Predicting Electromagnetic Noise in Induction Motors NGUYEN MINH KHOA Master of Science Thesis Stockholm, Sweden 2014 Predicting Electromagnetic Noise in Induction Motors NGUYEN MINH KHOA Master’s Thesis in Scientific Computing (30 ECTS credits) Master Programme in Computer simulation for Science and Engineering (120 credits) Royal Institute of Technology year 2014 Supervisor at ABB was Romain Haettel Supervisor at KTH was Michael Hanke Examiner was Michael Hanke TRITA-MAT-E 2014:48 ISRN-KTH/MAT/E--14/481--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci Abstract Induction motors are used in many applications, especially in our daily modes of transportation such as trains, cars, buses, trucks, etc. As a consequence, they contribute a significant amount of noise in the environment, which can cause serious problems on the human well-being. This thesis aims to simulate induction motors in order to predict the noise produced by the electromagnetic forces generated in the motors. Simulations can help the engineers with a noise-efficient design. In this work, COMSOL1 Multiphysicsr 4.4 software is used for the modeling. The two squirrel-cage induction motors are studied. Motor1 is a benchmark design while Motor3 models a device manufactured at ABB2 . Under no load condition, both motors show results in accordance with the theory of induction motors. The results of Motor3 obtained with COMSOL Multiphysics are compared with Adept (an in-house software of ABB for modeling motors). Acoustics modeling was done for Motor1. As expected, noise in Motor1 is contributed by the electromagnetic force and resonance with natural frequencies of the motor. 1 2 http://www.comsol.se (Accessed 25 Aug 2014). http://new.abb.com/se (Accessed 25 Aug 2014). Referat Prediktering av Elektromagnetiskt Ljud i Induktionsmotorer Induktionsmotorer används för många tillämpningar, till exempel, i transportmedel såsom bilar, bussar, lastbilar, etc. Som en följd bidrar de till en betydande mängd buller som kan påverka människornas välbefinnande och även hälsa. Denna avhandling syftar till att simulera motorer för att förutsäga den ljudnivå som genereras på grund av de elektromagnetiska krafterna i motorer. Simuleringar kan hjälpa ingenjörerna med en effektivare design av motorer med avseende på ljud. I detta arbete är COMSOL Multiphysicsr 4.4 den mjukvara som används för modellerin av två asynkronmotorer. Motor1 är ett riktmärke konstruktion och Motor3 modeller en anordning tillverkars i ABB. Utan last visar båda motorerna resultaten i enlighet med teorin om asynkronmotorer. Resultaten som beräknades på Motor3 med hjälp av COMSOL Multiphysics jämförs med Adept (en programvara utvecklad av ABB för modellering motorer). Akustiska beräkningar gjordes på Motor1. Som väntat bullernivåer bidragit av storleken av den elektromagnetiska kraften och den resonans med egenfrekvenser av motorns. Preface This project is a collaboration between ABB in Sweden and KTH Royal Institute of Technology, Stockholm. I would like to thank Dr. Romain Haettel from ABB for the opportunity and support throughout the project; Prof Michael Hanke from KTH for organizing the meetings to help me be on time with the schedule; Prof Jesper Oppelstrup from KTH and Dr. Anders Daneryd from ABB for their comments and advice. I appreciate very much their time and expertise in regular meetings to help me overcome the doubts and obstacles. Their enthusiasm and passion for research were an inspiration and motivation for me to complete this thesis. Stockholm, August 2014. Minh Khoa, Nguyen Contents List of Figures List of Tables List of Abbreviations and Acronyms 1 Introduction 1.1 Background and Motivations . . . . . . . . . . . . . . . . . . . . . . 1.2 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . 2 Induction motors and their sources of noise 2.1 Principles of induction motors . . . . . . . . . . . . 2.1.1 Geometry . . . . . . . . . . . . . . . . . . . 2.1.2 Power supply . . . . . . . . . . . . . . . . . 2.1.3 Stator slot layout and connection . . . . . . 2.1.4 Operation . . . . . . . . . . . . . . . . . . . 2.2 Noise from induction motors . . . . . . . . . . . . . 2.2.1 Noise and Sound . . . . . . . . . . . . . . . 2.2.2 Electromagnetic source of noise in induction 1 1 2 . . . . . . . . . . . . . . . . 3 3 3 6 7 10 11 11 12 in Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 14 15 16 16 18 18 20 20 22 4 Modeling electromagnetic noise in induction motors with COMSOL Multiphysics 4.1 Step 1: Modeling the operation of an induction motor . . . . . . . . 24 24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . motors 3 Modeling electromagnetic noise in induction motors 3.1 Overview: Three steps of modeling . . . . . . . . . . . 3.2 Finite element method . . . . . . . . . . . . . . . . . . 3.3 Step 1: Modeling the operation of an induction motor 3.3.1 Electromagnetics . . . . . . . . . . . . . . . . . 3.3.2 Stator circuit . . . . . . . . . . . . . . . . . . . 3.3.3 Rotor circuit . . . . . . . . . . . . . . . . . . . 3.3.4 Rotation . . . . . . . . . . . . . . . . . . . . . . 3.4 Step 2: Fourier transform of the electromagnetic force 3.5 Step 3: Modeling the Acoustic-Structure Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 25 28 28 28 30 30 30 31 5 Results and Discussion 5.1 Motor1 . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Motor Operation . . . . . . . . . . . . . . 5.1.2 Acoustics-Structure Interaction in 2D . . 5.1.3 Acoustics-Structure Interaction in 3D . . 5.2 Comparison between COMSOL Multiphysics and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adept for Motor3 34 34 34 37 43 46 6 Conclusion and Future improvements 6.1 Conclusion . . . . . . . . . . . . . . . . . . 6.2 Future improvements . . . . . . . . . . . . . 6.2.1 Current issues . . . . . . . . . . . . . 6.2.2 Suggestions for future improvements . . . . 49 49 50 50 50 4.2 4.3 4.1.1 Parameters for the Solver . . . . . . . . . . . . . . . . . . . 4.1.2 Rotating Machinery, Magnetic interface . . . . . . . . . . . 4.1.3 Electrical Circuit interface for the stator and rotor circuits 4.1.4 Global ODEs and DAEs interface for the rotation equation Step 2: Fourier transform of the electromagnetic force . . . . . . . Step 3: Modeling Acoustics-Structure Interaction . . . . . . . . . . 4.3.1 Geometry and Parameters . . . . . . . . . . . . . . . . . . . 4.3.2 Acoustics-Structure Interaction, Frequency interface . . . . 4.3.3 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices 50 A Matlab codes 51 Bibliography 53 List of Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 Main components of an induction motor. . . . . . . . . . . . . . . Rotor cage (squirrel cage) of an induction motor. . . . . . . . . . . 2D cross section of Motor1 . . . . . . . . . . . . . . . . . . . . . . Wire, Coil, Winding. . . . . . . . . . . . . . . . . . . . . . . . . . . Three-phase sinusoidal voltage supply. . . . . . . . . . . . . . . . . Winding diagram (single layer) for Motor1. . . . . . . . . . . . . . The stator circuit of Motor1 . . . . . . . . . . . . . . . . . . . . . . The stator circuit of Motor3 . . . . . . . . . . . . . . . . . . . . . . Winding diagram (double layer) for Motor3 . . . . . . . . . . . . . Stator-Airgap and Stator-Surrounding Air boundary in 2D model. . . . . . . . . . . 4 4 5 5 6 7 8 8 9 12 3.1 3.2 3.3 Three steps of the modeling in this thesis. . . . . . . . . . . . . . . . . . Rotor Circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An example of a result from Fast Fourier Transform in MATLAB. . . . 15 19 21 4.1 4.2 4.3 Relationship of the COMSOL Multiphysics interfaces used in Step 1. . . Middle-airgap boundary of Motor1 . . . . . . . . . . . . . . . . . . . . . The geometry in Step 3 for Motor1 (modeling Acoustics-Structure Interaction in 2D). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Meshing for the Acoustics-Structure Interaction model in 2D. . . . . . . Meshing for the Acoustics-Structure Interaction model in 3D. . . . . . . 25 26 4.4 4.5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 . . . . . . . . . . . . . . . . . . . . B-H curve of the core iron in Motor1. . . . . . . . . . . . . . . . . . . . Result: Rotational velocity of Motor1. . . . . . . . . . . . . . . . . . . . Result: Torque on the rotor of Motor1 . . . . . . . . . . . . . . . . . . . Result: Phase Currents of the Stator circuit in Motor1. . . . . . . . . . Result: Magnetic flux density and Magnetic vector potential at t=0.4259s. Result: A few Eigenmode shapes of Motor1 in 2D. . . . . . . . . . . . . Result: Total Electromagnetic force (magnitude) acting on the stator at different frequencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Result: Sound Power and Sound Power Level of Motor1 for the 2D model. 5.9 Result: Motor1 in 2D - Displacement field of the stator, Sound pressure and Sound intensity fields in the surrounding air. . . . . . . . . . . . . . 5.10 Result: A few Eigenmodes of Motor1 for the 3D model. . . . . . . . . . 31 32 33 35 35 36 36 37 39 40 41 42 44 5.11 Result: Sound Power and Sound Power Level of Motor1 for the 3D model. 5.12 Result: Motor1 in 3D - Displacement field of the Stator, Sound pressure and Sound intensity fields of Motor1 at 4200Hz. . . . . . . . . . . . . . . 5.13 Result: Comparison of the Phase Current between COMSOL Multiphysics and Adept for Motor3. . . . . . . . . . . . . . . . . . . . . . . . 5.14 Result: Comparison of the Rotational Speed between COMSOL Multiphysics and Adept for Motor3. . . . . . . . . . . . . . . . . . . . . . . . 45 46 47 48 List of Tables 3.1 Relationship between the fft result and its input signal. . . . . . . . . . 21 4.1 4.2 4.3 25 27 4.5 4.6 Parameters for the solver in Step 1. . . . . . . . . . . . . . . . . . . . . Physics setting in COMSOL Multiphysics in Step 1. . . . . . . . . . . . Steps to establish the communication between MATLAB and COMSOL Multiphysics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Format of the text file storing the amplitudes of the radial Maxwell Stress Tensors in frequency domain. . . . . . . . . . . . . . . . . . . . . . . . . Modeling parameters for Motor1 in Step 3. . . . . . . . . . . . . . . . . Physics setting in COMSOL Multiphysics for Motor1 in Step 3. . . . . . 5.1 5.2 5.3 5.4 Starting and Operating Conditions for Motor1. . . . . . . . . . . . Result: Eigenfrequencies of the stator structure in 2D. . . . . . . . Result: Noise level and electromagnetic forces for a few frequencies Result: Eigenfrequencies of the stator structure in 3D. . . . . . . . 34 38 41 43 4.4 . . . . . . . . . . . . 29 30 30 32 List of Abbreviations and Acronyms 2D two dimensions 3D three dimensions acsl Acoustic-Solid Interaction, Frequency DAE Differential algebraic equation Eq Equation FEM Finite element method Fig Figure ODE Ordinary differential equation rmm Rotating Machinery, Magnetic rms root mean square rpm revolutions per minute WHO World Health Organization Chapter 1 Introduction Induction motors play an important role in our lives today. They are found in everyday’s modes of transportation: cars, trains, elevators, etc. Because of its usefulness and close proximity to human beings, one serious issue needs to be addressed: Noise. Noise is literally defined as unwanted sound. Although it is a subjective experience of human beings, it is undeniable that noise has a significant affect on our physical and mental well-being. Physical problems caused by noise include hearing impair or even deafness. Mental problems can be sleep disturbance, annoyance, etc. Due to the serious impact of noise, The World Health Organization (WHO) has guidelines on the noise control [22]. Some governments have even imposed regulations on this matter such as the United Kingdom with the Noise Act 1996 [1]. ABB, as a producer in motors, is concerned about this issue in its products. Therefore, the goal of this thesis, as an ABB’s initiative, is to simulate one of ABB motors in order to predict the noise level it produces. This will help the engineers with the design of a noise-efficient induction motor. 1.1 Background and Motivations Modeling induction motors in three dimension (3D) and time domain is very timeand computation-consuming. However, a quasi-3D model can be accomplished reasonably by combining the finite element and lumped element modeling. In [2] and [13], the quasi-3D modeling of induction motors is presented in detail from equation setup, matrix assembly to finite element solution. Finite element method (FEM) is combined with electrical circuits to simulate the operation of an induction motor in time domain. COMSOL Multiphysics is a commercial software for modeling complex scientific and engineering problems. Its strength is the ability to couple different types of physics in one model such as: thermodynamics, acoustics, electromagnetics, structural mechanics, etc. Many studies have been successfully done with Comsol Multiphysics. A frequency analysis of induction motors in COMSOL Multiphysics is performed 1 CHAPTER 1. INTRODUCTION in [7]. The same analysis in time domain with the software is found in [8] and [16]. In [14], a noise analysis of transformers is done with COMSOL Multiphysics. The transition from time domain to frequency domain for noise analysis is clearly shown. Adept, an in-house software from ABB, is highly specialized for analyzing ABB motors. It is capable of modeling induction motors in both frequency domain and time domain. Inspired by those works, this thesis aims to fulfill two tasks: - Employ the quasi-3D modeling approach to model the operation of induction motors in time domain with a transition to frequency domain for noise analysis. - Compare the results between COMSOL Multiphysics and Adept on modeling the motor operation. 1.2 Organization of the thesis The organization of the thesis is as follows: Chapter 1 introduces the background, motivation and organization of the thesis. Chapter 2 presents some motor terminology, the principles of induction motors and their electromagnetic source of noise. Chapter 3 gives an overview of the modeling steps. The underlying physics, assumptions and equations are also presented. Chapter 4 shows the implementations in COMSOL Multiphysics. Chapter 5 presents the results and discussion. Chapter 6 gives a conclusion and suggestions for future improvements. 2 Chapter 2 Induction motors and their sources of noise 2.1 2.1.1 Principles of induction motors Geometry The induction motors studied in this thesis contain 4 main components (Fig 2.1) - Stator: this static part has slots on its inner surface. Coils are fitted onto these slots. The stator core is made up by many sheets of iron stacked one upon another to reduce eddy-current losses. Each sheet of iron is called a laminate and the technique of making such layered structure is called lamination. The terminals of the coils are connected to an external power supply. - Rotor: this rotating part is composed of a core and a cage. The rotor core is also laminated to reduce eddy-current losses. The rotor cage (squirrel cage) has parallel conductor bars fixed together at two ends by the end rings (Fig 2.2a). - Shaft: this is also a rotating part. The rotation of the rotor is transferred to the shaft. Once the shaft is loaded, the rotation motion is transformed into linear motion. - Fan: this part is for cooling the heat generated by the motor. Fig 2.3 shows the 2D cross section of Motor1, a benchmark configuration in this study. Motor1 has 28 rotor bars and 36 stator slots. The geometry of Motor1 is built from [17]. Motor1 has the length of 100mm. Figure 2.4 depicts what wires, coils, coil sides and overhang windings (or end windings) are. In Motor1, wires are made of copper. A coil is a long wire wound into many turns. A coil side is the part of a coil that occupies one slot in the stator core region. Overhang windings refer to the parts of a coil outside the stator core region. Figure 2.4b shows the difference between coil sides and overhang windings. Once a coil is excited by a power supply, the currents in two coil sides of a coil are always in opposite directions. 3 CHAPTER 2. INDUCTION MOTORS AND THEIR SOURCES OF NOISE Figure 2.1: Main components of an induction motor. (a) Rotor bars and End rings (b) An end-ring segment (highlighted) is the part of an end ring which is between two rotor bars Figure 2.2: Rotor cage (squirrel cage) of an induction motor. 4 CHAPTER 2. INDUCTION MOTORS AND THEIR SOURCES OF NOISE (b) One quarter of the cross section of Motor1. (a) 2D cross section of Motor1. Figure 2.3: 2D cross section of Motor1 (a) A single wire. (b) Different parts of a coil. (c) Winding of a Coil on the Stator. Figure 2.4: Wire, Coil, Winding. 5 CHAPTER 2. INDUCTION MOTORS AND THEIR SOURCES OF NOISE 2.1.2 Power supply The power supply for the motors in this study is a three phase voltage supply. A 3phase voltage supply comprises 3 sinusoidal voltage sources (VA , VB and VC ) which differ from one another by 120 degrees (= 2π/3 rad) in phase, for example: VA = Vp sin(ωt + ϕ0 ) V = V sin(ωt + ϕ + 2π/3) p 0 B V = V sin(ωt + ϕ + 4π/3) p 0 C (2.1) where Vp is the peak voltage (in V ), t time (in s), and ω angular frequency (in rad/s). ϕ0 , ϕ0 + 2π/3, ϕ0 + 4π/3 are initial phases (in rad) in phase A, B and C, respectively. The angular frequency ω (in rad/s), frequency f (in Hz) and period T (in s) of a sinusoidal signal are related to one another by: ω = 2πf = 2π/T (2.2) Fig 2.5 shows the 3-phase voltage supply in Eq 2.1 with time Figure 2.5: Three-phase sinusoidal voltage supply, Vp = 220 V, f = 50 Hz, ϕ0 = 0 rad. 6 CHAPTER 2. INDUCTION MOTORS AND THEIR SOURCES OF NOISE 2.1.3 Stator slot layout and connection Two types of windings are often seen on the stator: single layer and double layer. Here are the descriptions of the winding in Motor1 (single layer) and Motor3 (double layer). Motor1 Fig 2.6 shows 36 slots of Motor1 with 6 labels AP1, AN1, BP1, BN1, CP1, CN1. The letters P (positive) and N (negative) indicate the opposite directions of the current in a coil. In other words, P and N indicate two coil sides of a coil. A,B and C indicate the phases from the power supply. Number "1" indicates the branch number in the phase circuit. The following rules apply for slots with the label AP1: - all wires in each slot are connected in series. - all slots are connected in series. - the current is in the positive direction. - these slots are connected to branch 1 of phase A of the power supply. - AP1 slots are connected in series with AN1 slots. These rules apply similarly to the other labels. Figure 2.6: Winding diagram (single layer) for Motor1. 7 CHAPTER 2. INDUCTION MOTORS AND THEIR SOURCES OF NOISE The stator circuit for Phase A of Motor1 is shown in Fig 2.7. The resistances of the coil sides AP1 and AN1 are lumped into one resistor. Overhang windings are represented by one resistor and one inductor. The induced voltage (or back electromotive force) across the coil sides caused by the electromagnetic field is represented by a voltage source. Figure 2.7: Electrical circuit for phase A of Motor1. Motor3 Fig 2.9 shows a 2D cross section of Motor3 (ABB motor). Motor3 has 38 rotor bars and 48 stator slots. Each slot is divided into two parts. The part that faces the airgap belongs to the inner layer and the other part belongs to the outer layer. Hence, Motor3 has double layer winding. In Motor3, the circuit of each phase has two branches as indicated by numbers "1" and "2" in the stator slots’ labels (Fig 2.9). Fig 2.8 shows the circuit for phase A. Phase B and C have similar circuit connections. Figure 2.8: Electrical circuit for phase A of Motor3. 8 CHAPTER 2. INDUCTION MOTORS AND THEIR SOURCES OF NOISE Figure 2.9: Winding diagram (double layer) for Motor3 9 CHAPTER 2. INDUCTION MOTORS AND THEIR SOURCES OF NOISE 2.1.4 Operation The principle of an induction motor and its formula are summarized here. More details and explanations can be found in [19]. Motor1 is powered by a 3-phase sinusoidal voltage supply. This power supply excites the coils wound on the stator slots, the consequence of which is the induced magnetic field inside the motor. This magnetic field rotates at a speed called synchronous speed which can be calculated by: nss = 120 × f nP (2.3) where nss is the synchronous speed (in revolutions per minute or rpm), f the frequency of the power supply (in Hz), nP the number of poles of the machine. As a side note, rpm is converted to rad/s as such: 1 rpm = 1 revolution/minute = (2π radians)/(60 seconds). The configuration of the stator winding and its connection to the power supply determine the number of poles and the rotation direction of the motor. The induced magnetic field cuts across the rotor bars, which induces currents in the bars because the bars together with the end rings make a closed circuit. The bars are excited and generate a magnetic field around themselves which opposes the magnetic field due to the stator coils (Lenz’s law). The interaction between the magnetic field due to the stator coils and the magnetic field due to the rotor bars produces a torque to rotate the rotor. Hence, the rotation of the rotor is induced. That is why this type of motor is called induction motor. After a while, the rotor would rotate at the synchronous speed. However, once its speed matches the synchronous speed, no currents are induced in the bars. As a result, no force acts on the rotor and the rotor slows down. When it slows down, the currents in the rotor bars start rising due to more magnetic fluxes crossing the rotor bars, torque acts on the rotor again and the rotor speeds up. At no load condition and no energy losses incurred during the operation, the mean rotational velocity of the rotor will be equal to the synchronous speed. Otherwise, the rotor always rotates at a speed less than the synchronous speed. This characteristic is represented by slip, nss − nrs s= × 100 (2.4) nss where s is the slip (in %) and nrs is the rotor speed (in rpm) Because induction motors usually have slip, they are also called asynchronous motors. 10 CHAPTER 2. INDUCTION MOTORS AND THEIR SOURCES OF NOISE 2.2 2.2.1 Noise from induction motors Noise and Sound Noise is unwanted sound and sound is the human perception of mechanical vibrations. Three components are needed to make a noise: the source of noise, the medium where the noise can propagate and a receiver. Due to the physiology of the human ear drums, we perceive sound in terms of loudness and pitch. Hence, the sound pressure and frequency are the interesting physical quantities. When the sound is composed of different frequencies, the root mean square (rms) value of the sound pressure (in P a) is calculated as: p= q p21 + p22 + ... (2.5) where p1 , p2 , ... are the sound pressure of the 1st, 2nd, ... frequency, respectively. Sound pressure is the force of sound on a surface area which is perpendicular to the direction of the force. Sound power level, Sound pressure level and Sound intensity level are quantities that are usually used to evaluate noise. The significant difference among them is that sound power level characterizes the source of sound while the sound pressure level and sound intensity level characterize the effect of sound. Hence, sound pressure level and sound intensity level depend on the distance from the source while sound power does not. These quantities are calculated by the formula [15]: p2 = p2ref I LI = 10 log10 Iref P LP = 10 log10 Pref Lp = 10 log10 p 20 log10 | pref | (2.6) where Lp , LI , LP are sound pressure level, intensity level and power level, respectively. Their units are in decibel (dB). p, I, P are the the rms values of sound pressure (in P a), intensity (in W/m2 ) and power (in W ). pref = 2×10−5 P a , Iref = 10−12 W/m2 , Pref = 10−12 W are the reference values for sound pressure, intensity and power, respectively. Sound intensity is defined as sound power flowing across a unit area. Hence, sound power can be calculated by an integration of rms value of sound intensity over a closed surface near the source of sound, as I P = I · dS 11 (2.7) CHAPTER 2. INDUCTION MOTORS AND THEIR SOURCES OF NOISE 2.2.2 Electromagnetic source of noise in induction motors Three main sources of noise from an induction motor are [9]: mechanical sources (frictions from bearings, brushes), aerodynamic sources (flow of air) and electromagnetic sources (electromagnetic field). The electromagnetic field exerts a force on the stator core. The vibration of the stator thereby affects the surrounding air. The vibration of the tiny molecules in the air is propagated to the perceivers (human beings, animals, sensors, etc). In induction motors, the total electromagnetic force F acting on the stator can be expressed as follows I F= S σ M ST · dS = Z σ M ST · dS + S1 Z σ M ST · dS (2.8) S2 where σ M ST is Maxwell stress tensor, S the outward normal to the surface of the stator. S, the total surface of the stator, is composed of two surfaces: S1 the stator surface facing the air gap and S2 the stator surface facing the surrounding air (Fig 2.10). The term σ M ST · dS will be called the Maxwell stress tensor in radial direction or radial Maxwell stress tensor from now on in this thesis. The air surrounding the stator has high magnetic reluctivity. Almost no magnetic fluxes, and hence no forces are present there. Therefore, the integration over S2 can be ignored in Eq 2.8. We are left with only the integration over the StatorAirgap Boundary S1 . Figure 2.10: Stator-Airgap and Stator-Surrounding Air boundary in 2D model. In this thesis, Maxwell stress tensor σ M ST is used as the electromagnetic force acting on the stator. This stress tensor σ M ST can be calculated as [10], 1 1 1 M ST σij = 0 (Ei Ej − δij E 2 ) + (Bi Bj − δij B 2 ) 2 µ0 2 12 (2.9) CHAPTER 2. INDUCTION MOTORS AND THEIR SOURCES OF NOISE In 2D, i and j refer to the coordinates x and y. E(= q Ex2 + Ey2 ) and B(= q Bx2 + By2 ) are the magnitudes of the electric field strength and the magnetic flux density, respectively. 0 , µ0 are the permittivity and permeability of free space. δij is Kronecker delta, from which δxx = δyy = 1 and δxy = δyx = 0. As a result, M ST = −σ M ST = 1 (E 2 − E 2 ) + 1 (B 2 − B 2 ) σxx x y yy x y 2 0 2µ0 M M ST = (E E ) + 1 (B B ) σxy ST = σyx 0 x y x y µ0 (2.10) When only the magnetic field is present, which is largely true for motors, M ST = −σ M ST = 1 (B 2 − B 2 ) σxx yy x y 2µ0 M ST = σ M ST = 1 (B B ) σxy x y yx µ0 13 (2.11) Chapter 3 Modeling electromagnetic noise in induction motors in Theory 3.1 Overview: Three steps of modeling The modeling is broken down into three consecutive steps (Fig 3.1): - The first step is Modeling the operation of an induction motor in COMSOL Multiphysics. This is done in 2D and time domain. Hence the cross section of a motor is modeled. This step follows the lines of [8]. The rotor rotates according to the induced speed. - The second step is done in MATLAB1 . The electromagnetic force (the Maxwell stress tensor in radial direction in this thesis) obtained from the first step is transformed to frequency domain in order to prepare for the third step. - The third step is Modeling the Acoustics-Structure interaction in COMSOL Multiphysics. This is done in frequency domain. The electromagnetic force obtained from step 2 is applied on the stator for each frequency. Air is filled around the stator for noise investigation. This step can be done in 2D or 3D. In the 3D model, the stator is extruded from the 2D model into the third dimension by the length of the stator. The reasons for three steps are: - Modeling the transient behavior of the motor in 3D is very time- and computationconsuming. As shown later, the implemented 2D model in this thesis is already time-consuming. - The motor reaches the steady state very quickly (in less than a second for the motors in this study). In addition, they exhibit harmonic behavior at steady state (as shown in Chapter 5), which makes it possible for the frequency analysis. 1 http://www.mathworks.se/products/matlab/ (Accessed 25 Aug 2014). 14 CHAPTER 3. MODELING ELECTROMAGNETIC NOISE IN INDUCTION MOTORS IN THEORY Figure 3.1: Three steps of the modeling in this thesis. 3.2 Finite element method Finite element method (FEM) is a popular method to find an approximate solution for differential equations. The general steps to perform FEM are: - Establish the governing equations of the problem. These are usually differential equations. - Formulate a weak representation for the governing equations. - Choose the basis for the solution (the approximate solution is a linear combination of this basis). - Rewrite the problem in terms of matrices instead of differential operators. - Solve the linear algebra problem. Details of the FEM can be found in resources and textbooks such as [11] and [21]. In this thesis, COMSOL Multiphysics already handles the necessary steps to perform the FEM. Therefore, in the following sections, only the physics, governing equations and assumptions are presented. 15 CHAPTER 3. MODELING ELECTROMAGNETIC NOISE IN INDUCTION MOTORS IN THEORY 3.3 3.3.1 Step 1: Modeling the operation of an induction motor Electromagnetics Maxwell’s equations The electromagnetic field in time and space is described by Maxwell’s equations as ∂B ∂t ∂D ∇×H=J+ ∂t ∇·D=ρ ∇×E=− ∇·B=0 (3.1) (3.2) (3.3) (3.4) where E is the electric field strength (in V /m), H the magnetic field strength (in A/m), B the magnetic flux density (in W b/m2 ), D the electric flux density (or current displacement, in C/m2 ), J the current density (in A/m2 ) and ρ the electric charge density (in C/m3 ). Equations 3.3 and 3.4 are Gauss’ law for electrics and magnetics, respectively. Eq 3.2 is Maxwell-Ampere’s law and Eq 3.1 is Faraday’s law. Because the motor does not operate at high frequency, ∂D ∂t << J; and hence, Eq 3.5 which is Ampere’s law is used instead of Eq 3.2, ∇×H=J (3.5) Constitutive equations Constitutive equations (or material equations) describe the relations of the electromagnetic field with the materials D = E (3.6) H = νB (3.7) J = σE (3.8) where (permittivity, in F/m), ν (reluctivity, in m/H) and σ (electrical conductivity, in S/m) are material-dependent quantities. Eq 3.8 is also known as Ohm’s law For nonlinear B − H relationship, Eq 3.9 is used instead of Eq 3.7 H = f (|B|) (3.9) B=∇×A (3.10) Solution for the field Eq 3.4 allows us to write 16 CHAPTER 3. MODELING ELECTROMAGNETIC NOISE IN INDUCTION MOTORS IN THEORY where A is a vector field. A is in fact called magnetic vector potential. Substitute Eq 3.10 into Eq 3.1, one gets ∂A )=0 (3.11) ∂t The last equation allows us to define a scalar field φ (also called electric scalar potential) such that ∇ × (E + ∂A − ∇φ ∂t Substitute Eq 3.7 and 3.8 into Eq 3.5 to obtain E=− (3.12) ∇ × (νB) = σE (3.13) Substitute Eq 3.10 and 3.12 into the last equation, we have ∇ × (ν∇ × A) = −σ ∂A − σ∇φ ∂t (3.14) In the last equation, the term σ ∂A ∂t represents the eddy current source and the term σ∇φ represents the external excitation source. The advantage of modeling the electromagnetic field in 2D is that, A and J only contain the component in the third dimension, i.e. A = Az ez (3.15) J = Jz ez (3.16) with ez the unit vector pointing in the third dimension. Here are a few necessary assumptions: - The laminations of the stator and rotor cores are perfect. Therefore, the eddy current can be ignored in the domains of the stator and rotor cores. - The coils are made of thin wires so that the eddy current is ignored in the coils as well. With those assumptions, Eq 3.14 can be written for each domain of the motor as follows Nw Iw ez /Sw −σ ∂A + σV e /l b z b ∂t ∇ × (ν∇ × A) = ∂A −σ ∂t 0 in stator slots in rotor bars in shaft elsewhere (cores, airgap) (3.17) where Nw , Iw , Sw are the number of turns, current (in A) and cross-sesction area (in 2 m ) of the stator slots, respectively. Vb is the voltage (in V ) across a rotor bar and lb (in m) is the length of a bar. 17 CHAPTER 3. MODELING ELECTROMAGNETIC NOISE IN INDUCTION MOTORS IN THEORY 3.3.2 Stator circuit Applying Kirchoff’s law of voltage for the circuit of Phase A in Fig 2.7, we have dIA (3.18) dt where Vinduced is the induced voltage (in V ) across the AP1 and AN1 coil sides. Rcs is the resistance of the coil sides (in Ω). RW and LW are the resistance (in Ω) and inductance (in H) of the end windings, respectively. IA is the current in phase A (in A). VA = Vinduced + Rcs IA + RW IA + LW The induced voltage is calculated as Vinduced = NAP 1 lcs SAP 1 dAz dS dt Z − AP 1 slots NAN 1 lcs SAN 1 dAz dS dt Z (3.19) AN 1 slots where NAP 1 and NAN 1 are the number of wires in AP1 and AN1 slots. SAP 1 and SAN 1 are the cross-section area of a single AP1 slot and AN1 slot, respectively. lcs is the length of a coil side (= length of the stator) and Az the magnitude of the magnetic vector potential in z-direction. Hence, Eq 3.18 becomes NAP 1 lcs VA = SAP 1 Z dAz dS dt − AP 1 slots NAN 1 lcs SAN 1 Z dAz dS + dt AN 1 slots (3.20) dIA Rcs IA + RW IA + LW dt Eq 3.20 when written for all phases gives all the equations for the stator circuit. 3.3.3 Rotor circuit The assumptions here are: - No defects in the bars or rings. - All bars are the same in shape and size. - The rings are uniform and the same in shape and size. Each end-ring segment (see Fig 2.2b) of the rotor is modeled with a resistance and an inductance as in [8]. The rotor circuit is shown in Fig 3.2. By Kirchhoff’s voltage law, the voltage equation for the loop of ir,n is Vbar,n−1 − Vbar,n + 2Rs (ibar,n−1 − ibar,n ) + 2Rr ir,n + 2Lr 18 dir,n =0 dt (3.21) CHAPTER 3. MODELING ELECTROMAGNETIC NOISE IN INDUCTION MOTORS IN THEORY Figure 3.2: Rotor Circuit. Rr and Lr are resistance and inductance of each ring segment, Rs is the resistance of the part of a bar outside the rotor core. Iinduced,n denotes the induced current in the nth bar. where Vbar,n denotes the voltage accross the nth bar in the rotor core region. Similarly for the loop of ir,n+1 , Vbar,n − Vbar,n+1 + 2Rs (ibar,n − ibar,n+1 ) + 2Rr ir,n+1 + 2Lr dir,n+1 =0 dt (3.22) Subtract Eq 3.21 from Eq 3.22 gives (2Vbar,n − Vbar,n−1 − Vbar,n+1 ) + 2Rs (2ibar,n − ibar,n−1 − ibar,n+1 ) d(ir,n+1 − ir,n ) +2Rr (ir,n+1 − ir,n ) + 2Lr =0 dt (3.23) Since ibar,n = ir,n+1 − ir,n (Kirchhoff’s current law), the last equation can be rewritten as (2Vbar,n − Vbar,n−1 − Vbar,n+1 ) + 2Rs (2ibar,n − ibar,n−1 − ibar,n+1 ) dibar,n +2Rr ibar,n + 2Lr =0 dt We have from Eq 3.8 and Eq 3.12 for a rotor bar, J = σE = −σ Z Sbar JdS = −σ Z ∂Az − σ∇φ ∂tZ ∂Az dS − σ ∂t Sbar ∇φ dS Sbar ibar = −σ Z Sbar 19 ∂Az Vbar dS + ∂t Rbar (3.24) CHAPTER 3. MODELING ELECTROMAGNETIC NOISE IN INDUCTION MOTORS IN THEORY Z ibar,n = −σ Sbar,n Vbar,n ∂Az dS + ∂t Rbar (3.25) where Rbar is the resistance of a rotor bar (the part inside the core region), Sbar the cross-section area of a bar and Az is the magnitude of the magnetic vector poR ∂A tential in z-direction. The induced current in the nth bar is the term (σ ∂t dS) Sbar,n in the equation. Eq 3.24 together with Eq 3.25 when written for all bars gives all the equations of the rotor circuit. 3.3.4 Rotation The electromagnetic torque Trotor (in N m) which is responsible for the rotation can be calculated from Maxwell stress tensor σ M ST (in N/m2 ) as such I r × (σ M ST · n) dS Trotor = (3.26) Srotor where r is the vector pointing from the center of the rotor to a point on the rotor surface. Srotor is the rotor surface and n is the normal vector pointing out from the rotor surface. In the 2D model, the integration is done along the rotor perimeter Γrotor as such I r × (σ M ST · n) dΓ Trotor = lrotor (3.27) Γrotor where lrotor is the length of the rotor (in m). According to Newton’s second law, the motion of the rotor can be expressed by either one of the following equations, ωr dω = Trotor − Tload dt d2α J 2 = Trotor − Tload dt J (3.28) (3.29) where J is the moment of inertia of the rotor (in kg · m2 ), ωr the rotational velocity (in rad/s), α the angular displacement (in rad) and Tload the load torque (in N · m). 3.4 Step 2: Fourier transform of the electromagnetic force In Step 1, The Maxwell stress tensor at each location on the stator-airgap surface is a time-dependent signal. It must be transformed into frequency domain for the frequency analysis in Step 3. The fft function in MATLAB, which performs Fast Fourier Transform algorithm, is used to do this task. Its syntax in MATLAB is output = fft(input) where input and output are arrays of the same size. 20 CHAPTER 3. MODELING ELECTROMAGNETIC NOISE IN INDUCTION MOTORS IN THEORY Fig 3.3 shows an example of the result from fft for a random signal in MATLAB. The output and input are both arrays of 25 values. The output (the last two plots on the right) contains two pieces of information (amplitudes and phases) for a spectrum of frequency. However, the spectrum of frequency is not calculated by the fft function in MATLAB. This is why the array indexes are shown on the x-axes in the figure. The user has to deduce himself the frequency range and frequency resolution from the sampling duration and sampling interval (or time step) of the input signal. Table 3.1 is very helpful for this task. In the output signal in Fig 3.3, the first value corresponds to 0 Hz and the rest of the values are symmetric with respect to the center value. This is because the input is a real-valued signal. Hence, the meaningful result of the fft for a real-valued signal lies in one half of the spectrum. A comprehensive textbook on Fourier Transform and Fast Fourier Transform algorithm can be found in [18]. Figure 3.3: Result of fft in MATLAB. The input signal is an array of 25 values. input sampling time duration = tdur sampling time interval = ∆t output = fft(input) frequency resolution = 1/tdur 1 frequency range = [0, 2∆t ] (Nyquist theorem) Table 3.1: Relationship of the input and output signals when using fft, where input is a real-valued signal. As an example, if the input signal is sampled for the duration of tdur = 5 s at the time interval ∆t = 0.1 s, using table 3.1, the output will have the frequency 1 1 ) and the frequency range of [0 Hz, 5 Hz] (5 Hz = 2×0.1s ). resolution of 0.2 Hz(= 5s 21 CHAPTER 3. MODELING ELECTROMAGNETIC NOISE IN INDUCTION MOTORS IN THEORY 3.5 Step 3: Modeling the Acoustic-Structure Interaction In induction motors, electromagnetic forces cause the vibration of the stator. The displacement of the stator from its original position causes the surrounding air to vibrate. The physical interaction between the stator and surrounding air is described by the following equations [6] ρS ü = ∇ · σ + FV , in the stator domain (3.30) 1 1 p̈ + ∇ · (− (∇p − qd )) = Qm , in the air domain (3.31) 2 ρA c ρA −n · (− 1 (∇p − qd )) = −n · ü ρA σ · n = pn , at the stator-surrounding air boundary (3.32) , at the stator-surrounding air boundary (3.33) Eq 3.30 is derived from Newton’s second law of motion with ∇ · σ being the surface force, FV the body force and ρS ü the inertia force. u is the displacement vector, ρS the density of the stator and σ the stress tensor. In Eq 3.31, p is the sound pressure, c the speed of sound in the air and ρA the density of air. qd and Qm are dipole source and monopole source, respectively. Eq 3.32 and 3.33 specify the continuity of the acceleration and pressure in the normal direction n at the Stator-Surrounding Air boundary. With the following assumptions: - Neither dipole nor monopole sources are present in the air (qd and Qm are zeros) - The air density is unchanged with time and space (ρA = const) The above equations can be rewritten as such: ρS ü = ∇ · σ + FV , in the Stator domain (3.34) p̈ − c ∇ · (∇p) = 0 , in the Air domain (3.35) , at the Stator-Surrounding Air boundary (3.36) , at the Stator-Surrounding Air boundary (3.37) 2 1 n · (∇p) = −n · ü ρA σ · n = pn In frequency domain, they become −ρS ω 2 u = ∇ · σ + FV , in the Stator domain (3.38) −ω 2 p − c2 ∇ · (∇p) = 0 , in the Air domain (3.39) 1 n · (∇p) = −ω 2 n · u ρA σ · n = pn , at the Stator-Surrounding Air boundary (3.40) , at the Stator-Surrounding Air boundary (3.41) where ω(= 2πf ) is the angular frequency (in rad/s) and f is the frequency (in Hz). 22 CHAPTER 3. MODELING ELECTROMAGNETIC NOISE IN INDUCTION MOTORS IN THEORY Maxwell stress tensor is applied as surface force at the Stator-Airgap Boundary. The region inside the stator is supposedly filled by the airgap and rotor. Hence, it is not modeled. In addition, the above equations require Young’s modulus and Poisson’s ratio of the structure to be able to solve. Young modulus relates the stress with strain while Poisson’s ratio relates the strains in different directions. 23 Chapter 4 Modeling electromagnetic noise in induction motors with COMSOL Multiphysics 4.1 Step 1: Modeling the operation of an induction motor Four interfaces of Comsol Multiphysics are used for this step: - Rotating Machinery, Magnetic (rmm): models the electromagnetic field and rotation. The inputs are the currents from the stator coils and voltages across the rotor bars. This module solves for the magnetic vector potential A. The induced voltages across the coils, induced currents in the bars and axial torque on the rotor can be derived from A. - Electrical Circuit (the stator circuit): uses lumped element modeling. A 3phase sinusoidal voltage supply and induced voltages across the coils are the inputs to this interface. The outputs are the coil currents. - Electrical Circuit (the rotor circuit): uses lumped element modeling. The inputs are the induced currents in the bars. Outputs are the bar voltages. - Global ODEs and DAEs: defines Eq 3.29. The output is the rotation angle. The inputs are the axial torque (calculated from the rmm module) and load torque. Fig 4.1 summarizes the relationships of these interfaces for modeling the operation of an induction motor in this thesis. 4.1.1 Parameters for the Solver Table 4.1 shows the parameters for modeling Motor1 in Step 1. If the solution does not converge or takes much time to converge in the first time step, the author suggests to create another Time Dependent Study with the Method set to Initialization only instead of Generalized alpha, then use the result of this Study as the initial values for the study shown in table 4.1. 24 CHAPTER 4. MODELING ELECTROMAGNETIC NOISE IN INDUCTION MOTORS WITH COMSOL MULTIPHYSICS Figure 4.1: Relationship of the COMSOL Multiphysics interfaces used in Step 1. Vinduced , Iinduced and Torque are derived from the magnetic vector potential A. Dimension Study type Method Jacobian update Discretization 2D Time Dependent Generalized alpha with a fixed time step at 0.1 milisecond on every iteration quadratic elements Table 4.1: Parameters for the solver in Step 1. 4.1.2 Rotating Machinery, Magnetic interface In order for this model to work in COMSOL Multiphysics, the geometry of the motor must be segregated into two separate parts as shown in Fig 4.2. One part contains the rotor and half of the air gap while the other part contains the stator and the other half of the air gap. When building the geometry, an assembly must be formed to create an Identity Pair in COMSOL Multiphysics. This technique is shown in the example of Generator in 2D in [3]. In order to prepare for step 2, the Stator-Airgap Boundary (Fig 2.10) should be defined in COMSOL Multiphysics. This can be done by creating a Selections>Explicit node under Definitions node. Table 4.2 describes the physics settings for this module. 25 CHAPTER 4. MODELING ELECTROMAGNETIC NOISE IN INDUCTION MOTORS WITH COMSOL MULTIPHYSICS Figure 4.2: Middle of the airgap in Motor1 is shown. The whole geometry is segregated into two parts at this line. 26 CHAPTER 4. MODELING ELECTROMAGNETIC NOISE IN INDUCTION MOTORS WITH COMSOL MULTIPHYSICS Domain or Boundary Stator Slot Domains COMSOL Multiphysics Node Multi-Turn Coil Rotor Bar Domains Single-Turn Coil Shaft, Stator Core and Rotor Core Domains Ampere’s Law Rotor Domain (Rotor Core + Bar + Shaft) Stator Core Domain Force Calculation Rotor Domain (Rotor Core + Bar + Shaft) and Half of the Airgap Domain Stator outer Boundary Prescribed Rotation Middle Boundary of the Airgap (Identity Pair) Continuity Force Calculation Magnetic Insulation Comments Reversed Current Direction nodes are added for domains AN, BN and CN. The coil excitation is configured as Circuit(current) the coil excitation is configured as Circuit(voltage) the constitutive relation of the magnetic field is set to HB curve to use the nonlinear curve enables calculation of the axial torque acting on the rotor enables calculation of electromagnetic forces acting on the stator. This is important for Step 2 rotate the rotor and half of the airgap at the prescribed angle Air has high reluctivity. Almost no magnetic fluxes cross this boundary into the air. Hence this condition is reasonable ensures the continuity in the magnetic vector potential A at the segregation line Table 4.2: Physics setting in COMSOL Multiphysics in Step 1. 27 CHAPTER 4. MODELING ELECTROMAGNETIC NOISE IN INDUCTION MOTORS WITH COMSOL MULTIPHYSICS 4.1.3 Electrical Circuit interface for the stator and rotor circuits Phase A circuit of Motor1 is shown in Fig 2.7. The circuits in phase B and C can be constructed in a similar manner. In the stator circuit, the External I vs U components in COMSOL Multiphysics represent the coils which are coupled with the Rotating Machinery, Magnetic (rmm) interface. In the rotor circuit, the External U vs I components represent the bars coupled with the rmm interface. These nodes act as voltage sources for the former and current sources for the latter. It is from the author’s experience that errors occur when there are no resistors in series with the External I vs U components or no resistors in parallel with the External U vs I components. Therefore, in the rotor circuit, a resistor of large value (1020 Ω) is connected in parallel with each External U vs I component. In [20], the formula to determine the parameters for both the stator and rotor circuits are presented. However, in this thesis, the end winding effects in Motor1 are not considered. Hence, the resistance and inductance of the end windings are set to zeros. In the rotor circuit of Motor1(Fig. 3.2), the part of the bars outside the rotor core is not considered. Hence its resistance RS = 0 Ω. The end-ring inductance LR is also assumed to be 0 H. In Motor3, these parameters are derived from Adept (ABB software for modeling induction motors). 4.1.4 Global ODEs and DAEs interface for the rotation equation Two physics nodes are available in the rmm interface of COMSOL Multiphysics to rotate the rotor domain: Prescribed Rotation and Prescribed Rotational Velocity. As stated in the documentation of COMSOL Multiphysics 4.4 [4], the Prescribed Rotational Velocity node only works for constant rotational speed. Therefore, this node is not applicable for this thesis. In order for the model to work with Prescribed Rotation, Eq 3.29 is implemented instead of Eq 3.28 and the Global ODEs and DAEs interface is used to implement this equation. The moment of inertia of the rotor is calculated by defining Mass Properties under the Definitions node for the rotor. The load torque is set to zero, i.e. no load. 4.2 Step 2: Fourier transform of the electromagnetic force Because the initial state of an induction motor happens very fast (see Chapter 5), the noise in that state is not interesting in this study. Therefore, only the radial Maxwell stress tensor in the steady state along the Stator-Airgap Boundary are extracted from the result of Step 1. The operation of the motor in step 1 has to be simulated until the steady state is present for a certain amount of time. The length of the time duration and time interval of the extracted data will determine 28 CHAPTER 4. MODELING ELECTROMAGNETIC NOISE IN INDUCTION MOTORS WITH COMSOL MULTIPHYSICS the frequency range and resolution according to Table 3.1. After the time duration and the time interval is decided upon, the procedure for Step 2 is as follows - Establish communication between MATLAB and COMSOL Multiphysics. - Extract the radial Maxwell stress tensor from COMSOL Multiphysics into MATLAB for the prescribed time range, transform them into frequency domain and export the output to text files. Table 4.3 shows the steps to establish the communication between MATLAB and COMSOL Multiphysics 1. Run COMSOL 4.4 with MATLAB. Wait for the process to complete, i.e. until the command prompt in MATLAB window appears. 2. Open COMSOL Multiphysics 4.4. 3. In COMSOL Multiphysics, click File>Client Server>Connect to Server. The window "Connect to Server" pops up. Fill in ’localhost’ for Server, username and password. Click OK to connect. 4. When COMSOL Multiphysics successfully connects to localhost, click File>Open to open the model in Step 1. Table 4.3: Steps to establish the communication between MATLAB and COMSOL Multiphysics. The codes in MATLAB for extracting the Maxwell stress tensor on the StatorAirgap Boundary, transforming them to frequency domain and writing the result to text files can be found in Appendix A. These codes require the function GetFrequencySpectrum whose definition is also shown in Appendix A. Two output files are necessary. The first file stores the amplitudes of the radial Maxwell stress tensors in frequency domain. The second file stores the phases. Table 4.4 shows the format of the file for storing the amplitudes. In each row, the first two columns indicate x and y coordinates on the Stator-Airgap Boundary, third column the frequency, and the last two columns the amplitudes in x and y directions, respectively. f1 , f2 , ..., fN is the frequency spectrum obtained from Fast Fourier Transform. The file for storing the phases has a similar format with the only difference that the last two columns indicate the phases. 29 CHAPTER 4. MODELING ELECTROMAGNETIC NOISE IN INDUCTION MOTORS WITH COMSOL MULTIPHYSICS X1 X1 .... X1 X2 .... X2 X3 .... Y1 Y1 .... Y1 Y2 .... Y2 Y3 .... f1 f2 ... fN f1 ... fN f1 ... Ax (X1 , Y1 , f1 ) Ax (X1 , Y1 , f2 ) ...... Ax (X1 , Y1 , fN ) Ax (X2 , Y2 , f1 ) ...... Ax (X2 , Y2 , fN ) Ax (X3 , Y3 , f1 ) ...... Ay (X1 , Y1 , f1 ) Ay (X1 , Y1 , f2 ) ...... Ay (X1 , Y1 , fN ) Ay (X2 , Y2 , f1 ) ...... Ay (X2 , Y2 , fN ) Ay (X3 , Y3 , f1 ) ...... Table 4.4: Format of the text file storing the amplitudes of the radial Maxwell Stress Tensors in frequency domain. 4.3 Step 3: Modeling Acoustics-Structure Interaction This part can be done in 2D or 3D. For the 2D model, the geometry of the stator is taken from Step 1. For the 3D model, the stator geometry is extruded into the third dimension according to the length of the stator. 4.3.1 Geometry and Parameters Table 4.5 shows the parameters for this step. Dimension Study type Discretization 2D or 3D Frequency domain Quadratic elements Table 4.5: Modeling parameters for Motor1 in Step 3. Fig 4.3a shows the stator and the air in the surrounding for the 2D model. The model has an outermost layer of air defined as Perfectly Matched Layer (PML). The PML works as a sound absorbing layer to effectively reduce the sound reflection at the PML boundary [12]. Traditionally, the air domain has to be made very large so that the sound reflection is minimized. The PML layer helps to reduce the size of the air domain significantly. The thickness of the surrounding air layer is made to be equal to the radius of the stator so that the largest incident angle of sound on the PML boundary is 30 degrees (Fig. 4.3b). The PML layer is defined by adding Perfectly Matched Layer node under Definitions node in COMSOL Multiphysics. In the setting for the PML, Rational stretching type, scaling factor of 1 and scaling curvature parameter of 1 are used. 4.3.2 Acoustics-Structure Interaction, Frequency interface The physics interface Acoustic-Solid Interaction, Frequency (acsl) in COMSOL Multiphysics is used to complete this step. Table 4.6 shows the domain and boundary 30 CHAPTER 4. MODELING ELECTROMAGNETIC NOISE IN INDUCTION MOTORS WITH COMSOL MULTIPHYSICS (a) Material of the domains for Motor1 in step 3. (b) Illustration of incident angle of sound on the PML boundary. Rstator and Rair are the radius of the stator and the thickness of the surrounding air layer, respectively. Figure 4.3: The geometry in Step 3 for Motor1 (modeling Acoustics-Structure Interaction in 2D). definition in COMSOL Multiphysics. The domain inside the stator is supposedly filled with airgap and the rotor. Hence, this domain is excluded from the interface and meshing. The results in the text files from Step 2 must be imported into COMSOL Multiphysics. The node Interpolation under Definition>Component Couplings makes this possible. 4.3.3 Meshing As suggested in the COMSOL Multiphysics example for studying Acoustic Scattering off an Ellipsoid [5], the element size must not exceed 1/6 of the wavelength everywhere in the model to have good accuracy. The PML thickness seems not to matter because it is scaled numerically according to the wavelength in the model, however the element size in the PML must strictly obey this rule as well. The wavelength is related to the frequency by this formula λ= 31 c f (4.1) CHAPTER 4. MODELING ELECTROMAGNETIC NOISE IN INDUCTION MOTORS WITH COMSOL MULTIPHYSICS Domain or Boundary Stator Domain Stator-Airgap boundary COMSOL Multiphysics Node Linear Elastic Material Boundary Load Boundary between the PML and surrounding air layer Boundary between the Stator and the surrounding Air Far-Field Calculation Free & Normal Acceleration Comments The imported Maxwell stress tensor in frequency domain is applied on this boundary if far field results are interesting only use this if one wants to remove the influence of the air pressure on the stator Table 4.6: Physics setting in COMSOL Multiphysics for Motor1 in Step 3. where λ is the wavelength (in m), c in this case is the speed of sound in the air (in m/s) and f is the frequency (in Hz). It is also advised in [5] to have at least 6 elements across the PML to have an effective absorption. Fig 4.4 and Fig 4.5 show the meshing of Motor1 in 2D and 3D respectively for Step 3. Figure 4.4: Meshing for the Acoustics-Structure Interaction model in 2D. Swept mesh of 6 elements is used for the PML. 32 CHAPTER 4. MODELING ELECTROMAGNETIC NOISE IN INDUCTION MOTORS WITH COMSOL MULTIPHYSICS Figure 4.5: Meshing for the Acoustics-Structure Interaction model in 3D. Swept mesh of 6 elements is used for the PML. 33 Chapter 5 Results and Discussion 5.1 5.1.1 Motor1 Motor Operation Operating and Starting conditions The starting and operating conditions for Motor1 are shown in Table 5.1 Power supply Initial phase value Number of poles Operating condition Starting condition 3 phase, peak voltage = 220V , frequency = 50 Hz A (0 rad), B (2π/3 rad), C(4π/3 rad) 4 poles No load Stand still (zero velocity) and all other initial values are zeros Table 5.1: Starting and Operating Conditions for Motor1. Fig 5.1 shows the B-H curve of the core irons in Motor1. B-H data is imported and interpolated by COMSOL Multiphysics. The extrapolation (red lines) is set to constant. Rotational velocity and Slip Fig 5.2a shows the development of the rotational velocity in Motor1 over the first 0.6 second. The motor starts to stabilize after 0.3 second. A closer look at the last 0.3 second is shown in Fig 5.2b. The motor exhibits a harmonic pattern at the steady state and the steady-state speed is very close to the synchronous speed 1500 rpm (see Eq 2.3 for calculation of the synchronous speed). The negative sign indicates the rotational direction of the motor. It is observed in Fig 5.2b that the mean value of the steady-state speed is around 1499.7 rpm while the synchronous speed is 1500rpm. Hence, using Eq 2.4, the slip 34 CHAPTER 5. RESULTS AND DISCUSSION Figure 5.1: B-H curve of the core iron in Motor1. (a) Rotational velocity of Motor1 from 0 second to 0.6 second. (b) Zoom-in of the Rotational velocity of Motor1 for the last 0.3 second. Figure 5.2: Rotational velocity of Motor1. × 100 = 0.02 %. The slip is not zero because of is found to be very small 1500−1499.7 1500 the eddy current losses in the shaft and the rotor bars. Torque Fig 5.3 shows the development of the axial Torque acting on the rotor. At steady state, as expected, the torque centers around zero value. Phase currents Phase currents in the stator circuit are shown in Fig 5.4. It is clearly seen that the currents start to stabilize and exhibit harmonic patterns after 0.2 second. 35 CHAPTER 5. RESULTS AND DISCUSSION Figure 5.3: Torque on the rotor of Motor1. Figure 5.4: Currents in three phases of the Stator circuit in Motor1. 36 CHAPTER 5. RESULTS AND DISCUSSION Figure 5.5: Magnetic flux density and Magnetic vector potential at t=0.4259s. The color bar is for the Magnetic flux density and the other bar is for the Magnetic vector potential. Magnetic field The magnetic flux density and magnetic vector potential at t = 0.4259s are shown in Fig 5.5. Four poles on the stator can be clearly seen in this figure. The maximum flux density (norm) is marked at 2.56 T (color bar) which is higher than the maximum value of the flux density (2.4 T ) in Fig 5.1. This may be due to the discontinuity of the B-H curve at the tails in Fig 5.1. 5.1.2 Acoustics-Structure Interaction in 2D Natural modes of vibration In order to obtain the natural modes of the stator, a model is created in COMSOL Multiphysics with the Solid Mechanics interface. In this model, only the stator is present, i.e. air is not present and no forces are applied on the stator. The model is then solved for the eigenfrequencies (or natural frequencies). Table 5.2 shows the first few eigenfrequencies at two different mesh resolutions. The third column indicates the difference in percentage of the values in the second column compared to those in the first column. In the second column, although the mesh resolution is improved by 2 times, all the frequencies change by less than 0.003 percent. Hence, the eigenfrequencies in the second column are reasonably accurate. The eigenfrequencies usually come in pairs because of the symmetry of the motor in x and y directions. Fig 5.6 shows the shapes of some frequencies in the second column of Table 5.2. 37 CHAPTER 5. RESULTS AND DISCUSSION 27858 elements 1607.67988 1607.68091 4243.57661 4243.58255 7363.4644 7363.47132 10133.42935 10133.43674 11573.60819 11805.97428 11805.99965 56130 elements 1607.67498 1607.67427 4243.56118 4243.5623 7363.41425 7363.42001 10133.2785 10133.28985 11573.59357 11805.69039 11805.70814 % 0.000304787 0.0004130173 0.0003636084 0.0004771911 0.0006810653 0.0006968181 0.0014886372 0.0014495576 0.0001263219 0.00240463 0.0024691683 Table 5.2: Eigenfrequencies (in Hz) of the stator structure in 2D. The first two columns show the frequencies at two different mesh resolutions. The third column shows the difference in percentage of the second column compared to the first. 38 CHAPTER 5. RESULTS AND DISCUSSION (a) EigenFrequency = 1607.67 Hz (b) EigenFrequency = 4243.56 Hz (c) EigenFrequency = 7363.41 Hz (d) EigenFrequency = 10133.27 Hz (e) EigenFrequency = 11573.59 Hz (f) EigenFrequency = 11805.69 Hz Figure 5.6: A few Eigenmode shapes of Motor1 in 2D. 39 CHAPTER 5. RESULTS AND DISCUSSION Sound Power and Sound Power Level The Maxwell stress tensor in radial direction applying on the Stator-Airgap Boundary (Fig 2.10) is extracted in the range [0.5s, 0.6s] because the speed of the motor is most stable in this time span (see Fig 5.2b). The time step in Step 1 is 0.0001s. Therefore, the radial Maxwell stress tensor has a frequency range of [0Hz, 5000Hz] 1 and frequency resolution of 10Hz (= 0.6s−0.5s ). These values are derived using Table 3.1. The total electromagnetic force (magnitude) applying on the stator at different frequencies is shown in Fig 5.7. Three largest electromagnetic forces are present at the frequencies: 4900Hz (826N), 4400Hz (470N) and 4200Hz (429N). Figure 5.7: Total Electromagnetic force (magnitude) acting on the stator for frequency range [100Hz,5000Hz] and frequency step 10Hz. The following results come from the 2D model in Step 3. No damping is assigned to the material of the stator. The sound power of Motor1 is calculated by using Eq 2.7. The sound power and sound power level of Motor1 are shown in Fig 5.8a and Fig 5.8b, respectively. The results in Fig 5.8 were done for two different mesh resolutions (at 29706 and 107172 elements) to show their accuracy. The figure clearly shows that the results at the two mesh resolutions are roughly the same. 40 CHAPTER 5. RESULTS AND DISCUSSION (a) Sound Power (b) Sound Power Level Figure 5.8: Sound Power and Sound Power Level of Motor1 for frequency range [100Hz, 5000Hz] and frequency step 10Hz. The amount of noise, in theory, depends on two factors: - The magnitude of the electromagnetic force, and - The resonance with the natural frequency of the structure, i.e. the proximity of the applying frequency to the natural frequency of the structure. The following discussion uses information from Fig 5.8, Fig 5.7 and Table 5.3. Frequency (Hz) 1600 Noise level (dB) 72.8 Force (N) 142 4200 4240 92 79 429 15.8 4400 4900 78.7 74 470 826 Remarks close to natural frequency 1607.67Hz 3rd largest force close to natural frequency 4243.56Hz 2nd largest force Largest force Table 5.3: Noise level and electromagnetic forces for a few frequencies. Let us first examine three frequencies where there are most electromagnetic forces: 4900Hz (826N), 4400Hz (470N) and 4200Hz (429N). Among these three frequencies, 4200Hz gives the greatest noise level (92dB) and 4900Hz gives the smallest noise level (74dB). This is because 4200Hz is the nearest frequency while 4900Hz is the farthest frequency from the natural frequency at 4243.56Hz. Therefore, it can be concluded that the resonance contributes more to the noise level than the applying electromagnetic force for these three frequencies. At 4240Hz which is the closest frequency to 4243.56Hz, the electromagnetic 41 CHAPTER 5. RESULTS AND DISCUSSION force is very small (15.8N) whereas the noise is large (79dB). Hence, the noise at 4240Hz is contributed mostly by its resonance with the natural frequency. Although 4200Hz is not as close to the natural frequency 4243.56Hz as 4240Hz, its noise level is the greatest in the frequency spectrum. This is due to the combination of large electromagnetic force and the resonance with natural frequency. Noise at 1600Hz is the highest (72.8dB) in the range [100Hz,4000Hz] (Fig 5.8b) although its electromagnetic force is moderate (142N). Therefore, the noise here is also contributed by both the electromagnetic force and its resonance with natural frequency at 1607.67Hz. In conclusion, the above observations help to determine whether the noise at a particular frequency is caused mostly by the electromagnetic force or the resonance with natural frequency of the stator or both of them. Fig 5.9 shows the displacement field of the Stator, the sound pressure field and sound intensity field in the surrounding air at 4200Hz. The displacement field matches the mode shape at 4243.56 Hz in Fig 5.6. Figure 5.9: Displacement field of the stator, Sound pressure and Sound intensity (arrows) fields in the surrounding air at 4200Hz. The color bars on the left and right are for the Displacement (mm) and Sound pressure (Pa), respectively. 42 CHAPTER 5. RESULTS AND DISCUSSION 5.1.3 Acoustics-Structure Interaction in 3D Natural modes of vibration In order to find the natural modes of vibration for the stator in 3D, a model is created in COMSOL Multiphysics with the Solid Mechanics interface. Neither air is present nor any forces act on the stator. The model is solved for the eigenfrequencies. Table 5.4 shows a few eigenfrequencies. The first two columns show the eigenfrequences at two different mesh resolutions. The third column shows the difference in percentage of the values in the second column compared to those in the first column. In the second column, the eigenfrequencies do not change significantly (less than 0.4 %) even though the mesh resolution is improved by 4 times. Hence, the eigenfrequencies on the second column are reasonably accurate. The eigenfrequencies also come in pairs due to the symmetry of the stator in x and y directions. Fig 5.10 shows a few mode shapes for the second column in Table 5.4 190549 elements 1597.9317 1598.03439 2353.49366 2353.52882 4220.8718 4220.90193 5322.01036 5322.11795 7340.33569 7340.61622 8435.84386 8436.37463 8499.30579 8499.39159 736937 elements 1594.03765 1594.07132 2348.66958 2348.67275 4209.59546 4209.62692 5309.4732 5309.51257 7313.59082 7313.83342 8407.4027 8407.47022 8492.5918 8492.60222 % 0.2436931441 0.2479965403 0.2049752707 0.206331444 0.2671566571 0.2671232402 0.2355718827 0.2368489409 0.3643548624 0.3648576522 0.337146591 0.3426164824 0.0789945693 0.0798806588 Table 5.4: Eigenfrequencies (in Hz) of the stator structure in 3D. The first two columns show the frequencies for two different mesh resolutions. The third column shows the difference in percentage of the second column compared to the first. Comparing Fig 5.6 (Eigenmode shapes in 2D) and Fig 5.10 (Eigenmode shapes in 3D), one can see that the eigenfrequencies are smaller in the 3D model for the same mode shapes. This is due to the fact that the strain and stress in the third dimension restrict the vibration in 2D. Moreover, new mode shapes (the second column in Fig 5.10) are present between the regular modes (the first column in Fig 5.10) when moving from 2D to 3D. 43 CHAPTER 5. RESULTS AND DISCUSSION (a) EigenFrequency = 1594.03 Hz (b) EigenFrequency = 2348.66 Hz (c) EigenFrequency = 4209.59 Hz (d) EigenFrequency = 5309.47 Hz (e) EigenFrequency = 7313.59 Hz (f) EigenFrequency = 8407.40 Hz Figure 5.10: A few Eigenmodes of Motor1 for the 3D model. 44 CHAPTER 5. RESULTS AND DISCUSSION Sound Power and Sound Power Level The following results come from the 3D model in Step 3. No damping is assigned to the material of the stator. The meshing of the stator in this model follows the meshing which gives the results in the first column of Table 5.4. Even though the meshing is as such, the number of degrees of freedom was 900912 and it took half an hour to calculate the result for one single frequency with the available computer in this thesis work. Therefore, due to time constraints, the results presented here were done for [1000Hz,4950Hz] with the frequency step of 50Hz. Significant noise at other frequencies may be missing due to this coarse frequency step. Fig 5.11a shows the sound power calculated by using Eq 2.7 and Fig 5.11b shows the corresponding sound power level. The magnitude of the electromagnetic force at each frequency can be seen in Fig 5.7. Similar to the 2D model, noise is most prominent (91 dB) at 4200 Hz. However, the electromagnetic force at 4200Hz is 429N which is the second largest (the largest electromagnetic force is 826N at 4900Hz). Hence, the noise at 4200Hz is contributed by both the electromagnetic forces and its resonance with the natural frequency at 4209.59Hz. In the 3D model, noise at 1600Hz does not stand out as in the 2D model. Perhaps, it is also due to the fact that stress and strain in the third dimension damp the vibration, and hence the noise level of the 2D model in general. (a) Sound Power (b) Sound Power Level Figure 5.11: Sound Power and Sound Power Level of Motor1 (3D model) for frequency in [1000Hz, 4990Hz] step by 50Hz. Fig 5.12 shows the Displacement, Sound pressure and Sound intensity fields at 4200Hz. The Displacement field matches the mode shape at 4209.59 Hz in Fig 5.10. 45 CHAPTER 5. RESULTS AND DISCUSSION Figure 5.12: Displacement field of the stator, Sound pressure and Sound intensity (arrows) fields in the surrounding air of Motor1 at 4200 Hz. The color bars on the left and right are for the Sound pressure (Pa) and Displacement (mm), respectively. 5.2 Comparison between COMSOL Multiphysics and Adept for Motor3 The comparison between COMSOL Multiphysics and Adept is done for the operation of Motor3. The 2D cross section of Motor3 was introduced in Fig 2.9. All parameters in the COMSOL Multiphysics model are derived from the Adept model. The supply power is 3-phase sinusoidal voltage source with the peak voltage at 316.80 V and frequency at 75 Hz. The operating condition is no load and everything starts from zero. Motor3 is a 4-pole motor. Fig 5.13 shows the comparison between COMSOL Multiphysics and Adept for the current in phase A. In steady state, the peak current in COMSOL Multiphysics is 31.1 % (= 100 × 118A−90A ) greater than that in Adept. However, both curves 90A seem to be in phase with each other. Fig 5.14 shows the comparison of the rotational speed. In steady state, Adept has a rotational speed close to the synchronous speed 2250Hz, in fact, slightly higher than the synchronous speed!!! The slip in Adept is very close to 0% while rpm the slip in COMSOL Multiphysics is around 0.0311% (= 100 × 2250 rpm−2249.3 ). 2250 rpm The ripples in the curves are very different, which indicates that their frequency spectra will be different. The reason, in the author’s opinion, accounting for the differences in the results between the two programs is that Adept takes into account the lamination effect, temperature rise effect and the end effect of the electromagnetic field while the model in COMSOL Multiphysics in this thesis does not. 46 CHAPTER 5. RESULTS AND DISCUSSION (b) Closer view in the steady state (a) For the first 1 second Figure 5.13: Comparison of the current in Phase A between COMSOL Multiphysics and Adept for Motor3. 47 CHAPTER 5. RESULTS AND DISCUSSION (a) For the first 1 second (b) Closer view in the steady state Figure 5.14: Comparison of the Rotational Speed between COMSOL Multiphysics and Adept for Motor3. 48 Chapter 6 Conclusion and Future improvements 6.1 Conclusion The technique on how to predict the electromagnetic noise for induction motors using COMSOL Multiphysics was laid out in this thesis. Three steps are necessary. First, an induction motor is simulated in time domain until the steady state. The radial Maxwell stress tensor along the stator-airgap boundary during the steady state is converted to frequency domain. The results of the radial Maxwell stress tensor in frequency domain are applied on the stator-airgap boundary in the AcousticsStructure interaction model for noise analysis. Motor1 and Motor3 showed the results in accordance with the theory of induction motors. COMSOL Multiphysics and Adept were compared for the results of Motor3 (ABB motor). The slip in COMSOL Multiphysics is around 0.0311 % and in Adept is around 0%. The ripples in the curves of rotational speed during the steady state from both programs are very different, which indicates different harmonic patterns. For the current in phase A during the steady state, the peak value in COMSOL Multiphysics is 31.1% greater than that in Adept while the curves are in phase with each other. The difference could be due to the fact that some physical properties are modeled in Adept whereas they are not modeled in COMSOL Multiphysics in this thesis. The Acoustics-Structure interaction was analyzed for Motor1 for the frequencies up to 5000Hz in 2D and 3D. As expected, noise is influenced by two factors: the magnitude of the electromagnetic force and the resonance with the natural frequency of the structure. 49 CHAPTER 6. CONCLUSION AND FUTURE IMPROVEMENTS 6.2 6.2.1 Future improvements Current issues Here are a few current issues: - Step 1 of the modeling process is time-consuming. It took 25 hours to compute the first 1 second of Motor3 operation in COMSOL Multiphysics while it took only 4 hours for the same model in Adept. Moreover, a smaller frequency resolution in Step 3 requires the model in Step 1 to be simulated for a longer time span. Therefore, future improvements on the speed up for Step 1 should have highest priority. - From the author’s experience, coarser meshing in Step 1 reduces the number of degrees of freedom but does not guarantee a computational speed up. This is because Step 1 uses Newton method to solve the nonlinear problem due to nonlinear B-H relationship. Fewer number of degrees of freedom may require more number of Newton iterations to reach the convergence condition, which makes the computation slower. - Other physical effects are not considered in the current model in COMSOL Multiphysics such as: lamination effect, temperature rise effect, end effect of the electromagnetic field, etc. - The 3D simulation in Step 3 is still very challenging because it requires a good mesh resolution to have accurate results. Consequently, more memory is needed and computation time is spent. 6.2.2 Suggestions for future improvements There is room for improvement concerning the Computation time and the Model accuracy. Here are a few suggestion for future improvements: - Reduce the geometry by taking advantage of the symmetry of the motor. This may improve the computation speed but may not be applicable for modeling motors with defects. - It is from the author’s experience that, COMSOL Multiphysics runs three time faster for constant rotational velocity by using Prescribed Rotational Velocity node. Perhaps, the inefficiency of the Prescribed Rotation node can be improved in the future. - Include the eddy current or skin effect for the stator coils. - Include the lamination effect for the cores in Step1 and/or Step3 - Include the temperature effects. The motor generates a lot of heat during operation, which changes some material constants. - Find an accurate model to determine the resistance and inductance for the end windings and end-ring segments. - Design a smooth B-H curve. - Include the mechanical sources of noise and/or aerodynamic sources of noise. 50 Appendix A Matlab codes 1 3 5 7 9 %F i r s t Step : G e t t i n g Maxwell s t r e s s t e n s o r data from Comsol % t s t a r t and tend mark t h e time where t h e s i g n a l s a r e t o be e x t r a c t e d t s t a r t = 0.5001; % in seconds tend = 0 . 6 0 0 0 ; % in seconds tstep = 0.0001; %i n s e c o n d s n1 = c e i l ( t s t a r t / t s t e p ) ; % This s o l u t i o n number c o r r e s p o n d s t o t s t a r t n2 = c e i l ( tend / t s t e p ) ; % This s o l u t i o n number c o r r e s p o n d s t o tend d a t a s e t = ’ d s e t 1 ’ %must match t h e S o l u t i o n i n Comsol boundary = ’ s e l 1 1 ’%must match t h e S t a t o r −Air b o u n d a r i e s d e f i n e d i n Comsol model=M o d e l U t i l . model ( ’ Model2 ’ ) ; %t h e p o i n t e r t o t h e model i n Comsol 11 13 15 17 19 21 %Get t h e s t r e s s t e n s o r i n x−d i r e c t i o n on t h e boundary a t t s t a r t temp=mpheval ( model , ’rmm. nTx_Fstator ’ , ’ D a t a s e t ’ , d a t a s e t , ’ solnum ’ , n1 , ’ s e l e c t i o n ’ , boundary ) ; [ pd_pu , index , ~ ] = uni que ( temp . p ’ , ’ rows ’ ) ; % pd_pd s t o r e s t h e l o c a t i o n s o f u niqu e nodes num_nodes = s i z e ( pd_pu , 1 ) ; % number o f un ique nodes . Nodes mean locations N = n2−n1 +1; % number o f s a m p l e s s t r e s s X t=z e r o s ( num_nodes ,N) ; % s t r e s s X t h o l d s t h e s t r e s s t e n s o r i n x− direction s t r e s s Y t=z e r o s ( num_nodes ,N) ; % s t r e s s X t h o l d s t h e s t r e s s t e n s o r i n y− direction f o r k=n1 : n2 pd_bx = mpheval ( model , ’rmm. nTx_Fstator ’ , ’ D a t a s e t ’ , d a t a s e t , ’ solnum ’ , k , ’ s e l e c t i o n ’ , boundary ) ; pd_by = mpheval ( model , ’rmm. nTy_Fstator ’ , ’ D a t a s e t ’ , d a t a s e t , ’ solnum ’ , k , ’ s e l e c t i o n ’ , boundary ) ; [ pd_pu_x , index_x , ~ ] = u n iqu e ( pd_bx . p ’ , ’ rows ’ ) ; [ pd_pu_y , index_y , ~ ] = u niqu e ( pd_by . p ’ , ’ rows ’ ) ; 23 25 27 end s t r e s s X t ( : , k−n1+1) = pd_bx . d1 ( index_x ) ’ ; s t r e s s Y t ( : , k−n1+1) = pd_by . d1 ( index_y ) ’ ; 29 51 APPENDIX A. MATLAB CODES 31 33 35 37 %Second s t e p : Transform time domain t o f r e q u e n c y domain and w r i t e t o text f i l e s a m p l i n g _ d u r a t i o n = N∗ t s t e p ; % i n s e c o n d s df = 1.0/ sampling_duration ; % frequency r e s o l u t i o n numfre = c e i l (N/ 2 ) ; % number o f f r e q u e n c i e s f r e _ r a n g e = ( 0 : numfre −1) ’ ∗ d f ; % f r e q u e n c y r a n g e ampFilename = ’ TensorAmplitude . t x t ’ ; % t h i s f i l e s t o r e s t h e a m p l i t u d e s phaFilename = ’ TensorPhase . t x t ’ ; % t h i s f i l e s t o r e s t h e p h a s e s f c l o s e ( f o p e n ( ampFilename , ’w ’ ) ) ; % c r e a t e a blank f i l e f c l o s e ( f o p e n ( phaFilename , ’w ’ ) ) ; % c r e a t e a blank f i l e 39 41 f o r i =1:num_nodes % i t e r a t e o v e r a l l nodes %C a l c u l a t e t h e f o u r i e r t r a n s f o r m f o r t h e s t r e s s a t node l o c a t i o n pd_pu ( i , : ) a m p l i t u d e = z e r o s ( numfre , 5 ) ; % 5−column data phase = z e r o s ( numfre , 5 ) ; % 5−column data 43 45 % F i l l i n t h e f i r s t 3 column i n amp and pha % The f i r s t 3 columns i s [ x , y , f r e q u e n c y ] a m p l i t u d e ( : , 1 : 3 ) = [ repmat ( pd_pu ( i , : ) , numfre , 1 ) f r e _ r a n g e ] ; phase ( : , 1 : 3 ) = a m p l i t u d e ( : , 1 : 3 ) ; 47 49 51 53 55 end [ a m p l i t u d e ( : , 4 ) phase ( : , 4 ) ] = GetFrequencySpectrum ( s t r e s s X t ( i , : ) ) ; % compute a m p l i t u d e and phase o f s t r e s s X t a t t h e l o c a t i o n pd_pu ( i ,:) [ a m p l i t u d e ( : , 5 ) phase ( : , 5 ) ] = GetFrequencySpectrum ( s t r e s s Y t ( i , : ) ) ; % compute a m p l i t u d e and phase o f s t r e s s Y t a t t h e l o c a t i o n pd_pu ( i ,:) d l m w r i t e ( ampFilename , amplitude , ’−append ’ , ’ d e l i m i t e r ’ , ’ ’ ) ; d l m w r i t e ( phaFilename , phase , ’−append ’ , ’ d e l i m i t e r ’ , ’ ’ ) ; Listing A.1: MATLAB codes to extract Maxwell stress tensor, transform it to frequency domain and write the results to text files. 1 3 5 7 f u n c t i o n [ amplitude , phase ] = GetFrequencySpectrum ( s i g n a l ) % compute t h e f o u r i e r t r a n s f o r m o f t h e s i g n a l N = l e n g t h ( s i g n a l ) ; % g e t t h e number o f s a m p l e s i n t h e s i g n a l r e s u l t = f f t ( s i g n a l ) ∗ 2/N; % n o r m a l i z e t h e f o u r i e r t r a n s f o r m r e s u l t r e s u l t = r e s u l t ( 1 : c e i l (N/ 2 ) ) ; % t a k e o n l y t h e f i r s t h a l f o f t h e spectrum a m p l i t u d e = abs ( r e s u l t ) ; phase = a n g l e ( r e s u l t ) ; Listing A.2: Definition for the function GetFrequencySpectrum. The function ransforms a real-valued signal from time domain to frequency domain. 52 Bibliography [1] The National Archives. Noise Act 1996. HMSO, 1996. http://www. legislation.gov.uk/ukpga/1996/37/pdfs/ukpga_19960037_en.pdf (Accessed 13 Jul, 2014). [2] Antero Arkkio. Analysis of Induction Motors Based on the Numerical Solution of the Magnetic Field and Circuit Equations. PhD thesis, Helsinki University of Technology, 1987. [3] COMSOL. AC/DC Module Model Library Manual, Comsol version 4.4, Nov 2013. [4] COMSOL. AC/DC Module User’s Guide, Comsol version 4.4, Nov 2013. [5] COMSOL. Acoustics Module Model Library Manual, Comsol version 4.4, Nov 2013. [6] COMSOL. Introduction to the Acoustics Module, Comsol version 4.4, Nov 2013. [7] R. Escarela-Perez, E. Melgoza, and E. Campero-Littlewood. Time-harmonic modeling of squirrel-cage induction motors: A circuit-field coupled approach. In Proceedings of the COMSOL Conference 2008 Boston, 2008. [8] J. Güdelhöfer, R. Gottkehaskamp, and A. Hartmann. Numerical calculation of the dynamic behavior of asynchronous motors with comsol multiphysics. In Proceedings of the 2012 COMSOL Conference in Milan, 2012. [9] Jacek F. Gieras, Chong Wang, and Joseph Cho Lai. Noise of Polyphase Electric Motors. Taylor & Francis Group, LLC, 2006. [10] David J. Griffiths. Introduction to electrodynamics. Prentice Hall, 3rd edition, 1999. [11] David V. Hutton. Fundamentals of Finite Element Analysis. McGraw-Hill, 2004. [12] Steven G. Johnson. Notes on perfectly matched layers (pmls), Mar 2010. http: //math.mit.edu/~stevenj/18.369/pml.pdf (Accessed 15 Jul, 2014). 53 BIBLIOGRAPHY [13] Sami Kanerva. Simulation of Electrical machines, Circuit and Control systems using Finite element method and System simulator. PhD thesis, Helsinki University of Technology, 2005. [14] Moritz Kreutzer. Modelling of core noise from power transformers. Master’s thesis, KTH Royal Institute of Technology, 2011. [15] Kelvin C. Maliti. Modelling and Analysis of Magnetic Noise in Squirrel-Cage Induction Motors. PhD thesis, KTH Royal Institute of Technology, 2000. [16] Javier Martinez, Anouar Belahcen, and Antero Arkkio. A 2d fem model for transient and fault analysis of induction machines. Przeglad Elektrotechniczny, 88(7B):157–160, 2012. [17] David Meeker. Induction motor example, Aug 2004. http://www.femm.info/ wiki/InductionMotorExample (Accessed 10 Jul, 2014). [18] E. Oran Brigham. The Fast Fourier Transform. Prentice-Hall, Inc., 1974. [19] P.C. Sen. Principles of Electric Machines and Power Electronics. John Wiley & Sons, 1997. [20] Jonathan Sprooten. Finite element and Electrical circuit modelling of faulty Induction machines - Study of Internal effects and Fault detection techniques. PhD thesis, Universite Libre de Bruxelles, 2007. [21] Gilbert Strang. Computational Science and Engineering. Wellesley-Cambridge Press, 2007. [22] World Health Organization, Geneva. Guidelines for Community Noise, 1999. http://whqlibdoc.who.int/hq/1999/a68672.pdf?ua=1 (Accessed 13 Jul, 2014). 54 TRITA-MAT-E 2014:48 ISRN-KTH/MAT/E—14/48-SE www.kth.se

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