A Study Of Brushless Doubly-Fed (Induction) Machines
A Study of BrushlessDoubly-Fed (Induction)MachinesC ONTRIBUTIONS IN M ACHINE A NALYSIS , D ESIGN AND C ONTROLADISSERTATION SUBMITTED FOR THE DEGREE OFD OCTOROFP HILOSOPHYPaul C. RobertsSeptember 2004(revised January 2005)Emmanuel CollegeUniversity of Cambridge
To God, Ruth and my parents.I will lift my eyes to the mountains;From where shall my help come?My help comes from the LORD,Who made heaven and earth.Psalm 121
AbstractThe Brushless Doubly-Fed Machine (BDFM) shows commercial promise as a variable speed drive orgenerator. However, for this promise to be realised the design of the machine must be improved beyond that proposed to date. This dissertation contributes towards this goal through machine analysis,design and control.A generalised framework is developed for a coherent and rigorous derivation of models for a wideclass of BDFMs, of which machines with ‘nested-loop’ design rotors are a subset. This frameworkis used to derive coupled circuit, d-q axis, sequence components and then equivalent circuit modelsfor the class of machines. Proofs are given for all derivations, exploiting the circulant propertiesof the mutual inductance matrices. The coherence between the different models allows parameterscalculated for the coupled circuit model to provide parameter values for the other models.A method of model order reduction is proposed for the class of BDFMs with ‘nested-loop’ rotors,and examples given of the efficacy of the procedure. The reduction method allows parameter values tobe computed for a simple equivalent circuit representation of the machine. These calculated parametervalues, and those for other BDFM rotor designs are verified by experimental tests on a prototypeBDFM.The significance of particular equivalent circuit parameters is investigated from the model. Series rotor inductance terms are found to have a significant and direct effect on machine performance.These terms are shown to relate directly to the design of the rotor, and are quantified using the previously developed framework. Seven different rotor designs, including a new BDFM rotor design, areconsidered to show how the values of these parameters change.An experimental method of parameter estimation is developed for the equivalent circuit model,and the relationship between these parameters and the parameters in other forms of the model derived.The experimental method is shown to be applicable both to standard induction machines and to BDFMmachines, yielding accurate results in each case.A synchronous reference frame model for the class of BDFMs is derived and is used to analyse thestability of the machine via a linearized model. Practical methods for the design of PID controllersare proposed to stabilise the machine using voltage source inverters. Results are presented fromexperimental implementations which show a significant improvement in performance over previouslypublished results.The non-linear control technique, feedback linearization, is applied to the BDFM and shown tohave some robustness to modelling errors, in a realistic simulation. An initial attempt at implementation of the scheme is reported. Preliminary results are encouraging, and warrant further investigation.Keywords: ac machines, BDFM, Brushless Doubly Fed Machines, control, coupled circuits, dqaxis, equivalent circuits, feedback linearization, model reduction, parameter estimation, synchronousreference framei
AcknowledgementsFirst, and foremost, I would like to express my gratitude to my supervisors doctors Richard McMahon,Jan Maciejowski and Tim Flack whose insight and enthusiasm have greatly improved this dissertation.I would like to thank colleagues past and present for making work much more enjoyable. Somepleasant memories include, the tea-time Bridge sessions, the Poker evenings, lunch with Dan andhitch-hiking to and from an international conference.I would like to gratefully acknowledge the help of Ehsan Abdi-Jalebi and Xiaoyan Wang in theoperation of the BDFM test rig. In particular their assistance in the collection of the experimentaldata used in figures 5.8, 5.9, 5.10, 5.11, 6.3 and 6.4. Thanks are also due to Ehsan for proof readingthis dissertation.I am most grateful for assistance of Mark Barrett, John Grundy, the electricians, carpenters,welders and other support staff in preparing and setting up the prototype BDFM test rig. Thanksare also due to Davor Dukic, Iskandar Samad and George Makrides for their assistance in the development of the instrumentation hardware for the test rig. I am very grateful for the assistance ofcomputing support staff, particularly John Sloan and Patrick Gosling.This work has been funded by the Engineering and Physical Sciences Research Council. A highpowered inverter was generously donated by Semikron UK Ltd. Cambridge Silicon Radio (CSR) Ltd.kindly supplied a number of Bluetooth models. FKI Energy Technology particularly its subsidiariesLaurence, Scott & Electromotors (LSE) Ltd. and Marelli Motori SpA provided the prototype machineframe and manufactured the prototype rotors described in chapter 5. I would like to thank theseinstitutions for their support. In particular, thanks to Peter Tavner, formerly of FKI and Terry Gallantof LSE for their continued interest in the BDFM, and to David Hargreaves of CSR.I would like to thank my parents for a lifetime of support and for continually fuelling my enthusiasm for tinkering with anything I could get my hands on.Finally, I would like to thank my wife, Ruth, for her unflagging support throughout, and seeminglyendless patience with me.iii
ivAcknowledgementsAs required by the University Statute, I hereby declare that this dissertation is not substantiallythe same as any that I have submitted for a degree or diploma or other qualification at any otheruniversity. This dissertation is the result of my own work and includes nothing which is the outcomeof work done in collaboration, except where specifically indicated. This dissertation is 64,277 wordsin length, and contains 83 figures.Paul RobertsEmmanuel CollegeSeptember 2004
ContentsAbstractiAcknowledgementsiiiNotation & Terminologyxi1 Introduction11.1Evolution of the BDFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.2Description of the BDFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71.2.1Machine Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71.2.2Synchronous mode of operation . . . . . . . . . . . . . . . . . . . . . . . .81.2.3Potential Applications for the BDFM . . . . . . . . . . . . . . . . . . . . .10Approach of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101.3.1BDFM Model Development . . . . . . . . . . . . . . . . . . . . . . . . . .111.3.2Use of BDFM models to investigate performance, new rotor designs and the1.31.3.3estimation of machine parameters . . . . . . . . . . . . . . . . . . . . . . .12Analysis of stability of the BDFM and the development of control strategies .132 BDFM Coupled Circuit Modelling and Parameter Calculation152.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .152.2Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .162.3General Electrical Machine Coupled-Circuit Model . . . . . . . . . . . . . . . . . .182.3.1Torque Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19Calculation of Parameters for Electrical machines . . . . . . . . . . . . . . . . . . .202.4.1Magnetic Flux Density due to Single Coil . . . . . . . . . . . . . . . . . . .212.4.2Mutual (and self) Inductance of Single Coils . . . . . . . . . . . . . . . . .242.4.3Calculation of spatial harmonic components of mutual inductance . . . . . .252.42.5Mutual Inductance of Machine Windings. . . . . . . . . . . . . . . . . . . . . . .272.5.1Calculation of rotor and stator inductance matrices . . . . . . . . . . . . . .282.5.2Leakage inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28v
viCONTENTS2.5.32.6Resistance calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29Some properties of machine windings . . . . . . . . . . . . . . . . . . . . . . . . .292.6.1Mutual inductance between two stator windings . . . . . . . . . . . . . . . .332.6.2Mutual inductance between two stator windings where coils groups are not inseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36BDFM Rotor Mutual Inductance terms . . . . . . . . . . . . . . . . . . . . . . . . .392.7.1Rotor-rotor mutual inductance matrices . . . . . . . . . . . . . . . . . . . .402.7.2Rotor-Stator mutual inductance matrix . . . . . . . . . . . . . . . . . . . . .422.8Effect of Slotting on Mutual inductance terms . . . . . . . . . . . . . . . . . . . . .432.9BDFM Coupled-Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .442.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .472.73 d-q Transformed Model493.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .493.2The d-q-0 state transformation matrix . . . . . . . . . . . . . . . . . . . . . . . . .503.3Transformation to d-q-0 axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .543.3.1Determination of d-q model rotor current from bar currents . . . . . . . . . .64Model order reduction for Nested-loop rotor . . . . . . . . . . . . . . . . . . . . . .653.4.1Model Reduction Techniques. . . . . . . . . . . . . . . . . . . . . . . . .663.4.2New BDFM Rotor State Reduction Technique . . . . . . . . . . . . . . . . .753.5Simulation comparison of different BDFM model reduction techniques . . . . . . . .783.6Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .833.44 Equivalent Circuit Model and its Implication for BDFM Performance874.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .874.2Conversion to Symmetrical Components . . . . . . . . . . . . . . . . . . . . . . . .884.3Steady-state Equivalent Circuit Representation . . . . . . . . . . . . . . . . . . . . .954.4Equivalent circuit for the BDFM with a single set of rotor coils . . . . . . . . . . . .984.4.1Physical interpretation of parameters in the per-phase equivalent circuit model 1004.4.2Development of Torque Equations from the Equivalent Circuit . . . . . . . . 1044.4.3Phasor diagram for rotor branch circuit . . . . . . . . . . . . . . . . . . . . 1084.5Equivalent Circuit Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . 1084.6Equivalent Circuit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.7Magnetic Loading for the BDFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.8Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175 Possible Rotor Designs and Evaluation5.1119Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
CONTENTS5.2viiRotor Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.2.1Rotor 1 - the ‘nested-loop’ design rotor . . . . . . . . . . . . . . . . . . . . 1215.2.2Rotor 2 - the new double layer design rotor . . . . . . . . . . . . . . . . . . 1225.2.3Rotor 3 - isolated loop rotor . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.2.4Rotor 4 - isolated loop rotor with coils removed . . . . . . . . . . . . . . . . 1265.2.5Rotor 5 - 6 bar squirrel cage rotor design . . . . . . . . . . . . . . . . . . . 1275.2.6Rotor 6 - wound rotor design . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.2.7Rotor 7 - standard squirrel cage rotor . . . . . . . . . . . . . . . . . . . . . 1285.3Experimental and Calculated Torque-Speed Curve Results . . . . . . . . . . . . . . 1295.4Calculated Rotor Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.5Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.65.5.1Experimental Torque-Speed Curves . . . . . . . . . . . . . . . . . . . . . . 1325.5.2Calculated machine parameters from tables 5.1 and 5.2 . . . . . . . . . . . . 134Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356 BDFM Parameter Identification1376.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.2Parameter Extraction Optimization Method . . . . . . . . . . . . . . . . . . . . . . 1386.36.2.1General Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . 1386.2.2Application to BDFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139Parameter Estimation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1436.3.1Comparison of data obtained using estimated parameter values with experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1446.3.2Comparison of estimated to calculated parameter values . . . . . . . . . . . 1476.3.3Comparison of estimated parameter values to manufacturer’s parameter values 1496.4Relationship of extracted parameters to the d-q axis model . . . . . . . . . . . . . . 1516.5Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1537 Modelling for Control of the BDFM1557.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1557.2Synchronous reference frame model . . . . . . . . . . . . . . . . . . . . . . . . . . 1567.2.1Transformation from the rotor reference frame to the synchronous referenceframe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1587.2.2Evaluation of component matrices in the synchronous reference frame . . . . 1607.3Synchronous Reference Frame Model Equilibrium Conditions . . . . . . . . . . . . 1637.4Linearization of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1667.4.1Conclusions for the Linearized Model . . . . . . . . . . . . . . . . . . . . . 1687.4.2Simplification of Linearized Model . . . . . . . . . . . . . . . . . . . . . . 168
viiiCONTENTS7.4.3Linearization Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1697.5Simulated Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1717.6Open-loop Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1757.7Closed-loop ‘stator 2 phase angle control’ . . . . . . . . . . . . . . . . . . . . . . . 1787.7.17.87.9Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179RRControl when ω2 dt ( p1 p2 )θr ω1 dt . . . . . . . . . . . . . . . . . . . . 180Future work on linear model based control . . . . . . . . . . . . . . . . . . . . . . . 1817.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1838 Feedback Linearization for the BDFM8.1Feedback Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1888.1.18.28.38.4187Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188Application to the BDFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1918.2.1BDFM model in terms of flux linkages . . . . . . . . . . . . . . . . . . . . 1918.2.2Control strategy 1: Speed Only Regulation . . . . . . . . . . . . . . . . . . 1938.2.3Control Strategy 2: Speed and Flux Regulation . . . . . . . . . . . . . . . . 194Towards A Practical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 1958.3.1Zero Dynamics and Idealized FBL Stability . . . . . . . . . . . . . . . . . . 1968.3.2Practical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2019 Conclusions and Future Work2059.1Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2059.2Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2089.2.1Analysis of the BDFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2089.2.2BDFM Machine Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2109.2.3Stability analysis and control, including parameter estimation . . . . . . . . 210A Mathematics213A.1 Trigonometric Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213A.2 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213B Prototype Machine Stator and Rotor Design Details229B.1 Prototype machine frame details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230B.2 Prototype Machine Stator Windings . . . . . . . . . . . . . . . . . . . . . . . . . . 230B.2.1Machine Winding Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . 230B.2.2Stator Winding details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231B.3 Rotor 1: Nested-loop Rotor Design Details . . . . . . . . . . . . . . . . . . . . . . . 237B.3.1Rotor-rotor inductance terms . . . . . . . . . . . . . . . . . . . . . . . . . . 237
CONTENTSB.3.2ixRotor-Stator inductance details . . . . . . . . . . . . . . . . . . . . . . . . . 239B.4 Rotor 2: New Double Layer Rotor Design Details . . . . . . . . . . . . . . . . . . . 239B.4.1Rotor-rotor inductance terms . . . . . . . . . . . . . . . . . . . . . . . . . . 239B.4.2Rotor-Stator inductance details . . . . . . . . . . . . . . . . . . . . . . . . . 241B.5 Rotor 3: Isolated loop rotor design . . . . . . . . . . . . . . . . . . . . . . . . . . . 243B.5.1Rotor inductance terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243B.6 Rotor 4: Isolated loop design rotor with one set of loops removed . . . . . . . . . . . 245B.6.1Rotor inductance terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245B.7 Rotor 5: 6 bar cage rotor design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247B.7.1Rotor-rotor inductance terms . . . . . . . . . . . . . . . . . . . . . . . . . . 247B.7.2Rotor-Stator inductance details . . . . . . . . . . . . . . . . . . . . . . . . . 248B.8 Rotor 6: Wound Rotor Design Details . . . . . . . . . . . . . . . . . . . . . . . . . 249B.8.1Rotor-rotor inductance terms . . . . . . . . . . . . . . . . . . . . . . . . . . 249B.8.2Rotor-Stator inductance details . . . . . . . . . . . . . . . . . . . . . . . . . 250B.9 Rotor 7: Standard Squirrel Cage Rotor Details . . . . . . . . . . . . . . . . . . . . . 251B.9.1Rotor-rotor inductance terms . . . . . . . . . . . . . . . . . . . . . . . . . . 251B.10 Machine slot utilisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253C Leakage Inductance and Effective Air Gap257C.1 Leakage Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257C.1.1Slot and Tooth-top Permeance . . . . . . . . . . . . . . . . . . . . . . . . . 257C.1.2Overhang Permeance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258C.1.3Zig-zag Permeance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259C.1.4Leakage flux per coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259C.2 Effective air gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262D Previous BDFMs267E Experimental Apparatus269E.1 Apparatus Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269E.1.1xPC Target PC and peripheral boards . . . . . . . . . . . . . . . . . . . . . 270E.1.2Torque Transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270E.1.3DC load motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272E.1.4Inverter Output Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272E.1.5Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272E.1.6Position and Speed Measurements . . . . . . . . . . . . . . . . . .
Who made heaven and earth. Psalm 121. Abstract The Brushless Doubly-Fed Machine (BDFM) shows commercial promise as a variable speed drive or generator. However, for this promise to be realised the design of the machine must be improved be- . values, and those for other BDFM rotor designs are veried by experimental tests on a prototype BDFM.
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