BORANG PENGESAHAN STATUS TESIS

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1 UNIVERSITI TEKNOLOGI MALAYSIA PSZ 19:16 (Pind. 1/97) BORANG PENGESAHAN STATUS TESIS JUDUL: DIRECT TORQUE CONTROL OF INDUCTION MOTOR DRIVES USING SPACE VECTOR MODULATION (DTC - SVM) SESI PENGAJIAN: 2004/2005 Saya ZOOL HILMI BIN ISMAIL (HURUF BESAR) mengaku membenakan tei (PSM/Sajana/Dokto Falafah)* ini diimpan di Peputakaan Univeiti Teknologi Malayia dengan yaat-yaat kegunaan epeti beikut: 1. Tei adalah hakmilik Univeiti Teknologi Malayia. 2. Peputakaan Univeiti Teknologi Malayia dibenakan membuat alinan untuk tujuan pengajian ahaja. 3. Peputakaan dibenakan membuat alinan tei ini ebagai bahan petukaan antaa intitui pengajian tinggi. 4. **Sila tandakan ( ) SULIT TERHAD (Mengandungi maklumat yang bedajah keelamatan atau kepentingan Malayia epeti yang temaktub di dalam AKTA RAHSIA RASMI 1972) (Mengandungi maklumat TERHAD yang telah ditentukan oleh oganiai/badan di mana penyelidikan dijalankan) TIDAK TERHAD Diahkan oleh (TANDATANGAN PENULIS) (TANDATANGAN PENYELIA) Alamat Tetap: Lot 316, Kampung Kubang Tin, Jln. Ail, D. Nik Rumzi Nik Idi Melo, Kota Bhau, Kelantan. Nama Penyelia Taikh: 11 Novembe 2005 Taikh: 11 Novembe 2005 CATATAN: * Potong yang tidak bekenaan. ** Jika tei ini SULIT atau TERHAD, ila lampikan uat dai pihak bekuaa / oganiai bekenaan dengan menyatakan ekali ebab dan tempoh tei ini pelu dikelakan ebagai SULIT atau TERHAD. Tei dimakudkan ebagai tei bagi Ijazah Dokto Falafah dan Sajana ecaa penyelidikan, atau dietai bagi pengajian ecaa keja kuu dan penyelidikan, atau Lapoan Pojek Sajana Muda (PSM).

2 I heeby declae that I have ead thi thei and in my opinion thi thei i ufficient in tem of cope and quality fo the awad of the degee of Mate of Engineeing (Electical - Mechatonic and Automatic Contol). Signatue : Name : D. Nik Rumzi Nik Idi. Date : 11 Novembe 2005

3 DIRECT TORQUE CONTROL OF INDUCTION MOTOR DRIVES USING SPACE VECTOR MODULATION (DTC-SVM) ZOOL HILMI ISMAIL A poject epot ubmitted in patial fulfillment of the equiement fo the awad of the degee of Mate of Engineeing (Electical - Mechatonic & Automatic Contol) Faculty of Electical Engineeing Univeiti Teknologi Malayia NOVEMBER, 2005

4 ii I declae that thi thei entitled Diect Toque Contol of Induction Moto Dive Uing Space Vecto Modulation (DTC-SVM) i the eult of my own wok and the mateial which ae not the eult of my own wok have been clealy acknowledged. Signatue : Name : Zool Hilmi Imail Date : 11 Novembe 2005

5 iii Specially dedicated fom Abe Long to my beloved mothe, fathe, bothe, ite and a pecial fiend who have encouaged, guided and inpied me thoughout my jouney of education

6 iv ACKNOWLEDGEMENT I would like to take thi oppotunity to expe my deepet gatitude to my poject upevio, D. Nik Rumzi Nik Idi who ha peitently and deteminedly aited me duing the whole coue of thi poject. It would have been vey difficult to complete thi poject without the enthuiatic uppot, inight and advice given by him. My outmot thank alo goe to my family who ha given me uppot thoughout my academic yea. Without them, I might not be the peon I am today. My pecial gatitude to Aia Pujol, a hi thei ha been my guidance and giving ome idea to uppot my poject. It i of my geatet thank and joy that I have met thee people. Thank you.

7 v ABSTRACT Diect Toque Contol i a contol technique ued in AC dive ytem to obtain high pefomance toque contol. The conventional DTC dive contain a pai of hyteei compaato, a flux and toque etimato and a voltage vecto election table. The toque and flux ae contolled imultaneouly by applying uitable voltage vecto, and by limiting thee quantitie within thei hyteei band, de-coupled contol of toque and flux can be achieved. Howeve, a with othe hyteei-bae ytem, DTC dive utilizing hyteei compaato uffe fom high toque ipple and vaiable witching fequency. The mot common olution to thi poblem i to ue the pace vecto depend on the efeence toque and flux. The efeence voltage vecto i then ealized uing a voltage vecto modulato. Seveal vaiation of DTC-SVM have been popoed and dicued in the liteatue. The wok of thi poject i to tudy, evaluate and compae the vaiou technique of the DTC-SVM applied to the induction machine though imulation. The imulation wee caied out uing MATLAB/SIMULINK imulation package. Evaluation wa made baed on the dive pefomance, which include dynamic toque and flux epone, feaibility and the complexity of the ytem.

8 vi ABSTRAK Sitem kawalan tenaga putaan ecaa teu adalah teknik kawalan yang digunakan dalam pemacu item au ulang-alik dimana ia betujuan mencapai kawalan tenaga putaan yang lebih baik. Sitem kawalan yang ada ekaang ini tedii daipada pembanding hiteei, penafian fluk dan tenaga putaan dan juga jadual pemilihan vekto voltan. Fluk dan tenaga putaan dapat dikawal ecaa eentak dengan mengenakan vekto voltan yang euai dan menghadkan kuantiti-kuantiti ini dalam bataan yang telah ditetapkan, maka kawalan tenaga putaan dan fluk ecaa beaingan dapat dicapai. Walaubagaimanapun, pengunaan pembanding hiteei boleh menghailkan iak tenaga putaan yang tinggi di amping peubahan yang tidak menentu dalam fekueni penuian. Biaanya, penyeleaian untuk maalah ini adalah dengan menggunakan uangan vekto (pace vecto) yang begantung kepada fluk dan tenaga putaan. Voltan ujukan kemudiannya diealiaikan menggunakan pemodulat vekto voltan. Bebeapa kaedah DTC-SVM telah dicadangkan dan dibincangkan dan pelakanaan tuga untuk pojek ini adalah untuk mengkaji, menilai dan membuat pebandingan ecaa imulai bagi bebeapa teknik DTC-SVM yang diaplikaikan tehadap moto indukto. Simulai dijalankan dengan menggunakan pakej MATLAB/SIMULINK. Penilaian dibuat bedaakan peihal petai pemacu yang mana tedii daipada dinamik untuk tenaga putaan, kebolehlakaan, dan keumitan dalam item.

9 vii TABLE OF CONTENTS CHAPTER TITLE PAGE TITLE DECLARATION DEDICATION ACKNOWLEDGEMENT ABSTRACT ABSTRAK TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS LIST OF APPENDICES i ii iii iv v vi vii xi xii xvi xx 1 INTRODUCTION 1.1 Oveview of Induction Moto Aim of the Reeach Poject Scope of Wok Poject 6

10 viii 1.4 Thei Outline 7 2 INDUCTION MOTOR MODEL 2.1 Equation of Induction Moto Model Voltage Equation Applying Pak Tanfom Voltage Matix Equation Space Phao Notation Cuent Space Phao Flux Linkage Space Phao The Space Phao of Stato and Roto Voltage Space Phao Fom of the Moto Equation Toque Expeion Intoduction Deduction of the Toque Expeion by Mean of Enegy Conideation 35 3 DIRECT TORQUE CONTROL: PRINCIPLES AND GENERALITIES 3.1 Induction Moto Contolle Voltage /Fequency Vecto Contol Field Acceleation method Diect Toque Contol Pinciple of Diect Toque Contol Intoduction 40

11 ix DTC Contolle DTC Schematic Paamete Detuning Affect 50 4 DIRECT TORQUE CONTROL SPACE VECTOR MODULATION (DTC SVM) 4.1 Intoduction Vaiou Diect Toque Contol Space Vecto Modulation (DTC-SVM) DTC-SVM with Cloed-Loop Toque Contol DTC-SVM with Cloed-Loop Flux Contol DTC-SVM with Cloed-Loop Toque and Flux Contol Opeating in Cateian Coodinate 64 5 ANALYSIS AND COMPARISON 5.1 Intoduction Simulink Model Equation Ued in Model Conventional Diect Toque Contol DTC-SVM with Cloed-Loop Toque Contol DTC-SVM with Cloed-Loop Flux Contol DTC-SVM with Cloed-Loop Toque and Flux Contol Opeating in Cateian Coodinate Simulated Reult Inteim Concluion 87

12 x 6 CONCLUSION AND RECOMMENDATION FOR FUTURE WORK 6.1 Concluion Recommendation fo Futue Wok 94 REFERENCES 96

13 xi LIST OF TABLES TABLE NO. TITLE PAGE 3.1 Geneal Selection Table fo Diect Toque 43 Contol, being k the ecto numbe 3.2 Look up table fo Diect Toque Contol. FD/FI: 44 flux deceae/inceae. TD/=/: toque deceae/equal/inceae. S x : tato flux ecto. Φ : tato flux modulu eo afte the hyteei block. τ : toque eo afte the hyteei block 5.1 Po and con of the imulated contol cheme 89

14 xii LIST OF FIGURES FIGURE NO. TITLE PAGE 1.1 Oveview of induction moto contol method Co-ection of an elementay ymmetical 10 thee-phae machine 2.2 Equivalence phyic tanfomation Co-ection of an elementay ymmetical 18 thee-phae machine, with two diffeent fame, the D-Q axi which epeent the tationay fame fixed to the tato, and α-β axi which epeent otating fame fixed to the oto 2.4 Stato-cuent pace phao expeed in 26 accodance with the otational fame fixed to the oto and the tationay fame fixed to the tato 2.5 A magnitude epeented by mean of the vecto, 28 and it angle efeed to the thee diffeent axe. The thee diffeent axi ae: D-Q fixed to the

15 xiii tato, α-β fixed to the oto whoe peed i w m and finally the geneal efeence fame epeented by mean of the axi x-y whoe peed i equal to w g 3.1 Stato flux vecto locu and diffeent poible 43 witching voltage vecto. FD: flux deceae. FI : flux inceae. TD: toque deceae. TI: toque inceae 3.2 Diect Toque Contol Schematic Refeence and etimated flux elation Block cheme of DTC-SVM with cloed-loop 56 toque contol 4.3 Roto flux etimato block diagam Block cheme of DTC-SVM with cloed-loop 60 flux contol 4.5 Block diagam to detemine the efeence value 61 of the tato flux vecto 4.6 Stato flux component in ynchonou efeence 62 fame 4.7 Stato flux etimato block diagam Roto flux etimato block diagam Schematic of the tato-flux-oiented contol of 65 an induction machine with a pace vecto PWM invete

16 xiv 4.10 Stato magnetizing-cuent pace phao and 68 oto-cuent pace phao 4.11 Stato flux etimato block diagam Conventional Diect Toque Contol Schematic 75 uing SIMULINK/MATLAB 5.2 DTC-SVM cheme with cloed-loop toque 77 contol chematic uing SIMULINK/MATLAB 5.3 DTC-SVM cheme with cloed-loop flux contol 78 chematic uing SIMULINK/MATLAB 5.4 DTC-SVM cheme with cloed-loop toque and 80 flux contol opeating in Cateian coodinate-tato flux-oiented contol chematic uing SIMULINK/MATLAB 5.5 Dynamic epone fo conventional DTC Dynamic epone fo DTC-SVM with 82 cloed-loop toque contol 5.7 Dynamic epone fo DTC-SVM cheme with 82 cloed-loop flux contol 5.8 Dynamic epone fo tato field oiented 83 contol 5.9 Toque epone fo conventional DTC Toque epone fo DTC-SVM cheme with 84 cloed-loop toque 5.11 Toque epone fo DTC-SVM cheme with 85 cloed-loop flux contol

17 xv 5.12 Toque epone fo tato field oiented contol Stato flux path in d-q plane fo conventional 86 DTC 5.14 Stato flux path in d-q fo DTC-SVM cheme 86 with cloed-loop toque 5.15 Stato flux path in d-q fo tato field oiented 87 contol

18 xvi LIST OF SYMBOLS a opeato. i i(t) - Roto cuent pe phae. i - Space phao of the oto cuent expeed in the oto efeence fame. i' - Space phao of the oto cuent expeed in the tato efeence fame. i i(t) - Stato cuent pe phae. i - Space phao of the tato cuent expeed in the tato efeence fame. i' - Space phao of the tato cuent expeed in the oto efeence fame. L m - Thee phae magnetizing inductance. L - Total thee phae oto inductance. L - Roto elf-inductance. L 1 - Leakage oto inductance. L m - Roto magnetizing inductance. L - Total thee phae tato inductance.

19 xvii L - Stato elf-inductance. L m - Stato magnetizing inductance. L 1 - Leakage tato inductance. M - Mutual inductance between oto winding. M - Mutual inductance between tato winding. M - Maximal value of the tato-oto mutual inductance. p - Deivation opeato. P - Pai of pole. R - Roto eitance. R - Stato eitance. - Slip. 1/ - Integato opeato. Te - Intantaneou value of the electomagnetic toque. T - Intant toque efeed to the nominal toque and in pc pecentage. T=Tz - Sampling time. u i(t) - Roto voltage pe phae. u - Space phao of the oto voltage expeed in the oto efeence fame. u' - Space phao of the oto voltage expeed in the tato efeence fame. u i(t) - Stato voltage pe phae. u - Space phao of the tato voltage expeed in the tato efeence fame.

20 xviii u' - Space phao of the tato voltage expeed in the oto efeence fame. ω m - Mechanical peed ω - Geneal peed g ω - Roto pulation ω - Stato pulation ρ - Phae angle of the oto flux linkage pace phao with epect to the diect-axi of the tato efeence fame. ρ - Phae angle of the tato flux linkage pace phao with θm - Stato to oto angle. θ - Roto angle. θ - Stato angle. epect to the diect-axi of the tato efeence fame. ψ i (t) - Flux linkage pe oto winding. ψ - Space phao of the oto flux linkage expeed in the oto efeence fame. ψ' - Space phao of the oto flux linkage expeed in the tato efeence fame. ψ i(t) - Flux linkage pe tato winding. ψ - Space phao of the tato flux linkage expeed in the tato efeence fame. ψ' - Space phao of the tato flux linkage expeed in the oto efeence fame. α/β - Diect- and quadatue-axi component in the oto efeence fame.

21 xix d/q - Roto diect- and quadatue-axi component in the tato efeence fame. D/Q - Stato diect- and quadatue-axi component in the tato efeence fame. g - Geneal efeence fame. m - Magnetizing. - Roto a,b,c - Roto phae. Ref - Refeence. - Stato. A,B,C - Stato phae. x/y - Diect- and quadatue-axi component in geneal efeence fame o in pecial efeence fame. x - Co vecto poduct. * - Complex conjugate.

22 xx LIST OF APPENDICES APPENDICES TITLE PAGE A Matlab Function of Induction Moto Model 101 B Voltage Vecto Selection Table in Matlab file 106

23 1 CHAPTER 1 INTRODUCTION 1.1 OVERVIEW OF INDUCTION MOTOR The induction moto have moe advantage ove the et of moto. The main advantage i that induction moto do not equie an electical connection between the tationay and the otating pat of the moto. Theefoe, they do not need any mechanical commutato (buhe), leading to the fact that they ae maintenance fee moto. Beide, induction moto alo have low weight and inetia, high efficiency and a high oveload capability. Theefoe, they ae cheape and moe obut, and le pove to any failue at high peed. Futhemoe, the moto can wok in exploive

24 2 envionment becaue no pak ae poduced. Taking into account all of the advantage outlined above, the induction moto mut be conideed a the pefect electical to mechanical enegy convete. Howeve, mechanical enegy i moe than often equied at vaiable peed, whee the peed contol ytem i not an inignificant matte. The only effective way of poducing an infinitely vaiable induction moto peed dive i to upply the induction moto with thee phae voltage of vaiable fequency and vaiable amplitude. A vaiable fequency i equie becaue the oto peed depend on the peed of the otating magnetic field povided by the tato. A vaiable voltage i equied becaue the moto impedance educe at the low fequencie and conequently the cuent ha to be limited by mean of educing the upply voltage.[1][2] Induction moto ae alo available with moe than thee tato winding to allow a change of the numbe of pole pai. Howeve, a moto with eveal winding i moe expenive becaue moe than thee connection to the moto ae needed and only cetain dicete peed ae available. Anothe altenative method of peed contol can be ealized by mean of a wound oto induction moto, whee the oto winding end ae bought out to lip

25 3 ing. Howeve, thi method obviouly emove mot of the advantage of the induction moto and it alo intoduce additional loe. By connecting eito o eactance in eie with the tato winding of the induction moto, poo pefomance i achieved.[2][33] Hitoically, eveal geneal contolle have been developed: Scala contolle: Depite the fact that Voltage-Fequency (V/f) i implet contolle, it i the mot widepead, being in the majoity of the indutial application. It i known a a cala contol and act by impoing a contant elation between voltage and fequency. The tuctue i imple and it i nomally ued without peed feedback. Howeve, thi contolle doe not achieve a good accuacy in both peed and toque epone, mainly egading to the fact that the tato flux and toque ae not diectly contolled. Even though, a long a the paamete ae identified, the accuacy in the peed can be 2% (except in a vey low peed), and the dynamic epone can be appoximately aound 50m.[3][4] Vecto Contolle: In thee type of contolle, thee ae contol loop fo contolling both the toque and the flux.[5] The mot widepead contolle of thi type ae the one that ue vecto tanfom uch a eithe Pak o Ku. It accuacy can each value uch a 0.5% egading the peed and 2% egading the toque, even when at tand till. The main diadvantage ae the huge computational capability equied and the compuloy good identification of the moto paamete.[6]

26 4 Field Acceleation Method: Thi method i baed on the maintaining the amplitude and the phae of the tato cuent contant, whilt avoiding electomagnetic tanient. Theefoe, the equation can be implified aving the vecto tanfomation, which occu in the vecto contolle. Thi technique ha achieved ome computation eduction, thu ovecoming the main poblem with vecto contolle and allowing thi method to become an impotant altenative to vecto contolle.[8][10] Figue 1.1: Oveview of induction moto contol method.[11][9]

27 5 Diect toque contol (DTC) ha emeged ove the lat decade to become one poible altenative to the well-known Vecto Contol of Induction Machine. It main chaacteitic i the good pefomance, obtaining eult a good a the claical vecto contol but with eveal advantage baed on it imple tuctue and contol diagam.[7] DTC i aid to be one of the futue way of contolling the induction machine in fou quadant.[1][11] In the DTC, it i poible to contol diectly the tato flux and the toque by electing the appopiate invete tate. Thi method till equied futhe eeach in ode to impove the moto pefomance, a well a achieve a bette behavio egading envionment compatibility (Electo Magnetic Intefeence and Enegy), that i deied nowaday fo all indutial application. 1.2 AIM OF THE RESEARCH PROJECT The main objective of thi poject i to tudy on the vaiou technique of diect toque contol (DTC) baed on Space Vecto Modulation (DTC-SVM) applied to induction moto dive ytem. With DTC-SVM, it i poible to achieve fixed witching fequency and low toque ipple, hence ovecoming the majo dawback of conventional DTC. Thi poject will imulate and pefom analyi on ome of the peent DTC-SVM dive uing MATLAB/SIMULINK imulation package.

28 6 The conventional DTC i fitly analyzed and poved by mean of MATLAB/SIMULINK imulation. Then, the vaiou technique of diect toque contol baed on Space Vecto Modulation will be peented and alo the po and con of the peent DTC-SVM contol tategie will be highlighted. 1.3 SCOPE OF WORK PROJECT The poject i divided into thee tage. Thi i to enue that the poject i conducted within it intended bounday and i heading to the ight diection to achieve it objective: The fit tage i to tudy on the woking pinciple of the diect toque contol of induction moto dive that utilize hyteei compaato and to undetand on the limitation of thi conventional contol technique. Secondly, it will concentate on pefoming the imulation on the vaiou type of DTC-SVM fo induction moto dive ytem. The thid tage of the poject i to analyze on the pefomance of the vaiou contol technique of DTC-SVM baed on the MATLAB/SIMULINK imulation eult.

29 7 1.4 THESIS OUTLINE Thi ection will give an outline of the tuctue of the thei. The following i an explanation fo each chapte. Chapte 2 dicue a mathematical model of cage oto induction moto. Diffeent way of implementing thee model ae peented. The element of pace phao notation ae alo intoduced and ued to develop a compact notation. Then, all the model equation will be applied on the futhe chapte. Chapte 3 i devoted to intoduce diffeent Diect Toque Contol (DTC) tategie. Thi chapte ummaize diffeent induction moto contolle, uch a the vey well known vecto contol and V/Hz. The pinciple of DTC ae thooughly dicued and peented. Chapte 4 deal with diffeent kind of Diect Toque Contol with Space Vecto Modulation (DTC-SVM) contol technique. All the baic pinciple and detail deivation of voltage efeence fo each contol cheme ae dicued within thi chapte. Actually, the compaion between each contol algoithm aleady can be obeved on thi chapte.

30 8 Chapte 5 give the analyi and tate the diffeence between conventional DTC, DTC-SVM with toque contol, DTC-SVM with flux loop contol and DTC-SVM with toque and flux loop contol in tem of toque epone, contol technique, tato flux tajectoy and etc. Chapte 6 peent the concluion and ecommendation fo futue wok. Finally, all C-pogamming ued in the imulation ae lited in the appendixe.

31 9 CHAPTER 2 INDUCTION MOTOR MODEL 2.1 EQUATION OF THE INDUCTION MOTOR MODEL A dynamic model of the machine ubjected to contol mut be known in ode to undetand and deign vecto contolled dive. Due to the fact that evey good contol ha to face any poible change of the plant, it could be aid that the dynamic model of the machine could be jut a good appoximation of the eal plant. Nevethele, the model hould incopoate all the impotant dynamic effect occuing duing both teady tate and tanient opeation. Futhemoe, it hould be valid fo any change in the invete upply uch a voltage o cuent.[6] Such a model can be obtained by mean of eithe the pace vecto phao

32 10 theoy o two-axi theoy of electical machine. Depite the compactne and the implicity of the pace phao theoy, both method ae actually cloe and both method will be explained. Fo implicity, the induction moto conideed will have the following aumption: Symmetical two-pole, thee phae winding. The lotting effect ae neglected. The pemeability of the ion pat i infinite. The flux denity i adial in the ai gap. Ion loe ae neglected. The tato and the oto winding ae implified a a ingle, multi-tun full pitch coil ituated on the two ide of the ai gap. Figue 2.1: Co-ection of an elementay ymmetical thee-phae machine.[33]

33 VOLTAGE EQUATIONS The tato voltage will be fomulated in thi ection fom the moto natual fame, which i the tationay efeence fame fixed to the tato. In a imila way, the oto voltage will be fomulated to the otating fame fixed to the oto. In the tationay efeence fame, the equation can be expeed a follow: dψ A (t) u A (t)=ri A (t)+ dt u B dψ B(t) (t)=r i B(t)+ dt dψ C(t) u C(t)=R i C(t)+ dt (2.1) (2.2) (2.3) Simila way expeion can be obtained fo the oto: dψ a (t) u a (t)=r i a (t)+ dt dψ b(t) u b(t)=ri b (t)+ dt dψ c(t) u c(t)=ri c (t)+ dt (2.4) (2.5) (2.6) The intantaneou tato flux linkage value pe phae can be expeed a: Ψ A =Li A +Mi B+Mi C +Mcoθmi a +Mco(θ m +2π/3)i b +Mco(θ m +4π/3)i c (2.7)

34 12 Ψ B=Li B+Mi A +Mi C +Mcoθmi b +Mco(θ m +2π/3)i c +Mco(θ m +4π/3)i a Ψ C=Li C+Mi A +Mi B +Mcoθmi c +Mco(θ m +2π/3)i a +Mco(θ m +4π/3)i b (2.8) (2.9) In a imila way the oto flux linkage can be expeed a follow: Ψ a =Li a +Mi b +Mi c +Mco(-θ m)i A +Mco(-θ m +2π/3)i B+Mco(-θ m +4π/3)i C (2.10) Ψ b =Li b +Mi a +Mi c +Mco(-θ m)i B+Mco(-θ m +2π/3)i C +Mco(-θ m +4π/3)i A (2.11) Ψ c =Li c +Mi a +Mi b +Mco(-θ m)i C+Mco(-θ m +2π/3)i A +Mco(-θ m +4π/3)i B (2.12) Taking into account all the peviou equation, and uing the matix notation in ode to compact all the expeion, the following expeion i obtained: R +pl pm pm pmcoθm pmcoθm1 pmcoθ m2 ia pm R +pl pm pmcoθm2 pmcoθm pmcoθm1 i B ua u B u C pm pm R +pl pmcoθm1 pmcoθm2 pmcoθ = m ic ua pmcoθm pmcoθm1 pmcoθm2 R +pl pm pm ia u b b pm coθm2 pm coθ i m pm coθm1 pm R +pl pm u c pm i coθ c m1 pm coθm2 pm coθm pm pm R +pl (2.13)

35 APPLYING PARK S TRANSFORM. In ode to educe the expeion of the induction moto equation voltage given in equation 2.1 to equation 2.6 and obtain contant coefficient in the diffeential equation, the Pak tanfom will be applied.[1] Phyically, it can be undetood a tanfoming the thee winding of the induction moto to jut two winding, a it i hown in figue 2.2 Figue 2.2: Equivalence phyic tanfomation. In the ymmetical thee-phae machine, the diect- and the quadatue-axi tato magnitude ae fictitiou. The equivalencie fo thee diect (D) and quadatue (Q) magnitude with the magnitude pe phae ae a follow: ua 1/ 2 coθ -inθ u0 u B =c. 1/ 2 co(θ-2π/3) -in(θ-2π/3). ud u C 1/ 2 co(θ+2π/3) -in(θ+2π/3) uq (2.14)

36 14 ua 1/ 2 coθ -inθ u0 u B =c. 1/ 2 co(θ-2π/3) -in(θ-2π/3). ud u C 1/ 2 co(θ+2π/3) -in(θ+2π/3) uq (2.15) whee c i a contant that can take eithe the value 2/3 o 1 fo o-called non-powe invaiant fom o the value 2 / 3 fo the powe-invaiant fom a it i explained in ection Thee peviou equation can be applied a well fo any othe magnitude uch a cuent and flux. Notice how the expeion 2.13 can be implified into a much malle expeion in by mean of applying the mentioned Pak tanfom. ud R+pL -Lpθ i plm -L m(pθ +P. ωm) u Lpθ R+pL L (pθ +P. ω ) pl i =. u pl -L (pθ -P. ω ) R +pl L pθ i Q m m m Q α m m m α uβ L m(pθ-p. ωm) plm Lpθ R+pL iβ D (2.16) whee L=L-M L=L-M, L=L-M L=L-M, L =(3/2)M m L =(3/2)M m

37 VOLTAGE MATRIX EQUATIONS If the matix expeion 2.16 i implified, the eultant matixe ae obtained a hown in equation 2.17, 2.18 and 2.19.[1] Fixed to the tato. It mean that ω = 0 and conequently ω = - ω m. ud R+pL i 0 plm 0 u 0 R +pl 0 pl i =. u pl P.ω L R +pl P.ω L i Q m Q d m m m m d u -p.ω q mlm plm -P.ωmL R +pl iq D (2.17) Fixed to the oto It mean that ω = 0 and conequently ω = - ω m. ud R+pL -LP ωm plm -LP m ωm i u Q LPωm R +pl i LmPωm pl m =. u d plm 0 R +pl 0 i u 0 pl q m 0 R +pl i D Q d q (2.18)

38 Fixed to the ynchonim It mean that ω = ω. ud R+pL L i ω plm -Lmω u Lω R +pl L ω pl i =. u pl -L ω R +pl -L ω i Q m m Q d m m d uq Lmω plm Lω R +pl iq D (2.19) 2.2 SPACE PHASOR NOTATION Space phao notation allow the tanfomation of the natual intantaneou value of a thee-phae ytem onto a complex plane located in the co ection of the moto. In thi plane, the pace phao otate with the angula peed equal to the angula fequency of the thee-phae upply ytem. A pace phao otating with the ame angula peed, fo example, can decibe the otating magnetic field. Moeove, in the pecial cae of the teady tate, whee the upply voltage i inuoidal and ymmetic, pace phao become equal to thee-phae voltage phao, allowing the analyi in tem of complex algeba.[1][11].

39 17 In ode to tanfom the induction moto model, in natual co-odinate, into it equivalent pace phao fom, the 120 opeato i intoduced: j 2π / 3 2 j 4π / 3 a = e, a = e (2.20) Thu, the cuent tato pace phao can be expeed a follow: 2 i =c. 1.i A (t)+a.i B(t)+a i C(t) (2.21) The facto c, take uually one of the two diffeent value eithe 2/3 o (2/3). The facto 2/3 make the amplitude of any pace phao, which epeent a thee phae balanced ytem, equal to the amplitude of one phae of the thee-phae ytem. The facto (2/3) may alo be ued to define the powe invaiance of a thee-phae ytem with it equivalent two-phae ytem.[1] CURRENT SPACE PHASORS In thi ection the induction machine aumption intoduced in ection will be ued. Figue 2.4 peent the model of the induction machine with two diffeent fame, the D-Q axi which epeent the tationay fame fixed to the tato, and the α-β axi which epeent otating fame to the oto.

40 18 Figue 2.3: Co-ection of an elementay ymmetical thee-phae machine, with two diffeent fame, the D-Q axi which epeent the tationay fame fixed to the tato, and α-β axi which epeent otating fame fixed to the oto The tato cuent pace phao can be expeed a follow: 2 2 θ i =. i A(t)+ai B(t)+a i C(t) = i e 3 j (2.22) Expeed in the efeence fame fixed to the tato, the eal-axi of thi efeence fame i denoted by D and it imaginay-axi by Q. The elation between the thee-phae component and the D-Q two-axi component i a follow: i =i D(t)+j.i Q(t) (2.23) o: 2 2 Re(i ) = Re.( i A+ai B+a i C) = i 3 D

41 Im(i ) = Im.( i A+ai B+a i C) = i 3 Q (2.24) The elationhip between the pace phao cuent and the eal tato phae cuent can be expeed a follow: 2 2 Re(i ) = Re. i A+ai B+a i C = i Re( a i ) = Re. a i A+i B+ ai C = i Re( a.i ) = Re. ai A+ a i B+i C = i 3 A B C (2.25) In a imila way, the pace phao of the oto cuent can be witten a follow: 2 2 jα i =. i a(t) + ai b(t) + a i c(t) = i e 3 (2.26) Expeed in the efeence fame fixed to the oto, the eal axi of thi efeence fame i denoted by α and it imaginay-axi by β. The pace phao of the oto cuent expeed in the tationay efeence fame fixed to the tato can be expeed a follow: jθ j ( α + θ m i' ) = i e = i e (2.27) The elation between the cuent oto pace phao and the α-β two-axi i a follow:

42 20 i = i α (t) + j.i β (t) (2.28) o: 2 2 Re(i ) = Re.( i a+ai b+a i c) = i Im(i ) = Im.( i a+ai b+a i c) = i 3 α β (2.29) The elationhip between the pace phao cuent and the eal tato cuent can be expeed a follow: 2 2 Re(i ) = Re. i a+ai b+a i c =i Re( a i ) = Re. a i a+i b+ ai c =i Re( ai ) = Re. ai a+ a i b+i c =i 3 α c b (2.30) The magnetizing cuent pace-phao expeed in the tationay efeence fame fixed to the tato can be obtained a follow: i i N i' (2.31) e m = + N e

43 FLUX LINKAGE SPACE PHASOR In thi ection the flux linkage will be fomulated in the tato phao notation accoding to the diffeent efeence fame Stato flux-linkage pace phao in the tationay efeence fame fixed to the tato. Simila to the definition of the tato cuent and the oto cuent pace phao, it i poible to define a pace phao fo the flux linkage a follow: 2 2 ψ = ( ψ A+aψ B+a ψc ) 3 (2.32) If the flux linkage equation 2.7, 2.8, 2.9 ae ubtituted in equation 2.32, the pace phao fo the tato flux linkage can be expeed a follow: ( ) ( ) ( ) 2 ( coθ co( θ 4 π / 3) co( θ 2 π / 3) ) 2 ( θ θ π θ π ) 2 ( coθ co( θ 2 π / 3) co( θ 4 π / 3) ) ia L + am + a M + ib al + M + a M + ic a L + M + am ia M m + am m + + a M m + + Ψ = 3 + ib am co m + M co( m + 2 / 3) + a M co( m + 4 / 3) + + ic a M m + am m + + M m + (2.33) Developing the peviou expeion 2.33, it i obtained the following expeion:

44 ( ). ( ) ( ) 2 ( coθ co( θ 4 π / 3) co( θ 2 π / 3) ) 2 b ( θm θm π θm π ) 2 2. c ( coθm co( θm 2 π / 3) co( θm 4 π / 3) ) i L + am + a M + ai L + am + a M + a i L + a M + am 2 + ia M m + am m + + a M m + Ψ = 3 + ai. M co + a M co( + 2 / 3) + am co( + 4 / 3) + a i M + a M + + am + A B C (2.34) And finally, expeion 2.34 can be expeed a follow: 2 2 ( ) ( ) Ψ = L +am +a M i + M coθ +am co(θ +4π/3)+a M co(θ +2π/3) i m m m jθ ( ) ( ) m ( ) = L -M i +1.5coθ M i = L -M i +1.5e M i = L -M i +1.5M i ' m = L i +L i ' (2.35) m whee L i the total thee-phae tato inductance and the Lm i the o-called thee-phae magnetizing inductance. Finally, the pace phao of the flux linkage in the tato depend on two component, being the tato cuent and the oto cuent. Once moe, the flux linkage magnitude can be expeed in two-axi a follow: ψ = ψ D + j.ψ Q (2.36) Whee it diect component i equal to: ψ = L.i + L.i (2.37) D D m d And it quadatue component i expeed a:

45 23 ψ Q = L.i Q + L m.i q (2.38) The elationhip between the component i d and i α and i q and i β may be intoduced a follow: j θ i' m = i d + j.i q = i. e (2.39) The compactne of the notation in the pace phao nomenclatue compaed to the two-axi notation in 1.1 i noticeable Roto flux-linkage pace phao in the otating efeence fame fixed to the oto. The oto flux linkage pace phao, fixed to the oto natual fame ca be defined a follow: 2 ψ = 3 2 ( a b ) ψ +aψ +a ψc (2.40) If the flux linkage equation 2.10, 2.11, 2.12 ae ubtituted in equation 2.40, the pace phao fo the oto flux linkage can be expeed a follow: ( 2 ) ( 2 ) ( 2 ) 2 A ( m m m ) 2 B ( m m m ) 2 C ( m m m ) ia L +am +a M +ib al +M +a M +ic a L +M +am + 2 +i M coθ +am co(θ +4π/3)+a M co(θ +2π/3) + Ψ = 3 +i am coθ +M co(θ +4π/3)+a M co(θ +2π/3) + +i a M coθ +am co(θ +4π/3)+M co(θ +2π/3) (2.41)

46 24 By e-aanging the peviou expeion 2.41, it can be expeed a: Ψ = 3 +a.i ( 2 ) ( 2 ) 2 ( 2 ) 2 a ( m m m ) 2 2 b ( M coθ m +a M co(θ m +4π/3)+aM co(θ m +2π/3) ) 2 c ( m m m ) i L +am +a M +a.i L +am +a M +a i L +a M +am 2 +i M coθ +am co(θ +4π/3)+a M co(θ +2π/3) a b c +a.i M coθ +a M co(θ +4π/3)+aM co(θ +2π/3) (2.42) And finally: 2 2 ( ) ( ) Ψ = L +am +a M i + M coθ +am co(θ +2π/3)+a M co(θ +4π/3) i m m m -jθ ( ) ( ) m ( ) = L -M i +1.5co(-θ )M i = L -M i +1.5e M i = L -M i +1.5M i ' m = Li +Lmi ' (2.43) whee i i the tato cuent pace phao expeed in the fame fixed to the oto. Again, the flux linkage magnitude can be expeed in the two-axi fom a follow: ψ = ψ α + j.ψ β (2.44) Whee it diect component i equal to: ψ α = L.i α + L m.i α (2.45) And it quadatue component i expeed a: ψ β = L.i β + L m.i β (2.46)

47 Roto flux-linkage pace phao in the tationay efeence fame fixed to the tato. The oto flux linkage can alo be expeed in the tationay efeence fame uing j. θ m the peviouly intoduced tanfomation e, and can be witten a: ( ) ψ' = ψ + j.ψ = ψ.e = ψ + j.ψ e m (2.47) jθ m j ( α + θ ) d q α β The pace phao of the oto flux linkage can be expeed accoding to the fixed co-odinate a follow: j θ ψ' m = L.i' + L m.i 'e = L.i + L m.i (2.48) The elationhip between the tato cuent efeed to the tationay fame fixed to the tato and the otational fame fixed to the oto i a follow: i = i'.e j θm -jθm i.e = i' (2.49) whee i = i + j.i D Q i' = i + j.i α β (2.50) Fom figue 2.5, the following equivalencie can be deduced: i = i.e jθ i' = i'.e = i'.e = i.e jα j ( θ - θ m ) - j θm (2.51)

48 26 Figue 2.4: Stato-cuent pace phao expeed in accodance with the otational fame fixed to the oto and the tationay fame fixed to the tato Stato flux-linkage pace phao in the otating efeence fame fixed to the oto. Similaly than ection, it can be deduced the following expeion: ( ) ψ' = ψ.e = L.i + L.i '.e = L.i ' + L.i (2.52) -jθm -j θm m m

49 THE SPACE PHASORS OF STATOR AND ROTOR VOLTAGES The pace phao fo the tato and the oto voltage can be defined in a imila way like the one ued fo othe magnitude u =. u A(t) + ai B(t) + a i C(t) = u D+ j.u Q= ua- ub- u C + j ub-u u =. u (t) + a.i (t) + a i (t) = u + j.u =. u - u - u + j.. u -u ( ) ( ) 2 a b c α β a b c b c C (2.53) whee the tato voltage pace phao i efeed to the tationay efeence fame and the oto voltage pace phao i efeed to the otating fame fixed to the oto.[1] Povided the zeo component i zeo, it can alo be aid that: u u u A B C =Re u ( ) 2 ( ) ( ) =Reau =Re au (2.54) Equivalent expeion can alo be obtained fo the oto.

50 SPACE-PHASOR FORM OF THE MOTOR EQUATIONS The pace phao fom of the voltage equation of the thee-phae and quadatue -phae mooth ai-gap machine will be peented. The equation will be expeed in a geneal otating efeence fame, which otate at a geneal peed, ω g and then to the efeence fame fixed to the tato, oto, and ynchonou peed Space phao voltage equation in the geneal efeence fame If the voltage in the figue 2.6 i the tato cuent, then it fomulation in the pace phao fom i a follow [11]: - j θ i g g = i.e = i x+ j.i y (2.55) Figue 2.5: A magnitude epeented by mean of the vecto, and it angle efeed to the thee diffeent axe. The thee diffeent axe ae: D-Q fixed to the tato, α-β fixed to the oto whoe peed i ω m and finally the geneal efeence fame epeented by mean of the axi x-y whoe peed i equal to ω g.

51 29 In a imila way and fo othe magnitude, it can be witten the following equation: - j θg g x y u = u.e = u + j.u g - j θg ψ = ψ.e = ψ + j.ψ x y (2.56) whee the magnitude ae the voltage pace phao and the tato flux linkage epectively. Howeve, if the magnitude in the figue 2.6 i fo intance the oto cuent, it pace phao notation will be: -j.(θg-θ m) g x i = i e = i + j.i y (2.57) and fo othe magnitude: - j (θg-θ m) g x y u = u.e = u + j.u ψ = ψ.e g - j (θg-θ m) = ψ + j.ψ x y (2.58) Manipulating the peviou equation yield the following tato and oto pace phao voltage equation in the geneal efeence fame. jθg ( g ) d ψ e dψ u.e = R.i e + = R.i e + + j.e ωψ dt dt j θg jθg jθg g jθg g g g g g j.(θg-θ m) ( ) d ψ e j (θg-θ m) j.(θg-θ m) u g.e = R.i g.e + dt dψ = R.i.e + e + j.e (ω -P.ω ).ψ dt j(θg-θ m) j ( θg-θ m) g j(θg-θ m) g g m g (2.59)

52 30 Simplifying equation 2.59, it will eult a equation dψ u = R.i + + j.ω.ψ dt g g g g g dψ u = R.i + dt g g g g + j.(ω -P.ω ).ψ m g (2.60) Whee, the flux linkage pace phao ae: ψ g = L.i g + L m.ig ψ g = L.i g + L m.i g (2.61) Uing the two-axi notation and the matix fom, the voltage equation can be epeented by: ux R+pL -Lω g plm -Lmωg ix u y L i ωg R+pL Lω m g pl m y =. u pl L (P.ω -ω ) R +pl L (-ω +P.ω ) i x m m m g g m x uy L m(ωg-p.ω m) plm L (wg-p.w m) R +pl iy (2.62) Space-phao voltage equation in the tationay efeence fame fixed to the tato If ω g =0, equation 2.18 can be expeed a equation 2.67.

53 31 ud R+pL i 0 plm 0 u 0 R +pl 0 pl i =. u pl P.ω L R +pl P.ω L i Q m Q d m m m m d uq -p.ωmlm plm -P.ωmL R +pl iq D (2.63) The tato voltage pace phao can be expeed a follow: u= R.i+ dψ dt (2.64) The oto voltage pace phao can be expeed a follow: - j θm -jθm u'.e = R.i' e + -jθm ( ) d ψ' e dt dψ' u' = R.i ' + + j.p.ω ψ' dt m And the flux linkage pace phao can be expeed a follow: (2.65) ψ = L.i + L m.i' ψ' = L.i'+ L m.i (2.66) Space-phao voltage equation in the otating efeence fame fixed to the oto If ω =ω g m, the matix expeion obtained i 2.67, being equal to the expeion 2.18.

54 32 ud R+pL -LPω i m plm -LmPωm u LPω R+pL LPω pl i =. u pl 0 R +pl 0 i Q m m m m Q d m d u 0 pl q m 0 R +pl iq D (2.67) The tato voltage pace phao can be expeed a follow: dψ' u' = R.i' + dt m + j.p.ω ψ' (2.68) The oto voltage pace phao can be expeed a follow: u= R.i+ dψ dt (2.69) And the flux linkage pace phao can be expeed a follow: ψ' = L.i' + L.i m ψ = L.i + L m.i' (2.70) Space-phao voltage equation in the otating efeence fame fixed at ynchonou peed If ωg = ω, the matix expeion obtained i 2.71, being equal to the expeion 2.19.

55 33 ud R+pL Lω i plm -Lmω u L ω R+pL Lω pl i =. u pl -L ω R+pL -Lω i Q m m Q d m m d uq Lmω plm Lω R+pL iq D (2.71) The tato voltage pace phao can be expeed a follow: u = R.i + dψ g g g g dt + j.ω ψ (2.72) The oto voltage pace phao can be witten a follow: dψg u g = R.i g + + j.(ω-p.ω m ).ψ g (2.73) dt And the flux linkage pace phao can be expeed a follow: ψ g = L.i g + L m.i g ψ g = L.i g + L m.i g (2.74)

56 TORQUE EXPRESSIONS Intoduction The geneal expeion fo the toque i a follow: t = c.ψ x i' (2.75) e whee the c i a contant, ψ and i' ae the pace phao of the tato flux and oto cuent epectively, both efeed to the tationay efeence fame fixed to the tato. The expeion given above can alo be expeed a follow: t e = c.ψ. i.inγ (2.76) whee γ i the angle exiting between the tato flux linkage and the oto cuent. It follow that when γ=90 the toque obtained i maximum and it expeion i exactly equal to the one fo the DC machine. Howeve, in DC machine the pace ditibution of both magnitude i fixed in pace, thu poducing the maximum toque fo all diffeent magnitude value. Futhemoe, both magnitude can be contolled independently o epaately. In an AC machine, howeve, it i much moe difficult to ealize thi pinciple becaue both quantitie ae coupled and thei poition in pace depend on both the tato and oto poition. It i a futhe complication that in quiel-cage machine, it i not poible to monito the oto

57 35 cuent, unle the moto i pecially pepaed fo thi pupoe in a pecial laboatoy. It i impoible to find them in a eal application. The each fo a imple contol cheme imila to the one fo DC machine ha led to the development of the o-called vecto-contol cheme, whee the point of obtaining two diffeent cuent, one fo contolling the flux and the othe one fo the oto cuent i achieved DEDUCTION OF THE TORQUE EXPRESSION BY MEAN OF ENERGY CONSIDERATIONS Toque equation i being deduced by mean of enegy conideation.[1] Theefoe, the tating equation i a follow: P mechanic = Pelectic- P lo - P field (2.77) Subtituting the peviou powe fo it value, the equation can be expeed a follow: * dψ * ( ) ( ) 3 2 * 2 dψ' * t.ω e = Re u.i - R. i - Re i + Re u'.i' -R. i' -Re i' 2 dt dt (2.78)

58 36 Since in the tationay efeence fame, the tato voltage pace phao u can only balanced by the tato ohmic dop, plu the ate of change of the tato flux linkage, peviou expeion can be expeed a follow: * * ( ) ( ) t.ω =.Re - j.ωψ'i' = - ω.re j ψ' i' = - ω.ψ' x i' e (2.79) Expeing the equation in a geneal way fo any numbe of pai of pole given: 3 t= e -.P.ψ' x i' 2 (2.80) If equation 2.66 and 2.35 ae ubtituted in equation 2.80, it i obtained the following expeion fo the toque: t = e 3.P.ψ x i 2 (2.81) If the poduct i developed, expeion 2.81 i a follow: 3 e D Q Q t= P(ψ.i -ψ.i D ) 2 (2.82) Finally, diffeent expeion fo the toque can be obtained a follow: t e= - P(Li' +Lmi ) x i' = - PLmi x i' = - PLmi' x i L 3L 3 L t= - P (L i'+li) x i'= - P ψ x i' = - P ψx ψ' 2 L 2L 2 L L - L m m m e m 2 m (2.83)

59 37 CHAPTER 3 DIRECT TORQUE CONTROL: PRINCIPLES AND GENERALITIES 3.1 INDUCTION MOTOR CONTROLLERS Voltage /Fequency Thee ae too many diffeent way to dive an induction moto. The main diffeence between them ae the moto pefomance and the viability and cot in it eal implementation. Depite the fact that Voltage /Fequency (V/Hz) i the implet contolle, it i the mot widepead, being in the majoity of the indutial

60 38 application. It i known a a cala contol and act impoing a contant elation between voltage and fequency. The tuctue i vey imple and it i nomally ued without peed feedback. Howeve, thi contolle doe not achieve a good accuacy in both peed and toque epone mainly due to the fact that the tato flux and the toque ae not diectly contolled. Even though, a long a the paamete ae identified, the accuacy in the peed can be 2% (except in a vey low peed) and dynamic epone can be appoximately aound 50m.[3][4][33] Vecto contol. Accoding to thee type of contolle, thee ae contol loop fo contolling both the toque and the flux.[5] The mot widepead contolle ae the one that ue vecto tanfom uch a eithe Pak o Ku. It accuacy can each value uch a 0.5% egading the peed and 2% egading the toque, even in tand till. The main diadvantage ae the huge computational capability equied and the compuloy good identification of moto paamete.[6][33]

61 Field Acceleation method. The equation ued in thi method can be implified avoiding the vecto tanfomation. It i achieved ome computational eduction, ovecoming the main poblem in the vecto contolle and then becoming an impotant altenative fo the vecto contolle.[10][6][8] Diect Toque Contol. In diect toque contol (DTC), it i poible to contol diectly the tato flux and the toque by electing the appopiate invete witching tate [12][11]. It main featue ae a follow: Diect toque contol and diect tato flux contol. Indiect contol of tato cuent and voltage. Appoximately inuoidal tato fluxe and tato cuent High dynamic pefomance even at locked oto. Thi method peent the following advantage: Abence of co-odinate tanfom. Abence of mechanical tanduce.

62 40 Cuent egulato, PWM pule geneation, PI contol of flux and toque and co-odinate tanfomation ae not equied. Vey imple contol cheme and low computation time. Reduced paamete enitivity. Vey good dynamic popetie. Although, ome diadvantage ae peent: High toque ipple and cuent ditotion Low witching fequency of tanito with elation to computation time Contant eo between efeence and eal toque 3.2 PRINCIPLES OF DIRECT TORQUE CONTROL Intoduction A it ha been intoduced in expeion 1.81, the electomagnetic toque in the thee-phae induction machine can be expeed a follow [11][13]:

63 3 uu u t e = PΨ x i 2 (3.1) whee Ψ uu i the tato flux, i i the tato cuent(both fixed to the tationay efeence fame fixed to the tato) and P the numbe of pai of pole. The peviou equation can be modified and expeed a follow: 41 t 3 uuu u = P Ψ. i.in(α - ρ ) 2 e (3.2) whee ρ i the tato flux angle and α i the tato cuent one, both efeed to the hoizontal axi of the tationay fame fixed to the tato. If the tato flux modulu i kept contant and the angleρ i changed quick ly, then the electomagnetic toque i diectly contolled. The ame concluion can be made by uing anothe expeion fo the electomagnetic toque. Fom equation 1.83, next equation can be witten a: 3 L uuu uuu m ' t = P Ψ 2 x Ψ.in(ρ - ρ ) e 2 L L L m (3.3) The fact that the oto flux can be aumed contant i tue a long a the epone time of the contol i much fate than the oto time contant. A long a the tato flux modulu i kept contant, then the electomagnetic toque can be apidly changed and contolled by mean of changing the angle (ρ - ρ ).[7][14]

64 DTC Contolle The way to impoe the equie tato flux i by mean of chooing the mot uitable Voltage Souce Invete tate. If the ohmic dop ae neglected fo implicity, then the tato voltage impee diectly the tato flux in accodance with the following equation: uuu d dt o Ψ = u uuu Ψ = u t (3.4) (3.5) Decoupled contol of the tato flux modulu and the toque i achieved by acting on the adial and tangential component epectively of the tato flux-linkage pace vecto in it locu. Thee two component ae diectly popotional (R =0) to the component of the ame voltage pace vecto in the ame diection. Figue 3.1 how the poible dynamic locu of the tato flux, and it diffeent vaiation depending on the VSI tate choen. The poible global locu i divided into ix diffeent ecto ignaled by the dicontinuou line.

65 43 Figue 3.1: Stato flux vecto locu and diffeent poible witching voltage vecto. FD: flux deceae. FI : flux inceae. TD: toque deceae. TI: toque inceae. In accodance with Figue 3.1, the geneal Table 3.1 can be witten. It can be een fom Table 3.1 that the tate V k and V k+3, ae not conideed in the toque becaue they can both inceae (fit 30 degee) o deceae (econd 30 degee) the toque at the ame ecto depending on the tato flux poition.[12] VOLTAGE VECTOR INCREASE DECREASE Stato Flux V k, V k+1, V k-1 V k+2, V k-2, V k+3 Toque V k+1, V k+2 V k-1, V k-2 Table 3.1: Geneal Selection Table fo Diect Toque Contol, being k the ecto numbe. Finally, the DTC claical look up table i a follow:

66 44 Φ τ S 1 S 2 S 3 S 4 S 5 S 6 TI V V V V V V FI T= V 2 V 3 V 4 V 5 V 6 V 1 TD V 0 V 7 V 0 V 7 V 0 V 7 TI V 3 V 4 V 5 V 6 V 1 V 2 FD T= V 7 V 0 V 7 V 0 V 7 V 0 TD V 5 V 6 V 1 V 2 V 3 V 4 Table 3.2: Look up table fo Diect Toque Contol. FD/FI: flux deceae/inceae. TD/=/: toque deceae/equal/inceae. S x : tato flux ecto. Φ : tato flux modulu eo afte the hyteei block.τ : toque eo afte the hyteei block. The ecto of the tato flux pace vecto ae denoted fom S 1 to S 6. Stato flux modulu eo afte the hyteei block ( Φ ) can take jut two value. Toque eo afte the hyteei block (τ ) can take thee diffeent value. The zeo voltage vecto V 0 and V 7 ae elected when the toque eo i within the given hyteei limit, and mut emain unchanged.

67 DTC Schematic. In Figue 3.2, a poible chematic of the Diect Toque Contol i hown. A it can be een, thee ae two diffeent loop coeponding to the magnitude of the tato flux and toque. The efeence value fo the flux tato modulu and the toque ae compaed with the actual value, and the eulting eo value ae fed into the two-level and thee-level hyteei block epectively. The output of the tato flux eo and the toque eo hyteei block, togethe with the poition of the tato flux ae ued a input of the look up table (ee table 3.2). The poition of the tato flux i divided into ix diffeent ecto. In accodance with the Figue 3.2, the tato flux modulu and toque eo tend to be eticted within it epective hyteei band. It can be poved that the flux hyteei band can affect to the tato cuent ditotion in tem of low ode hamonic and the toque hyteei ban affect the witching fequency.[7] The DTC equie the flux and toque etimation, which can be pefomed a it i popoed in Figue 3.2 chematic, by mean of two diffeent phae cuent and the tate of the invete.

68 46 Figue 3.2: Diect Toque Contol Schematic. Howeve, flux and toque etimation can be pefomed uing othe magnitude uch a two tato cuent and the mechanical peed, o two tato cuent again and the haft poition.[14][12] Stato flux and toque etimato uing ω m, i A and i B magnitude. Thi etimato doe not equie co-odinate tanfom. It i ued the moto model fixed to the tationay efeence fame fixed to the tato. Fitly, all thee-phae cuent mut be conveted into it D and Q component. By mean of the Pak tanfomation defined in equation 1.14, it can be aumed: i D = c.1.5.i A

69 47 i = c. 3/2.(2.i + i ) (3.6) Q B A If oto cuent i extacted fom equation 2.65 and ubtituted into equation 2.66, it can be aid: L L Pω m Ψ'.(1+p. j )=L i R R (3.7) m And if the expeion 3.7 i aanged: Ψ'.(R +p.l )=L R.i +j. Ψ'.L.P.ω (3.8) m m Expanding the peviou equation into it D and Q component, it obtained: Ψ'.(R +p.l )=L R.i +j. Ψ '.L.P.ω (3.9) D m D Q m Ψ'.(R +p.l )=L R.i +j. Ψ '.L.P.ω (3.10) Q m Q D m And taking into account that thi expeion will be evaluated in a compute it hould be expeed in z opeato. Theefoe doing the z tanfom of equation 3.9 and 3.10 the following equation ae obtained: R - Tz L R - T z L 1-e Ψ' D. z-e = ( LmR.i D +j.ψ' Q.L.P.ω R m ) (3.11) R - Tz L R - T z L 1-e Ψ' Q. z-e = ( LmR.i Q +j.ψ' D.L.P.ω R m ) (3.12) And in time vaiable:

70 48 R R - Tz L 1 - e Ψ' k =Ψ' k - 1.e + i (k-1).p.ψ (k-1).wm(k-1) - Tz L ( ) ( ) ( L R. +L ' ) D D m D Q R R R - Tz L 1 - e Ψ' k =Ψ' k - 1.e + i (k-1).p.ψ (k-1).wm(k-1) - Tz L ( ) ( ) ( L R. +L ' ) Q Q m Q D R (3.13) (3.14) Finally, tato flux can be obtained a follow: L Ψ k =i k +Ψ' k ( ) ( ) ( ) D D D Lm L L m (3.15) L Ψ k =i k +Ψ' k ( ) ( ) ( ) Q Q Q Lm L L m (3.16) Toque i obtained uing equation Stato flux and toque etimato uing V dc, i A, i B magnitude. In cae that eno-le diect toque contol i deied, neithe oto peed no oto poition ae available. In ode to obtain an etimation of the tato flux pace vecto, two poible method may be applied: An etimation that doe not equie peed o poition ignal may be ued. The moto peed may be etimated and fed into a flux etimato.

71 49 Stato flux and toque etimation baed on the tato voltage equation doe not equie peed o poition infomation when tationay co-odinate ae applied. Thu, fom the VSI tate and having the intantaneou value of the V dc, it can be deduced the voltage in each phae. Once the voltage and the cuent value ae calculated and meaued epectively, they ae tanfomed in D and Q component by mean of Pak tanfomation. Finally fom equation 2.64 it can be aid: ( ) ψ = u -R.i dt (3.17) and expeing thi equation in z opeato by mean of the z tanfom: -1 z ( ) 1-z ψ =.T. u - R.i (3.18) -1 Expeing the peviou equation in time and in it D and Q component: ΨD ( k) -ΨD ( k - 1 ) = T. u D (k-1) - T.R.i D(k-1) ΨQ ( k) -ΨQ ( k - 1 ) = T. u Q (k-1) - T.R.i Q(k-1) (3.19) It may be deduced that the tato voltage pace vecto component ae deived fom the invete intenal witch etting. Thi fact avoid the meauement of the tato voltage pule. In pactice, the D.C. link voltage i meaued, thu the D and Q component of the tato voltage pace phao can be deived. It hould be noted that a co-odinate tanfom i not equied. Howeve, the accuacy of the etimation i limited due to the open loop integation that can lead to lage flux etimation eo.[12]