dynamic model
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MEP 1522 ELECTRIC DRIVES Dynamic Model of Induction Machine WHY NEED DYNAMIC MODEL? • In an electric drive system, the machine is part of the control system elements • To be able to control the dynamics of the drive system, dynamic behavior of the machine need to be considered • Dynamic behavior of of IM can be described using dynamic model of IM WHY NEED DYNAMIC MODEL? • Dynamic model – complex due to magnetic coupling between stator phases and rotor phases • Coupling coefficients vary with rotor position – rotor position vary with time • Dynamic behavior of IM can be described by differential equations with time varying coefficients DYNAMIC MODEL, 3-PHASE MODEL Magnetic axis of phase B ibs a b’ c’ Magnetic axis of phase A ias b c ics Magnetic axis of phase C a’ Simplified equivalent stator winding DYNAMIC MODEL – 3-phase model stator, b rotor, a rotor, b r stator, a rotor, c stator, c DYNAMIC MODEL – 3-phase model Let’s look at phase a Flux that links phase a is caused by: • Flux produced by winding a • Flux produced by winding b • Flux produced by winding c DYNAMIC MODEL – 3-phase model Let’s look at phase a • Flux produced by winding b The relation between the currents in other phases • Flux produced by winding c and the flux produced by these currents that linked phase a are related by mutual inductances DYNAMIC MODEL – 3-phase model Let’s look at phase a as as,s as,r Lasias Labsibs Lacs ics Mutual inductance between phase a and phase b of stator Las,ariar Las,bribr Las,cr icr Mutual inductance Mutual inductance Mutual inductance Mutual inductance between phase a of stator between phase a and phase between phase a of stator a of stator and phase a ofbetween rotor phase c of stator phase c of rotor and phase b ofand rotor DYNAMIC MODEL – vabcs = Rsiabcs + d(abcs)/dt 3-phase model - stator voltage equation vabcr = Rrriabcr + d(abcr)/dt v abcs v as v bs v cs v abcr v ar v br v cr • • i abcs i as i bs i cs i abcr i ar i br i cr - rotor voltage equation abcs as bs cs abcr ar br cr abcs flux (caused by stator and rotor currents) that links stator windings abcr flux (caused by stator and rotor currents) that links rotor windings DYNAMIC MODEL – 3-phase model abcs abcs,s abcs,r abcr abcr,r abcr,s Flux linking stator winding due to stator current abcs ,s L as L abs L acs L abs L bs L bcs L acs i as L bcs i bs L cs i cs Flux linking stator winding due to rotor current abcs , r L as ,ar L bs,ar L cs ,ar L as , br L bs, br L cs , br L as ,cr i ar L bs,cr i br L cs ,cr i cr DYNAMIC MODEL – 3-phase model Similarly we can write flux linking rotor windings caused by rotor and stator currents: Flux linking rotor winding due to rotor current abcr , r L ar L abr L acr L abr L br L bcr L acr i ar L bcr i br L cr i cr Flux linking rotor winding due to stator current abcr ,s L ar ,as L br,as L cr ,as L ar , bs L br, bs L cr , bs L ar ,cs i as L br,cs i bs L cr ,cs i cs DYNAMIC MODEL – 3-phase model Combining the stator and rotor voltage equations, é ê ê ê ê ê ê ê ê ë é vas ù ê Rs + pLas ú vbs ú ê pLabs ú ê vcs ú ê pLacs =ê ú var pLar,as ê ú vbr ú ê pLbr,as ê vbc úû ê pLcr,as ë pLabs pLacs pLas,ar pLas,br pLas,cr Rs + pLbs pLbcs pLbs,ar pLbs,br pLbs,cr pLbcs Rs + pLcs pLcs,ar pLcs,br pLcs,cr pLar,bs pLar,bs Rr + pLar pLar,br pLar,cr pLbr,bs pLbr,cs pLbr,ar Rr + pLbr pLbr,cr pLcr,bs pLcr,cs pLcr,ar pLcr,br Rr + pLcr p is the derivative operator , i.e. p = d/dt ùé úê úê úê úê úê úê úê úê úûë ias ù ú ibs ú ú ics ú iar ú ú ibr ú icr úû DYNAMIC MODEL – 3-phase model Magnetic axis of winding y y x y’ Magnetic axis of winding x Windings x and y, with Nx and Ny number of turns, separated by α x’ It can be shown that the mutual inductance between winding x and y is æ rl öæ p ö Lxy = = mo N x N y ç ÷ç ÷ cos a Ix è g øè 4 ø y xy DYNAMIC MODEL – 3-phase model æ rl öæ p ö Lxy = = mo N x N y ç ÷ç ÷ cos a Ix è g øè 4 ø y xy • To get self inductance for phase a of the stator let Nx = Ny = Ns and α = 0 æ rl öæ p ö Lam = mo N ç ÷ç ÷ è g øè 4 ø 2 s • If we consider the leakage flux, we can write the self inductance of phase a of the stator Las = Lam + Lls Las Self inductance of phase a Lam Magnetizing inductance of phase a Lls Leakage inductance of phase a DYNAMIC MODEL – 3-phase model Due to the symmetry of the windings, Lam = Lbm = Lcm , Hence Las = Lbs = Lcs = Lms + Lls • The magnetizing inductance Lms, accounts for the flux produce by the respective phases, crosses the airgap and links other windings • The leakage inductance Lls, accounts for the flux produce by the respective phases, but does not cross the airgap and links only itself DYNAMIC MODEL – 3-phase model • It can be shown that the mutual inductance between stator phases is given by: Labs Lbcs L acs 2 rl oNs L abs Lbcs L acs o cos120 g 4 2 rl oNs Lms 2 g 8 DYNAMIC MODEL – 3-phase model The mutual inductances between stator phases can be written in terms of magnetising inductances abcs ,s L ms L ls L ms 2 L ms 2 L ms 2 L ms L ls L ms 2 ias ibs ics L ls L ms 2 L ms 2 L ms DYNAMIC MODEL – 3-phase model æ rl öæ p ö Lxy = = mo N x N y ç ÷ç ÷ cos a Ix è g øè 4 ø y xy Self inductance between rotor windings is when Nx = Ny = Nr and α = 0 2 æ rl öæ p ö æ N 2 ö æ ö æ ö æ ö Lmr = mo N ç ÷ç ÷ = ç r ÷ mo N s2 ç rl ÷ç p ÷ = ç N r ÷ Lms è g øè 4 ø è N s2 ø è g øè 4 ø è N s ø 2 r • The mutual inductances between rotor phases can be written in terms of stator magnetising inductances Y abcr,r é 2 2 2 æ ö æ ö æ ö N N L N L ê r Lms + Lls - ç r ÷ ms - ç r ÷ ms ç ÷ ê èN ø è Ns ø 2 è Ns ø 2 s ê 2 2 2 ê æ N r ö Lms æ Nr ö æ N r ö Lms = ê -ç ÷ -ç ÷ ç ÷ Lms + Lls N 2 N è sø è sø è Ns ø 2 ê ê 2 2 2 æ ö æ ö æ ö N L N L N ê r ms - ç r ÷ ms ç r ÷ Lms + Lls ê -ç N ÷ 2 è sø è Ns ø 2 è Ns ø êë ù ú ú úé i úê as úê ibs úê úêë ics ú ú úû ù ú ú ú úû DYNAMIC MODEL – 3-phase model The mutual inductances between the stator and rotor windings depends on rotor position æ rl öæ p ö Lxy = = mo N x N y ç ÷ç ÷ cos a Ix è g øè 4 ø y xy Self inductance between phase a of rotor and phase a of stator is when Nx = Ns , Ny = Nr and α = θr æ Nr ö æ Nr ö æ ö æ rl öæ p ö 2 rl æ p ö Las,ar = mo N s N r ç ÷ç ÷ cosq r = ç ÷ mo N s ç ÷ç ÷ cosq r = ç ÷ Lms cosq r è g øè 4 ø è Ns ø è Ns ø è g øè 4 ø DYNAMIC MODEL – 3-phase model The mutual inductances between the stator and rotor windings depends on rotor position abcs ,r abcr ,s cos r Nr L ms cos r 2 3 Ns cos r 2 3 cos r Nr L ms cos r 2 3 Ns cos r 2 3 cos r 2 cos r cos 2 r 3 3 cos r 2 cos r cos 2 r cos r 2 3 cos r 2 3 cosr 3 3 i i br i cr cos r 2 3 cos r 2 3 cos r ar i i bs i cs as DYNAMIC MODEL – abcs ,r cos r Nr L ms cos r 2 3 Ns cos r 2 3 3-phase model cos r 2 cos r cos 2 r 3 3 cos r 2 3 cos r 2 3 cosr i i br i cr ar DYNAMIC MODEL – abcs ,r cos r Nr L ms cos r 2 3 Ns cos r 2 3 3-phase model cos r 2 cos r cos 2 r 3 3 cos r 2 3 cos r 2 3 cosr stator, b rotor, a rotor, b r stator, a rotor, c stator, c i i br i cr ar DYNAMIC MODEL, 2-PHASE MODEL • It is easier to look on dynamic of IM using two-phase model. This can be constructed from the 3-phase model using Park’s transformation Three-phase There is magnetic coupling between phases Two-phase equivalent There is NO magnetic coupling between phases DYNAMIC MODEL – 2-phase model • It is easier to look on dynamic of IM using two-phase model. This can be constructed from the 3-phase model using Parks transformation stator, b stator, q r rotor, a rotor, b r rotating rotating rotor, r rotor, stator, a r stator, d rotor, c stator, c Three-phase Two-phase equivalent DYNAMIC MODEL – 2-phase model • It is easier to look on dynamic of IM using two-phase model. This can be constructed from the 3-phase model using Parks transformation stator, q r rotating rotor, However coupling still exists between r stator and rotor, rotor windings stator, d Two-phase equivalent DYNAMIC MODEL – 2-phase model • All the 3-phase quantities have to be transformed to 2phase quantities • In general if xa, xb, and xc are the three phase quantities, the space phasor of the 3 phase systems is defined as: 2 x x a ax b a 2 x c 3 , where a = ej2/3 x x d jxq 1 1 2 2 x d Rex Re x a ax b a 2 x c x a x b x c 2 2 3 3 1 2 x b x c x q Imx Im x a ax b a 2 x c 3 3 DYNAMIC MODEL – 2-phase model • All the 3-phase quantities have to be transformed into 2-phase quantities is 2 ia aib a 2ic 3 aib 2 ia 3 is ids jiqs 2 ai b 3 q i ds d ia is 2 a ic i qs 2 2 a ic 3 is 1 1 2 2 ids Re is Re ia ai b a 2ic ia i b ic 2 2 3 3 1 2 ib ic iqs Im is Im ia ai b a 2ic 3 3 DYNAMIC MODEL – 2-phase model • The transformation is given by: i d 32 13 i 0 1 3 q 1 i o 13 3 1 3 1 3 1 3 i a i b i c idqo = Tabc iabc The inverse transform is given by: iabc = Tabc-1 idqo For isolated neutral, ia + ib + ic = 0, i.e. io =0 DYNAMIC MODEL – 2-phase model IM equations : 3-phase vabcs = Rsiabcs + d(abcs)/dt 2-phase vdq = Rsidq + d(dq)/dt vabcr = Rrriabcr + d(abr)/dt v = Rrri + d( )/dt stator, q r rotating rotor, rotor, r stator, d DYNAMIC MODEL – 2-phase model vdq = Rsidq + d(dq)/dt Express in stationary frame Express in rotating frame v = Rri + d( )/dt where, dq dqs,s dqs,r dqs ,s r ,r L dd L qd L L r ,r r ,s L dq i ds L qq i qs L ir L ir dqs,r L d L d ir i L L q r q r ,s L d L q ids i L L q qs d DYNAMIC MODEL – 2-phase model Note that: Ldq = Lqd = 0 Ldd = Lqq L = L = 0 L = L The mutual inductance between stator and rotor depends on rotor position: Ld = Ld = Lsr cos r Lq = Lq = Lsr cos r Ld = Ld = - Lsr sin r Lq = Lq = Lsr sin r DYNAMIC MODEL – 2-phase model stator, q r rotating rotor, rotor, r stator, d Ld = Ld = Lsr cos r Lq = Lq = Lsr cos r Ld = Ld = - Lsr sin r Lq = Lq = Lsr sin r DYNAMIC MODEL – 2-phase model In matrix form this an be written as: v d R s sLdd 0 vq v sLsr cosr v L sr sinr 0 R s sLdd sLsr sinr sLsr cosr sLsr cosr sLsr sinr R r sL 0 sLsr sinr isd sLsr cosr isq ir 0 R r sL ir • The mutual inductance between rotor and stator depends on rotor position DYNAMIC MODEL – 2-phase model stator, q rotor, q The mutual inductance can be made independent from rotor position by expressing both rotor and stator in stator, d rotor, d the same reference frame, e.g. in the Both stator and rotor stationary reference frame rotating or stationary r Magnetic path from stator linking the rotor winding independent of rotor position mutual inductance independent of rotor position DYNAMIC MODEL – 2-phase model How do we express rotor current in stator (stationary) frame? ir 2 ira airb a 2irc 3 ir is known as the space vector of the rotor current In rotating frame this can be written as: i r i r e jr In stationary frame it be written as: qs ir irq r r ird ds irs i r e jr ir e j ird jirq DYNAMIC MODEL – 2-phase model If the rotor quantities are referred to stator, the following can be written: v sd R s sLs v sq 0 v rd sLm v rq r L m 0 R s sLs r L m sLm sLm 0 R r ' sLr r L r 0 isd sLm isq r L r ird R r ' sLr irq Lm, Lr are the mutual and rotor self inductances referred to stator, and Rr’ is the rotor resistance referred to stator Ls = Ldd is the stator self inductance Vrd, vrq, ird, irq are the rotor voltage and current referred to stator DYNAMIC MODEL – 2-phase model It can be shown that in a reference frame rotating at g, the equation can be written as: v sd R s sLs v sq gL s v rd sLm v rq (g r )L m gL s R s sLs (g r )L m sLm sLm gL m R r ' sLr (g r )L r gL m isd sLm isq (g r )L r ird R r ' sLr irq DYNAMIC MODEL Space vectors IM can be compactly written using space vectors: g d s v sg R s isg jg sg dt sg L s isg L m i rg d rg rg L r irg L m isg 0 R r i rg dt j( g r ) rg All quantities are written in general reference frame DYNAMIC MODEL Torque equation Product of voltage and current conjugate space vectors: vs is* 2 2 v as av bs a 2 v cs i as a 2i bs ai cs 3 3 It can be shown that for ias + ibs + ics = 0, 2 Re v i v as ias v bs i bs v cs ics 3 * s s Pin = ( v as ias + v bs ibs + v cs ics ) = 3 Re[ vs is* ] 2 DYNAMIC MODEL Torque equation 3 3 3 * Pin = Re[ vs is ] = Re[ (v d + jv q )(id - jiq )] = [ v d id + vq iq ] 2 2 2 If v d v v q and 3 t Pin i v 2 i d i i q Pin = ( v as ias + v bs ibs + v cs ics ) = 3 Re[ vs is* ] 2 DYNAMIC MODEL Torque equation The IM equation can be written as: [] [] [ ] [] v = R i+ L si+ G wr i+ F w g i The input power is given by: pi = 3 t 3 i V = [ it [ R] i + it [ L]si + it [G ] w r i + it [ F] w g i] 2 2 Power Losses in winding resistance Rate of change of stored magnetic energy Power Mech power associated with g – upon expansion gives zero DYNAMIC MODEL Torque equation p mech = w m Te = v sd R s sLs v sq gL s v rd sLm v rq (g r )L m 3 t i [G ] w r i 2 gL s R s sLs (g r )L m sLm 0 0 0 r L m 0 0 r L m 0 sLm gL m R r ' sLr (g r )L r 0 0 0 r L r 0 isd 0 isq r L r ird 0 irq gL m isd sLm isq (g r )L r ird R r ' sLr irq DYNAMIC MODEL Torque equation p mech = w m Te = isd 3 isq mTe 2 ird irq t 3 t i [G ] w r i 2 0 0 L misq L r irq r L i L i m sd r rd We know that m = r / (p/2), Te = 3p L m ( isq ird - isd irq ) 22 DYNAMIC MODEL Torque equation Te = but ( ) 3p L m Im is ir'* 22 s L s is L m ir L m ir s L s is Te = ( 3p Im is ( y*s - L s i*s ) 22 ( ) ) 3p 3p * 3 p \ Te = Im is ys T ys ´ is e i= T = L i i i e m ( sq rd sd rq ) 2 2 22 22 DYNAMIC MODEL Simulation Re-arranging with stator and rotor currents as state space variables: isd 1 isq ird L2m L r L s irq R sL r 2 r L m R sL m r L mL s r L2misq R sL r r L mL s R sL m R rLm r L mL r R rL s r L r L s r L mL r isd R r L m isq 1 2 r L r L s ird L m L r L s R r L s irq The torque can be expressed in terms of stator and rotor currents: Te = 3p L m [ isq ird - isd irq ] 22 Te - TL = J dw m dt L r 0 Lm 0 0 L r v sd 0 v sq Lm Which finally can be modeled using SIMULINK:
