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Chapter 1 Introduction 1.1 Introduction Variable speed drives that drive three phase motors are ubiquitous components of industrial machines that help save energy and optimize systems. Traditional scalar control techniques for variable speed operation of three phase electric motors offer simple implementation but limit the performance that can be achieved. With a scalar drive, algorithm limitations can mean that meeting dynamic response specifications requires choosing a larger motor and a larger drive that complements the larger motor. This tends to drag down the efficiency while resulting in a more expensive system. Control schemes have been developed to overcome this problem by squeezing out more performance out of the same motor. This allows designers to properly size motors and drives, lowering cost and results in a more efficient system overall. Industry automation is mainly developed around motion control systems in which controlled electric motors play a crucial role as the heart of system. Therefore, the high performance motor control systems contribute, to a great extent, to the desirable performance of automated manufacturing sector by enhancing the production rate and the quality of the products. In fact the performance of modern automated systems, defined in terms of swiftness, accuracy, smoothness ad efficiency, mainly depends on the motor control strategies [1] - [3].The recent developments of the power electronics industry resulted in a considerable increase in the power that can be manipulated by the semiconductor devices. In spite of that, the maximum voltage supported by these devices remains the major obstacle in medium and high voltage applications. For such applications multilevel inverters have been proposed [2] - [4]. Multilevel inverters present lower harmonic distortion of the output voltages when compared to standard two-level inverters operating at the same switching frequency. Newly developed permanent magnet synchronous (PMS) motors with high energy permanent magnet materials particularly provide fast dynamics, efficient operation and very good compatibility with the applications if they are controlled properly. However, the AC motor control including control of PMS motors is a challenging task due to very fast motor dynamics and highly nonlinear models of the machines. Therefore, a major part of motor control development consists of deriving motor mathematical models in suitable forms. There are two competing control strategies for AC motors i.e. vector control (VC), it is also known as Field Oriented Control (FOC) and direct torque control (DTC). Permanent magnet synchronous motors (PMSM) are widely used in low and mid power applications such as computer peripheral equipments, robotics and adjustable speed drives and electric vehicles. The growth in the market of PMSM motor drives has demanded the need of simulation tool capable of handling motor drive simulations. Simulations have helped the process of developing new systems including motor drives, by reducing cost and time. Simulation tools have the capabilities of performing dynamic simulations of motor drives in a visual environment so as to facilitate the development of new systems. 1 In this work, the simulation of a field oriented control of PMSM is developed using Simulink. The field oriented control is one of the high performance control strategies for AC machine. The FOC scheme has been realized successfully in the induction motor drives. The aim of the project is to study the implementation of the Field Oriented Control (FOC) in permanent magnet synchronous motor (PMSM). 1.2 Historical Background Almost 41 years ago, in 1971 F. Blaschke presented the first paper on Field-Oriented Control (FOC) for induction motors. Since that time, the technique was completely developed and today is mature from the industrial point of view. Today field oriented controlled drives are an industrial reality and are available on the market by several producers and with different solutions and performance. Fifteen years later, a new technique for the torque control of induction motors was developed and presented by I. Takahashi and Noguchi as direct torque control (DTC) [5]. Seventeen years later it was developed by M. Depenbrock as direct self control (DSC) [6]. Since the beginning, the new technique was characterized by simplicity, good performance and robustness. Using DTC or DSC it is possible to obtain a good dynamic control of the torque without any mechanical transducers on the machine shaft. Thus, DTC and DSC can be considered as “sensor less type” control techniques. The basic scheme of DSC is preferable in the high power range applications, where a lower inverter switching frequency can justify higher current distortion. The name direct torque control is derived by the fact that, on the basis of the errors between the reference and the estimated values of torque and flux, it is possible to directly control the inverter states in order to reduce the torque and flux errors within the prefixed band limits. Unlike FOC, DTC does not require any current regulator, coordinate transformation and pulse width modulated (PWM) signals. In spite of its simplicity, DTC allows a good torque control in steady-state and transient operating conditions to be obtained. The problem is to quantify how good the torque control is with respect to FOC. In addition, this controller is very little sensible to the parameters detuning in comparison with FOC. On the other hand, it is well known that DTC presents some disadvantages that can be summarized in the following points: 1) Difficulty to control torque and flux at very low speed; 2) High current and more torque and flux ripple; 3) More harmonic content in stator current; 3) Variable switching frequency behavior; 4) High noise level at low speed; 5) Lack of direct current control. 2 1.3 Objectives of this work The objectives of this thesis are: i) To model the permanent magnet synchronous motor (PMSM). ii) To design the model of field oriented control system (FOC) iii) To design the model of speed controller. iv) To analyze the performance of the designed model and the control system. v) To control the torque and flux of PMSM. vi) To control the amplitude and the rotating speed of the stator flux linkage by the selection of proper stator voltage vectors. 1.4 Scope of Thesis The scope of work for this thesis: i) PMSM with saliency is considered. ii) Simulation is performed using MATLAB simulink. iii) The performance of FOC is discussed based on simulation results. iv) The performance of Continuous-Time Proportional Integral (PI) and Discrete-Time Proportional Integral (PI) controllers are discussed based on simulation results. 1.5 Research Methodology Our project work has been an organized sequential simulation work. First of all detailed study of Permanent Magnet Synchronous Motor (PMSM) and Field Oriented Control (FOC) method has been done. Emphasis on Interior Permanent Magnet Synchronous Motor (IPMSM) has been given for modeling the PMSM block. For it pole saliency has been considered. Different torque control method has been studied especially Field Oriented Control method for IPMSM. The characteristics of FOC and other vector methods are compared. After that the performance has been checked. By using Matlab Simulink model of FOC was constructed. The model of FOC was designed with mathematical equation of FOC. To control the speed of IPMSM the different speed controlling methods were studied. By comparing the characteristics two models were selected for the better performance. Continuous- Time Proportional Integral (PI) and Discrete-Time Proportional Integral (PI) controller blocks were used for the simulation purpose. The mathematical expression converted into simulation model by Matlab Simulink. Then the overall Simulink model has been simulated and results were analyzed with different parameters. Conclusions were made depending on the different simulation graphical results. 3 1.6 Introduction to this Thesis The theory of permanent magnet synchronous motor (PMSM) is presented in chapter 2. The mathematical model of PMSM is showed in chapter 3. The details of FOC of IPMSM has been presented in chapter 4. In chapter 5, different speed control methods and the detailed information on Continuous-Time PI controllers has been discussed. Chapter 6 presents the detailed information on Discrete-Time PI controllers. The simulation results are showed with little description in chapter 7. And chapter 8 ends with the discussion and conclusion. 4 Chapter 2 Interior Permanent Magnet Synchronous Motor 2.1 Introduction Permanent magnet synchronous machines describe a special class of synchronous machines that has permanent magnets fitted in the rotor. A permanent magnet synchronous machine is basically an ordinary AC machine with windings distributed in the stator slots so that the flux created by the stator current is approximately sinusoidal and uses permanent magnets to produce the air gap magnetic field rather than using electromagnets. In permanent magnet (PM) synchronous motors, permanent magnets are mounted inside or outside the rotor. Unlike DC brush motors, permanent magnet synchronous motor requires a “drive” to supply commutated current. This is obtained by the voltage source inverter (VSI) fitted to the motor windings. By energizing specific windings in the stator, based on the position of the rotor, a rotating magnetic field is generated. Currents in the stator windings are switched in a predetermined sequence and hence the permanent magnets that provide a constant magnetic field on the rotor follow the rotating stator magnetic field at constant speed. This speed is dependent on the applied frequency and pole number of the motor. Since the switching frequency is derived from the rotor, the motor cannot lose its synchronism. The current is always switched before the permanent magnets catch up; therefore the speed of the rotor is directly proportional to the current switching rate [1]. Permanent magnet synchronous motors have received alternative as a energy converting device. The motor does not operate without suitable controller from the nominal supply system. In this chapter description of the different components such as permanent magnet motors, position sensors, inverters and current controllers of the drive system are discussed. A review of permanent magnet materials and classification of permanent magnet motors is also given. A detailed mathematical model of the system is also provided at the end of the chapter. 2.2 Magnetic materials Ferromagnetic materials can be divided into soft and hard magnetic materials. This is depending on the magnitude of the magnetic field strength H that is required to saturate the material. The soft magnetic materials can be described by their relative permeability μr. The relative permeability is a measure of how easy the material can be magnetized by an external field H. The magnetic flux density B in the material is given by: B 0 r H (2.1) Where, μ0 is the permeability of free space. For a soft magnetic material μ r is high and the magnetic field strength required for saturation is low. This means that the area enclosed by the hysteresis loop is small and thus the hysteresis losses are low. This makes a soft magnetic material suitable as lamination in electrical machines where the lamination is used to complete the flux path. 5 2.2.1 Properties of permanent magnet materials The properties of the permanent magnet material will affect directly the performance of the motor and proper knowledge is required for the selection of the materials and for understanding PM motors Permanent magnet materials used to create an air-gap flux in electrical machines are hard magnetic materials. They have a low relative permeability ( μr ≈ 1) and require a high magnetic field strength for saturation, but they then keep this magnetization. Permanent magnet materials are usually graded by their maximum energy product. (BH)max This is twice the maximum energy that can be stored within a permanent magnet material in the absence of any externally applied field. The magnet with the highest energy product will have the smallest volume for producing a specific flux density. The remanence flux density is a measure of the materials magnetization without any external field. A high remanence flux density is required in order to obtain a high energy product. The flux density within a permanent magnet can be expressed by the following idealized expression [1]: B Br 0 r H (2.2) Another important quality is the materials coercivity HcJ. It is a measure of the reverse field needed to drive the magnetization to zero after being saturated. A high coercivity means that the magnet can sustain a high demagnetization field. The remanence flux density is temperature dependent, it will decrease for increased o temperatures. Normally the remanence is given in reference to 25 C. Br Br (250 C).(1 B r T ) (2.3) where αBr (< 0) is the reversible temperature coefficient and ΔT is the deviation from the reference temperature. Thus it is important to know the magnets operating temperature. Further on, the coercivity is also decreasing for an increased temperature. Thus the permanent magnets can be more easily demagnetised when they are warm. It is possible to assign an approximately constant value to the reversible temperature coefficient of Br . But the reversible temperature coefficient of HcJ is not as easy to predict. Instead, the actual demagnetization curves can be used to study this parameter. 2.2.2 Permanent Magnet Materials The earliest manufactured magnet materials were hardened steel. Magnets made from steel were easily magnetized. However, they could hold very low energy and it was easy to demagnetize. In recent years other magnet materials such as Aluminum Nickel and Cobalt alloys (ALNICO), Strontium Ferrite or Barium Ferrite (Ferrite), Samarium Cobalt (First generation rare earth magnet) (SmCo) and Neodymium Iron-Boron (Second generation rare earth magnet) (NdFeB) have been developed and used for making permanent magnets. 6 The rare earth magnets are categorized into two classes: Samarium Cobalt (SmCo) magnets and Neodymium Iron Boride (NdFeB) magnets. SmCo magnets have higher flux density levels but they are very expensive. NdFeB magnets are the most common rare earth magnets used in motors these days. 2.2.3 Samarium Cobalt magnets Samarium cobalt magnets, SmCo has been mostly used in electrical machines before the introduction of the NdFeB-magnets. SmCo-magnets can resist a very high operating temperature and have good temperature stability over a large temperature range. Due to its temperature stability, the material is very suitable in sensor applications. The material can also be suitable in electrical machines if the operating temperature is very high. But the SmCo-magnets are very brittle and quite expensive, approximately 20-30 % more expensive then NdFeB-magnets [1]. Characteristics of the SmCo-magnets are [1]: - relatively high remanence, Br ≤ 1,15 T o -1 - reversible temp. coeff. of Br , 20-150 C, -0.045 ≤ αBr ≤ -0.035 % K - good energy density, (BH)max ≤ 240 kJ/m3 o - high temperature stability, up to 350 C (continuously) o - high Curie temperature, approximately up to 800 C - high coercivity, HcJ < 2400 kA/m 2.2.4 Neodymium Iron Boron magnets Neodymium iron boron, NdFeB is the most commonly used material in high power density permanent magnet machines. This is due to its high energy product which reduce the magnet volume required for a specific airgap flux density. NdFeB-magnets have a high remanence flux density and a high coercivity at relatively high temperatures. But they have a low Curie temperature and are not as temperature stable as the SmCo-magnets are. NdFeB is also more sensitive to humidity than SmCo and should therefore be coated with a protection layer. Characteristics of the NdFeB-magnets are [1]: - high remanence, Br ≤ 1,4 T o -1 - reversible temp. coeff. of , Br , 20-150 C, -0.12 ≤ αBr ≤ -0.11 % K - high energy density, (BH)max ≤ 385 kJ/m3 o - good temperature stability, up to 200 C (continuously) o - low Curie temperature, approximately up to 350 C - high coercivity, HcJ < 3260 kA/m A flux density versus magnetizing field for these magnets is illustrated in Fig. 2.1. 7 1.5 Br μo M NdFeB 1.0 B(Tesla) NdFeB at SmCo 120⁰C 0.5 MH C FERRIT E MHc AlNIC O -1000 -500 0 H(kA/m) Fig. 2.1: Flux Density versus Magnetizing Field of Permanent Magnetic Materials. 2.3 Permanent Magnet Synchronous Motor A permanent magnet synchronous motor (PMSM) is a motor that uses permanent magnets to produce the air gap magnetic field rather than using electromagnets. These motors have significant advantages, attracting the interest of researchers and industry for use in many applications due to its decreased loss and higher torque production rate. 2.4 Permanent Magnet Synchronous Motor Drive System The motor drive system consists of four main components, the PM motor, inverter, control unit and the position sensor. The components are connected in a basic drive system as shown in Fig. 2.2. 8 Ia Inverter PM Motor Position Ib Load Ic Sensor + DC Ia Source - Ib Ic Control Controller Rotor Position Input Fig. 2.2: Drive System schematic 2.5 Role of Position Sensor Operation of permanent magnet synchronous motors requires position sensors in the rotor shaft when operated without damper winding. The need of knowing the rotor position requires the development of devices for position measurement. There are four main devices for the measurement of position, the potentiometer, linear variable differential transformer, optical encoder and resolvers. The ones most commonly used for motors are encoders and revolvers. Depending on the application and performance desired by the motor a position sensor with the required accuracy can be selected 2.6 Features of Permanent Magnet Machines The use of permanent magnets for the construction of electrical machines brings the following benefits [8]: 1. No electrical energy is absorbed by the field excitation system and thus there are no excitation losses, 2. Increase in the efficiency, 3. Higher torque and/or output power per volume than when using electromagnetic excitation, 4. Higher magnetic flux density in the air gap, 5. Better dynamic performance than motors with electromagnetic excitation, 6. Simplification of construction and maintenance, 7. Reduction of prices for some type of machines. 2.7 Classification of Permanent Magnet Machine Based on Field Flux There are generally three types of PM-machines [9]. The different types are: Radial, Axial and Transversal flux machines. 9 2.7.1. Radial Flux Permanent Magnet machines The radial flux permanent magnet machine is the classic type and most common. It is quite similar to other AC-machines and is also used in mostly the same areas. The rotor can have buried or surface mounted magnets, the poles can be skewed, have pole shoes etc. The stator is quite similar to other classical AC-machines (induction and synchronous) both for windings and tooth shape. It is common, but not always necessary, to use semi closed slots. Magnetic slot wedges is also an option. Two layer fractional windings are mostly used even though the simplicity of concentrated windings has begun to be appreciated. The active materials; copper, magnets, iron, sheet metal, converting the mechanical energy to electric or visa versa, are placed along the air gap. For radial flux permanent magnet machine with large diameter this means that the active material becomes a thin shell around the air gap thus most of the volume to the machine is air or supporting structures to transfer the torque to or fro the shaft to the rotor rim. Since the force is acting at a large radius a high torque is produced [11]. 2.7.2. Axial Flux Permanent Magnet Machine Axial flux permanent magnet machines are magnetized in the axial direction. The air gap is radial to the shaft. Therefore, compared to radial flux permanent magnet machine, the length of axial flux permanent magnet machine (La) is equal to the thickness of active materials in the radial flux permanent magnet machine (Dg) and the thickness (Do-Di) of the axial flux permanent magnet machine is equal to the length of radial flux permanent magnet machine (Lr) as illustrated in Fig. 2.3 [11]. Fig. 2.3: Comparison of the diameter of active material in radial and axial machines. Given the same outer diameter and the same force per area in the air gap, the axial flux permanent magnet machine has a lower torque per volume of active material due to the fact that much of the force is working on a smaller radius and thus producing less torque. The great advantage of the axial flux permanent magnet machine over the radial flux permanent magnet machine is the possibility to use the volume of the machine more effectively and that the power density (W/m3) gets higher. 10 The axial flux permanent magnet machine usually has a disc shaped design, with large diameter and short length, which are useful in different applications. Several discs can be connected in series and make a multi disc machine. As for radial flux permanent magnet machine the rotor in an axial flux permanent magnet machine can be made with surface mounted or buried magnets. The stator has also the same possibilities for different design of teeth and windings, but it is somewhat more common to use concentrated windings. It is also used ironless stators; this reduces iron losses, makes it possible to cool the winding more effectively and also eliminates the attractive forces between the magnets in the rotor and the iron in the stator, but increases the need for magnets [12]. Fig. 2.4: Ironless axial and radial flux machines Fig. 2.4 presents the out look of axial and radial flux machines (ironless) with an outer rotor conFig.uration. 2.7.3 Transverse Flux Permanent Magnet Machine Transverse flux permanent magnet machine is the most complex and is the least equal to classic machine design. It can be single sided [13] or double sided [14] respectively with one or two wound rings of with iron cores to lead the magnetic flux around the copper. The transverse flux permanent magnet machine has a rotor with either buried or surface mounted permanent magnets. The transverse flux permanent magnet machine has a very high force and power density and therefore has been considered very promising in application with high torque and low speed. One major problem with transverse flux permanent magnet machine is that to achieve this high force density, the synchronous reactance can grow very high which in turn makes the converter expensive. Another factor is the cost due to the many parts in a transverse flux permanent magnet machine. 11 2.8 Classification of Permanent Magnet Machine Based on Flux Density Distribution PM motors are classified on the basis of the flux density distribution and the shape of current excitation. They are PMSM and PM brushless DC motors (BLDC) [15].The PMSM has a sinusoidal-shaped back EMF and is designed to develop sinusoidal back EMF waveforms. They have the following: 1. Sinusoidal distribution of magnet flux in the air gap, 2. Sinusoidal current waveforms, 3. Sinusoidal distribution of stator conductors. BLDC has a trapezoidal-shaped back EMF and is designed to develop trapezoidal back EMF waveforms. They have the following: 1. Rectangular distribution of magnet flux in the air gap, 2. Rectangular current waveform, 3. Concentrated stator windings. 2.9 Types of Permanent Magnet Radial Field Motors In PM motors, the magnets can be placed in two different ways on the rotor. Depending on the placement they are (i) Surface mounted PMSM (SPMSM) (ii) Interior PMSM (IPSM) 2.9.1 SPMSM Surface PM motors have a surface mounted permanent magnet rotor. Each of the PM is mounted on the surface of the rotor, making it easy to build, and specially skewed poles are easily magnetized on this surface mounted type to minimize cogging torque. This conFig.uration is used for low speed applications because of the limitation that the magnets will fly apart during high-speed operations. These motors are considered to have small saliency, thus having practically equal inductances in both axes. The permeability of the permanent magnet is almost that of the air, thus the magnetic material becoming an extension of the air gap. For a surface permanent magnet motor Ld Lq The rotor has an iron core that may be solid or may be made of punched laminations for simplicity in manufacturing. Thin permanent magnets are mounted on the surface of this core using adhesives. Alternating magnets of the opposite magnetization direction produce radially 12 directed flux density across the air gap. This flux density then reacts with currents in windings placed in slots on the inner surface of the stator to produce torque. Fig. 2.5, 2.6 and 2.7 shows the placement of the magnet. Fig. 2.5: Surface Permanent Magnet Motor. Fig. 2.6: Surface PM (SPM) Fig. 2.7: Surface inset PM (SIPM) Synchronous Machine Synchronous Machine 13 Fig. 2.8: Structure and flux linkage of SPMSM Fig. 2.8 illustrates schematically the structure and flux linkage of a SMPMSM. When the motor saliencies are taken into account, the conventional model of PMSM is inaccurate since the inductances of the motor are no longer constants and will be the function of the rotor position and stator current. 2.9.1.1 Surface-mounted PM machines Surface mounted permanent magnets (SMPM) is the most common rotor conFig.uration for PM machines. The magnets are placed on the rotor surface as shown in Fig. 2.9. Fig. 2.9: Cross section of a quarter of an 8-pole rotor with surface mounted magnets 14 The magnets are glued onto the rotor surface and fixed by a carbon or glass fiber bandage. In relation to other PM concepts, the surface mounted machines are easy to manufacture and consequently the construction cost is lower. The main drawback of these machines is the exposition of the permanent magnets to demagnetization fields. The magnets are also subject to high centrifugal forces. But at moderate peripheral speeds, the use of a glass fiber bandage is usually enough to withstand these forces. As the permeability of the magnets is almost the same as the permeability of air, the d-axis and the q-axis reluctances are equal. Hence, the SMPM machines have no saliency and the torque is produced by the interaction between the stator currents and the magnets only. 2.9.1.2 Inset PM machines In inset PM machines, the magnets are mounted on the rotor surface as shown in Fig. 2.10. But the magnets are sunken in the rotor core, offering better protection than in a SMPM machine. However, a bandage is often required in order to withstand the centrifugal forces. Due to the iron between the magnets, the q-axis reluctance is lower than the d-axis reluctance. This is known as salient poles. Due to the saliency, a reluctance torque is created in addition to the torque from the magnets. Fig. 2.10: Cross section of a quarter of an 8-pole rotor with inset magnets 15 2.9.2 IPMSM Interior PM motors have interior mounted permanent magnet rotor as shown in Fig.:2.11 and Fig.: 2.12. Each permanent magnet is mounted inside the rotor. It is not as common as the surface- mounted type but it is a good candidate for high-speed operation. There is inductance variation for this type of rotor because the permanent magnet part is equivalent to air in the magnetic circuit calculation. These motors are considered to have saliency with q-axis inductance greater than the d-axis inductance ( Ld Lq ). Fig. 2.11: Interior PM (IPM) Fig. 2.12: Interior PM with Synchronous Machine circumferential orientation Synchronous Machine 2.9.2.1 Buried PM machines Another PM concept is to bury the magnets in the rotor core as shown in Fig. 2.13. This is referred to as buried PM machines. The magnets are better protected against demagnetization fields and mechanical stress. There are many different possibilities for the placement of magnets in the rotor. The magnets can be placed in such a way that the flux generated by the magnets in the rotor is concentrated and thus high air-gap flux densities can be achieved [3]. As the inset PM machines the buried PM machines have salient poles. A main drawback of the buried PM machines is the difficult manufacturing process and thus the high production cost. 16 Fig. 2.13: Cross section of a quarter of an 8-pole rotor with buried magnets 2.10 Features of Interior Permanent Magnet Synchronous Motors The following are the significant features of the permanent magnet synchronous motor [16]: 1. High efficiency, 2. High torque to inertia ratio, 3. High torque to volume ratio, 4. High air gap flux density, 5. High power factor, 6. High acceleration and deceleration rates, 7. Simplicity, ruggedness and compact structure, 8. Linear response in the effective input voltage, 9. Lower maintenance cost. 2.11 Applications of Permanent Magnet Synchronous Motors The field of applications of permanent magnet synchronous motor is as follows [16]: 1. Chemical industry, 2. Texturing plants, 3. Electrical household appliances 4. Robotic automation as shown in Fig. 2.14, 5. Escalators as shown in Fig. 2.15 etc. 6. Ship propellers and engines as shown in Fig. 2.16, 17 7. Transport system as shown in Fig. 2.17 Fig. 2.14: Industrial robot Fig. 2.15: Elevator hoisting machine Fig. 2.16: Ship propeller Fig. 2.17: Automobile system 18 2.12 Comparison between the Features of PMSM and Induction Motor (IM) Comparison between PMSM and IM are summarized in Table 2.1. Table 2.1 Comparison between PMSM and IM [17]. PMSM IM Excellent dynamics with proper Smooth torque possible control High efficiency High speed operation possible High torque/volume Low price High pull-out torque possible Simple construction Benefits Good heat dissipation Durable Good overloading capacity Several suppliers available Expensive Complicated control Danger of demagnetization of the Always runs at lagging power factor magnet Drawbacks Poor field weakening Low efficiency with lighter loads 2.13 Advantages of PMSM over IM The following are the advantages of PMSM over IM [17]: 1. Has better overloading capacity, 2. Higher torque per volume ratio, 3. Higher torque per current ratio, 4. Allows more compact construction, 5. Requires smaller inverter, 6. Higher overall efficiency, 7. Effective use of reluctance torque, 8. Smaller losses, 9. Higher power density, 10. More reliable, 11. Better and more accurate torque response etc. These are the principal reasons after the choice of PMSM for this thesis work. 19 2.14 Summery This chapter presents a theoretical review of permanent magnet motors drives which includes permanent magnet materials, classification of the permanent magnet motors, the construction and advantages and finally ends with the comparison between PMSM and IM supporting our choice of PMSM for this thesis work. 20 Chapter 3 System Model 3.1 Introduction This chapter deals with the detailed modeling of a permanent magnet synchronous motor. In a permanent magnet synchronous motor (PMSM) where the inductances vary as a function of the rotor angle, the two-phase (d-q) equivalent circuit model is most suitable solution to analyze and design the controller of multiphase electrical machines because of its simplicity and intuition. This theory is now applied in the analysis of other types of motors including PM synchronous motors, induction motors etc. In this section, an equivalent two-phase circuit model of a three-phase PM synchronous machine (interior and surface mount) is used in order to clarify the concept of the space vectors, transformation and the relation between three-phase quantities and their equivalent two-phase quantities. 3.2 Transformation and Dynamic d-q Modeling Fig. 3.1: Transformation between stationary axis and rotational axis. The dynamic d-q modeling is used for the study of motor during transient and steady state. It is done by converting the three phase voltages and currents to αβ variables to d-q variables transformation as shown in Fig. 3.1. Converting the phase voltages variables Vabc to Vαβ variables in rotor reference frame the obtained from 21 1 .5 .5 Va Vs 2 0 3 3 Vb (3.1) Vs 3 2 2 Vc The α-β axes components can be converted in d-q axes reference frame as follows: Vsd cos sin Vs (3.2) Vsq sin cos Vs 3.3 Nomenclature vd , vq d - and q – axis stator voltage id , iq d - and q – axis stator current icd , icq d - and q – axis core loss current imd , imq d - and q – axis magnetizing current d , q d - and q – axis stator flux linkage af the flux linkage due to the rotor magnets linking the stator R s , Rc stator and core loss resistances Ld , Lq d - and q – axis stator inductance Te , TL electromagnetic and load torques Jm moment of inertia Bm friction co-efficient P number of pole Pn number of poles pair ( = P/2) rotor flux linkage r rotor electrical angular velocity rm rotor mechanical angular velocity 3.4 Dynamic Model of PMSM without Core Loss Detailed modeling of PM motor drive system is required for proper simulation of the system. The d-q model has been developed on rotor reference frame as shown in Fig. 3.2. At any time t, the rotating rotor d-axis makes and angle θr with the fixed stator phase axis and rotating stator mmf makes an angle α with the rotor d-axis. Stator mmf rotates at the same speed as that of the rotor. Throughout the derivation of the two-phase (d-q) mathematical model of PMSM, the following assumptions are made [21]: 22 1. Stator windings produce sinusoidal MMF distribution. 2. Space harmonics in the air-gap are neglected. 3. Air-gap reluctance has a constant component as well as a sinusoidal varying component. 4. Balanced three-phase sinusoidal supply voltage is considered. 5. Eddy current and hysteresis effects are neglected. 6. Presence of damper windings is not considered; 7. Magnetic current is linear Fig. 3.2: Representation of different reference frame PM synchronous motors used today rarely have that kind of configuration. Comparing a primitive version of a PMSM with wound-rotor synchronous motor, the stator of a PMSM has windings similar to those of the conventional wound-rotor synchronous motor which is generally three-phase, Y-connected, and sinusoidally distributed. However, on the rotor side instead of the electrical-circuit seen in the wound- rotor synchronous motor, constant rotor flux provided by the permanent magnet in/on the rotor should be considered in the d-q model of a PMSM. 23 3.4.1 Voltage and current equation (a) d-axis equivalent circuit (b) q-axis equivalent circuit Fig. 3.3: Equivalent circuit of IPMSM without considering core loss voltage equation: vd Rsid d r q (3.3) v q Rs iq q r d (3.4) where, indicates d/dt. Flux Linkages are given by: d Ld id af (3.5) q Lq i q (3.6) Developed Torque : 3 Te P[af iq ( Ld Lq )iq id ] (3.7) 2 The mechanical dynamic expression: J m r Bm r Pn (Te TL ) (3.8) 24 Steady-State Equation: In steady-state condition all derivative terms are zero, so Vd R s I d r q (3.9) Vq R s I q r d (3.10) Te Bm r TL (3.11) State-Equations: Using equations (3.3), (3.4), (3.5), (3.6) and (3.8), we can write the state-space equation for IPMSM without considering core loss: d 1 id id [ R s i d r Lq i q v d ] (3.16) dt Ld d 1 iq iq [ Rs iq r Ld id r af v q ] (3.17) dt Lq d 1 r r [ Bm r Pn Te PnTL ] (3.18) dt Jm 3.5 Dynamic Model of PMSM with Considering Core Loss The maximum torque per ampere vector control was derived under the assumption that the core loss may be neglected. Unfortunately, the ideal IPMSM does not exist practically. It is important to consider core loss for precise torque control in industrial application. The effects of core loss due to eddy current have been investigated for electrical machine. The effects of core loss deteriorate the performance of torque control. Consequently, the complexity arises in vector control of electrical drives taking core loss into account. 25 Fig. 3.4 (a) d-axis equivalent circuit with considering core-loss Fig. 3.4 (b) q-axis equivalent circuit with considering core-loss 3.5.1 Voltage and Current equation With considering core loss the flux linkages are given by: d Ld imd af (3.19) q Lq imq (3.20) Voltage equation : From equation (3.3) and (3.20) , we get v d Rs i d d r q 26 v d Rs id Ld imd r Lq imq (3.21) From equation (3.4) and (3.19) , we get v q Rs iq q r d v q Rs iq Lq imq r ( Ld imd af ) (3.22) where, indicates d/dt. From equivalent circuit imd id icd (3.23) imq iq icq (3.24) Developed torque : Te Pn [ af imq ( Ld Lq )imq imd ] (3.25) Mechanical dynamic expression: J m r Bm r Pn (Te TL ) (3.26) From d-axis equivalent circuit, we get Rc icd Ld imd Lq r imq (3.27) After solving this equation, we get Ld L icd imd q r imq (3.28) Rc Rc From q-axis equivalent circuit Rc icq Lq imq Ld r imd r af (3.29) After solving we get Lq Ld 1 icq imq r imd r af (3.30) Rc Rc Rc 27 Steady-State Equation: In steady-state condition all derivative terms are zero, so from Eq. (3.3) and (3.4) Vd Rs I d r q Rs I d r Lqimq Rs ( I md I cd ) r Lqimq (3.31.1) Vd Rs ( I md I cd ) r Lq imq (3.31.2) Vq R s I q r d (3.32.1) Vq Rs ( I mq I cq ) r Ld I md r af (3.32.2) I md I d I cd (3.33) I mq I q I cq (3.34) Lq I cd r I mq (3.35) Rc Ld 1 I cq r imd r af (3.36) Rc Rc Lq I cd r ( I q I cq ) (3.37) Rc Ld 1 I cq r ( I d I cd ) r af (3.38) Rc Rc Te ( Bm / Pn ) r TL (3.39) Derivation of State-space Equations: From d-axis equivalent circuit we get v d Rs id Rc icd (3.40) With the help of equation (3.23) , we can get d R Lq v imd imd ( s )imd r imq d dt Rsc Ld Ld Rsc Ld (3.41) R s Rc Rsc Where , Rc From q-axis equivalent circuit we get 28 v q R s i q Rc icq (3.42) With the help of equation (3.24), we can get d R L af vq imq imq ( s )imq d r imd r dt Rsc Lq Lq Lq Rsc Lq (3.43) R s Rc Rsc Where : Rc Rearranging equation (3.26) we get Te J m r Bm r TL J m r Bm r Pn (Te TL ) But if we consider it for mechanical angle then we can write Te J m rm Bm rm TL (3.44) We know P m e 2 r rm Pn From equation (3.44), we have J Te m r Bm r TL Pn Pn (3.45) After solving we get d B P r r m r n [Te TL ] dt Jm Jm (3.46) Thus the state space model of IPMSM with considering core loss is [10] d R Lq v imd imd ( s )imd r imq d (3.41.1) dt Rsc Ld Ld Rsc Ld d R L af vq imq imq ( s )imq d r imd r (3.43.1) dt Rsc Lq Lq Lq Rsc Lq d B P r r m r n [Te TL ] (3.46.1) dt Jm Jm In our simulation work, we used the model of IPMSM with consideration of core loss to design the discrete-time PI controller. 29 The d-axis stator voltage Eq. (3.41.1) can be written: dimd Rsc Ld Rs imd Rsc Lq r imq v d (3.47) dt Let, u d Rsc Lq r i mq v d (3.48) where, v d u d Rsc Lq r imq (3.49) dimd Rs 1 i md ud (3.50) dt Rsc Ld Rsc Ld Rs 1 Let, Ad , Bd , then R sc Ld R sc Ld dimd Ad imd Bd u d (3.51) dt The Laplace transformation of Eq. (3.51) can be written as follows: si md ( s) Ad imd ( s) Bd u d ( s) Bd imd ( s ) u ( s) s Ad d imd ( s ) Bd (3.52) u d ( s ) s Ad The q-axis stator voltage Eq. (3.43.1) can be written: dimq Rsc Lq Rs imq Rsc Ld r imd Rsc af r v q (3.53) dt Let, u q Rsc Ld r imd Rsc af r v q (3.54) where, v q u q Rsc Ld r i md Rsc af r (3.55) dimq Rs 1 imq uq (3.56) dt Rsc Lq Rsc Lq Rs 1 Let, Aq , Bq , then Rsc Lq R sc Lq dimq Aq imq Bq u q (3.57) dt 30 The Laplace transformation of Eq. (3.57) can be written as follows: si mq ( s ) Aq imq ( s ) Bq u q ( s ) Bq imq ( s) s A u q q ( s) imq ( s) Bq u q ( s) s A q (3.58) Using Eqs. (3.52) and (3.58), the current controller can be designed. 3.6 Summary In this chapter the mathematical model of the permanent magnet synchronous motor for both cases, without considering core loss and considering core loss, has been discussed. This chapter gives the Clark’s transformation used for the dynamic d-q modeling of the PMSM. A number of equations necessary for the mathematical modeling of PMSM have been discussed that includes voltage and current equations, torque and speed equations and the state equations. 31 Chapter 4 Vector Control of PMSM 4.1 Introduction The torque ripple of the PMSM can bring the problems such as speed fluctuation, resonance and acoustic noise, and then affect the above equipment's reliability, accuracy and degree of comfort. So it is important to suppress the torque ripple. In the traditional drive system operated in a high- speed region, it is not necessary to consider this problem because the torque pulsation is filtered out by the inertia of the motor and load. However, the torque ripple is caused by the structure imperfectness of PMSM such as the space harmonics and the unbalanced there-phase windings. Techniques of suppressing the torque ripple, which causes the vibration, could be generally divided into two major methodologies. The first class is based on the optimum design of the motor such as skewing, PM arc pole width, dummy slots or teeth and fractional slot pitch winding. A special winding technique has been adopted as another way of reducing the torque pulsation. The short-pitched or fractional slot pitch windings provide a good capability of rejecting the torque harmonics. However, these methods have their drawbacks. They require more complex stator or rotor construction, the average torque generally decreases and the leakage inductance and stray losses are slightly higher. These methods are based on the finite- element technique taking into account the structure of the motor. In the analytical technique, the accuracy could be improving the torque ripple with the help of increasing of the computational capacity. However, if more precise vibration suppression is required, suppressing the torque ripple by control is needed. A second class of torque minimization techniques is based on active control schemes based on different controller techniques, which adjust the stator current behaviors in order to obtain a smooth torque. Most of the methods utilize the additional feed- forward compensation voltage/current so that the torque ripple is reduced. 4.2 Different Types of Control Techniques There are some common techniques used for DTC as shown in Fig. 4.1 depending of motors variable frequency. Variable Frequency Control Techniques of PMSM can be divided into 1. Scalar Control 2. Vector Control 4.2.1 Scalar Control Scalar control is based on relationships valid in steady state. Only magnitude and frequency of voltage, current, etc. are controlled. Scalar control is used e.g. where several motors are driven in parallel by the same inverter. 32 Volts/Hertz control is among the simplest control schemes for motor control. The control is an open-loop scheme and does not use any feedback loops. The idea is to keep stator flux constant at rated value so that the motor develops rated torque/ampere ratio over its entire speed range. Scalar control is based on relationships valid in steady-state conditions. Amplitude and frequency of the controlled variables are considered. 4.2.2 Vector Control The problem with scalar control is that motor flux and torque in general are coupled. This inherent the coupling affects the response and makes the system prone to instability if it is not considered. In vector control, not only the magnitude of the stator and rotor flux is considered but also their mutual angle. These relationships are valid even during transients which is essential for precise torque and speed control. Vector based control method can be classified 1. Field Oriented Control (FOC) 2. Direct Torque Control (DTC) Fig. 4.1: Some common control techniques 4.3 Limitations of traditional scalar control Electric motors are a prime mover of choice, and account for over half of U.S. electricity consumption, so the potential cost and energy savings through increasing the performance and efficiency of electric motors used in industrial applications is significant. Most of the motors in variable-speed drives are alternating current (AC) induction motors. Scalar control is based on a very simple control strategy: the voltage and frequency applied to the motor is changed to change the speed of the motor. 33 To run the motor at various frequencies, the frequency of the three-phase sinusoidal drive is varied, and the voltage applied to the motor is varied in proportion. This changes the speed of the rotating magnetic flux in the motor, and in turn changes the speed of the machine. This results in a rotating magnetic field, rotating at the synchronous speed, typically 1800 rpm or 3600 rpm for 2 pole or 4 pole per phase machines with 60Hz excitation. These motors operate relatively efficiently at the synchronous speed when the voltage drop across the stator is nominal. For example, to reduce the operating speed to say half the nominal speed for a 208V/60Hz, 1800 rpm machine, the frequency would change to 30Hz and the voltage to 104V. This would result in the machine running at 900 rpm. A significant number of industrial applications benefit greatly from variable speed operation, and variable speed drives are increasingly adopted in a large variety of industrial equipment. Since in a constant V/F control there is no overt effort to maintain the alignment between the stator and rotor flux, oscillations and current spikes can occur during rapid transients. An aggressive speed regulator might accelerate the stator flux too quickly, disturbing the flux alignment in the machine. This may actually reduce the instantaneous torque produced, and also temporarily reduce the back-emf in the motor windings, resulting in a current inrush. In an open loop operation scenario, the induction machine self aligns to a new equilibrium. With an aggressive closed loop speed regulator this mechanism results in torque and current oscillations. One way to avoid these transients is to limit the transient demands. To do this the regulator may be detuned, limiting performance. Another option can be to use a larger machine, with a higher torque capability. The larger machine may indeed allow the system to respond to the transient though it follows that with scalar control driving a larger motor, the power converters may need to be oversized to handle the torque requirements of the transient and surge currents. 4.4 Comparison of DTC with FOC 4.4.1 Features Important Features of FOC: This control method implies the following properties of the control: 1. Speed or position measurement or some sort of estimation is needed 2. Torque and flux can be changed reasonably fast, in less than 5-10 milliseconds, by changing the references 3. The step response has some overshoot if PI control is used 4. The switching frequency of the transistors is usually constant and set by the modulator 5. The accuracy of the torque depends on the accuracy of the motor parameters used in the control. Thus large errors due to for example rotor temperature changes often are encountered. 6. Reasonable processor performance is required, typically the control algorithm has to be calculated at least every millisecond. 34 Important Features of DTC: This control method implies the following properties of the control: 1. Torque and flux can be changed very fast by changing the references. 2. High efficiency & low losses - switching losses are minimized because the transistors are switched only when it is needed to keep torque and flux within their hysteresis bands. 3. The step response has no overshoot. 4. No coordinate transforms are needed; all calculations are done in stationary coordinate system. 5. No separate modulator is needed; the hysteresis control defines the switch control signals directly. 6. There are no PI current controllers. Thus no tuning of the control is required. 4.4.2 FOC versus DTC In DTC, field orientation is achieved without feedback using advanced motor theory. Furthermore, a position encoder measuring speed or position of the motor shaft is not required. The DTFC scheme is known to produce a quick and robust response in ac drives due to the low motor parameter sensitivity of the stator voltage equation in estimating stator flux. However during steady state, pulsations of torque, flux and current may occur. Some techniques such as Space Vector Pulse Width Modulation (SV-PWM), realizing a given voltage state phasor in a PWM inverter, have been suggested to reduce the acoustic noise, torque, flux and current pulsations at steady state. Another problem of this approach is the operation at low speeds. The integration for determining the flux (voltage model) is difficult when the back-EMF voltage approaches the stator resistance voltage drop and it is therefore sensitive to stator resistance variations. For solving this problem in high performance DTC drives, continuous estimation of stator resistance is required. Other approaches to achieve DTC have been proposed and tested such as neuro-fuzzy structures. DTC, in its traditional form, results in a non-constant inverter switching frequency, which may result in high inverter/motor losses. In fact, there is no really steady state in DTC. Techniques that fix these problems result in parameter dependent solutions. On the contrary, the FOC can directly control the current and therefore the input current of the motor contains fewer harmonics compared to the DTC technique. Lower harmonic content of the FOC leads to a better efficiency. However, the modulator stage adds to the signal processing time and therefore limits the level of torque response time possible from the PWM drive. Both torque control techniques have been implemented and tested. Summarizing, the DTC provides a better dynamic torque response whereas the FOC provides a better steady state behavior. 35 Table: 4.1: Comparison between FOC and DTC FOC DTC Transformation Present Void Dynamics High High Robustness Robust Robust Speed sensor Necessary Not Necessary Parameter sensitivity Big Average Control close Necessary PWM Not of PWM Decouplage Necessary Orientation Natural Torque regulator and Flux Three stator regulator Regulators regulator (hysteresis) Behavior down speed Good Not good 4.5 Field Oriented Control (FOC) Vector control (also called Field Oriented Control, FOC) is one method used in variable frequency drives to control the torque (and thus finally the speed) of three-phase AC electric motors by controlling the current fed to the machine The vector control philosophy started to be developed around 1970. Several types of vector control are possible: rotor-oriented, rotor-flux-oriented, stator-flux-oriented and magnetizing- flux-oriented. The final objective of the vector control philosophy is to be able to control the electromagnetic torque in a way equivalent to that of a separately excited dc machine: Field- oriented control enables control over both the excitation flux-linkage and the torque-producing current in a decoupled way. However, only the rotor-flux-oriented control yields complete decoupling. Choosing a different flux orientation may outweigh the lack of complete decoupling for some special applications. Here, only the rotor-flux-oriented type of control, also termed “Field-Oriented Control” (FOC), is considered. FOC can be implemented as indirect (feed-forward) or direct (feedback) depending on the method used for rotor flux identification. The direct FOC determines the orientation of the air-gap flux by use of a hall-effect sensor, search coil or other measurement techniques. However, using sensors is expensive because special modifications of the motor are required for placing the flux sensors. Furthermore, it is not possible to directly sense the rotor flux. Calculating the rotor flux from a directly sensed signal may result in inaccuracies at low speed due to the dominance of stator resistance voltage drop in the stator voltage equation and inaccuracies due to variations on flux level and temperature. 36 In case of induction machines, the indirect method is based on reconstruction (estimation) approaches employing terminal quantities such as voltage and currents in a motor model to calculate the flux position. Indirect FOC does not have inherent low-speed problems and is therefore preferred in most applications. The difference between rotor and flux speed depends on the slip frequency being almost proportional to the generated electromagnetic torque. In the case of PMSM, the rotor flux linkage is inevitably fixed to the rotor position. The goal of FOC is to maintain the amplitude of the rotor flux linkage Ψr at a fixed value, except for field-weakening operation or flux optimization, and only modify a torque-producing current component in order to control the torque of the ac machine. This control strategy is based on projections. Electromagnetic torque is produced by the interaction of stator flux linkages and stator currents (or rotor flux and rotor current), and can be expressed as a complex product of the flux and current space phasors. In order to gain a complete decoupling of torque and flux, the current phasor is transformed into two components of a rotating reference frame: A flux producing component id, aligned with the d-axis representing the direction of the rotor flux phasor, and a torque-producing component iq, aligned with the q-axis perpendicular to the rotor flux. In this way, a linear relation between torque and torque producing current is achieved and the torque in the ac machine could be expressed as Tel = c Ψr iq. Thus, the electromagnetic torque generated by the motor can be controlled by controlling the q- axis current. This is equivalent to the torque control of a separately excited dc machine. As shown later, the rotor flux can be controlled directly by controlling the d-axis current. FOC provides fast dynamic response due to the independent torque and flux control of the ac machine making the control accurate in every operation point (steady state and transient). 4.6 Working Principle of FOC The stator phase currents are measured and converted into a corresponding complex (space) vector. This current vector is then transformed to a coordinate system rotating with the rotor of the machine. For this the rotor position has to be known. Thus at least speed measurement is required the position can then be obtained by integrating the speed. Then the rotor flux linkage vector is estimated by multiplying the stator current vector with magnetizing inductance Lm and low-pass filtering the result with the rotor no-load time constant Lr/Rr, that is the ratio of the rotor inductance to rotor resistance. Using this rotor flux linkage vector the stator current vector is further transformed into a coordinate system where the real x-axis is aligned with the rotor flux linkage vector. Now the real x-axis component of the stator current vector in this rotor flux oriented coordinate system can be used to control the rotor flux linkage and the imaginary y-axis component can be used to control the motor torque. We also can explain in the following way: 37 The basic idea behind the field oriented control is to manage the interrelationship of the fluxes to avoid the issues mentioned above, and to squeeze out the most performance from the motor. To understand how this works, we first look at the structure of the motor. A three-phase motor incorporates windings that are displaced by 120 degrees (or a fraction of that) along the stator. Feeding the windings with three voltages separated in phase by one-third of a cycle produces a rotating magnetic field. A conceptual representation is shown in Fig. 4.2. Fig. 4.2: Generation of a rotating magnetic field and torque in an induction machine. The rotor in an induction machine consists of a closed circuit. Most often, a squirrel cage rotor is used, which has conductor bars shorted together with thick end rings. When the stator magnetic field sweeps the rotor, an emf is induced in the rotor circuit, and produces a current. The current produces its own magnetic field, and this induced magnetic field interacts with the stator magnetic field to producing mechanical force upon the rotor. 4.7 How FOC works FOC of permanent-magnet synchronous motors starts by measuring two of three phase currents, ia and ib. Only two current sensors are needed as third phase current is calculated from i a + ib + ic = 0. The Clarke Transform converts three-phase currents onto a two-axis plot to create variables Iα and Iβ. As viewed from the perspective of the stator, Iα and Iβ are time-varying quadrature- current values. The two-axis plot is rotated to align with the rotor flux using transformation angle θ. Angle θ is calculated during the last iteration of the control loop. This conversion provides i d and iq 38 variables from Iα and Iβ. The Park Transform aligns quadrature currents id and iq with the rotating plot. Both id and iq remain constant during steady-state conditions. id, iq and the reference values for each generate error signals. The id reference controls rotor- magnetizing flux while the iq reference governs motor torque output. These error signals are fed to proportional-integral (PI) controllers whose outputs are sent to the motor as voltage vectors Vd and Vq. Vd and Vq are rotated back into the stationary reference frame using the transformation angle to obtain quadrature voltage values, Vα and Vβ. Vα and Vβ are then mathematically transformed back into three phase-voltages Va, Vb, and Vc, which determine the new PWM duty-cycle. The new coordinate transformation angle is then estimated using Vα, Vβ, Iα, and Iβ as inputs. The new commutation angle guides the FOC algorithm in placing the next voltage vector. Of course, to determine the required phase voltages it’s necessary to know the current position of the rotor relative to the three phase windings. An encoder or resolver could supply that information, but at added cost and complexity. However, you can estimate the rotor position using motor currents and voltages. The FOC algorithm is made up of three operational states: motor stopped; open-loop FOC; and sensorless FOC. When the start button is pressed, the algorithm first enters the initialization state, where variables are set to initial values. To accurately estimate rotor position, the motor must run at a minimum speed to return a useful back-EMF value. An open-loop start-up procedure helps get the motor up to minimum speed. At motor start-up, sinusoidal voltages from the DSC spin the rotor. Current components for torque (iq) and flux production (id) are manipulated by the FOC based on the model only because there is not enough back-EMF at this point. Current components id and iq are controlled during motor start-up to keep torque constant. The start-up procedure accelerates the rotor by using an internal ramp function to increment angle θ every control cycle until the motor reaches minimum speed. Once at minimum speed, the speed controller is added to the execution thread and the algorithm switches over to sensorless FOC operation. Here the desired speed is continuously read from an external voltage reference to set motor rpm. Fault inputs, as well as the start/stop button, are monitored continuously. Any fault in the controller stops the motor and returns the algorithm to the “Motor Stopped” state. Many vendors offer FOC algorithms for free. Usually the source code is downloadable from the Web or comes packaged with a motor-control development board. Graphical-user interfaces in some development software present operation feedback for quick analysis and let engineers quickly change program variables for testing. 39 Fig. 4.3: Basic torque control scheme of FOC for ac motor drives. 4.8 Field Oriented Control of IPMSM 4.8.1 FOC of IPMSM without considering core loss According to the working principle, the constraint of FOC for IPMSM without considering core loss is Id 0 (4.1) According to the constraint of FOC of IPMSM, from equations (3.5),(3.6) and (3.7), we get d af (4.2) q Lq I q (4.3) af I q 3P Te (4.5) 2 2 Now from equation (3.3) and (3.4), we get 40 V d r q (4.6) Vq R s I q r af (4.7) From the above Eq. (4.5), it is seen that the torque can be controlled by changing the stator q- axis current since the d-axis flux is constant. From the desired torque we can calculate the desired stator q-axis current from Eq. (4.5) as follows: 2 2 Te Iq 3 P af (4.8) Using Eqs. (4.6) and (4.7), the desired stator d-axis and q-axis voltages can be calculated by using Eqs. (4.2), (4.3) and (4.8) in terms of desired stator d-axis and q-axis currents. However, to obtain the better performance in transient condition, we use two current controllers to obtain the desired stator voltages rather than to calculate from Eqs. (4.6) and (4.7). 4.8.2 Mathematical Expression of FOC of IPMSM with considering core loss 4.8.2.1 FOC with Id=0 According to the working principle of FOC, I md I cd (4.9) Te Pn [ af I mq ( Ld Lq ) I mq I md ] (4.10) Using Eqs. (3.23) and (3.24), the torque equation and the d-axis flux equation become: Te Pn [ af ( I q I cq ) ( Ld Lq )( I q I cq )( I cd )] (4.11) d Ld I cd af (4.12) It is seen from Eqs. (4.11) and (4.12), considering the core loss the d-axis flux is not constant since this term not only depends on the constant permanent magnet flux but also depends of the d-axis core-loss current and the torque cannot be control by varying only q-axis stator current. Moreover, the FOC becomes more complex. So, the precise flux and torque control is not possible by using the conventional FOC strategy that means by setting the stator d-axis current equal to zero. To simplify the FOC strategy, it is better to consider the d-axis magnetizing current equal zero, which is described in the next section. 41 4.8.2.2 FOC with Imd=0 In vector control, I md 0 (4.13) Lq I cd r I mq Rc (4.14) 1 I cq r af Rc (4.15) So from equation (3.5), we get d af (4.16) From Eq. (4.10) Te Pn [ af I mq ] (4.17) Rs Lq Vd r I mq r Lq imq Rc (4.18) Rs Vq Rs I mq r af r af Rc (4.19) From the above Eq. (4.17), it is seen that the torque can be controlled by changing the magnetizing q-axis current since the d-axis flux is constant. From the desired torque we can calculate the desired magnetizing q-axis current from Eq. (4.17) as follows: Te I mq Pn af (4.20) Using Eqs. (4.18) and (4.19), the desired stator d-axis and q-axis voltages can be calculated by using Eq. (4.20) in terms of desired magnetizing q-axis currents. However, to obtain the better performance in transient condition, we use two current controllers to obtain the desired stator voltages rather than to calculate from Eqs. (4.18) and (4.19). Using Eqs. (3.46) and (4.17), the mechanical dynamic expression of IPMSM with considering core loss can be written as follows: 42 d r Bm Pn2 P r af imq n TL (4.21) dt Jm Jm Jm Using Eq. (4.21), the speed controller can be designed. Let, Aω Bm / J m , Bω ( Pn2 / J m )af , BL ( Pn / J ) , then dωr Aωωr BωWΤ BLTL (4.22) dt The Laplace transformation of Eq. (4.22) can be written as follows: sωr ( s ) Aω ωr ( s ) Bω imq ( s ) B L TL ( s ) Bω BL ωr ( s ) imq ( s ) T (s) s Aω s Aω L Assuming that the change of load torque is a step function (i.e. the change of load torque is zero), the open loop transfer function between speed and imq is given as follows: ωr ( s) Bω (4.23) imq ( s) s Aω Using Eqs. (4.23), the speed controller can be designed. 4.9 Summary This chapter begins with the introduction of the control scheme of two major control techniques, FOC and DTC. Then we describe traditional scalar control limitation. Then we discuss the comparison of FOC and DTC. After that the whole details of FOC has been discussed. Then we showed mathematical representation of FOC control scheme. 43 Chapter 5 Controller Design Based on Continuous PI Controller 5.1 Introduction Many applications, such as robotics and factory automation, require precise control of speed and position. Speed Control Systems allow one to easily set and adjust the speed of a motor or machine. The control system consists of a speed feedback system, a motor/machine, an inverter, a controller and a speed setting device. A properly designed feedback controller makes the system insensible to disturbance and changes of the parameters. The purpose of a motor speed controller is to take a signal representing the demanded speed, and to drive a motor at that speed. Closed-loop speed control systems have fast response, but become expensive due to the need of feed back components such as speed sensors. The operation of the controller must be according to the speed range. For operation up to rated speed it will operate at constant torque region and for speeds above rated speed it will operate in flux-weakening region. 5.2 Different Types of Speed Control There are several types of speed controllers available but most common Industrial Automatic Controllers are: 1. Two position or on-off Controller 2. Proportional Controller (P) 3. Integral Controller (I) 4. Proportional Integral Controller (PI) 5. Proportional Derivative Controller (PD) 6. Proportional Integral derivative Controller (PID) 7. Integral Proportional Controller (IP) 5.3 Comparison of PI with other controllers The comparisons of PI controller with other controllers are given below: 1. The P controller shows a relatively high maximum overshoot Mp/(Kp >0), a long setting time as well as steady-state error (e∞). 2. The I controller has a higher maximum overshoot than the PI controller due to the slowly starting I behavior , but no steady-state error. 3. Like the P controller, the Proportional-Integral (PI) algorithm computes and transmits a controller output (CO) signal every sample time, T, to the final control element (e.g., valve, variable speed pump). The computed CO from the PI algorithm is influenced by the controller tuning parameters and the controller error, e (t). 4. Integral action enables PI controllers to eliminate offset, a major weakness of a P controller. 44 5. PI controller fuses the properties of the P and I controller. It shows the maximum overshoot and setting time similar to the P controller but no steady-state error. 6. The real PD controller has a smaller maximum overshoot due to the faster differential (D) action compared with the controller types mentioned in P, I and PI controller. Also in this case a steady-state error is visible, which is smaller than in the case of the P controller. 7. The PID controller fuses the properties of a PI and PD controller. It shows the maximum overshoot than the PD controller and has no steady-state error due to the integral (I) action. 8. PI controllers have two tuning parameters to adjust. While this makes them more challenging to tune than a P controller, they are not as complex as the three parameter PID controller. Comparing with other controllers it has been found that PI controller eliminates the problems regarding P and I controllers. The integral control mode is that it responses relatively slowly to an error signal and that it initially allows a large deviation at the instant the error is produced which can lead to system instability and cyclic operation. For this reason the integral control mode is normally used with another controller. PI control is not as complex as PID controller. Again PID and PD controller are not used in machineries. So for these reasons in this thesis PI controller was used as the speed controller. 5.4 Proportional Integral (PI) Controller The PI (proportional plus integral) controller function is the most frequently used controller function in practical applications. The PI controller stems from a PID controller with the D-term (derivative) deactivated. The D-term is often deactivated because it amplies random (high-frequent) measurement noise, causing abrupt variations in the control signal. The continuous-time PI controller function is as follows: K t u (t ) K c e(t ) c e( )d (5.1) Ti 0 Where, u is the control signal (the controller output), e = r - y is the control error, where r is the reference or set-point and y is the process output variable (process measurement), Kc is the controller gain, and Ti is the integral time. Kc and Ti are the controller parameters which are to be tuned. In most practical applications the continuous-time PI controller is implemented as a corresponding discrete-time algorithm based on a numerical approximation of the integral term. Typically, the sampling time of the discrete-time controller is so small - compared to the dynamics (response-time or time-constant) of the control system - that there is no significant difference between the behavior of the continuous-time PI controller and the discrete-time PI controller. 45 5.4.1 Function of Proportional Term As with the P controller, the proportional term of the PI controller, Kc·e(t), adds or subtracts from CObias based on the size of controller error e (t) at each time t. As e (t) grows or shrinks, the amount added to CObias grows or shrinks immediately and proportionately. The past history and current trajectory of the controller error have no influence on the proportional term computation. Fig. 5.1 illustrates this idea for a set point response. The error used in the proportional calculation is shown on the plot: At time t = 25 min, e (25) = 60–56 = 4 At time t = 40 min, e (40) = 60–62 = Fig. 5.1: Graphical representation of P controller Recalling that controller error e(t) = SP – PV, rather than viewing PV and SP as separate traces as we do above, we can compute and plot e (t) at each point in time t. Below, in Fig. 5.2 is the identical data to that above, only that it is recast as a plot of e (t) itself. It is to be noticed that in the plot above, PV = SP = 50 for the first 10 min, while in the error plot below, e (t) = 0 for the same time period. 46 Fig. 5.2: P controller error This plot is useful as it helps us to visualize how controller error continually changes size and sign as time passes. 5.4.2 Function of Integral term The major advantage of integral controller is that they have the unique ability to return the controlled variable back to the exact set point following a disturbance. While the proportional term considers the current size of e(t) only at the time of the controller calculation, the integral term considers the history of the error, or how long and how far the measured process variable has been from the set point over time. Integration is a continual summing. Integration of error over time means that we sum up the complete controller error history up to the present time, starting from when the controller was first switched to automatic. Controller error is e(t) = SP – PV. In the plot in Fig. 5.3, the integral sum of error is computed as the shaded areas between the SP and PV traces. Each box in the plot has an integral sum of 20 (2 high by 10 wide). If we count the number of boxes (including fractions of boxes) contained in the shaded areas, we can compute the integral sum of error. So when the PV first crosses the set point at around t = 32, the integral sum has grown to about 135. We write the integral term of the PI controller as: K c 32 Kc Ti 0 e(t )dt Ti (135) (5.2) 47 Fig. 5.3: Sum up of controller error Since it is controller error that drives the calculation, we get a direct view of the situation from a controller error plot as shown below in Fig. 5.4: Fig. 5.4: Integral summing up of error Note that the integral of each shaded portion has the same sign as the error. Since the integral sum starts accumulating when the controller is first put into automatic, the total integral sum grows as long as e(t) is positive and shrinks when it is negative. At time t = 60 min on the plots, the integral sum is 135 – 34 = 101. The response is largely settled out at t = 90 min and the integral sum is then 135 – 34 + 7 = 108. 48 5.4.3 Algorithm of PI controller While different vendors cast what is essentially the same algorithm in different forms, here we explore what is variously described as the dependent, ideal, continuous, position form: KI Ti co CObias K p e(t ) e(t )dt (5.3) where, CO = controller output signal CObias = controller bias or null value; set by bump less transfer as explained below e(t) = current controller error, defined as SP – PV SP = set point PV = measured process variable KI = controller gain, a tuning parameter Ti = reset time, a tuning parameter The PI controller has one outer speed loop. The speed error EN= (r * r ) between the reference speed NR and the actual speed N of the motor is fed to the PI controller and the Kp and KI are the proportional and integral gains of the PI controller. The output of the PI controller E1 acts as a current reference command to the motor, C1 is a simple proportional gain in the current loop and kch is the gain of the GTO thyristor chopper, which is used as the power controller. 5.5 Design PI controller to Control the Speed of IPMSM In our work, we have to design three controller to control the speed (outer loop) and the magnetizing d-axis current and q-axis current (inner loop) controllers. Using Eqs. (3.52), (3.58) and (4.23), the general form of plant equation can be written as follows: B a( s) u ( s) (5.4) s A Here, a is actual value and u is the output input of plant. u(t ) K P e(t ) K I e(t )dt (5.5) K K u(s) K P I [r (s) a(s)] K P I e( s) (5.6) s s Here, r is the reference value, Kp is proportional constant and KI is the integral constant. 49 5.5.1 Speed controller based on PI controller: According to Eqs. (5.4) and (5.5), the expression of PI speed controller is given as follows: K imq ( r r ) K P ( r r ) I * * * s (5.7) where, imq* = output of speed controller ωr* = reference speed ωr = actual value of speed of PMSM K Pω = proportional gain of speed controller KIω = integral gain of speed controller Fig. 5.5: Schematic of continuous time PI speed controller. The control objective is to make the motor speed follow the reference input speed change by designing an appropriate controller. The proportional-integral (PI) controller is used to reduce or eliminate the steady-state error between the measured motor speed ( r ) and the reference speed ( r *) to be tracked. When the state of the reference frame changes there is a high error in motor speed and for this reason there is a quicker response given by the PI controller which causes an overshoot. This overshoot hampers the efficiency as well as it effects the production. 5.5.2 d-axis magnetizing current controller based on PI controller: K Id u d (imd imd ) K Pd (imd imd ) * * s (5.8) where, ud = output of d-axis magnetizing current controller imd* = reference of d-axis magnetizing current imd = actual value of d –axis magnetizing current KPd = proportional gain of d-axis magnetizing current controller KId = integral gain of d-axis magnetizing current controller 50 The block diagram of d-axis magnetizing current controller based on PI controller is shown in Fig. 5.6. Fig. 5.6: Schematic of continuous time PI d-axis magnetizing current controller. 5.5.3 q-axis magnetizing current controller based on PI controller: K Iq u q (imq imq ) K Pq (imq imq ) * * s (5.9) Where, uq = output of q-axis magnetizing current controller * imq = reference of d-axis magnetizing current imq = actual value of q-axis magnetizing current KPq = proportional gain of q-axis magnetizing current controller KIq = integral gain of q-axis magnetizing current controller The block diagram of q-axis magnetizing current controller based on PI controller is shown in Fig. 5.7. Fig. 5.7: Schematic of continuous time PI q-axis magnetizing current controller. 5.5.4 Calculation of Desired Stator Voltages of IPMSM The d-axis and q-axis stator voltage can be calculated from the output of current controllers according to Eqs. (3.49) and (3.55) which is give in a block diagram as show in Fig. 5.8. 51 (a) d-axis stator voltage calculation. (b) q-axis stator voltage calculation. Fig. 5.8: Block diagram to calculate the stator voltages from the output of PI controller. 5.6 Summary This chapter discusses about the importance of precise control of speed and position of the machines in different industrial applications and the importance of precise and efficient speed controller. It then introduces different types of speed controlling systems emphasizing on proportional integral (PI) controller and illustrating its detailed working principle and functional parts along with necessary equations and algorithms. We also designed d-axis magnetizing current and q-axis magnetizing current controller. At the end of this chapter, we have shown how to calculate both d-axis and q-axis stator voltage circulation. 52 Chapter 6 Speed Control Based on Discrete Time PI Controller 6.1 Introduction In previous chapter we discussed about the details of Continuous Time PI controller. Now, as a part of our experiment, we are going to discuss about Discrete Time PI controller. PI controllers are universally known because of their flexibility combined with the relatively easy tuning. A routine is then presented that implements the discrete time representation of the PI controller. 6.2 PI/PID Controller Design from the Time domain Before discussing the design of digital control algorithms, let us consider discrete equivalents of analog controllers. Analog Control refers to the design and implementation of controllers in the continuous domain. This includes electronic controllers, which although discrete in nature, implements control by emulating the continuous nature of analog control strategies. A typical example is the electronic PI/PID algorithm. There are a number of ways by which this common and versatile controller can be implemented in discretised form. Consider the ideal PID controller written in the continuous time domain form: Kc t de(t ) u (t ) K c e(t ) e(t )dt K cTd dt u0 Ti 0 (6.1) To discretise the controller, we need to approximate the integral and the derivative terms to forms suitable for computation by a computer. From a purely numerical point of view, we can use: de(t ) e(t ) e(t 1) t t dt Ts and e(t )dt Ts e(i) 0 0 The discretised PID algorithm is therefore: K c Ts t K c Td (e(t ) e(t 1)) u (t ) K c e(t ) Ti e(i) i 0 Ts u0 (6.2) 53 which is now in the form of a difference equation, suitable for coding in an appropriate programming language. This particular form of the PID algorithm is known as the 'positional' PID controller, because the control signal is calculated with reference to a base level, uo. 6.3 Structures of Positional and Velocity PID Algorithms Although the structures of the positional and velocity PID algorithms appear very different, they are in fact related. From equation (6.3), time shifting back one sampling interval, we obtain K c Ts t 1 K c Td (e(t 1) e(t 2)) u (t 1) K c e(t 1) Ti e(i) i 0 Ts u0 (6.3) Subtracting this from the original, we end up with the velocity form, i.e. K cTs KT u(t ) u (t 1) K c [e(t ) e(t 1)] e(t ) c d [e(t ) 2e(t 1) e(t 2)] Ti i Ts (6.4) 6.4 Implementation and Performance of Discrete PID Controllers 6.4.1 Commissioning discrete PID Controllers To calculate control, the positional PID requires knowledge of uo, which is some steady-state control output level. This may have implications during commissioning. With the velocity form however, the previous control can be set to any reasonable arbitrary level, and hence commissioning is simpler. 6.4.2 Integral Windup A more important aspect is the use of a summation to calculate the contribution of the integral term. This can lead to problems leading to a phenomenon known as integral windup, causing long periods of overshoots in the controlled response. This can be caused by a poorly tuned controller or the controller output is overly constrained (for some safety reasons or inappropriately sized final control element say), or a combination of both factors. What happens is this. Say the controlled process has a positive gain and a positive set-point change occurs. The controller will then try to reduce the error between set-point and output, which is initially positive. The integral component will sum these positive errors to generate the necessary integral action. An overshoot occurs, whereupon the errors become negative. However, the direction of the control signal will not change to compensate if the sum of previously positive error dominates, in which case the overshoot becomes prolonged. The direction of control action will change only when the contribution of negative errors cancels the accumulated positive errors sufficiently. This phenomenon is known as integral windup or reset windup. 54 This phenomenon is illustrated in the Fig. 6.1 and 6.2 (a) Output without control constraints (b) Unconstrained control signal Fig. 6.1 The following set of plots show the effects of integral windup, when the output of the controller is subject to a constraint. 55 (a): Output with control constraints (b): Constrained control signal Fig. 6.2 With the velocity PID algorithm, because it does not make use of a sum of errors to generate integral action, the problem of integral windup will not occur, even if the control signal is constrained. 6.4.3 Controller tuning All the tuning procedures for continuous time PID controllers apply. These range from the empirical Ziegler-Nichol ultimate gain method, to the recipe based integral of error criteria. With recipe based methods, a factor of 0.5Ts is usually added to the process dead-time to account for the delay caused by the sampler. 56 6.4.4 Choice of Sampling Interval Another important aspect in sampled data control systems is the choice of sampling intervals. With electronic controllers that emulate continuous time algorithms, this choice is simple: sample as fast as possible. This is because of the approximations that are used to generate the difference equations describing the controllers. Smaller sampling intervals mean that the properties of the underlying controller design will be less distorted, hence more predictable and better performances. A good example is the discretised PID controllers. They perform best when sampling intervals are small. However, too fast a sampling is wasteful of resources. the cost of implementation will increase because more capable components must be installed a DCS typically has many hundreds of input-output channels to administer. The functioning of the DCS will degrade significantly if every control loop is to sample at the highest frequency possible. fast sampling intervals will mean that high frequency components such as noise will also be captured in the signal, and this is not necessarily beneficial to the performance of the control loop. If the sampling interval is too long, then signal loss will occur. An extreme case is the phenomenon known as 'aliasing'. Signal aliasing refers to the situation whereby the sampled versions of two very different signals are indistinguishable. As illustrated in the Fig. 6.3 below, the sampled representation of a step (dotted horizontal line) and a periodic wave are identical. Fig. 6.3: sample representation step and a periodic wave 57 Many rules-of-thumb regarding the choice for sampling times for different types of loops have been published, including the following recommendations: Flow loops 1s Level loops 5s Temperature Loops 30s to 10 mins These are however, merely rough guidelines. The choice of an appropriate sampling interval should be based on the dynamics of the process being controlled. The sampling operation must return the key dynamic characteristics of the process. From experience, a sampling interval of approximately 10% of the dominant time constant works well in practice. The position of the poles and zeros of the discrete transfer functions depend on the sampling interval used. Although this characteristic is not of great significance in PID type algorithms, it becomes important when discrete process models are used directly in the design of digital controllers. We shall cover this in more detail when we discuss model based digital controller design. 6.5 Design PI controller to Control the Speed of IPMSM Equation (5.4) can be written as follows: du(t ) de(t ) KP K I e(t ) (6.5) dt dt Here, e(t) = r(t) – a(t), r(t) is the reference signal and a(t) is the actual signal. Using Euler first approximation u (k ) e(k ) KP K I e( k ) Ts Ts u (k ) K P e(k ) K I Ts e(k ) u (k ) K P r (k ) K P a(k ) K I Ts e(k ) (6.6) If the reference signal is considered as a step function then r (k ) 0 u (k ) K I Ts e(k ) K P a(k ) (6.7) The present value of controller output can be written as u(k ) u (k ) u(k 1) (6.8) 58 6.5.1 Speed controller based on Discrete-Time PI controller: According to Eqs. (5.7) and (6.7), the expression of discrete-time PI speed controller is given as follows with schematic block diagram in Fig. 6.4. imq (k ) K I Ts [r (k ) r (k )] K P [r (k ) r (k 1)] * * (6.9) Fig. 6.4: Schematic of discrete time PI speed controller 6.5.2 d-axis magnetizing current controller based on Discrete-Time PI controller: The mathematical expression of discrete-time PI d-axis magnetizing current controller is given as follows with schematic block diagram in Fig. 6.5. ud (k ) K IdTs [imd (k ) imd (k )] K Pd [imd (k ) imd (k 1)] * (6.10) Fig. 6.5: Schematic of discrete time PI d-axis magnetizing current controller 59 6.5.3 q-axis magnetizing current controller based on Discrete-Time PI controller: The mathematical expression of discrete-time PI q-axis magnetizing current controller is given as follows with schematic block diagram in Fig. 6.6. uq (k ) K IqTs [imq (k ) imq (k )] K Pq [imq (k ) imq (k 1)] * (6.11) Fig. 6.6: Schematic of discrete time PI q-axis magnetizing current controller The d-axis and q-axis stator voltage can be calculated from the output of current controllers according to Eqs. (3.49) and (3.55). 6.6 Summary This chapter discusses the details of Discrete-Time PI Control. It then introduces with structures of positional and velocity PID algorithms, implementation and performance of Discrete PID Controllers, details of sampling interval. Then we designed speed controller, d-axis and q-axis magnetising current controller with detailed working principle and functional parts along with necessary equations and algorithms. 60 Chapter 7 Simulation and Results 7.1 Introduction In Chapter 2, 4, 5 and 6, the PMSM, FOC, continuous PI based speed control and discrete time PI based speed control systems were explained respectively in detail. In this chapter the simulation models of FOC of IPMSM with considering core loss programmed in Matlab/Simulink has been discussed. The simulation models are presented and the results are discussed. The PMSM parameters used in this project are given in Table 7.1. These parameters were taken from reference [13]. Based on the proposed model and the parameters, the dynamic performance of the Interior Permanent Magnet Synchronous Motor Field oriented control system with known initial values is simulated. The simulation works has been done for different operating conditions of IPMSM with considering core loss. Table 7.1.The list of motor parameters used in the simulation [9] Stator resistance, Rs 5.8 Ω Core loss resistance, Rc 100 Ω Rated speed, Nm 1500 rpm Rotor flux constant, λaf 0.533 WB Motor Moment of Inertia, J 0.00529 kg-m2 Sampling Time, St 0.00100 Number of pole pairs, Pn 2 Mechanical torque, Tm 6 Nm d - axis stator inductance, Ld 44.8e-03 q – axis stator inductance, Lq 102.7e-03 7.2 Matlab/Simulink Model for IPMSM with Considering Core Loss The IPMSM motor with considering core loss is modeled in MATLAB/Simulink using the modeling equations 3.19 to 3.46 derived in chapter 3. The overall model which is used in Matlab/Simlink is shown in Fig. 7.1. 61 Fig. 7.1: Overall Simulation Model of IPMSM Fig. 7.2: Subsystem of PMSM Fig. 7.2 shows subsystem to calculate the reference speed in rad/sec from revolution per minute. 62 Fig. 7.3: Controller Subsystem of IPMSM Fig. 7.3 shows the internal modelling of the controller subsystem of Fig. 7.1. Fig. 7.4: Speed Controller (continuous) Fig. 7.4 shows the simulink model of speed controller based on Continuous-Time PI system. This model is made by using the equation (5.7). Meaning of parameters are listed below: imqr = output of speed controller Wm* = reference speed Wm = actual value of speed of PMSM KPw = proportional gain of speed controller KIw = integral gain of speed controller 63 Fig. 7.5: Speed Controller (discrete) Fig. 7.5 shows the simulink model of speed controller based on Discrete-Time PI system. This model is made by using the equation (6.9). Meaning of parameters are listed below: imqr = output of speed controller Wm* = reference speed Wm = actual value of speed of PMSM KPw = proportional gain of speed controller KIw = integral gain of speed controller st = sampling time 1/z = delay Fig. 7.6: q-axis current controller (continuous) Fig. 7.6 shows the simulink model of q-axis current controller based on Continuous-Time PI system. This model is made by using the equation (5.9). Meaning of parameters are listed below: uq = output of q-axis magnetizing current controller imqr = reference of q-axis magnetizing current imq = actual value of q-axis magnetizing current KIq = integral gain of q-axis magnetizing current controller KPq = proportional gain of q-axis magnetizing current controller 64 Fig. 7.7: q-axis current controller (discrete) Fig. 7.7 shows the simulink model of q-axis current controller based on Discrete-Time PI system. This model is made by using the equation (6.11). Meaning of parameters are listed below: uq = output of q-axis magnetizing current controller imqr = reference of q-axis magnetizing current imq = actual value of q-axis magnetizing current KIq = integral gain of q-axis magnetizing current controller KPq = proportional gain of q-axis magnetizing current controller st = sampling time 1/z = delay Fig. 7.8: d-axis current controller (continuous) Fig. 7.8 shows the simulink model of d-axis current controller based on Continuous-Time PI system. This model is made by using the equations (5.8). Meaning of parameters are listed below: 65 ud = output of d-axis magnetizing current controller imqr = reference of d-axis magnetizing current imd = actual value of d-axis magnetizing current KId = integral gain of d-axis magnetizing current controller KPd = proportional gain of d-axis magnetizing current controller Fig. 7.9: d-axis current controller (discrete) Fig. 7.9 shows the simulink model of d-axis current controller based on Discrete-Time PI system. This model is made by using the equations (6.10). Meaning of parameters are listed below: ud = output of d-axis magnetizing current controller imqr = reference of d-axis magnetizing current imd = actual value of d-axis magnetizing current KId = integral gain of d-axis magnetizing current controller KPd = proportional gain of d-axis magnetizing current controller st = sampling time 1/z = delay 7.3 Simulation Results Based on Continuous-Time PI Controller In our simulation the speed is stepped up from 500 rpm to 1000 rpm at 1.0 second and the load torque is stepped up from 50% to 100% of its rated value at 3 second. The gain of PI controller to control the d-axis magnetizing current are: KPd = 8.4195 , KId = 1.0665e+03 . And q-axis magnetizing current are: KPq = 26.7970 , KIq = 2.4448 . The following figures are shown for transient responses of speed and torque for three different values of gain of PI speed controller. Value-1: we have considered the following values KPw = 0.09 KIw = 0.0001 66 Using these values we have found following response Fig. 7.10: performance in Continuous-Time PI Controller In the above result (Fig. 7.10), there is no overshotting, no steadystate error and actual speed never match to reference speed. This is the main problem of this case. This is not excepted. Value-2: we consider the following values KPw = 0.09 KIw = 0.1 Using these values we have found following response Fig. 7.11: performance in Continuous-Time PI Controller 67 In the above result (Fig. 7.11), there is slight overshotting, no steadystate error and actual speed matches to reference speed very slowly. This is also not excepted. Value-3: we got the following values from Matlab Simulation KPw = 0.0744 KIw = 0.5583 Using these values we have found following response Fig. 7.12: performance in Continuous-Time PI Controller In the above result (Fig. 7.12:), there is a significant overshotting, no steadystate error but actual speed matches to reference speed in a short period of time. This is also not excepted due to overshotting. To overcome the above problem we consider Discrete PI Controller 7.4 Simulation Results Based on Discrete-Time PI Controller In our simulation the speed is stepped up from 500 rpm to 1000 rpm at 1.0 second and again the speed is stepped down from 1000 rpm to 500 rpm at 5.0 second. And the load torque is stepped up from 50% (3 Nm) to 100% (6 Nm) of its rated value at 3 second. 68 Fig. 7.13: performance in Discrete-Time PI Controller In the above result (Fig. 7.13), there is no overshotting, no steadystate error and actual speed matches to reference speed in no time 7.4.1 Change in magnetising current Fig. 7.14: d-axis and q-axis magnetising current 69 From Fig. 7.14 and 7.15, we can verify result as we considered Imd=0. This circumstance matches with simulation. Fig. 7.15: magnetising current Fig. 7.16: close view of magnetising current 7.4.2 Change in stator voltage and stator current In Fig. 7.17 and 7.19, we have shown graphical representation of stator voltage and stator current 70 Fig. 7.17: graphical representation of stator voltage Fig. 7.18: close view of stator voltage Fig. 7.19: graphical representation of stator current 71 Fig. 7.20: close view of stator current 7.5 Summary This chapter it illustrated the total block Simulink model PMSM as well as the Simulink model blocks of the subsystem like the magnetising current controller and speed controller. In every cases the transient responses has been considered. This simulation results are then analyzed to evaluate the performance of Field Oriented Controlled method of PMSM. 72 Chapter 8 Discussions and Conclusions 8.1 Discussion Our project is a complete organized simulation work on speed control of permanent magnet synchronous motor using field oriented control method. In our work we have first studied in detail about permanent magnet synchronous motor, learned about their features, characteristics and operation principle. We emphasized mostly on the special features of the interior permanent magnet synchronous motor and its advantages over commonly used induction motor that influenced our choice of the machine as the medium of implementation of FOC. Next we studied about the mathematical model of PMSM, from where we used different equations for voltage, current, torque, flux and speed equations that were later used in simulation processes. There are two different cases (a) without considering core-loss and (b) considering core-loss. For the case of without considering core-loss, significant error occurs. To optimize this we choose the case which is considering core-loss. For speed controlling of PMSM, different speed controlling methods were studied in detail at first; principally their working functions, special features, equations and compatibility. Emphasis was given to the Field Oriented Control method and its advantages over other controlling method which influenced our choice for FOC. The next part of our work involved the detailed study of different speed controller and magnetizing current controller which were necessary to improve the performance of PMSM operation system. The functions and characteristics of those speed controllers such as P, I, PID and PI (both Continuous and Discrete) were studied in detail and their performances were compared. Finally Continuous-Time PI controller and Discrete-Time PI controller were chosen for simulation purpose. The last part of our work was the analysis of different simulation results of different controlling processes. The results were obtained by varying different parameters and there by optimized parameter values were determined. The simulation results were thoroughly analyzed and conclusions were drawn on the basis of these results. Finally some discussions were given on the limitations of our work and further improvement on our work. 73 8.2 Limitations Our project is mainly simulation based work instead of practical implementation due to the cost effect of PMSM. In actual implementation no machine is 100% efficient It is also hard to maintain a constant difference between the actual and reference values. 8.3 Future Works Further development of our work can be done on several sectors such as: 1. Developing better controlling technique with more accurate output using fuzzy logic. 2. Further development can be done using this model with Modified Discrete-Time PI Controller. 3. Error can be reduced by further works on hysteresis comparator. 4. 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