ANALYSIS, MEASUREMENT AND ESTIMATION OF THE CORE
LOSSES IN ELECTRICAL MACHINES
A Dissertation
Presented to
The Graduate Faculty of The University of Akron
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
Burak Tekgun
December, 2016
ANALYSIS, MEASUREMENT AND ESTIMATION OF THE CORE
LOSSES IN ELECTRICAL MACHINES
Burak Tekgun
Dissertation
Approved: Accepted:
Adviser Interim Department Chair Dr. Yilmaz Sozer Dr. Joan Carletta
Committee Member Interim Dean of the College Dr. Igor Tsukerman Dr. Donald P. Visco Jr.
Committee Member Dean of the Graduate School Dr. Alexis De Abreu-Garcia Dr. Chand Midha
Committee Member Date Dr. Dane Quinn
Committee Member Dr. Alper Buldum
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ABSTRACT
Energy efficiency has become one of the main concerns as a result of growing energy demands.
Since electric machines are one of the major energy consumers, their efficiency is critical. The losses in electric machines consist of copper loss, mechanical loss, and core loss. According to previous studies, core losses constitute 20%-25% of the total losses in sinusoidal voltage fed machines. These losses further increase when the machine is fed with pulse width modulated variable speed drives. The complex nature of core losses and the measurement limitations complicate the core loss analysis in electric machines due to the non-uniformity of flux densities.
Moreover, some additional effects such as minor hysteresis loops, high-frequency slot harmonics, and DC bias further increase the complexity of the analysis. Currently, core loss estimations are performed in finite element analysis (FEA) software packages using flux density waveforms in each element of the model. These methods are based on Steinmetz equation and the loss separation principle, with the coefficients of the respective nonlinear equations determined using the core loss measurement data under certain conditions. When it comes to the electric machine core loss estimation, these models result in more than 100% estimation error in some cases, even at rated conditions. Although there are mathematical hysteresis models that can accurately estimate the total core losses, these models are very complex, they require detailed material information, and they are hard to integrate into the FEA software. Therefore, a better method that can accurately estimate the core losses in electric machines with a less complicated procedure is highly desirable.
In this dissertation, core loss measurements and estimations based on the actual flux density waveforms of the electric machines are investigated. An approach has been developed to estimate the core losses under any operating condition. The study is organized as three related tasks. The
iii first one is to develop a test system that can generate the desired flux density waveforms in a core under test (CUT) with/without a DC offset. This task includes hardware and control algorithm development sub-tasks. The second task is to obtain the flux density waveforms in the regions that have a uniform loss distribution in the machine, and then, estimate the total core loss by generating the flux density waveforms in a toroidal core made out of the same material as is used in the machine. The final task is to measure the core losses at the load conditions. After completing all three tasks, the results show that the core loss estimation technique developed in this dissertation decreases the estimation error to less than 10%.
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ACKNOWLEDGEMENTS
I would like first to express special thanks to my advisor, Dr. Yilmaz Sozer, for his unlimited support, help and continuous encouragement throughout my doctoral study.
I also would like to give my thanks to my dissertation committee members, Drs. Igor
Tsukerman, Alexis De Abreu-Garcia, Dane Quinn, and Alper Buldum for their valuable comments and suggestions to improve the quality of the dissertation.
Many thanks to our distinguished sponsor ABB for funding our project and, Drs, Rich
Schiferl, Stephen D. Umans, Boris Shoykhet, Parag Upadhyay, and Steven Englebretson for their valuable comments, suggestions.
My thanks also go to the Ministry of National Education of Turkey for the financial support of my graduate studies.
Finally yet importantly, I want to thank my wife, my parents, and all my colleagues for their support and encouragement throughout my doctoral study.
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TABLE OF CONTENTS
Page
LIST OF TABLES ...... ix
LIST OF FIGURES ...... x
CHAPTER I. INTRODUCTION ...... 1
II. LITERATURE REVIEW ...... 5
2.1. Factors Influencing Iron Losses ...... 5
2.2. Overview of the Core Loss Models ...... 7
2.2.1. Steinmetz Equation Based Models ...... 7
2.2.2. Loss Separation Models ...... 10
2.2.3. Hysteresis Models ...... 13
2.3. Measurements and Accuracy ...... 16
2.4. Core Loss Test Setup and Control ...... 22
2.5. Conclusions ...... 26
III. PROPOSED CORE LOSS ESTIMATION METHOD ...... 27
3.1. Accuracy of Existing Estimation Methods ...... 27
3.2. Typical Flux Density Waveforms of an IPMSM ...... 29
3.3. Effect of Loading ...... 31
3.4. Proposed Core Loss Estimation Method...... 32 vi 3.5. Flux Regulation for the Core Loss Tester ...... 33
3.6. Core Loss Test Setup ...... 34
3.6.1. Hardware ...... 35
3.6.2. Control Mechanism ...... 36
3.6.3. System Modeling ...... 37
3.6.4. Proposed Controller Design ...... 40
3.7. Conclusions ...... 43
IV. FLUX CONTROLLED CORE LOSS TEST SETUP DEVELOPMENT ...... 44
4.1. Control Design and Implementation with the Simulation Platform ...... 44
4.2. Core Loss Test Setup Hardware Development ...... 49
4.3. Experimental Results ...... 58
4.4. Conclusions ...... 61
V. CORE LOSS ESTIMATION OF AN IPMSM ...... 62
5.1. Determination of Loss Regions ...... 62
5.2. Determination of Flux Density Waveforms of the IPMSM ...... 64
5.3. Core Loss Tests under Actual Flux Density Waveforms ...... 67
5.4. Total Core Loss Estimation ...... 79
5.5. Conclusions ...... 82
VI. CORE LOSS MEASUREMENT OF AN IPMSM ...... 83
6.1. abc to dq0 Transformation ...... 84
6.2. dq Model of an IPMSM ...... 85
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6.3. Parameter Determination of an IPMSM ...... 89
6.4. Core Loss Measurement of an IPMSM ...... 91
6.4.1. Development of Motor Test Setup ...... 92
6.4.2. Motor Non-Idealities and Performance Tests ...... 94
6.4.3. Motor Test Results ...... 96
6.4.4. Levenberg-Marquardt Recursive Parameter Estimation ...... 97
6.4.5. Core Loss Resistance Estimation ...... 98
6.5. Conclusions ...... 100
VII. SUMMARY AND FUTURE WORK ...... 101
7.1. Summary ...... 101
7.2. Future Work ...... 104
REFERENCES ...... 105
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LIST OF TABLES
Table Page
1. System parameters...... 44
2. Core loss per mass data for each stator loss region at the unloaded condition...... 79
3. Core loss per mass data for each stator loss region at rated loading condition...... 80
4. Core loss per mass data for each rotor loss region at rated loading condition...... 81
5. Motor test results for unloaded and rated load conditions at 1800rpm...... 96
ix
LIST OF FIGURES
Figure Page
2.1. Existing core loss models ...... 7
2.2. Preisach hysteresis model ...... 13
2.3. Structure of the B-H loop core loss measurement method ...... 17
2.4. (a) Epstein, (b) toroid, and (c) single sheet core loss testers ...... 19
2.5. Typical core loss tester system ...... 22
3.1. (a) Pure sine wave and (b) sine wave with harmonics ...... 28
3.2. Meshes in the FEA simulations ...... 28
3.3. Stator flux density waveforms of an IPMSM...... 29
3.4. Rotor flux density waveforms of an IPMSM ...... 30
3.5. Flux paths of a 48 slot, 8-pole IPMSM ...... 31
3.6. Proposed core loss estimation procedure ...... 32
3.7. Sinusoidal flux density generation in a toroidal transformer with voltage and flux control structures...... 34
3.8. Proposed core loss test system ...... 35
3.9. Simplified control structure ...... 37
3.10. Equivalent circuit model ...... 38
3.11. Simplified equivalent circuit model ...... 38
3.12. Proposed flux controller ...... 42
4.1. Flux density under periodic distortion and instantaneous error variation ...... 46
4.2. (a) Error convergence of sense coil voltage in every cycle, (b) error variation ...... 47
4.3. Corresponding output waveforms in steady state condition...... 47
x
4.4. Primary winding current and primary winding voltage, sense coil voltage and calculated flux density waveform at 100 Hz ...... 48
4.5. Primary winding current and primary winding voltage, sense coil voltage and calculated flux density waveform at 400 Hz with 1.4 T DC offset ...... 48
4.6. Primary winding current and primary winding voltage, sense coil voltage and calculated flux density waveforms 1000 Hz fundamental ...... 49
4.7. H-bridge and half bridge topologies ...... 49
4.8. The H-bridge inverter made out of two SiC MOSFET modules ...... 51
4.9. Right-hand rule ...... 52
4.10. Parallel and coplanar sheet orientations ...... 52
4.11. Field generated by the current passing through a conductor...... 53
4.12. Fields canceling each other with the parallel and coplanar plate overlapping ...... 53
4.13. Parallel and coplanar plate inductance approximations ...... 53
4.14. DC bus and the capacitor bank ...... 54
4.15. (a) DC bus design drawing, (b) completed DC bus (top), and (c) (side) ...... 55
4.16. Gate drive circuitry ...... 56
4.17. The control board ...... 56
4.18. Complete system schematic ...... 57
4.19. Complete implemented core loss test system...... 57
4.20. Primary winding voltage, primary winding current and the secondary winding voltage waveforms at 125 Hz. (blue: primary winding voltage; purple: primary winding current; red: secondary winding voltage) ...... 58
4.21. Flux density, sense coil voltage, corresponding primary winding current and primary winding voltage waveforms at 250 Hz ...... 59
4.22. Flux density waveform at the yoke of an IPMSM, corresponding output signals at 80 Hz . 60
4.23. Flux density waveform at the rotor of an IPMSM, corresponding sense coil voltage to generate the flux density waveform, primary winding current and voltage waveforms at 480 Hz fundamental frequency ...... 60
5.1. Stator core loss distribution and the selected regions ...... 63
xi
5.2. Rotor loss distribution using color scales that have maximum limit core loss per volume value (a) 124000 W/m3, (b) 12400 W/m3, and (c) 4100 W/m3 (d) Rotor core loss regions. 63
5.3. Stator flux density waveforms of an IPMSM at unloaded conditions ...... 64
5.4. Rotor flux density waveforms of an IPMSM at unloaded conditions ...... 65
5.5. Stator flux density waveforms of an IPMSM at unloaded conditions ...... 66
5.6. Rotor flux density waveforms of an IPMSM at unloaded conditions ...... 67
5.7. Prepared toroidal transformers from the stator and rotor laminations...... 68
5.8. Flux density waveform of the stator sensor S1 and the corresponding secondary winding voltage...... 69
5.9. Reference and generated secondary winding voltage ...... 69
5.10. Reference and generated flux density waveform of S1 ...... 70
5.11. Reference and generated flux density waveform of S5 ...... 70
5.12. Reference and generated flux density waveform of S6 ...... 71
5.13. Flux density waveform of the motor taken from the rotor sensor R11 ...... 71
5.14. Reference and generated flux density waveform of S1 at rated loading conditions ...... 72
5.15. Reference and generated flux density waveform of S2 at rated loading conditions ...... 72
5.16. Reference and generated flux density waveform of S3 at rated loading conditions ...... 73
5.17. Reference and generated flux density waveform of S4 at rated loading conditions ...... 73
5.18. Reference and generated flux density waveform of S5 at rated loading conditions ...... 73
5.19. Reference and generated flux density waveform of S6 at rated loading conditions ...... 74
5.20. Reference and generated flux density waveform of S7 at rated loading conditions ...... 74
5.21. Reference and generated flux density waveform of R1 at rated loading conditions ...... 74
5.22. Reference and generated flux density waveform of R2 at rated loading conditions ...... 75
5.23. Reference and generated flux density waveform of R3 at rated loading conditions ...... 75
5.24. Reference and generated flux density waveform of R4 at rated loading conditions ...... 75
5.25. Reference and generated flux density waveform of R5 at rated loading conditions ...... 76
5.26. Reference and generated flux density waveform of R6 at rated loading conditions ...... 76
xii
5.27. Reference and generated flux density waveform of R7 at rated loading conditions ...... 76
5.28. Reference and generated flux density waveform of R8 at rated loading conditions ...... 77
5.29. Reference and generated flux density waveform of R9 at rated loading conditions ...... 77
5.30. Reference and generated flux density waveform of R10 at rated loading conditions ...... 77
5.31. Reference and generated flux density waveform of R11 at rated loading conditions ...... 78
5.32. Reference and generated flux density waveform of R12 at rated loading conditions ...... 78
5.33. Reference and generated flux density waveform of R13 at rated loading conditions ...... 78
6.1. Three phase reference axes and rotating dq reference frame...... 84
6.2. IPMSM (a) q and (b) d axis equivalent circuit models in rotor reference frame...... 88
6.3. Winding resistance measurement procedure ...... 89
6.4. Lq and Ld inductances of an IPMSM ...... 90
6.5. Inductance measurement circuit...... 90
6.6. Phase inductance vs. rotor position ...... 91
6.7. The connection diagram of the motor test system ...... 93
6.8. Motor under test and its power electronic driver ...... 94
6.9. Motor test bench ...... 94
6.10. d and q axis inductance variation versus the phase current ...... 95
6.11. IPMSM (a) d and (b) q equivalent circuits with core loss components...... 95
6.12. Trajectories of the estimated parameters ...... 99
6.13. Estimated and measured output variables (a) torque (b) efficiency ...... 99
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CHAPTER I.
INTRODUCTION
Energy efficiency has become one of the main concerns worldwide due to the rapid growth of global energy consumption. Electric machines dominate electric energy consumption. Therefore, the efficiency improvement of machines has gained significant interest for the past few decades.
Over the years, electric machine efficiencies have enhanced in parallel with the design techniques.
Better understanding of loss components allowed researchers to analyze and reduce the losses, and consequently, to improve the machine efficiencies [1]–[3]. In the last decade, the increased popularity of electrical vehicles, as well as government regulations concerning electric machine efficiency [4], further stimulated the research in this area. Several methods have been developed to improve efficiency and reduce the losses, especially electromagnetic iron losses as being dominant over other loss components. The complexity of the dynamic behavior of core losses and the measurement limitations make the analysis challenging in electric machines due to the non- uniformity of flux densities. Moreover, some additional effects such as magnetic field distortion, rotor AC fields, rotating magnetic fields, high-frequency slot harmonics, and minor hysteresis loops caused by pulse width modulation (PWM) inverters increase the complexity of the core loss analysis [5]. These additional losses are often lumped together into “stray load losses” that, instead of being calculated, are rather simply assumed to be certain percentage of the motor output power.
In some cases, stray load losses are found to be up to 2% of the motor output power. Prediction and consequent reduction of these losses is extremely important for the optimal design of highly efficient motors and generators.
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From a physical perspective, the losses in ferromagnetic materials are all based on Joule heating
[6]. The spin relaxation losses are negligible in electric machines [4], because their effect is only noticeable at the MHz frequency ranges and above [7]. Thus the two loss components, hysteresis and eddy current losses are caused by the same physical phenomena: any change in magnetization causes the domain walls to move, which induces eddy currents resulting in Joule heating [4], [6],
[8]. It is a fact that hysteresis losses occur even at zero frequency. While macroscopic magnetization changes slowly, the magnetization inside the domain changes rapidly and thus induces eddy currents [6], [9]. Therefore, separating losses into loss types such as hysteresis losses, eddy current losses and excess losses is an engineering-oriented empirical approach. Instead of explaining the physical phenomena, this is just an attempt to separate various physical effects in electromagnetic systems based on the frequency and flux density [4]. Such empirical approaches are typical in FEA due to their simple structure. However, the loss estimation ability of these models becomes poor for electric machines, where flux densities have more than one harmonic component and are non- uniform throughout the machine. These models are typically fit into a non-linear approximating function for certain conditions, and whenever these conditions change, the estimation accuracy is dramatically reduced.
The objective of this research is to develop a method that can accurately estimate the core losses in electric machines during the design stage. The analysis of the core losses requires both measurements and the technical data of the material that is going to be used in the machine. The analysis becomes more realistic and accurate when it is performed with the same flux densities that exist in the material. In order to perform this analysis under realistic conditions, three objectives are considered in this research:
. To develop a core loss test system which is able to generate any kind of flux density
waveform in the core under test (CUT). The test system includes hardware
development and the control algorithm development and implementation using a
digital signal processor (DSP).
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. To estimate the total core losses of the machine based on the actual flux waveforms
that exist in various parts of the machine with the developed test system.
. To develop a core loss measurement technique for electric machines in order to verify
the accuracy of the proposed core loss estimation method.
The outline of the thesis is as follows. Chapter II provides a review of the core loss models, core loss measurements, and the core loss testers. In addition, the single-phase inverter control strategies that are used for the new core loss test unit are investigated. In Chapter III, the proposed core loss estimation method for electric machines is explained and the development of a new core loss tester that controls the flux density in the magnetic core under test is presented. Chapter IV first presents the control design, simulation and flux controller, followed by the hardware development and experimental results. In Chapter V, an interior permanent magnet motor’s core loss is estimated through the proposed estimation method. First, the loss regions of the motor are determined and then, all of the flux density waveforms from these regions are generated in a toroidal transformer using the flux controller we developed. The transformer is made of the same material used in the motor. Therefore, the core losses are estimated for unloaded and rated loading conditions. Chapter VI goes through the basics of the direct and quadrature (dq) axis motor model, shows how to determine the electrical parameters of an interior permanent magnet motor, and discusses the non-idealities of the motor that arise due to the saturation, temperature variation, and loading. Additional losses arise from the aforementioned non-idealities along with spatial harmonics, DC offset flux, etc. These additional losses contribute to the overall core losses in the motor. Hence, the total core loss, assumed to be the loss resulting from subtracting the copper and mechanical losses from the total loss, is compared with the estimated core losses. Moreover, the non-idealities are modeled in the equivalent circuit model and the core losses are represented as two additional resistors for each circuit. The unknown core loss resistors are determined using a recursive least squares estimation algorithm. The comparison of the total core loss determined
3 through performance tests and the estimation conducted with the proposed method, verified that the proposed method has less than 10% estimation error at the rated loading condition. Finally, a summary of the dissertation, conclusions and possible future extensions are provided in Chapter
VII.
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CHAPTER II
LITERATURE REVIEW
There are various core losses estimation methods provided in the literature. These methods can be used during the design stage of an electric machine. Before proceeding to the detailed review of the iron loss models, factors that effect the core losses during the manufacturing process and assembly will be reviewed in the following section. Later, the development of the core loss models is investigated in detail in Section 2.2. In Section 2.3, core loss measurement techniques and accuracy considerations are reviewed. Existing core loss testers and their waveform generation methods are investigated. Control methods of single-phase inverters are reviewed in Section 2.4 to select an appropriate control method for the core loss tester.
2.1. Factors Influencing Iron Losses
The investigation of the core losses requires the measurement and technical data of the electrical steel (silicon steel) that is used in the electric machine. A close examination shows that the core losses vary even between the same catalog type of electrical steels due to contamination during the manufacturing, different alloy composites, grain sizes, non-isotropic effects, and a lack of information of the measured magnetic properties [10]. For low accuracy core loss models, these variations are negligible and not considered. However, the maximum guaranteed values of the magnetic properties and losses are determined from the catalog values. The core loss data provided by the manufacturers work well for low accuracy core loss models. However, in case the exact values are needed, especially for machines in and above the MW range, manufacturers provide certificates for the lamination rolls from the tests performed after manufacturing.
5
The influences on core losses that occur during the manufacturing process are very important and need to be investigated deeply. The analytical loss models provide reasonable results and agree with the loss measurement data; however, the estimation accuracy when it comes to the loss determination of the final assembled machine is considerably reduced.
The magnetic properties of the materials are affected from cutting and punching. This process creates inhomogeneous stresses inside the material causing the hysteresis curve to differ in various regions of the steel sheets. This effect also depends on the alloy composite and the grain size, which are actually the main factors that influence the hysteresis curve, especially between 0.4 T to 1.5 T
[11], [12]. The regions affected from the cutting and punching process extend to up to 1 cm from the cutting edge. The permeability of these regions are relatively low [13], [14], which results in a higher core loss, especially in the motors that have parts smaller than 1 cm in width [13]. For this kind of motors, the effect of punching should be considered in the core loss calculations [15]. An expression can be found in [15] to calculate the core losses in the teeth of an induction machine. A method that takes into account the cutting-edge effects in design tools is given in [16], [17]. Another factor to be considered during the core loss measurements is that the tested steel sheet width should be adjusted when machining small parts [4], [18], [19]. For small motors, the magnetic properties influenced by cutting and punching can be reduced with a stress-relief annealing process after machining the sheets. However, this annealing process may damage the thin electrical insulation layer on the sheet and cause a short circuit between the laminations.
Degradation of the magnetic properties can also occur during the stacking and welding process.
Welding the steel sheets together causes short circuits between the laminations and therefore core losses increase [12], [20]. Magnetic degradation effects due to the manufacturing are investigated in [21] for an interior permanent magnet synchronous motor (IPMSM). An empirical approach to estimate the effects of the manufacturing process is developed in [22] for an induction motor.
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2.2. Overview of the Core Loss Models
Quantifying electromagnetic iron losses has been a common interest of both physicists and engineers. Physicists` approaches are mainly based on the development of mathematical models that explain the dynamics of the hysteresis curve of the material, while the engineers` approaches are based on empirical models that use some excitation and geometric quantities. The efforts of both researchers have led to the development of numerous core loss models. Figure 2.1. Existing core loss models illustrates the most commonly used models for core loss calculations.
Core Loss Models
Steinmetz Equation Based Loss Separation Based Models Mathematical Hysteresis Models Models
Modified Generalized Natural Eddy Current Anomalous Friction Like Steinmetz Steinmetz Steinmetz Separation into Loss Surface Viscosity Energy Based Factor for Hysteresis Equation Equation Equation Classical and Model Based Model Vector Model Excess Loss Model (MSE) (GSE) (NSE) Excess Losses
Improved Generalized Steinmetz Equation (iGSE)
Figure 2.1. Existing core loss models.
2.2.1. Steinmetz Equation Based Models
Charles P. Steinmetz put the first approach forward in 1894 [23] to calculate iron losses. In this approach, the core losses are expressed as an exponential function depending on the peak magnetic flux density and the frequency as follows:
ˆ Pfe C SE f B (1)
where, Pfe is the time averaged iron loss per volume, f is the frequency, Bˆ is the peak flux density level, and CSE, α, and β are named as the Steinmetz coefficients, which are determined by fitting the model to the loss measurement data. The major drawback of this method is that it is only valid for sinusoidal excitation for a limited frequency and flux density levels. Currently, there are models
7 developed that take arbitrary waveforms into account using the same concept with some modifications on the model.
The flux density waveforms are not always sinusoidal in various applications such as power converters and inverters where DC premagnetization and the high-frequency components exist. It is a well-known fact that, the Steinmetz coefficients vary with frequency. To overcome this problem, several modifications were proposed on the Steinmetz equation to extend the model. One of these models is the Modified Steinmetz Equation (MSE) [8], which can calculate the core loss caused by arbitrary waveforms. In this model, an equivalent frequency expression is introduced.
The equivalent frequency is a function of the macroscopic re-magnetization rate [24], which is proportional to the rate of change of the flux density (dB/dt). The equivalent frequency is calculated using the following formula.
2 2 T dB f dt (2) eq 22 B 0 dt where, ΔB is the difference between the maximum and minimum flux density quantities in a period of the alternating flux density. Substituting (2) into (1) gives the following expression.
1 ˆ PCfBffeSEeqr (3) where, fr is the remagnetizing frequency, CSE, α, β, are the fitting coefficients. With this method, core loss caused by arbitrary waveforms, even DC premagnetization can be considered with an additional correction factor. The major drawback of this method is that its accuracy decreases even with a small fundamental frequency variation. An upgraded version of MSE, called Generalized
Steinmetz Equation (GSE), is developed in [25]. In this method, the core losses are calculated using the flux density and the rate of change of flux density as:
T 1 dB P C B(t) dt (4) fe SE T0 dt where T is the signal period. The DC premagnetization is intuitively considered in this model without requiring a correction factor or measurements, which is the advantage of the GSE over
8
MSE. In addition, the equivalent frequency or equivalent amplitude can be derived with the GSE to be applied to the classical Steinmetz Equation.
The GSE has an accuracy limitation when more than one peak appear throughout the period
[4], especially when the third, or close to the third, harmonic becomes significant. In order to eliminate this problem, another method, called improved Generalized Steinmetz Equation (iGSE), has been introduced in [26]. In this method, a recursive algorithm separates the flux waveform into minor and major loops and then calculates the iron loss separately with the following expression.
T 1 dB PCBdt (5) fe, xSE Tdt0 here, ΔB is determined for each minor loop. The disadvantage of this method is that it is not sensitive to the premagnetization.
Another approach based on Steinmetz equation, reported in [27], is called the Natural
Steinmetz Equation. This method also uses ΔB, but mostly focuses on triangular waveforms such as the ones in DC-DC converters. Instead of splitting the waveforms into major and minor loops, the equation is directly applied to the full period.
BdB C T Pdt SE (6) fe 2 Tdt0
In summary, the approaches based on the Steinmetz equation and Steinmetz parameters is that of predicts the core losses practically without the need of the measured material loss data or BH curve. The Steinmetz parameters are mostly supplied by the manufacturers or can be easily extracted from the manufacturers` loss measurement data. The major drawback of these approaches is the variation of the Steinmetz parameters with the frequency and the DC premagnetization, which makes them less accurate compared to the hysteresis loss models. Recently, an approach based on the Steinmetz parameter variation as a function of the premagnetization was proposed in [28], where the authors suggested that the magnetic material manufacturers supply the Steinmetz parameters vs. the DC premagnetization tables for their products. Among the mentioned Steinmetz
9
Equation models, only MSE is tested for electrical steel, others are developed mainly for ferrite cores.
2.2.2. Loss Separation Models
The second approach is presented as an extension of the first approach in [4], where the core loss is split into hysteresis losses and dynamic eddy current losses, as shown in the following expression:
ˆˆ222 PPPCffehystechystec BCfB (7) where, Physt is the average hysteresis loss per volume, Pec is the average dynamic eddy current loss per volume, Chyst and Cec are the loss coefficients. In this method it is assumed that, at low frequencies, the hysteresis loss is proportional to the hysteresis loop of the magnetic material and the dynamic Eddy current loss is calculated using Maxwell equations, as follows:
2 2 dBt d dt (8) P ec 12 where, d is the thickness of the lamination, ρ is resistivity, and ϒ is the density of the material. The method has good accuracy for some Nickel-Iron alloys. However, its accuracy decreases when it comes to steel alloys (Silicon-Iron alloys) [4]. In order to improve the estimation accuracy for electrical steel, a correction factor called anomalous loss (excess loss) factor was introduced in
[29]. With this correction factor, the model was extended to the following structure:
ˆˆ22 2 PPPCfehysta echysta f BC ec f B (9)
where, a is the anomalous loss factor, defined by the following expression [29]
Pec, meas a (10) Pec, calc
10
The loss separation method was further improved by incorporating an additional term, called excess or anomalous loss component, into the original formulation. Similarly to (9), the new loss component is also a function of the flux density and the frequency, as follows:
ˆˆˆ2221.51.5 PPPPCffehystecexchystecexc BCfBCfB (11) where, Pexc is the averaged excess loss per volume, and Cexc is the excess loss coefficient.
Considering the excess loss in (11) depends on empirical factors, a new theory and a statistical model are developed by Bertotti [30]. In this model, magnetic objects are introduced to calculate the core losses. The loss factor Cexc is formed based on the physical description of the active magnetic objects and the domain wall motion [30]–[33] as follows :
Cexc SV G 0 (12)
where, S is the cross-sectional area of the steel sheet, V0 is the statistical distribution coefficient that characterizes the local coercive field distribution (coercive field is a measure of the capability of a magnetic material to withstand an external magnetic field without being demagnetized) based on the grain size (particle size) [32]. In (12) is the conductivity and G is the eddy current damping factor. The model only holds if the skin effect is neglected [34]. Further analysis on the properties of the Bertotti`s statistical model coefficients is presented in [35].
There are more analytical iron loss models developed after Steinmetz approximations. In general, these models separate the losses into static and dynamic losses. One of them is the rotational loss model [35]–[40] based on the rotating magnetization on the magnetic materials. The most used loss model that considers the rotational losses is the loss separation model after the magnetizing process. In this model, rotational losses, higher order harmonic losses, and linear magnetizing losses are combined to calculate the core losses [4], as follows:
Pfe b1 P lm b 2 P rot b 3 P hf (13)
where, Plm is the linear magnetizing loss, Prot is the rotational iron loss, Phf is the loss caused by higher order harmonics. Further, b1, b2, and b3 are the loss coefficients that are obtained from some
11 geometric quantities and curve fitting. In electric machines, the rotational magnetizing only occurs at the tooth tips close to the air gap. The remaining parts, such as the stator yoke and the upper section of the teeth, are subject to the linear magnetizing. Hence, the linear magnetizing losses always dominate the rotational magnetizing losses. Therefore, after the magnetizing process the loss separation model does not provide many advantages in electric machines [4].
In another approximation [41], the rotational magnetizing and Bertotti`s loss separation model are combined by introducing a correction factor to incorporate the rotational losses into the Bertotti model; that is,
ˆˆˆ222 a3 Paffe BaaBfB214 (14) where, a1 = Cec (excess loss coefficient), a3 and a4 are the high order loss factors which make the model representation more accurate at large fields. Further, a2 = Chyst k, where k is defined as:
B kr11min (15) Bmax here, r is the rotational hysteresis factor, Bmin and Bmax are the minimum and maximum flux density values in one period.
Another method developed for permanent magnet synchronous machines is presented in [42].
This method takes the magnetizing from different directions into account by separating the core losses in the stator teeth and the yoke of the machine. The flux density waveforms of the stator teeth and the yoke are assumed to be piecewise linear. This allows the rate of change of the flux density to be expressed in terms of the number of poles and phases.
Fourier analysis is also used in core loss calculations [43], [44]. The loss is calculated using the harmonic components of the magnetic field and magnetic field density; that is,
ˆˆ Pffeiii i B H sin (16) i1
12
ˆ ˆ th where, f is the fundamental frequency, i is the order, Bi and Hi are the peak values of the i
ˆ ˆ harmonic component, and i is the angle between Bi and Hi . As an extension to this model, a higher order harmonics correction factor is added to the model in [4].
The Preisach model [45] and the Jilles-Atherton model [46], [47] are the models that use the actual material properties such as the BH loop to calculate the core losses accurately even for arbitrary waveforms. Since these mathematical models mimic the actual material behavior, DC offset and the minor hysteresis loops are intuitively considered in the models. The following section gives a detailed explanation of mathematical hysteresis models.
2.2.3. Hysteresis Models
As stated earlier, the models based on Steinmetz Equation and loss separation are engineering- oriented approaches. Rather than explaining the physical phenomena they attempt to separate the various physical influences on electromagnetic systems based on the frequency and flux density.
Models that are more accurate use the material properties and the full hysteresis curves to explain the physics behind the electromagnetic losses, which is the hysteresis behavior. The origin of the first physical hysteresis model can be tracked back to the landmark research of Preisach in 1935
[45], [48]. It reproduces the one-dimensional hysteresis curve from an infinite number of parameters [48], [49] as shown in Figure 2.2.
y(t) n =3
y(t) 1 n R x(t) α1β1 x(t) α β R 1 1 α2β2 α1 - 1 β1 x(t) . y(t) x(t) . y(t) y(t) . α2 β2 R αnβn y(t) x(t)
x(t) α3 . β3 . x(t) .
Figure 2.2. Preisach hysteresis model.
13
Since its introduction, the Preisach model has remained known in the electromagnetic area and has led to considerable research in the field. However, the accuracy of the resulting hysteresis curve is not enough to represent the material’s behavior when it comes to expressing the losses in and the forces on the material [50]. Energy balance needs to be provided in the model. Since the model lacks the real interpretation of the energy, additional modifications have been introduced into the model [51], [52]. Jiles and Atherton have put another well-known and well-documented hysteresis model forward in 1986 [53]. The basic assumption in the model includes the real material model considering the energy balance of the electromagnetic system. There are several other hysteresis- based iron loss models in the literature with several modifications and improvements. Although these models require more measurements and data about the electromagnetic material, they result in better accuracy than simple Steinmetz Equation or loss separation models. Among other hysteresis models, some of them come forward in terms of applicability [4]. The latter include the
Dynamic Preisach model, Magneto-dynamic Viscosity Based hysteresis model, Loss Surface model, Friction Like Hysteresis model, and Energy Based Hysteresis model, some of which will be discussed next in detail.
As an extension of the original Preisach model, the dynamic Preisach model [48], [54], [55] introduces a rate dependent factor for each elementary Preisach loop of the hysteresis model. The delay time between the instantaneous flux density and the magnetic field has been taken into account to find the rate dependent function, which makes it possible to mimic the effect of enlargement of the hysteresis loop while the frequency is increasing. Therefore, the model is able to represent the excess losses by using a dipole function that can vary from -1 to 1, instead of switching between -1 and 1, as was the case in the original Preisach model. The dipole functions are determined by the material properties and dynamic hysteresis measurements. The model is successfully applied to electric machines using FEA in [56], [57].
14
The loss surface model is another hysteresis model introduced in 2000 [58], where the magnetic field is calculated as a characteristic surface function. The surface function S is represented as a combination of static and dynamic parts; that is,
dBdB SHBHBHB,,statdyn (17) dtdt
dB where, B is the magnetic flux density and is the rate of change of the magnetic flux density dt with respect to time. In this model, the magnetic field H is connected to the sheet surface with the flux density in the thickness of the sheet. The rate independent Preisach model is used to model the static part of the magnetic field using the first-order reversal curves and the static hysteresis loop measurements. In order to model the dynamic part of the magnetic field, the static field is subtracted and then, two linear analytical equations that are connected with a second order polynomial are used to describe the high and low values of the rate of change of the magnetic flux density. The
Loss Surface Model is used in the commercial software by Cedrat Flux for various electric machines [59].
Another hysteresis model is the Magneto-dynamic Viscosity based model [60]. Similar to the loss surface method, it uses the rate independent Preisach hysteresis model. The delay time between the magnetic flux density and the magnetic field is modeled with a viscous type differential equation. The shape of the dynamic part of the loop is determined with this differential equation.
This model relies on the static hysteresis loop, the thickness of the first-order reversal curves sheet, and the material resistivity.
Another approach is the Friction-like Hysteresis model, which is based on the dry friction –like pinning [61], [62]. The model combines the properties of the Preisach and Jilles-Athertorn models.
The basic assumption in this model is that the magnetizing can be expressed as a superposition of the influence of a finite number of pseudoparticles. The free energy of each individual particle is assumed to be the summation of a convex component that represents the relationship between the
15 magnetic field and the flux density. The ripple component represents the effect of the magnetic domain wall movements that cause the minor hysteresis loops.
Friction behavior is introduced in [50] as another method for iron loss modeling. In this approach, the hysteresis is modeled based on energy balance where the magnetic power dissipation is represented with a friction-like force. Therefore, the stored magnetic energy and the dissipated energy can be calculated instantaneously. The model can be applied to the 3D numerical analysis due to its vectorial structure [63].
In summary, there is a wide range of available core loss models for estimating the core losses in electric machines. The Steinmetz Equation based models and the loss separation models are the most common models used for core loss estimations due to their simple structure and straightforward calculation procedures. These models provide easy integration with the FEA software and rough core loss estimation. On the other hand, the mathematical hysteresis models result in much higher accuracy. However, these models require knowledge of the material properties and measurement data performed under various conditions. Moreover, the integration of these models into the FEA simulations is not straightforward and simulations take much longer time to run.
2.3. Measurements and Accuracy
Many researchers have proposed different methods for accurate measurement of core losses
[64], [65]. The thermal approach [66] uses an isolated chamber to measure the temperature difference between the inlet and outlet coolant. Although this method is considered as universal, the separation of copper losses is difficult and time-consuming. Another method was proposed for high-frequency measurements [67]. It uses the resonance of the capacitor and the inductor wound on the core under test conditions. Hence, the power taken from the source has only active components, which are copper and core losses. However, the method is only valid for sinusoidal voltages. The most common method is the four wire core loss measurement presented in [68], also
16 called as BH loop measurement. In this method, the core is wound as a transformer with two windings: the primary winding is used for the excitation and the secondary for sensing the induced voltage. Core loss can be calculated by integrating the product of the measured voltage and current passing through the primary winding. Although, this method has some drawbacks, such as excluding the winding loss from the measured loss and the phase discrepancy due to the current sensing resistance parasitic, wound transformer parasitic, or the physical limitations of the measurement devices [69], [70], it is considered to be the best way to measure core losses due to its speed and accuracy [71].
The BH loop method, which is widely used [28], [72], employs two windings placed on the core under test (CUT). Figure 2.3. Structure of the B-H loop core loss measurement method.
vs(t)
N2
N1
i(t)
Figure 2.3. Structure of the B-H loop core loss measurement method.
The primary winding is for excitation, and the secondary is for sensing the induction. Integrating the sensed winding voltage gives the flux density as:
1 T B()() t v d (18) 0 s NA2 e where, N2 is the number of turns of the secondary winding and Ae is the effective core cross- sectional area of the CUT. The sensed current from the primary winding gives the magnetic field strength H,
N1 i t Ht (19) l e
17 where, N1 is the number of turns of the primary winding and le is the magnetic path length. The loss per unit volume can be represented as:
P f H d B (20) V which may also be written as
T N fitvtdt 1 P 0 12N (21) 2 VAl ee where, P is core loss, V is the volume of the core, f is frequency, and vs(t) is the sensed voltage from the secondary winding. The discrete form of Equation (21) can be described as:
1 K N ikvk 1 P KN 12 (22) 0 2 VA l ee where, K is the number of samples in one period.
The core loss measurements performed by magnetic electrical steel manufacturers are provided for a few operating points such as 50 Hz - 60 Hz at 1 T, 1.5 T, etc. This data is used to obtain the
BH curve, which can then be imported into the FEA software for estimating the core losses. There are several core loss measurement standards for manufacturers to follow [73]–[75]. Although, different manufacturers follow different standards and nomenclature for their product, the material properties such as core loss, resistivity, and permeability are considered to be the same for the same kind of steel. To increase the electrical resistance and permeability and thus decrease the losses, silicon and aluminum are alloyed to the material. Non-oriented steel contains 0.5 – 3.25% of silicon,
0.005% of carbon, and up to 0.5% of aluminum. Material properties such as magnetostriction level and Curie temperature, which are determined by the alloy percentages, are used to define the material grade standardization by the American Iron and Steel Institute (AISI). According to the
AISI standard, the material grade is the number that is written next to the M letter. Since low-grade materials have the highest silicon percentage, they have the highest resistivity and permeability and
18 thus the lowest core losses. The lamination thickness also influences the material performance by reducing the eddy current losses.
There are three different types of core loss testers; namely, Epstein, toroid, and single sheet testers as shown in Figure 2.4. The American Society of the International Association for Testing and Materials (ASTM) has standards for material preparation, experimental setup, and test methods for all the testers mentioned above [75]–[77].
(a) (b) (c)
Figure 2.4. (a) Epstein, (b) toroid, and (c) single sheet core loss testers.
According to the ASTM standards, the steel sheets are cut into ring-shaped laminations and stacked together. First, the secondary winding is placed on the core then the primary winding wound on top of it. The number of turns and the stack size determine the required current and voltage levels applied to the primary winding at a given frequency. The secondary winding may have multiple tap leads to adjust the number of turns to allow for frequency variations. Low- frequency measurements are made by connecting multiple search (secondary, tertiary, etc.) windings in series to improve the accuracy, while the high-frequency measurements are made with only a single search winding. To test motor laminations, a toroid tester is the most suitable core loss tester because of its closed magnetic loop and the similarity between its geometric structure and that of a real electric motor. A disadvantage of the toroid tester is that the sample preparation is time-consuming.
19
In an Epstein tester, the magnetic material is cut into strips, stacked, secondary and primary windings are wound on the frame, and then stacked laminations are inserted into the frames.
Inserting large number of laminations helps to reduce the reluctance at low frequencies, hence, the accuracy increases. Although the Epstein tester is a common tester, it has some drawbacks. The major drawback of this tester is that there is an air gap between the frames, which increases both the reluctance and leakage. In addition, the preparation of the laminations is also time consuming.
The single sheet tester is the most common tester used by steel manufacturers due to its easy to test structure; it tests a single sheet at a time. The single sheet tester needs to be used with one of the previously mentioned testers because it requires calibration after each measurement. In addition, the standard single sheet tester measures the flux only at the center of the sheet, hence, it does not fully represent the material. In order to overcome this problem, there is a customized version of the single sheet tester that measures the flux through the steel sheet [13].
Although, the outcome of the measurements from each core loss tester is expected to be the same, the actual measurements differ. The research in [78]–[80] compares the performance of the different testers according to the different standards. The research shows that the toroid tester gives higher core loss values than the other testers at all frequencies. This fact is explained by the influence of the cutting while the laminations are being prepared, which result in low permeability and higher core losses. It should be noted that the core loss tester is used for repeatability and to allow the reader to compare the results with those from the other sources.
For a valid and accurate core loss measurement, having a proper understanding of the various measurement errors is vital. Some of the aspects which affect the measurement system accuracy are the phase discrepancy between the measured voltage and current waveforms (due to the poor frequency response of the current sensor), the core temperature, etc.
The phase shift between the voltage and current measurement is actually the main source of error, especially at high frequencies. Even a slight acquisition delay coming from one of the sensors could give inaccurate results. Usually, the current probe has a delay and a poor frequency response
20 that should be compensated for before starting the core loss calculation. The percentage measurement error can be expressed as [68]
cos cos E 100 (23) cos where, E is the percent error, θ is the actual phase shift between the sensed voltage and current waveforms, and α is the error in the actual phase angle measurement. The measurement angle α
can be described as a function of a delay time td and the applied frequency f as follows [68]:
o ftd 360 (24)
According to [28], when Equation (24) is substituted into (23) and solved for td , an acceptable delay time can be obtained, with an accuracy of ± 4 % for up to a 20 kHz signal error.
Capacitive couplings also lead to errors and should be carefully avoided. The researchers [81] argue the fact that the capacitances due to the primary and secondary windings (inter-capacitance), the capacitances between the turns (self-capacitance), and the capacitance between the windings and the core would increase if the core is grounded. Therefore, the core should not be grounded. In addition, a lower number of turns is also preferable to decrease the inter-capacitances. To be able to produce a uniformly distributed flux density, the primary winding should be wound all over the core. There is a tradeoff between lowering the number of turns, which decreases the inter- capacitances and pushing higher currents to achieve the desired magnetic field intensity levels to saturate the core.
Another important aspect is the temperature variation of the core. The subsequent measurements should be performed at a constant temperature (25o C). Performing the test inside an oil chamber helps keep the temperature under control; however, each individual test should be done in a short amount of time to get the highest accuracy from the measurements. An automated excitation system is very helpful in getting the measurements done in very short amount of time.
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2.4. Core Loss Test Setup and Control
All of the core loss testers mentioned above use a single-phase signal generator in their tests.
Typically, the signal generator is a voltage controlled single phase PWM inverter with/without an
LC filter connected to the output as shown in Figure 2.5.
Figure 2.5. Typical core loss tester system.
For most cases, the tests require to reach and sometimes exceed a flux density of 2 T, which saturates the core under test [2], [82]. Hence, the output voltage waveform becomes distorted due to the nonlinear behavior of the load. The output voltage is generated through an LC filter and the voltage across the filter capacitor changes according to the voltage drop on the filter inductor based on the current drawn from the output. A feedback control system should be included in the controller to compensate for intensive load variations. The following literature review on voltage control of single-phase inverters mostly focuses on the uninterruptable power supply (UPS) that has the same signal generation method with the core loss tester.
Various control structures have been developed for voltage controlled PWM inverters. In early stages, a carrier-modulated PWM that compares the duty ratio with triangular or saw-tooth waveforms [83] was very popular. Recently, microcomputer based techniques have been introduced, where the PWM signal is generated based on certain performance criteria [84]–[86].
The criteria mostly include harmonic elimination [87], [88]. Having a slow regulation response and
22 the phase displacement between the reference and the generated signals are the major drawbacks of these methods [89].
Improvements in digital signal processing technology has lead researchers to improve the voltage and current control techniques for PWM inverters [90]–[99]. Some of these control techniques rely on the output voltage measurement and apply proportional-integral (PI) [91], deadbeat, sliding mode, and DQ controls [92], [93]. Other ways to control the PWM inverters are based on the combination of multiple feedbacks such as output voltage, filter inductor current, filter capacitor current, output current, etc., and structure the control in multiple loops. Either each loop can be controlled with PI controllers, or only the voltage loops can be controlled with a PI controller while the current loops can be controlled with relatively faster controllers such as sliding mode controllers [94].
The control techniques based on instantaneous feedback controls may achieve a good dynamic response to the disturbances if they are properly designed [100]. Deadbeat control provides a very fast dynamic response, however, it is vulnerable to parameter variations [101]–[103]. In addition, it has a poor controller performance while supplying the nonlinear loads (crest load) [101].
Techniques based on the sliding mode control, superior to the previously mentioned methods in terms of being robust against internal and external disturbances, having fast dynamic response, and simple implementation features, are presented in [104], [105]. The major drawbacks of the sliding mode control are not having zero steady state error ability, variable switching frequency, and undesirable oscillations having a finite frequency, also known as chattering, exist. Moreover, the dynamic response of the controller is poor during load transients because the constant sliding gain used in the sliding surface causes the sliding line to be static. This problem is solved by introducing a time varying sliding gain [106]. However, even though the static sliding line problem may be solved, the error convergence is not completed in a finite time. Another improvement is made by introducing a nonlinear sliding surface to ensure zero steady state error [107]; however, the variable switching frequency issue is not solved with this modification. An alternative sliding
23 mode control method solves the variable switching problem by smoothing the control law in a narrow boundary layer [108]. That way, the pulse width modulator generates constant frequency
PWM signals. Even though the smoothed control law degrades the performance of the controller, it keeps the output voltage error at an acceptable rate.
Boundary control is another control structure based on a geometric approach. The method uses the natural frequency of the power stage to obtain a logarithmic function in order to approximate the ideal switching surface [109]–[111]. Although the boundary control method has a good dynamic response, it suffers from the variable switching frequency issue [101].
Another voltage control method is proposed based on the internal model principle [112]. The internal model principle theory states that zero steady state error can be achieved if the models corresponding to the reference and disturbances are included in the model. The controller has an infinite gain at the specified frequency.
Multi-loop control strategies have also been used successfully in voltage control [96], [113].
Although, they have a good dynamic response, these methods lack the ability to compensate the distortions caused by the nonlinear loads. In order to improve the voltage distortion problem, a feedforward controller, that tunes itself adaptively, is added to the controller in [114]. The main disadvantage of this method is that the implementation is very complex [101]. Another approach using multiple loops is the current decoupling approach [115]. In this approach, the load current is decoupled into active and reactive components using the instantaneous reactive power theory, to compensate the amplitude and phase of the output voltage. Although the control method has a good transient response, it uses the resistance and inductance values to calculate the equivalent output impedance, which makes it vulnerable to parameter variations. A single phase DQ control strategy along with a harmonic eliminating compensator is introduced in [116]. The method uses synchronous reference frame PI controller for regulating the output voltage. Additionally, it uses a current shaping controller on stationary reference frame based on the output capacitor current for active damping. This approach has very good THD levels and dynamic response; however, it is
24 mainly designed for sine wave generation and as such, it is not suitable for arbitrary signal generation.
Recently, the Lyapunov`s function based control strategy was introduced for the single phase
UPS system [101]. The Lyapunov’s function is formed based on the fact that, if the total energy dissipated continuously, the system states always go to the equilibrium point. That way the globally asymptotical stability is guaranteed with a steady state error. The steady state error is eliminated with an additional voltage loop. The major drawbacks of this control method are that the control is sensitive to the measurement noise because it uses derivations of the measured quantities and that the design stage is time-consuming.
The predictive current control technique [117] is another control mechanism. This technique has also very good dynamic response. These control schemes are also effective reducing distortions of the output [118]. However, the current prediction cannot be applicable for a wide range of loads
[119]. Another drawback of these control techniques is that they require complex calculations in every control cycle and are therefore not suitable for the systems that require fast computation times.
Repetitive control (RC) structure was first proposed for a UPS application in 1988 [91]. The total harmonic distortion is usually the result of the periodic disturbances that occur at least twice in a cycle. The repetitive controller guarantees the zero steady state error for all the harmonic frequencies greater than twice the sampling frequency. The disadvantages of the conventional repetitive controllers are hard to stabilize the controller and the dynamic response of the controller is poor for fluctuating loading conditions. In order to improve the RC performance, a feedback controller is added to the RC controller [120]. This way, the feedback controller provides the fast response and robustness while the RC provides high accuracy tracking by periodic learning [121],
[122]. In the conventional form of the RC, the periodic signal generator delays the signal with a fundamental period of N samples. Hence, it occupies N memory cells. This structure eliminates the harmonics below the Nyquist frequency by introducing a finite gain for the harmonics. The long
25 delay time (N samples) makes the system inferior to the feedback controllers in terms of having a fast dynamic response [120]. The dynamic response of the controller has been improved by introducing the odd harmonic RC [123], [124]. The odd harmonic RC only delays the signal N/2 samples and thus only N/2 memory cells are required for the control. Clearly, the odd harmonic RC only cancels out the odd harmonics while the repeating even harmonics which would cause stability issues and distortion. This problem has been overcome with the dual mode RC which eliminates both the even and odd harmonics. This structure uses N memory cells, N/2 for the even-harmonic signal generator and N/2 for the odd-harmonic signal generator [120]. The dual mode RC improves the dynamic response and has a faster error convergence ability without increasing the complexity.
2.5. Conclusions
In this chapter, a literature survey on core loss models, core loss measurements, testers, and control of the single-phase signal generators is provided. Existing core loss models are classified into three types as Steinmetz equation based models, loss separation based models, and mathematical hysteresis models. Advantages and drawbacks of these models are investigated. It was found that the Steinmetz equation and loss separation based models have accuracy problems for electric machines. Mathematical hysteresis models are accurate but are too complex to be integrated into the finite element analysis software. In addition, existing core loss measurement methods and testers are investigated to allow the selection of a suitable tester and measurement methods. Finally, control methods of single-phase signal generators are investigated to develop an appropriate control mechanism based on the application requirements.
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CHAPTER III
PROPOSED CORE LOSS ESTIMATION METHOD
This Chapter presents a new core loss estimation method for electric machines. The following section discusses why the existing core loss estimation methods are not suitable for the electric machines, and how the flux waveforms are varying on different regions of the electric machines.
The features of an interior permanent magnet synchronous motor’s flux density waveforms are investigated as an example in Section 3.2. The new core loss estimation method is proposed in
Section 3.3. This method requires controlling the flux waveforms in the core under test; therefore,
Section 3.4 discusses the necessity of controlling flux. Section 3.5 gives the details of the core loss test setup, which includes the hardware, control mechanism, system modeling and the proposed controller structure.
3.1. Accuracy of Existing Estimation Methods
As reviewed in Chapter II, there is a broad range of core loss estimation methods developed for various purposes. Among other models, the models based on Steinmetz equation (SE) and loss separation (LS) are very popular, as well as very practical for rough core loss estimations. However, the estimation error varies depending on the application. In some cases, the error exceeds 100% of the estimation. These approaches are based on curve fitting an equation to the experimental loss data, and so the fitting is performed at a certain condition. That means the model gives an accurate loss estimation at the conditions under which the loss data is taken. For example, if the loss data that is fitted to the model equation is taken from the waveform given in Figure 3.1.a, the loss estimated for the waveform shown in Figure 3.1.b will almost be the same. However, they are
27 different in reality. Both waveforms have 2 T peak amplitude at 60 Hz; however, the second waveform has harmonic components which will result in a higher core loss.
Figure 3.1. (a) Pure sine wave and (b) sine wave with harmonics.
Mathematical hysteresis-based loss models are much more complex and accurate as well.
However, these models require more knowledge about the material and testing. Even though they result in higher accuracy, most of the time the integration of hysteresis loss models into the FEA is very complex and simulations are time-consuming.
Figure 3.2. Meshes in the FEA simulations.
In FEA, the calculations are performed in each meshed element as shown in Figure 3.2. The total core loss is calculated by the summation of the estimated losses from each element. Typically,
28 around a hundred thousand elements exist in a mid-size motor simulation, and the number of elements increases for simulations that are more realistic. As a result, even a slight estimation error accumulates leading to a very high error in the total core loss estimation.
3.2. Typical Flux Density Waveforms of an IPMSM
Many factors have an influence on the flux density waveforms in the machine like motor type, motor geometry, winding structure, slot openings, rotor speed, loading, etc. For instance, flux density variation for different regions of an IPMSM from the stator and the rotor are given in Figure
3.3 and Figure 3.4.
Figure 3.3. Stator flux density waveforms of an IPMSM.
If the waveforms are observed in detail, the waveform of the Sensor 1 (S1) has high-frequency components in addition to the fundamental 80 Hz. This is due to the position of the sensor as, being close to the air gap; it includes harmonics caused by the air gap flux, rotor slots, instant field variations at the magnet ends, etc. On the Sensor 2 (S2), the waveform has less high-frequency harmonic components; however, it goes more into the saturation region due to the stator tooth getting narrower in that region. Sensor 3 (S3) and Sensor 4 (S4) give similar waveforms having even
29 lower high-frequency components. At the yoke of the motor, the flux density decreases because the flux path widens out as seen from the Sensors 5 (S5) and Sensor 6 (S6).
Figure 3.4. Rotor flux density waveforms of an IPMSM.
The flux density waveforms of the rotor differ from the stator. While the stator flux density waveforms have the same fundamental component with the excitation current, the fundamental frequency of the rotor waveforms varies with the rotor speed, loading, and motor type. In an induction machine, rotor flux frequency is equal to the slip frequency, while in a synchronous machine, it reduces to zero and turns out to be a DC quantity. In addition, a high-frequency component exists due to the stator slot openings. To illustrate this point, Figure 3.5 presents how the flux path is completed in the IPMSM given above. The flux tends to pass through the path where the rotor is linked to the stator through the air gap. However, the air gap distance changes while the linked flux is switching from one stator tooth to the next because of the stator slot
30 openings. This air gap variation causes the flux density waveform to fluctuate at a certain frequency. This frequency is a function of the number of stator slots, the number of poles and the excitation frequency. For example, the frequency of the highest harmonic component at the rated conditions for the 48 slot, 8-pole IPMSM goes to 960 Hz while the excitation frequency is 80 Hz.
Figure 3.5. Flux paths of a 48 slot, 8-pole IPMSM.
3.3. Effect of Loading
According to the loading conditions, the flux density waveforms will also change, as it is expected. The standard motor performance measurements state that the core loss measurement is performed at unloaded conditions, and this loss is referred to as the classic iron loss [125]. The classical iron loss does not change with loading as long as the excitation frequency is constant.
However, the core losses increase with loading due to the field and flux increase in order to generate the required rotational force. Therefore, the core loss measurement also becomes inaccurate when the machine is loaded. Often, this core loss difference is combined with some other unexplained loss components and referred to as stray losses, which are not calculated but merely estimated as a percentage of the motor output power. In some cases, the stray loss goes up to 2% of the motor
31 output power. The analysis and reduction of these losses are crucial for next generation higher efficiency electric machines.
3.4. Proposed Core Loss Estimation Method
The flux density waveforms in electric machines are not uniform over the geometry, as various regions contain different flux density waveform properties. While some of the regions have waveforms at the same frequency as that of the excitation frequency, some of them have additional harmonic components, and others have DC biased high-frequency waveforms. If the waveforms are looked in detail, they can be more diversely classified. The SE and LS approaches suffer from the disadvantage of lacking accuracy when it comes to loss estimations where the flux density waveforms are arbitrary. Although, the mathematical hysteresis models are accurate, they are too complex to be used in FEA simulations. In this dissertation, a core loss estimation procedure is proposed which is believed to be both more realistic than the SE and LS based approaches, and less complex than the mathematical hysteresis based models. The proposed model is a combination of
FEA and core loss measurements; the estimation procedure is given in Figure 3.6.
Tools Method Signal generator, BH loop core loss Mapping the loss toroid tester measurement regions and 2 determining the flux density waveforms BH loop core loss measurement 1 FEA Iprimary Vsecondary
Signal Generator Toroid Tester Hardware
Control Feedback
Controlling flux density Analysis of the core
losses under actual Controller 3 flux density waveforms Flux density controlling core loss test system
Determination of loss Total core loss per volume/mass for 5 estimation 4 each region
Figure 3.6. Proposed core loss estimation procedure.
32
The method begins with the FEA of the electric machine. Then, the flux density waveforms are obtained from different regions where the loss is approximately uniform. The third step is to generate the same flux density waveforms in a core, made out of the same material used in the machine, and analyze the core losses under these waveforms. In order to do that, some tools are required; namely, a signal generator and a toroid core loss tester. The core loss tester is needed to have the ability to control the AC and DC flux densities in the CUT. Therefore, as a fourth step, the core losses are measured with the BH loop method and associated with the mass/volume of the material for each region. It should be mentioned that since the BH loop method is used for the core loss measurements; great care is taken for the phase discrepancy between the measured quantities.
A precision resistor and a differential voltage probe are used for the current measurements to avoid any delay on the current waveforms. The fifth step is to use the loss data and the volume or mass of the regions to estimate the total core losses. The verification of the estimation will be done with the motor core loss tests.
3.5. Flux Regulation for the Core Loss Tester
In this work, in contrast with the conventional voltage controlled signal generator, a flux controlled signal generator is proposed. In order to point out the necessity of the flux control, consider a sinusoidal flux density generation case. Figure 3.7 shows a sinusoidal flux density generation in a toroidal transformer with voltage and flux control structures.
Although it is expected that a sinusoidal flux will be generated in the core when the primary winding voltage is a sine wave, because the magnetizing inductance Lm is saturable, when it is saturated, the secondary voltage Vsense becomes distorted, and so does the flux density waveform.
On the other hand, the flux controller structure compensates the saturation effect from the primary winding, and consequently, the generated secondary voltage Vsense and the flux density become perfectly sinusoidal, as it is desired. Similarly, the non-sinusoidal flux density waveforms determined from the various parts of the motor can be generated using the same tool, allowing the
33 core losses to be analyzed under real operating conditions. Clearly, this makes the flux controlling core loss tester a necessity.
Figure 3.7. Sinusoidal flux density generation in a toroidal transformer with voltage and flux control structures.
In conclusion, as a practical engineering approach, this method answers the question of how a particular core material would behave under certain flux density conditions. Consequently, the loss estimation becomes more realistic.
3.6. Core Loss Test Setup
The core loss test setup consists of the hardware, control mechanism, system modeling and the proposed controller design. A single-phase H-bridge inverter is selected as the hardware to generate the desired flux density waveforms. In addition, a controller is proposed for controlling the flux density in the CUT. In order to determine the controller parameters, the test system is modeled, thereby; defining the system as a whole.
34
3.6.1. Hardware
In order to generate the desired flux density waveforms, a hardware consisting of a high bandwidth signal generator and a toroid tester are needed. The proposed test system, shown in Figure 3.8, is mainly designed for the BH loop core loss test, which was explained in Chapter II. In this method, the DC input voltage is processed with the H-bridge inverter and the signal, at the desired frequency and shape, is applied to the toroid transformer. Then, the flux density and flux intensity are calculated instantaneously from the measured quantities Vs and I1.
Signal Generator I1 L + + + N N V Vi C VC 1 2 s VDC - CDC - -
I1 VC Vs
Control and Interfacing Board
Figure 3.8. Proposed core loss test system.
The flux linkage λ is calculated through the integration of the voltage, measured from the sense coil, and then the flux density B can be calculated by the following expression:
1 B V t dt s (26) NA2 core
where, N2 is the number of turns on the sense coil, Acore is the cross-sectional area of the core, Vs is the induced voltage on the sense winding.
The signal generator includes a single-phase PWM inverter and a filter connected to the output of the inverter to smooth out the pulsating voltage. A detailed explanation for the selection of the single-phase inverter topology, selection of the switching devices, the design of the signal generator, and the design of the output filter will be given in Chapter IV.
35
3.6.2. Control Mechanism
Due to the nonlinear behavior of the core, even the desired voltage shape applied to the primary winding does not generate the desired flux density. For example, a sinusoidal output voltage itself will not be able to generate a sinusoidal flux when the core is saturated. In most cases, the core is saturated to reach required flux density levels beyond 2 T. Controlling only the voltage of the main winding will not result in the desired flux through the core. Direct flux measurement provides better control, but still requires higher performance from the controller especially during saturation, which brings us to the point where the controller structure to be used need to be decided.
The system needs to be analyzed before the final controller structure decision. During the core loss test, the load does not change as long as the frequency is kept constant. Frequency variation effects the toroid transformer`s DC resistance due to the skin effect, and the magnetizing inductance, especially at frequencies above 1 kHz. Hence, it can be said that the non-periodic disturbances occur only when the frequency changes, consequently, the load changes. On the other hand, the disturbances caused by the magnetic saturation occur periodically at every cycle. Thus, a periodic compensator is required, as part of the controller, to eliminate the periodic disturbances.
However, a periodic compensator alone will not be able to perform the best under changing load conditions; therefore, another feedback compensator is also required. Accordingly, the controller structure is required to combine a periodic and a feedback controller. In addition, a feedforward controller is included into the system to reduce the transient time because a short settling time is important to avoid the temperature rise in the core. In order to control the DC bias field, a feedback controller that controls the average primary winding current is also included into the system. The reference field is found from the CUT static BH curve, and then the reference amplitude is calculated from the active length of the toroid and the number of turns of the primary winding.
Figure 3.9 presents the simplified block diagram of the proposed flux control structure.
The digital control law, which is composed of four terms, is given as:
36
u(k)=uff(k) +ufb(k)+ uper(k)+ udc(k) (27)
where uff(k), ufb(k), uper(k), and udc(k) are respectively the feedforward, feedback, periodic, and DC offset controllers. The feedforward controller gain is selected based on the toroid transformer turns ratio. The feedback controller is a conventional feedback controller which can be selected as PI, deadbeat, state feedback, etc., which is utilized to improve the dynamic response of the system while improving the stability margins. In this work, the state feedback structure is used. Along with the fast dynamic response feature, the state feedback gains can be optimally calculated.
Digital Controller
DC Flux Controller
BDC,ref HDC,ref IDC,ref u (k) ltor Feedback dc N1 Compensator BH Curve
Feedforward uff(k)
VS, ref Compensator u (k) u(k) PLANT BAC,ref _ d _ Feedback fb N2A 2πf (Inverter, LC filter, dt core Compensator toroid transformer) Periodic uper(k)
AC Flux Controller Controller VC, IL, VS
V , I , V C L S
Figure 3.9. Simplified control structure.
3.6.3. System Modeling
In order to calculate the control law, the system model needs to be developed. The capacitor voltage, the filter inductor current, and the sensed coil voltage have been selected as state variables, and plant model has derived accordingly. Figure 3.10 shows the equivalent circuit model of the core loss test system.
37
1=0 R2 L2 =0 L IL Io R1 L1 I N1:N2 I2
+ + + +
Vi Vc C Lm V1 V2 Vs
- - - -
Inverter and LC Filter Toroid Transformer
Figure 3.10. Equivalent circuit model.
Here, L and C are the output filter inductor and capacitor, R1 and Ll are the primary winding resistance and leakage inductance, Lm is the magnetizing inductance, R2 and L2 are the secondary winding resistance and inductance.
The core loss test does not require any loading on the secondary winding; therefore, the secondary winding parameters are not included in the model and thus Vs = V2. Hence, the first state equation will be as follows:
11 iVVL (28) LLci
The primary winding leakage inductance is much smaller (about 10 - 30 times) than the magnetizing inductance. Also, there is no load on the secondary. Hence, Lm and L1 are combined for simplicity. The simplified circuit representation is shown in Figure 3.11.
L IL Io R1 N1:N2
+ + +
Vi Vc C Lm+L1 V1 Vs
- - -
Figure 3.11. Simplified equivalent circuit model.
The following equations may be readily derived from the simplified equivalent circuit:
38
11 N1 ViVVc LCs (29) CR CNR121 C
NNR221 1 ViVVs LCs (30) N11111 CN R CR CLL m
Using the equations above, the state space system can be defined as follows:
xAxButt =+ t (31) yCxtt = where,
T xtitVtVt Lcs (32)
ut Vi t (33)
yt Vs t (34)
1 00 L 11N A 1 (35) CR CN R C12 1 NNR221 1 N11 CN 111 R CR CLL m
T 1 B 00 (36) L
C 0 0 1 (37)
The corresponding discrete time model can be driven as:
39
xABkkk1=x+u dd (38)
y Ckk x = (39)
ATs Ad e (40)
-1 ATs BABd =-I e (41)
T (42) xkikVkVk Lcs where, k is the sample number and Ts is the sampling time.
3.6.4. Proposed Controller Design
Having modeled the test system in state space form, a control law consisting of the state feedback controller, a periodic controller, and the feedforward controller can be designed. First, an optimal state feedback controller is designed. The control input signal is the voltage applied from the converter, defined as:
ukVk i (43)
The control can be expressed as:
uKx()kk = (44)
Here, K is the optimal control coefficients vector, and it can be calculated as linear quadratic regulator law.
K KKKp2,1 p 2,2 p 2,3 (45)
The cost function that will be used to find the optimal control can be expressed as
JxTT Qx u Ru dt (46) 0
40 where Q and R are positive semi-definite and positive definite matrices, respectively.
Correspondingly, the linear time invariant (LTI) static state feedback control can be calculated as:
K R=- B M-1T (47) where, M 0 is the solution of the following algebraic Riccati equation:
AMMAMBRBMQT-1T +-+= 0 (48)
The second component of the overall control law is the periodic controller. The periodic controller guarantees that the plant output tracks the reference signal without having a steady state error, as long as the closed loop system that generates the periodic signal is stable. In this work, periodic disturbances consist of more than one harmonic component. Hence, the periodic controller should be able to include the model of these signals. The following periodic controller transfer function can generate periodic signals with harmonic components of the fundamental frequency:
Na uzKzperp 1 (49) ezQzz 1 a where, e(z) is the z-transform of the error e(k)= r(k)-y(k), Kp1 is the gain of the periodic controller,
N is the time advance step size, a is the number of delayed samples caused by the low-pass filter, and Q(z) is the transfer function of a low-pass filter. The low pass filter has to be included into the system to avoid the accumulation of the high-frequency measurement noise, that way overall robustness of the system is also improved. The cutoff frequency of the low-pass filter needs to be selected as high as possible to improve the secondary winding voltage THD. If the cutoff frequency is low, the controller will not be able to react to the harmonic components that have higher frequencies than the cutoff frequency. The gain of the periodic controller affects the convergence time; that is, the higher the gain is the faster the convergence. However, increasing the gain makes the system more vulnerable to the high-frequency measurement noise and may cause instability.
Therefore, the gain of the periodic controller should be limited to maintain a stable operation.
41
The detailed structure of the controller is presented in Figure 3.12. It is designed such that the induced flux waveform will track the reference with a zero steady state error and a small settling time. The reference voltage of the sense (secondary) coil is calculated using (26). The error between the reference and the measured sensed coil voltages is fed through the periodic compensator. The periodic controller brings the system closer to the point where the error is minimal. The state feedback is taken from the states compensating the non-periodic disturbances, providing a better tracking performance. The feedforward path is added to the controller to obtain faster response during transitions.
The DC offset flux density controller determines the DC field strength from the pre-determined static BH curve and then the reference DC current is calculated using the following equation:
HlDCtor IDC (50) N1
A conventional PI controller regulates the average current level, and hence the DC flux density level is controlled. The DC flux will not be reflected to the secondary coil; therefore, it will not affect the AC flux controller in steady state conditions.
DC Offset Controller H I BDC,ref DC,ref l DC,ref tor PI N + Controller BH Curve - <>
IL K K Vc p2,1 + + p2,2 + + Vs Kp2,3 + u FB PER Q(z) z-N + +
Vs,ref - + BAC,ref * d * e N2Acore2πf Kp1 dt + +
AC Flux Controller N1/N2 FF
Figure 3.12. Proposed flux controller.
42
3.7. Conclusions
In this Chapter, the proposed core loss estimation method is introduced and explained in detail.
In order to perform the proposed method, a new core loss tester needs to be developed. The new tester should be able to control AC and DC flux densities in the core being tested to allow the core loss measurements to be performed under the actual flux density waveforms of the motor. This chapter also provides the details of the new tester’s hardware, control mechanism, plant model, and the controller design.
43
CHAPTER IV
FLUX CONTROLLED CORE LOSS TEST SETUP DEVELOPMENT
In this chapter, the flux controlled core loss tester is designed based upon the plant parameters.
In order to verify the effectiveness of the control algorithm, the system is simulated at various conditions. Then, the hardware development of the tester is provided in detail and the experimental results obtained for various flux density waveforms are presented.
4.1. Control Design and Implementation with the Simulation Platform
A complete model of the proposed flux controller test system is obtained using the procedure given in Chapter III. Table 1 shows the system parameters used in the simulation and the experiments.
Table 1. System parameters.
Parameter Value Parameter Value
Ts [µsec] 6.667 L1 [µH] 300
L [µH] 85 Lm [mH] 3.75
C [µF] 0.1 N1 [#turns] 400
R1 [Ω] 0.18 N2 [#turns] 30 2 -5 ltor [m] 0.678 Acore [m ] 2.28x10 Using Equations (35), (36), (37), the system, control, and output matrices are calculated as follows:
0 1.1765 104 0 1.176510 4 7 7 8 A 1 10 5.5556 10 7.4074 10 , B 0 , C 0 0 1 (51) 5 6 7 7.5 10 4.1667 10 5.5556 10 0
The corresponding discretized system, control, and output matrices are determined using (39) and (40).
44
0.6397 9.083 121.4 0.02617 A 21.31 119 1578 , B 1.64 , C 0 0 1 d d d (52) 1.598 9.001 119.3 0.123
The optimal control parameters are calculated using (22) and (23).
3 K 26.9935145.23091.942110 (53)
The low pass filter in the periodic controller is selected as a finite impulse response (FIR) filter.
The transfer function of the FIR filter is as follows:
NN22 Qzhkxnkhnz n (54) nNnN11
To avoid excessive delays, the FIR filter`s order is kept as small as possible. Higher order filters perform better, especially when filtering low frequencies; however, in this application, the high- frequency noise can be filtered with a lower order filter. Therefore, the order of the filter selected as 4, while the cut-off frequency is selected as 10 kHz. The following coefficients are calculated using the MATLAB Signal Processing Toolbox.
h 0.03150.23890.45920.23890.0315 (55)
A 4th order FIR filter causes a 2-sample delay, hence, α is set to 2. As mentioned above, the gain Kp1 defines the convergence speed of the filter, and it can be selected between 0 and 1.
However, high-frequency measurement noise limits the gain. In simulations, there is no measurement noise and thus the gain is kept at 0.95, but in the experiments, this value goes down to 0.1 in some cases.
To verify the effectiveness of the proposed control system, a set of simulations have been performed under periodic disturbances caused by the magnetic saturation in the Matlab/Simulink environment. The toroid transformer is modeled with a saturating core using the pre-determined static B-H curve to simulate the transformer under most realistic conditions. The skin effect due to
45 the frequency variation is neglected. The flux density waveform, at 400 Hz, and the instantaneous error variation, due to a reference amplitude of 2.2 T, are shown in Figure 4.1. This Flux density waveform requires that a trapezoidal voltage waveform be generated at the secondary coil. While a trapezoidal reference is a challenging waveform to generate, it is a good example to show the performance of the controller when the reference waveform has rapid variations and constant regions under high saturation currents. The error convergence of the sensed coil voltage, provided in Figure 4.2, clearly illustrates how the sensed coil voltage reaches a trapezoidal shape progressively. The generated flux waveform is quite close to the reference. As the error gets to be minimum, the commanded voltage Vi, the primary winding voltage VC, and the current IL are presented in Figure 4.3. As shown in this figure, although the secondary voltage is trapezoidal, the inverter output and the primary winding voltages are far from being trapezoidal, which shows the necessity of the flux control scheme. Such a control cannot be performed precisely using the voltage control structure. In conventional voltage control methods, the reference output voltage is estimated using the filter and the toroidal core parameters to generate the desired flux density. However, the estimation accuracy is highly dependent on the parameters which are varying dramatically with the operating conditions. The proposed method generates the desired flux density waveforms with only two feedbacks without requiring the system parameters. Hence, its tracking performance is not degraded by parameter variations.
Figure 4.1. Flux density under periodic distortion and instantaneous error variation.
46
(a) (b) Figure 4.2. (a) Error convergence of sense coil voltage in every cycle, (b) error variation.
[A]
L
[V], I [V],
i
[V], V [V],
c
V
Figure 4.3. Corresponding output waveforms in steady state condition.
The control operation is tested at various output waveforms with different frequencies and DC offset levels. Figure 4.4 through Figure 4.6 show the primary winding voltage and current, the sensed coil voltage, and the magnetic flux densities at each given frequency. As shown, the proposed controller is able to generate reference flux waveforms through the core at various ranges of frequencies and DC offset levels. Figure 4.4 presents the results at 100 Hz without having a DC offset. Whenever a pure sinusoidal reference flux waveform was commanded, the proposed controller generated the flux density waveform with a THD value of 0.2%.
Figure 4.5 shows a 400 Hz flux density waveform that can be generated with a trapezoidal voltage on the secondary winding, while a 1.4 T DC offset exists.
47
Figure 4.4. Primary winding current and primary winding voltage, sense coil voltage and
calculated flux density waveform at 100 Hz.
Figure 4.5. Primary winding current and primary winding voltage, sense coil voltage and
calculated flux density waveform at 400 Hz with 1.4 T DC offset.
As previously mentioned, the flux density waveforms consist of high-frequency components; therefore, the system is tested with a 1 kHz fundamental frequency waveform that includes the 3rd,
5th, and 7th harmonic components. Figure 4.6 shows the flux and the corresponding output waveforms. The tracking performance of the proposed controller is satisfactory.
48
Figure 4.6. Primary winding current and primary winding voltage, sense coil voltage and
calculated flux density waveforms 1000 Hz fundamental.
4.2. Core Loss Test Setup Hardware Development
The topology that will be used for the single-phase inverter can be either half-bridge or H- bridge (full-bridge) topologies shown in Figure 4.7.
+ V DC S1 S1 S3 CDC 2 VDC VDC + - + VO VO - - C + DC V DC S2 S2 S4 CDC 2 -
Half bridge inverter H-bridge inverter
Figure 4.7. Half-bridge and H-bridge topologies.
The H-bridge topology has an important feature that makes it superior to the half-bridge topology. The H-bridge inverter requires half of the DC bus voltage that is required for the half bridge inverter to generate the same voltage amplitude. Since, the voltage levels required for the high frequencies are already high; using a half-bridge inverter makes the required DC bus voltage
49 even higher, consequently, the switching device voltage rating increases as well. Therefore, the H- bridge topology is more suitable for the test system.
The inverter is designed considering several design specifications. FEAs show that the frequency of flux waveforms may go up to 10 kHz depending on the speed, type, and geometry of the machine. Moreover, a DC component exists especially in the rotor flux waveforms of synchronous machines. Generating such waveforms accurately require switching frequencies to be as high as possible. The switching devices need to be switching at a frequency which is enough to produce a 10 kHz sine wave while maintaining the current and voltage capabilities. Conventionally, the switching frequency is selected based on the maximum allowable THD level at the rated conditions. For the 3% maximum allowable THD level at the maximum frequency, the switching frequency is determined to be 150 kHz with the help of the circuit simulations. In this research, signal generation, testing, control, and measurements were initially performed with the toroid transformers that were provided from the sponsoring company. Since the tester is going to be used for a wide range of loads; it is not possible to determine the maximum requirements for the active and passive components. Therefore, the current and voltage requirements were determined based on the parameters of these transformers as 100 A and 600 V. Therefore, the test system is supposed to generate the desired signals up to a 10 kHz frequency with a 150 kHz switching frequency, having 100 A and 600 V maximum current and voltage limits, respectively.
The switching device is one of the vital components and, as such, great care is required for the selection of the type of switching device and technology. The key features of the selected switch are that it must meet the current and voltage requirements, and be able to switch at 150 kHz with a low switching loss. The switching frequency requirement dictates the use of metal oxide semiconductor field effect transistors (MOSFET), instead of isolated gate bipolar junction transistors (IGBT). Silicon (Si) and silicon carbide (SiC) are the technologies to be compared for the final decision. Si technology is a well-known technology with an almost 70 year-history while the SiC technology is an emerging technology that was commercialized in 2011 [126]. The features
50 of inherent radiation-resistance, high-temperature operating capacity, high blocking voltage and power handling capacity, high power efficiency make the SiC MOSFET superior to the Si counterpart. With over 20% efficiency improvements [127], SiC MOSFETs has become the choice for most next generation power devices and high-power density drives. Therefore, the SiC
MOSFET was selected to be the active switching device, and specifically, the 100 A, 1200 V half- bridge SiC MOSFET (CAS100H12AM1) module from CREE Inc. [128] is selected due to its compact structure. This half-bridge module includes two series connected SiC MOSFETs and two anti-parallel SiC Schottky diodes in it. Two of these modules are used to build the H-bridge inverter as shown in Figure 4.8.
1
HEATSINK
2
Gate1 Gate2 Gate1 1 3
SiC Mosfet Module SiC Mosfet Module 2 Gate2
3
Figure 4.8. The H-bridge inverter made out of two SiC MOSFET modules.
Since the switches perform fast switching at high power levels, it is very important to reduce the parasitic inductance between the DC bus capacitors and the MOSFET modules to decrease the ringing effect and excessive overshoots; it is also challenging to maintain a ripple free DC bus. The parasitic inductance is the main problem to overcome. Two fundamental discoveries help explaining the inductance occurrence. The Ampere’s Law given in following form.
BJ (25) where, B is the magnetic field, is the permeability of the conductor, and J is the electric current density. This expression states that the gradient of the magnetic field is equal to the product of the
51 material permeability and the current density, which is also known as the right-hand rule, as shown in Figure 4.9.
B I
Figure 4.9. Right-hand rule.
According to Lenz`s Law, an electric circuit with an inductance induces a voltage which opposes the change on current, that is,
dI VL (26) dt
Here, the main problem is that the magnetic field increases the inductance. Clearly, canceling out the field is a reasonable remedy. There are two ways to cancel out the magnetic field and the inductance on the DC link; namely, stacking the conductors vertically by forming them in parallel and placing the conductors side by side in the same horizontal plane, as shown in Figure 4.10.
Parallel Plate Coplanar Plate
Figure 4.10. Parallel and coplanar sheet orientations.
Both methods help to reduce the inductance. The resulting inductance can be roughly calculated using Ampere`s Law. Before proceeding to the calculations, it is expedient to illustrate how the fields cancel each other. Figure 4.11 shows the field generated by the current passing through a conductor from the end and side views. Figure 4.12 shows how the fields are simply canceling each other with the parallel and coplanar plate overlapping.
52
X X X X
Figure 4.11. Field generated by the current passing through a conductor.
X X
Figure 4.12. Fields canceling each other with the parallel and coplanar plate overlapping.
The inductance per unit length can be calculated from the given dimensions and the expressions shown in Figure 4.13.
w d=w+t t h w L h µ L µ -1 ( d ) m w cosh m π w Figure 4.13. Parallel and coplanar plate inductance approximations
An analysis given in [129], [130] shows that the resulting inductance per unit length for the coplanar arrangement is almost six times higher than that of the parallel plate arrangement. Hence, the parallel plate arrangement is the perfect match for the test system DC bus. Therefore, a printed circuit board has been used as a laminated bus bar. Distributed low equivalent series resistance
(ESL) capacitors are connected through the PCB between the DC supply and the switches. In addition, a variety of capacitors, of different capacitance range, have been placed close to the
MOSFET modules for further elimination of the parasitic inductance, ringing, and overshoots. This ensures not only a steady bus voltage but also that the ringing and overshoots are minimized. The
53 calculated parasitic inductance turned out to be less than 40 nH. The design drawing and the actual picture of the DC bus and the capacitor bank are given in Figure 4.14 and Figure 4.15 a, b and c.
Figure 4.14. DC bus and the capacitor bank.
The test system includes an LC filter connected to the output of the inverter to filter out the undesired pulsating waveforms. As a rule of thumb, the cutoff frequency is traditionally selected as half of the switching frequency. For further noise elimination, in this application, the cutoff frequency is selected as one-third of the switching frequency, which is 50 kHz.
54
HEATSINK
0
15
n
1
1
0
15
.
.
n
5
5
n
n
50
0
10
p
50
0
10
p
u
u
u 2 . 1 u 2 . 1
50
50
u
u
+
-
(a)
(b) (c)
Figure 4.15. (a) DC bus design drawing, (b) completed DC bus (top), and (c) (side).
Gate driver circuits have been designed for each switch individually, using commercially available gate driver chips from IXYS. The gate driver has 3 kV galvanic isolation and capable of providing 30 A peak currents for a small amount of time. The MOSFETs were switched with an external 10 Ω resistor to ensure fast turn on and turn off. The complete gate driver circuitry is given in Figure 4.16.
55
Figure 4.16. Gate drive circuitry.
In addition, a control circuit that includes interfacing and fault protection circuits is designed for the current and voltage measurements, signal conditioning, interfacing with the DSP, and hardware protection based on the measurements. The completed control board is shown in Figure
4.17.
Figure 4.17. The control board.
Since the core loss measurements are performed with the BH loop method, where the phase discrepancy between the measured quantities is a major issue. In order to avoid the phase discrepancy problem, a precision resistor is used for the primary winding current measurements.
The complete system schematic is given in Figure 4.18, while the implemented setup is given in Figure 4.19.
56
S1 S3 VDC + L IL I1 - CDC + + S4 S2 N1 N2 Vs Vi C Vc
Inverter - - S1 S2 S3 S4 LC Filter Rm l Acore tor Toroid Transformer Gate Driver Circuit IL DSP (TMS320F28377D) Vc
Power Vs
(24V,5V, 3.3V, Fault Protection Signal Conditioning Signal Conditioning ±15V) Circuit
Control Circuit & Gate Drivers
Figure 4.18. Complete system schematic.
Figure 4.19. Complete implemented core loss test system.
57
4.3. Experimental Results
To verify the flux controller performance, the test has been performed with a 125 Hz sine wave reference that has 0.84 T peak amplitude and a 1.27 T DC offset. The flux density that exceeds 2.1
T peak value and the AC component has approximately 1.2% THD. The corresponding primary winding voltage, primary winding current, and the secondary winding voltage waveforms are given in Figure 4.20. As mentioned before, to both keep the secondary winding voltage (red) sinusoidal and to reach the DC offset level, the primary winding voltage should be generated as shown in the blue waveform in Figure 4.20.
Figure 4.20. Primary winding voltage, primary winding current and the secondary winding voltage waveforms at 125 Hz. (blue: primary winding voltage; purple: primary winding current;
red: secondary winding voltage).
Figure 4.21 presents the experimentally obtained steady state flux density waveform, the sensed coil voltage, as well as the corresponding non-sinusoidal primary winding voltage and current waveforms at 250 Hz. The tester can generate a pure sin wave flux with a 1.01% THD, even if the peak amplitude of the primary current reaches up to 135 A.
58
Figure 4.21. Flux density, sense coil voltage, corresponding primary winding current and
primary winding voltage waveforms at 250 Hz.
Figure 4.22 shows the generation of an actual stator flux density waveform at the yoke of an
IPMSM observed in the FEA of the motor. The corresponding sense coil voltage, the primary winding current and voltage waveforms to generate the mentioned flux density waveform are also provided in Figure 4.21. The fundamental frequency of the flux waveform is 80 Hz and it has up to the 11th higher order harmonic components without DC offset. The system is able to track such a waveform with an acceptable tracking error as shown in Figure 4.22.
The flux densities in the rotors of the IPMSMs have a DC offset component as they operate at synchronous speed. In addition to the DC offset, high-frequency AC components exist due to the stator slots passing the rotor and spatial magnetomotive force (MMF) harmonics of the stator winding. A reference signal has been generated based on a flux density waveform observed in the rotor of an IPMSM that has a fundamental 480 Hz AC component and a 1.2 T DC offset. It also includes even and odd higher order harmonic components up to the 19th order, which is 9.12 kHz.
59
Figure 4.23 presents the flux density waveform generated in the core and corresponding primary and secondary winding waveforms.
Figure 4.22. Flux density waveform at the yoke of an IPMSM, corresponding output signals at 80
Hz.
Figure 4.23. Flux density waveform at the rotor of an IPMSM, corresponding sense coil voltage to generate the flux density waveform, primary winding current and voltage waveforms at 480 Hz
fundamental frequency.
60
4.4. Conclusions
The proposed system keeps the core loss computation procedure simple while preserving plausible accuracy level. In core loss measurement, it is important to apply the flux waveforms that actually exist in various parts of the electric machines. The proposed flux control scheme consists of two independent controllers for the AC and DC components of the flux. The AC flux controller incorporates both feedback and feed-forward compensation, along with the periodic controller. The
DC flux controller controls the DC offset level using the static BH curve of the core material. The combination of the controllers exhibits satisfactory result in tracking the reference flux signal. The
THD of the experimentally generated sinusoidal waveforms are less than 1.3%, which is a measure of the tracking accuracy. Moreover, the controller is able to track the waveforms that have higher order harmonics to up to 10 kHz and DC offset.
A high bandwidth system is implemented with a 60-kVA H-bridge SiC inverter operating at 150 kHz switching frequency and a dual-core processor from Texas Instruments (TMS320F28377D), which completes the control cycle in less than 4 µsec.
Testing the magnetic materials under the actual waveforms that exist in various parts of the electric machines provides the advantage of observing and analyzing the core loss behavior in any condition. Hence, these losses can be estimated more accurately.
61
CHAPTER V
CORE LOSS ESTIMATION OF AN IPMSM
In this chapter, the core loss estimation of an IPMSM is presented. A 10 hp, 4-pole, 36 slots,
1800-rpm IPM motor is selected to analyze the proposed core loss estimation method. Section 5.1 shows the determination of the core loss regions of the motor. Section 5.2 presents the flux density waveforms obtained for unloaded and loaded conditions. The core loss tests are performed under the resulting waveforms and the core loss measurement results for both conditions are presented in
Section 5.3. Finally, the total core loss is estimated in Section 5.4.
5.1. Determination of Loss Regions
The motor core loss regions are determined using FEA of the motor under rated conditions.
The actual core loss distribution on the motor will be assumed to be similar to that estimated using one of the existing core loss estimation methods. Specifically, here the Bertotti loss method is used because it considers the flux density rate of change; hence, high-frequency effects can be distinguished easily. The estimated core loss using Bertotti’s method is displayed using a color scale in Figure 5.1.
Determining the rotor core loss regions is a little different than for the stator because the core loss per volume values differ from each other significantly. Therefore, the core loss color map is not as clear as is the case in the stator for distinguishing the regions from each other. In order to see the loss distribution through the rotor, different color scales are used, as shown in Figure 5.2.a, b, and c. The loss regions are determined as shown in Figure 5.2.d, as it can be clearly seen from these figures.
62
S7
S S S S S 1 2 3 4 S5 6
(a) (b)
Figure 5.1. Stator core loss distribution and the selected regions.
(a) (b)
R7
R12 R6 R8
R9
R5 R13
R10 R11
R4
R
3 R2 R1
(c) (d)
Figure 5.2. Rotor loss distribution using color scales that have maximum limit core loss per volume value (a) 124000 W/m3, (b) 12400 W/m3, and (c) 4100 W/m3. (d) Rotor core loss regions.
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5.2. Determination of Flux Density Waveforms of the IPMSM
After determining the motor core loss regions, the flux density waveforms of the motor for the unloaded conditions have been obtained from the FEA. When the motor is unloaded, the input power includes only the stator copper loss, the friction and windage loss, and the core loss. The copper loss can be calculated using the stator winding resistance and the current passing through them. The friction loss can be determined from the motor datasheet. The core loss can be measured by performing the unloaded condition test. A detailed explanation of the parameter determination and performance tests are given in Chapter VII. For the unloaded condition, the input power should be equal to the summation of the copper, friction – windage, and core losses. This allows the input current to be calculated and applied to the FEA model of the motor. The resulting flux density waveforms are presented in Figure 5.3 and Figure 5.4.
S7
S1 S2 S3 S4 S5 S6
Figure 5.3. Stator flux density waveforms of an IPMSM at unloaded conditions.
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While the stator flux density waveforms are zero-mean AC signals with a 60 Hz fundamental and higher order harmonics, the rotor waveforms contain a 1080 Hz AC fundamental and higher order harmonics, in addition to a DC offset. The 1080 Hz fundamental frequency is the result of the stator teeth passing through the rotor poles. The DC offset varies in different regions depending on the flux path and the rotor geometry.
Figure 5.4. Rotor flux density waveforms of an IPMSM at unloaded conditions.
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Taking a closer look at the above waveforms reveals that, the stator Sensor 1 (S1) has a 60 Hz frequency fundamental. This waveform becomes distorted and additional high-frequency components are added to the 60 Hz waveform when the motor is loaded. Because the position of the sensor is close to the air gap, the waveform includes harmonics caused by the air gap flux, rotor slots, instantaneous field variations at the magnet ends, etc. On the stator Sensor 2 (S2), the waveform amplitude increases due to the narrowing flux path, which increases the saturation of the core material. Similarly, this waveform also includes high-frequency harmonics when the motor is loaded. However, the high-frequency component is not as significant as in the S1 sensor. The Sensor
3 (S3) waveform has similar properties to those of S2. Sensor 4 (S4) measures similar output to those of the Sensor 3 (S3) having even lower high-frequency components. At the yoke of the motor, the flux density decreases because the flux path widens, as seen from Sensors 5 (S5) and 6 (S6).
The same procedure is followed for the determination of the flux density waveforms for the rated conditions. The current level is set to produce the rated torque and the waveforms are taken as presented in Figure 5.5.
S7
S1 S2 S3 S4 S5 S6
Figure 5.5. Stator flux density waveforms of an IPMSM at unloaded conditions.
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Figure 5.6. Rotor flux density waveforms of an IPMSM at unloaded conditions.
It can be clearly seen from the figures that the amplitudes of the fundamental and harmonic components increase with load, which leads to a significant increase for the core losses.
5.3. Core Loss Tests under Actual Flux Density Waveforms
Once the waveforms are obtained, the proposed method requires generating the same flux waveforms in the core made of the same material used in the machine. Hence, two toroidal transformers have been wound on the stator and rotor electrical steels as shown in Figure 5.7. The toroid made out of the stator material has 264 turns on the primary and 30 turns on the secondary windings. The measured magnetizing inductance is 2.21 mH, the leakage inductance is 200 µH,
67 and the DC resistance is 330 mΩ. The second toroid is made of the steel used in rotor laminations.
To reach the desired DC offset magnetic field levels, the number of turns in the primary winding has been increased to 300 and the core size decreased. The sample has three individual secondary windings to adjust the secondary winding voltage amplitude. The magnetizing inductance, leakage inductance, and the DC resistance of the second toroid are 14.9 mH, 340 µH, and 2.11 Ω, respectively.
(a) (b)
Figure 5.7. Prepared toroidal transformers from the stator and rotor laminations.
The flux controlled core loss tester is used to generate the flux density waveforms obtained from the FEA. For a given flux density waveform, the core loss tester generates the required voltage waveform on the secondary winding. Figure 5.8 shows the flux density and the corresponding secondary voltage waveforms.
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Figure 5.8. Flux density waveform of the stator sensor S1 and the corresponding secondary winding voltage.
Figure 5.9 and Figure 5.10 present the reference and generated waveforms of the secondary winding voltage and the flux density waveforms for the stator sensor S1.
Figure 5.9. Reference and generated secondary winding voltage.
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Figure 5.10. Reference and generated flux density waveform of S1. Similarly, other flux density waveforms from the regions given in Figure 5.3 are generated.
Sensors S5 and S6 waveforms are given in Figure 5.11 and Figure 5.12, respectively, as examples.
Figure 5.11. Reference and generated flux density waveform of S5.
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Figure 5.12. Reference and generated flux density waveform of S6.
The rotor flux density waveforms in Figure 5.4 contain a DC offset and high-frequency components. Because these waveforms are taken at the unloaded conditions, the high-frequency components are not very significant except for some of the regions. These high-frequency components become more important when the motor is loaded, which causes the rotor core loss to increase rapidly. Figure 5.13 shows the flux density waveform of the motor taken from the rotor sensor R11, which exhibits the largest high-frequency oscillation. The waveform taken from the same region at the rated conditions has a DC offset component that exceeds 2.4 T with a high- frequency component of 0.1 T in amplitude.
Figure 5.13. Flux density waveform of the motor taken from the rotor sensor R11.
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The waveforms obtained from the FEA at the rated loading conditions are generated in the test system as well. Figures 5.14 through 5.19 present the reference and generated flux density waveforms for the stator core loss regions.
Again, the rotor waveforms contain different levels of DC offsets and high-frequency components that have higher amplitudes than the ones in unloaded condition. Therefore, the core loss in the rotor increases significantly when the motor is loaded. Figure 5.20 through Figure 5.32 show the reference and generated flux density waveforms of the rotor core loss regions at the rated loading condition.
Figure 5.14. Reference and generated flux density waveform of S1 at rated loading condition.
Figure 5.15. Reference and generated flux density waveform of S2 at rated loading condition.
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Figure 5.16. Reference and generated flux density waveform of S3 at rated loading condition.
Figure 5.17. Reference and generated flux density waveform of S4 at rated loading condition.
Figure 5.18. Reference and generated flux density waveform of S5 at rated loading condition.
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Figure 5.19. Reference and generated flux density waveform of S6 at rated loading condition.
Figure 5.20. Reference and generated flux density waveform of S7 at rated loading condition.
Figure 5.21. Reference and generated flux density waveform of R1 at rated loading condition.
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Figure 5.22. Reference and generated flux density waveform of R2 at rated loading condition.
Figure 5.23. Reference and generated flux density waveform of R3 at rated loading condition.
Figure 5.24. Reference and generated flux density waveform of R4 at rated loading condition.
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Figure 5.25. Reference and generated flux density waveform of R5 at rated loading condition.
Figure 5.26. Reference and generated flux density waveform of R6 at rated loading condition.
Figure 5.27. Reference and generated flux density waveform of R7 at rated loading condition.
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Figure 5.28. Reference and generated flux density waveform of R8 at rated loading condition.
Figure 5.29. Reference and generated flux density waveform of R9 at rated loading condition.
Figure 5.30. Reference and generated flux density waveform of R10 at rated loading condition.
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Figure 5.31. Reference and generated flux density waveform of R11 at rated loading condition.
Figure 5.32. Reference and generated flux density waveform of R12 at rated loading condition.
Figure 5.33. Reference and generated flux density waveform of R13 at rated loading condition.
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5.4. Total Core Loss Estimation
Core loss values per unit mass for each region are measured with the toroid tester. The core loss per mass data for the stator loss regions, along with the mass of each region, are given in Table
2.
Table 2. Core loss per mass data for each stator loss region at the unloaded condition.
Loss Region Core loss [W/kg] Mass of the region [kg]
1 9.5 0.5361
2 10.4 1.0722
3 10.4 1.0722
4 7.4 1.0186
5 4.3 1.1901
6 7.6 8.6874
7 9.4 1.0293
Total stator core loss 115.5 W
The estimated total stator core loss is 115.5 W. Similarly, total no load rotor core loss is calculated as 6 W by adding up the losses of the rotor regions. Hence, the total no-load core loss is estimated at 121.5 W, which is very close to the measured value of 125 W using the IEEE motor loss segregation standard.
Using the proposed method the total core loss of the motor is estimated accurately when the motor is unloaded (classical iron losses). The results are compared with those of the method used in the Flux 2D FEA software package; that is, Bertotti’s core loss model, in which the total core loss is calculated as 67 W.
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Using the measured loss for the flux density waveforms given in Figures 5.14 through 5.33,
Table 3 and Table 4 are created. The estimated total core loss is 150 W, which includes a 135.5 W loss in the stator and a 14.5 W loss in the rotor.
Table 3. Core loss per mass data for each stator loss region at rated loading condition.
Loss Region Core loss [W/kg] Mass of the region [kg]
1 0.5361 15.6
2 14.6 1.0722
3 14.6 1.0722
4 9 1.0186
5 5.2 1.1901
6 8.1 8.6874
7 10.1 1.0293
Total stator core loss 135.5 W
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Table 4. Core loss per mass data for each rotor loss region at rated loading condition.
Loss Region Core loss [W/kg] Mass of the region [kg]
1 3 0.53
2 0.4 0.55
3 0.41 0.81
4 1.45 1.10
5 0.65 1.26
6 0.265 0.38
7 2.5 0.40
8 0.6 0.65
9 0.35 1.02
10 3.2 1.11
11 0.55 0.71
12 2.1 0.68
13 1.5 0.49
Total stator core loss 14.5 W
The Bertotti loss model results in total losses of 106 W at the loaded condition, in contrast with the proposed model which gives 150 W. At present, there is no standard method for measuring the core losses at the rated conditions to verify the accuracy of the proposed method. Accordingly,
Chapter VI presents a method for determining the core loss resistances on the equivalent circuit using a recursive estimation algorithm, and the motor performance test results for verifications.
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5.5. Conclusions
This chapter presented the core loss estimation procedure of a 10 hp IPMSM using the proposed core loss estimation method. The core loss of the machine is estimated to be 121 W for unloaded conditions and 150 W for rated loading conditions. The results are compared with the Bertotti core loss estimation model results, which are 67 W and 106 W, respectively.
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CHAPTER VI
CORE LOSS MEASUREMENT OF AN IPMSM
In this Chapter, the core loss measurement of an IPMSM is investigated. In order do that, the dq model of the IPMSM is derived, explained, and the electrical parameters of the motor are determined. Motor performance tests are conducted under various loading conditions for determining the total loss of the motor, which will be used for segregating the losses. The power dissipated by the motor winding resistances (copper loss) can be accurately calculated using the winding DC resistance and the phase currents. The mechanical losses are provided by the manufacturers, which are linearly proportional to the speed. The rest of the loss is the core loss, which is dissipated by the magnetic materials of the motor. In the conventional approach, the core loss is considered as load independent; however, the FEAs show that loading increases the amplitude of the flux and its high-frequency harmonic components. This causes saturation on the magnetic material, variation on the dq axis inductances. Hence, the core loss increases. This loss difference is often referred to as stray loss, which is not calculated but simply estimated as 1% -
2% of the motor output power. Because the proposed method considers all the harmonic components in the motor, there is no need to separate these additional losses as stray losses. Based on this information, the core loss is modeled using four resistors connected on both the d and q axis equivalent circuits. Using the measured quantities and the equivalent circuit model, the core loss resistances are estimated using the recursive least squares algorithm.
Section 6.1 presents the basics of the dq transformation. Then, the motor dq model is derived and explained in Section 6.2. The IPMSM parameter determination tests are explained and performed for the 10 hp IPMSM in Section 6.3. The motor tests are performed and the core loss
83 resistances connected to the d and q axes circuit models are estimated using a nonlinear least squares algorithm in Section 6.4.
6.1. abc to dq0 Transformation
The dq0 transformation (also called Park transformation) is the direct, quadrature, and zero transformation introduced by R. H. Park in 1929 [131]. This transformation is a mathematical transformation that simplifies the analysis of three phase circuits by rotating their reference frame.
The zero axis is the summation of all three quantities, which results in zero in a balanced system.
The calculations can easily be carried out using the DC quantities before performing the inverse transform to calculate the actual three-phase quantities. The dq transformation is widely used for simplifying the analysis of three phase inverters, synchronous motors, induction motors, as well as for control purposes. It is especially useful for transforming the variable inductances into constant quantities in synchronous machines.
Im fb
ω q
θ Re fa
d fc
Figure 6.1. Three phase reference axes and rotating dq reference frame.
Figure 6.1 shows the three-phase axes and the rotating dq axes. The dq frame rotates with the angular speed ω, and the q axis is aligned with the phase a when the reference frame angle θ is
84 zero. Using trigonometric relations the d and q axes quantities (fqs and fds) can be calculated as follows:
222 ffffqsasbscscoscoscos (27) 333
222 ffffqsasbscssinsinsin (28) 333 here, f represents any of the quantities in the abc or dq frame. Subscript s denotes that the quantities are stator quantities. The zero sequence is expressed as follows, and it is zero in the balanced systems,
1 ffff (29) 0sasbscs3
The summation of the (27) and (28) after multiplying (28) by –j gives the space vector, referred to as the rotating reference frame, as follows
22 22jj ffjff ef ef eeff a f ajj 33 2 qdqdabcabc (30) 33
2 2 j j where, ae 3 and ae2 3 . Hence, the space vector form of the dq quantity can be written in the following format:
j fefqdabc (31)
6.2. dq Model of an IPMSM
The voltage equations of an IPMSM can be derived in the rotating reference frame using the relations given above. The complex representation of the voltage equation on the abc frame is
vabcs r s i abcs p abcs (32)
where, vabcs is the stator voltage, iabcs is the stator current, rs is the winding resistance, abcs is the stator flux linkage in the abc reference frame, and p denotes the derivative. The rotor angle is the
85 same as the synchronous angle and is represented as θr. Therefore, converting the voltage vector from the abc frame to the dq frame is basically equivalent to multiplying it by e jr , as follows:
jjjrrr veriepeabcssabcsabcs (33)
Applying the chain rule to the Equation (33) the voltage equation can be rewritten as
jjjjrrrr verabcss iepeje abcsabcsrabcs (34)
The voltage vector referred to the rotor reference frame can be written in the following form:
vr r i r p r j r qds s qds dqs r qds (35) where, superscript r denotes the quantities of the rotor reference frame.
Similarly, the flux linkage vector can be transformed into th dq reference frame as follows:
jr qdsabcs e
jr 33jjjjrrrr * 2 2 LLieLlsAabcsB ieeee abcsm 22 33 j (36) jjrr* 2 LLieLlsAabcsB iee abcsm 22 j 33rr* 2 LLiLlsAqdsBqdsm ie () 22 where, it can be observed from (36) that the position dependent inductances have disappeared from the flux linkage equation in the rotor reference frame.
The magnetizing inductances on the d and q axes can be written in the following form [132]:
3 LLL (37) md2 A B
3 LLL (38) mqAB 2
Therefore, the flux linkage equation given in (36) can be rewritten using the magnetizing inductances; that is,
LLLL j md mqrr md mq * 2 qdsL ls i qds L B() i qds m e (39) 22
86 where, Lls is the winding leakage inductance, Lmq and Lmd are the q and d axis magnetizing inductances.
The scalar form of the dq voltage vector is more useful for further derivations. The following complex representations can be used to expressing the relevant vectors.
rrr vqdsqsds v jv (40)
rrr i iqdsqsds ji (41)
rrr qdsqsds j (42)
The real and the imaginary parts of the voltage and the flux linkage equations are equated on both sides to obtain the scalar forms as follows:
rrrr vrqss ip qsqsrds (43)
r r r r vds r s i ds p ds r qs (44) where,
rr qsqqs Li (45)
rr dsddsmLi (46)
Here, m is the permanent magnet flux, and the q and d axis inductances Lq and Ld are defined as
LLLqlsmq (47)
LLLd ls md (48)
The scalar form of the voltage equation can be obtained, by substituting (47) and (48) into (43) and
(44) and assuming the permanent magnet flux is constant as following.
r r r r vqs r s i qs pL q i qs r L d i ds r m (49)
rrrr vrdss ipL dsd dsriL i q qs (50)
Finally, the equivalent circuit of the IPMSM is determined as shown in Figure 6.2.
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r r ωrLdi qs ωrλm ωrLqi qs ir r Lls ir r Lls qs s + - + - ds s - +
r r λm v qs Lmq v ds Lmd If= Lmd
(a) (b)
Figure 6.2. IPMSM (a) q and (b) d axis equivalent circuit models in rotor reference frame.
The electromagnetic torque produced by the motor can be derived using the power balance equation. The instantaneous power can be written as
22 iirr 333r rr rrr 22 qsds Pveqs qsdsiv dssir ir qss ipLpL dsdq ( )( ) 22222 Copper loss Rateof changeof stored energy (51)
3 r rr r r dsii qsr qs ds 2 Torque production
For a lossless system, the output power is equal to the electromechanical power, which is equal to the multiplication of the shaft torque times the mechanical shaft speed ωrm; that is,
2 (52) rmr n where, n is the number of poles of the motor. Therefore, the torque equation can be written as follows:
3 n r r r Te m i qs L q L d i ds i qs (53) 22
For a lossy system, the shaft torque can be calculated by subtracting the additional losses from the produced electromagnetic power. These additional losses can be considered the core loss, friction and windage losses, and stray loss. The manufacturers usually provide the friction and windage loss, which are linearly proportional to the speed. Thus, they can be accurately estimated based on the speed. The core loss and the stray losses are often combined together and represented as an additional resistance on the equivalent circuit. Stray losses are the losses that are not related to a
88 specific source but to a combination of various sources such as harmonics, field distortion, additional core loss, etc. [133]. In this work, the stray loss is combined with the core loss because of the majority of the stray losses are the additional core losses that increase with loading [131],
[133]. The estimated core losses are compared with these values.
6.3. Parameter Determination of an IPMSM
The IPMSM electrical parameters are required for modeling and simulating the motor. The methods determining the winding resistance, q and d axis inductances and magnet flux are given in this section.
The stator winding resistance can be determined by performing a DC test. The windings are excited with a DC source and 10-20% of the rated current is passed through the motor windings.
The voltage and the current values are recorded and the resistance is calculated using Ohm`s law, as shown in Figure 6.3.
Idc rs Vdc =2rs V + Idc dc - L L rs rs L
Figure 6.3. Winding resistance measurement procedure.
A multi-meter can be used for measuring winding resistances greater than 10 Ω. However, if the resistance is lower than that, the measurement will not be accurate.
The synchronous inductances (Lq and Ld) of the IPMSM are not equal to each other due to the saliency of the machine. The d and q axes inductances change with the rotor position. The inductance measured from the phase windings when the rotor is aligned with the q axis is Lq; when the rotor is aligned with the d axis is Ld as shown in Figure 6.4. Thus, measurements of the inductance as it varies with the rotor position can be done with an LCR-meter.
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d axis q axis
q axis d axis
Lq Ld
Figure 6.4. Lq and Ld inductances of an IPMSM.
The measured inductance of the circuit given in Figure 6.5 is 3/2 times the phase inductance. The phase inductance will be a sinusoidal signal having n/2 times the rotor mechanical frequency, with an offset, as shown in Figure 6.6. A
I rs
+ L V - dc L rs rs L B C
Figure 6.5. Inductance measurement circuit.
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Figure 6.6. Phase inductance vs. rotor position.
The maximum and the minimum values of this curve give the Lq and Ld inductances, respectively [134].
6.4. Core Loss Measurement of an IPMSM
The classical iron loss can be measured with the no load test. This loss is considered to be independent of the load at a constant frequency. However, the validity of this approach has been disproved through the FEA simulations. Nonetheless, this fact is not been experimentally verified due to the lack of standard core loss measurement methods in loaded conditions. The core loss measurement can be possible with a technique that is based on vector control of the machine, as reported in [135]. In this method, the machine is operated as a motor and generator using the same amplitude q and d axis currents, which are responsible for the field and the torque generation. The performance data is taken at steady state conditions. The steady state circuit that has two core loss resistances connected to each axis equivalent circuit is used for analysis. The core loss resistances are calculated based on some mathematical manipulations. This method is only suitable when the control of both the field (id) and torque (iq) currents is possible; that is, when a custom design regenerative variable speed drive (VSD) is used. In a motor test setup that does not have such a
VSD for controlling the motor under test, this method is not applicable. Hence, a similar approach
91 that does not require controlling the dq axis currents is proposed. This method also has core loss resistances on both of the dq axis circuits. In addition, extra resistors are inserted in the circuit model in order to have more freedom representing the core losses. This approach is used in various studies to represent the stray losses [125], [136], [137]. Because the proposed method considers all of the harmonic components in the motor, there is no need to separate these additional losses as stray losses. The core loss is the remaining loss found after subtracting the copper and mechanical losses from the total loss. Hence, the extra resistors represent the additional core loss as it increases with the loading. The estimations performed at various loading conditions can be verified through the experiments. In order to perform these studies, a new motor test system is necessary. The next section gives the details of the motor test setup.
6.4.1. Development of Motor Test Setup
The motor test setup consists of a dynamometer, a torque sensor, the motor to be tested, shaft couplings, an energy analyzer, an oscilloscope, and appropriate regenerative power electronic drive systems. The dynamometer consist of an induction motor; speed and direct torque controlled power electronic drivers, and a torque sensor. The torque sensor, from Magtrol, is capable of measuring up to 40 Nm torque with less than 0.1% error for speeds between 0 to 6000 rpm. The Magtrol 2410 readout provides the analog torque and speed outputs. The supplied active and reactive powers, currents and voltages, and the output power are required to be measured with a power analyzer for performance calculations. The Power analyzer used in the test system is the WT3000 from
Yokogawa Electric Corp., which is capable of analyzing the performance of the motor with four independent built-in current and voltage sensors of up to 30 A RMS current and 600 V RMS voltage, respectively. Moreover, it has torque and speed analog inputs that can be connected to the analog output of the torque sensor’s readout for calculating the shaft power. An oscilloscope needs to be used to store the voltage and current waveforms for calculating the currents and voltages in the dq reference frame. The scope used in the system is a Tektronix digital oscilloscope with current
92 and differential voltage probes. All of the mentioned equipment can be connected to a computer through USB communication, hence, the measurements and control commands are sent from the computer to the devices and vice versa. The connection diagram of the test setup is presented in
Figure 6.7.
The motor under test is a 10 hp, 1800 rpm, 460 V IPMSM from ABB, Baldor Reliance Inc.
Similarly, the motor driver is from the same company. Further, the voltage and power ratings are matched to that motor. The motor and its power electronic driver are shown in Figure 6.8, while the built motor test bench is presented in Figure 6.9.
Control & Measurements Control & Measurements Measurements
Computer
Drive Measurements Drive
SPEED 1798 SET 1800 IA 5 IB 5 SP EED 1798 IC 5 Power Analyzer SET 1800 IA 5 OK CNC ENT IB 5 Oscilloscope IC 5 Voltages OK CNC ENT Me as. Phase A Phase B P hase C
SPEED 1798 1798 1 798 IA 5 5 5 IB 5 5 5 IC 5 5 5 Currents P 500 5 00 5 00 Q 120 1 20 120 S 515 515 515 PHI 75 Torque 40
OK CNC ENT
Torque Sensor Readout Excitation Excitation TQ: 30.2 Nm SP: 1800 rpm
Torque & Speed
Motor Torque Sensor Dynamometer
Figure 6.7. The connection diagram of the motor test system.
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Figure 6.8. Motor under test and its power electronic driver.
Figure 6.9. Motor test bench.
6.4.2. Motor Non-Idealities and Performance Tests
The performance tests are required for observing the motor behavior under various conditions to verify the accuracy of the determined parameters and understand the non-idealities of the motor.
For instance, the dq axis inductances are determined when the motor is unloaded, without having
94 any excitation on the windings. Ideally, these parameters should be constant in any condition, but that is not actually the case. These parameters vary with the loading condition, due to the saturation of the magnetic materials used in the motor. The q axis inductance of the motor under test is 164 mH, and it drops down to 91 mH at the rated condition, while the d axis inductance decreases from
24 mH to 22 mH. Figure 6.10 shows the Ld and Lq inductance variations versus the phase current.
0.20
0.15
0.10 Ld 0.05 Lq
Inductance[mH] 0.00 0 2 4 6 8 10 12 Phase Current [A]
Figure 6.10. d and q axis inductance variation versus the phase current.
Another variable parameter is the winding resistance, which varies with temperature. It varies linearly with temperature, as seen from[138].
RTRT TR0 1 (54)
where, RT0 is the resistance at temperature T0, T is the temperature variation from T0 to the
operating temperature, and R is the temperature coefficient of copper.
Another non-ideality is the core loss. These losses are represented as resistors, connected to the d and q axis equivalent circuits as shown in Figure 6.11.
r r ω λ ωrLqi qs r ωrLdi qs r m L r Lls i r Radd,q ls i ds rs Radd,d qs s + - + - - +
r r R Rc,d v qs c,q Lmq v ds Lmd If
(a) (b)
Figure 6.11. IPMSM (a) d and (b) q equivalent circuits with core loss components.
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The performance tests are conducted with the motor test system that was built. The power analyzer data is captured for the estimation process. Since the copper loss and the mechanical
(friction and windage) losses can be accurately determined, the rest of the loss is the core loss. The core loss resistances given in Figure 6.11 can be estimated using a recursive least squares algorithm.
6.4.3. Motor Test Results
The motor is tested using the built motor tests setup at rated speed while the motor is unloaded and at the rated load. Input-output powers, phase currents, shaft torque and speed are measured.
Since the speed is kept constant, the friction and windage losses (Pmech) are constant. The copper loss (Pcu) can be calculated using the winding resistor (rs = 0.62 Ω) and the phase current. The difference between the input and output powers is the total loss (Ptotal,loss ), and the core loss (Pcore) is the subtraction of the mechanical and copper losses from the total loss. The IPMSM and the induction machine are both operated in volt per hertz control mode. Table 5 presents the results.
Table 5. Motor test results for unloaded and rated load conditions at 1800rpm.
Unloaded Rated Load
I [A] 7 10.6
Pmech [W] 65 65
Pcu [W] 91 209
Ptotal,loss [W] 281 434
Pcore [W] 125 160
Pcore,est [W] 121 150
The results verifies the proposed core loss estimation method’s error is under 10 %. For the unloaded condition, estimated and measured core loss are 121 W, and 125 W; for the rated loading condition, 150 W and 160 W.
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6.4.4. Levenberg-Marquardt Recursive Parameter Estimation
Since it is not possible to obtain the dq axes core loss resistances while the motor is loaded, a recursive estimation algorithm is used to determine these parameters based on the tests performed at various loading conditions. Of the large variety of estimation algorithms available in the literature
[139]–[143], the Levenberg – Marquardt (LM) algorithm is among the most reputable ones [144].
The LM algorithm iteratively locates the minimum of a multivariable function that is expressed as a sum of squares of real-valued non-linear functions. LM can be considered as a combination of steepest descent with Gauss-Newton methods [145]. If the current approximation is far from the true solution, the slow but reliable steepest descent method is employed; otherwise, the Gauss-
Newton method is used.
Let the function f map a parameter vector pm to an estimated measurement vector
n xfpx(), . Given an initial parameter estimate p0 and an initial vector x , the task is to find the vector p* that best satisfies the functional relation f , meaning that it minimizes the squared distance T , where xxˆ . The starting point of the LM algorithm is a linear approximation
to f in the neighborhood of p. For small p , a Taylor series expansion leads to the approximation
(55) fpfpJ pp
where J is the Jacobian matrix. With an initial guess p0, the iterative procedure produces a series of
* vectors p1, p2, …, etc., converging toward a local minimizer p for f. Hence, one seeks p that
minimizes the quantity x f px ppp f p JJ . This p is the solution to the standard linear least squares problem:
TT (56) JJJ p
The matrix JJT is an approximation to the second order derivatives matrix. The LM method solves a small variation of (56) also known as the augmented normal equations
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T (57) NJp where the off-diagonal terms of N are interchangeable with the corresponding terms of JJT .
N J J T gives the diagonal elements for random 0 . The idea of adjusting the diagonal ii ii
T terms of JJ is called damping, and is called the damping term. After computing p the algorithm checks whether the updated parameter leads to error reduction. If yes, then the update is accepted and the estimation process repeats with a decreased damping term. Otherwise, the damping term is increased, (57) is solved again and the iteration continues until a value of that reduces the error is found. This process continues until an acceptable update to the parameter vector is found.
The Matlab Control and Estimation Tool GUI [146] provides a straightforward interface and simplifies the estimation process. At various loading conditions, the experimental data for the d
r r and q axis currents, the shaft torque, and motor efficiency (i qs, i ds, Tshaft, η) are recorded. The simulation is built in the Matlab / Simulink environment considering all of the parameter variations based on the phase currents and temperature. The calculated torque and efficiency of the motor are compared with the experimental data to predict the dq axes core loss resistances Rc,q, Rc,d, Radd,q, and Radd,d. The error between the measured and the simulated output data has been evaluated according to the LM algorithm and the parameters were updated for the next iteration accordingly.
The method succeeded and convergence was achieved. The results are provided in the following section.
6.4.5. Core Loss Resistance Estimation
In this section, the core loss resistance estimation results are presented. The simulation of the motor dq model is built in the Matlab / Simulink environment considering all of the non-idealities of the motor. Measured dq axis currents from the performance tests are given to the simulation as inputs; these inputs are responsible for the dq axis inductance variations, copper losses, iron losses,
98 and the generated torque. The output of the simulation is the torque and the efficiency. The performance data is obtained at three different loading conditions. This data is compared with the simulation outputs, then, the error is calculated. The estimation algorithm adjusts the parameters iteratively to reduce the error to zero. Estimated parameter trajectories during this process are presented in Figure 6.12. The estimated and the actual torque and efficiency plots at the steady state conditions are given in Figure 6.13.
Figure 6.12. Trajectories of the estimated parameters.
Figure 6.13. Estimated and measured output variables (a) torque (b) efficiency.
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It can clearly be observed that the estimation is successful and that convergence is achieved, the estimation error is less than 1 % at the maximum point.
6.5. Conclusions
The core loss estimation through the flux controlled core loss tester gave 121 W at the unloaded condition and 150 W at the rated condition. The performance tests are conducted at unloaded, partially loaded, and fully loaded conditions. With the help of the power analyzer, the input three phase voltages and currents are measured, then, the input powers are calculated. The speed and the torque of the motor shaft are also measured and fed to the power analyzer for calculating the output power. Hence, the efficiencies and the total losses are determined. Since the rotor resistance is known, the copper losses are calculated using (51). The mechanical (friction and windage) losses are also known at the rated speed, and since these losses are proportional to the speed, the mechanical loss coefficient is determined. Thus, the mechanical losses are calculated by multiplying the speed and the mechanical loss coefficient. Finally, the remaining loss is the core loss of the machine.
The measured core loss at the unloaded condition is 125 W while the estimated core loss with the proposed method is 121 W; the measured core loss at the rated condition is 160 W and the estimated core loss with the proposed method is 150 W. Regardless of the measurement errors on the performance tests and the core loss tests, the proposed core loss estimation method has less than
10 % estimation error for the given IPMSM.
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CHAPTER VII
SUMMARY AND FUTURE WORK
7.1. Summary
This dissertation proposes a novel core loss estimation method for electric machines based on the actual flux density waveforms in various regions of the motor. Conventionally, the Steinmetz equation or loss separation based core loss models are used for core loss estimations in finite element analysis (FEA) software packages due to the relatively simple and straightforward calculation procedures. However, these models lack accuracy when it comes to the estimation of core losses in electric machines, where the flux density waveforms are not spatially uniform. This is mainly due to the fact that conventional methods are based on curve fitting an equation to the experimental loss data, under certain conditions, which renders them inaccurate at different operating conditions. Thus, they result in a high estimation error which, in some cases, is more than
100%. There are mathematical hysteresis models that can accurately calculate the core losses under any set of loading conditions. However, these models are complex and require a great deal of knowledge about the material to be tested. Moreover, it is difficult to integrate these sophisticated models into finite element analysis software. In Chapter II, a detailed review of existing core loss methods is provided. In addition, factors affecting core losses are explained, while core loss measurement methods, core loss testers and their control are presented.
The proposed core loss estimation method uses the predetermined flux density waveforms from the FEA. Various motor regions contain different flux density waveform properties in electric machines. While some of the regions have waveforms at the same frequency as that of the excitation frequency, some of them have additional harmonic components, and still others have DC biased
101 high-frequency waveforms. To determine the actual core loss, the magnetic material used in the motor should be tested with the same flux density waveforms. This requires special testing equipment that is able to control the AC and DC flux densities on the core being tested. In Chapter
III, the proposed core loss method and the flux control concept are introduced. The development of the flux controlling core loss tester is presented. The challenging task here is to control the system under heavy disturbance. The source of the disturbance is the toroidal transformer, which is basically a non-linear inductor. When saturation occurs, the primary winding current increases rapidly, hence, the tracking performance of the controller degrades because of the abrupt current variation. Keeping in mind the periodic nature of this disturbance, a periodic controller to compensate for the output signal error is used. In addition to the periodic controller, a state feedback, and a feedforward controllers are added to the system to suppress non-periodic disturbances, and improve stability and settling time. The required hardware and control mechanism determination, the plant model derivation and controller development are provided in
Chapter III, as well.
In Chapter IV, the flux controller designed, based on the control law given in Chapter III using the plant parameters, is implemented and then the system is simulated for performance analysis.
First, the performance of the designed flux-controlling core loss tester is validated with sinusoidal flux density waveforms. It was found that the generated sinusoidal flux density has less than 1% total harmonic distortion at high power levels. This implies that the flux control performance is satisfactory under heavy periodic disturbances. Later, the system is simulated at various amplitudes, frequencies, and harmonic components and its performance is verified. After verifying the controller performance, a detailed hardware development procedure including the single-phase inverter topology selection, switching device selection, DC bus circuit and control circuit design considerations are presented. In order to generate higher order harmonics, the switching speed needs to be kept high at high power levels. State of the art silicon carbide (SiC) MOSFET technology is used for flux generation. In contrast with their silicon (Si) counterparts, SIC
102
MOSFETs have the ability to block higher voltages, switch at higher frequencies, and operate at higher temperatures with higher efficiencies. However, these MOSFETs require extra care with regard to parasitic inductances, which could cause large oscillations, high-frequency ringing, and consequently electromagnetic interference (EMI) problems. The hardware development section covers all the design considerations for the EMI. Finally, the experiments are successfully performed at various operating conditions, which include some of the actual flux density waveforms obtained from the FEA of an IPMSM motor.
After developing and verifying the performance of the new core loss tester, core loss estimation is performed for a 10 hp IPMSM in Chapter VI. As described before, the estimation procedure starts with the FEA and the core loss regions, where the core loss is uniformly distributed, are determined. The flux density waveforms are obtained from each region. These waveforms are then generated in the toroidal core made of the same material used in the motor. The core loss per volume/mass information is measured and the total loss of the motor is calculated for unloaded and loaded conditions.
Chapter VII presents the rotating dq reference frame equivalent circuit in order to determine the electrical parameters of the motor. The dq model of the motor is derived and the electrical parameters are determined through parameter determination tests. In order to verify core loss estimations, the motor performance tests are necessary. Measurement of the core loss is straightforward when the motor is unloaded. The total loss consists of copper, friction and windage, and core losses. The copper loss is calculated using the winding resistance and the phase current.
The friction loss is provided by the manufacturer and is proportional to the rotor speed; the remaining loss is the core loss. When the motor is loaded, the effect of saturation, spatial harmonics, etc. cause the core loss to increase. Conventionally, this additional loss is called stray loss. The stray loss is usually estimated as a percentage (1-2%) of the motor output power. Since the proposed core loss estimation method considers all the harmonics and saturation effects, the additional loss is not separated from the core loss. Therefore, the loss after subtracting the copper, friction and
103 windage losses is the total core loss as it was in the unloaded condition. Both unloaded and loaded condition total core loss estimation using the proposed method is compared with the motor test results and the estimation error was found to be less than 10 %. The error stems from measurement errors caused by the measurement equipment, bandwidth limitations of the developed core loss tester, and from neglecting the eddy current losses in the permanent magnets.
Moreover, this core loss behavior is represented in the equivalent circuit with four resistances that are inserted into the d and q axis circuit models; one in series to the main current path and the other one in parallel after the winding resistance. These resistances are estimated using the
Levenberg-Marquardt nonlinear recursive least squares estimation method. The data used for the estimation is obtained through performance tests conducted at various loading conditions.
Convergence was achieved and thus the core loss resistances were determined successfully.
7.2. Future Work
The proposed method can be used to determine the additional core loss variations for various size and type of motors.
Dynamic load variations during the performance tests can be used for distinguishing the losses better.
Even though the proposed estimation method accurately estimates the core loss under any conditions, conducting experiments for various loading conditions is time-consuming. Instead of performing steady state measurements, instantaneous power and flux measurements can be used for developing a more accurate empirical core loss model using the flux controlled core loss tester.
A lookup table and an equation based core loss estimation method can be developed. Any possible waveforms and corresponding loss data for a certain material can be stored in a look-up table for estimating the core losses.
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