# POLITECNICO DI MILANO

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1 POLITECNICO DI MILANO Scuola di Ingegneria Industriale e dell Informazione Corso di Laurea Magistrale in Ingegneria Elettrica DOUBLE STAR PM MACHINE: Mathematical Modelling, Simulation And Control Supervisor: Prof. Franceso Castelli Dezza Prof. Marco Mauri By: Arslan Shabbir Matriculation#: Anno Accademico

2 Abstract: The thesis work presented deals with electrical machine with phases greater than 3 and this thesis is presented especially with the approach to mathematically model a Double Star Electrical Motor. Double star machine has two sets of the phase winding that are mechanically displaced with thirty degrees. In the current era the development in power electronics technology let us to develop more and more multi-phase machine owing to their great benefits in great power supply with a relative less machine size. This thesis work is on a double star permanent synchronous motor. The thesis first part is about the introduction of multiphase machine, why they are becoming more and more given emphasis. The second part is about the step by step approach to mathematically model the double permanent magnet synchronous motor. The modular mathematically modelling approach can be applied to other multiphase machine then the one being considered in this project. The mathematical modelling involves many machine concept, along with linear algebra, since the double star permanent magnet synchronous motor has magnetic coupling between two winding sets and the rotor. The model also refers to a 9x9 induction matrix. Later the closed loop form of double star Permanent Magnet Sychronous Machine ( PMSM) is formed so that the mathematical model can be simulated. The simulation of this model is done in Simulink. Many custom build level-2 sfunction are being used in order to realize the mathematical model being presented in this thesis. Later the PWM control strategy is presented for this double star PMSM that deals both with the starting of motor and then during normal torque operation.

3 Summary Generally saying multiphase machine and in particular double star electrical machine are of increasing importance in today electrical drive system as far as high power and reliability of electrical machine operation is concerned. a. Power Rating Enhancement; More than three phase Invertor model: We can think of an inverter-fed: increasing its power rating above certain limits, which exceed the maximum capability of a single inverter unit, necessarily implies using more than one inverter and a natural consequence of such choice is that the machine stator needs to be equipped with more than three phases b. Reliability & Fault Tolerance: For increasing number of phase then three, the fault tolerance becomes an important requirement and economic/safety consideration. Summarizing, double star machine are of increasing importance because they exhibit various advantages over counter 6 phase machine like: Less torque ripple and improved performance Very redundant electrical machine structure that can operate even in case of one faulty converter More power rating of electrical drives because of using two power invertors. Thus, have capability to imply two power supplies. Usage of three phase invertor modules then six phase inverters Chapter 1 is a brief introduction of multiphase machine general capabilities. The chapter relies mostly on the sizing equations of the stator and rotor winding. Also in this chapter an introduction is being drawn why multiphase machine got more and more importance in present time especially the main advantages of using such technology is being elaborated. Chapter 2 is the introduction of the double star motor that is going to be mathematically modelled, simulated and controlled in the later part of thesis. This chapter

4 also illustrates some applicative cases of double star machine from automobile, rail traction, ship to aircraft applications. Chapter 3 is based on the mathematical modelling of double star synchronous motor. A step by step approached is made. The model takes the basic machine equations and then use particular Park s Transformation, inductive matrix transformation and formation to the machine closed loop form. This close loop form is very important if we have to run this mathematical model in the simulation environment. Chapter 4 is based on the development of the control strategy of this double star machine. The machine is being controlled with PWM technique owing to the development of speed and torque control. Chapter 5 the simulation work is being presented and explained. Th machine data of a typical 100KW electrical machine is taken. Simulation environment used is Simulink in which many custom build level 2 s-functions are being used owing to the complexity of mathematical model. At the end an appendix, abbreviations and references is given.

5 Table of Contents List of Figures MULTIPHASE ELECTRICAL MACHINE HISTORICAL DEVELOPMENT OF MULTIPHASE MACHINE AND POWER INVERTORS CONSTRUCTION OF MULTI STAR MACHINE: Sizing Equations for Multi Star Machine INTRODUCTION TO DOUBLE STAR MACHINE THE CASE OF A 2MW ALTERNATOR FOR SHIPBOARD POWER GENERATION DOUBLE STAR PMSM APPLICATION IS AEROSPACE FOR BETTER FAULT TOLERENCE MATHEMATICAL MODELLING OF DOUBLE STAR PMSM ROTOR CONFIGURATIONS MACHINE VECTOR SPACE MODELLING ASSUMPTIONS/HYPOTHESIS: STEP BY STEP MATHEMATICAL MODELLING APPROACH: Step1:Vector Space Model Explanation of θr and β Generic dq0 Modeling of Double Star Machine: Direct Axis inductance (Ld) Quadrature Axis inductance (Lq) Finding Ldouble(l) Finding Ldouble(ag) Electromagentic Torque Calculation & Forming Close Loop Equation: CONTROL STRATEGY QUADRATURE AXIS CURRENT CONTROL STRATEGY Block Diagram to Frame Control Strategy...55

6 4.2 FINDING K3 AND K4 PARAMETERS Simulink Block Diagram For iq current Regulation Pulse Width Modulation Of VSI SIMULATION OF MATHEMATICAL MODEL IN MATLAB/SIMULINK MACHINE SIMULATONS WITHOUT ELECTRICAL DRIVE Torque speed Curve for 5Nm, 50Nm, 100Nm, 200Nm: Working of double star under one Fautly Three phase Current waveforms Current Torque Speed Curve At Various Loading Condition Double Star PMSM Simulation with Different Machine Data MACHINE SIMULATONS WITH ELECTRICAL DRIVE MODEL PULSE WIDTH Modulation In the Drive Model...81 APPENDIX...86 CALCULATION OF DIRECT AXIS INDUCTANCE (Ld), QUADRATURE AXIS INDUCTANCE (Lq) AND ROTOR INDUCTANCE (Lf)...86 SYMBOLS AND ABBREVIATIONS...89 REFERENCES...93

7 List of Figures Figure 1.1: Phase arrangement scheme for multi star machine with N stator windings displace by 2 π/n electrical degrees apart Figure 2.1 Schematic Generation view of a 2MW alternator for shipboard power generation Figure 2.3: Output DC Current Figure 2.4: Output Electromagnetic Torque Figure 3.1: Double Star Motor Stator and Rotor Winding Configuration Figure 3.2: Double Star PMSM Source Voltage and Loading Schematic Configuration Figure 3.2: Magnetic Material B Vs H Curve Figure 3.3: Working Point of Magnetic Hard Material Figure 3.4: Magnetic Material Maximum Energy Product Figure: 3.5 Double Star PMSM Vector Space Model Configuration Figure 3.6: 9x9 [L dq0 ] matrix obtained after simulation. Figure 4.1: Block Diagrm To Frame For Control Strategy On Quadrature Axis Current Figure 4.2: Equivalent dq axis representation of quadrature axis circuit of Stator Winding Figure 4.3: Bode plot of Open Transfer Function of B(s) Figure 4.4: Phase Margin Evaluation by plot K 3 and K 4 pu plot with phase margin Figure 4.5: Open Loop and Cloop Transfer Functions For the q-axis circuit Figure 4.6: Nyquist Plot to Check The Stability Criterian of Close Loop Transfer Function Figure 4.7: Bllock Diagram For iq Current Regulation Figure 4.8: Figure4.9: Switching Logic For Voltage Source Inverter Voltage Source Inverter Block Diagram

9 Figure 5.19: Pulse width modulation that controlled VSI do for the first 3 seconds (Machine reach the steady state speed) Figure 5.20Pulse width modulation that controlled VSI do for the first 10 seconds Figure 5.21 Pulse width modulation that controlled VSI do for the first 0-25 seconds

10 1.0 MULTIPHASE ELECTRICAL MACHINE Multiphase machines are getting increasingly common in electric power industry. The current growth in electrical energy consumption and high power electrical applications led to the use of multiphase machines for power segmentation. The main advantage we get by such machine is in increase of the frequency of torque ripples which can be more easily filtered by the mechanical load. In addition, the multiplication of the number of phases offers an increased reliability and allows to operate with one or more phases in default. Also, the recent progresses in the areas of power electronics and digital control have allowed the development of variable speed drives with multiphase machines. However, these machines present some control difficulties because of their strongly coupled nonlinear models, and certain quantities are not accessible or measurable. Thus, some control techniques such as field-oriented control (FOC) or direct torque control (DTC) requires knowledge of the magnetic (rotor or stator) flux. The flux can t be measured unless a flux sensor is mounted inside the machine. A brief history of multiphase machine is presented along with the developments that makes their use possible in modern electrical systems. The applications of this kind of machines are: locomotive traction, industrial high-power applications, electric and hybrid-electric vehicles (propulsion, integrated starter/alternator concept, and others), the concept of the moreelectric aircraft and wind turbine generation. Their main benefits in high power electrical system such as less per phase rating and better sinusoidal profile of air-gap Flux density is in particular explained at the end of this chapter.

11 1.1 Historical Development of Multiphase Machine And Power Invertors From the late nineteenth century through the middle of the twentieth century, DC-to-AC power conversion was accomplished using rotary converters or motor-generator sets (M-G sets). In the early twentieth century, vacuum tubes and gas filled tubes began to be used as switches in inverter circuits. So during that time multiphase machine is not beneficial at all and even the concept of the machine is missing. The origins of electromechanical inverters explain the source of the term inverter. Early AC-to-DC converters used an induction or synchronous AC motor direct-connected to a generator (dynamo) so that the generator's commutator reversed its connections at exactly the right moments to produce DC. A later development is the synchronous converter, in which the motor and generator windings are combined into one armature, with slip rings at one end and a commutator at the other and only one field frame. The result with either is AC-in, DC-out. With an M-G set, the DC can be considered to be separately generated from the AC; with a synchronous converter, in a certain sense it can be considered to be "mechanically rectified AC". Given the right auxiliary and control equipment, an M-G set or rotary converter can be "run backwards", converting DC to AC. Hence an inverter is an inverted converter Since early transistors were not available with sufficient voltage and current ratings for most inverter applications, it was the 1957 introduction of the thyristor or silicon-controlled rectifier (SCR) that initiated the transition to solid state inverter circuits. And the history of multiphase machine starts from 1960 but the problem arises due to six step mode of three phase inverter. The machine gets an attraction since with n phase machine the lowest frequency of torque ripple is of order 2n+1. Thus the concept of more phases and lest torque ripple and better efficiency of the machine was evolved. Efforts were made in the development of five phase and six-phase variable-speed drives supplied from both voltage source and current source inverters. The other main historical reasons for early developments of multiphase drives, are better fault tolerance and the possibility of splitting the motor power (current) across a higher number of phases. By increasing the number of phases it is also possible to increase the torque per rms

12 ampere for the same volume machine. Improvement of noise characteristics and reducing the stator copper loss are other advantages of multi-phase systems. This advantage of multiphase machines is nowadays somewhat less important since pulse width modulation (PWM) of voltage source inverters (VSIs) enables control of the inverter output voltage harmonic content. 12-pulse line-commutated inverter circuit with commutation by SCR let us to realize Pulse width Modulation. The commutation requirements of SCRs are a key consideration in SCR circuit designs. SCRs do not turn off or commutate automatically when the gate control signal is shut off. They only turn off when the forward current is reduced to below the minimum holding current, which varies with each kind of SCR, through some external process. For SCRs connected to an AC power source, commutation occurs naturally every time the polarity of the source voltage reverses. SCRs connected to a DC power source usually require a means of forced commutation that forces the current to zero when commutation is required. The least complicated SCR circuits employ natural commutation rather than forced commutation. With the addition of forced commutation circuits, SCRs have been used in the types of inverter circuits described above. Since IGBT are available in higher voltage and current ratings they have become the preferred switching components for use in inverter circuits as they can be turned off by means of control signals. However, for the multiphase machine the converter ratings cannot be increased over a certain range due to the limitation on the power rating of semiconductor devices. One solution to this problem is use multi star machines can be used as an alternative to multiphase machine thus multi-level converters. In multi star machines, by dividing the required power between multiple phases, more than the conventional three, higher power levels can be obtained and power electronic converters with limited power range can be used to drive the multi-phase machine. Whether it is better to use multi star machines or multi-level converters is debatable and in fact it is extremely application dependent. Insulation level is one of the limiting factors that can prohibit the use of high voltage systems. Therefore, multi star machines that employ converters operating at lower voltage level are preferred. The main benefit for the increasing use of multi star machine is detailed as:

13 a) Less per phase rating. Suppose, we have to supply Power P Megawatts to a Load. The three phase power supply will supply P/3 Megawatts to each winding, so we have same P/3 rating for each invertor phase. Under greater P the invertor rating becomes inadequate. Such greater load supply can be achieved by increasing the per phase invertor rating or by implying multiple phase machine. Conventionally multiphase machine available are 5 phase, 6 phase, 12 phase and so on. Another way of lowering per phase load power is to employ multiple (N) three phase machine. In this research thesis, dual star (N=2) machine is used. One of the main benefit of employing dual star rather than traditional six phase machine is in: Invertor Designing: Two Three Phase Invertor units are used in Dual Star three phase Machine whereas, for conventional six phase machine we have to use six phase invertors. Less torque ripple (is explained later in detail in simulation result) Improved power rating of components because of less transients in current and voltage. Increased Reliability: Machine operation under complete failure of one of the two 3-phase winding. i.e. in case of one faulty phase, the machine would can continue its normal operation. Whereas, the control strategy and operating parameters of another phase are not required to be changed. But the operation is degraded with reduce power. Proposal: Increasing the number of star further increase the faulty tolerance and reliability of operation with less degradation of output power rating. Note: In literature; N=2 (tells about star configuration) n=3 ( 3 phase machine). E.g. N=4 and n=5 would be 4 star 5-phase electrical machine. Concluding, different multiple star configuration with different multiple phase winding scheme can be used in order to achieve different loading condition, machine sizing/dimension and invertor rating.

14 b) Air-gap Flux density with better sinusoidal Profile: We shall have better air gap flux density because of mutual cancelation of different harmonic rotating fields by different phase sets in dual star machine. This can be elaborated through simulation results that air-gap flux density occupies almost a sinusoidal profile. Better sinusoidal flux density profile leads to less flux pulsation in rotor circuit that implies reduce eddy current losses. (very important consideration in high speed electrical machine because with speed, eddy current losses tend to grow) Better torque profile (with reduce amplitude but increased frequency of harmonic) due to flux harmonic profile owing to double star winding topology. 1.2 Construction of Multi Star Machine: The available types of multi star machines are in principle the same as conventional electrical machine. There are induction and synchronous multi star machines, where a synchronous machine. They have design of distributed stator winding that gives near-sinusoidal MMF distribution and supplied with sinusoidal currents the exception is the permanent magnet synchronous machine with trapezoidal flux distribution and rectangular stator current supply, known as brushless DC machine, or simply BDCM) Nevertheless, spatial MMF distribution is never perfectly sinusoidal and some spatial harmonics are inevitably present. A stator winding can be designed to yield either near-sinusoidal or quasi-rectangular MMF distribution, by using distributed or concentrated windings, for all ac machine types. Near sinusoidal MMF distribution requires use of more than one slot per pole per phase. As the number of phases increases it becomes progressively difficult to realize a near-sinusoidal MMF distribution. For example, a five-phase four-pole machine requires a minimum of 40 slots for this purpose, while in a seven-phase four-pole machines at least 56 slots are needed (for a threephase four-pole machine the minimum number of slots is only 24). In both stator winding designs, there is a strong magnetic coupling between the stator phases. If the machine is a permanent magnet synchronous machine, then concentrated winding design yields a behavior similar to a BDCM. A permanent magnet multiphase synchronous machine can also be of so called modular design where an attempt is made to minimize the coupling between stator phases (a three-phase permanent magnet machine may be designed in

15 the same manner, but the most important benefit of modular design, fault tolerance, is then not exploited to the full extent). It should be noted that the spatial flux distribution in permanent magnet synchronous machines is determined by the shaping of the magnets. Stator Winding Stator windings of an n-phase machine can be designed in such a way that the spatial displacement between any two consecutive stator phases equals α=2 π/n, in which case a symmetrical multiphase machine results. This will always be the case if the number of phases is an odd prime number. However, if the number of phases is an even number or an odd number that is not a prime number, stator winding may be realized in a different manner, as k windings having i sub phases each (where n=i k ). In such a case, the spatial displacement between the first phases of the two consecutive sub phase windings is α=π/n, leading to an asymmetrical distribution of magnetic winding axes in the cross section of the machine (asymmetrical multiphase machines). Figure 1.1: Phase arrangement scheme for multi star machine with N stator windings displace by 2 π/n electrical degrees apart

16 1.2.1 Sizing Equations for Multi Star Machine Although characterized by a variety of possible phase arrangements, multi-phase windings can be treated in the same way from machine sizing viewpoint, under the only hypothesis that each pole span encompasses exactly as many phase belts as the phases are (n). This hypothesis is verified in the vast majority of multi-phase designs; only those designs are not covered where successive phases are shifted by 360n electrical (1.1) where ΦP is the flux per pole, Ns the number of series-connected turns per phase, Kw the winding factor and f the rated frequency. Quantities ΦP, Ns and f can in turn be expressed as follows: (1.2) Where, the number of turns in the stator winding can be expressed as: (1.3) And the electrical frequency can be expressed as following with the mechnical speed and pole number (1.4) In terms of: the average flux density Bg in the air-gap, the useful core length L, the machine average diameter D at the air-gap, the number of pole pairs np, the number of slots per pole per phase q (possibly fractional), the number of series-connected turns per coil Nt, the number of parallel paths per phase b and the speed nrpm in revolutions per minute. A further design figure, called the output coefficient C, can be also introduced to describe the degree of utilization of the machine volume (roughly proportional to D2L ) in terms of useful machine torque (proportional to P nrpm, where P is the rated active power):

17 (1.5) Substitiuting equations from 1.1 to 1.5 we get the following machine sizing expression (1.6) Where the slot pitch expression is given as: (1.7) Substituting the slot pitch expression into the voltage sizing expression: (1.8) The slot pitch can be alternatively expressed in terms of current density σs and the electric loading λs: (1.9) Where σs and λs is given as following: (1.10) (1.11)

18 Where, As and Kf are the cross section area and the filling factor of a stator slot. Thus, finally we have the following sizing equation: (1.12) where k is a non-dimensional constant whose value only depends on the units used to express the other quantities. Coefficient in the first brackets does not depend on the winding structure, but only on the magnetic, thermal and electrical loading of the machine; therefore, for machines of homogeneous design in terms of thermal class, insulation technology, cooling system effectiveness, etc. The above sizing equations draw the relationship between the following design quantities: machine power (P) and voltage (V) ratings; winding structure in terms of slot cross-section area (As), number of turns per coil (Nt), number of phases (n) and number of parallel ways per phase (b). This sizing equation also tells if the power rating P increases while the voltage V below a certain level, this naturally leads to decrease the number of turns per coil Nt, which may result in the need for: increasing the slot cross-section area As; increasing of the number b of parallel ways per phase; the increase of the number of phases n.

19 2.0 INTRODUCTION TO DOUBLE STAR MACHINE Electromagnetic design and Dimensioning of double star machine obeys typical Electrical Machine Construction equations and activities; for an ordinary 6 phase equally distributed wound machine. The main consideration of Double Star Machine is not in Dimensioning or Design but rather in the analysis of how machine behaves under: Steady State, Transients (Machine startup, External Loading ) or some abnormal conditions. The electrical construction and sizing equations of a typical electrical machine is expressed in the previous chapters; whereas the mathematical modelling and control strategy of such machine will be expressed in the later part of the thesis. This chapter is based on applications of some applications of double star Motor to draw a picture that where in Electrical Power system this electrical entity fits. 2.1 The case of a 2MW alternator for shipboard power generation On-board ship power generation is one of the most promising contexts for developing new generation of isolated DC distribution networks. This trend is particularly spread in the military area, as many of the most important navies in the world are fostering the development of Medium Voltage DC Integrated Power Systems (MVDC IPS). In this scenario, the Italian Navy, through the General Direction of Naval Armaments and Weapons (NAVARM), awarded a contract for the realization of the so called Naval Package, a demonstrator of an innovative generation system for naval ships. The realization of the NP is part of an extended program of Italian Navy aimed at obtaining experimental results in the field of shipboard MVDC IPS for naval applications. The NP project involved the realization of 2-MW generation system prototypes suitable for supplying a 3000 V shipboard DC network using a 6300-rpm gas turbine (GT) as the prime mover (Fig. 3-1). This 6300 rpm alternator coupled to the GT through a gear-box. The main characteristic data of this wound rotor double star motor is given in the following page :

20 Alternator rotor type Number of alternator phases GT/ alternator coupling Wound rotor 2x3 Epicycloidal gearbox Number of alternator poles 4 Alternator speed Alternator frequency Rectifying power electronics 6300 rpm 210 Hz Diode bridges The schematic of generation system and the the component is given as: Figure 2.1 Schematic Generation view of a 2MW alternator for shipboard power generation

21 The stator winding of the generator have a multiphase arrangement consisting of 2 threephase sets connected to power-electronics rectifier. The 2 rectifiers are connected in cascade and their series closed on the DC load. The generator design is substantially like that of large turbo-alternators, with a solid-steel round rotor on whose surface slots are milled to accommodate concentric excitation coils. The main challenge in such generator development is scaling down a turbo-alternator design (tailored on very large machine sizes, with rotor lengths of up to several meters) to the relatively compact dimensions of a generator whose external frame does not exceed 1.5 meters of length. The use of a solid steel rotor technology with end-winding retaining ring was dictated by the high rotor peripheral speed (machine rated speed is 6300 rpm), which discouraged the use of a salient-pole design, although this would be much more usual for the machine dimensions. The stator has two three-phase windings, displaced by 30 electrical degrees apart, each connected to a diode rectifier. The reasons for using a dual star design instead of a single threephase one relate to performance and fault tolerance; on the other side, the reason for not exceeding number of stator than two was to avoid excessive complications, size and cost increases due to the need for additional cablings, power electronics and auxiliary equipment. The advantages offered by the dual star arrangement compared to the ordinary three phase design in terms of performance and fault tolerance can be understood by comparing the voltage and current waveforms obtained for the system configurations of the dual star healthy configuration and the dual star configuration with a faulty diode with the single star (threephase) healthy configuration and the single star configuration with a faulty diode. In the following page the simulation results from the Prof. Alberto research the graphs are presented. That graphs cleatly expicts abouts the faulty rectifier waveform and electromagnetic waveforms in a simple three phase machine and the doubel star machine. The doubel star motor gives less ripple in the output torque wave forms.

22 Figure 2.3: Output DC Current Figure 2.4: Output Electromagnetic Torque

23 Similar considerations can be made about the rotor field current, whose ripple reflects the magnitude of the harmonic rotating fields which originate in machine air-gap. The smaller field ripple amplitude observed in the case of the dual star generator is the consequence of the better air-gap field harmonic content which characterizes this configuration, where no 5th and 7th order rotating fields are present. 2.2 Double Star PMSM Application is Aerospace For Better Fault Tolerance In most applications, the failure of a drive has a serious effect on the operation of the system. In some cases, the failure results in lost production whereas in some others it is very dangerous to human safety. Therefore, in life dependent application it is of major importance to use a drive which continues operating safely under occurrence of a fault. The major faults which can occur within a machine or converter are considered as: winding open circuit winding short circuit (phase to ground or within a phase) winding short circuit at the terminals power device open circuit, power device short circuit and DC link capacitor failure. In order to limit the short circuit current, the machine should have a sufficiently large phase inductance and in order to avoid loss of performance in healthy phases in faulty condition, mutual inductance between the phases should be small. These two points are required for the reliability of the system. With the increasing development of aircraft, new more electric and all-electric aircraft in the aviation sector have attracted increasingly more attention. The typical characteristic of the more-electric or all-electric aircraft is that part or all pneumatic and hydraulic systems are replaced by electrical drive systems, which can reduce the running cost, aircraft s volume and weight, and fuel cost and improve the reliability and maintainability of the aircraft. It is estimated that the weight and fuel cost of an aircraft can be reduced by 10% and 9% adopting more-electric and all-electrical aircrafts, respectively. More-electric aircraft is a transitional scheme from

24 conventional aircraft to all-electric aircraft. At present, the mainly more-electric aircraft includes the European Airbus A380, American Boeing B787 and Lockheed Martin F 35. The current research shows that the design of the electrical drive system is one of the key technologies of more-electric aircraft. Compared with the conventional hydraulic fuel pump system, the electrical drive system can not only improve the system efficiency and the flexibility of variable speed control but also reduce the weight and volume of the system in the aircraft fuel pump system.

25 3.0 MATHEMATICAL MODELLING OF DOUBLE STAR PMSM The modeling of Double star machine is done with a quite traditional method of dq0 transformation of already employed by Nelson and Krause in the 1970s. Vector-Space Decompostion (VSD) method is used for modeling machine variables under idealized conditions. Goal: To extend the theory of well known Park s dq0 theory for double star electrical machine modeling. The aim is to find an explicit impression of dual star machine in dq0 transformed coordinates that is given as function of physical inductances (computed either by experimental procedure or by modeling the machine by Finite element analysis). Then the Electromagentic Torque is calculated and close loop equation is formed. Double Star arrangement consist of splitting the stator winding into two identical sections displaced by 30 electrical degrees apart. Figure 3.1: Double Star Motor Stator and Rotor Winding Configuration As previously stated, the benefit of using such split configuaration is that the 2 winding sections can be supplied by the same already experimentally/commercially working three-phase invertors

26 Figure 3.2: Double Star PMSM Source Voltage and Loading Schematic Configuration 3.1 Rotor configurations The rotor is the same as its three phase counterparts and it is made with permanent magnets, in particular interior permanent magnets (IPM) and surface permanent magnets (SPM) manufacture techniques are used. The rotor configuration is better explained on the basis of the magnetic material of which it is constructed. Some of the magnetic being used for the manufacturing of rotors alongwith their properties is listed in the following table: Table 3.1: Properties of Some Hard Magnetic Materials

27 Further the materials operating characteristic can be be referred to the following flux density characteristic: Figure 3.2: Magnetic Material B Vs H Curve Note: We work in the second quadrant since the external magnetic field try to demagnetize the magnetic material. The working point of a hard material is defined as following: Figure 3.3: Working Point of Magnetic Hard Material

28 Considering the above operating condition of a magnetic material we define the magnetic voltage law in the air gas as following: H m h m = H g h g = Ug Whereas, the flux balance can be defined by the following equation: B m A m = B g A g = φg The above equations can be combined as: B m H m A m h m = B g 2 μ0 A g g = U g φg becomes By defining Vm = Am x hm and Vg = Ag x g the PM and air gap volume respectively, equation Ultimately the magnetic volume required can be given as: The minus sign in equation occurs because the PM working magnetic strength Hm is negative, due to the operation in the 2nd quadrant. Equation 3.6 shows that, if the air gap volume Vg and flux density Bg are defined, the PM volume Vm is in inverse proportion with respect to the product Bmx Hm: this quantity is an energy per unit volume W/m3, and for this reason it is called energy product. Thus, it seems to be advisable that the PM working point be in that portion of the characteristic where the product Bmx Hm assumes its maximum value, since in this point (with fixed value of Bg, Hg, g, Ag) the PM volume has a minimum. In a similar way, since the energy stored in the air gap is expressed by: 3.0.6

29 If the PM volume is defined, the air gap energy is the higher, the higher is the energy product Bmx Hm. But we must investigate where is the point with maximum value of the product Bmx Hm. Figure 3.5 shows the behavior of the product Bmx Hm as a function of B. Usually, on the B H plane, together with the demagnetization curves, also the hyperbola loci of constant Bmx Hm are reported; this allows to read the Bmx Hm value in each point, and in particular its maximum value (Bmx Hm )max. The same figure shows that the point with (Bmx Hm )max is on the knee of the PM curve (this usually occurs for materials which have a sharp knee). In applications where, during the operation, the PM is subjected to varying demagnetizing fields, and/or the geometrical sizes of the circuit change, and/or the temperature varies, the PM working point can move, and may position itself below the knee, causing the PM irreversible demagnetization. Therefore, as regards the PMs whose knee is in the 2nd quadrant, it is advisable to make them work not in the point (Bmx Hm )max, but in a point shifted on the right, in order to avoid the cited demagnetization risk. Figure 3.4: Magnetic Material Maximum Energy Product PM motors are broadly classified by the direction of the field flux. The first field flux classification is radial field motor meaning that the flux is along the radius of the motor. The second is axial field motor meaning that the flux is perpendicular to the radius of the motor. Radial field flux is most commonly used in motors and axial field flux have become a topic of interest for study and used in a few applications.

30 PM motors are classified on the basis of the flux density distribution and the shape of current excitation. They are PMSM and PM brushless motors (BDCM). The PMSM has a sinusoidal-shaped back EMF and is designed to develop sinusoidal back EMF waveforms. They have the following features: Sinusoidal distribution of magnet flux in the air gap Sinusoidal current waveforms Sinusoidal distribution of stator conductors. BDCM has a trapezoidal-shaped back EMF and is designed to develop trapezoidal back EMF waveforms. They have the following features: Rectangular distribution of magnet flux in the air gap Rectangular current waveform In PM machines, the magnets can be placed in two different ways on the rotor. Depending on the placement they are called either as surface permanent magnet motor or interior permanent magnet motor. Surface mounted 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 configuration 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 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.

31 3.2 Machine Vector Space Modelling The vector space modelling of the phase variable model can be represented by the following basic vector notation( V = R s I+p φ ). where is R s is the stator winding resistance matrix, p is the time derivative operator, and φ is the flux linkage vector. 3.3 Assumptions/Hypothesis: a) Uniform Air-gap b) Sinusoidal Winding Distribution c) Neglection of space harmonics in the air gap due to winding distribution d) Space harmonics in air gap due to rotor saliency are neglected. e) Leakage flux distribution is independent of rotor position f) Magnetic saturation effect is neglected g) Both stator winding structures are considered identical This vector space model can be expressed by the following diagram: q axis θr θ a 1 s d axis

32 Note: Propose Modeling/Future Work: VSD method extension for machine topology about nonuniform air-gap and non-sinusoidal winding distribution evaluated through finite element analysis. 3.4 Step By Step Mathematical Modelling Approach: The mathematical modeling starts with the dq0 transformation to decouple the stator circuit form the rotor one. So, the modelling goes on with following steps: Step1: To represent two three phase voltage/current set into rotating space vectors plus homopolar component. Thus we have two space vectors projected on a couple of dq axis that is elaborated in the figure 3.1. Three phase voltages in dq0 aixs: V d1 V 1 = [ V q1 ] = T 1 [ V 01 V a1 V b1 V c1 ] = T 1 V abc1 (3.1.1a) V d2 V 2 = [ V q2 ] = T 2 [ V 02 V a2 V b2 V c2 ] = T 2 V abc2 (3.1.1c) V rd 0 V r = [ V rq ] = T r [ 0 ] = T r V rabc V 0r V f (3.1.1c) Three phase currents in dq0 aixs: i d1 i 1 = [ i q1 ] = T 1 [ i 01 i a1 i b1 i c1 ] = T 1 i abc1 (3.1.2a) i d2 i 2 = [ i q2 ] = T 2 [ i 02 i a2 i b2 i c2 ] = T 2 i abc2 (3.1.2b)

33 i rd 0 i r = [ i rq ] = T r [ 0] = T r i rabc i r0 i f (3.1.2c) Where, T 1 = ( 2 cos (θ r ) sin(θ r ) 0 1 1/2 1/2 3 ) [ sin(θ r ) cos(θ r ) 0] [ 0 3/2 3/2] / 2 1/ 2 1/ 2 (3.1.3a) T 2 = ( 2 cos (θ r α) sin(θ r α) 0 1 1/2 1/2 3 ) [ sin(θ r α) cos(θ r α) 0] [ 0 3/2 3/2] (3.1.3b) / 2 1/ 2 1/ 2 T r = ( 2 cos (β) sin(β) 0 1 1/2 1/2 3 ) [ sin(β) cos(β) 0] [ 0 3/2 3/2] (3.1.3c) / 2 1/ 2 1/ 2

34 3.4.2 Explanation of θ r and β q axis θr θ a 1 s d axis Figure: 3.5 Double Star PMSM Vector Space Model Configuration Here a 1s axis & a 2s axis are rotating with θ r such that for a 1s axis is aligned with dq axis while a 2s axis is shifted by α degrees. θ r is mathematically expressed as: t θ r = ω(γ)dγ + θ r (0) 0 And a r axis is moving in reference to dq axis by velocity (ω- ω m ) leading to angle β that is mathematically given as: t β = (ω(γ) ω m )dγ + θ r (0) 0 + θ m (0) Further we define the derivatives that are used in future calculations:

35 T 1 = ω ( 2 sin (θ r ) cos(θ r ) 0 1 1/2 1/2 3 ) [ cos(θ r ) sin(θ r ) 0] [ 0 3/2 3/2] (3.1.4a) / 2 1/ 2 1/ 2 T 2 = ω ( 2 sin (θ r α) cos(θ r α) 0 1 1/2 1/2 3 ) [ cos(θ r α) sin(θ r α) 0] [ 0 3/2 3/2] (3.1.4b) / 2 1/ 2 1/ 2 T r = (ω ω m ) ( 2 sin (β) cos(β) 0 1 1/2 1/2 3 ) [ cos(β) sin(β) 0] [ 0 3/2 3/2] (3.1.4c) / 2 1/ 2 1/ 2 expressed as: Generic dq0 Modelling of Double Star Machine: Three phase voltages present on stator winding and rotor can be mathematically V abc1 R [ V abc2 ] = [ 0 3 R 0 3 ] [ V rabc R r i abc1 i abc2 i rabc ] [ φ 1abc φ 2abc] (3.1.5) φ rabc Pre-Matrix multiplication equation with already defined transformation in equation (3.1.1a, 3.1.1b, 3.1.1c) to get dq0 voltage sets, i.e.: T [ T ] [ T r V abc1 V abc2 T R i abc1 T ] = [ 0 3 T ] [ 0 3 R 0 3 ] [ i abc2 ] [ 0 3 T ] [ V rabc T r R r i rabc T r Simplifying right hand side of equation 6; implies: V 1 T R i abc1 T [ V 2 ] = [ 0 3 T ] [ 0 3 R 0 3 ] [ i abc2 ] [ 0 3 T ] [ V r T r R r i rabc T r φ 1abc φ 2abc] (3.1.6) φ rabc φ 1abc φ 2abc] (3.1.7) φ rabc expression: i abc1 Finding [ i abc2 ] from equation 3.1.2a, 3.1.2b, 3.1.2c result in following mathematical i rabc 1 i abc1 T 1 [ i abc2 ] = [ i rabc T T r i 1 ] [ i 2 Substituting equation 8 into equation 7: ir ] (3.1.8)

36 V 1 T R [ V 2 ] = [ 0 3 T ] [ 0 3 R 0 3 ] [ V r T r R r T T T r T ] [ 0 3 T ] [ ir T r i 1 ] [ i 2 φ 1abc φ 2abc] (3.1.9) φ rabc Consider the following expression: T R [ 0 3 T ] [ 0 3 R 0 3 ] [ T r R r T T T r R Matrix rule, that ABA 1 =A if B is a diagonal Matrix. Since, [ 0 3 R 0 3 ] is a diagonal R r matrix, implies following simplification: T R [ 0 3 T ] [ 0 3 R 0 3 ] [ T r R r T T T r ] R ] = [ 0 3 R R r ] (3.1.10) Substituting equation 10 into equation 9, V 1 R i 1 T [ V 2 ] = [ 0 3 R 0 3 ] [ i 2 ] [ 0 3 T ] [ V r R r ir T r Or: φ 1abc φ 2abc φ rabc V 1 R i 1 T 1 φ 1abc [ V 2 ] = [ 0 3 R 0 3 ] [ i 2 ] [ T 2 φ 2abc] (3.1.11) V r R r ir T r φ rabc ] Now, in order to simplify [ φ 1abc φ 2abc φ rabc ] we first define the dq0 flux of the machine, i.e. φ 1 T φ 1abc T 1 φ 1abc [ φ 2 ] = [ 0 3 T ] [ φ 2abc ] = [ T 2 φ 2abc ] (3.1.12) φ r T r φ rabc T r φrabc Now considering flux only due to 1 set of stator three phase winding: φ 1 = T 1 φ 1abc φ 1 = d/dt(t 1 φ 1abc ) = T 1 φ 1abc + T 1 φ 1abc

37 i.e T 1 φ 1abc = φ 1 T 1 φ 1abc (3.1.13a) Similarly, T 2 φ 2abc = φ 2 T 2 φ 2abc T r φ rabc = φ r T r φ rabc Substituting a, b, c into equation : (3.1.13b) (3.1.13c) V 1 R [ V 2 ] = [ 0 3 R 0 3 ] [ V r R r φ 1 T 1 φ 1abc ] [ φ 2 T 2 φ 2abc ] ir φ r T r φ rabc i 1 i 2 Simplifying here the more generic model to a more particular double star electrical motor generic model: Or: V 1 R [ V 2 ] = [ 0 3 R 0 3 ] [ V r R r φ 1 T 1 φ 1abc ] [ φ 2 T 2 φ 2abc ] ir φ r T r φ rabc i 1 i 2 V 1 R i 1 φ 1 [ V 2 ] = [ 0 3 R 0 3 ] [ i 2 ] [ φ 2 V r R r ir φ r T 1 φ 1abc ] + [ T 2 φ 2abc ] (3.1.14) T r φ rabc From equation substitute the value of φ 1abc, φ 2abc, φ rabc into equation 14 to get the phasor flux equation into dq0 coordinates, i.e. : V 1 R i 1 φ 1 [ V 2 ] = [ 0 3 R 0 3 ] [ i 2 ] [ φ 2 V r R r ir φ r T 1 T 1 1 φ 1 ] + [ T 2 T 1 2 φ 2 ] (3.1.15) T r T 1 r φ r Again Considering equation , φ 1 T φ 1abc [ φ 2 ] = [ 0 3 T ] [ φ 2abc ] φ r T r φ rabc

38 i.e. : T = [ 0 3 T ] [L] [ T r T ] = [ 0 3 T ] [L] [ i abcr T r i 1abc i 2abc φ 1 T [ φ 2 ] = [ 0 3 T ] [L] [ φ r T r T 1 1 T T T r T T r i 1 ] [ i 2 i 1 ] [ i 2 ir ] ir ] (3.1.16) Where, [L] here is a 9x9 matrix, that can be mathematically expressed as: L 1a,1a L 1a,1b L 1a,1c L 1a,2a L 1a,2b L 1a,2c L 1a,ra L 1a,rb L 1a,rc L 1b,1a L 1b,1b L 1b,1c L 1b,2a L 1b,2b L 1b,2c L 1b,ra L 1b,rb L 1b,rc L 1c,1a L 1c,1b L 1c,1c L 1c,2a L 1c,2b L 1c,2c L 1c,ra L 1c,rb L 1c,rc L 2a,1a L 2a,1b L 2a,1c L 2a,2a L 2a,2b L 2a,2c L 2a,ra L 2a,rb L 2a,rc L 2b,1a L 2b,1b L 2b,1c L 2b,2a L 2b,2b L 2b,2c L 2b,ra L 2b,rb L 2b,rc L 2c,1a L 2c,1b L 2c,1c L 2c,2a L 2c,2b L 2c,2c L 2c,ra L 2c,rb L 2c,rc L ra,1a L ra,1b L ra,1c L ra,2a L ra,2b L ra,2c L ra,ra L ra,rb L ra,rc L rb,1a L rb,1b L rb,1c L rb,2a L rb,2b L rb,2c L rb,ra L rb,rb L rb,rc [ L rc,1a L rc,1b L rc,1c L rc,2a L rc,2b L rc,2c L rc,ra L rc,rb L rc,rc ] Here the script used in inductance defining is as: L 1a,1a means the inductance between winding set 1 phase a and winding set 1 phase a, i.e. self-inductance. L 1a,2a means the inductance between winding set 1 phase a and winding set 2 phase a L 1a,ra means the inductance between winding set 1 phase a and rotor winding phase a a hypothetical one. The 9x9 [L] matrix can be expressed in 3x3 matrix with each matrix element being a 3x3 matrix. Such defining helps to handle the[l] more easily: Where, L 11 L 12 L 1r [L] = [ L 21 L 22 L 2r ] (3.1.17) L r1 L r2 L rr L 11 is the 3x3 inductance matrix between stator 3phase winding 1. L 22 is the 3x3 inductance matrix between stator 3phase winding 2. L 1r is the 3x3 inductance matrix between stator 3phase winding 1 and rotor

39 L 2r is the 3x3 inductance matrix between stator 3phase winding 2 and rotor L double = [ L 11 L 12 L 21 L 22 ] (3.1.18) Further we find L 1r, L rr & L double with different approaches. Here, referring to appendix a particular electrical machine case that deals with how to find values of L f, L d, L q and L m. These variables are found by approach given in book Electromechanical Energy Conversion with Dynamics of Machines by R.D. Begamudre. The approach deals with finding the d axis and q aixs reactance on the basis of a particular synchronous machine data. The data required is: Inside diameter of armature bore (D) Length of air gap (g) Net iron length in armature (L) Number of slots per pole per phase (q) Ratio of pole arc/pole pitch Machine operating data Slot pitch angle (γ) Distribution factor (k d = sin ( qγ )(q sin 2 (γ)) 2 Coil span angle (τ) Coil span factor (k c = sin ( τ 2 )) Turns in series per phase of armature (N ph ) Number of turns in field winding (N f ) Fundamental component of flux crossing air gap; produced per ampere of field current ( φ 1 ) Total flux linkage of field coil due to ref. current (φ f ) Unit permeance of air gap ( μ 0 (P 0 + P 2 cosθ) )

40 Synchronous machine details from the book Electromechanical Energy Conversion with Dynamics of Machines Flux density in air gap because of Mmf of field windings (N f ): B = N f μ 0 /g Fundamental component of flux density (B 1 )under the consideration that flux density is uniform over the pole arc and zero beyond: π/2 B 1 = 4/π sinθdθ Flux per pole crossing the air gap and linking the armature phase is expressed as: φ f = φ 1 + φ 1f flux in armature φ 1 = B av area under one pole = 2 π B 1 πdl P where, (φ 1f = 15% φ 1 )i.e. percentage of fundamental Resulting voltage generated in armature phase will be given as: E 0 = 4.44fk c k d N ph φ 1 Self-Inductance of Field Winding (L f = N f φ f ) Direct Axis inductance (L d ) (3.1.19a) Direct Axis inductance is described in terms of direct axis armature reactance (X d = ωl d ). The armature reactance is because of fundamental component of flux crossing the air gap in the direct axis exactly across the polar axis. Armature mmf centered on direct axis from the book Electromechanical Energy Conversion with Dynamics of Machines

41 Under fictious 1A current through phase a the mmf of one concentrated short-pitch coil will be rectangular and of magnitude: 2N eq = 2N c k c N c = N ph /(P q) (N c is number of turns per coil) The mmf because of all coil under one pole: F a = 2N ph k c k d /P Fundament component of one pole mmf: F 1ph = 4 π F a Amplitude of the fundamental of the rotating mmf: F 1 = 3 2 F 1ph = 6 2 π Air gap Flux Density: B(θ) = F 1cosθ (P g 0 + P 2 cos2θ) μ 0 Air gap Flux Density Fundamental Component: B 1 (θ) = F 1 g μ 0(P 0 + P 2 2 )cosθ Flux Crossing the airgap: φ ad = 2 π B 1 πdl Armature reactance in direct axis: X ad = φ ad φ 1 E 0 Where, E 0 = 4.44fk c k d N ph φ 1 P N ph P k ck d Direct axis synchronous reactance: X d = 1. 15X ad (3.1.19b) Since, X d = X ad + X l and X l is the leakage flux that does not flow in the main magnetic circuit crossing the airgap Quadrature Axis inductance (L q ) Direct Axis inductance is described in terms of quadrature axis armature reactance (X q = ωl q ). The armature reactance is because of fundamental component of flux crossing the air gap in the quadrature axis exactly across the inter-polar axis. Armature mmf centered on quadrature axis (inter-polar axis)

43 L double = [ L 11 L 12 L 21 L 22 ] The inductance between stator winding can be defined in terms of Leakage inductance and air gap inductances, i.e. : L double = L double (l) + L double (ag) (3.1.21) Where, L double (l) is the leakage inductance of stator winding L double (ag) is the mutual inductance between two phases located at different phase angles Finding L double (l) In order to find L double (l) we use the conventional winding scheme and then define the mapping function that will lead to work between the two winding sets: Winding schematic and defining the mapping function W 2x3 Mapping a dual star winding into a conventional 6 phase winding scheme

44 Stator phase variables of double star machine (considering symmetry): y A1 y A2 y B1 y 2xABC = y B2 y C1 [ y C2 ] Stator phase variables of conventional 6 phase machine (considering symmetry) : y c = y 0 y 1 y 2 y 3 y 4 [ y 5 ] By consider the winding scheme layout in consideration the star phase variable of double star machine can be mapped onto stator phase variable of conventional 6 phase machine if pre multiplied by W 2x3. i.e. : y c = W 2x3 y 2xABC Here the function W 2x3 can be easily evaluated and mathematically expressed as: W 2x3 = (3.1.22) [ ] This mapping function W 2x3, we shall use later to map the inductance matrix of conventional 6 phase machine to double star machine. The main reason of working with conventional 6 phase machine is high degree of symmetry in conventional winding scheme that ultimately helps to model double star machine leakage inductance matrix more easily. Leakage inductance matrix of a conventional 6 phase Machine: The leakage inductance matrix of a conventional 6 phase machine can be expressed into following expression: L (l) c,6 = l 0 l 1 l 2 0 l 2 l 1 l 1 l 0 l 1 l 2 0 l 2 l 2 l 1 l 0 l 1 l l 2 l 1 l 0 l 1 l 2 l 2 0 l 2 l 1 l 0 l 1 [ l 1 l 2 0 l 2 l 1 l 0 ] (3.1.23)

45 The equation 23 can be validated by experimental setup and also by theory regarding the symmetrical structure of conventional 6 phase machine. Supportive theoretical viewpoint for equation * 1. The assumption that all phases are geometrically identical and that their leakage inductance does not depend on the rotor position necessarily implies that the mutual leakage inductance depends only on their mutual displacements. This accounts for the elements on each diagonal (representing the mutual inductances between equally-distanced phases) to be equal. 2. Furthermore, let us take a generic index j with 1 j n 1 and consider phase j and phase n j. Looking at the conventional phase arrangement (Fig. 14-3), it is immediately seen that phases j and n j have the same displacement from phase 0 but opposite conventional directions. This implies that their mutual inductances with respect to phase 0 have equal magnitude and opposite sign, that is: Experimental Validation setup: In this experimental setup, rotor is removed and one phase of double star is excited and voltage is measure in all the other windings and the leakage inductance is measured for consequent windings by the following expression: l excitation phase,j = E k 2πfL Where, j can be of any winding ( ) but not of the excitation phase. After finding L (l) c,6 of equation from three parameters l 0, l 1, l 2 we would be able to find the L double (l) by employing the mapping function W 2x3 with the following mathematical expression: (l) (l) L c,6 = t W2x3 L double W2x3 i.e. : L (l) double = W t (l) 2x3 L c,6w2x3 (3.1.24) *Above stated comments are taken from Phd Thesis MODELING AND ANALYSIS OF MULTIPHASE ELECTRIC MACHINES FOR HIGH-POWER APPLICATIONS by Alberto Tessarolo

46 3.4.7 Finding L double (ag) If the effects of magnetic saturation are neglected and space harmonics in the air gap due to winding distribution are neglected, the magnetizing term takes the following well known form: Magnetizing Inductance= L d+l q 2 Where, L d is direst axis inductance L q is quadrature axis inductance cos(φ Ѱ) + L d L q cos (2(θ 2 r φ+ѱ )) 2 Two phases are located at angle θ = φ and θ = Ѱ θ r is the rotor reference angle The reference to the above general expression can be found from the Phd Thesis MODELING AND ANALYSIS OF MULTIPHASE ELECTRIC MACHINES FOR HIGH-POWER APPLICATIONS by Alberto Tessarolo Conclusively, taken into account the above well known form for magnetizing inductance in air gap, the L double (ag) takes the following mathematically expression: L double (ag) = W 2x3 t ( L d+l q 2 + L d-l q 2 1 cos(α) cos(2α) cos(3α) cos(4α) cos(5α) cos(5α) 1 cos(α) cos(2α) cos(3α) cos(4α) cos(4α) cos(5α) 1 cos(α) cos(2α) cos(3α) cos(3α) cos(4α) cos(5α) 1 cos(α) cos(2α) cos(2α) cos(3α) cos(4α) cos(5α) 1 cos(α) [ cos(α) cos(2α) cos(3α) cos(4α) cos(5α) 1 ] cos(2θ r ) cos(2θ r -α) cos(2θ r -2α) cos(2θ r -3α) cos(2θ r -4α) cos(2θ r -5α) cos(2θ r -α) cos(2θ r -2α) cos(2θ r -3α) cos(2θ r -4α) cos(2θ r -5α) cos(2θ r -6α) cos(2θ r -2α) cos(2θ r -3α) cos(2θ r -4α) cos(2θ r -5α) cos(2θ r -6α) cos(2θ r -7α) cos(2θ r -3α) cos(2θ r -4α) cos(2θ r -5α) cos(2θ r -6α) cos(2θ r -7α) cos(2θ r -8α) cos(2θ r -4α) cos(2θ r -5α) cos(2θ r -6α) cos(2θ r -7α) cos(2θ r -8α) cos(2θ r -9α) [ cos(2θ r -5α) cos(2θ r -6α) cos(2θ r -7α) cos(2θ r -8α) cos(2θ r -9α) cos(2θ r -10α)] )W 2x3 (3.1.24) Finally, from equation and we get the value of L double (l) and L double (ag) to evaluate the value of L double by the following expression: L double = L double (l) + L double (ag)

47 Finally to get L dq0double from L double first we define the function T 2x3 using the T 1 and T 2 functions from equation 3.1.3a and 3.1.3b respectively. So, T 2x3 = [ T T 2 ] (3.1.25) L dq0double can be mathematically expressed as: L dq0double = T 2x3 L double T 2x3 1 (3.1.26) Again considering equation 7, L 11 L 12 L 1r [L] = [ L 21 L r1 L 22 L r2 L 2r ] L rr We define the function T 3x3 using the T 1, T 2 and T 3 functions from equation 3a, 3b and 3c respectively. (An alternative way to find all the inductance in a conventional winding scheme inductance matrix is to use Finite Element Analysis to simulate the machine structure. Then, apply mapping function W 2x3 and transfer function T 3x3 to find [L dq0 ] ). T T 3x3 = [ T ] T r [L dq0 ] = T 3x3 [L]T 3x3 1 (3.1.27)

48 Example of [L dq0 ] found after Simulating: Machine Data: L0= L1=0.025 L2=0.01 Ld=0.48 Lq=0.2 Lf=0.01 r= J=0.91 B=0.01 Machine operated with 3 phase voltage with peak amplitude 400V and frequency 50Hz Figure 3.6: 9x9 [L dq0 ] matrix obtained after simulation.

49 Considering equation 16 again: φ 1 T [ φ 2 ] = [ 0 3 T ] [L] [ φ r T r T 1 1 And considering equation : T T r i 1 ] [ i 2 i 1 i 2 ] = [L dq0 ] [ ] (3.1.28) ir ir V 1 R i 1 φ 1 [ V 2 ] = [ 0 3 R 0 3 ] [ i 2 ] [ φ 2 V r R r ir φ r T 1 T 1 1 φ 1 ] + [ T 2 T 1 2 φ 2 ] T r T 1 r φ r Apply some fundamental matrix multiplication property on equation 15, V 1 R i 1 φ 1 [ V 2 ] = [ 0 3 R 0 3 ] [ i 2 ] [ φ 2 V r R r ir φ r 1 T 1 T 1 ] + [ T 2 T T r T r φ 1 ] [ φ 2 φ r ] (3.1.29) φ 1 Substituting the value of [ φ 2 ] from equation into equation , φ r V 1 R i 1 i 1 [ V 2 ] = [ 0 3 R 0 3 ] [ i 2 ] [L dq0 ] [ i 2 V r R r ir i r 1 T 1 T 1 ] + [ T 2 T T r T r i 1 ] [L dq0 ] [ i 2 (3.1.30) ir ] Here, R = r [ 0 1 0] where, r is one stator phase winding resistance R r = [ r d r q r f ] where, r f is the field circuit resistance

50 Since, i r current has no significant dq current and is supplied only by a constant current 0 of dc value i f ; i.e i r = [ 0]. If we carefully consider the following 9x9 matrix i f 0 Here, the lines underline with red line are multiplied with 0 of i r = [ 0]. So, in equation 30 for i f computation of V 1, V 2 they are independent from rotor machine data. Which leads to decoupling of stator circuit from that of rotor. Thus, finally we are able to write equation 30 in terms of following two decoupled equations: [ V 1 V 2 ] = [ R R ] [i 1 i2 ] L dq0double [ i 1 i 2 And ] + [ T 1 1 T T 2 T ] L dq0double [ i 1 ] (3.1.31a) i2 2 [V r ] = [R r ][i r ] [L dq0rr ][i r ] + [T r T r 1 ][L dq0 ][i r ] (3.1.31b) Equation 31a and 32b is simulated in Simulink/Matlab to calculate V 1, V 2, i 1, i 2, V r and i r.

51 3.4.8 Electromagnetic Torque Calculation & Forming Close Loop Equation: Electromagnetic power consumed in winding is described by following equation: P e = i s t V s The proof of above expression can be given as: = i dq t V dq (if T t T = 1) i dq t V dq = (Ti s ) t (TV s ) = i s t T t TV s Thus considering, P e = i dq t V dq Where, V dq = [ V 1 ] = [ R 0 i 1 3 V R ] [i 1 ] -L i2 dq0double [ i 2 Where, e dq is the rotor induced emf on the stator And ] + [ T -1 1*T T 2*T ] L dq0double [ i 1 ] e i2 dq 2 P e = i dq t V dq = i dq t [ R R ] [i 1 i2 ] - i 1 i t dq L dq0double [ ] +i dq i 2 t ([ T -1 1*T T 2*T ] L dq0double [ i 1 ] e i2 dq ) (3.1.32) 2 Equation can be further classified into following three power categories: Joul Losses = i dq t [ R R ] [i 1 i2 ]

52 i 1 Energy Stored in machine magnetic Structure = i t dq L dq0double [ ] i 2 Source Power converted into Mechanical Power: P em = i dq t ([ T 1*T T 2*T ] L dq0double [ i 1 ] e i2 dq ) (3.1.32) 2 Also, P em = ω m T em = ω p T em T em = p ω P em (3.1.33) Where, ω m is the mechanical speed, ω is electrical frequency rad/s and p is no. of poles pair. Substituting equation 32 into equation 33, to find electromagnetic torque as: T em = p ω i dq t [ T 1 T T 2 T ] L dq0double [ i 1 ] p i i2 ω dq t e dq (3.1.33) 2 Where, o o p i ω dq t [ T 1 T T 2 T ] L dq0double [ i 1 ] is the Reluctance Torque i2 2 p ω i dq t e dq is the Torque due to interaction between stator & rotor mmf Equation 33 is used for electromagnetic torque calculation and is also simulated in the Matlab model. Further the external loading torque (T ext ) and electromagnetic torque interaction in double star PMSM can be given as:

53 T em T ext = J d ω dt p + B ω p Or d ω dt p = 1 J [(T em T ext ) B ω p ] In order to find we suppose, x= ω p, and apply the control system state space equation model to get ω. i.e.: x = 1 J [(T em T ext ) Bx] After solving, we get the following transfer function for x: Or x = T em T ext B + Js ω = p T em T ext B+Js (3.1.34) Equation 34 for ω helps to make the closed form expression of dq0 model of double star motor from physical machine inductance parameters and through vector space modeling of input equations.

54 4.0 CONTROL STRATEGY 4.1 Quadrature Axis Current Control Strategy High Power multiphase machine working efficiency can be achieved with a high rated voltage source converter but it will put strong constraints on power electronic devices and limit their switching frequency. The concept of power segmentation has then emerged to allow the use of Voltage Source Inverter (VSI) with reduced size power electronic devices. There are different ways to achieve it. One of them is to use multi-level inverter fed multiphase machines. In our case of double star PMSM both the stars are being fed by their own VSI. A pulse width modulation (PWM) approached is used for this voltage source inverter. The phase delay of each of this VSI is being controlled in a way to achieve Torque-speed Control strategy. A CSI inverter is the dual of a six-step voltage source inverter. With a current source inverter, the DC power supply is configured as a current source rather than a voltage source. The inverter SCRs are switched in a six-step sequence to direct the current to a three-phase AC load as a stepped current waveform. CSI inverter commutation methods include load commutation and parallel capacitor commutation. With this method, the input current regulation assists the commutation. The CSI inverter commutation is realized through q-axis current control as under: v sq v q = L dq d dt i q ωl dq i d + Ri q v q v sq = L dq d dt i q + L dq i d Ri q d v q v sq ωl dq i d = L dq i dt q Ri q 4.1 Re-writing the above expression in Laplace domain: v q v sq ωl dq i d = sl dq i q Ri q 4.2

55 So, v q v sq ωl dq i d = (sl dq + R)i q Thus, i q = v q v sq ωl dq i d (sl dq +R) Block Diagram to Frame Control Strategy i q iq v sq + ωl dq i d v sq ωl dq i d 1 1 K 3 v q i 1 + st q s R + sl dq i q Figure 4.1 Block Diagrm To Frame For Control Strategy On Quadrature Axis Current In the above block diagram, first block is for the regulator, whose Proportional and Integral parameter will be found later. The second block is for the implemented inverter that is working at T s = 20ms. The inverter use pulse width modulation whose delay is found on the basis of reference i q values in the stator winding. The reference i q is found from the synchronous speed and torque requirement. Initially for machine startup of our 100KW model, the startup currents are very high and during steady state operation they are comparatively less. It s not efficient to use the same regulator for both parts. In practice its more or less same with the function given as: Regulator Gain = K(K 3 +K 4 /s) Where, K is quite small for machine startup operation. From the block diagram, we can write the following open loop transfer function: L(s) = K(K 3 + K 4 s )( 1 1+sT s )( 1 R+sL dq ) 4.4

56 following: This open loop function can be written in regulator gain and system transfer function as And L(s) = K ( K 3 + K 4 )B(s) ) 4.5 s L(s) = K (L 1 (s)) Finding K 3 And K 4 Parameters There are many methods to tune the regulator. A trivial method of using Bode Plot is elaborated here. For Bode Plot method we need we need to set two parameters: cutoff frequency "ω c " and phase margin "φ m ". L 1 (s) for unity gain and phase margin φ m can be given by the following two equations: L 1 (jω) = π + φ m = L 1 (jω) 4.8 Finally using equation 4.3 to 4.8 we are able to write L 1 (jω) in the K 3 and K 4 form: ( K 3 + K 4 ) B(jω) ) =1 jω Or, can be written as: ( K 3 + K 4 jω ) = 1 B(jω) ) And phase margin equation can be formulated as: 4.9 K 3 + K 4 jω + B(jω) = π + φ m 4.10 Combining equation 4.9 and 4.10 the final expression of K 3 and K 4 can be given as: K 3 = 1 B(jω) ) cos( π + φ m B(jω) ) 4.11 K 4 ω c = 1 B(jω) ) cos( π + φ m B(jω) ) 4.12

57 We plot the bode plot of transfer function B(s) and chose the values of K 3 and K 4 with the following criterion for stable system: phase margin φ m should be more than 60. And that φ m must satisfy the condition that both K3 and K4 have same signs We use the following quadrature axis circuit to find Lq and R values: Figure 4.2: Equivalent dq axis representation of quadrature axis circuit of Stator Winding As from the [L dq0 ] found after Simulating we have the following 6x6 matrix: Simplifying the circuit using L dq 1, 1 = ; L dq 2, 2 = L dq 3, 3 =

58 Thus, we have L q = mH and R= With these Lq and R values we make the Bode plot of B(s) and chose a cutoff frequency ω c 3dB lower the T t 1 where T t is the time constant of the above circuit T t = L q /R Figure 4.3: Bode plot of Open Transfer Function of B(s)

59 So, we get ω c =0.15 rad/s. So, we have to chose a value greater than that ω c. Now, after setting ω c in order to find phase margin φ m we plot K3 and K4 whenphase margin ranges over a given interval and choose value of φ m where K3 and K4 has same signs. Figure 4.4: Phase Margin Evaluation by plot K 3 and K 4 pu plot with phase margin

60 I chose the phase margin 65 degrees, since for that value both K 3 and K 4 have positive values. After selecting the phase margin 65 degrees and cutoff frequency 50Hz we find the following open Loop and close Loop transfer function response: Figure 4.5: Open Loop and Cloop Transfer Functions For the q-axis circuit And the values of the K 3 and K 4 corresponding to transfer function is found to be : K 3 = And K 4 =

61 At the end in order to get an idea that our close loop system, we do the Nyquist plot of the close loop transfer function. That is given as: Figure 4.6: Nyquist Plot to Check The Stability Criterian of Close Loop Transfer Function The plot shows that the system is stable for the chosen K 3 and K 4 values. These K 3 and K 4 values correspond to the steady state operation of machine, since we only get the already L dq0 matrix when the double machine starts rotating at synchronous speed. There are two ways to get the machine start like any normal synchronous machine, first de-excite the rotor winding rotate the machine by applying torque on prime mover and at synchronous speed we excite the rotor and do the external loading. The alternative way can be achieved by precisely tuning the regulator for the startup operation. The value of K for the steady operation can be tuned from 1 to 10 and it depend on how quick we want our machine to adjust to external torque. If K not rightly chosen and it is chosen very high for the stead state operation the stator winding current may become too high

62 leady to permanent damage to machine. Or if chose too low the machine might not be able to achieve the steady state value and will slow and slow ultimately speed gets to zero. Since the winding currents are too high during startup operation K value in the expression L(s) = K ( K 3 + K 4 s )B(s) ) Is chosen to be in range of to Like any machine the startup operation is quite sensitive and a small wrong collection of tuner may ultimately leads to very high current and the machine will not achieve the steady state operation Simulink Block Diagram For i q c current Regulation Figure 4.7: Bllock Diagram For iq Current Regulation

63 In this block diagram pidtag tells the time to switch between two two PI regulators, i.e. the time when machine starts up and the time when machine comes to steady state operaton. w is the current speed in rad/s of the machine and the thetar tells about the rotor position with respect to the reference as already depicted in the systematic diagram of double star machine. At the output of regulator we have iq reference value that we put to another block to find the reference i1 and i2 currents. (Note i1 and i2 are dq0 currents of stator winding 1 and stator winding 2). of pulses Pulse Width Modulation Of VSI The reference iq will make the commutation of VSI to control the duty cycle of its series A low duty cycle corresponds to the when no electrical power is required to be transfer to the machine that is we want to restrict the electromagnetic torque by controlling the i q current following in the stator winding. A high duty cycle on the other side corresponds to the time to deliver more electromagnetic torque by increasing the i q values. The PI regulator will provide high reference values for i q during that time. The modulation signal used is the actual current flowing in the iq winding. The PWM technique used is an intersective method i.e. when the reference iq signal is lower than the modulated signal (actual iq) the output signal is high and contrary otherwise. The schematic understanding of reference, modulating and output signal is given as following:

64 This Pulse width modulation can be referred to as a Schmitt Trigger that compares the referecne iq signal from the regualor with the sinusoidal signal from the stator winding current and adjust the duty cycle of voltage source inverter. Here the duty cycle is equal to zero the transferred electromagnetic torque would be minimum, while at 100% duty cycle the transferred electromagnetic torque corresponds to the maximum value. Each intermediate value determines a corresponding electromagnetic torque supply. A graphical representation of the switch control is shown below: Figure 4.8: Switching Logic For Voltage Source Inverter

65 The current iag, ibg, and icg fluctuate let the Vdc fluctuate between maximum and zero. The amplitude of the inverter output voltages is controlled by adjusting the amplitude of the sinusoidal control voltages. The ratio of the amplitude of the sinusoidal waveforms relative to the amplitude of the triangle wave is the amplitude modulation ratio. In systems the inverter supplies power to the inductive loads, so the inverter must source power in all four quadrants in order to realize the torque speed control. VSI SIMULINK BLOCK Figure4.9: Voltage Source Inverter Block Diagram Here the Vab is default Simulink block Controlled Voltage source that converts an input signal into equivalent voltage source. Whereas, the comparator block diagram is shown as below:

66 Figure 4.10: Block diagram of Comparator To Produce Input signal For Controlled Voltage Source Here the relay Output the specified 'on' or 'off' value by comparing the input to the thresholds of 0.05 and The output is our Vdc or -Vdc value. The results of this modulation, with ia reference, ia actual and va at the output of voltage source inverter can be viewed by the following simulation results:

67 5.0 SIMULATION OF MATHEMATICAL MODEL IN MATLAB/SIMULINK Goal: The implementation of mathematical model formulated before and to do numerical evaluation of such machine model with a particular machine data. Later some simulation results are discussed to show how the formulated model successfully simulates the performance of different multi-star electrical machines. Figure: 5.1 Block Diagram of the Simulink Double Star PMSM Model This Simulink model implements all the equations devised previously. In order to implement this model, so many custom blocks are made utilizing the Level-2 S function available in Simulink

68 5.1 Machine Simulations without Electrical Drive Machine Data: self and mutual inductance in range 1-4 mh, resistance of each winding is Ohm. Machine moment of inertia (J =0.41kgm^2) and Viscous co-efficient (B=0.01) Ideal 3 phase voltage souce 400V, 50Hz is applied with no offset phase at both windings. The important consideration here is that that a step external torque inforce hypothetically machine into steady state region. Also there is a big offshoot in electromagnetic torque just before getting into the steady operation. Later with the proposed control scheme the electromagnetic torque as well as this big overshoot in electromagnetic torque is controlled. The sole purpose of these simulations is to verify the mathematical model of double star PMSM that is developed in the simulation environment, especially with the various benefits that we can get with this machine that we found from the literature review.

69 Various simulations done for the machine operation without the controlled voltage source inverter is summarized as: 1. Torque speed Curve for: External torque: 5Nm, 50Nm, 100Nm, 200Nm 2. Flux, Current, Torque speed Curve for: External torque: 50Nm,200Nm 3. Working of double star under: Power supply to both 3 phase winding Power supply to only one 1 phase winding 4. Current waveform 50/200 Nm loading torque 5. Effect of Loading of 100/200Nm torque at 100/200 rad/s loading speed 6. Effect of loading torque 10,50,100 at 0rad/s with different machine data

70 5.1.1 Torque speed Curve for 5Nm, 50Nm, 100Nm, 200Nm: Figure 5.2 Double Star PMM Torque Speed For the case 5Nm Figure 5.3 Double Star PMM Torque Speed For the case 50Nm

71 Figure 5.4 Double Star PMM Torque Speed For the case 100Nm Figure 5.5 Double Star PMM Torque Speed For the case 200Nm

72 5.1.2 Working of double star under one Faulty Three phase Figure 5.6 Power supply to both 3 phase winding at ext torque 100Nm at loading speed 200 rad/s Figure 5.7: Power supply to only one 1 phase winding, other tripped/short circuited at ext torque 100Nm at loading speed 200 rad/s

73 Figure 5.8: Power supply to both 3 phase winding at ext torque 200Nm at loading speed 200 rad/s Figure 5.9: Power supply to only one 1 phase winding, other tripped/short circuited at ext torque 200Nm at loading speed 200 rad/s Important Conclusion: The Double Star PMSM satisfy the theoratical model that the machine can work under faulty one 3 phase. The only consideration is that the torque ripple increases or can be failure in very worse loading case.

74 5.1.3 Current waveforms 50 Nm loading torque at speed 200 rad/s, the current waveform is during the instant there is external loading to almost the instant the speed get approx. constant Figure 5.10 Current wavefor of phase 1a of stator winding at Loading Figure 5.11 Current wavefor of phase 1a of stator winding

75 5.1.4 Current Torque Speed Curve At Various Loading Condition (Discrete simulation mode with sample time 50e-3) Figure 5.12 Double Star PMM Flux Current Torque Speed For the case 50Nm Figure 5.13 Double Star PMM Flux Current Torque Speed For the case 100Nm

76 Figure 5.14 Double Star PMM Flux Current Torque Speed For the case 50Nm Loading Torque Figure 5.15 Double Star PMM Stator winding a 3 phase current

77 5.1.5 Double Star PMSM Simulation with Different Machine Data Machine data: l 0_l1_l2_lm-in mH-r-0.02, J=4, B=0.01 Voltage ABC peak value: 2KV Initital torque loading: 100Nm Speed Torque curve

78 Machine data: l 0_l1_l2_lm-in20-40mH-r-0.02, J=2, B=0.01 Voltage ABC peak value: 400V Initital torque loading: 400Nm Flux, Current, Torque, Speed Curve 100Nm Torque ext. 0rad/s : Flux Current curve wrt time for torque ext 400Nm

79 5.2 Machine Simulations with Electrical Drive Model The implementation VSI by putting a control strategy on iq current helps us to simulate the double star PMSM. A pulse width modulation (PWM) approached is used for this voltage source inverter. The phase delay of each of this VSI is being controlled in a way to achieve Torquespeed Control strategy. The simulation results depicted below satisfy this control strategy Figure 5.16: Double Star PMSM Simulation with Electrical Drive Here, Blue curve represents the speed that is regulated at 300 rad/s Dotted Purple Curve Represents the Electrical Torque Red Curve Represent the External Torque, that is initially zero but at 5s a continously increasing Ramp function si applied. Control Cirucuit Functioning: The control circuit theoratically explained in chapter 4 simulation is evaluated. As we can see from the figure 5.13 the speed is gradually regualted to 300 rad/s. There is no offshoot in external torque. It is quite regulated. Both the regulators, one for the machine startup and one

80 for the steady state operation are working fine. The machine performance if compared in figure 5.13 with the uncontrolled case of figure 5.1 to 5.12 is compared we found quite big comparison. Here we find no big offshoost of electromagnetic torque. The only hactic part i see is in the simulation time. Uncontorlled Double star PMSM model simulates in second while this control circuit tooks quite a lot of time. Especially the following 25 second simulation of double star machine model almost took 20 minutes. Figure 5.17: Double Star PMSM Simulation with Electrical Drive from 0 to 25s Tuning the regulator more precisely with a bit erro margin even yields more better results, since if we see in figure 5.13 we see some electromagnetic torque ripples. The following curve represents more torque speed regulated results:

81 5.2.1 PULSE WIDTH Modulation In the Drive Model Figure 5.18 Pulse width modulation that controlled VSI do during Electrical Drive startup Note: The simulation in figure 5.14 is just 1 sec from 0-1. The correspoding machine speed or electromagnetic torque can be referred to figure 5.13

82 Figure 5.19 Pulse width modulation that controlled VSI do for the first 3 seconds (Machine reach the steady state speed) Figure 5.20Pulse width modulation that controlled VSI do for the first 10 seconds (Machine reach the steady state speed and also an external Torque ramp function is applied here)

83 Figure 5.21 Pulse width modulation that controlled VSI do for the first 0-25 seconds Figure 5.16 present the whole picture for the simulation time that how the electrical drive model works. The green curve in figure 5.16 is the actual current waveform that flows in stator winding 1 phase a, the blue curve gives the reference values. In the stator the reference curve is almost zero. But when the machine gets to steay state and external loading torque is applied this reference current values increases. If the Loading torque more the reference current adjust its value to supply the external loading requirements and also regulate the electromagnetic torque to reduce its possible ripple level. There is high frequency ripple transient still there with very less relative harmonic content ultimatley for the better functionality of double star PMSM.

84 0 to 10 second represent of figure 5.16 with green line as output VSI voltage for phase 1a. In the above both plots, Green curve represents the VSI phase v a voltage level Blue curve represents the reference current for the VSI controlled inverter Orange (yellow ochre) curve represents the actual current flowing in phase a of 1 st winding.

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