Comparison of FOC and DTC in an application of a screw joint emulator

(1)

IN THE FIELD OF TECHNOLOGY DEGREE PROJECT

DESIGN AND PRODUCT REALISATION AND THE MAIN FIELD OF STUDY MECHANICAL ENGINEERING, SECOND CYCLE, 30 CREDITS STOCKHOLM SWEDEN 2020,

Comparison of FOC and DTC in an application of a screw joint

emulator

AXEL FYRESKÄR JOEL GREBERG

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF INDUSTRIAL ENGINEERING AND MANAGEMENT

(2)

(3)

Comparison of FOC and DTC in an application of a screw joint emulator

AXEL FYRESKÄR, JOEL GREBERG

Master’s programme in Engineering Design, Mechatronics Track Date: July 13, 2020

Supervisor: Bengt Eriksson Examiner: Hans Johansson

School of Industrial Technology and Management Host company: Atlas Copco

Swedish title: En jämförelse av FOC och DTC i en applikation av en skruvförbandsemulator

(4)

(5)

Examensarbete TRITA-ITM-EX 2020:453

En jämförelse av FOC och DTC i en applikation av en skruvförbandsemulator

Axel Fyreskär Joel Greberg

Godkänt 2020-07-10

Examinator

Hans Johansson

Handledare

Bengt Eriksson

Uppdragsgivare

Atlas Copco Industrial Technique

Kontaktperson

Pär Augustsson

Sammanfattning

Atlas Copco tillverkar bland annat åtdragningsverktyg för skruvförband. Syftet med dessa fästelement är att producera en klämkraft mellan två delar och hålla dem ihop. Ett av de vanligaste sätten att uppskatta klämkraften är genom att mäta vridmoment, varför det är viktigt att verktyget producerar det förväntade vridmomentet och kan upptäcka avvikelser. En vanlig metod för detta är långtidstester, vilket resulterar i miljontals åtdragningar.

Normala skruvförband är inte lämpade för detta ändamål, eftersom slitage ändrar

skruvförbandets egenskaper över tid. Istället föreslogs en skruvförbandsemulator baserad på en synkronmotor med permanenta magneter (PMSM). Denna rapport undersöker möjligheten till en sådan emulator och utvärderar prestandan för två motorstyrningsalgoritmer, Field Oriented Control (FOC) och Direct Torque Control (DTC), i denna specifika applikation. Genom att modellera systemet i Simulink och Matlab kan simuleringar utföras och analyseras för att jämföra de två motorstyrningsalgoritmerna. Den bäst presterande algoritmen implementerades sedan digitalt på ett fysiskt system för att bekräfta resultaten.

Det visades att ett skruvförband kunde emuleras med en PMSM. FOC visade sig vara mer lämpad för denna applikation, med lägre vridmomentsbrus än DTC på bekostnad av responstid.

Prestandan hos skruvförbandsemulatorn konstaterades vara tillfredsställande för långtidstester.

iii

(6)

iv

(7)

Master of Science Thesis TRITA-ITM-EX 2020:453

Comparison of FOC and DTC in an application of a screw joint emulator

Axel Fyreskär Joel Greberg

Approved

2020-07-10

Examiner

Hans Johansson

Supervisor

Bengt Eriksson

Commissioner

Atlas Copco Industrial Technique

Contact person

Pär Augustsson

Abstract

Atlas Copco produces, among other things, tightening tools for screw joints. The purpose of these fastening elements is to produce a clamping force between two parts, holding them together. One of the main ways to estimate the clamping force is by measuring torque, hence it has to be made sure that the tightening tool consistently produces the requested torque and detects anomalies. One of the methods used for this is long term testing, resulting in millions of tightenings.

Normal screw joints are not suited for this purpose, as material fatigue changes the characteristics over time. Instead, a screw joint emulator based on a Permanent Magnet Synchronous Motor (PMSM) was proposed. This thesis explores the possibility of such an emulator, and evaluates the performance of two motor control algorithms, Field Oriented Control (FOC) and Direct Torque Control (DTC), in this specific application. By modeling the system in Simulink and Matlab, simulations could be performed and analyzed in order to compare the two motor control algorithms. The best performing algorithm was then implemented digitally on a physical system in order to confirm the results.

It was shown that a screw joint could be emulated with a PMSM. FOC proved to be more suited in this application, with lower torque ripple than DTC at the cost of response time. The performance of the PMSM screw joint emulator was concluded to be satisfactory for long term testing.

v

(8)

vi

Acknowledgments

We would like to thank our industry supervisor at Atlas Copco, Pär Augusts- son, for always giving us a helping hand when needed. We would also like to thank our supervisor Bengt Eriksson, for in-depth discussions about the project and its direction. We are grateful to our examiner Hans Johansson for valuable meetings at the beginning of the project, regarding complex theoret- ical problems we were facing.

At last, we would like to thank everyone else who helped with this project by producing parts, finding solutions, or generally providing support for the project.

(9)

List of Figures

1.1 Illustration of a typical screw joint [1]. . . 3

1.2 Photo of an Atlas Copco nutrunner in the ETV STR series. . . 4

1.3 Illustration of the four reference tightenings. . . 7

1.4 Illustration of an idealized reference tightening. . . 8

2.1 The two-step tightening process. . . 13

2.2 PMSM with an internal rotor and surface-mounted permanent magnets. . . 15

2.3 Equivalent circuit of a PMSM. . . 16

2.4 Stator fixed abc-coordinate system of a PMSM. . . 18

2.5 Stator fixed αβ-coordinate system of a PMSM . . . 19

2.6 Rotor fixed dq-coordinate system. . . 20

2.7 Schematic of an inverter using Metal Oxide Semiconductor Field Effect Transistors (MOSFETs) . . . 21

2.8 Illustration of possible voltage vectors and the three motor phases direction. . . 22

2.9 Overview of the Field Oriented Control (FOC) algorithm. . . . 23

2.10 Example of a possible voltage vector Vref to be generated with the Space Vector Pulse Width Modulation (SVPWM) algorithm. 24 2.11 Switching times using SVPWM [11]. . . 25

2.12 Flux linkage sectors used in DTC . . . 28

2.13 Block diagram of closed-loop flux estimator with PI controller. 31 2.14 Sine encoder outputs . . . 33

3.1 Simulink model used for verification of the motor block’s inertia. 38 3.2 Result of inertia verification. . . 39

3.3 Simulink model used for verification of the motor block’s torque. 39 3.4 Result of torque verification. . . 40

3.5 Torque reference generator. . . 42

ix

(12)

x LIST OF FIGURES

3.6 Overview of the inverter model, used both for FOC and Direct

Torque Control (DTC). . . 42

3.7 Overview of the common parts of both FOC and DTC models. 43 3.8 Top-level overview of the four main blocks of DTC. . . 44

3.9 Simulink implementation of the DTC flux estimator. . . 44

3.10 Simulink implementation of the DTC torque estimator. . . 44

3.11 Simulink implementation of the vector selection for DTC . . . 45

3.12 Tightening regions of interest. . . 46

3.13 Side by side Torque-time plot of simulated tightenings with DTC and FOC, 2 Nm soft joint. . . 47

3.14 Side by side Torque-time plot of simulated tightenings with DTC and FOC, 2 Nm stiff joint. . . 48

3.15 Side by side Torque-time plot of simulated tightenings with DTC and FOC, 5 Nm soft joint. . . 48

3.16 Side by side Torque-time plot of simulated tightenings with DTC and FOC, 5 Nm stiff joint. . . 48

3.17 Magnitude of the control signal from the FOC controller, ex- pressed in the αβ-frame, during a 5 Nm soft tightening. . . 49

3.18 DC-bus current for the FOC controller during a 5 Nm soft tightening. . . 50

3.19 DC-bus current for the DTC controller during a 5 Nm soft tightening. . . 50

4.1 Simplified schematic of the hardware connections . . . 54

4.2 Timing of SVPWM period and control loop . . . 55

4.3 Illustration of the tightening phase state machine. . . 56

4.4 Plot of real screw joint tightening and PMSM screw joint to 5 Nm, soft joint. . . 57

4.5 Plot of real screw joint tightening and PMSM screw joint to 5 Nm, stiff joint. . . 58

4.6 Plot of real screw joint tightening and PMSM screw joint to 2 Nm, soft joint. . . 58

4.7 Plot of real screw joint tightening and PMSM screw joint to 2 Nm, stiff joint. . . 59

4.8 Comparison of real screw joint tightening and PMSM screw joint for a stiff 5 Nm joint. . . 60

5.1 Same 2 Nm stiff tightening as in figure 4.7, with the addition of markers. . . 62

5.2 Comparison of simulated FOC and DTC DC-bus currents. . . 62

(13)

LIST OF FIGURES xi

5.3 Side by side comparison of the implemented PMSM screw joint and the simulated FOC screw joint. . . 63

(14)

List of Tables

1.1 Risk analysis table. . . 10

2.1 Possible switching states . . . 21

2.2 List of available voltage vectors . . . 22

2.3 Voltage vector switching sequence [11]. . . 26

2.4 Switching table for two-level hysteresis of both torque and flux. 29 3.1 Tightening parameters for the simulations. . . 45

3.2 Measurements done in each of the regions of the tightenings. . 47

3.3 Performance measurements and statistics for simulated DTC tightenings. . . 51

3.4 Performance measurements and statistics for simulated FOC tightenings. . . 51

3.5 T-test results . . . 52

4.1 Motor parameters for the AKM-33E PMSM used. . . 54

xii

(15)

Nomenclature

PMSM Permanent Magnet Synchronous Motor MTPA Maximum Torque Per Ampere

RTFE Robust Torque Flux Estimator DTC Direct Torque Control

FOC Field Oriented Control PWM Pulse Width Modulation BLDC Brushless Direct Current

IGBT Insulated-Gate Bipolar Transistor

MOSFET Metal Oxide Semiconductor Field Effect Transistor SVPWM Space Vector Pulse Width Modulation

SPWM Sinusoidal Pulse Width Modulation BEMF Back-Electro Magnetic Force ISR Interrupt Service Routine ADC Analog-Digital Converter FPU Floating Point Unit

DMA Direct Memory Access SNR Signal To Noise Ratio RMS Root Mean Square

FPGA Field Programmable Gate Array

1

(16)

(17)

Introduction

1.1 Background

In order to securely mate two components, one of the most common methods is the use of screw joints. Screw joints use a screw or bolt with external threads and a nut with internal threads. In some cases, the nut is replaced by internal threads in one of the mating components. The components to be mated are placed between the screw and nut (if used), together forming the complete screw joint. By applying torque and rotating either the screw or nut while holding the other still, the screw joint is tightened. A clamping force is produced, holding the components together [1]. An illustration of a screw joint can be seen in figure 1.1.

Figure 1.1: Illustration of a typical screw joint [1].

Atlas Copco is a global supplier of, among other things, tightening tools, compressors, and generators [2]. One of the most common types of tightening tools is called a nutrunner, which is an assembly tool used for tightening screw joints. Nutrunners can be powered either pneumatically or electrically, with

3

(18)

4 CHAPTER 1. INTRODUCTION

the latter being what the term will refer to from here on. A photo of a nutrunner is available in figure 1.2.

Figure 1.2: Photo of an Atlas Copco nutrunner in the ETV STR series.

The System Solution Verification team within Atlas Copco is responsible for, among other things, testing and ensuring that their tools perform as expected. In doing so, it is crucial to externally verify all data reported by the tool. Externally, in this case, refers to only using the tool’s external interfaces available to the end-user, and not making any changes to the firmware or other internals of the tool. External verification is done in order to keep the tests as close to real-world usage as possible. These kinds of tests often need to be automated and performed continuously over a long period of time, resulting in millions of cycles. Due to the changes in the friction of the screw joint, and high-cycle fatigue as a result of the repeated stress cycles, it is not feasible to utilize actual screw joints for this purpose [3][4]. The current state of the art utilizes hydraulic solutions that are both prohibitively expensive and bulky and not suited for long term testing, as they were developed for calibration purposes [5].

The System Solution Verification team would like a cheaper, alternative system to the one mentioned above, in order to allow for multiple tests to be performed simultaneously in an office environment. The research, development, and implementation of such a system will be the main focus of this master thesis.

The system should allow tightening strategies of different torque levels and tightening angles, as it would then be possible to test a range of different tools with a variety of different screw joints. Other teams within Atlas Copco have shown interest in such a product, including the Data-Driven Services team, who would like to use the data generated by the system for deep learning purposes in order to develop a predictive maintenance application for the tools.

Such an approach would have the possibility of reducing the required maintenance, resulting in saved resources - both economic and environmental. Such a solution would remove the need for fixed service intervals, and the tools would only be serviced as needed. With increased knowledge of the tightening tool’s behavior during tightening, another possible benefit would be higher

(19)

CHAPTER 1. INTRODUCTION 5

joint quality assurance, meaning fewer joints with insufficient clamping force as a result of improper tightening.

1.2 Purpose

This thesis is based on the idea that a controlled braking system can be used in long term testing of tightening tools in order to reproduce the characteristics of a real screw joint, as seen from the tightening tool’s perspective. From preliminary research done at Atlas Copco, a PMSM was shown to be a viable option for the braking system (from here on referred to as a PMSM screw joint) and will be used in this thesis.

The PMSM screw joint shall allow for varying torque and tightening angles (joint stiffness) in order to replicate a range of screw joints. It is desirable to be able to replicate both stiff and soft screw joints, as differences in material hardness and geometry produce a screw joint with a different set of characteristics, and as a result, putting new demands on the PMSM screw joint. The tightening tool was configured for a two-step strategy throughout this thesis, as it is one of the most used strategies in the industry and contains most of the dynamic characteristics seen during a regular screw joint tightening. The two-step tightening strategy is further explained in section 2.1.

By proving that the PMSM screw joint can replicate the behavior of real screw joints of varying stiffness and target torques when tightened with the two-step strategy, the PMSM screw joint should be able to mimic a wide range of different screw joints. Further discussion about this is available in section 1.3. The tightening tool chosen to be used throughout the thesis was the ETV STR31- 05-10, which has a maximum torque of 5 Nm, and a maximum speed of 3070 rpm. The reason for choosing this particular tool was mainly that the PMSM screw joint is intended to be mounted in a smaller office area, putting high demands on size, noise levels, and the safety of lower torque levels. It was also regarded that if it is possible to develop a fully working PMSM screw joint for this tool, it would likely be possible to adapt this solution for more powerful tools in the future.

In order to control a PMSM, it is common to use a control algorithm that takes a reference torque as input and produces the switching logic for the inverter.

In many applications, further control loops are cascaded with the torque control algorithm for speed and position control. Two of these control algorithms,

(20)

FOC and DTC, will be modeled and simulated with the help of Matlab and Simulink. Their performance in this specific application will then be evaluated. FOC was chosen as it is considered the industry standard for PMSM control, while DTC promises faster response times. The control algorithm with the best performance will then be implemented on a physical PMSM screw joint.

To clearly specify the main focus of this master thesis project, a research ques- tion has been formulated as follows:

Given a two-step tightening strategy, how well can a PMSM emulate physical screw joints of varying torque targets and stiffness with respect to torque over time, and what are the implications on this when using FOC or DTC control?

1.3 Method Description

First, data (torque-time and angle graph) from four reference tightenings of physical screw joints will be collected using the nutrunner’s built-in torque transducer and data logging tools. The reference tightenings will be performed at two levels of torque (2 and 5 Nm), and at two levels of stiffness (target torque at roughly 150^◦ and 700^◦), for a total of four tightenings. A joint is commonly defined as stiff when target torque is reached in less than 180^◦, and soft when target torque is reached above 360^◦. The four tightenings will define an operational zone for the PMSM screw joint, see figure 1.3.

(21)

Figure 1.3: Illustration of the four reference tightenings, together defining an operational zone for the PMSM screw joint.

The reference tightenings will then be used as a model to create ideal torque-time graphs that can be used as a reference for the control algorithms throughout the project. The ideal torque-time graph is found by identifying critical points such as snug point and first target torque, and then linearizing between them, see figure 1.4. Linearization can be done since an ideal screw joint, up until the joint materials’ yield points are reached, will behave linearly.

(22)

Figure 1.4: Illustration of a reference tightening in orange and the corresponding ideal tightening in blue.

The development phase will include both modeling and simulations of the system, as well as a digital implementation of one of the control algorithms on hardware. By calculating the deviation from the ideal torque graphs for both of the control algorithms, an objective measure of the control algorithms’ performance in this specific application can be made. Deviation will be calculated with multiple measurements, as one single measurement is not enough to cap- ture the behavior of the systems. Because of this, the regions of high and low dynamics will be analyzed individually. High dynamics refers to the regions where the speed of the tool quickly changes, such as stopping at the first or sec- ond target torque, while low dynamics is considered to be all regions where the speed of the tool is kept constant. Signal to noise ratio and Root Mean Square (RMS) noise will be used to evaluate performance in the slow dynamics region, while fall times will be used in the fast dynamics regions. The performance of the simulated control algorithms can then be compared using an independent t-test with two samples. The independent t-test uses multiple measurements from two groups, also known as samples. The measurement’s mean and standard deviation is then calculated for each of the samples sepa-

(23)

rately. The t-value is then calculated as t =

¯X1− ¯X2

qs²₁

n1 +_n^s²²

2

(1.1)

whereX¯_1,2 are the mean values of the measurements, s1,2 the standard devi- ations, and n1,2 the number of measurements from each sample. A table can then be used to find the p-value corresponding to the calculated t-value. The p-value will then determine if there is a statistically significant difference between the two samples [6]. In the case of this thesis, the two algorithms will be the samples for analysis. For each performance measure in each region, one t-test will be performed in order to evaluate the two hypothesizes:

1. H0 : µ₁ = µ₂ 2. H1 : µ₁ 6= µ₂

The t-test will either refute or support H⁰, resulting in a conclusion on whether FOC or DTC has any significant advantage over the other in the region cur- rently evaluated. In the case that H0 is supported by the majority of the t-test (no significant difference between the two algorithms), other aspects such as implementation complexity, hardware price, and power consumption can be considered in order to conclude which control algorithm is most suited. The best-suited control algorithm will then be implemented on hardware.

1.4 Delimitations

Due to time constraints in relation to the accuracy of the results, only the algorithm that shows the most promising results in the simulations will be implemented digitally on the physical system. The project will, as mentioned above, focus on one single type of tightening, namely the two-step method.

As the tightening tool is a complex system of both hardware and software, its software and mechanical properties will not be modeled in the simulation environment. Instead, a noise-free version of the velocity for the four types of reference tightenings will be used in the simulations. A more detailed discussion of the reasoning behind this decision and the impact of using a forced speed instead of fully modeling the tool is available in chapters 3 and 5.

In some two-step tightenings, different rotational velocities are used for the

(24)

different stages of the tightening. In this project, all stages will share the same target velocity in order to minimize the complex dynamics that arise when rapidly changing velocity.

The project will not attempt to replicate imperfections in the screw joints, such as non-constant prevailing torque (see section 2.1), non-linear torque/angle re- lations, dirt, or other factors that could impact the behavior of the screw joint and the tool.

In order to ensure that the project will be reasonable to finish within the as- signed 20 weeks, motors and the associated hardware was purchased instead of developed.

1.5 Risk Analysis

It is inherent to every project that certain risks will arise and have to be han- dled correctly in order for the project to keep on track, both in time and budget use. It is, therefore, essential to critically assess what the significant risks for the project might be, so that these can be actively mitigated. A brainstorm- ing session resulted in a list of project-related risks that were analyzed with respect to their likelihood and the severity of the consequences, according to table 1.1.

Risk Likelihood Consequence Sum

Research takes longer than expected 1 3 4

Development takes longer than expected 3 3 6

Testing takes longer than expected 1 3 4

Difficulties in finding suitable hardware 1 9 10

Parts do not arrive as scheduled 3 3 6

Parts arrive with faults 1 9 10

Parts do not perform as expected 3 3 6

Parts fail during development/testing 9 9 18

Hardware does not perform as simulated 9 3 12

Control system does not perform as simulated 9 3 12

Control system interference with the tool 3 3 6

Unable to get adequate performance due to hardware limitations 3 3 6 Chosen evaluation metrics do not describe system performance comprehensively 3 3 6

Table 1.1: Risk analysis table.

As can be seen in the table above, the three main topics to be aware of are related to hardware failure during testing or development, hardware not

(25)

performing as simulated, and the control system not performing as simulated.

While there were other topics with severe consequences, they were deemed unlikely to occur and will not be discussed further in this report. For all risks, extra time slots have been added to the planning in order to ensure that possible delays in the individual phases do not push the project end date past the planned deadline. For the three main topics, a short discussion on possible preventative and mitigating actions follows below.

There is a high risk of hardware failure during the development and testing phase of the project, and the consequences following a failure of multiple components might be severe. Hence, essential safety-steps must not be left out while handling the hardware during critical phases. The project members should strive to, when applicable, test individual sub-systems separately in a controlled environment before assembling a larger system, to avoid having a single faulty component or configuration ending up causing more damage than necessary. Furthermore, time should be allocated to ensure that even if a sub- system would fail during testing, there would be enough time to order, ship, and implement a new one without delaying the whole project. In subsystems where risk is extra high, and the economic aspect allows for it, it might be preferable to order more than one unit from the start. That way, the waiting time associated with ordering new parts only when the first one has already failed can be avoided.

The second risk regarded very likely to happen concerns the hardware not performing as in the simulations. As the simulations always differ slightly from the physical system, this is expected, but still considered a risk. Depending on the magnitude of the error, it might impact the performance enough to change the results. Care must be taken to ensure enough realism in the simulations so that the real-world tests do not differ in such a significant way that the hardware choices are impacted. If, for example, the simulation would show a 5 Nm motor to be enough, but real-world tests later show that motor to be too soft, extra time and money would have to be spent on a new motor having to be or- dered, shipped and implemented. Another way of mitigating issues relating to hardware performance is to strive to choose components with enough performance overhead to allow for a certain degree of differences between real-world and simulations. Such choices often come at an economical cost. However, if a delay can be avoided, it is likely to be beneficial in the end compared to ordering a new one, having to factor in shipping times and labor.

(26)

The third risk discussed in this chapter relates to the accuracy of the simulation, but this time from a control theory perspective. If the control system is not implemented identically on hardware, or if it acts on input data that does not closely resemble the one in the simulations, the results will most likely differ. Again, a bit of variation is expected, but if the error is large enough to re- quire different hardware in the real world, it might cause delays. Modeling the control system with reasonable parameters, including control loop frequencies and accuracy of sensor data and noise, is crucial to be able to make a realistic choice of hardware and software parameters for the solution. How the control algorithms are implemented in software on the real system is crucial to ensure accurate performance measurements, and the project members should aim to test and verify each software component individually before assembling the whole system, much like in the hardware development.

1.6 Ethical considerations

By increasing the accuracy and amount of data available about the wear of the tools, Atlas Copco would have the possibility of reducing the required maintenance, resulting in saved resources - both economic and environmental. How- ever, it could be debated whether or not the resources saved would be canceled out by the energy consumed by the PMSM screw joint.

With Atlas Copco’s customers in the automotive and aerospace industry in mind, another possible benefit would be higher joint quality assurance, reducing the risk of defective joints with all its associated risks in the mentioned industries.

(27)

Frame of reference

This chapter aims to outline and explore the theory needed in order to under- stand the modeling, simulation, and implementation of a PMSM screw joint on hardware.

2.1 Screw joints and tightening

The thesis will focus on one single tightening tool with a two-step tightening strategy, see figure 2.1.

Figure 2.1: Two-step tightening process. The rotational velocity can be seen in red, and the torque in blue. In this figure, ω^ris the rotational velocity, T is the torque, and t is the time from the start of the tightening process.

Rundown is defined as the part up until the screw head touches the joint.

During the rundown, constant speed is held. The torque required to rotate the screw during rundown is low and usually called prevailing torque. The tool

13

(28)

14 CHAPTER 2. FRAME OF REFERENCE

considers the rundown phase as completed when the torque required to ro- tate the screw rises above the prevailing torque; a point called the snug point.

After rundown, it is common that a lower speed is held up until first target torque. First target torque is usually set to 40-60% of the final target torque.

Once first target torque is reached, the tightening stops in order to allow for relaxation (stress equalization) in the joint. Lastly, the tool tightens up to fi- nal target torque, after which it stops completely. During these phases, the tool is operating in speed-control mode, while the torque required to hold that speed determines the start and end of a phase. The materials of the screw joint, and their respective Young’s modulus, changes the characteristics of the joint, mainly in terms of stiffness. The stiffness is quantified as a stiffness constant, k_slope, and is measured in Nm/degree. A joint with materials of higher elasticity constants make for a softer joint, while lower elasticity constants make for a stiffer joint. With a softer joint, a larger rotation of the screw or nut is needed to achieve a given target torque.

2.2 PMSM

The PMSM is a three-phase electrical motor, with permanent magnets attached to its rotor, and phase windings in its stator, 120 degrees apart. The precise construction varies, as the rotor could be either internal or external, the number of permanent magnets (poles) varies, and the design and exact placement of the poles varies. By pushing current through the stator windings, a magnetic field, also known as flux, is generated that interacts with the magnetic field from the permanent magnets on the rotor. By controlling the current through the windings, and hence the magnetic field, a torque can be produced, causing the rotor to spin. The PMSM has sinusoidally distributed windings, meaning the Back-Electro Magnetic Force (BEMF) generated by spinning the motor will take the shape of sine-waves, as opposed to a Brushless Direct Current (BLDC) motor, where the BEMF has a trapezoidal shape. Sinusoidal BEMF makes for smoother operation with less torque ripple, provided that the commutation of the phases are performed correctly. Phase commutation with FOC and DTC is performed differently and will be further explained in sections 2.4.1 and 2.5, respectively. An illustration of a PMSM is available in figure 2.2.

(29)

CHAPTER 2. FRAME OF REFERENCE 15

Figure 2.2: Example of a PMSM with an internal rotor and surface-mounted permanent magnets (2 pole pairs). [7]

2.2.1 Mathematical modeling

The PMSM can mathematically be described as



 va

v_b v_c



=





Rs 0 0

0 R_s 0

0 0 R_s







 ia

i_b i_c



+





dψa

dψdtb

dψdtc

dt



 (2.1)

where vabcare the phase voltages, Rsis the individual phase resistance (half of the measured resistance between two phases), iabc the phase currents, and ψ_abcthe flux linkage. The flux linkages are caused by stator currents and the permanent magnets and can be described by



 ψ_a ψ_b ψc



=





L_aa L_ab L_ac L_ba L_bb L_bc Lca Lcb Lcc







 i_a i_b ic



+ ψ_m





cos(θ_el) cos(θ_el−^2π₃ ) cos(θel+^2π₃ )



 (2.2)

where Laa, Lbb, Lcc are called self inductances, Lab, Lac, Lba, Lbc, Lca, and L_cb are called mutual inductances, ψm is the flux linkage generated by the permanent magnets, and θel the rotor’s electrical angle, further explained in section 2.2.2. The flux linkage generated byt he permanent magnets can be calculated from

ψ_m = 2K_t

3N (2.3)

where Ktis motor torque constant and N the number of pole pairs. Equation 2.1 can be derived from an equivalent circuit of the PMSM, seen in figure 2.3.

(30)

In this figure, the flux linkages have been replaced by the induced voltages eabc, also known as BEMF. Such a substitution can be done since the derivative of the flux linkage is equal to the BEMF, as seen in equation 2.4 below.

dψabc

dt = e_abc (2.4)

Figure 2.3: Equivalent circuit of a PMSM.

By using the Clarke and Park transformations, further described in section 2.2.2, equation 2.1 takes a new form in the αβ-coordinate system:

vα = Rsiα+dψ_α

dt (2.5)

v_β = R_si_β +dψ_β

dt (2.6)

In the dq-coordinate system, the same equations become v_d= R_si_d+ dψ_d

dt − ω_elψ_q (2.7)

v_q= R_si_q+dψ_q

dt + ω_elψ_d (2.8)

where ωel is the electrical angular speed of the rotor. ψd and ψq are the magnetic flux linkage in the d and q axis and can be calculated as

ψ_d= L_di_d+ ψ_m (2.9)

(31)

ψ_q = L_qi_q (2.10)

where Ld,q are the direct and quadrature inductance components. By substi- tuting equation 2.9 and 2.10 into equation 2.7 and 2.8, respectively, the direct and quadrature component voltages can now be described as

v_d= R_si_d+ di_d

dt L_d− ω_elL_qi_q (2.11) v_q = R_si_q+ di_d

dt L_q+ ω_el(L_di_d+ ψ_m) (2.12) At last, the electrical torque produced by the motor is described by

T_e = 3

2N (ψ_m+ (L_d− L_q)i_d)i_q (2.13) The electrical torque can be split up into two parts, magnet torque T^m and reluctance torque Tr

T_e = T_m+ T_r (2.14)

where the magnet torque is

T_m = 3

2N ψ_mi_q = K_ti_q (2.15) and the reluctance torque is

T_r= 3

2N (L_d− L_q)i_di_q (2.16) The motor’s saliency ratio,

ρ = L_q

L_d (2.17)

determines the amount of reluctance torque produced at any given operational point [8]. For motors with surface mounted poles/permanent magnets (such as the one used in this thesis), L^q = L_d= L_sare all equal, meaning a saliency ratio of one, also known as a non-salient machine. As a result, no reluctance torque is produced - and the electromechanical torque is purely dependant on the quadrature axis current, and any direct axis current results in pure power losses.

2.2.2 Coordinate transformations

In order to better understand and control the motor, it is useful to define a few coordinate systems, first of which is the stator fixed abc-system, aligned with

(32)

each of the stator windings. This coordinate system can be used to describe both winding voltages and currents. As an example, the current in each winding (~i_a, ~i_b, ~i_c) can be summed, and together create the stator current vector,

~i.

~i = ~i_a+ ~i_b+ ~i_c (2.18) This is illustrated in figure 2.4, together with the abc-coordinate system.

Figure 2.4: The stator fixed abc-coordinate system of a PMSM, with an example current or voltage vector.

As it is redundant to have three coordinate axes in a 2D-plane, this coordinate system can be reduced into the two-axis αβ-coordinate system using the Clarke transform. When aligning the α-axis with the a-axis, this transform is given by



 α β 0



= 2 3







1 ⁻¹₂ ⁻¹₂ 0

√ 3 2

−√ 3

q 2 1 2

q1 2









 a b c



 (2.19)

and the inverse transform as



 a b c



=







1 0 1

−¹₂

√ 3

2 1

−¹₂ −

√3

2 1









 α β 0



 (2.20)

(33)

The resulting coordinate system overlaid with the abc-coordinate system can be seen in figure 2.5.

Figure 2.5: The stator fixed αβ-coordinate system of a PMSM, with an example current or voltage vector.

With the rotor rotating with a mechanical speed of ω^r, the electric angular frequency is

ω_el= ω_r· N (2.21)

where N is the number of pole pairs. Similarly, the electric angular position of the rotor is found using

θel= θr· N (2.22)

where θ^ris the mechanical angular position of the rotor. The second transformation, called the Park transform, uses the rotor’s electric angular position to produce a rotor fixed coordinate system, dq. The Park transform is given by



 d q 0



=





cos(θ_el) sin(θ_el) 0

−sin(θ_el) cos(θ_el) 0

0 0 1







 α β 0



 (2.23)

and the inverse Park transform by



 a b c



=





cos(θ_el) −sin(θ_el) 1 cos(θ_el− ^2π₃ ) −sin(θ_el−^2π₃ ) 1 cos(θ_el+ ^2π₃ ) −sin(θ_el+^2π₃ ) 1







 d q 0



 (2.24)

(34)

As seen in equation 2.15, the torque produced by the motor is directly proportional to the q-axis current multiplied with the torque constant, K^t. As a result, the stator current vector, ~i, should be aligned with the q-axis in order to produce the highest possible amount of torque for a given stator current.

The dq-coordinate system is illustrated in figure 2.6.

Figure 2.6: The rotor fixed dq-coordinate system. Here, θrrepresents the rotor mechanical angle, and N, the number of pole pairs is equal to one. Because of the motor having one pole pair, θ^el = θ_r[9].

By combining the Clarke and Park transform, one can go directly from abc- coordinates to dq-coordinates. The equation for this transformation is given in equation 2.25 [10].



 d q 0



= 2 3





cos(θel) cos(θel− ^2π₃ ) cos(θel+ ^2π₃ ) sin(θ_el) sin(θ_el−^2π₃ ) sin(θ_el− ^2π₃ )

1 2







 a b c



 (2.25)

By designing the current controller in this coordinate system, and then doing the inverse Park and Clarke transforms, abc-phase voltage commands can be acquired and sent to the inverter in order to apply the required phase voltages to the motor.

2.3 Inverters

To drive a PMSM, an inverter is often used to create an alternating current from a DC power supply. It can be built using transistors, MOSFETs, or Insulated-

(35)

Gate Bipolar Transistors (IGBTs). The DC voltage supply can be acquired from a battery, a bench power supply, or from a rectified AC voltage source.

The DC power supply has a voltage, V^bus, and is applied across the switching components of the inverter, as seen in figure 2.7.

Figure 2.7: Schematic of an inverter using MOSFETs

The switching components are switched in pairs such that no two switching components connected to the same motor phase are conducting at the same time - which would create a short circuit of the DC power supply. As an example, with s1closed (conducting), s2is open (not conducting). By opening and closing the switches, +Vbusand GN D can be applied to the motor terminals.

In order to accommodate for the transient phase of the switching components, a dead time is commonly inserted between switches. The risk of shorting phases and damaging components is thus reduced. A full table of all possible switch states is shown in table 2.1.

Phase High/Low Switch States

a - high S1−Closed S2−Open a - low S1−Open S2−Closed b - high S3−Closed S4−Open b - low S3−Open S4−Closed c - high S5−Closed S6−Open c - low S5−Open S6−Closed

Table 2.1: Possible switching states

As the two switches/gates of a phase are always inverted (S1 = Closed ⇒ S₂ = Open, and the other way around), this table can be condensed into a short list, using only the state of the upper gate, see table 2.2.

(36)

Voltage vector Phase states (abc)

V0 000

V1 100

V2 110

V3 010

V4 011

V5 001

V6 101

V7 111

Table 2.2: List of available voltage vectors

For easier understanding, this can be drawn as a graph showing the directions of the voltage vectors, see figure 2.8.

Figure 2.8: Illustration of possible voltage vectors and the three motor phases direction.

Out of the eight possible voltage vectors, V1 trough V6 are called active, and V0 and V7 has a vector sum of zero (as seen in figure 2.8) and are called zero vectors. Using a Pulse Width Modulation (PWM) signal to control the switches, any voltage vector within the hexagon shown in figure 2.8 can be applied to the motor phases, which in turn means that the phase currents, and thereby the torque, can be controlled on a finer level [8].

(37)

2.4 FOC

FOC is a vector-based control method for PMSMs, using the Clarke-Park transformation, see equation 2.25, in order to control the currents in the dq- coordinate system. A sensing method for the rotor position is hence required.

Sensing can be done either by using an encoder or estimating the position using the voltages induced in the phases when spinning the motor (BEMF). An overview of the FOC algorithm can be seen in figure 2.9.

Figure 2.9: Overview of the FOC algorithm.

As shown in 2.2.1, for a PMSM with a saliency ratio of one, the electromechanical torque produced is purely dependent on the quadrature current. As a result of this, the direct current reference, i^∗dis usually set to zero (when flux field weakening is not required for higher speed operation). The quadrature current reference, i^∗q is set to

i^∗_q = T^∗

K_t (2.26)

where T^∗ is the reference torque. The current controller is traditionally based on two simple PI-controllers, one for each current reference, i^∗d and i^∗q. The current controller generates voltage references, vd^∗and vq^∗, that are then transformed back into abc-coordinates using the inverse Clarke-Park transformation. These voltages are sent to a PWM generator, running an algorithm to generate the gate signals needed in order to produce the requested voltages.

This process is known as commutation. Examples of such algorithms are SVPWM and Sinusoidal Pulse Width Modulation (SPWM). SVPWM is further described in section 2.4.1 and will be used in this thesis.

Two of the phase currents are measured using current sensors, further explained in section 2.7, and the third current can then be calculated with the

(38)

help of Kirchoff’s law

ia+ ib+ ic = 0 (2.27)

The phase currents are then transformed into the dq-coordinate system and used as feedback for the current controller [8].

2.4.1 SVPWM

SVPWM is an algorithm used to generate a voltage vector of the desired magnitude and direction (V^ref) from the available voltage vectors of a three-legged inverter, see figure 2.10.

Figure 2.10: Example of a possible voltage vector V^ref to be generated with the SVPWM algorithm.

As seen in figure 2.10, the possible voltages to apply are all within the hexagon. It can also be seen that the maximum voltage magnitudes are not equal in all directions. By limiting the applied voltage vectors to be within the circle with radius V^{ref max}, these non-linearities are avoided. To apply V^ref, SVPWM switches between the two adjacent voltage vectors in order to achieve the angle, α, of Vref. The zero vectors are used in the switching sequence to reduce the magnitude of the produced voltage vector. In figure 2.10, the used

(39)

voltage vectors would be V1, V2, and the zero vectors V0 and V7. Switching times for the example above can be seen in figure 2.11 below.

Figure 2.11: Switching times using SVPWM [11].

In this figure, the author used the notation U 0-U 7 for voltage vectors and T for one PWM-period, referred to in this report as V 0-V 7 and Tpwm, respectively. Here, t0is the on-time for the zero vectors (split equally between V0and V₇), t¹the on-time for the first active vector, and t²the on-time for the second active vector. Tpwm the sampling period, and Tmod the sum of the on-times t1

and t2. ∆t is half of Tpwm. Note here that the sampling period and the PWM period are the same, but offset with half a period with respect to eachother.

Mathematically, the on-time for each voltage vector can be found using

t₀ = T_pwm− T_mod (2.28)

t₁ = A |V_ref| cos(α) − t₂

2 (2.29)

t₂ = B |V_ref| sin(α) (2.30)

Constants A and B are calculated with A = (3

√3

2π)T_pwm (2.31)

B = 3

πT_pwm (2.32)

(40)

These equations, combined with the required inverter gate states for each of the voltage vectors, see table 2.2, also gives the on-times for the inverter gates.

There are multiple options for determining the exact order in which the switching happens. It is common to split the sampling period into two equal halves, where the switching sequence is reversed in the second half of the sampling period. A valid switching sequence could then be V⁰, V¹, V², V⁷, V⁷, V², V¹, V⁰ [11]. A full table for the switching sequences used in this thesis is available in table 2.3.

Sector Vector sequence

1 V0, V¹, V², V⁷, V⁷, V², V¹, V⁰ 2 V0, V³, V², V⁷, V⁷, V², V³, V⁰ 3 V0, V3, V4, V7, V7, V4, V3, V0

4 V0, V5, V4, V7, V7, V4, V5, V0

5 V0, V5, V6, V7, V7, V6, V5, V0

6 V0, V¹, V⁶, V⁷, V⁷, V⁶, V¹, V⁰

Table 2.3: Voltage vector switching sequence [11].

2.5 DTC

DTC is a different vector control-based strategy for controlling a PMSM, where the torque and flux of the motor are estimated using measured phase currents and voltages. As opposed to FOC, DTC does not rely on PWM for voltage excitation. Instead, classical DTC uses a bang-bang approach, where as soon as the estimated torque or flux is outside of a predefined deadband from their respective reference values, a voltage vector providing maximum compensation in the opposite direction is selected. This type of controller is called a hysteresis controller. However, some proposed DTC algorithms use SVPWM similarily to FOC [12]. Below, the basic working principle of classical DTC will be explained.

The phase-currents have to be measured and transformed into the αβ coordinate system using the Clarke transformation in order to estimate the torque and flux. The phase voltages in αβ coordinates are also needed, which can be either measured or estimated using the known states of the three inverter legs along with the bus voltage of the driver. The estimated flux is attained by in- tegrating the difference between the estimated phase voltages and the voltage drop across the winding resistances, as in equation 2.33 below.

(41)

ψαβ,estimated = ψ_m_αβ(θ_el) + Z

(v_αβ − R_si_αβ)dt (2.33)

where ψαβ,estimatedis the estimated flux, vαβ are the estimated voltages, Rsis the stator resistance, iαβ are the measured currents, and ψmαβ(θ_el) the maximum magnetic flux in the alpha and beta direction calculated as

ψ_m_α(θ_el) ψ_m_β(θ_el)

= ψ_mcos(θ_el) sin(θ_el)

(2.34) The magnitude and angle of this vector, ψαβ,estimated, is the stator flux magnitude and stator flux angle.

From the output of the flux estimator, the torque can be estimated using the following formula,

T_est = 3

2N (ψ_αi_β− ψ_βi_α) (2.35) where T^est is the estimated torque. The estimated torque and flux values are then compared to their respective references (T^∗and ψs^∗), and the error is fed into the hysteresis controller. By dividing the voltage vector plane into six regions and placing the estimated flux vector in this plane, the corresponding sector can be calculated, see Figure 2.12.

(42)

Figure 2.12: Flux linkage sectors used in DTC

In the figure above, ψαβ,estimated is the flux vector, S denotes the sector number, and V denotes the voltage vector in each direction. Around the magnitude of ψαβ,estimated, the flux reference and dead-band can be seen. For a given sector, four different voltage vectors can be chosen, depending on whether torque and flux should be increased or decreased. A voltage vector perpen- dicular to the flux vector should be chosen in order to increase torque, and to increase flux magnitude, a voltage vector in the same direction as the flux vector should be chosen. In the example above, where ψαβ,estimated is located in sector two, vectors V3(increase torque and flux), V4(increase torque, decrease flux), V6 (decrease flux and torque), and V1 (decrease torque, increase flux) can be chosen. A table of the voltage vector selection is available in table 2.4 below.

(43)

Sector number

1 2 3 4 5 6

Flux Torque

∆ψ_s= 1 ∆T = 1 V₂(110) V₃(010) V₄(011) V₅(001) V₆(101) V₁(100)

∆T = 0 V₆(101) V₁(100) V₂(110) V₃(010) V₄(011) V₅(001)

∆ψ_s= 0 ∆T = 1 V₃(010) V₄(011) V₅(001) V₆(101) V₁(100) V₂(110)

∆T = 0 V₅(001) V₆(101) V₁(100) V₂(110) V₃(010) V₄(011)

Table 2.4: Switching table for two-level hysteresis of both torque and flux.

In this table, ∆ψs and ∆T denote whether the flux and torque should be increased (1), or decreased (0). After each vector, the three numbers represent the states for each inverter phase. The voltage vector selection can also be made using algorithm 1.

Algorithm 1: DTC voltage vector selection algorithm [8]

if ∆ψ_s= 1 then if ∆T = 1 then

Selected vector ← Vk+1; else

Selected vector ← Vk−1; end

else

if ∆T = 1 then

Selected vector ← V^k+2; else

Selected vector ← Vk−2; end

end

where k is the current sector of the flux vector ψs. As DTC compensates for an error in either flux or torque with full voltage as opposed to a PWM regulated voltage, it needs to run at a relatively high control loop frequency in order to minimize torque ripple from overly aggressive compensation [13].

While standard DTC has many benefits in some areas, it lacks in others. High torque ripple, the need for fast hardware to run the control loop, errors in estimated torque and flux, and needing high-quality current sensors for feedback are just a few of the drawbacks. There have been numerous improvements proposed to make DTC more robust, often by improving the flux estimator.

Further alternatives to equation 2.33 will be presented in section 2.5.1.

(44)

2.5.1 Flux estimators

Flux can not be measured directly and is instead estimated from system parameters and measurements of other quantities. As a result, flux estimation can be troublesome in many cases. In general, there are two approaches to flux estimation, voltage-model based, and current-model based. Voltage-model based approaches have the advantage of not needing the electrical angle of the rotor, as it is approximated using the BEMF. However, at low speeds, the BEMF amplitude is low, which negatively impacts the reliability of the estimated electrical angle. Because of this, current-based models are more suitable for low-speed operation [14]. There are examples of these two approaches being combined, using current-based models for low speed, and voltage-based models for high-speed operation [15]. The voltage-model based approaches covered in this chapter are the pure integrator from equation 2.33, the low-pass filter, and the PI-controller, while Robust Torque Flux Estimator (RTFE) is the only current-model based approach discussed.

The pure integrator approach described by equation 2.33 needs both an accurate voltage measurement or estimator, current measurement, and knowledge of the phase resistance. The phase resistance changes as a function of parameters such as winding temperature and inverter resistances. The transfer function of the pure integrator can be seen in equation 2.36 below

G_integrator = 1

s (2.36)

where G^integrator is the transfer function and s the Laplace operator. As can be seen, the transfer function has a pole in s = 0, which indicates that it is unstable. As a result, any error in Rs or voltage and current measurements/estimations will cause an integral drift, and any error in ψmαβ(θ_el) will cause a DC-offset, making the flux estimator unreliable. In turn, this could cause a selection of the wrong voltage vector. Multiple alternatives to pure integration have been proposed in order to combat this.

Instead of using a pure integrator, a low-pass filter as seen below could be used in its place

G_lowpass = 1

1 + sT (2.37)

where Glowpass is the low-pass transfer function, and T is equal to the inverse of the cut off frequency of the filter. Glowpasshas a pole in s = −1/T , and as T cannot be negative, the pole will always be negative, resulting in the low-pass

(45)

filter always being stable. The low-pass filter would remove the DC-offset and take care of the integral drift at the cost of phase shift, making the flux estimator slower. The cut-off frequency needs to be chosen according to a trade-off between phase shift and motor operating speeds. A low cut-off frequency increases phase shift but decreases the operating speed’s lower limit, while a high cut-off decreases phase-shift and increases the operating speed’s lower limit [14].

Another alternative is to use a PI-controller in order to make a closed-loop estimator and stabilize the previously unstable integrator. The block diagram of such an estimator can be seen in figure 2.13 below.

Figure 2.13: Block diagram of closed-loop flux estimator with PI controller.

The reference flux magnitude, |ψs^∗| is used as a reference for the PI controller, with an angle set to be the same as the estimated flux vector ψ^αβ, see equation 2.38 below.

ψ_{P I}^∗ = |ψ_s^∗| e^jθ^ψαβ (2.38) The error is then calculated as the difference between this new vector and the estimated flux vector. This method of flux estimation is supposed to eliminate integral drift and handle incorrect values of Rs [16].

A fourth approach is the current-model based RTFE, where torque and flux estimation is done in the rotor fixed dq-reference frame. Since the estimations are done in the dq-reference frame, the torque can be estimated using equation 2.15. Reliance on flux estimation for an accurate torque estimation is thus eliminated. The flux estimation can be done using equation 2.9 and 2.10. The estimated flux vector is then transformed back into the stator fixed αβ-reference frame, from which point the rest of the DTC algorithm can be calculated as described in section 2.5 [15][17]. Due to the low-speed performance current-based models gives, this flux estimator will be used in this

Comparison of FOC and DTC in an application of a screw joint emulator

Comparison of FOC and DTC in an application of a screw joint

emulator

AXEL FYRESKÄR JOEL GREBERG

Comparison of FOC and DTC in an application of a screw joint emulator

AXEL FYRESKÄR, JOEL GREBERG

Sammanfattning

Abstract

Acknowledgments

Contents

List of Figures

List of Tables

Nomenclature

Introduction

1.1 Background

1.2 Purpose

1.3 Method Description

1.4 Delimitations

1.5 Risk Analysis

1.6 Ethical considerations

Frame of reference

2.1 Screw joints and tightening

2.2 PMSM

2.2.1 Mathematical modeling

2.2.2 Coordinate transformations

2.3 Inverters

2.4 FOC

2.4.1 SVPWM

2.5 DTC

2.5.1 Flux estimators