Indirect Efficiency Determination and Parameter Identification of Permanent-Magnet Synchronous Machines Vom Fachbereich Elektrotechnik und Informationstechnik der Technischen Universität Darmstadt zur Erlangung des akademischen Grades eines Doktor-Ingenieurs (Dr.-Ing.) genehmigte Dissertation von Dipl.-Ing. Björn Deusinger geboren am 28. März 1986 in Frankfurt am Main Referent: Prof. Dr.-Ing. habil. Dr. h.c. Andreas Binder Korreferent: Prof. Dr.-Ing. Martin Doppelbauer Tag der Einreichung: 16. Juni 2020 Tag der mündlichen Prüfung: 9. November 2020 D 17 Darmstadt 2020 Deusinger, Björn: Indirect Efficiency Determination and Parameter Identification of Permanent- Magnet Synchronous Machines Darmstadt, Technische Universität Darmstadt Jahr der Veröffentlichung der Dissertation auf TUprints: 2021 URN: urn:nbn:de:tuda-tuprints-189155 URI: https://tuprints.ulb.tu-darmstadt.de/id/eprint/18915 Tag der mündlichen Prüfung: 09.11.2020 Veröffentlicht unter CC BY-SA 4.0 International https://creativecommons.org/licenses/ https://tuprints.ulb.tu-darmstadt.de/id/eprint/18915 https://creativecommons.org/licenses/ Erklärungen laut Promotionsordnung § 8 Abs. 1 lit. c PromO Ich versichere hiermit, dass die elektronische Version meiner Dissertation mit der schrift- lichen Version übereinstimmt. § 8 Abs. 1 lit. d PromO Ich versichere hiermit, dass zu einem vorherigen Zeitpunkt noch keine Promotion ver- sucht wurde. § 9 Abs. 1 PromO Ich versichere hiermit, dass die vorliegende Dissertation selbstständig und nur unter Verwendung der angegebenen Quellen verfasst wurde. § 9 Abs. 2 PromO Die Arbeit hat bisher noch nicht zu Prüfungszwecken gedient. i Vorwort Die vorliegende Arbeit ist im Rahmen meiner Tätigkeit als wissenschaftlicher Mitarbeiter am Institut für Elektrische Energiewandlung der Technischen Universität Darmstadt ent- standen. Ich möchte mich an dieser Stelle bei allen bedanken, die zum Gelingen meines Promotionsvorhabens beigetragen haben. Hierbei gilt mein Dank in erster Linie meinem Doktorvater Prof. Dr.-Ing. habil. Dr. h.c. Andreas Binder für seine engagierte Betreuung und seine wertvollen fachlichen und methodischen Anregungen. Ebenfalls danke ich Prof. Dr.-Ing. Martin Doppelbauer, dem Inhaber der Professur Hybridelektrische Fahrzeuge am Karlsruher Institut für Technologie (KIT) für die freundliche Übernahme des Korreferats dieser Arbeit. Die Untersuchung des vorgestellten Messverfahrens fand in enger Abstimmung mit dem nationalen Normungsgremium DKE/K 311 statt. Ich danke allen beteiligten Mitglie- dern für die gute Zusammenarbeit – insbesondere Dr.-Ing. Axel Möhle (Siemens AG) und Dr.-Ing. Christian Lehrmann (Physikalisch-Technische Bundesanstalt Braunschweig) so- wie den Vorsitzenden Prof. Dr.-Ing. Bernd Ponick (Leibniz Universität Hannover) und Prof. Dr.-Ing. Martin Doppelbauer (KIT). Die zugehörigen Forschungsprojekte wurden vom Bundesministerium für Wirtschaft und Energie im Rahmen der Förderprogramme In- novation mit Normen und Standards (INS) und Wissens- und Technologietransfer durch Patente und Normen (WIPANO) gefördert, wofür ich mich ebenfalls bedanke. Des Wei- teren gilt mein Dank allen Firmen, die Testmaschinen zur Verfügung gestellt und mich bei den Messungen unterstützt haben. Besonderer Dank gilt Dr.-Ing. Olaf Körner, Dr.-Ing. Hayder Al-Khafaji und Frederick Claudino (Siemens Mobility GmbH) für die erfolgreiche gemeinsame Projektdurchführung. Ich danke meinen wissenschaftlichen Kolleginnen und Kollegen Jeongki An, Maximilian Clauer, Daniel Dietz, Nicolas Erd, Bogdan Funieru, Yves Gemeinder, Robin Köster, Marcel Lehr, Xing Li, Oliver Magdun, Gael Messager, Alexander Möller, Sascha Neusüs, Kersten Reis, Omid Safdarzadeh, Jinou Wang und Martin Weicker herzlich für die ergie- bigen fachlichen Diskussionen, die gute Zusammenarbeit und die stets freundschaftliche Gemeinschaft. Vielen Dank ebenfalls an alle Studenten, die im Rahmen ihrer Abschluss- arbeiten bei meinen Forschungstätigkeiten mitgewirkt haben. ii Ferner danke ich herzlich allen technischen und administrativen Mitarbeiterinnen und Mit- arbeitern am Institut für Elektrische Energiewandlung: Klaus Gütlich für die zahllosen elektrischen Installationen und Prüfstandsaufbauten, Herbert Moschko für die Hilfe bei messtechnischen Fragestellungen, Andreas Fehringer und Markus Lohnes für die umfang- reichen mechanischen Auf- und Umbauarbeiten an den Testmaschinen und Prüfständen sowie Sabine Waldhaus, Anette Gallinat und Annette Siler für die Unterstützung bei der Projektverwaltung. Zum Schluss möchte ich mich besonders bei meiner Familie bedanken – vor allem bei meinen Eltern Monika und Alfred Deusinger, die durch die fortwährende Unterstützung und Förderung meiner wissenschaftlichen Ausbildung maßgeblich zum Erreichen meiner beruflichen Ziele beigetragen haben. Björn Deusinger im Juni 2021 iii Abstract This thesis presents a method for efficiency determination of inverter-fed permanent- magnet synchronous machines by summation of the individual losses. For electrically excited synchronous machines there are already standardized methods, as for high effi- ciency values the direct procedure of input/output measurement is too inaccurate. With the novel method for machines in the base speed range, the individual losses are deter- mined at the no-load experiment and the removed rotor test and recalculated for rated operation. The prerequisites and the analysis of the experiments are described. To evalu- ate the procedure, measurement series are performed on five different permanent-magnet synchronous machines with a rated power of 45 kW . . . 160 kW. The indirect efficiency is compared with a direct measurement. Also for four of the five test machines finite element simulations are carried out to prove the assumptions of the proposed method. The result is, that for four of the test machines a good accordance with deviations below 0.5 % at rated operation is reached. But because of the special design of the fifth test machine the losses and thus the efficiency show deviations of about 1 %. In total, the indirect procedure is usable as an adequate replacement for typical kinds of permanent-magnet synchronous machines with distributed windings and especially for big rated powers, where the direct method is too imprecise. In addition to the efficiency determination the thesis also shows, how to determine the no-load voltage, the short-circuit current, and the synchronous in- ductance by using the described experiments. iv Kurzfassung Diese Arbeit präsentiert ein Verfahren, um den Wirkungsgrad von umrichtergespeis- ten Permanentmagnet-Synchronmaschinen im Einzelverlustfahren zu ermitteln. Für elek- trisch erregte Synchronmaschinen sind bereits seit langem derartige Verfahren standardi- siert, da für sehr hohe Wirkungsgrade die direkte Messung aus Eingangs- und Ausgangs- leistung zu ungenau ist. Mit Hilfe des neuartigen Verfahrens für Maschinen im Grunddreh- zahlbereich werden die Einzelverluste im Leerlaufversuch sowie im Bohrungsfeldversuch bestimmt und für den Bemessungsbetrieb umgerechnet. Die Anforderungen an die Versu- che und deren Auswertung werden beschrieben. Zur Evaluation des Messverfahrens wer- den Versuchsreihen an fünf unterschiedlichen Permanentmagnet-Synchronmaschinen mit Bemessungsleistungen von 45 kW . . . 160 kW durchgeführt, wobei letztlich der indirekte Wirkungsgrad mit einer direkten Messung verglichen wird. Außerdem werden für vier der fünf Testmaschinen Finite-Elemente-Simulationen durchgeführt, um die Annahmen, welche für das Verfahren getroffen werden, zu überprüfen. Es zeigt sich, dass für vier der Testmaschinen bei Bemessungsbetrieb eine gute Übereinstimmung mit Abweichun- gen kleiner als 0.5 % erzielt werden kann. Die fünfte Testmaschine hat aufgrund ihrer Konstruktion ein spezielles Verlustspektrum, sodass hier die Methode Abweichungen von ca. 1 % zeigt. Insgesamt empfiehlt sich die indirekte Methode aber als adäquater Ersatz für die direkte Messung bei typischen Permanentmagnet-Synchronmaschinen mit verteil- ten Wicklungen und vor allem bei großen Bemessungsleistungen, wo die direkte Methode zu ungenau ist. Neben der Wirkungsgradbestimmung wird gezeigt, wie sich anhand der beschriebenen Versuche zusätzlich die Leerlaufspannung, der Kurzschlussstrom sowie die Synchroninduktivität der Testmaschinen lässt. v Table of contents 1. Introduction 1 2. Efficiency determination of electrically excited synchronous machines 5 2.1. Method A – Direct measurement: Input-Output . . . . . . . . . . . . . . 6 2.2. Method B – Summation of losses with load test . . . . . . . . . . . . . . 6 2.2.1. No-load losses . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.2. Load-dependent losses . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.3. Excitation losses . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.4. Efficiency calculation . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.5. Distinction to permanent-magnet synchronous machines . . . . . 9 2.2.6. Determination of the synchronous inductance . . . . . . . . . . . 9 2.3. Method C – Summation of losses without load test . . . . . . . . . . . . 10 2.4. Method D + E – Back-to-back test . . . . . . . . . . . . . . . . . . . . . 11 2.5. Method F – Zero power factor test . . . . . . . . . . . . . . . . . . . . . 11 2.6. Method G – Summation of losses except the additional load losses . . . . 11 2.7. Inverter-fed machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3. The permanent-magnet synchronous machine (PMSM) 13 3.1. Parameters and equivalent circuit . . . . . . . . . . . . . . . . . . . . . . 13 3.2. Losses in the stator winding . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2.1. Frequency-independent losses . . . . . . . . . . . . . . . . . . . 18 3.2.2. Frequency-dependent losses . . . . . . . . . . . . . . . . . . . . 18 3.2.3. Total losses and temperature dependency . . . . . . . . . . . . . 23 3.3. Losses in the motor lamination . . . . . . . . . . . . . . . . . . . . . . . 24 3.4. Losses in the permanent magnets . . . . . . . . . . . . . . . . . . . . . . 26 3.5. Friction and windage losses . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.6. Additional losses due to inverter supply . . . . . . . . . . . . . . . . . . 28 4. Experiments 31 4.1. Generator no-load test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2. Motor no-load test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 vi Table of contents 4.3. Removed rotor test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.4. Full load test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.5. Generator short-circuit test . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.6. Reactive current test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5. Indirect efficiency determination of PMSMs 45 5.1. Method description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.2. Drawbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6. Tested machines 53 6.1. Fractional-slot tooth-coil winding . . . . . . . . . . . . . . . . . . . . . 53 6.1.1. Test machine M1 . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.1.2. Test machine M2 . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.2. Integer-slot distributed winding . . . . . . . . . . . . . . . . . . . . . . . 57 6.2.1. Test machine M3 . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.2.2. Test machine M4 . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.3. Fractional-slot distributed winding . . . . . . . . . . . . . . . . . . . . . 59 6.3.1. Test machine M5 . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7. Measurements 61 7.1. Test bench . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7.1.1. Conventional setup . . . . . . . . . . . . . . . . . . . . . . . . . 61 7.1.2. Removed rotor setup . . . . . . . . . . . . . . . . . . . . . . . . 64 7.2. Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7.3. Measurement results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.3.1. Generator no-load test . . . . . . . . . . . . . . . . . . . . . . . 67 7.3.2. Motor no-load test . . . . . . . . . . . . . . . . . . . . . . . . . 71 7.3.3. Removed rotor test . . . . . . . . . . . . . . . . . . . . . . . . . 73 7.3.4. Load operation and efficiency test . . . . . . . . . . . . . . . . . 75 7.3.5. Generator short circuit test . . . . . . . . . . . . . . . . . . . . . 81 7.3.6. Reactive current test . . . . . . . . . . . . . . . . . . . . . . . . 83 7.4. Results of other authors . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 8. Finite element simulations 87 8.1. Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 vii Table of contents 8.2. Generator no-load operation . . . . . . . . . . . . . . . . . . . . . . . . 89 8.2.1. Model description . . . . . . . . . . . . . . . . . . . . . . . . . 90 8.2.2. Generator no-load flux density and voltage . . . . . . . . . . . . 94 8.2.3. Iron losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.2.4. PM losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 8.3. Removed rotor operation . . . . . . . . . . . . . . . . . . . . . . . . . . 101 8.3.1. Model description . . . . . . . . . . . . . . . . . . . . . . . . . 101 8.3.2. Current-depending losses . . . . . . . . . . . . . . . . . . . . . . 103 8.3.3. Iron losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 8.4. Load operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 8.4.1. Model description . . . . . . . . . . . . . . . . . . . . . . . . . 110 8.4.2. Current-depending losses . . . . . . . . . . . . . . . . . . . . . . 111 8.4.3. Iron losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 8.4.4. PM losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 8.4.5. Total losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.5. Efficiency calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 9. Conclusion 121 List of Tables 125 List of Figures 127 Bibliography 134 Appendix A. Test machine data 143 Appendix B. Measurement setup and device data 145 Appendix C. Measurement results 148 Appendix D. Calculation of measurement uncertainty 167 Appendix E. Simulation results 172 Appendix F. Iron loss determination in JMAG 199 Appendix G. Overestimation of iron losses at load 202 viii Notations and Symbols Notations ao number of parallel branches per phase − ai number of parallel sub-conductors − aT number of horizontal conductors per slot − A area, cross section m2 b width m B magnetic flux density T BR magnetic remanence flux density T d diameter m dE penetration depth m f electric frequency Hz fPWM inverter switching frequency Hz h height m I current A k time harmonic order (inverter) − kR resistance increase factor (current displacement) − ksc short-circuit ratio − kV iron loss increase due to manufacturing − kϑ resistance increase factor (temperature) − K total resistance increase factor − l length m L inductance H ms number of stator phases − M torque Nm n rotational speed s−1 n number − Nc number of turns per coil − ix Notations and Symbols Ns number of turns per phase − P power W q number of slots per pole and phase − Qs number of stator slots − r radius m R ohmic resistance Ω Re Reynolds number − T electric period (1/ f ) s u(xi) uncertainty of the value xi [xi] uc(y) combined uncertainty of the measurand y = f (x1,x2, . . . ,xN) [y] U electric voltage V V volume m3 X reactance Ω αe pole coverage ratio % α20 temperature coefficient related to 20 °C 1/K β ∗ current angle °el. γ position angle °el. δ air gap width m ε measured or calculated error − η efficiency − ϑ temperature °C ∆ϑ temperature rise K κ electric conductivity 1/(Ωm) µ space harmonic order (rotor) − µ magnetic permeability Vs/(Am) ν space harmonic order (stator) − ν kinematic viscosity m2/s ξ reduced conductor height − ρ mass density kg/m3 τp pole pitch m x Notations and Symbols ϕ phase angle °el. ϕ skin effect factor − ψ proximity effect factor − Ψ magnetic flux linkage amplitude Vs Subscripts 0 no-load 1 fundamental ad additional avg average b winding overhang br brush B bore field, removed rotor calc calculated crit critical Cu copper d direct axis d dissipative loss dir direct div division e equivalent ec eddy current el electrical elem elements em electromagnetic ex excess exc excitation ext external fr friction Fe iron xi Notations and Symbols Ft Foucault gen generator h main (voltage) h horizontal Hy hysteresis i inner in input ind indirect L layer LL line-to-line m mechanical M magnet max maximum meas measured mot motor N rated o outer out output p pole, back EMF P Potier (reactance) q quadrature axis Q slot r rotor R resistance s stator sc short-circuit sh sheet sim simulated syn synchronous T conductor U,V,W phase U, V, W xii Notations and Symbols v vertical w windage x reactance x,y,z Cartesian coordinates z axial direction of cylinder coordinates δ air gap ρ radial direction of cylinder coordinates ϕ tangential direction of cylinder coordinates σ leakage Abbreviations AC alternating current DC direct current DCM DC machine DFIG doubly-fed induction generator FFT Fast Fourier Transform PM permanent magnet PMSM permanent-magnet synchronous machine PWM pulse width modulation RMS root mean square VSI voltage source inverter cond. conductor el. electrical const. constant N. of number of p.p. percentage point var. variable vs. versus xiii 1. Introduction Due to the high demand on motor and generator efficiency, permanent magnet syn- chronous machines are used increasingly in different applications, e.g. big wind gener- ators up to 6 MW rated power, industrial drives with high efficiency class IE4 ([10], [16], [28], [29]), or motors for (hybrid) electric cars. If the machine efficiency is higher than 95 %, the effort for accurate efficiency determination is rather big, as a high measurement accuracy is required. So the experimental validation of the designated machine efficiency values is a big challenge for the machine manufacturers. The methods of efficiency determination can be classified into two main categories: the direct measurement and the indirect electrical and mechanical measurement. For the direct method a full-load input/output power measurement is required. The machine efficiency η is calculated via η = Pout Pin , (1.1) where Pout is the output power and Pin the input power. On the other hand, for indirect efficiency determination, the loss components are deter- mined individually and are summed up to the total losses Pd. With these losses the effi- ciency is calculated via η = Pout Pin = Pin −Pd Pin . (1.2) Besides these two methods there is also the possibility of using a calorimetric measure- ment according to IEC 60034-2-2 [26], but with even more technical effort (e.g. [1], [14], [34]). Therefore this method is not addressed in this work. For big electrical machines, an accurate direct efficiency determination is difficult for sev- eral reasons. The first problem is the need for a load test with rated power, which requires an adequate load machine with the same power rating. Furthermore, big generators for power plants (mainly electrically excited synchronous machines) with high output power (above several MW) are often finally assembled directly at the plant. So a measurement 1 Introduction is only possible there. As the rated efficiency of these machines is usually in the range of 95 % and higher, a high measurement accuracy is very important, as the following example [63] shows. Example 1 (Direct efficiency determination): The electrical machine is operated in motor operation at the three-phase sinu- soidal grid. The mechanical output power is measured by a speed and torque transducer to Pout,meas = 2π ·n ·M , (1.3) where n is the mechanical rotational speed and M is the shaft torque. The elec- trical input power is measured via Pin,meas = √ 3 ·ULL · Is · cosϕ , (1.4) where ULL is the stator line-to-line voltage, Is the stator current and cosϕ is the power factor. The true machine efficiency η is 95 %, and the measurement accuracy ε is 0.2 %. As a worst-case assumption, the output power is measured due to ε too high and the input power too low: ηmeas = Pout,meas Pin,meas = Pout · (1+ ε) Pin · (1− ε) = η · 1+ ε 1− ε = 0.95 · 1.002 0.998 = 0.9538 . The efficiency ηmeas/η is determined too high by 0.38 p.p. A comparison of the true losses Pd and the measured losses Pd,meas Pd = (︃ 1 η −1 )︃ ·Pout = 0.0526 ·Pout , Pd,meas = (︃ 1 ηmeas −1 )︃ ·Pout,meas = 0.0484 ·Pout,meas shows a deviation of Pd,meas/Pd = 0.92, i.e. the losses are measured too low by 8 %. If the same measurement accuracy is applied to the indirect measurement procedure, the error will be reduced significantly, as described in the next example [63]. 2 Introduction Example 2 (Indirect efficiency determination): The input power Pin,meas is measured due to ε too high and the losses Pd,meas are determined too low in worst-case. Then the efficiency is calculated to ηmeas = Pout,meas Pin,meas = Pin,meas −Pd,meas Pin,meas = 1− Pd,meas Pin,meas = 1− Pd Pin · 1− ε 1+ ε = 1− (1−η) · 1− ε 1+ ε = 1−0.05 · 0.998 1.002 = 0.9502 ∼= η = 0.95 The efficiency ηmeas/η is determined too high by 0.02 p.p. In this case the de- viation between the true and the measured efficiency is with 0.02 p.p. very low. The deviation between Pd,meas and Pd is 0.4 %. Therefore for electrically excited synchronous machines and induction machines there have been standardized indirect efficiency determination methods for many years [35]. If the machine’s rated power is higher than 1 MW, the indirect methods are mandatory [14], [25]. As the losses are determined individually, adequate testing procedures and machine models are necessary to calculate the correct efficiency. A brief explanation of these measures for electrically excited synchronous machines is given in Chapter 2. This method is not directly applicable for permanent-magnet synchronous machines. The only standardized option is currently a direct efficiency measurement. This thesis presents and evaluates a novel procedure to determine the machine efficiency indirectly without the need of a full-load measurement. Therefore three experiments are presented to calculate the efficiency for sinusoidal operation and inverter operation: The motor no-load test, the generator no-load test, and the removed rotor test. Also the short-circuit test and the reac- tive current test are explained to identify the synchronous inductance and the short-circuit current of the machine. With help of measurements of five different permanent-magnet synchronous test machines up to a rated power of 160 kW (S1 operation according to IEC 60034-1 [24]) in motor and generator operation the indirect determination procedure is evaluated and compared to analytical and numerical simulation results. The basic principle of the proposed method was first introduced in [69] and extended in [63], [66]. It was continuously evaluated in cooperation with the national standardization committee DKE/K 311, which is responsible for the standardization process of rotating electrical machines. Since then several publications of the Physikalisch-Technische Bun- desanstalt (PTB) Braunschweig (the German national metrology institute) also deal with 3 Introduction the evaluation and application of the proposed method for motors with a rated power of 7.5 kW [40], [61], [62]. There a good accordance between the indirect method compared with direct measurements is reached. The results are briefly given in Section 7.4. This thesis explains the proposed method in detail, while the equivalent circuit of the permanent-magnet synchronous machine is used to describe the relevant losses, which are of big impact for industrial drives operated in the base speed range, where no field weaken- ing is applied (Chapter 3 to Chapter 5). The five test machines and the measurement setups for the relevant experiments are described in Chapter 6 and Chapter 7, while Chapter 7 also gives the results of the comparison between measured indirect and direct efficiency. In Chapter 8 finite element simulations are performed on four of the test machines to compare the simulation values with the measurements and to proof the applicability of the proposed method and of the calculation assumptions. Chapter 4 also describes, how together with further experiments the parameters of the ma- chines are analyzed with respect to no-load voltage, short-circuit current, and synchronous inductance. 4 2. Efficiency determination of electrically excited synchronous machines For electrically excited synchronous machines several testing methods for efficiency de- termination are established and approved by IEC standard 60034-2-1 [25]. This chapter briefly explains the different testing procedures with the focus on the two indirect methods B and C, that are mandatory for bigger synchronous machines. The three preferred methods that comply to the national energy efficiency regulations are: Method A – Direct measurement: Input-Output 1, Method B – Summation of losses with load test , Method C – Summation of losses without load test . In addition, there are four more methods available, that are not compliant to national energy efficiency regulations, but may be used for field testing: Method D – Dual-supply-back-to-back , Method E – Single-supply-back-to-back , Method F – Zero power factor with excitation current determined from Potier, ASA, or Sweden diagram [30] , Method G – Summation of losses with load test except the additional load losses . These methods are only mentioned for the sake of completeness and have rather small impact on this work. It is assumed, that the primary winding is located at the stator. For machine topologies with a rotating primary winding, the terms stator and rotor have to be exchanged. 1At the present situation the only applicable method also for permanent-magnet synchronous machines. 5 2.1. Method A – Direct measurement: Input-Output 2.1. Method A – Direct measurement: Input-Output For a machine frame size below or equal 180 mm the direct measurement is applied, as the expected efficiency values are low enough to be determined accurately. The full load test requires a second machine of the same power rating and a torque transducer to determine the mechanical power Pm = 2π · n ·M. The electrical power Pel = √ 3 ·ULL · Is · cosϕs is measured at the same time. The direct efficiency in motor operation ηmot and in generator operation ηgen is calculated via ηmot = Pm,out Pel,in +Pext , (2.1) ηgen = Pel,out Pm,in +Pext , (2.2) where Pext are the external losses of the excitation system. The machine shall be in thermal equilibrium, i.e. the stator winding temperature does not change by more than 2 K in one hour. This procedure is at the present situation the only method, that may also be applied for permanent-magnet synchronous machines. Therefore in this work, the direct measurement is conducted to compare the results to the proposed indirect efficiency determination. The procedure is slightly different, as no excitation losses occur and the permanent-magnet synchronous machine is usually operated at inverter operation. The exact procedure is explained in Section 4.4. 2.2. Method B – Summation of losses with load test For machines with a frame size above 180 mm and a rated power up to 1 MW this indirect efficiency determination procedure is applied. Again a drive equipment for full load is required to determine the excitation losses at rated operation. The total losses of the synchronous machine can be divided into three major groups: • No-load losses: Iron losses PFe and friction and windage losses Pfr+w 6 2.2. Method B – Summation of losses with load test • Load-dependent losses: Stator copper losses PCu= and additional load losses Pad • Excitation losses Pexc 2.2.1. No-load losses To determine the no-load losses, a no-load test is performed. The machine is driven at constant rated speed either by an auxiliary drive at open terminals (generator no-load) or uncoupled (motor no-load). By variation of the excitation current If, the characteristics Us0(If), Is0(If) in case of motor no-load, and P0(If) are obtained. The no-load power P0 is determined as the mechanical input power in case of generator no-load. A minimum number of seven different voltage values between 30% . . .110% of the rated stator voltage UsN is required. With knowledge of the stator winding resistance Rs per phase (ms phases) at the correct stator winding temperature ϑCu the iron losses and friction and windage losses PFe+fw are calculated together as PFe+fw = P0 −ms ·Rs · I2 s0 −Pexc0 . (2.3) The excitation losses Pexc0 depend on the excitation system and are discussed below. By plotting the iron losses and friction and windage losses over the square of the stator voltage, the extrapolated value of the curve PFe+fw(U2 s0) for U2 s0 = 0 shows the constant amount of friction and windage losses Pfr+w. The variable part of the curve is associated with the iron losses PFe, which are therefore known at rated stator voltage (Figure 2.1). 2.2.2. Load-dependent losses At first a full load test of the machine is performed at rated excitation, and rated stator volt- age and current. The measurement shall start after the machine is in thermal equilibrium (machine over-temperatures ∆ϑ ≤ 2K in one hour). After the electrical measurement the stator winding resistance Rs= at DC operation and thus the stator winding temperature ϑCu is determined, when compared to Rs= at cold machine. The stator copper losses are calculated via PCu= = ms ·Rs= · I2 sN . (2.4) 7 2.2. Method B – Summation of losses with load test 0 0 Squared no-load voltage N o- lo ad ir on an d fr ic tio n/ w in da ge lo ss es Pfr+w U2 sN PFe Figure 2.1.: Exemplary curve of the no-load loss separation of an electrically excited syn- chronous machine Afterwards the steady-state short-circuit test is carried out to determine the additional load losses. Therefore, like at the no-load test, the machine is driven by the auxiliary drive for different excitation states, but now with all terminals short-circuited. The mechanical input power Pm,sc = 2π · n ·Msc is measured by a torque transducer, while the excitation current is adjusted to let the rated stator current IsN flow as short-circuit current in the stator winding. The rated additional load losses Pad,N are calculated as the difference between the mea- sured mechanical power and the stator copper losses PCu=, friction and windage losses Pfr+w and excitation losses Pexc: Pad,N = Pm,sc −PCu=−Pfr+w −Pexc . (2.5) As the additional load losses (e.g. due to current displacement) are depending on the square of the current, the loss value for different operation points is calculated via Pad = Pad,N · (︃ Is IsN )︃2 . (2.6) 2.2.3. Excitation losses The excitation losses of synchronous machines depend on the excitation system, that is used to supply the required DC excitation. Several different options are available, like 8 2.2. Method B – Summation of losses with load test shaft driven or external exciters with slip-rings or brushless excitation with a rotating diode bridge. Therefore the excitation losses Pexc may include: • Losses in the field winding Pf • Losses in the brushes Pbr of the slip-ring system • External losses in the excitation system Pext The required DC field excitation current at rated conditions is determined during the full load test at thermal equilibrium with Method B. Extended calculation examples are given in IEC 60034-2-1 [25], but are not addressed here. 2.2.4. Efficiency calculation Finally the total losses Pd are summed up to Pd = PFe +Pfr+w +PCu=+Pad +Pexc . (2.7) The indirect efficiency is then determined according to (1.2) 2.2.5. Distinction to permanent-magnet synchronous machines The described procedures of no-load and short-circuit measurement both require a variable excitation to adjust the magnetic flux to a) rated flux in no-load operation and b) minimum flux (mainly stray flux) in short-circuit operation. This is not possible due to the constant permanent-magnet excitation. On the other hand, no excitation losses occur in permanent- magnet synchronous machines. Therefore, in order to perform an indirect efficiency determination also for permanent- magnet synchronous machines, adequate alternative testing procedures have to be found to measure and separate the load-independent and load-dependent losses. 2.2.6. Determination of the synchronous inductance According to IEC 60034-4 [30], the synchronous inductance Ld can be calculated using the no-load and short-circuit characteristic. Therefore the short-circuit ratio ksc is deter- 9 2.3. Method C – Summation of losses without load test mined according to Figure 2.2 via ksc = Is,sc(If0) IsN = If0 If,sc . (2.8) From this, the synchronous inductance Ld results as Ld = 1 ωs ZN ksc = 1 ωs UsN Is,sc(If0) , (2.9) where ZN =UsN/IsN is the nominal impedance of the synchronous machine. 0 0 UsN, IsN U0, Is,sc If0 If,sc If Is,sc(If) U0(If) ≡ ksc ≡ 100% Figure 2.2.: No-load and short-circuit characteristic of an electrically excited synchronous machine 2.3. Method C – Summation of losses without load test For even bigger machines with a rated power above 1 MW the previous method is not longer applicable due to the too complicated full load test. Instead of determining the rated excitation current from measurement at rated conditions, now IfN is calculated using the no-load and short-circuit characteristic. Additionally an over-excitation test at zero power factor has to be carried out: The machines is operated either as motor with zero active power or as generator with zero mechanical shaft power, while the excitation is increased further, until the rated stator current is reached. With this excitation current, the Potier reactance XP is determined according to IEC 60034-4 [30]. The Potier reactance is considered as an artificially increased leakage reactance to represent the increased voltage drop due to maximum saturation of the excitation poles. 10 2.4. Method D + E – Back-to-back test To determine the rated excitation current IfN, three geometrical construction methods ex- ist: Potier diagram, ASA diagram, and Sweden diagram. The details of the construction are not addressed here, but are described in IEC 60034-4 [30]. With knowledge of the rated excitation current, the excitation losses Pexc are calculated. For the other loss components, the procedure is equivalent to the previously described Method B. 2.4. Method D + E – Back-to-back test Both methods D and E require two identical machine units. One machine is operated as motor, the other one as generator at rated stator voltage and current. For the efficiency determination of the tested machines 50 % of the total measured losses are taken into account. At Method D each machine is fed separately, while at Method E both machines are coupled electrically and fed by one single source. 2.5. Method F – Zero power factor test This Method F approximates the load losses with help of the zero power factor test at motor operation and the calculated excitation losses like at Method C. To adjust the stator current to rated current a variable voltage source is required at the stator side. Again, the efficiency is determined by summation of the individual losses, while the input power is calculated from the rated values of stator voltage, current, and power factor. 2.6. Method G – Summation of losses except the additional load losses In this procedure the short-circuit test is omitted. Therefore, the additional load losses cannot be determined, and the resulting efficiency values are less accurate. The calculation itself is performed like explained above for Method B. 11 2.7. Inverter-fed machines 2.7. Inverter-fed machines The previously mentioned methods are only usable for machines operated at sinusoidal voltage sources. As nowadays variable-speed drives are increasingly utilized, the addi- tional losses at inverter-feeding and the efficiency in more operation points than only at rated speed and torque are addressed. Here IEC 60034-2-3 [27] describes interpolation procedures to determine the efficiency of AC drives for a wider operation range by mea- surement of seven fixed operation points P1 . . . P7 (Table 2.1), see also [15]. Table 2.1.: Operation points according to IEC 60034-2-3 [27] n/nN M/MN P/PN P1 0.9 1 0.9 P2 0.5 1 0.5 P3 0.25 1 0.25 P4 0.9 0.5 0.45 P5 0.5 0.5 0.25 P6 0.5 0.25 0.125 P7 0.25 0.25 0.0625 Again for these seven operation points, the direct measurement from input and output power is presently applied for permanent-magnet synchronous machines. 12 3. The permanent-magnet synchronous machine (PMSM) The permanent-magnet synchronous machine is widely used in industrial, energy or trac- tion applications, where a high efficiency is needed. Several stator and rotor designs are established, while two main types have to be mentioned: Surface-mounted magnets and buried magnets (Figure 3.1). With surface-mounted magnets usually the magnetic reluc- tance in d-axis and q-axis is equal, whereas for buried magnets it is possible to significantly increase this difference by the specific magnet arrangement and the use of flux barriers. d q d q d q a) b) c) Figure 3.1.: Different section examples of permanent-magnet synchronous machines: a) Surface-mounted rotor magnets, b) buried rotor magnets, c) buried rotor magnets with increased reluctance difference Lq/Ld [65] 3.1. Parameters and equivalent circuit The presented indirect method for efficiency determination is mainly based on the elec- trical equivalent circuit per phase of a permanent-magnet synchronous machine. There are already standardized procedures for induction machines, based on the equivalent cir- cuit. The idea of calculating the efficiency this way is therefore well-accepted. Figure 3.2 shows the equivalent circuit of a permanent-magnet synchronous machine without reluc- tance difference in d- and q-axis for constant speed n. The parameters are explained in Table 3.1. The inner air gap power Pδ is built up by the stator current Is the back EMF Up. According 13 3.1. Parameters and equivalent circuit Rs=+∆RsIs jXsσ= jωsLsσ jXdh= jωsLdh RFeU s Ux Uh Up Figure 3.2.: Equivalent circuit per phase of a permanent-magnet synchronous machine without reluctance difference in d- and q-axis (Xdh ∼= Xqh) to the electromagnetic energy conversion the electromagnetic torque Mem is created. Due to the rotor rotation several rotor losses occur: the mechanical friction and windage losses Pfr+w (Section 3.2), the eddy current and hysteresis losses in the rotor lamination due to sub- and super-harmonic field waves PFe,r (Section 3.3), and the eddy current losses in the rotor magnets PM (Section 3.4). If e.g. motor operation is assumed, the shaft torque will be: M = Mem − (︁ Pfr+w +PFe,r +PM )︁ /(2π ·n) . (3.1) In this case the rotor losses are not present in the equivalent circuit at first. They may be but are approximated by calculating the equivalent iron resistance RFe from the stator iron losses PFe,s (Section 3.3) and the rotor iron losses PFe,r +PM. Then the calculated shaft torque is M = Mem. The electromagnetic torque of the machine at q-current operation (Is = Isq, Isd = 0) (Figure 3.3) is calculated via [5] Mem = ms · p · Ψp√ 2 · Is , (3.2) where ms is the number of stator phases 2, p is the number of pole pairs and Ψp is the magnetic flux linkage amplitude due to the rotor permanent-magnet excitation. The AC resistance Rs∼ represents both the DC copper losses PCu= and the additional eddy 2In the most common case of three-phase machines ms = 3. 14 3.1. Parameters and equivalent circuit Table 3.1.: Electrical parameters per phase of a permanent-magnet synchronous machine according to Figure 3.2 Us Stator phase voltage Ux Reactance voltage Uh Induced voltage in the stator winding due to the resulting air gap field as sum of stator and rotor fundamental field wave (main voltage) Up Induced voltage in the stator winding due to the magne- tized rotor (back EMF) Is Stator phase current Rs∼ = Rs=+∆Rs AC winding resistance per phase (as sum of the DC resis- tance Rs= and additional losses due to current displace- ment ∆Rs) RFe Equivalent iron resistance to take the stator iron losses into account Xsσ = 2π fs ·Lsσ Stator leakage reactance ( fs: Stator frequency, Lsσ : Sta- tor leakage inductance) Xdh = 2π fs ·Ldh Main reactance of d-axis (Ldh: Main inductance of d- axis) current losses in the stator winding ∆PCu due to the induction effect of the AC stray slot field ( f = fs, Section 3.2). For the calculation of the equivalent iron resistance RFe from measurements it is assumed that the main flux as well as the stator stray flux contribute to the stator iron losses. In the stator yoke this assumption is more or less correct, in the teeth it is less accurate, as the main flux passes the teeth in normal direction whereas the stray flux is crossing perpendicularly with linear increasing intensity along the tooth axis towards the bottom of the slots. For the small stray flux at the winding overhang the flux path is even more complex. Nevertheless also the stray flux has an impact on the stator iron losses due to the induced eddy currents. Therefore the equivalent iron resistance RFe is arranged like in Figure 3.2 and not between the main reactance Xdh and the leakage reactance Xsσ [41]. The equivalent circuit is valid for sinusoidal voltages and currents and Ld = Lq. If the machines is fed by a voltage source inverter, the stator voltage Us is replaced by Us,k (k-th 15 3.1. Parameters and equivalent circuit d-axis q-axis Up jXdhIsjXsσ IsRs · Is Uh U s Ux Is=Isq ϕ Ψ p · Figure 3.3.: Phasor diagram per phase of a permanent-magnet synchronous machine at load with q-current operation (Xqh = Xdh) voltage harmonic of the inverter output voltage, frequency fs,k = k · fs), while Up is zero (∀k ̸= 1). This leads to harmonic stator currents Is,k and additional losses due to inverter feeding (Section 3.6). For machines with reluctance difference in d-axis and q-axis (Figure 3.4) the torque cal- culation in (3.2) extends to Mem = ms · p · (︃ Ψp√ 2 · Isq +(Ld −Lq) · Isd · Isq )︃ , (3.3) where (Ld −Lq) · Isd · Isq is the amount of reluctance torque. In case that the inductance Lq = Lqh + Lsσ in q-axis is smaller than Ld in d-axis, a negative d-current Id has to be set for positive torque Mem and the current angle β ∗ (Figure 3.4) is positive between 0° . . .90°el. in motor operation and between 180° . . .90°el. in generator operation. The phasor diagram (Figure 3.4) shows, that this operation is already a kind of field weakening due to Isd < 0, although the machine is operated in the base speed range. The following subsections give a brief overview of the loss components, that are relevant for this work. A deeper view into the nature of losses and their calculation is given e.g. in [5], [43], [48]. 16 3.2. Losses in the stator winding d-axis q-axis · U sUx Up jXq · Isq jXd · IsdRs · Is Isq Isd Is ϕ β ∗ Ψ p Figure 3.4.: Phasor diagram per phase of a permanent-magnet synchronous machine at load with positive current angle β ∗ = 10°el. 3.2. Losses in the stator winding For slow turning permanent-magnet synchronous machineswith a high number of poles and a high rated torque usually the copper losses 3 in the stator winding are the dominant loss component. For AC machines these losses have to be divided into two major com- ponents: The DC copper losses PCu=, which are independent from the frequency of the stator current and the additional stator losses ∆PCu, which strongly depend on the stator frequency fs. Both components together are expressed as PCu∼, where PCu∼ = PCu=+∆PCu . (3.4) In general the stator copper losses depend on the stator winding resistance. With the number of stator phases ms the losses are calculated as PCu∼ = ms ·Rs∼ · I2 s = ms · (Rs=+∆Rs) · I2 s , (3.5) where, like in (3.4), also the stator winding resistance is split into a DC component Rs= and an amount of increased resistance ∆Rs due to the additional stator copper losses. Both 3The term copper losses and the subscript Cu is used for each kind of conductor material. 17 3.2. Losses in the stator winding components of the stator resistance also depend on the stator winding temperature θ in different manners, which will be described in the following sections. Usually a reference temperature of 20 °C is assumed and therefore defined by the subscript cold, while the (variable) winding temperature under load is much higher and is defined as warm. 3.2.1. Frequency-independent losses The DC component of the cold stator resistance depends on the material conductivity and the conductor geometry and is calculated via Rs=,cold = Ns · lCu κ20°C ·ACu ·ao ·ai , (3.6) with the number of turns per phase Ns, the total conductor length per turn (stack length plus winding overhang) lCu, the number of parallel branches per phase ao and parallel strands per turn ai, the electric conductivity κ20°C at 20°C 4, and the conductor cross section per strand ACu. In the typical temperature range of electrical machines, the electric conductivity changes linearly with the conductor temperature ϑCu. Therefore the warm resistance is calculated via Rs=,warm = [1+αCu · (ϑCu −20°C)] ·Rs=,cold = kϑ ·Rs=,cold , (3.7) where αCu is the linear temperature coefficient 5, and kϑ is defined here as the linear DC resistance increase factor. The linear resistance increase of copper windings is shown in Figure 3.5. 3.2.2. Frequency-dependent losses At AC feeding additional losses due to current displacement occur in the stator winding. This effect is considered by the corresponding increase of stator resistance and is described as follows. 4For copper windings a conductivity of κ20°C = 56MS/m is typical. 5αCu = 0.392%/K 18 3.2. Losses in the stator winding 0 20 40 60 80 100 120 140 160 180 0.8 1 1.2 1.4 1.6 1.8 Stator winding temperature ϑCu / °C R s= ,w ar m /R s= ,c ol d Figure 3.5.: Linear temperature dependency of the electrical resistance of copper Second order current displacement The slot leakage flux crosses the slot perpendicularly. If an AC current with frequency fs is flowing in the stator winding, circulating eddy currents in axial direction are induced in the conductors. According to Lenz’s rule, these currents must flow in the specific direction to create an additional magnetic field, which partially compensates the original inducing leakage field [5]. As a result, the eddy currents add up with the original impressed stator current in the upper part of the conductor, while they cancel out in the lower conductor part. Therefore the resulting current flows in a smaller area at the top side of the conductor (skin effect). This effect is called second order current displacement. Also an interference between the magnetic field of adjacent conductors occurs leads to additional current dis- placement in the conductor area (proximity effect). As the conducting area reduces, the conductor resistance is increased. The current density decreases exponentially along the conductor cross-section in slot axis direction, while the penetration depth dE ∝ √︄ 1 π f µκ (3.8) depends on the current frequency f , the conductor material permeability µ , and the mate- rial conductivity κ . The permeability is usually near to µ0 as well for copper (diamagnetic material) as aluminum (paramagnetic material). The electric conductivity strongly de- 19 3.2. Losses in the stator winding pends on the conductor temperature ϑCu, which leads to a decreasing current displacement for rising temperature. For calculation, the concepts of Field and Emde [19], [20] are applied, which are used e.g. in [3], [36], [41]. At first, an artificial slot geometry is considered, where the conductors are equally dis- tributed (Figure 3.6b). Conductors with round cross section are transformed to equivalent rectangular conductors with the width bT and the height hT. The number of vertical con- ductor layers is called mT, while the number of conductors side by side is aT. bQ aT = 6 mL = 8 hL = 2 ·hT ai aT = 6, ai = 1 mT = 16 hT bT a) b) Figure 3.6.: Simplified exemplary slot model for the determination of current displace- ment: a) first and second order current displacement effect: 8 turns per slot with 12 parallel strands per turn, b) only second order current displacement effect: 16 turns per slot with one strand per turn 20 3.2. Losses in the stator winding With the reduced conductor height ξ for this geometry ξ = hT dE = hT · √︄ π · f ·µ0 ·κCu(ϑ) · aT ·bT bQ (3.9) the second order current displacement factors are determined: ϕ(ξ ) = ξ · sinh(2ξ )+ sin(2ξ ) cosh(2ξ )− cos(2ξ ) , (3.10) ψ(ξ ) = 2ξ · sinh(2ξ )− sin(2ξ ) cosh(2ξ )+ cos(2ξ ) , (3.11) where ϕ(ξ ) describes the influence of the skin effect and ψ(ξ ) the influence of the prox- imity effect between adjacent conductors. In (3.9) aT ·bT is the total width of conductors, which is evaluated in relation to the slot width bQ. With (3.10) and (3.11) the resistance increase factor kR2 = ϕ(ξ )+ m2 T −1 3 ·ψ(ξ ) (3.12) is determined. The current displacement is assumed to occur only in the axial machine section with the iron length lFe due to the AC slot stray flux, whereas the flux density in the winding overhang is much smaller. Therefore (nearly) no resistance increase occurs in the winding overhang (length lb), and the increase factor kR2 reduces to kR2 = kR2 · lFe + lb lFe + lb . (3.13) First order current displacement In order to reduce second order current displacement, the conductor height has to be re- duced. Therefore ai parallel sub-conductors per turn are introduced (Figure 3.6a). The actual arrangement especially of round conductors is usually not exactly known, so that assumptions about the location of the conductors in the slot are necessary for calculation. If no further measures are applied, the AC slot leakage flux again induces eddy currents in the parallel connection of the ai sub-conductors, which leads to equalizing currents between the parallel sub-conductors. So the resistance is also increased, as if the bundle 21 3.2. Losses in the stator winding was one "big" conductor with the accumulated cross section ACu = ai · bT · hT. But the equalizing currents have to flow along the total length per sub-conductor to the electric parallel connection point, so these losses are smaller than in one "big" conductor. This increase of losses is called first order current displacement. In order to effectively reduce these circulating equalizing currents, the sub-conductors, also called conductor strands, have to be transposed along the machine’s axial length in one slot or between different slots. If all possible permutations of transpositions were applied, the resulting induced eddy currents as equalizing currents would cancel out to zero. For big machines this is done by using Roebel bars. However, for round wires usually a random unpredictable transposition is present, which in most cases is rather ineffective. Therefore in the following formulas do not consider any transpositions. These calculations are only valid, if the sub-conductor cross-section is small enough, so that no excessive second order current displacement occurs. Again the reduced conductor height is calculated, but now the height of the total conductor layer hL per turn is taken into account, and the length between two connection points of the ai parallel sub-conductors is – in worst case – only one turn length 2 · (lFe + lb): ξ = hL dE = hL · √︄ π · f ·µ0 ·κCu(ϑ) · lFe lFe + lb · aT ·bT bQ . (3.14) With the current displacement factors (3.10) and (3.11), applied to the circulating equaliz- ing current effect, in combination with (3.14) the first order current displacement increase factor of the resistance is determined: kR1 = ϕ(ξ )+ m2 L −1 4 ·ψ(ξ ) . (3.15) AC resistance increase Both first order and second order resistance increase factors (3.13, 3.15) are combined to the resulting AC resistance increase factor kR. Thus the AC resistance Rs∼ is determined via Rs∼(ϑ) = kR(ϑ) ·Rs=(ϑ) = (︁ kR1(ϑ)+ kR2(ϑ)−1 )︁ ·Rs=(ϑ) . (3.16) 22 3.2. Losses in the stator winding The factor kR(ϑ) decreases with increasing conductor temperature due to the decreasing κCu(ϑ). 3.2.3. Total losses and temperature dependency By combining the DC and AC resistance factors kϑ and kR(ϑ) to K = kϑ · kR(ϑ) = kϑ · (︁ kR1(ϑ)+ kR2(ϑ)−1 )︁ , (3.17) the AC resistance Rs=,warm at a given conductor temperature ϑCu is determined. The total losses PCu∼ are hence PCu∼ = ms · kR ·Rs=,warm · I2 s = ms ·K ·Rs=,cold · I2 s . (3.18) As the conductivity κCu decreases with rising temperature ϑ , the DC component of the stator resistance is increased (Figure 3.5), while the AC component is decreased according to increased penetration depth (3.8). For rather small kR(ϑ = 20°C) < 2, the increasing trend of the DC component is dominant (Figure 3.7) – exemplary shown for test machine M2 (see Chapter 6) at rated stator frequency fsN = 133.3Hz. 0 20 40 60 80 100 120 140 160 180 0.8 1 1.2 1.4 1.6 1.8 2 K = kϑ · kR kR kϑ Stator winding temperature ϑCu / °C R es is ta nc e in cr ea se fa ct or Figure 3.7.: Calculated AC resistance increase. Example: Test machine M2 at rated stator frequency fsN = 133.3Hz For higher kR(ϑ = 20°C) > 2, a loss minimum occurs at temperatures ϑCu > 20°C (Figure 3.8) – exemplary shown for test machine M2 at increased stator frequency fs = 2.5 · fsN = 333.3Hz. 23 3.3. Losses in the motor lamination 0 20 40 60 80 100 120 140 160 180 0.5 1 1.5 2 2.5 3 3.5 K = kϑ · kR kR kϑ ϑCu,opt Stator winding temperature ϑCu / °C R es is ta nc e in cr ea se fa ct or Figure 3.8.: Calculated AC resistance increase. Example: Test machine M2 at the stator frequency fs = 2.5 · fsN = 333.3Hz. Minimum of AC stator resistance at ϑCu,opt = 125°C 3.3. Losses in the motor lamination Due to the operation principle of electromagnetic machines, high-permeable ferromag- netic iron is necessary to guide the magnetic flux. If the varying electromagnetic fields of the electrical machine enter the iron, losses due to induced eddy currents and changing magnetization with corresponding hysteresis losses occur. In order to reduce the eddy currents, the magnetically active iron parts are made of electrically insulated laminated stacks. Classical empirical calculation approaches like Steinmetz [55] or Bertotti [4] usually di- vide the total iron losses PFe into up to three loss groups: hysteresis losses PFe,Hy ∝ f ·B2, classical eddy current losses PFe,ec ∝ f 2 ·B2, and excess losses PFe,ex ∝ f 3/2 ·B3/2, which are identified with anomalous eddy current losses. For each component loss coefficients are determined by measurement e.g. with a standardized Epstein frame or single sheet tester. Several extended semi-analytic or analytic loss models, like [21], [22], [33], [54], exist, that allow a loss calculation especially for high-speed machines, where due to the high frequencies the iron losses are the main loss component. The manufacturers of steel sheets for electrical machines often provide iron loss coeffi- cients at defined testing parameters, but do not give the parameters of each of the three 24 3.3. Losses in the motor lamination above noted individual loss groups. For non-grain-oriented steel sheets, which are mainly used in electrical machines, the excess losses are usually less then 10 % of PFe. Hence, only hysteresis and classical eddy current losses are used as approximation of the total iron losses as PFe = kV · (︁ PFe,Hy +PFe,Ft )︁ . (3.19) Each loss group depends with a limit of B < 2.3T roughly on the square of the magnetic flux density B. Here the Foucault losses PFe,Ft ∝ f 2 ·B2 are the approximation of the total eddy current losses. The eddy current reaction field is for f < 1kHz usually neglected, giving the dependency ∝ f 2. By using finite element software, a separation into these two loss groups (3.19) is possible, if no detailed separate loss coefficients are available. The processes of punching or laser cutting, stacking, and welding lead to increased hys- teresis losses, to decreased magnetic permeability and increased eddy current losses due to an increased effective electrical conductivity of the iron stack. This results e.g. from crystal changes, low-resistive bridges between sheets at the cutting edges and welding spots, or local surface damages [37], [38], [47], [59], [60]. The loss increase is considered by the post-processing factor kV in (3.19) and may differ for the individual sections of the motor. In this work the factor will be determined from the difference of measured and simulated no-load iron losses as one separate parameter for each section. At the permanent-magnet synchronous machine the air gap flux density Bδ correlates via the magnetic flux density B ∝ Bδ in the iron with the reactance voltage Ux. Therefore a square dependency PFe ∝ U2 x is assumed in the following chapters. For the stator and the rotor of a permanent-magnet synchronous machine different travel- ing and pulsating field waves are responsible for the iron losses in the steel sheets. The stator lamination is mainly induced by the fundamental rotating traveling field wave, through the combination of the stator field and the first order rotor field µ = 1 at syn- chronous speed and to a small amount by higher harmonics µ > 1. The rotor is rotating synchronously with the stator field. Therefore the fundamental stator field wave ν = 1 cannot induce the rotor. But due to harmonic effects other field waves 25 3.4. Losses in the permanent magnets ν ̸= 1 may enter the rotor and produce losses like (3.19). At generator no-load operation, when the stator field is zero, the rotor can only be induced by the modulation effect of the rotor field due to the stator slot openings. The frequency fQ of the inducing field wave due to this effect with the order µQ in the rotor is determined by fQ = Qs ·n = Qs · fs p = Qs p · fs , (3.20) with the number of stator slots Qs, the mechanical rotor speed n, the number of pole pairs p, and the fundamental electrical stator frequency fs of stator voltage (and current). The rotor frequencies fQ are also caused by the slot harmonic field waves νQ of the stator at load. So, if the stator current system Is excites the stator field, further harmonics ν ̸= 1 and especially the slot harmonics νQ can induce the rotor. The rotor frequencies fr depend on the harmonics ν of the stator field wave by: fr = |1−ν | · fs . (3.21) For integer-slot windings and the double-layer tooth-coil winding with q = 1/2, which all have no sub-harmonic field waves, the lowest value is ν = 1. For all other fractional slot windings there exist long-wave-length sub-harmonic fields |ν | < 1, e.g. for q = 1/4 one sub-harmonic with ν =−1/2. 3.4. Losses in the permanent magnets Like the rotor iron, also rotor magnets are induced by harmonic air gap field components, e.g. due to the pulsating field changes at the stator slot openings. This leads to eddy current losses PM in the magnet volume. Several publications (e.g. [6], [8], [13], [23], [42], [44], [46], [49]) deal with analytical calculation approaches to determine these mag- net losses. Some of these method are based on planar representations of the cylindrical field problem, where a stator current sheet with variable amplitude and frequency is ap- plied, or on extended equivalent circuit models for higher harmonics. On the other hand three-dimensional, or less accurate two-dimensional, numerical simulations are possible to determine the magnet losses. 26 3.5. Friction and windage losses The eddy currents can effectively be reduced by using magnet segments instead of massive magnets per pole ([18], [53]). This is especially true for surface-mounted rotor magnets. For buried magnets, the surrounding rotor iron guides the magnetic flux around the mag- nets due to the much higher magnetic iron permeability (µFe ≫ µM), so that the magnet losses are small, if the rotor iron is not saturated too much. But then the iron losses in the rotor iron surface are increased. The machines, that are investigated in this work, do all have segmented rotor magnets (Chapter 6). Also the rotational speed n is rather low. Therefore the magnet losses are considered to be rather small compared to the total machine losses and are not calculated analytically here. At the numerical simulations (Chapter 8) they are of course considered. 3.5. Friction and windage losses In each rotating machine friction and windage losses occur due to a) the dominating air movement inside the machine as air friction and fan losses (in case of shaft-mounted fans) and b) the much smaller inner friction in the mechanical bearings. These losses strongly depend on the machine’s mechanical speed n and therefore have a big impact on high- speed machines. The friction and windage losses are mainly load-independent (apart from a load-dependent component of the bearing friction losses) and are already fully present at no-load operation. In order to calculate the air friction and fan losses in detail, complex non-linear fluid dynamics simulations are required, but there exist also empirical formulas to approximate the losses. In this work, due to the rather low speeds (≤ 3000min−1), small shaft diameter, and grease-lubricated high-quality low-friction ball bearings, the bearing friction losses are small and therefore neglected. For a cylindrical rotor the surface is usually very smooth and the air friction losses at low speed are of small magnitude. The losses are roughly estimated [11], [41] via: Pfr+w,air = cf ·π ·ρair · (2π ·n)3 · r4 ro · lFe (3.22) with use of cf = 0.035 ·Re−0.15 (3.23) 27 3.6. Additional losses due to inverter supply and Re = 2π ·n · rro · δ νair , (3.24) where ρair is the temperature-dependent air mass density, rro is the rotor outer radius, lFe the active iron length, δ is the mechanical air gap, νair is the kinematic viscosity of air, and Re is the dimensionless Reynolds number. These formulas hold true for turbulent air flow Re > Recrit, with Recrit ∼= 1000. If a shaft-mounted fan is used to cool the machine, the fan power has to be provided internally by the electrical motor power. The fan power leads to increased friction and windage losses. Like before, an empirical formula is used to estimate the fan losses at turbulent air flow [50]: Pfr+w,fan = 20 · rro · (lFe +0.15) · (2π · rro ·n)2 . (3.25) In (3.25), Pfr+w,fan is in [W], rro is in [m], lFe is in [m], and n is in [1/s]. The shaft-mounted fan losses – if present – usually exceed the air friction losses signifi- cantly. 3.6. Additional losses due to inverter supply At voltage source inverter operation the feeding voltage is not longer sinusoidal. Depend- ing on the modulation procedure higher harmonics with ordinal number k and RMS values Us,k of the stator voltage occur, while Us,1 is the fundamental voltage. The voltage har- monics lead to stator current harmonics Is,k with the frequency fs,k = k · fs = ωs,k/(2π). The equivalent circuit (Figure 3.9) differs from Figure 3.2 as the back EMF Up is con- sidered zero for all voltage harmonics k > 1. Considering a three-phase stator winding system, harmonics of ordinal number k, that generate fundamental field waves ν = 1, that rotate in the same direction as ν = 1 for k = 1, are of positive order k > 0. Those, that rotate into opposite direction, can be denoted by negative k. Hence, we get |k| > 1 and k > 0 or k < 0. 28 3.6. Additional losses due to inverter supply Rs=+∆Rs,kIs,k jωs,k ·Lsσ jωs,k ·Ldh RFe,kU s,k Figure 3.9.: Equivalent circuit per phase of a permanent-magnet synchronous ma- chine at inverter feeding for k > 1 and Ldh ∼= Lqh, Lsσd ∼= Lsσq = Lsσ . RFe,k ≫ ωs,k(Lsσ +Ldh) = ωs,kLd and ωs,k = |k|ωs The harmonic current Is,k is approximately calculated [5] via Is,k ∼= Us,k√︂ R2 s∼,k +(kωs)2 ·L2 d ∼= Us,k |k|ωs ·Ld , (3.26) with Rs,k = Rs=+∆Rs,k. As, in contrast to induction machines, at synchronous machines the total synchronous in- ductance is limiting the harmonic components and not only the small stray inductance Lsσ , the values Is,k are small, and the stator current is nearly sinusoidal even for moderate switching frequencies fPWM of the feeding voltage source inverter. The difference be- tween the RMS value of the total current Is and the RMS value of the fundamental current Is,1 should therefore be small with respect to the rated current IsN. Nevertheless the harmonic currents Is,k (RMS values) lead to a) additional losses in the stator winding with rather high frequencies, described by ∆Rs,k · I2 s,k, b) additional fast rotating fundamental air gap field waves (ν = 1) with positive or negative speed nsyn,k = k · fs/p. These field harmonics induce the stator and rotor and cause additional iron losses and magnet losses, described by RFe,k [7], [46], [58]. According to Figure 3.9 the values Is,k, |k| ≠ 1, are independent of the machine power proportional to Up · Is,1 · cosϕ i with cosϕ i = ∠(Up, Is,1). Hence, the sum of additional losses is nearly load-independent and depends only on the amount of voltage harmonics 29 3.6. Additional losses due to inverter supply and thus on the modulation degree ma of the voltage source inverter: ma = ÛLL,1 ( √ 3/2) ·UDC , 0 ≤ ma ≤ 1 , (3.27) where ÛLL,1 is the amplitude of the fundamental line-to-line inverter output voltage and UDC is the DC link voltage of the inverter. Figure 3.10 [41] shows an exemplary calculated Fourier spectrum of the most dominant voltage harmonics for synchronous pulse width modulation with a ratio of switching frequency vs. fundamental frequency fPWM/ fs = 15. The maximum harmonic content and therefore the maximum additional losses are ex- pected at a modulation degree between 0.5 . . . 0.75. Figure 3.10.: Calculated fundamental and harmonic voltage amplitudes of the inverter out- put voltage at PWM with fPWM/ fs = 15 in dependence of the modulation degree ma [41] 30 4. Experiments This chapter describes the experiments for the indirect efficiency determination: The gen- erator and motor no-load tests, and the removed rotor test. Furthermore the direct effi- ciency determination method is explained for comparison. In addition, the short-circuit test and the reactive current test may be used for parameter identification of the equivalent circuit. Each experiment requires specific testing equipment, which is listed at the beginning of each section. The measurement setup for the electrical stator parameters voltage, current, and power factor is considered obligatory for each experiment. 4.1. Generator no-load test REQUIREMENTS: Auxiliary drive, torque transducer VARIABLE: Mechanical stator speed n MEASUREMENT: Stator voltage Us, shaft torque M Like for electrically excited synchronous machines, at the generator no-load test the ma- chine is operated with open stator winding terminals and the rotor is driven by an auxiliary drive. But now there is no variable excitation but a variable speed operation. Both ma- chines are coupled by a torque-transducer to measure the no-load shaft torque M0. The measurement setup is visualized in Figure B.1a. The stator voltage at the open stator winding terminals represents the no-load load volt- age U0, which itself is nearly equal to the back EMF (Us = U0 ∼= Up, Figure 4.1) as the stator current Is is zero and the equivalent iron resistance RFe is much bigger then the synchronous reactance Xd = Xdh +Xsσ = ωs ·Ld. Due to the proportionality between the RMS value of the back EMF Up and the corresponding frequency fs and thus the mechan- ical speed n, the no-load voltage U0 rises linearly with n (Figure 4.2). The value of the voltage at a given speed can be used as an indicator for the correct magnet temperature, as the no-load voltage depends linear on the air gap flux density and therefore on the magnet 31 4.1. Generator no-load test jXsσ jXdh RFeU s=U0 Ux Uh Up Figure 4.1.: Equivalent circuit per phase at the generator no-load test (Xdh ∼= Xqh ≪ RFe) remanence, which depends on the magnet temperature (Chapter 3). Hence, the experiment should be carried out for the warm machine at temperatures similar to the steady-state load condition. 0 0 Speed N o- lo ad vo lta ge U0 nN n Figure 4.2.: Exemplary curve of generator no-load voltage over speed With help of the torque transducer the mechanical input power from the auxiliary drive is determined to: Pm,in,0 = 2π ·n ·M0 . (4.1) This is performed for different rotor speeds n. The mechanical input power covers the friction and windage losses Pfr+w as well as the no-load iron losses PFe,0 in stator and rotor and the eddy current magnet losses PM,0. As a result the no-load losses depend on the rotor speed n with Pm,in,0 ∝ nx,x = 2 . . .3, (Figure 4.3). 32 4.1. Generator no-load test 0 0 Speed N o- lo ad lo ss es P0 nN n Figure 4.3.: Exemplary curve of generator no-load losses over speed (subscript "m,in" suppressed for readability) For the further calculations the iron losses are required separately, but a clear separation of Pm,in,0 between the mechanical losses and the electromagnetic losses is only possible if the experiment is carried out additionally with a non-magnetized rotor. This might be done during the production process of the machine. If no non-magnetized rotor is available, the friction and windage losses have to be approximated analytically as shown in Section 3.5. As long as these losses are small compared to the total losses, the influence of the approximation on the efficiency value is small. The generator no-load experiments require a high precision torque measurement of typ- ically a) small torque range 0 ≤ M ≤ M0,max ≪ MN and b) small measurement error of below 0.5%. To overcome offset errors of the torque transducer, the test should be per- formed both in clockwise and counterclockwise rotation direction. The respective mea- sured losses are then to be averaged to the final no-load losses. This procedure is only possible, when no shaft-mounted fan is applied, which has a direction-dependent loss characteristic. Fans with simply radial fan blades allow this test procedure. 33 4.2. Motor no-load test 4.2. Motor no-load test REQUIREMENTS: Inverter-fed variable drive VARIABLE: Mechanical rotor speed n MEASUREMENT: Fundamental and harmonic stator voltage Us,1,Us,k, cur- rent Is,1, Is,k, and power factor cosϕs,1,cosϕs,k For machines with inverter feeding the variable speed under motor no-load is possible. Again the no-load losses are to be determined for different rotor speed values n. To achieve this, the machine is operated uncoupled, hence without a load machine, at the feeding voltage source inverter. The electrical input power of the machine Pel,in,0 is measured by a poly-phase power analyzer at the motor stator winding terminals with the ability to separate the fundamental losses Pel,in,0,1(k = 1) from the total no-load losses (sum over all k-values). At the measurement of the total losses no inverter output filter shall be applied to cover all relevant signal frequencies of order k [2]. By doing this, the additional losses due to inverter feeding Pel,in,0,ad are determined (Figure 4.4): Pel,in,0,ad = Pel,in,0 −Pel,in,0,1 . (4.2) The fundamental no-load losses Pel,in,0,1(k = 1) contain – like for the generator case – the no-load iron losses PFe,0 and friction and windage losses Pfr+w as well the I2R losses PCu,0 due to the fundamental no-load current Is,0,1. Therefore, if a separation of the fundamental and harmonic losses is not possible by the power analyzer, the additional losses due to inverter supply might be calculated via: Pel,in,0,ad = Pel,in,0 −ms ·Rs∼ · I2 s,0,1 −Pm,in,0 , (4.3) where PCu,0 = ms ·Rs∼ · I2 s,0,1 are the copper losses in the stator resistance at the actual stator winding temperature. A similar procedure is mentioned in IEC 60034-2-3 [27]. Here the additional losses shall be determined by subtracting the total losses, obtained by feeding the test motor by a true sinusoidal voltage source, such as a special synchronous generator. The measured fundamental voltage at the motor terminals is the no-load voltage Us,1 =U0, which is almost as big as the back EMF Up, because the stator current Is,1 = Isq to over- 34 4.3. Removed rotor test 0 0 Speed N o- lo ad lo ss es P0,1 P0 nN P0,ad n Figure 4.4.: Exemplary curve of fundamental and total motor no-load losses over speed (subscript "el,in" suppressed for readability) come the no-load losses and thus the voltage drop Xq · Isq is very small. The no-load voltage U0 depends linearly on the rotor speed n. This allows a definition of the additional losses due to inverter feeding over to the voltage (Figure 4.5). As explained in Section 3.6, the maximum of the additional losses is located between ma = 0.5 . . . 0.75, hence roughly between 50 % and 75 % of the maximum inverter output voltage at ma = 1, if no over- modulation is used. As the total stator inductance Ld or Lq is smoothing the stator current, the current waveform is nearly sinusoidal. The difference between the total and the funda- mental stator current Is and Is,1 is rather small for machines with a big stator inductance, e.g. for tooth-coil windings. This difference directly influences the amount of additional losses due to inverter feeding Pel,in,0,ad. 4.3. Removed rotor test REQUIREMENTS: Variable sinusoidal poly-phase AC source VARIABLE: Amplitude and frequency of stator current Is MEASUREMENT: Stator voltage Us and stator power factor cosϕs, stator winding temperature ϑ At the removed rotor test, the current-depending I2R losses in the stator winding are to be determined. These losses include the frequency-independent DC losses PCu= as well as the frequency-dependent AC losses ∆PCu (Section 3.2). Therefore an AC feeding of the 35 4.3. Removed rotor test 0 0 No-load voltage A dd iti on al no -l oa d lo ss es P0,ad UN U0 Figure 4.5.: Exemplary curve of the additional no-load losses due to voltage source in- verter feeding over the no-load voltage (subscript "el,in" suppressed for read- ability) stator winding is necessary. As the additional losses due to voltage source inverter feeding are already determined during the motor no-load test, the fed currents and voltages have to be sinusoidal. This can be achieved by a variable sinusoidal poly-phase AC source, e.g. a rotating converter (Section 7.1.2) or a well filtered static converter. Since the (average) temperature of the stator winding has a big impact on each kind of I2R loss (Section 3.2), the winding temperature has to be monitored or determined right after the measurement and should fit to the designated winding temperature at load operation. The removed rotor test is already included in IEC 60034-4 [30] for determination of the leakage inductance and has been used for measuring the stray-load losses in poly-phase inductions machines in [3]. When the rotor of a permanent-magnet synchronous machine is removed, the air gap in- creases to the whole stator inner diameter. This significantly reduces the main reactance Xdh to the bore field reactance XsB (Figure 4.6). Nevertheless the resulting magnetic field is not zero, i.e. also iron losses PFe,B in the motor lamination and eddy current losses in massive metal parts may occur. Therefore the end shields of the motor have to be re- mounted after the rotor removal to cover also all loss components there. Like before, the iron losses are represented by RFe in the equivalent circuit. With the measured electrical input power Pel,in,B during the removed rotor test at different 36 4.3. Removed rotor test Rs=+∆Rs( fs)Is jXsσ= jωsLsσ jXsB= jωsLsB RFe( fs)U s Ux,B Figure 4.6.: Equivalent circuit per phase at the removed rotor test at frequency fs (XsB ≪ Xdh: bore field reactance) stator frequencies fs the loss balance is: Pel,in,B = PCu=+∆PCu +PFe,B . (4.4) To calculate the current-depending losses, the iron losses PFe,B have to be subtracted from the electrical input power Pel,in,B: PCu=+∆PCu = Pel,in,B −PFe,B . (4.5) 0 0 Squared stator current I2 R lo ss es PCu∼ I2 sN fs I2 s Figure 4.7.: Exemplary curve of current-depending losses over squared stator current for different stator frequencies at removed rotor test Especially for machines with big leakage inductance (like tooth-coil windings with several sub- and super-harmonic field waves) the iron losses at the removed rotor test are not 37 4.4. Full load test negligible. The calculation method for estimating the iron losses is presented in Chapter 5. As the current-depending I2R losses PCu∼ = PCu=+∆PCu for different stator frequencies fs depend on the square of the stator current Is, the family of curves PCu∼ = f (︁ I2 s , fs )︁ is linear with I2 s (Figure 4.7) for constant stator winding temperatures ϑCu. 4.4. Full load test REQUIREMENTS: Inverter-fed variable drive, auxiliary drive as brake for up to 150 % of the rated power, torque transducer VARIABLE: Mechanical speed n, torque M MEASUREMENT: Fundamental and harmonic stator voltage Us,1,Us,k, cur- rent Is,1, Is,k, and power factor cosϕs,1,cosϕs,k To perform a direct efficiency determination a full load test is required. The tested machine is coupled with a braking or driving device, which is usually a second electrical machine with the same power and torque rating. As the mechanical output power in motor op- eration and the mechanical input power in generator operation is needed to calculate the direct efficiency, a torque transducer is, if possible, utilized for speed and torque sensing. Otherwise a separate speed measurement is needed, or the speed is calculated from the fundamental electrical stator frequency fs) . For big electrical machines, like generators for wind or water power plants, the demand of a second machine of the same size may lead to logistical problems, as the machine is often mounted only on-site and not in the manufacturer halls. So on-site there is no load machine available. Besides the mechanical measurement also the electrical parameters voltage, current, and power factor are needed to calculate the electrical in-/output power. Here usually a three-phase power analyzer is used. In this work, the full load test is performed to compare the calculated direct and indirect efficiency values for different torque and speed values in motor and generator operation. The full-load test covers all explained loss components of the permanent-magnet syn- chronous machine (Chapter 3), but requires a high measurement accuracy of less than 0.5 % of the electrical power Pel and the mechanical power Pm = 2π ·n ·M. For increased accuracy several measurements of different load M/MN are performed to allow an aver- 38 4.5. Generator short-circuit test aging of the measurement values. The direct efficiency in motor operation ηmot and in generator operation ηgen is calculated similar to (2.1) and (2.2) via ηmot = Pm,out Pel,in , (4.6) ηgen = Pel,out Pm,in . (4.7) With the fundamental values of the stator voltage Us,1, the stator current Is,1, and the power factor cosϕs,1 also the direct efficiency value for sinusoidal operation is calculated via ηmot,1 = Pm,out Pel,in,1 = Pm,out ms ·Us,1 · Is,1 · cosϕs,1 , (4.8) ηgen,1 = Pel,out,1 Pm,in = ms ·Us,1 · Is,1 · cosϕs,1 Pm,in , (4.9) where cosϕs,1 ∈ [−1 . . .0) in generator operation and cosϕs,1 ∈ (0 . . .1] in motor opera- tion. 4.5. Generator short-circuit test REQUIREMENTS: Auxiliary drive, torque transducer VARIABLE: Mechanical rotor speed n MEASUREMENT: Stator current Is, shaft torque M At the generator short-circuit test the machine is driven by an auxiliary drive like at the generator no-load test, but with all stator terminals short-circuited. Therefore the stator voltage Us is zero. The setup is similar to the generator short-circuit test of the electrically excited synchronous machine (Section 2.2.2), but here not the excitation current but the mechanical speed n is variable to change the current values of the machine. The back EMF Up is changing linearly with the speed, as shown at the no-load test. As the equivalent iron resistance RFe is usually much bigger than the stator resistance RFe ≫ Rs=+∆Rs = Rs∼, the sinusoidal stator short-circuit current Is,sc is mainly limited by the stator winding re- 39 4.5. Generator short-circuit test Rs=+∆Rs( fs)Is,sc jXsσ= jωsLsσ jXdh= jωsLdh RFe( fs)U s = 0 Ux Uh Up( fs) Figure 4.8.: Equivalent circuit per phase at the generator short-circuit test (Xdh ∼= Xqh) sistance Rs∼ and the synchronous reactance Xd (Figure 4.8). The stator current is thus calculated via Is,sc ∼=− Up Rs∼+ jXd , (4.10) Is,sc = |Is,sc|= Up√︂ R2 s∼+X2 d . (4.11) At low speed values the synchronous reactance Xd = ωsLd is rather small, leading to a fast linear rise of the stator short-circuit current with increasing speed n, while for higher speeds, and thus higher angular frequencies ωs = 2π fs the influence of the stator resistance descends. Then the stator short-circuit current is constant: Is,sc = Isc ∼= Up Xd = 1√ 2 ωs ·Ψp ωs ·Ld = 1√ 2 Ψp Ld . (4.12) Therefore, for ωsLd ≫ Rs∼( fs), the synchronous inductance can be calculated via Ld ∼= Up ωs · Isc . (4.13) With the decomposition of the stator short-circuit current Is,sc = Isd,sc + j Isq,sc according to [5] into Isd,sc = ωsLq ·Up R2 s∼+ωsLd ·ωsLq , Isq,sc = Rs∼ ·Up R2 s∼+ωsLd ·ωsLq , (4.14) a comparison of measured and calculated short-circuit current is possible. This allows the 40 4.5. Generator short-circuit test determination of the inductances via curve fitting. 0 0 Speed St at or cu rr en t Isc nN n Figure 4.9.: Exemplary curve of the short-circuit stator current over speed at the generator short-circuit test The short-circuit torque Msc = 3 ·U2 p 2π ·n · Rs∼ R2 s∼+X2 d (4.15) rises approximately linearly with n for low speeds until, like before, the increasing syn- chronous reactance Xd leads to a hyperbolic descending torque curve again [5] (Fig- ure 4.10). For machines with different d- and q-axis inductance (Ld ̸= Lq), the short-circuit torque depends on the d-axis reactance Xd = ωsLd and on the q-axis reactance Xq = ωsLq via Msc = 3 ·U2 p 2π ·n · Rs∼ · (R2 s∼+X2 q ) (R2 s∼+XdXq)2 . (4.16) The maximum short-circuit torque can exceed the rated torque by far, so that the short- circuit experiment has to be carried out with adequate torque measurement devices of a torque range 0 ≤ M ≤ M̂sc. If the short-circuit test cannot be performed due to the high short-circuit torque or the lack of an auxiliary drive, the reactive current test can be used to determine the synchronous inductance of the permanent-magnet synchronous machine. 41 4.6. Reactive current test 0 0 Speed Sh or t- ci rc ui ts ha ft to rq ue M̂sc nNn ≪ nN n Msc Figure 4.10.: Exemplary curve of the torque over speed at the generator short-circuit test 4.6. Reactive current test REQUIREMENTS: Inverter-fed variable drive VARIABLE: Stator current angle ϕs, Mechanical rotor speed n MEASUREMENT: Fundamental stator voltage Us,1, stator current Is,1, and stator power factor cosϕs,1 When the permanent-magnet synchronous machine is driven by a voltage source inverter, the synchronous inductance Ld can be determined instead of (4.13) by the reactive current test. The initial setup is the same as for the motor no-load operation (Section 4.2). The machine is driven without any coupled load, while the speed n is variable by the inverter output voltage. The stator current angle ϕs is defined as ∠(U s, Is,1). Again the fundamen- tal stator terminal voltage Us,1 is the no-load voltage U0. By increasing the current angle ϕs to nearly −90 °el., an additional negative d-current can be impressed by the inverter, which leads to field weakening. The small q-current remains constant, as the torque re- quirements do not change at constant speed. Therefore an additional voltage drop jXd · Isd occurs, reducing the stator voltage Us,1. Of course the stator voltage cannot be reduced to zero in motor operation, but the extrapolation of the nearly linear graph Is,1 = f (Us,1) to Us,1 = 0 yields to the maximum current, which is approximately the short-circuit current Isc (Figure 4.11). The inclination of the linear curve changes for different speeds, as the no-load voltage changes proportionally, but all curves have the same final value Isc. The 42 4.6. Reactive current test stator current cannot be increased any further than Isc. 0 0 Stator voltage St at or cu rr en t Isc U03U02U01 n3n2n1 Us,1 Is,1 Extrapolation Figure 4.11.: Exemplary curve at n = const. for different speeds n1 < n2 < n3 of the stator current Is,1 over the stator voltage Us,1 at the reactive current test Figure 4.12 compares the circuit diagrams of the mentioned experiments. In a) the small motor no-load current Isq is omitted for readability. With rising d-current Isd the stator voltage Us,1 gets smaller (b). At zero voltage theoretically the short-circuit case (c) is obtained. Therefore the reactive current experiment might also be called the motor short- circuit experiment. With the no-load voltage U0 at Isd = 0 the synchronous inductance is determined at fixed n and ωs via Ld ∼= 1 ωs ∆Us ∆Is ∼= U0 ωs · Isc . (4.17) 43 4.6. Reactive current test a) b) c) d-axis q-axis UpU s,1 jXq · Isq ϕs ∼= 0 Is,1 = Isq d-axis q-axis Up U s,1 jXd · Isd Isd Is,1 Isq ϕs jXq · Isq d-axis q-axis Up jXd · Isd Is,sc = Isd U s = 0 motor no-load reactive current generator short-circuit Figure 4.12.: Comparison of phasor diagrams at Rs ∼= 0 of a) motor no-load (Is,1 = Isq ≪ IN), b) reactive current (ϕs ∼=−90°el.), c) generator short- circuit experiment 44 5. Indirect efficiency determination of PMSMs This chapter describes the procedure, how to determine the efficiency of permanent- magnet synchronous machines by summation of losses. This method shall be used for machines operated in the base speed range, where a d-q-current operation may be used to utilize also the reluctance torque component. Investigations concerning the field weak- ening range are not part of this thesis. Similar to method B for electrically excited syn- chronous machines (Section 2.2) the machine losses are divided into three groups: • Voltage-depending losses – Iron losses PFe,0 and additional no-load losses Pad,0 ∼= PM,0 at sine wave volt- age operation – Additional losses due to inverter feeding Pel,in,0,ad • Current-depending losses PCu∼ at sinusoidal current supply – Stator DC copper losses PCu= – Additional stator losses due to current displacement ∆PCu = Pad,1,s • Friction and windage losses Pfr+w The current-depending additional rotor losses Pad,1,r are neglected. The individual loss groups are determined with help of the previously described experiments: Motor and generator no-load test and removed rotor test. For a given (rated) load operation point with the fundamental stator voltage Us,1, the fun- damental stator current Is,1, the fundamental power factor cosϕs,1, and thus the electrical power Pel = msUs,1Is,1cosϕs,1, the efficiency is calculated as follows. 45 5.1. Method description 5.1. Method description Step I: Iron losses at load Depending on which no-load experiment was carried out to determine the fundamental no-load losses (either motor or generator no-load test), the sum of iron and friction and windage losses at no-load as f (n) are determined for generator no-load operation as PFe,0 +Pad,0 +Pfr+w ∼= PFe+M,0 +Pfr+w = Pm,in,0 (5.1) and for motor no-load operation as PFe,0 +Pad,0 +Pfr+w ∼= PFe+M,0 +Pfr+w = Pel,in,0 −ms ·Rs∼ · I2 s,0,1 . (5.2) After subtracting the friction and windage losses Pfr+w, the no-load iron losses PFe,0 +Pad,0 ∼= PFe+M,0 = f (n) are determined. At load operation the stator field superimposes to the no-load field of the rotor magnets. This typically leads to increased iron losses PFe. As shown in Section 3.3, these losses depend on the square of the total magnetic flux and thus on the square of the resulting induced reactance voltage Ux in the stator winding. According to Figure 3.2 this voltage is determined via Ux =U s −Rs∼ · Is . (5.3) Therefore the iron losses PFe at load operation are calculated from the no-load iron losses as PFe = PFe,s +PFe,r+M ∼= PFe+M,0 · (︃ Ux U0 )︃2 . (5.4) For this calculation the value of the stator winding resistance per phase Rs,∼ has to be known at the actual stator winding temperature. For machines with a small amount of ad- ditional stator I2R losses ∆PCu (see below), the DC value of the stator resistance Rs=,warm, which may be determined by a resistance measurement according to IEC 60034-4 [30], or by using an appropriate temperature sensor to measure the average stator winding temper- ature ϑCu, can be used. If the resistance change due to current displacement is significant, the exact value of Rs∼,warm should be used at this step. However, usually the total voltage 46 5.1. Method description drop at the stator winding resistance is rather small compared to the reactance voltage Ux. From a critical point of view (5.4) is only an approximation of the total iron losses. At load operation the sub- or super-harmonic components of the stator field lead to increased rotor iron losses and magnet losses Pad,1,r = PFe,r+M, but these losses are not included in PFe+M,0. This means, the total iron and magnet losses at load and sine wave stator current may be underestimated. Another inaccuracy occurs, if the friction and windage losses are not known from a measurement with a non-magnetized rotor, and if their calculation is omitted, assuming that these losses are small. Then in (5.1) and (5.2) the losses PFe,0 are overestimated. This systematic error is small, if the reactance voltage Ux at load is not differing much from the no-load voltage U0. Step II: Current-depending losses at load According to (4.5) the current-depending losses at the removed rotor test with a stator sine wave current system are determined by subtracting the iron losses PFe,B from the measured electrical input power. These iron losses are determined in the same manner as shown before for load operation. Therefore again the reactance voltage Ux,B (Figure 4.6) has to be calculated via (5.3). As the amount of current displacement effect is not known yet, hence Rs∼ is not known, an iterative calculation process of Rs∼ might be necessary at this step. Afterwards the iron losses are calculated via PFe,B ∼= PFe+M,0 · (︃ Ux,B U0 )︃2 . (5.5) As the no-load losses PFe+M,0 also cover the rotor losses PFe,r+M,0 at no-load, which are of course missing during the removed rotor test, the iron losses PFe,B may be overestimated. Due to the big "equivalent" air gap δ = dsi/2 the reactance voltage Ux,B and thus the iron losses PFe,B are depending on the stator leakage reactance per phase. After the iron losses PFe,B are subtracted from Pel,in,B, the current-depending losses at sine wave load PCu∼ = f (I2 s , fs) are determined: PCu∼ = Pel,in,B −PFe,B . (5.6) 47 5.1. Method description Step III: Friction and windage losses at load Due to the nature of the mechanical friction and windage losses, the losses Pw = Pw(n) are load-independent and vary only with different rotor speed values n for fixed machine ge- ometry parameters. The bearing friction losses Pfr may also depend to a certain extend on the load torque M. Usually the bearing friction losses are much smaller than the windage losses, especially for rotors with shaft-mounted fan. Hence Pfr +Pw = Pfr+w = Pfr+w(n) are assumed also load-independent. Therefore the same calculations (Section 3.5) – or measurement results with non-magnetized rotor – like at no-load operation are applied. Step IV: Efficiency at sine wave operation The previously described three loss components are already present at sinusoidal voltage and current operation. Therefore the efficiency at sine wave operation is given by (5.7) for motor operation and (5.8) for generator operation: ηmot,1 = msUs,1Is,1cosϕs,1 −PFe −PCu∼−Pfr+w msUs,1Is,1cosϕs,1 , (5.7) ηgen,1 = msUs,1Is,1cosϕs,1 msUs,1Is,1cosϕs,1 +PFe +PCu∼+Pfr+w . (5.8) In both equations PFe +PCu∼+Pfr+w = Pd,1 (5.9) is corresponding to the sum of the losses at fundamental voltage and current at inverter supply. As the permanent-magnet synchronous machine is usually inverter driven, the additional losses due to inverter operation Pel,in,0,ad have to be considered to calculate the total effi- ciency at inverter operation. Step V: Additional losses due to inverter feeding Here only voltage source inverters are investigated, which form the major part of feeding inverters for permanent-magnet synchronous machines. During the motor no-load test the additional losses due to inverter feeding Pel,in,0,ad = f (U0) are determined via (4.2). These 48 5.2. Drawbacks losses are nearly load-independent for sufficient high switching frequencies fPWM of the inverter and depend only on the modulation degree ma of the inverter output voltage. Therefore for a given load point at the fundamental stator voltage Us,1, the losses are determined via Pel,in,0,ad(Us,1). Step VI: Efficiency at inverter operation Taking the additional losses due to inverter operation into account, (5.7) and (5.8) are extended by Pel,in,0,ad. Finally the total efficiency at voltage source inverter operation is given by (5.10) for motor operation and (5.11) for generator operation: ηmot = msUs,1Is,1cosϕs,1 −PFe −PCu∼−Pfr+w msUs,1Is,1cosϕs,1 +Pel,in,0,ad , (5.10) ηgen = msUs,1Is,1cosϕs,1 msUs,1Is,1cosϕs,1 +PFe +PCu∼+Pfr+w +Pel,in,0,ad , (5.11) with the total losses Pd at inverter operation: PFe +PCu∼+Pfr+w +Pel,in,0,ad = Pd,1 +Pel,in,0,ad = Pd . (5.12) The process of efficiency determination is visualized in Figure 5.1. 5.2. Drawbacks The load-dependent rotor losses PFe,r+M at sine wave current operation are neglected by the proposed method. Therefore the calculated efficiency will be fitting better to the di- rect efficiency value for machines with a small amount of stator winding sub- and super- harmonic field components. According to Step I, if the friction and windage losses are unknown or calculated impre- cise, the iron loss prediction at load is inaccurate. This holds true especially for machines with shaft-mounted fan, where the losses Pfr+w are relatively big. For external cooling systems the air friction losses are usually small except for special high-speed drives, and thus the error is also small. The assumption PFe ∝ U2 x is also influencing the efficiency calculation. Whereas in in- 49 5.2. Drawbacks Removed rotor test No-load test (generator/motor) Sine wave current- depending losses PCu∼(∝ I2 s ) Load-dependent iron losses PFe(∝ U2 x ) Friction and windage losses Pfr+w Additional losses due to inverter feeding Pel,in,0,ad Indirect efficiency at sine wave operation Indirect efficiency at inverter operation Figure 5.1.: Block diagram of the indirect efficiency determination of sine wave and volt- age source inverter operated permanent-magnet synchronous machines [66] duction machines the main field reactance is by 20 . . . 30 bigger than the stator leakage reactance, this ratio is much smaller with permanent-magnet synchronous machines, es- pecially with surface-mounted rotor magnets. Hence the stray flux contribution to the iron losses is considered. The influence of field harmonics of higher rotor order µ than the fundamental µ = 1 and the influence of variable iron saturation especially at load will disturb the assumption PFe ∝ U2 x to a certain extend. Of course the resulting air gap field at load as the interaction of the stator and rotor field with its additional load-dependent iron saturation will not be the same like at no-load and removed rotor operation. This effect also occurs for electrically excited synchronous machines, where the no-load and short-circuit test is used to determine the efficiency in- directly. Also here this influence cannot be considered. So, especially for machines with high magnetic utilization, the load-dependent saturation will lead to higher deviations from PFe ∝ U2 x . 50 5.2. Drawbacks The rotor removal of an already completed permanent-magnet synchronous machine is rather difficult due to the rotor permanent magnet forces especially for bigger machines. Therefore this method is primarily useful for the manufacturer, where the test equipment is available and the test can be performed during the production process before the machine is completed. In the same way the generator no-load test with a non-magnetized rotor is typically only possible during the manufacturing process. 51 6. Tested machines The goal of the thesis is the validation of the proposed method for different types of permanent-magnet synchronous machines in the power range ≤ 200kW, fitting to our laboratory conditions. Therefore five three-phase permanent-magnet synchronous ma- chines with quite different stator and rotor design have been chosen – each of them with a maximum efficiency of approximately 95 %, where also a direct efficiency determination is still possible with sufficient accuracy. This allows a direct efficiency determination as comparison. The first two test machines M1, M2 have a fractional-slot tooth-coil wind- ing, a higher number of 16 poles, a rather low rated speed of 1000 min−1, and high rated torque of 430 Nm. The two test machines M3, M4 have a distributed integer-slot single- layer winding, a smaller number of 6 . . .8 poles, and a lower rated torque of about 300 Nm. Test machine M5 has a fractional-slot distributed stator winding, a medium number of 12 poles, a rather high rated torque of 1019 Nm, and a rated speed of 1500 min−1. The rated power of the investigated test machines is in the range of 45kW . . .160kW. Each machine has a three-phase stator winding, NdFeB rotor magnets, and is designed to be drive