(3.3)

where is the permittivity, ε0 is the vacuum permittivity, r is the relative permittivity, σ is the electrical conductivity, µ is the permeability, µ₀ is the vacuum permeability, and µris the relative permeability.

In order to form an independent partial differential equation for electric field 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 Vpotis introduced, Equation 3.5 can be 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]:

∇²A

µ −σ∂A

∂t =−J+ σ∇V^pot (3.7)

The field distribution of magnetic potential and electric potential can be acquired by the numerical solution of Equation 3.7, deducing various physical quantities in electromagnetic field.

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CHAPTER 3. FEM MODELING OF MOTORS 19

3.2 Introduction to Maxwell

Numerical methods are applied to solve the partial differential equations in Section 4.1. Finite element method (FEM) and finite difference method (FDM) are the two main numerical methods for this issue. According to ˇCavka et al.

[30], in comparison with other numerical methods, FEM owns a favorable trait that neither the formulation nor the computer code needs to be changed when dealing with complex geometry and inhomogeneity material. It also usually creates symmetric and sparse matrix systems reducing the requirement of computer memory. These are the reasons why it is widely used in engineering. The solving process of FEM is divided into four steps in sequence [31]: 1) Region discretization: The solution region is discretized into subregions or elements with finite number, 2) Element analysis: A typical element is analyzed to acquire governing equation, 3) Element assemblage: All of the subregions or elements in the solution region are assembled, 4)Solution: The generated system of equations is solved.

Based on the theories mentioned above, many types of commercial FEM software have been developed for analyzing electromagnetic problems. Among them, Maxwell is one of the most outstanding software in the industry. It is extensively used in the analysis of electromagnetic components in industrial applications such as sensors, regulators, motors, transformers, and other industrial control systems. It was originally produced by ANSOFT in 2003.

Later, ANSOFT was acquired by ANSYS in 2008. Nowadays, Maxwell is a functioning module in ANSYS EM Suite and ANSYS provides some interface for Maxwell with other modules of ANSYS, such as Static Thermal Analysis, to do some co-simulation. The user-friendly interface of Maxwell is intuitive and easy to use. This software has the following advantages over other 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 as SolidWorks.

(3) It has powerful splitting mesh function. Manual splitting meshes and automatic splitting meshes are provided to users. The shape and density of the mesh 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:

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1. Select the modeling mode

There are two types of modeling mode in Maxwell, including Maxwell 2D, Maxwell 3D. In most cases of motor analysis, as the magnetization direction of the permanent magnet and the direction of the magnetic field generated by windings are in the same plane, modeling motors in Maxwell 2D could get a pretty precise result. The number of nodes generated in Maxwell 2D is also much smaller than that of Maxwell 3D, resulting in much faster calculation speed. However, Maxwell 2D cannot analyze the electromagnetic field distribution of motors in axial direction. Simulation in Maxwell 3D could get a more precise result and do more analysis with higher cost.

2. Select the solver

Maxwell 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 analysis and transient analysis respectively. The Eddy Current solver is mainly for the analysis of eddy effect and thermal effect.

3. Draw the motor

Although the way of drawing in Maxwell is similar to that of professional CAD software, Maxwell lacks some functions, such as crop. Thus, drawing a motor and importing it into Maxwell would be a great choice. Some motor spare parts with complicated geometry are difficult to be drawn by hand, such as windings. Maxwell also provides some User Defined Primitive models for these spare parts, as shown in Figure 3.1.

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Figure 3.1: Spare parts provided by User Defined Primitive.

As drawing a motor by hand is time-consuming and skilled, RMxprt in Maxwell could provide twelve kinds of motors automatically, as shown in Figure 3.2. Users just need to input the parameters of the researched motor without trivial drawing to get a 2D or 3D model. The generated model could be directly imported into Maxwell solvers. This function greatly facilitates the modeling of motors.

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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 the solution space and the motion space respectively.

4. Assign materials

A rich material library is provided by Maxwell. Also, new materials can be added to the material library by inputting parameters. It should be especially noted that the magnetization direction of permanent magnet material needs to be specified. The B-H curve of new silicon steel sheet material can be formed through inputting discrete points in that curve.

5. Assign boundary

In 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. The Help in this software shows that: The default boundary condition in Maxwell is Nature and Neumann. The magnetic field H continues across the boundary in Nature boundary condition, such as an interface between two objects. H is tangential to the boundary and flux cannot cross the boundary in Neumann boundary condition, such as the boundary of Region. Normally, the default boundary condition could meet the solving requirements.

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

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

6. Assign excitation

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

7. Assign solution parameters

Solution parameters are the outputs of simulation, including torque and force. In a motor, the rotor is compromised by several parts and all of them should be selected together while assigning solution parameters.

8. Split mesh

In Magnetostatic solver, an initial mesh is automatically generated before calculation. With the iterative calculations of solvers, the mesh is continually refined and become thicker until the solution accuracy is met. Such mesh is called adaptive mesh. Thus, it is not a must to split the mesh manually in the static solver. Also, adding manually split mesh would speed up meshing in Magnetostatic solver.

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

9. Setup of analysis

The analysis setup of Magnetostatic solver is to set up the maximum number 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 of Transient is to set up the simulation step size and duration.

10. Lookup result

When 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 other software, such as Matlab, for further post-processing.

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3.3 Co-simulation of Maxwell and Simulink

As a hybrid stepper motor is an actuator in digital products, it would always combine with a microprocessor and drive circuit so as to form an automation 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 option will automatically open the Simulink dialog for Maxwell link assignment, as shown in Figure 3.4. Now, the Simulink model can be edited with a prepared control algorithm. After editing the Simulink model, this model should be saved and closed. Lastly, the Simulink model can be imported to Maxwell by clicking Import Circuit in Figure 3.3. The following step is the same as normal Maxwell simulation process.

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

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Figure 3.4: The Simulink dialog for Maxwell link assignment.

This method owns several advantages, including simpleness, requiring less memory, and high reliability. However, the input port only can receive current signals, and the output port can only export voltage signals. In general, the application scope of this method is very narrow, as it cannot do some co-simulations whose control algorithms have position feedback.

The second method is based on the Simplorer module in ANSYS EM Suite. Simplorer is a powerful mechatronics simulation platform, as it provides plenty of physical models. A Simploerer model could integrate a Simulink model, a Maxwell model, and a mechatronics model together as a whole system, 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 Manager and then left-clicking the Set Symmetry Multiplier would open the window of Design Settings where the option Advanced Product Coupling should be enabled. The winding excitation is set as External.

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

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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 should be noted that this method needs to configure path settings in MATLAB. In the

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

Figure 3.7: The Simplorer interface in Simulink.

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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 condition. This method reduces the time of splitting mesh and computation, and the amount of stored data. However, in the process of applying periodic boundary conditions in this project, the software always reported an error and could not run. Therefore, the whole motor magnetic field is taken as the solution area.

As there is no hybrid stepper motor model provided by RMxprt, the 3D model of the hybrid stepper motor in this thesis was drawn manually. Figure 4.1 shows the 3D model of the motor. There are two identical permanent magnets in the rotor. Thus, this motor can be seen as a combination of two normal hybrid stepper motors. To simplify the analysis of the rotor, this model does not take the end winding into account.

29

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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 a ferromagnetic 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, as shown in Figure 4.1. The static analysis is conducted for this state. After 10 adaptive passes, the energy error is less than 2%. The mesh plot is shown in Figure 4.2 and this simulation took four hours. The calculated result of generated torque is 9.75 × 10⁻⁴Nm, which can be neglected as it is approximate to the real value 0 Nm. According to Figure 4.3 and Figure 4.4, the poles in 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 detent torque which takes it back to the initial position because of the uneven magnetic field density distribution. The area of stator near screws have larger magnetic field density, meaning that linking screws have an impact on the generated torque.

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CHAPTER 4. SIMULATION 31

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

(a) Top view (b) Bottom view

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

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Figure 4.4: The magnetic field density steamline plots when windings are not energized 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 Figure 4.5 and Figure 4.6 respectively. The contrast between the energized state and no energized state demonstrate that the windings enlarge the magnetic field density a lot. The comparison between the two streamline plots is not so evident because of the saturation effect.

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

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Figure 4.6: The magnetic field density steamline plots when the excitation current 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, greatly reducing the amplitude of the current in windings. The simulation exploring the back EMF is the first step to verify the effectiveness of the FEM model in the dynamic simulation of motors.

The scheme of the test rig is shown in Figure 4.7. The tested speed is 60 rpm, 120 rpm, and 180 rpm and the amplitude of the back EMF generated by the hybrid stepper motor is 4 V, 8 V and 12 V respectively. According to e = Ktv the maximum speed of the hybrid stepper motor cannot exceed 360r pm when the drive voltage is 24 V.

Figure 4.7: The scheme of the test rig.

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As Section 4.2 mentioned, the mesh of the transient simulation in Maxwell should be split by hand. The mesh of this transient simulation is shown in Figure 4.8. The results are displayed in Figure 4.9, Figure 4.10 and Figure 4.11 respectively. The results present that this model is valid. However, as the mesh is generated automatically after inputting mesh parameters, the shape and the size of the mesh cannot be defined by users, causing the mesh of some region is not split well. Maxwell cannot import mesh from professional mesh 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.

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4.4 Full-Step Control

The most convenient control algorithm for stepper motors is the open-loop control algorithm, divided into full-step control, half-step control, and microstepping control. After simplification and equivalence, the excitation schemes and the equilibrium positions for hybrid stepper motors are shown in Figure 4.12.

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 turns and only one phase at a moment is excited, this method demands minimal input energy in contrast with other methods. Half-step drive is an upgraded version of full-step control since an extra equilibrium position is added in the

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middle of the two equilibrium positions in full-step control. This provides the two phases with equal currents. As a result, there are eight equilibrium positions 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 previous researchers. When the two phases are excited by currents with different amplitude, the equilibrium position will be biased toward the stator pole with larger current amplitude. Based on this phenomenon, a series of equilibrium positions could be added between the two equilibrium positions in full-step control. The current in the two windings is stepwise changed according to the contours 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 current amplitude, and θre f is the referent mechanical degree. Using microstepping

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

Though the open-loop strategy only provides discrete positions, it reduces the 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 application is to explore the full-step control.

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

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

In summary, the full-step control cannot meet the requirement of industrial application where the load and the speed should be independently controlled.

Also, sharply speed variation, large noise, and low efficiency also limits its application. As half-step control and microstepping control are based on full- step control, they have the same limitations. Thus, a feedback control algorithm could be the solution for avoiding missing step in real-time.

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(a)

(b)

(c)

(d)

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

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(a)

(b)

(c)

(d)

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

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4.5 Vector Control

As Figure 4.12(c) displays, if the step size of microstepping is very tiny, the current of the windings can be identified as AC current. Therefore, control algorithms for AC motors can also be implemented in hybrid stepper motors.

Vector control, also known as Field Oriented Control (FOC), is one of the most commonly used control algorithms for AC motors. The vector control of hybrid stepper motors uses a transformation of the coordinate, called Park transformation, which transforms the currents in two windings under αβ coordinate system into two corresponding current components under dq system.

The vector analysis for the hybrid stepper motor in this thesis is shown in Figure 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 respectively, and the d-axis and q-axis are rotary and synchronized with the rotation of the rotor. The angle between these two coordinate systems is θe. It can be concluded from Figure 4.15 that the currents in winding A and B are converted from the stationary αβ coordinate system to the rotating dq coordinate system. The Park transformation for this hybrid stepper motor is :

id = iAcosθe+ iBsinθe

iq = −iAsinθe+ iBcosθe

(4.2)

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And the inverse Park transformation is:

iA = idcosθe− iqsinθe

iB = idsinθe+ iqcosθe

(4.3)

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

T = Kmiq (4.4)

where Kmis 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 the process of acquiring the position of a rotor. To solve this problem, the vector control is always utilized together with an Extended Kalman Filter.

In order to completely study the transient dynamic characteristics of hybrid stepper motors, it is obliged to simulate the vector control together with FEM simulation, which could be realized through co-simulation. The vector control algorithm for speed control has been developed by Ronquist and Winroth [3], Wallin [4], and Zhou [5]. Based on their work, the Simulink model and the Simplorer model in this simulation are shown in Figure 4.16 and Figure 4.17 respectively. The referent speed can be transformed into referent position by integration. Each referent position has a corresponding referent d and q value. In dq system, PI controllers are adopted for controlling the current. With Park transformation, the currents in dq system are transformed into corresponding currents in αβ system. Then, the PWM module and H-bridge transform the referent currents into corresponding voltage signal, which drives the motor.

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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 study the motor in steady-state. As the time step in this model is 0.05 ms and the simulation 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 error and the simulation is automatically stopped. The operation of the motor is not in steady-state. The waveforms of the current in two windings are shown in Figure 4.18. In this simulation, the current waveform profiles are not very close to a sinusoidal waveform, whereas the current waveform profiles in a real hybrid stepper motor are a sinusoidal waveform. Thus, this model should be modified in the future.

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Figure 4.18: The current waveforms of co-simulation.

Normally, a time step in Simulink is calculated very quickly if the Simulink is not coupled with other software. However, a time step in this 3D Simulink model would take several minutes. If the transient simulation could be finished, it would take several days. The calibration with the real system would take even longer. As a time step in 2D transient simulation takes only about 0.1 s, this co-simulation could be finished in several hours. Therefore, a 2D model should be developed in the future to meet the requirement of the co- simulation.

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Chapter 5 Conclusions and Future Work

5.1 Conclusion

In this thesis, a 3D FEM model is proposed for the transient simulation of a hybrid stepper motor. When the motor is energized, the current in windings enlarges the magnetic field density a lot. The transient model is verified by a back EMF experiment. The full-step control strategy has limitation in terms of control speed, and control torque. Therefore, in order to avoid the problems of full-step control, vector control is proposed as the solution. however, the co-simulation of Maxwell together with Simulink for vector control is not accomplished. But, the results from this thesis do conclude that it is indeed possible to perform co-simulation, which is very evident from the current waveforms as described in Section 4.5. In order to fully accomplish potential, one must overcome the limitations imposed by the computation time and the number of steps involved in the computation.

In conclusion, the transient simulation of hybrid stepper motors should be combined with a control algorithm to overcome the problem of missing step.

However, 3D FEM models need too much computation time and computation resources. Therefore, a 2D FEM model should be developed to meet the requirement 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 get a reliable dimension, the model is calibrated with experimental data by con- ducting numerous simulation. As the simulation is very time consuming, the calibration generally takes a long time.

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

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3D Finite Element Analysis of a Hybrid Stepper Motor

3D Finite Element Analysis of a Hybrid Stepper Motor

YUANYI FAN

3D Finite Element Analysis of a Hybrid Stepper Motor

YUANYI FAN

Abstract

Sammanfattning

Acknowledgement

Contents

Chapter 1 Introduction

1.1 Purpose

1.2 Scope and Challenge

1.3 Background

1.4 Previous Work/Literature Review

1.5 Contribution

1.6 Thesis Outline

Chapter 2

Hybrid Stepper Motor

2.1 Stepper Motor

2.2 Basic Structure and Working Principle

2.3 Detent Torque and Holding Torque

2.4 Motor Loss

2.4.1 Copper Loss

2.4.2 Iron Loss

Chapter 3

FEM Modeling of Motors

3.1 Theory of Electromagnetic Field

3.2 Introduction to Maxwell

3.3 Co-simulation of Maxwell and Simulink

Chapter 4 Simulation

4.1 3D Model

4.2 Static State

4.3 Back EMF

4.4 Full-Step Control

4.5 Vector Control

Chapter 5

Conclusions and Future Work

5.1 Conclusion