Dynamics and Control of Electrical Drives Piotr Wach Dynamics and Control of Electrical Drives ABC Author Prof.PiotrWach PolitechnikaOpolska InstituteofElectromechanicalSystemsandAppliedInformatics Mikołajczyka5str. 45-271Opole Poland E-mail:[email protected] ISBN978-3-642-20221-6 e-ISBN978-3-642-20222-3 DOI10.1007/978-3-642-20222-3 LibraryofCongressControlNumber:2011925364 (cid:2)c 2011Springer-VerlagBerlinHeidelberg Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthemate- rialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting, reproduction onmicrofilmor inanyother way, andstorage indatabanks. Dupli- cationofthispublicationorpartsthereof ispermittedonlyunder theprovisions oftheGerman CopyrightLawofSeptember9,1965,initscurrentversion,andpermissionforusemustalways beobtainedfromSpringer.ViolationsareliabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoes notimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Typeset&CoverDesign:ScientificPublishingServicesPvt.Ltd.,Chennai,India. Printedonacid-freepaper 987654321 springer.com To my dear wife Irena Acknowledgments The author would like to express warm gratitude to all who contributed in various ways so that this book finally has been completed in its form. Firstly, I would like to express remembrance and pay tribute to Prof. Arkadiusz Puchała (Academy of Mining and Metallurgy in Cracow), who was my principal tutor and supervisor of my Ph.D. thesis. However, our contacts terminated early, when he died quite young in 1973. Then I would like to thank warmly my colleagues from the Institute of Elec- tromechanical Systems and Applied Informatics, who encouraged me and created favorable conditions and nice, friendly atmosphere to work and research: Prof. Krystyna Macek-Kamińska – Director of the Institute, Prof. Marian Łukaniszyn – our present Faculty Dean as well as Prof. Jerzy Hickiewicz and Prof. Sławomir Szymaniec – partners in several research undertakings. Then I would like to thank Dr. Krzysztof Tomczewski, Dr. Ryszard Beniak, Dr. Andrzej Witkowski, Dr. Krzysztof Wróbel my former Ph.D students, work with whom gave me a lot of experience, exchange of ideas and excellent opportu- nity to discuss. Finally I would like to thank my dear son Szymon Wach for his very good – I am sure, translation of the book into English and Mr Eugeniusz Głowienkowski for preparation of those technical drawings that were not produced automatically by MAPLE™, as an outcome of computer simulations. Piotr Wach Notation Index a=2π/3 - phase shift between 3-phase symmetrical sine curves a=v& =&r& - acceleration vector δA,δA ,δA - virtual work, its mechanical and electrical component m e A - vector potential of a magnetic field A ,A - skew symmetric matrices: 2- and 3- dimentional 2 3 respectively B - magnetic induction vector C - electrical capacity D - viscous damping factor e - electromotive force (EMF) induced in k-th winding k E - total energy of a system f (K) - analytical notation of holonomic constraints function k F,F - vector of external forces, i-th component of this i vector f , f , f - frequency of voltage (current): feeding line, stator, L s r rotor g - acceleration vector of earth gravitation force g - number of branches of electric network h - number of holonomic constraints i=Q& - electric current as a derivative of electrical charge i ,i - excitation current, armature current f a I - symbolic value of sinusoidal current I - matrix of inertia of a rigid body i =[i i i ]T - vector of a 3-phase stator currents s s1 s2 s3 i =[i i ]T - vector of a 3-phase stator currents in a star connected s12 s1 s2 system i =[i i ]T - vector of a 3-phase rotor currents in a star connected r13 r1 r3 system i =[i i Ki ]T - vector of a m-phase rotor currents r r1 r2 rm i ,i - vectors of transformed stator, rotor currents in 0,u,v s0uv r0uv axes i ,i - vectors of transformed stator, rotor currents to u,v suv ruv axes X Notation Index i ,i - vector of transformed stator and rotor currents to sαβ rαβ α,β axes i ,i - vector of transformed stator and rotor currents to d,q sdq rdq axes i ,i - vector of transformed stator and rotor currents to x,y sxy rxy axes i =[i i ] - vector of stator currents transformed to the field sxyρ xρ yρ oriented x ,y axes ρ ρ J ,J - moment of inertia, moment of inertia of a motor’s s rotor j - current density vector J - Jacobi matrix of a transformation k - stiffness coefficient of an elastic element k ,k - magnetic coupling coefficients for stator and rotor s r windings k - pulse width factor in PWM control method u L,L ,L - Lagrange’s function, its mechanical and electrical m e component L ,L - leakage inductances of stator and rotor windings σs σr respectively L ,L ,L - self-inductance of stator and rotor windings, s r m magnetizing inductance L - angular momentum of a body m - number of bars (phases) in a squirrel-cage rotor winding m ,m - modulation coefficients: of an amplitude and fre- a f quency respectively m,M - mass of i-th particle and total mass of a body i M - moment of forces M ,M - main field inductance coefficients of stator, rotor s r phase windings M ,M - inductance matrices for a stator and rotor phase sph rph windings systems M ,M - matrices of mutual inductances between stator and ro- srph rsph tor phase windings M ,M ,M - matrices of mutual inductances between stator and s0uv r0uv sr0uv rotor phase windings transformed to 0,u,v axes N ,N - number of slots in a stator and rotor of electrical s r machine nh - number of nonholonomic constraints in a system p - number of pole pairs in electrical machine p - vector of momentum of mechanical system P - electric power P - generalized force acting along the k-th coordinate k Notation Index XI Q - electric charge of the k-th element characterized by k electrical capacity Q& =i - electric current in k-th winding as a derivative of re- k k spective charge Q,Q& - vector of electrical charges of a system, vector of electric currents q - number of pulses of power electronic converter q ,q& ,q&& - k-th generalized coordinate, - velocity, - acceleration k k k respectively q,q&,q&& - vectors of generalized coordinates, velocities, accel- erations of a system δq - vector of virtual displacements for generalized coor- dinates δq - virtual displacement for k-th generalized coordinate k R ,R - phase winding resistance for stator and rotor respec- s r tively r,r - radius-vector pointing i-th particle, radius-vector for i whole system in Cartesian coordinates δr - vector of virtual displacements of a system in Carte- sian coordinates R - vector of reaction forces of constraints in a system s - slip of an induction motor rotor motion in respect to magnetic field s - total number of degrees of freedom s ,s - number of mechanical, electrical degrees of freedom m e of a system S - action function of a system T,T ,T - kinetic energy, its electrical and mechanical compo- el me nent T′,T′,T′ - kinetic co-energy, its electrical and mechanical com- el me ponent T ,T - electromagnetic torque of a motor, load torque e l T ,T ,T - break torque, starting torque, rated torque of an induc- b st n tion motor T - period of a single pulsation sequence in PWM control p method T - friction force f T,T - orthogonal matrices of transformation for stator and s r rotor variables u,u - electric voltage, voltage supplied to k-th winding k u ,u - stator and rotor voltages respectively s r U ,U - rated voltage, stator rated voltage n sn U ,U ,U - phase to phase voltages in 3-phase electrical system 12 23 31 u ,u ,u - stator’s phase to phase voltages in 3-phase system s12 s23 s31 XII Notation Index U - phase voltage of stator’s winding sph U - feeding line voltage L 3 2 U = U - average value of rectified voltage in a 6 pulse 3-phase d0 π L system U - symbolic value of sinusoidal voltage U,U ,U - potential energy of a system, its mechanical and elec- me el trical component U ,U =[u ,u ,u ]T - stator’s voltage vector, stator’s phase voltages vector s sph s1 s2 s3 U ,U =[u ,u ,K,u ]T- vector of rotor voltages, vector of phase voltages of r rph r1 r2 rm rotor windings u - vector of stator voltages transformed to 0,u,v system s0uv of axes u - vector of stator voltages transformed to u,v system suv of axes u - vector of stator voltages transformed to x,y system sxy of axes u - vector of stator voltages transformed to x ,y field sxyρ ρ ρ oriented axes v=r& - vector of velocities of a system w - number of nodes of an electric network X - reactance of a winding X ,X ,X - reactance of a stator, rotor and magnetizing one s r m respectively (cid:523)=(χ,Kχ) - vector of coordinates in a primary coordinate system 1 n Z ,Z - number of stator’s, rotor’s teeth of SRM machine s r α,α - angles determining axis position of windings k l α ,α - switch on and switch off control angles of SRM on off machine γ - phase shift angle δξ - virtual displacement of k-th Cartesian coordinate in k unified coordinate system Ξ=(ξ,Kξ) - vector of Cartesian coordinates in unified system of 1 n coordinates ξ η - energy efficiency factor of a system θ - rotation angle r θ& =Ω - rotational speed of a rotor r r ρ - number of a magnetic field harmonic ρ - field orientation angle of x ,y axes (vector control) ρ ρ σ - leakage coefficient of windings φ - scalar potential of electromagnetic field