Lappeenranta University of Technology Faculty of Technology Department of Electrical Engineering

Transcription

1 Lappeenranta University of Technology Faculty of Technology Department of Electrical Engineering CONTROL OF PERMANENT MAGNET LINEAR SYNCHRONOUS MOTOR IN MOTION CONTROL APPLICATIONS MASTER S THESIS Examiners: Professor Juha Pyrhönen D.Sc. Markku Niemelä Supervisors: D.Sc. Markku Niemelä M.Sc. Mari Haapala Lappeenranta, May 20, 2009 Pavel Ponomarev Karankokatu 4 C Lappeenranta pavel.v.ponomarev@gmail.com

2 ABSTRACT Lappeenranta University of Technology Faculty of Technology Department of Electrical Engineering Pavel Ponomarev CONTROL OF PERMANENT MAGNET LINEAR SYNCHRONOUS MOTOR IN MOTION CONTROL APPLICATIONS Master s thesis pages, 50 figures, 7 tables and 2 appendices Examiners: Professor Juha Pyrhönen, D.Sc. Markku Niemelä Keywords: Linear motor, Permanent magnet motor, Direct Thrust Force Control (DTFC) The aim of the thesis is to study the principles of the permanent magnet linear synchronous motor (PMLSM) and to develop a simulator model of direct force controlled PMLSM. The basic motor model is described by the traditional two-axis equations. The end effects, cogging force and friction model are also included into the final motor model. Direct thrust force control of PMLSM is described and modelled. The full system model is proven by comparison with the data provided by the motor manufacturer.

3 Acknowledgments The work was carried out at Lappeenranta University of Technology (LUT) during the period from winter up to spring I would like to thank all people who made this work possible. I wish to express my deepest appreciation to my first supervisor professor Juha Pyrhönen and to my second supervisor D.Sc. Markku Niemelä for their guidance and support. I want to thank M.Sc. Mari Haapala for her help and guidance. Special thanks to Julia Vauterin who has made my life and study in Lappeenranta possible. I also wish to express my appreciation to my sister Malukhina Elena and her husband Malukhin Aleksey for their support. I am grateful to my parents and my sweetheart Julia for their love and support. Lappeenranta, May 2009 Pavel Ponomarev

4 TABLE OF CONTENTS List of Symbols and Abbreviations 5 1 Introduction Comparison of linear actuators Description of the motor system to be studied Direct thrust force control method Objectives and outline of the thesis 14 2 Modelling of a Permanent Magnet Linear Synchronous Motor Space vector theory, coordinate systems and transformations Basic motor model Nonlinearities and mechanical model Friction model Cogging force End-effect 29 3 Modelling of Direct Thrust Force Control of a PMLSM VSI voltage vectors Direct torque control Estimations 40 4 Simulations System overview Simulation results Conventional DTFC DTFC with MTPA strategy Results and Conclusions 53 References 55 Appendices 57 4

5 LIST OF SYMBOLS AND ABBREVIATIONS Symbols α C fr F disturb F friction F thrust, F T F thrust.est i d, i q i x, i y i s i sa, i sb, i sc k end L L d, L q L sσ m mov, m load p P elm R S fr U DC u s u s_est V fr phase shift operator Coulomb friction factor force of disturbances friction force thrust force thrust force estimation direct-axis and quadrature-axis currents currents in stationary xy-reference frame stator current vector stator current vector phase components end-effect coefficient inductance direct- and quadrature-axis inductances stator leakage inductance mover and load masses number of pole pairs electromagnetic power resistance Stribeck friction factor DC-link voltage stator voltage vector stator voltage vector estimation viscous friction factor γ κ ψ m ψ PM load angle angle between ABC- and xy-reference frames air gap flux linkage vector permanent magnet flux linkage 5

6 ψ s ψ sd, ψ sq ψ sx, ψ sy ω el φ τ τ v lin stator flux linkage vector stator flux linkage vector dq-reference frame components stator flux linkage vector xy-reference frame components electrical frequency flux linkage hysteresis comparator output signal thrust force hysteresis comparator output signal pole pitch linear velocity of the mover Abbreviations AC DC DFLC DTC DTFC FEM ITC LIM LM LSM MTPA PI PM PMLSM VSI alternating current direct current direct flux linkage control direct torque control direct thrust force control finite element method indirect torque control linear induction motor linear motor linear synchronous motor maximum thrust per ampere proportional-integral permanent magnet permanent magnet linear synchronous motor voltage source inverter 6

7 Chapter 1 Introduction Most of linear actuators have traditionally been made by means of rotating machines and special transmission devices such as ball rail, roller rail, linear shaft and ball screw systems. In these devices a rotating motion produced by rotating machines is converted into linear motion. Such a conversion decreases, and in many cases significantly, the efficiency of the whole system. These mechanical rotary-to-linear transmission devices also require a lot of maintenance which connected with wear outs of mechanical parts. Linear motors are electromagnetic devices which can produce linear motion without any intermediate gears, screws or crank shafts. The linear motion is obtained directly by electromagnetic forces. 1.1 Comparison of linear actuators A popular way to produce linear motion from a rotary motor is to use the belt and pulley system, figure 1.1. This system has limited thrust force capability due to tensile strength of the belt. It usually requires additional gear box to decrease rotational speed to be suitable for the pulley. Mechanical windup, backlash of the gearbox and belt stretching all these factors are contribute to inaccuracies in the system. Settling time of the system is very poor due to extensibility of the belt. And system characteristics become poorer with longer belts. 7

8 Figure 1.1. Belt and pulley system (Oriental Motor). Another way to achieve linear motion from rotary motor is to use rack and pinion system, figure 1.2. Such a system provides better thrust capability in comparison with the belt and pulley system. Pinion gear and required gearbox cause inaccuracies in the system, which will increase with wear of the system. Position accuracy and repeatability of the system is influenced by the backlash of the gears. Figure 1.2. Rack and pinion system (Oriental Motor). Screw system is also popular way to translate rotary motion into linear, figure 1.3. There are two screw systems: ball-screws and lead-screws. The very cheap lead-screw system has very low mechanical efficiency of about 50 percents. This system is not intended for high duty operational mode because of high wear. 8

9 Figure 1.3. Lead screw system (Oriental Motor). Linear motion electromagnetic devices allow high precision positioning. Performance is limited only by the resolution of the linear encoder and the stability of the mechanics. Since there is no backlash or mechanical windup, the linear motors have great repeatability and precision. Since the load is directly connected to the mover, the settling time of the system is very short. Performance characteristics remain unchanged even in long travel configurations. The reliability of the linear drives is much higher than the reliability of the linear actuator systems with rotating machines. The comparison of the properties of the most popular linear actuator systems is shown in the table 1.1. Table 1.1. Properties comparison table. Belt and pulley Rack and pinion Lead screw Ball screw Linear electrical motor Accuracy Speed range Travel range Thrust Friction Maintenance Life time Price Efficiency

10 1.2 Description of the motor system to be studied Linear motors (LMs) fall into two main types: linear synchronous motors (LSM) and linear induction motors (LIM). In linear induction motors the rotor magnetic field is produced by the induced current, when in linear synchronous motors rotor field is produced by the permanent magnets or by the independently generated current. Linear motors could be applied in many areas. Industrial automation systems are the main field of application of the linear motors. Linear induction motors have found applications in the following areas: conveyor systems; liquid metal pumping; material handling; low and medium speed trains. Linear synchronous motors are successfully utilized in any application where high precision motion control is required: machining tools; welding robots; laser scribing systems; industrial laser cutting systems; industrial robots. Linear motors are used for high-speed ground transportation as propulsion and levitation systems. Also extensive use of LMs could be found in building and factory transportation systems. LMs are applied as elevator hoisting machines, ropeless elevators in ultra-high buildings, horizontal transportation systems at factories. There are several types of LSMs which are classified according to the following items: Short primary / short secondary Ironcore / ironless Moving primary / moving secondary Tubular / flat Permanent magnet excitation / electromagnetic excitation Single sided / double sided Transverse flux / longitudinal flux Nowadays flat brushless permanent magnet (PM) linear synchronous motors are dominating. The main reasons for the PM-machine popularity are higher force density, efficiency and faster response due to existing magnetic field than in LIMs. 10

11 There are two design types of the flat PM linear synchronous motors: ironcore and ironless. The ironless motor has no iron parts in the forcer, so the motor has no attractive force or cogging which increases lifetime of the guideway bearings. Such a motor is ideal for smooth velocity control but has low force output. The ironcore flat linear PM motor has forcer with steel laminations. This configuration allows increasing force output of the motor due to focusing the magnetic field created by the windings. At the same time due to the strong attractive force between the ironcore armature and the magnetic track this type of motors characterized by increasing bearings wear. The strong cogging force causes difficulties in position control. In this thesis a flat single sided ironcore surface permanent magnet linear synchronous motor (PMLSM) with short moving primary is under observation, figure 1.4. This type of motors had been essentially developed for the factory automation. Figure 1.4. The construction of a flat PMLSM (Tecnotion 2008). A linear motor usually is a port of a bigger automation system. A complete linear motor system (see figure 1.4) usually consists at least of following parts: A mounting frame. A magnet track build up of magnet plates. 11

12 A set of linear guides or rails that support the slide. A positioning system consisting of a drive, a controller, a linear encoder for position feedback to the controller. A coil unit attached to the slide carried functional load. Safety end dumpers and switches to stop the movement in case of malfunctions. Cable chain to provide cabling to the coil unit. The practical range of the travel of movers of the mentioned type of linear motors is, generally, between 0.1m and 3m. For shorter lengths of travel, tubular configurations are better suited. For longer distance travels the path with PMs becomes rather expensive, and different machine types should be considered. Thrusts up to 10,000 N at speeds up to 5 m/s are common for that type of PMLSM. The first target of this M.Sc thesis is to study the principles of PMLSM. One traditional way to describe the construction of PMLSM is to cut and unroll a PM rotary synchronous motor to produce a flattened configuration. So we have a lying plate with permanent magnets and a stator winding which slides in a straight line. The slider which consists of a slotted armature and three-phase windings is termed moving primary, or mover, or slide, while the fixed magnet plate could be named as secondary, or exciter, or, somewhat confusingly, stator. The primary and the secondary are separated by a flat air gap. The constant value of the air gap is maintained by guide rail and bearings, Fig 1.5. Figure 1.5. The construction of a PMLSM (Hirvonen 2006). 12

13 When the coils of the mover are excited by 3-phase alternating current an armature magnetic field is generated. This field produces a travelling flux wave along the main axis of the linear motor. This flux wave interacts with the flux produced by the permanent magnets of the stationary fixed magnet plate. Interaction results in an electromagnetic force which moves the primary in positive or negative direction in dependence of the alignment of the permanent magnets along the magnet plate. In this case such a force is called thrust, or traction force, or propulsion force. The linear motion is obtained directly by electromagnetic force. 1.3 Direct thrust force control method The second target of this M.Sc thesis is to study direct thrust force control technique. The direct thrust force control (DTFC) is an implementation of direct torque control (DTC) of rotating AC machines for linear AC motors. DTC is an advanced drive technology used in variable frequency drives to control the torque (and consequently the speed) in three-phase AC electric motors. The idea of the DTC was introduced by Manfred Depenbrock in mid-1980s in Germany, and at the same time in Japan by Isao Takahashi and Toshihiko Noguchi. The idea was to influence on the flux linkage of the motor as directly as possible, and thus on the torque produced by the machine. The principal operation of the DTC is shown in the figure 1.6 as block diagram. 13

14 Figure 1.6. Direct Torque Control (ABB 1999). DTC uses motor electric torque and stator flux linkage directly as control variables, whether traditional field oriented current vector control uses currents as control variables to indirectly influence on the electric torque of the machine. For the current vector control the term ITC (Indirect Torque Control) is frequently used to contrast with the direct torque control. The DTC can be applied to any rotating field machine. Main advantages of the method are minimum torque response time, absence of voltage modulator block, possibility of sensorless control, control of the torque at low frequencies and relatively low annoying noise. The disadvantage is presence of some difficulties in the flux linkage integration. 1.4 Objectives and outline of the thesis The objectives of this work are as follows: Study the principles of PMLSM. Study the applicability of DTC in direct thrust force control DTFC. 14

15 Chapter 2 is dedicated to introducing a simulation model of a PMLSM. The modelling of a PMLSM is based on the space vector theory which is introduced in Section 2.1. The mathematical representation of a PMLSM is described in Section 2.2. Then a simplified mechanical model and linear motor nonlinearities are described in Section 2.3. At Chapter 3 a simulation model for direct thrust force control is described. The basic theory of the method is introduced along with the simulation model. Chapter 4 contains simulations. Section 4.1 contains description of some auxiliary controllers required for the simulations. The data obtained from the developed simulation model is compared with the data given by the manufacturer of the motor in Section 4.2. Also the influence of i d = 0 control is investigated. The conclusions reached in the work and results are discussed in Chapter 5. 15

16 Chapter 2 Modelling of a Permanent Magnet Linear Synchronous Motor 2.1 Space vector theory, coordinate systems and transformations In a three-phase machine, there is a phase shift of 120 electrical degrees between the different phases. So we can introduce the phase-shift operator α. 2π α e j 3 = (2.1) This operator could be applied to construct so called space vector of the stator current from instantaneous values of currents of different phases is = 2 α i () α () α () sa t + isb t + isc t 3 (2.2) This space vector illustrates the effect of the all three currents created together by the windings. The factor 2/3 allows using space vector with parameters of real equivalent circuit of the machine. Along with the stator current i s the voltage space vector u s, the space vector of the stator current linkage θ s and the flux linkage space vector ψ s could be constructed from the phase quantities us() t = α usa() t + α usb() t + α us () t 3 C (2.3) θs() t = 2 α θsa() t + α θsb() t + α θs () 3 t C (2.4) ψ s() t α sa() t α sb() t α sc() 3 ψ ψ ψ t (2.5) Now we have a vector representation created from three phase quantities which could be transformed into two-axis representation of the space vector. The stator windings 16

17 are fixed within the stator, so a two-axes stator-fixed orthogonal coordinate system could be introduced, figure 2.1. Figure 2.1. ABC and XY frames. Next, we determine the mathematical transformations from the three-phase A, B and C components into two-phase XY-axes components. The following equations are performed with stator (mover in linear motor s terminology) currents, but they also valid for voltages and fluxes. i o ( κ ) o ( κ ) ( κ ) ( κ ) i sa sx 2 cosκ cos cos i sb i = sy 3 sinκ sin sin isc o o, (2.6) where is an angle between A and X axes. If κ κ is set to zero we obtain next formula i i. (2.7) sa sx i sb i = sy i sc 2 2 The Matlab/Simulink block for this transformation is depicted next in the figure

18 Figure 2.2. ABC to XY transformation block. Reverse transformation from two-phase quantities into three-phase quantities i cosκ sin κ sa sx i sb = o o cos( κ ) sin ( κ ) i o o sy i sc cos( κ ) sin ( κ ) i. (2.8) If κ is set to zero we obtain next formula i 1 0 sa 1 3 isx i sb = 2 2 i sy i sc (2.9) And correspondent Matlab/Simulink block is depicted in the figure

19 Figure 2.3. XY to ABC transformation block. The model of a linear synchronous motor with permanent magnets will be introduced as a model in direct-quadrature (d-q) axes reference frame, figure 2.4. This frame rotates relative to the stationary fixed XY frame with the angular synchronous speed ω el. This ω el could be expressed from the linear velocity v lin using the equation π ω el = v lin. (2.10) τ All variables are expressed on orthogonal direct and quadrature axis which rotates with synchronous speed. d-axis is aligned along with the magnetic axis, and q-axis is aligned 90 degrees ahead in the direction of rotation, which traditionally assumed to be counter-clockwise. 19

20 Figure 2.4. dq reference frame. Next equations are used to convert space vectors from fixed XY-frame into rotating coordinate system of direct and quadrature axis: i () t = i cosθ + i sinθ, (2.11) d sx sy i ( t) = i sinθ + i cosθ, (2.12) q sx sy where θ is an angle between the X- and d- axes. Matlab/Simulink blocks for transformation are depicted in the figure 2.5. Figure 2.5. XY-dq transformation block. 20

21 Figure 2.6. dq-xy transformation block. 2.2 Basic motor model Control algorithms of AC motors frequently use the d-q axes model of AC machines. To derive a d-q axes model of the motor let us introduce stator voltage equations: u () t Ri d u () t Ri q dψ dt d = d + ωψ q, (2.13) dψ q = q + + ωψ d, (2.14) dt where u d and u q are the d-axis and q-axis components of the stator voltage space vector, i d and i q are the d-axis and q-axis components of the armature current vector, R is an armature phase resistance. The armature winding d-axis and q-axis flux linkages ψ d and ψ q in the previous equations are ψ d =L d i d + ψ PM, (2.15) ψ q =L q i q, (2.16) where L d and L q are the d-axis and q-axis armature inductances and ψ PM is the flux linkage of the permanent magnet. 21

22 The instantaneous power input to a three-phase armature is 3 P = uaa i + ubb i + ucc i = ( udd i + uqq i ), (2.17) 2 where u A, u B and u C are instantaneous phase voltages, i A, i B, and i C are instantaneous phase currents, and u d and u q are d- and q-axis voltage components, i d and i q are d- and q-axis current components. The power balance equation is obtained from equations of stator voltages (2.13) and (2.14). 2 dψ dψ d 2 q ui dd+ ui qq= Rid + id+ Riq + iq+ ωψ ( dq i ψqd i), (2.18) dt dt The last term accounts for the electromagnetic power of a single phase, two pole synchronous machine. For a three-phase machine 3 3 Pelm = ωψ ( diq ψqid ) = ωψ [ PM + ( Ld Lq ) id ] i q, (2.19) 2 2 where L d and L q are the armature inductances. The electromagnetic thrust F thurst of a PMLSM with p pole pairs is the electromagnetic power P elm in last equation divided by the linear velocity v lin in equation (2.10) and multiplied by p. 3 π 3 π F = thrust p ( ψdiq ψqid) p [ ψpm ( Ld Lq) id] q 2 τ = 2 τ + i. (2.20) The set of equations and 2.20 comprises the permanent magnet linear synchronous motor basic mathematical model. The Simulink block diagram of the motor model is depicted in the figure

23 Figure 2.7. Simulink motor model block. Next, the parameters given by the manufacturer of the motor (see Appendix 2) are presented in the table 2.1. PM flux linkage is calculated using motor force constant K and equation τ 2 F τ 2 thrust ψ PM = = K. (2.21) π 3p is π 3p Table 2.1. Parameters of the motor model. Symbol Value Parameter R 1.6 Ω phase resistance L d 13 mh d-axis inductance L q 13 mh q-axis inductance p 1 number of pole pairs τ m PM pole pitch K 93 N/A Motor force constant ψ PM Wb PM flux linkage The model corresponds to the linear synchronous motor with surface mounted PMs. The relative permeability of the permanent magnets is almost unity (1.04), which 23

24 means that permanent magnets are like air in the magnetic circuit and parameters of the magnetic circuit are equal in d- and q-axes. It causes in equal and quite low d- and q- axes inductances. 2.3 Nonlinearities and mechanical model In motion control applications the nonlinearities of a linear motor could bring significant tracking error or increase settling time. To avoid such effects this nonlinearities should be carefully modelled and taken into account while designing the control system. Next we introduce the mechanical model of the motor. The dynamic behavior of the linear motor system could be expressed by following equation: dv Fthrust () t = mtot + Fload() t + Ffriction ( v) + Fdisturb ( x), (2.22) dt where m tot is a sum of the mover mass m mov and the mass of the load m load ; v is a linear mover speed; F friction is a force which takes into consideration viscous, Stribeck and Coulomb effects; F load is an additional force produced by the load; F disturb is a force which accounts the effects of cogging and flux non-uniformity at the ends of the mover. Let us next consider all four elements of the right part of the equation The first term is indispensable part of any physical dynamic system associated with inertia. The second term is application related. The last two terms are discussed in the following three subsections. 24

25 2.3.1 Friction model In linear motors of observed type friction is very important due to significant attraction force between permanent magnets and iron parts of the mover. This attraction force should be considered for the mechanical design of PMLSM, in particular related to noise, vibration and linear bearing design. Next the friction model will be discussed. The friction force is composed of Coulomb, viscous and Stribeck effects and could be described by the figure 2.8. Figure 2.8. Viscous, Coulomb and Stribeck effects of friction. The Stribeck friction is the negatively sloped characteristics taking place at low velocities. The Coulomb friction results in a constant force at any velocity. The viscous friction opposes motion with the force proportional to the velocity. The friction force as a function of mover velocity: F ( v) = C sign( v) + V v+ S e kv sign( v ). (2.23) friction fr fr fr The parameters of the friction model used in the simulation model are represented in the table 2.2. The behavior of the friction model with these parameters is similar to 25

26 friction model described by Hirvonen (2006). However, exact values should be separately determined in every particular case, and should be proved by tests. Table 2.2. Parameters of the friction model. Symbol Value Parameter C fr 30 N Coulomb coefficient V fr 3 Ns/m viscous coefficient S fr 10 N Stribeck coefficient k 10 s/m Stribeck speed factor The Matlab/Simulink block for the friction model is shown in the figure 2.9. And friction model ramp response is shown in the figure Figure 2.9. Simulink block of the friction model. Figure Friction model ramp response. 26

27 2.3.2 Cogging force The last term of the equation 2.22 composed by two parts: cogging force and a force caused by end effects. F ( x) = F ( x) + F. (2.24) disturb cogging end_effect Cogging force is caused by interaction between the iron slots of the mover and the permanent magnets of the track. Cogging force is presented even when there is no motor current. Due to the slotted nature of the primary core, the cogging force is periodic and has two components related to the primary core length and PM pole pitching. The first component of cogging force could be reduced by modifying the core shape and optimizing the length of the mover. And the second component of the cogging force can be reduced substantially by skewing either the primary teeth or the permanent magnets (Jung et al. 1999). But these measures can reduce the maximum thrust force and the efficiency of the motor and increase the complexity of the motor structure. To achieve high positioning performance the cogging force should be carefully modelled and compensated. It should be mentioned that cogging force is very insignificant when configuration with aircore windings is used. But this configuration produces lower force output. The mathematical representation of the cogging force as a function of the mover position (Hirvonen 2006): [ ] F ( x) = K sin( ϕ x2π) A + A sin( ϕ x2π), (2.25) cogging s 1 r1 r2 2 where K s is a scaling factor, φ 1 and φ 2 are wavenumbers which are related to the τ and the primary core length, A r1 and A r2 are amplitudes of two harmonics. 27

28 The Simulink block diagram of the cogging force model according the mathematical model 2.25 is represented in figure 2.11 Figure Cogging force Simulink block. Parameters of the model are represented in the table 2.3. Table 2.3. Cogging force Simulink block parameters. Symbol Value Parameter K s 0.7 scaling factor φ m -1 1 st wavenumber φ m -1 2 nd wavenumber A r1 35 N 1 st harmonic A r2 15 N 2 nd harmonic Figure shows cogging force simulation results. 28

29 Figure Simulated cogging force End-effect End-effect is a special phenomena because of the limited length of the mover. Generally it is difficult to describe the end-effect with exact mathematical model (Jiefan et al. 2004). The powerful finite element method (FEM) should be applied to analyze the magnetic field to determine the role of the end-effect in any particular case. In practice many researchers use multiplying coefficient k end or function which takes into account the thrust reduction due to the end-effect (Gieras et al. 2000, 138). It is evident that the impact of the end-effect on the thrust is reduced with increasing number of poles of the mover. The value of the coefficient can be determined experimentally. F end_effect = ξ () t F. (2.26) T In this work the end-effect is described by the coefficient k end. The value of the coefficient is suggested to be Fend_effect = kendft. (2.27) 29

30 The Simulink block diagram of the dynamic model including friction, cogging force and end-effect is depicted on the figure Figure Dynamic model Simulink block. Table 2.4. Parameters of the dynamic model used in the simulations. Symbol Value Parameter m mov 4.8 kg mover mass m load 10.8 kg load mass 30

31 Chapter 3 Modelling of Direct Thrust Force Control of a PMLSM Direct thrust force control (DTFC) method is a DTC applied to the linear motors. The only difference between the DTC and the DTFC is that in the rotating field machines we deal with torque and angular velocity, whether in the linear machines these replaced by thrust force and linear velocity. Next the DTC will be observed in details. But first of all the concept of VSI voltage vectors should be introduced. 3.1 VSI voltage vectors At present, the voltage source inverter (VSI) drives are most widely used. The VSIs are applied to the control of all kinds of rotating field machines. There are mainly two types of VSI drives available: two-level and three level devices. The typical structure of the two-level VSI drive is shown in the figure 3.1 (Mohan et al. 2003). Figure 3.1. Two-level VSI drive. The voltage source inverter considered together with 3-phase winding can generate voltage vectors. The voltage vectors are defined by combinations of switch positions (S A, S B, S C ). For two-level VSI there are 2 3 possible combinations of power switches which can produce 6 active voltage vectors and 2 zero voltage vectors. Voltage vectors of a two-level VSI in a stationary XY-reference frame are presented in the figure

32 Voltage polarities at the three-phase terminals of the inverter are marked near the vectors. The voltage polarities are representing the power switch statuses of all three phases. Figure 3.2. Two-level VSI voltage vectors. The stator voltage space vector u s can take one of the following eight instantaneous values in dependence of polarities of A-, B- and C-phases which are shown in round brackets: u s (---) = u 0 = 0; u s (+--) = u 1 = U DC α 0 ; u s (++-) = u 2 = U DC (α 0 + α 1 ); u s (-+-) = u 3 = U DC α 1 ; u s (-++) = u 4 = U DC (α 1 + α 2 ); (3.1) u s (--+) = u 5 = U DC α 2 ; u s (+-+) = u 6 = U DC (α 2 + α 0 ); u s (+++) = u 7 = 0. The Simulink block diagram of the inverter is shown in the figure

33 Figure 3.3. VSI Simulink block diagram. 3.2 Direct Torque Control The Direct Torque Control is a continuation of the vector control of the AC motors. The foundations of the DTC were firstly described by I. Takahashi and T. Noguchi in middle 80s. And in middle 90s the first industrial DTC utilized drives were introduced by ABB. The main task of the DTC is to supply quick electromagnetic torque response of the motor. Unlike the vector control, where the torque is altered by the influence on the stator current vector, in the DTC the control variable is the stator flux linkage vector ψ s, figure 3.4 (because of that the better name for the DTC is Direct Flux Linkage Control, DFLC). The change of the flux linkage is achieved by the optimal switchings of the power switches of the VSI which feed the motor. 33

34 Figure 3.4. DTC block diagram (Luukko 2000). Let us next consider theoretical foundations of the DTC. For this purpose next equation should be introduced (Niemelä 1999, 13). 3 1 T = p ψ ψ sin( γ ), (3.2) L elm s m 2 sσ where L sσ is the stator leakage inductance, ψ m is the air gap flux linkage vector, and γ is an angle between the stator and the air gap flux linkages. This equation justifies the direct torque control. It shows that the motor torque is proportional to the sinus of the angle γ. Hence, if the magnitudes of the stator and air gap flux linkages are constant, the motor torque could be controlled by the variation of the angle γ. 34

35 ψ s() t = ( us Ri s s) dt+ ψ s0. (3.3) The flux linkage estimation equation 3.3 is the key element of the DTC. It gives a connection between the stator voltage vector u s and the flux linkage vector ψ s. This equation gives an opportunity to control the stator flux linkage vector ψ s, its position, and magnitude, by controlling the stator voltage vector u s. And consequently it allows to control the angle γ. It should be noticed that the winding voltage drop (term R s i s in eq. 3.3) is usually relatively small and could be neglected and next equation could be written. ψ s() t = ( us) dt+ ψ s0. (3.4) So the trajectory of the flux linkage moves in the direction of the applied voltage vector of the VSI with the speed proportional to the applied voltage. If one of the zero voltage vectors u 0 or u 7 is applied the locus of the flux linkage vector almost stands still because of the small value of the resistive winding voltage drop R s i s. This allows to freely control the rotating velocity of the flux linkage vector by changing the ratio between zero voltage vectors and active voltage vectors. Figure 3.5 shows the plane where the basic active VSI voltage vectors u 1 - u 6 are located in the stationary XY-axes reference frame. In the same stationary frame the flux linkage vector ψ s is shown. 35

36 Figure 3.5. Six sectors of the flux circle. The flux plane is divided into six sectors of equal sizes of 60 electrical degrees in such a way that the sectors are bisected by the basic VSI voltage vectors (Mohan 2001). Let us first consider the torque control. In PM synchronous machine the torque is proportional to the sinus of the angle γ between the PM flux linkage ψ PM (d-axis in the d-q reference frame) and the stator flux linkage ψ s. Torque could be effectively controlled by varying this angle. There are three possible situations. First, if the actual torque of the machine is too small, then the stator flux linkage vector must advance to increase the angle γ. It means that if we want to increase the torque we should choose voltage vector which is located ahead in the direction of rotation. For example, if the flux linkage vector is in the first sector, as it shown in the figure 3.5, then the voltage vectors u 2 and u 3 are suitable. 36

37 Second, if the actual torque is in tolerance borders, then the flux vector must keep its position, angle γ should not be changed. And we should select a zero voltage vector u 0 or u 7. Third, if the actual torque is too great, then the flux vector must go backward, angle γ should be decreased. It means that if we want to decrease the torque than the behind located voltage vector must be selected. For example, if the flux linkage vector is in the first sector, then the voltage vectors u 6 and u 5 are suitable. If the rotating speed of the motor is significant, then the zero voltage vectors could also be used to decrease torque. This logic can be implemented by the double hysteresis of the error signal as it shown in the figure 3.6. The output signal of the torque hysteresis comparator τ can possess three different values: 1, -1 and 0. Value τ = 1 corresponds to the instant when the torque increase is required; at τ = -1 the torque must be decreased; τ = 0 means that the torque is lying in the tolerance borders. Figure 3.6. Torque double hysteresis. 37

38 In the same time the magnitude of the flux vector must be kept in necessary borders. In each sector two voltage vectors can be used in both rotation directions to decrease or increase the flux linkage. The regulation of the flux linkage could be implemented with hysteresis of the flux linkage error signal as it shown in the figure 3.7. Figure 3.7. Flux linkage hysteresis. The output signal of the flux hysteresis comparator φ can possess two values 1 and 0. If the current magnitude of the stator flux linkage is significantly less than the reference value, then flux linkage magnitude should be increased which corresponds to φ = 1. But if the magnitude of the flux linkage is significantly bigger than the reference signal, then flux linkage magnitude should be decreased which corresponds to φ = 0. Signals from torque and flux linkage hysteresis controllers τ and φ as well as the position information θ (the sector number) about flux linkage vector are indexing elements of so-called optimal switching table, the core component of the DTC. This table provides an optimal selection of voltage vectors. Selected voltage vector next applies to the power switches and provides desired change of the flux vector to achieve optimal performance. The control of the power switches is adjusted only when the torque or absolute value of the flux linkage differs too much from the reference value. When the hysteresis limit is reached the next voltage vector is selected from the 38

39 optimal switching table to bring the flux linkage vector into the right direction. Figure 3.8 depicts an optimal switching table introduced by I. Takahashi and T. Noguchi (1986). Figure 3.8. Optimal switching table. The selection of the voltage vector is made so as to restrict the errors of the flux and torque within the hysteresis bands and to obtain the fastest torque response and highest efficiency at every instant. Typically the flux linkage reference signal for flux hysteresis controller is kept constant for the speed range below the rated speed. Suitable value for the flux linkage reference signal is flux linkage of the permanent magnets. If the motor intended to run above the rated speed the field weakening technique could be applied. The reference signal for the torque hysteresis controller is obtained from the speed PI type controller. The actual values of the torque and flux linkage are estimated in the estimator. The Simulink block diagram of the DTC controller is depicted in the figure 3.9. Figure 3.9. DTC controller model. 39

40 3.3 Estimations The actual values of the thrust force and the stator flux linkage for the hysteresis comparators are obtained by estimations. The estimation of the stator flux linkage could be performed by measurement of the stator voltage and current using expression 3.3. In the real drive there are no phase voltage measurement devices. Instead of direct measurement of the stator voltage the estimation of the stator voltage u s_est is used. Estimated voltage is constructed according to the information about the actual position of the power switches (S A, S B, S C ) and measured value of the DC-link voltage U DC by using formula us_est = udc( SAɑ + SBɑ + SCɑ 2 ). (3.5) 3 The Simulink block for the voltage estimation is depicted in the figure Figure Voltage estimation block. There is a valuable drawback in using formula 3.3 in flux estimation. Voltage integration is very sensitive for inaccuracies in parameters. Unaccounted voltage drop in power switches, stator current measurement errors, DC-link voltage measurement errors, and errors in the stator resistance estimation all this factors cause errors in the flux estimation. So estimated flux linkage could much deviate from the real value (see figure 3.14). Stator current measurement errors, DC-link voltage measurement errors, 40

41 stator resistance estimation error, inaccuracies in determination of PM flux linkage and d- and q-axes inductances - all these inaccuracies are included in the Simulink model (see Appendix 1). It should be mentioned that to increase accuracy of the flux estimation the temperature related resistance variation should be taken into account and resistance estimation model which uses information from a temperature sensor should be created. In this work, the temperature related stator resistance variations are neglected. One possible way to eliminate this error is to use current model for the flux linkage correction. Current model gives accurate estimation of the stator flux linkage, but needs accurate inductance model. Block diagram which illustrates calculation of the correction terms using current model is depicted in the figure Figure Flux linkage correction terms calculation using current model. Correction is occurred in the appointed time instances because current model calculation is a time consuming task. The Simulink block diagram of the corrector is depicted in the figure

42 Figure Corrector. These correction terms are then applied to the voltage model in the estimator, figure Figure Estimator. Figure 3.14 shows real and estimated flux linkage trajectories in XY coordinate system when estimated phase resistance is slightly bigger than real value and flux linkage correction is switched off. When the flux linkage correction is switched on the trajectories are almost identical. 42

43 Figure Real (left) and estimated (right) flux linkage trajectories without correction. The X-axes and Y-axes components of the flux linkage ψ x and ψ y are the output of the voltage model. These values are used to calculate thrust force estimation, flux linkage magnitude estimation, and the sector of the flux linkage vector position. Thrust force is calculated as a cross product of estimated flux linkage vector and measured motor current vector in accordance with next formula: 3 π Fthrust.est = p ( ψ xiy ψ yix ). (3.6) 2 τ The Simulink block diagram of the force estimation is depicted in the figure Figure Force estimation. Calculation block of flux linkage estimate modulus is depicted in the figure

44 Figure Calculation of flux linkage estimate modulus. The sector estimation block is shown in the figure Figure Sector estimation. 44

45 Chapter 4 Simulations 4.1 System overview The whole drive Simulink model is depicted in the figure 4.1. Inverter is assumed to be supplied from constant DC voltage. Figure 4.1. Drive model. The model of the whole system is represented in the figure 4.2. Figure 4.2. System model. A DTFC controller inverter unit requires several auxiliary controllers. The thrust force reference signal is either the thrust force reference from the speed controller or an external thrust force reference. Thrust force reference must be limited in order not to exceed the maximum allowed peak thrust of the motor and consequently the permitted 45

46 currents for the inverter. The basic algorithm of the speed controller is a PI control algorithm. The speed PI controller Simulink model with anti-windup which produces thrust force reference signal for the DTFC is represented in the figure 4.3. Figure 4.3. Speed PI controller. The values of the coefficients are represented in the table 4.1. The speed controller had been tuned experimentally by simulations. Table 4.1. Speed controller coefficients. Symbol Value Parameter speed_pid_p 6000 Proportional gain speed_pid_i 1000 Integration gain speed_saturation 760 N Saturation border Speed_antiwindup_gain 0.19 Antiwindup gain The speed reference signal is either the speed reference from the position controller or an external speed reference. The basic algorithm of the position controller is a PI control algorithm. The position PI controller Simulink model with anti-windup which produces speed reference signal for the speed controller is represented in the figure 4.4. Figure 4.4. Position PI controller. 46

47 The values of the coefficients are represented in the table 4.2. The position controller, as well as speed controller, had been tuned experimentally by simulations. Table 4.2. Position PI controller coefficients. Symbol Value Parameter position_pid_p 23 Proportional gain sposition_pid_i 3 Integration gain position_saturation 4.5 m/s Saturation border Position_antiwindup_gain 0.17 Antiwindup gain The flux linkage reference signal control block produces reference signal for the flux linkage hysteresis controller for the DTFC. Suitable value for the reference signal is the flux linkage magnitude of the permanent magnets. In order to increase speed range of the motor above the rated speed the field weakening technique could be implemented in this control block. Also some optimization strategies, like maximum thrust per ampere (MTPA) or i d = 0 control, could be implemented to increase efficiency of the drive in the speed range below the rated speed. In the surface mounted permanent magnet linear synchronous motor the direct-axis and quadrature-axis inductances are approximately equal, L d L q. In the steady state the thrust equation 2.20 can be simplified in the form 3 π Fthrust.est p ψ PMi q. (4.1) 2 τ This equation shows that the direct-axis current i d does not have any effect on the thrust. And for the given thrust force the minimum stator current and in most cases maximum efficiency are reached when i d = 0. This creates the basis for the i d = 0 control, which in linear synchronous drive with surface mounted PMs is equal to the MTPA strategy (Abroshan et al. 2008). To determine flux linkage reference signal next equation could be used. 47

48 ψ ( ) 2 2 Li ψ ( Li) + +. (4.2) s d d PM q q As the direct-axis current i d is equal to zero this equation transforms to ψ ψ + ( Li) 2, (4.3) 2 s PM q q where quadrature-axis current i q is obtained from the equation 4.1. The Simulink model of the i d = 0 controller which produces flux linkage reference signal for the DTFC is represented in the figure 4.5. Figure 4.5. i d = 0 flux linkage controller. 4.2 Simulation results In this section the simulation results are represented Conventional DTFC Let us first compare simulation model behaviour with the data provided by the manufacturer of the linear motor (see Appendix 2). Selected movement profile almost the same as in the data provided by the manufacturer. Figure 4.6 shows reference value and position response for the motor. 48

49 Figure 4.6. Position chart. Figure 4.7 shows reference value generated by the PI position controller and speed response for the motor. Figure 4.7. Speed diagram. Figure 4.8 shows acceleration of the mover. The acceleration diagram resembles the trust force produced by the mover with the account of end effects, cogging force and friction, figure

50 Figure 4.8. Acceleration diagram. Figure 4.9. Thrust force diagram. Figure 4.10 and figure 4.11 show currents during acceleration of the mover, currents at steady speed of 4.5 m/s, and deceleration currents. If we compare figure 4.11 with the current diagram provided by the manufacturer (see Appendix 2) we could see that the simulated current is slightly smaller. It could be the result of underestimation of end effects in simulation, or inexact cogging force and friction models (see Chapter 2.3). Also it could be because the manufacturer had used different from the DTC method to perform measurements. 50

51 Figure d- and q-axes currents. Figure Phase currents of the motor DTFC with MTPA control strategy During MTPA control, the reference signal of the flux linkage is adjusted by the way to produce required thrust force by minimum current. This strategy allows to increase efficiency of the motor and to minimize losses in copper. Figure 4.12 shows flux linkage reference signal adjustments during i d = 0 control. The permanent magnets flux linkage is also shown in the figure. 51

52 Figure Flux linkage reference signal, MTPA control. Figures 4.13 and 4.14 show currents during acceleration of the mover, currents at steady speed of 4.5 m/s, and deceleration currents, when maximum thrust per ampere control strategy is applied. Comparison of figures 4.10 and 4.13 gives us clear vision of advantages provided by i d =0 control technique. The same thrust force is achieved by smaller current. Figure d- and q-axes currents, MTPA control. Figure Phase currents of the motor, MTPA control. 52

53 Chapter 5 Results and Conclusions The objectives of this thesis were focused on the development of a simulation model of the direct thrust force controlled permanent magnet linear synchronous motor. A flat single sided ironcore surface permanent magnet linear synchronous type motor had been chosen for modelling. All the equations required for the basic motor model were introduced. The basic permanent magnet linear synchronous motor model represents conventional two-axes rotary type permanent magnet synchronous motor model. The end-effect, the specific for linear motors phenomenon, was accounted in enlarged motor model as well as friction and cogging force. Direct thrust force control of PMLSM was described. It represents the traditional direct torque control (DTC) applied to linear motors. All equations which justify DTC and consequently DTFC were introduced. The foundations of DTC were carefully described. The DTFC was modelled. Flux linkage correction was modelled using current model. The auxiliary modules for DTFC controller which are required for the speed and position control were introduced as well. These controllers represent simple PI regulators with anti-windup. The parameters for these controllers were experimentally tuned. The movement profile provided by the manufacturer was almost resembled by the simulation. It shows that PI speed and position controllers have sufficient performance and their tuning was performed properly. The simulation results were compared with the data provided by the manufacturer. Generally, results and data are equal. It proves the full system model. The simulated currents were slightly smaller than currents of provided motor data. It could be the result of underestimation of end effects in simulation, or inexact cogging force and 53

54 friction models. Also it could be because the manufacturer had used different from the DTC method to perform measurements. Also the implementation of maximum thrust per ampere control strategy was modelled. For the considered motor type the MTPA strategy and i d =0 control are the same. The results clearly show that MTPA strategy allows to decrease current required for the given thrust in comparison with conventional DTFC. There are few issues related for future development. Most significant one is transformation of continues simulation model into discrete-time system where calculation time and time required for analog-to-digital and back transformations could be taken into account. Field weakening must be modelled if speed above the rated speed of the motor should be considered. Another future work is to apply FEM to determine precise influence of end-effects. For modelling precision position control more accurate friction model and cogging force model should be used. Also these models should be included into control algorithm to reduce the influence of friction and cogging force. The position and speed PI controllers should be tuned analytically or even replaced by more intelligent ones to achieve better performance. Further work should also include tests with real linear motor to practically prove simulator model. Another potential direction of work is considering determination of initial angle between XY-reference frame and dq-reference frame. 54

55 REFERENCES ABB Technical Guide No.1: Direct Torque Control. [Online document]. [Accessed 8 March 2009]. Available at StdDrives/RestrictedPages/Marketing/Documentation/files/PRoducts/DTCTechGuide1.pdf Abroshan, M. & Malekian, K & Milimonfared, J. & Varmiab, B.A An Optimal Direct Thrust Force Control for Interior Permanent Magnet Linear Synchronous Motors Incorporating Field Weakening. IEEE, SPEEDAM, pages Boldea, Ion & Nasar, S.A Linear Motion Electromagnetic Devices. USA. Sheridan Books, Ann Arbor, MI. 269 p. ISBN Gieras, Jacek F. & Piech, Zbigniew J Linear Synchronous motors: Transportation and Automation Systems. USA. CRC press LLC. 327 p. ISBN Hirvonen, Markus On the Analysis and Control of a Linear Synchronous Servomotor with a Flexible Load. Dissertation. Lappeenranta University of Technology, Lappeenranta, Finland. 120 p. ISBN Jiefan, Cui & Chengyuan, Wang & Junyou, Yang & Lifeng, Liu Analysis of Direct Thrust Force Control for Permanent Magnet Linear Synchronous Motor. IEEE, Proceeding of the 5th World Congress on Intelligent Control and Automation, June 15-19, Hangzhou, P.R. China, pages Jung, In-Soung & Yoon, Sang-Baeck & Shim, Jang-Ho & Hyun, Dong-Seok Analysis of Forces in a Short Primary Type and a Short Secondary Type Permanent Magnet Linear Synchronous Motor. IEEE Transactions on Energy Conversion, Vol. 14, No. 4, December, pages

56 Luukko, Julius Direct Torque Control of Permanent Magnet Synchronous Machines Analysis and Implementation. Dissertation. Lappeenranta University of Technology, Lappeenranta, Finland. 172 p. ISBN Mohan, Ned Advanced Electrical Drives: Analysis, Control and Modeling using Simulink. USA. MNPERE. 184 p. ISBN Mohan, Ned & Undeland, Tore M. & Robbins, William P Power Electronics: Converters, Applications, and Design. Third edition. USA. John Wiley & Sons, Inc. 802p. ISBN Niemelä, Markku Position Sensorless Electrically Excited Synchronous Motor Drive for Industrial Use Based on Direct Flux Linkage and Torque Control. Dissertation. Lappeenranta University of Technology, Lappeenranta, Finland. 144 p. ISBN Oriental Motor. [Online Document]. [Accessed 17 April 2009]. Available at Pyrhönen, Juha Electrical Drives Lecture Notes. Lappeenranta University of Technology. Department of Electrical Engineering. Tecnotion A primer of Tecnotion linear motors. Version 2.1. Issue date: September Document nr [Online document]. [Accessed 10 March 2009]. Available at Products.php?Keuze=Techdocs 56

57 Appendix 1 _ Appendices Appendix 1 Simulation model m-file listing open('dtc_pmlsm.mdl'); F_load = 0 %N, load force t_load = 0 %s, start load time %========================================================== % sampling parameters %========================================================== sampling_base = 25e-6 % switching_sample = 25e-6 % mechanics_sample = 4*sampling_base %========================================================== % motor parameters %========================================================== % PSI_PM=tau/pi*2/3*F/I, where F/I - motor force constant PSI_PM = %Wb, flux linkage of the PM ( , 0.35, ) Ld = % H, d-axis inductance (m.b. *1.5) Lq = % H, q-axis inductance (m.b. *1.5) p = 1 %number of pole-pairs tau = % m, pole pitch Rs = 1.6 %Ohm, stator winding resistance %========================================================== % mechanical parameters %========================================================== M_coil_unit = 4.8 % kg, mover mass M_application = 10.8 % kg, load mass M_tot = M_coil_unit + M_application % Friction model: %C_fr*sign(speed) + V_fr*speed + S_fr*exp(- k_str*abs(speed))*sign(speed) C_fr = 30 % N, Coulomb coefficient S_fr = 10 % N, Stribeck coefficient V_fr = 3 % N/(m/s), viscouse k_str= 10 % s/m % End-effect: K_end = 0.01 % Cogging force model: % Ks*sin(wn1*x*2pi)[Ar1 + Ar2*sin(wn2*x*2pi)] Ks = -0.7 %scaling factor wn1 = 1/tau %1/m, 1st wavenumber 1/tau wn2 = 1/0.244 %1/m, 2nd wavenumber 1/l Ar1 = 25 %N, 1st harmonic Ar2 = 15 %N, 2nd harmonic Attraction_force = 3400 % N %========================================================== % inverter parameters %========================================================== U_DC = 560 %volts

58 _ Appendix 1 continued %========================================================== % control parameters %========================================================== V_max = 4.5 % m/s base speed Psi_hyst = 0.02*PSI_PM force_hyst = 60%0.02*T_peak theta_initial = 0 PSI_REF = PSI_PM %========================================================== % Id=0 and FW control and machine limitations %========================================================= I_constr = 8.2 %A %max continuous current V_base = 6 % m/s base speed PSI_base = %Wb %(U_DC-Rs*I_constr)/(p*pi*V_base/tau) F_base = 763% % 3/2*p*pi/tau*PSI_PM*I_constr F_peak = 1600 mtpa_sw= 1 % id=0 control on/off (1/0) %========================================================== % estimator model params %========================================================== PSI_PM_est = PSI_PM*0.98 %Wb, flux linkage of the PM Ld_est = Ld*1.01 % H, d-axis inductance (m.b. *1.5) Lq_est = Lq*1.01 % H, q-axis inductance (m.b. *1.5) Rs_est = Rs*1.1 %Ohm, stator winding resistance correction_sw = 1% Correction on/off (1/0) U_DC_meas = U_DC*0.99 K_corr = 1/switching_sample % flux correction coefficient corrector_sample = %s, correction interval(default s.) %========================================================== % position controller %========================================================== position_pid_p = 23 position_pid_i = 3 position_saturation = V_max %m/s, speed border pos_acc= %m Pos_antiwindup_gain = 0.17 speed_ref = 3 %========================================================== % speed controller %========================================================== speed_pid_p = 6000 speed_pid_i = 1000 speed_saturation = F_base% %N, Thrust (current) limitation speed_acc= %m/s Speed_antiwindup_gain = 0.19%0.02 %========================================================== clear switching_table %(force,psi,sector) %[-Psi 0 ; Psi+ 1] %goes back0 zerovector1 goes forward2 switching_table( :, :,1)=[5 7 3 ; 6 0 2]; % sector 0 switching_table( :, :,2)=[6 0 4 ; 1 7 3]; % sector 1 switching_table( :, :,3)=[1 7 5 ; 2 0 4]; % sector 2 switching_table( :, :,4)=[2 0 6 ; 3 7 5]; % sector 3 switching_table( :, :,5)=[3 7 1 ; 4 0 6]; % sector 4 switching_table( :, :,6)=[4 0 2 ; 5 7 1]; % sector 5

59 Appendix 2 _ Appendix 2 Motor data

60 Appendix 2 _

61 _ Appendix 2 continued

62 _ Appendix 2 continued

63 _ Appendix 2 continued

64 _ Appendix 2 continued