MATHEMATICAL MODELING OF INDUCTION MOTORS

37 CHAPTER 3 MATHEMATICAL MODELING OF INDUCTION MOTORS To tart with, a well-known technique called the SVPWM technique i dicued a thi form the bai of the mathematical modeling of IM. Furthermore, the d q dynamic model [7] uing Park tranformation, Kron dynamic model and the Stanley dynamic model i alo preented. The mathematical model preented in thi chapter i further ued to deign variable peed IM drive with variou type of controller, uch a the PI controller, the Mamdani-Fuzzy controller, the Takagi-Sugeno FL controller and the ANFIS controller, which are dicued in further chapter. 3.1 INTRODUCTION Deign of a controller i baed on tranfer function, which i further ued to control any parameter of the ytem, e.g. peed, torque, flux, etc. The mathematical model can be obtained by variou method, viz., from firt principle, ytem identification method, etc. Thi mathematical model may be a linear/non-linear differential equation or a tranfer function (in - or z-domain) or in tate pace form [5]. In general, the mathematical model of any IM can be modeled by variou method, viz., pace vector phae theory or the two-axi theory of electrical machine [70]. The model ued in our work

38 conit of SVPWM voltage ource inverter, IM, direct flux, torque peed control [16]. Main drawback of the coupling effect in the control of SCIM i that it give a highly over damped plant repone, thu making the ytem very low and luggih. Moreover, ince the order of the ytem i very high, the ytem uffer from intability a for thi higher order ytem, a controller of larger order hould be developed, which increae the implementation cot. The previouly mentioned problem can be olved by making ue of either vector control/field-oriented control or uing any other type of control method. Thee type of control trategie can control an IM like DC machine in eparately excited condition [7]. Of coure, the control of AC drive can exhibit better performance. Thu, due to the above-mentioned reaon, an IM model wa developed uing the concept of rotating d q field reference frame concept [16]. The power Fig 3.1: Power circuit connection diagram for the IM

39 Table 3.1: SCIM pecification HP 50 Speed Voltage Frequency 1800 rpm 460 V 50 Hz Phae 3 Pole 2 Type Squirrel Cage Type IM The pecification of the Squirrel Cage Induction Motor (SCIM) ued in our work, a in Table 3.1. The motor mathematical model, certain aumption have been conidered. The motor i ymmetrical 2-pole having 3-phae winding. The lotting effect and the iron loe are neglected. Note that in the deign and implementation of AC drive, a dynamic model of the IM i required to deign a controller to control the variou parameter. A very good control require an approximate model of the real time ytem. All dynamic effect of tranient/teady tate condition are conidered in the model. Furthermore, the dynamic model ued in the reearch work i valid for any change in the inverter' upply 3.2 REVIEW OF THE SVPWM TECHNIQUE Pule wih modulation (PWM) i ued in variou power electronic application. SVPWM i a popular technique ued in control of AC

40 motor drive. The triangular carrier wave i modulated by ine wave and the witching point of inverter are determined by the point of interection. The main application of PWM i in power electronic, for example in motor control. When generating analogue ignal, the diadvantage i that the PWM reolution rapidly decreae with the required ignal bandwih. Hence, to avoid the drawback of thi method, SVPWM could be ued for more ophiticated control of AC motor drive, a decribed in the further ub-ection. 3.2.1 Operational Principle of PWM Fig. - tapped grounded DC bu along with the principle of operation of PWM. Fig. 3.2: The ingle-phae circuit model of inverter The inverter output voltage i hown in Fig. 3.3 and i obtained a follow: 1. When the control voltage v cont i > V tr, V VAO 2 DC

2. When the control voltage 41 v i < V cont tr, V VAO 2 DC Fig. 3.3: PWM graphical concept of feature: PWM frequency i the ame a that of the frequency of V tr, the amplitude i controlled by the peak value of the voltage of v contr and the fundamental frequency i controlled by the frequency of v contr. Furthermore, the modulation index m of the inverter i given by V Peak value of VAO m contr (3.1) V / 2 tr V DC where (VA0 )1 i the fundamental frequency component of VA0. 3.2.2 Generation of Space Vector in PWM Control diagrammatically in Fig. 3.4. Here, S 1 to S 6 are 6 power witche that will determine the hape of the output and are controlled by the witching variable a, a, b, b, c and c repectively. When one of the

42 tranitor in the upper half of the inverter i in the on condition, i.e., when a, b or c i in logic 1 tate, the correponding tranitor in lower half of the inverter will be in off condition (logic 0 tate). Thu, the on and off tate of the tranitor S1, S3, S5 in the upper half of the inverter can be ued to determine the output voltage of the inverter. inverter The line-line (phae phae) voltage and line-neutral voltage relationhip can be expreed in the form of a matrix. The firt relationhip between witching variable vector and line-to-line voltage vector i a follow: Vab 1 1 0 a V bc V DC 0 1 1 b V ca 1 0 1 c (3.2) Furthermore, the relationhip between the witching variable vector and the phae voltage vector i given a follow:

43 Van 2 1 1 a VDC V bn 1 2 1 b 3 V cn 1 1 2 c (3.3) The eight poible combination of the on and off pattern for the 3 witche in the upper part of the PWM inverter i a in Fig. 3.4. The on and off tate of tranitor in lower part of inverter bridge are oppoite to that of the witche in the upper part of the inverter bridge. According to Equation (3.2) and (3.3), the 8 witching vector, output line to neutral voltage and the output line to line voltage in term of VDC are given in Table 3.2. Table 3.2: Switching vector with their voltage level Voltage Switching vector VLN VLL Vector a b c Van Vbn Vcn Vab Vbc Vca V0 0 0 0 0 0 0 0 0 0 V1 1 0 0 2-1 3 3-1 3 1 0-1 V2 1 1 0 1 3 V3 0 1 0-1 3 V4 0 1 1-2 3 1-2 3 3 0 1-1 2-1 3 3-1 1 0 1 3 V5 0 0 1-1 3-1 3 V6 1 0 1 1-2 3 3 1 3-1 0 1 2 3 0-1 1 1 3 1-1 0 V7 1 1 1 0 0 0 0 0 0 SVPWM generate le harmonic ditortion in the output voltage and/or current, which are applied to the three phae of an IM. Alo,

44 it provide more efficient ue of the upply voltage to the IM compared with the other method. To implement the SVPWM, the output voltage equation in abc reference frame are tranformed into the tationary d q reference frame that conit of 2 axe: the d-axi and the q-axe a hown in Fig. 3.5. Fig. 3.5: The relationhip b/w abc ref and tationary d q frame The relationhip between the two reference frame can be obtained a follow: f K f (3.4) dqo abc where 1 1 1 2 2 fd fa 2 0 3 3 2 2, dq0 q, abc b K 3 f f f f 1 1 1 2 2 2 f 0 f c (3.5) Here, f i conidered a current variable or a voltage variable. From Fig. 3.5, the tranformation given in Equation (3.5) can be conidered equivalent to an orthogonal projection of the vector [a b c] T onto the 2- dimenional perpendicular unit vector [1 1 1] T, which i in the

45 equivalent d q plane in a V 6 3-dimenional ytem. Becaue of thi tranformation, 6 non-zero vector and 2 zero vector could be generated, which in turn reult in 8 permiible witching tate. The 6 non-zero vector V 1 to V 6 which can be hown a the axe of a circle a in Fig. 3.6. Thu, thi feed the electric power to the load. The angle between any two adjacent non-zero vector i 60 0. The two zero vector V1 to V6 are placed at the origin of the hexagonal (circle) grid of vector. Thee eight baic vector in the grid are denoted by V0, V1, V2, V3, V4, V5, V6, and V7. The operator tranformation given in Equation (3.5) i ued to obtain the deired reference voltage Vref in the d q plane. The main objective of SVPWM technique ued to develop the mathematical model of the IM i to approximate the reference voltage vector Vref uing the eight poible witching tate. In general, the SVPWM concept can be implemented in 3 tep: Step 1: Find the voltage Vd, Vq, Vref and the angle Step 2: Find the time period (duration) T1, T2 and T0. Step 3: Find the witching time of each power tranitor (1 to 6). The eight permiible witching tate are hown in Fig. 3.6.

46 Fig. 3.6: Diagrammatic repreentation of the equence of the pace vector 3.3 THE DYNAMIC D-Q MODEL USING PARK S TRANSFORMATION In the previou modeling method, per phae equivalent circuit of the machine wa conidered. Thi wa valid only in the teady-tate condition. The equivalent circuit that wa ued for obtaining the mathematical model of the IM i hown in Fig. 3.7 and 3.8, repectively [7]. Fig. 3.7: IM Equivalent circuit in the d-axi frame

47 Fig. 3.8: IM Equivalent circuit in the q-axi frame In the current method of modeling and control of SCIM, epecially one with an adjutable AC drive, the machine hould have certain feedback loop. Therefore, the tranient behavior hould be taken into conideration in the machine model. Furthermore, highperformance drive control require a better undertanding of the vector/field-oriented control. Thi ection provide a better undertanding of the concept relating to the development of the d q theory. In the IM, the 3-phae rotor winding move with repect to the 3-phae tator winding. Generally, any machine model could be bet decribed by a et of non-linear differential equation with timevarying mutual inductance. However, uch a model tend to be very complex and the controller deign become further complex [7]. The mathematical model of the IM (the 3-phae IM) could be repreented by an equivalent 2-phae, where d, q, d r and q r correpond to the tator, rotor, direct and quadrature axe, repectively. Although thi model look quite imple, the problem of time-varying parameter till remain. Hence, to olve thi problem, R. H. Park, in 1920, developed the tranformation technique to olve the

48 problem of time-varying parameter. The tator variable are referred with repect to the rotor reference frame, which rotate at a ynchronou peed. The time-varying inductance that occur due to the interaction between the electric and magnetic circuit can be removed uing thi Park tranformation. Later, in 1930, H.C. Stanley demontrated that the time-varying inductance that appear in the v i equation of the IM due to electric and magnetic effect can be removed by tranforming the variable with repect to the fictitiou tationary winding. In thi cae, the rotor variable are tranformed to the tator reference frame. In thi thei, a dynamic machine model in ynchronouly rotating and tationary reference frame i preented, which i further ued to develop ophiticated controller to control the peed of the IM. The tator voltage equation formulated from tationary reference frame [7] are a follow: d A ( ) ( ) () t VA t RiA t (3.6) d B ( ) ( ) () t VB t RiB t (3.7) d C ( ) ( ) () t VC t RiC t (3.8)

49 The rotor voltage equation formulated to the rotating frame fixed to the rotor are a follow [7]: d ra () t Va ( t) Rrira ( t) (3.9) d rb () t Vb ( t) Rrirb ( t) (3.10) d rc () t Vc ( t) Rrirc ( t) (3.11) where the flux linkage related to the tator and rotor winding are given a [7] L i M i M i M co i A A B C r m ra 2 co 4 M co i M i 3 3 r m rb r m rc co 4 3 co 2 M i L i M i M i B A B C r m ra M co i M i 3 r m rb r m rc co 2 3 4 co M i M i L i M i C A B C r m ra M co i M i 3 r m rb r m rc co 2 3 co 4 M co i M i ra r m A r m B M i L i M i M i 3 r m C r ra r rb r rc (3.12) (3.13) (3.14) (3.15)

50 4 co 3 co 2 M co i M i rb r m A r m B M i M i L i M i 3 r m C r ra r rb r rc (3.16) 2 co 4 M co i M i 3 3 rc r m A r m B M co i M i L i M i r m C r ra r rb r rc (3.17) Note that L, M and i are, repectively, the elf-inductance, mutual inductance and the current referred to the tator and rotor winding. Subtituting Equation (3.12) (3.17) in Equation (3.6) (3.11) and further implifying, the equation for the tator and rotor can be written in the vector-matrix notation form a follow: R pl pm pm VA pm R pl pm V B pm pm R pl V C pm r com pm r com 1 pm r com2 Vra pm r co m2 pm r co m pm r V com 1 rb V pm r com 1 pm r com2 pm r co rc m pm r com pm r com 1 pm r co m2 ia pm r com2 pm r com pm r com 1 i B pm r com 1 pm r com2 pm r com ic Rr plr pm r pm r ira pm i r R r r rb r pl pm irc pm r pm r R pl r r (3.18)

51 The 3-phae tator and rotor voltage equation written in vectormatrix can be further tranformed into 2-phae tator and rotor voltage equation uing the well-known Park' tranformation. To obtain thi, a 3-phae SCIM with tationary axi a-b-c 120 0 apart i conidered. The 3-phae tationary reference frame variable a-b-c are tranformed into 2-phae tationary reference frame variable (d q ). Furthermore, thee 2-phae variable are tranformed into ynchronouly rotating reference frame variable (d e q e ) and vicevera. Let u aume that (d S q ) axe are oriented at an angle of θ. The direct axi voltage v d and quadrature axi voltage v q are further reolved into another type of component, viz., a-b-c, and finally, writing them in the vector-matrix notation form, we obtain co in 1 va vq 0 0 v b co 120 in 120 1 vd v c 0 0 co 120 in 120 1 vo (3.19) Taking the invere of the above, we obtain the following: 0 0 0 0 co co 120 co 120 v q 2 va v d in in 120 in 120 v b 3 v o 0.5 0.5 0.5 v c (3.20) where v o i added a the zero equence component, which may or may not be preent. Note that in the above equation, voltage wa

52 conidered a the variable. Similarly, the current and flux linkage equation can alo be tranformed into imilar equation. Note that if θ i et to zero, the q -axi will be aligned with the a-axi. Once the zero equence component are ignored, the tranformation equation can be implified a v a v (3.21) q 1 3 vb vq vd (3.22) 2 2 1 3 vc vq vd (3.23) 2 2 The invere equation are obtained a 2 1 1 vq va vb vc va (3.24) 3 3 3 1 1 vd vb vc (3.25) 3 3 The ynchronouly rotating d e q e axe rotate at ynchronou peed ωe with repect to the d q axe and the angle θe i equal to ωe. The 2-phae d q winding are tranformed into the hypothetical winding mounted on the d e q e axe. The d q axe voltage can be converted or reolved into the d e q e frame a follow: v v co v in (3.26) q q e d e

53 v v in v co (3.27) d q e d e The upercript e ha been dropped henceforth from the ynchronouly rotating frame parameter. Reolving the rotating frame parameter into a tationary frame, Equation (3.26) and (3.27) can be written a v v co v in (3.28) q q e d e v v in v co (3.29) d q e d e Let u aume that the 3-phae tator voltage are balanced and are given by v V co( t ) (3.30) a m e v co( 2 b Vm et ) (3.31) 3 v co( 2 c Vm et ) (3.32) 3 Subtituting equation (3.30) (3.32) in (3.24) (3.25) yield v V co( t ) (3.33) q m e v V in( t ) (3.34) d m e Subtituting equation (3.26) (3.27) in (3.33) (3.34) yield v V co (3.35) q m

54 v V in (3.36) d m Equation (3.33) and (3.34) how that vq and vd are balanced 2- phae voltage of equal peak value and the latter i at 90 0 angle phae lead with repect to the other component. Equation (3.35) and (3.36) verify that inuoidal variable in a tationary frame appear a DC quantitie in a ynchronouly rotating reference frame. Note that the tator variable are not necearily balanced inuoidal wave; in fact, they can be any arbitrary time function alo. The variable in a reference frame can be combined and repreented in a complex pace vector or phae a V v v jv qd q d, in Vm co et j et, ^ j j et V e e, m j 2 Ve r. (3.37) which indicate that the vector V rotate counter-clockwie at a peed of ωe from the initial (t = 0) angle of to the q e -axi. Equation (3.37) alo indicate that for a inuoidal variable, the vector magnitude i the peak value V ^ which i m 2 time the rm phae magnitude of V. The q e d e component can alo be combined into a vector form a follow:

55 v v jv qd q d, vq coe vd ine j vq ine vdco e je vq jvd e, Ve je. (3.38) Equation (3.38) can further be written uing invere notation a j e V v jv v jv e (3.39) q d q d From the concept of vector algebra, the vector magnitude in the tationary and rotating frame will be equal. In other word, Equation (3.39) can be re-written a ^ 2 2 2 2 V V v v v v (3.40) m q d q d The factor e jθ can be interpreted a a vector rotational operator (defined a a vector rotator or unit vector) that convert the rotating frame variable into tationary frame variable. coθe and inθe are the Carteian component of the unit vector. In Equation (3.38), e -jθ i defined a the invere vector rotator that convert the d q variable into d e q e variable. Vector V and it component are further projected on the rotating and tationary axe. The a-b-c variable can alo be expreed in the vector form. Subtituting equation (3.24) and (3.25) into Equation (3.37), we get

56 V v jv q d 2 1 1 1 1 va vb vc j vb vc 3 3 3 3 3 2 1 3 1 3 v 3 j v j v 2 2 2 2 2 2 va avb a v c 3 a b c (3.41) where 2 2 j j 3 2 3 a e & a e the two parameter a and a 2 are interpreted a unit vector aligned to the repective b and c axe of the machine. The reference axi correpond to the va-axi. Note that all the above tranformation done for the tator variable can alo be carried out for the rotor circuit variable. 3.4 DYNAMIC KRON EQUATION MACHINE MODEL For a 2-phae machine, we need to repreent both d q and dr qr circuit and their variable in a ynchronouly rotating d e q e frame. The tator circuit equation can be modeled a follow: v d R i (3.42) q q q v d R i (3.43) d d d where q and d are the q-axi and d-axi tator flux linkage, repectively. Equation (3.42) and (3.43) are further converted into d e q e frame a

v 57 d R i (3.44) q q q e d v d R i (3.45) d d d e q Note that in the above equation, all the variable are in rotating form. The lat term in equation (3.44) and (3.45) are the peed emf due to the rotation of the axe, i.e., when ωe=0, the equation revert back to the tationary form. The flux linkage in the d e and q e axe induce emf in the q e and d e axe with a leading angle of 90 0. If the rotor i not moving, i.e., ωr=0, then the rotor equation for a doubly fed wound rotor IM will be imilar to the tator equation and are given by v d (3.46) qr qr Rr iqr e dr v d R i (3.47) dr dr r dr e qr where all the variable and parameter are referred to the tator. Since the rotor move at a peed of ωr = 0, the d q axe fixed on the rotor move at a peed of ωe ωr relative to the ynchronouly rotating frame. Therefore, in d e q e frame, the rotor equation can be rewritten a follow: v d (3.48) qr qr Rriqr e r dr

v 58 d R i (3.49) dr dr r dr e r qr The flux linkage expreion in term of the current can be written imilar to that in equation (3.50) (3.55).Uing equation (3.50) (3.55) in equation (3.44) (3.49), the electrical tranient model of the IM in term of v and i i given in matrix form a in Equation (3.56); in the invere form, the i v matrix could be written a in Equation (3.57). q l q m q qr L i L i i (3.50) qr lr qr m q qr L i L i i (3.51) qm m q qr L i i (3.52) d l d m d dr L i L i i (3.53) dr lr dr m d dr L i L i i (3.54) dm m d dr L i i (3.55) R L L L L e m g m vq iq e L R L e Lm L m v dr i d Lm e r Lm Rr Lr e r Lr vqr iqr e r Lm e r Lr Rr L r v dr i dr (3.56) Or

59 1 R L e L Lm g Lm iq vq e L R L e Lm L m i d v dr Lm e r Lm Rr Lr e r Lr iqr vqr e r Lm e r Lr Rr L r i dr v dr (3.57) where i the Laplacian operator. For a ingle phae IM, uch a the cage motor, vqr = vd = 0. If the peed ωr i conidered to be contant (infinite inertia load), the electrical dynamic of the IM are given by a fourth-order linear ytem. Then, by deriving the input vq, vd and ωe, the current iq, id, iqr and idr can be olved from Equation (3.56). Note that the peed ωr cannot be treated a a contant and i related to the torque a d 2 dr Te TL J m TL J (3.58) p where TL i the load torque, J i the rotor inertia and ωm i the mechanical peed of the IM. In order to obtain a compact repreentation of the IM model, the equivalent circuit are expreed in the complex form. Multiplying Equation (3.45) by -j and adding it to Equation (3.44) give d vq jvd R iq jid q jd je q j d (3.59) d vqd Riqd qd je qd (3.60)

60 where vqd, iqd, etc. are the complex vector. Similar to the previou equation, the rotor Equation (3.48) (3.49) can be combined to repreent the rotor equation a d vqdr Rr iqdr qdr j e r qdr (3.61) Note that the teady-tate equation can alway be derived by ubtituting the time derivative component to zero. The final teadytate equation after making certain aumption can be obtained a v R I je (3.62) R 0 R r j e r (3.63) r where the complex vector have been ubtituted by the correponding rm phaor. Thee equation atify the teady-tate equivalent circuit hown in Fig. 3.7 and 3.8, repectively, if the parameter Rm i neglected. The development of torque i alo very important in the modeling of IM. Here, it will be expreed in a more general form, relating the d q component of the variable. From Equation (3.58), the torque can be generally expreed in the vector form a T e 3 P m I 2 2 r (3.64) Reolving the variable into d e q e component, we obtain

61 3 P T i i 2 2 e dm qr qm dr (3.65) Several other torque expreion can be derived from the above torque relation a follow: 3 P T i i 2 2 e dm q qm d (3.66) 3 P T i i 2 2 e d q q d (3.67) 3 P T L i i i i 2 2 e m q dr d qr (3.68) 3 P T i i 2 2 e dr qr qr dr (3.69) Equation (3.57), (3.58) and (3.65) give the complete mathematical model of the electro-mechanical dynamic of an IM in the ynchronou frame. The compoite ytem i of the fifth order and non-linearity of the model i evident. 3.5 DYNAMIC STANLEY EQUATION MACHINE MODEL The dynamic machine model in tationary frame can be derived imply by ubtituting ωe= 0 in Equation (3.57) or in Equation (3.44), (3.45), (3.48) and (3.49). The correponding tationary frame equation are given a follow:

v v 62 d R i (3.70) q q q d R i (3.71) d d d d 0 Ri r qr qr r dr (3.72) d 0 Ri r dr dr r qr. (3.73) where vqr = vdr = 0. The torque Equation (3.64) (3.69) can alo be written with the correponding variable in the tationary frame a follow: 3 P T i i 2 2 e dm qr qm dr (3.74) 3 P T i i 2 2 e dm q qm d (3.75) 3 P T i i 2 2 e d q q d (3.76) 3 P T L i i i i 2 2 e m q dr d qr (3.77) 3 P T i i 2 2 e dr qr qr dr (3.78) which give rie to the Stanley machine model Equation (3.42) (3.43) and (3.72)-(3.73) can eaily combined to derive the complex model a

v 63 d R i (3.79) qd qd qd d 0 Rr iqd qdr jr qd (3.80) where v qd = v q jv d, ψ qd = ψ q jψ d, i qd = i q ji d and ψ qdr = ψ qr jψ dr. The complex equivalent circuit in tationary frame can alo be developed further and i not hown here for the ake of convenience. Often, a per-phae equivalent circuit with r r and inuoidal variable can be obtained, which lead to Fig. 3.7 and 3.8 by omitting the parameter Lm of all the variou model explained in the different ection o far, the equation of the dynamic d-q model depicted in ection 3.3 i ued in thi reearch work.