AN INDUCTION MACHINE MODEL BASED ON ANALYTIC TWO-DIMENSIONAL FIELD COMPUTATIONS

SPEEDAM 004 June 16th - 18th, CAPRI (Italy) AN INDUCTION MACHINE MODEL BASED ON ANALYTIC TWO-DIMENSIONAL FIELD COMPUTATIONS Martn J. Hejmakers Electrcal Pwer Prcessng Unt, Delft Unversty f Technlgy Mekelweg 4, 68 CD Delft, The Netherlands, m.j.hejmakers@ew.tudelft.nl Abstract Ths aer resents a cherent descrtn f an nductn machne n whch twdmensnal analytcal feld analyss, a netwrk descrtn f the magnetc crcut, and the classcal equvalent crcut are cmbned. In ths descrtn, classcal machne arameters (man and leakage nductances) and quanttes lke the man and leakage fluxes take n a meanng fr the case f a large ar ga (tw-dmensnal feld analyss). Ths descrtn can be used t smlfy tw-dmensnal feld cmutatns fr creatng feld lts, whch are f great value fr teachng electrcal machnes. 1. - INTRODUCTION One f the frst sustns made n teachng electrcal machnes s that the ar ga s small cmared t ts radus. In ths way the ar-ga feld cmutatn becmes very easy because the feld nly has a radal cmnent. Hwever wth nly a radal feld, the statr cannt exert a trque n the rtr. Ths cntradctn s slved by susng that the frce s exerted n currents n wndngs n the ar ga (neglectng ther tangental feld). I wuld lke t shw that t s ssble t use a qute smle analytc tw-dmensnal feld analyss t shw the bendng f the feld lnes n the ar ga crresndng t the trque. Besdes, the ar-ga leakage flux s made vsble t. Because the cmutatn f the feld lt s very fast, t can als be used t llustrate henmena descrbed n, fr examle, 4. It s nt my ntentn t shw the students the tw-dmensnal feld analyss, althugh t s nt very cmlcated. I just want t have a clear mdel t make cmuter anmatns fr shwng dfferent henmena n nductn machnes. By makng the ar ga relatve large, these henmena becme very clear. Ths als hlds fr the ar-ga leakage flux. The result f ths artcle s a cherent descrtn f an nductn machne n whch tw-dmensnal analytcal feld analyss, a netwrk descrtn f the magnetc crcut, and the classcal equvalent crcut are cmbned. In ths descrtn, classcal machne arameters (man and leakage nductances) and quanttes lke the man and leakage fluxes take n a meanng fr the case f a large ar ga (tw-dmensnal feld analyss). Further, the magnetc equvalent crcut can be used t smlfy twdmensnal feld cmutatns fr creatng feld lts. We start wth a hyscal mdel f an nductn machne. Next, the magnetc feld s cmuted analytcally. We use the exressns fund t derve a magnetc netwrk and the crresndng equvalent crcut. In sectn 7, the flux lnkages frm the equvalent crcut are used n the rtr and statr vltage equatns. In ths way we have fund the equatns crresndng t the mdel f the nductn machne. In sectn 8 sme examles are gven.. - THE PHYSICAL MACHINE MODEL The machne mdel s based n the usual assumtns fr mdellng nductn machnes, excet that the ar-ga length desn t have t be nfntely small (fgure 1). The statr (s) wndng cnssts f three (a, b, and c) snusdally dstrbuted, nfntely thn wndngs alng the statr crcumference n the ar ga, the rtr (r) wndng f three such wndngs alng the rtr crcumference. The Fgure 1 z δ statr r s r r e ϕ 1 4 5 rtr A reresentatn f the nductn machne 4

SPEEDAM 004 June 16th - 18th, CAPRI (Italy) magnetc axes f these wndngs are dected n fgure fr a -le machne. The angle θ s the rtr stn angle. Fgure rb sb sc rc β θ ra sa The magnetc axes f a -le nductn machne As an examle, we descrbe the wndng f statr hase a, the wndng dstrbutn (number f cnductrs er metre) f whch s exressed as Z sa (ϕ)=ẑ s sn ϕ (1) where s the number f le ars and ϕ s the angle f the cylndrcal crdnate system whch s used (see the fgures 1 and ). All cnductrs f ne hase are sused t be cnnected n seres. Hence, the ttal number f turns f a hase wndng s N s =r s Ẑ s and we can wrte the wndng dstrbutn (1) as Z sa (ϕ)= N s sn ϕ () r s and the crresndng surface current densty as K sa (ϕ)= sa Ẑ s sn ϕ = N s sa sn ϕ () r s. - THE MAGNETIC FIELD Frst, the magnetc feld s cmuted analytcally. The tw-dmensnal feld analyss n cylndrcal crdnates used s qute ld already. In was mentned that ts text s a rernt f a text rgnally ublshed n 199. Fr the analyss f the magnetc feld, we nly cnsder the case n whch the magnetzatn drectn s alng the hrzntal axs. Later, we can rtate ths axs, whch results n a rtatn f the whle feld attern..1 - The feld equatn n cylndrcal crdnates Ths subsectn descrbes hw t cmbne (Maxwell s) equatns t btan ne artal dfferental equatn fr each regn n fgure 1, frm whch the magnetc feld n that regn can be slved. A mre cmrehensve descrtn may be fund n many text bks, fr examle, 1 and. Because we assumed the wndngs t be nfntely thn n the brders f the ar ga, the current densty s zer n a regn and Amere s law s gven by α H = 0 (4) where H s the magnetc feld strength. The relatn between the magnetc flux densty B and H s gven by B = µ H, where µ s the magnetc ermeablty, whch s sused t be cnstant n the regn. Usng ths n equatn (4) results n B = µ H = 0 (5) T smlfy the equatns and the creatn f feld lts, we use the magnetc vectr tental A: B = A. Substtutn f ths exressn n equatn (5) results n: ( A)=0 (6) Usng the gauge A = 0, ths may be wrtten as ( A)= ( A) A = A = 0 (7) Here, the feld quanttes are cnsdered t be ndeendent f z (tw-dmensnal feld analyss). Therefre, (7) can be wrtten as (n cylndrcal crdnates) ( 1 r A ) z + 1 A z r r r r ϕ = 0 (8) We may slve ths equatn by usng the methd f searatn f varables: { A z (r,ϕ)= (C α,k r k + D α,k r k )snkϕ k=1 } (9) (C β,k r k + D β,k r k )cskϕ The cnstants C α,k, D α,k, C β,k, and D β,k are determned by the bundary cndtns. Because we nly cnsder the case n whch the magnetzatn drectn s alng the hrzntal axs, the vectr tental s mrrr symmetrcal wth resect t the vertcal axs. As we may see n exressn (9), ths als means that the ceffcents wth subscrt β are zer. Therefre, we dn t use the subscrt α frm nw n. Further, the vectr tental lnearly deends n ts exctatn, the surface current denstes. Because they nly cntan sne waves wth sn ϕ, we nly use the term n the Furer seres wth k =. Besdes, we dvded the sace n cylndrcal regns (see fgure 1), whch are ndcated by the subscrt m (here, m=1...5): A z,m (r,ϕ)=ã z,m (r)snϕ (10) where à z,m (r)=c m r +D m r (11) The functn à z,m (r) may be seen as a knd f amltude. Hwever, t may have a negatve value. The radal flux densty and the tangental magnetc feld strength can be exressed as B r,m (r,ϕ)= 1 A z,m r ϕ = r Ãz,m(r)cs ϕ (1) H ϕ,m (r,ϕ)= 1 A z,m = H ϕ,m (r)sn ϕ (1) µ m r 44

SPEEDAM 004 June 16th - 18th, CAPRI (Italy) where H ϕ,m (r)= 1 dã z,m (r) µ m dr (14) Because all functns are snusdal and reresent ne magnetzatn drectn, we nly need the functns à z,m (r) and H ϕ,m (r).. - The bundary cndtns We can slve the set f equatns by means f the bundary cndtns. The frst s: B r,m (r,m,ϕ)=b r,m+1 (r,m+1,ϕ) where r,m s the radus f the uter brder f regn m and r,m+1 s the radus f the nner brder f regn m+1, whch are equal f curse (r,m =r,m+1 ). When we lk at equatn (1), we can see that we nly need the equatn fr à z : à z,,m = à z,,m+1 (15) where à z,,m s the value f à z at the uter brder f regn m and à z,,m+1 ths value at the nner brder f regn m+1. The secnd bundary cndtn s related t the tangental cmnent f the magnetc feld strength. Usng () fr the surface current densty, we get: H ϕ,m (r m,ϕ) H ϕ,m+1 (r m,ϕ)= K m (ϕ)= m Ẑ m sn ϕ Here, we als nly need the equatn fr H ϕ : H ϕ,,m + H ϕ,,m+1 = m Ẑ m (16) where H ϕ,,m s the value f H ϕ at the uter brder f regn m and H ϕ,,m+1 ths value at the nner brder f regn m+1.. - New frms fr the vectr tental We can drectly fnd the ceffcents C and D n the exressn fr à z ((11)) by means f the bundary cndtns (15) and (16) and the characterstc equatn fr the regn (14). Hwever, these ceffcents have n hyscal meanng and t may be useful t exress these ceffcents as functns f tw ther quanttes, esecally, when a cmbnatn f quanttes crresndng wth the bundary cndtns s used. Here, we chse the values f the magnetc feld strengths (usng (14) and (11)): H ϕ, = H ϕ (r )= ( ) Cr Dr µr H ϕ, = H ϕ (r )= ( Cr Dr ) (17) µr where stands fr nner and fr uter brder. Because we are nly cnsderng ne f the regns n ths subsectn, we may leave ut the subscrt m. Frm these equatns we can fnd exressns fr C and D. Usng these, we can wrte fr the vectr tental ((11)): à z (r)= µkε ( r ) r H ϕ, r + r r r r H ϕ, r + r r where we used the arameters ε and k accrdng t ε = r r r r + r r r r ; k = r r + r r Fr the bundary values f the vectr tental, we fnd: à z, = à z (r )= µ ε ( r H ϕ, kr H ϕ, ) (18) (19) à z, = à z (r )= µ ε ( kr H ϕ, r H ϕ, ) (0) We can fnd the values f H ϕ fr each brder frm the set f equatns cnsstng f the bundary cndtns (15) and (16) fr all brders and the characterstc equatns (0) fr all regns. We cannt use exressn (18) fr feld cmutatns n deal rn, because n that case µ = and H ϕ =0 are vald. Fr that reasn, we exress the ceffcents C and D as functns f the vectr tental à z nstead f the magnetc feld strength H ϕ. In ths way we get à z (r)= kε ( à z, r r r r + à z, r 4. - A MAGNETIC NETWORK r ) r r (1) The quanttes used n the set f equatns (15), (16) and (0) ( H ϕ and à z ) are nt drectly accessble frm an electrc crcut. When we use a magnetc netwrk, we get a better relatn wth electrc quanttes. Fr ths urse, we use the le flux (usng (1)) Φ(r)=l π π B r (r,ϕ)r dϕ = lã z (r) () where l s the length f the magnetc art f the machne. Usng ths, we can wrte the bundary cndtn (15) as Φ,m = Φ,m+1 () We can see ths equatn as the nde equatn f a netwrk reresentatn n whch the fluxes are seen as currents. Further, we ntrduce the magnetc vltage (usng (1)) U m (r)= π/ 0 H ϕ (r,ϕ)r dϕ = r H ϕ (r) (4) Here, subscrt m stands fr magnetc, and des nt dente the regn. Usng (4), we can wrte the bundary cndtn (16) as U m,,m +U m,,m+1 = F m,m (5) where we ntrduced the magnetmtve frce 45

SPEEDAM 004 June 16th - 18th, CAPRI (Italy) F m,m = r m mẑm = m N m (6) whch crresnds wth the maxmum f the enclsed current fr ne le tch (N m / s the number f cnductrs f ne le tch). The frst subscrt f F m,m stands fr magnetc and the secnd dentes the brder number. The radus r m s the radus f the brder between the regns m and m+1: r m = r,m = r,m+1. Usng () and (4), we fnd fr the bundary values ((0)): Φ = µlε (U m, ku m, )= U m, + U m, U m, R mσ R mm Φ = µlε (ku m, U m, )= U m, + U (7) m, U m, R mσ R mm where we ntrduced R mm = 1 µlεk ; R mσ = 1 µlε (1 k) (8) Here, the frst letter n the subscrts stands fr magnetc and the secnd dentes the man flux, resectvely the leakage flux. We can see (7) as the set f equatns reresentng a twrt (fgure ). S, we can see each cylndrcal regn as a tw-rt and the whle machne as a ladder netwrk f tw-rts, cnnected accrdng t the bundary cndtns () and (5). + U m, Φ - Fgure R m σ R m σ Φ R mm + U m, - The magnetc netwrk fr a regn We can fnd the values f U m frm the bundary cndtns () and (5) and the characterstc equatn fr the regn (7) r by the netwrk reresentatn (fgure ). 5. - AN EQUIVALENT CIRCUIT We can get a better relatn between the bundary quanttes U m and Φ and the electrc quanttes, when we cnvert the magnetc netwrk nt an equvalent crcut (5). Fr ths urse we relace the magnetc vltages U m, and U m, by, resectvely, the currents and. These currents are nt real currents, but currents reresentng the magnetc vltages: = U m, ; = U m, (9) Further, we relace the flux Φ by the flux lnkage λ. Befre ths relacement, we cnsder a snusdally dstrbuted, nfntely thn wndng at radus r 0 wth the magnetc axs n the drectn ϕ =α 0 /: Z 0 (r 0,ϕ)= N 0 r 0 sn(ϕ α 0 ) (0) The flux lnked wth ths wndng may be fund by λ(r 0 )=l π 0 A z (r 0,ϕ)Z 0 (r 0,ϕ)r 0 dϕ (1) Substtutng (0), (10), and (11) wth () nt ths exressn leads t λ 0 = π 4 N 0Φ 0 csα 0 () It shuld be nted that the factr π/4 s the wndng factr f a snusdally dstrbuted wndng. Nw, we ntrduce the flux lnkages whch relace the fluxes Φ and Φ : λ = π 4 Φ ; λ = π 4 Φ () In fact, these are the flux lnkages accrdng t () fr the case that N 0 =1 and α 0 =0. Usng (9) and (), the bundary cndtns () and (5) becme: λ,m = λ,m+1 ;,m +,m+1 = F m,m (4) Usng (9) and (), we can use the set f netwrk equatns (7) t create the equvalent crcut n fgure 4, n whch we ntrduced L m = π 1 =k π 4 R mm 4 µlε ; L σ = π 1 =(1 k) π µlε (5) 4 R mσ 4 λ Fgure 4 L σ L m L σ λ The equvalent crcut fr a regn When we cnsder the exressns (5) and fgure 4, we can see that the arameter k n fact s the culng factr. Fr the cmutatn f the feld n the regn, we can use the vectr tental ((1) wth () and ()): ) r + λ Ã z (r)= 4 kε π 4l ( λ r r r r r r r 6. - THE BASICS OF THE MACHINE MODEL (6) Befre dervng the machne-mdel equatns fr the case f three wndngs n the statr and three wndngs n the rtr, we frst cnsder the smler case f tw wndngs n the hrzntal axs n ths sectn. Here, we nly excte statr wndng sa and rtr wndng ra, whle the magnetc axs f last wndng s alng the hrzntal axs (θ = 0, see fgure ). S, there nly s a surface current densty at r = r = r r and at r = r = r s.in fact, we nw have a transfrmer wth thse wndngs. As we can see n fgure 1, ur nductn machne mdel cnssts f 5 regns, s that we have t lace 5 equvalent 46

SPEEDAM 004 June 16th - 18th, CAPRI (Italy) crcuts accrdng t fgure 4 n cascade. Hwever, n the secnd and the furth regn (r and s), the relatve ermeablty s assumed t be nfnte. S, the nductances n fgure 4 are nfnte t. Ths means that there are n currents n the crresndng equvalent crcuts and we get three searate regns (the frst, thrd, and ffth regn). Because there s n exctatn n the frst () and the ffth (e) regn, there are als n currents n these regns. The nly actve equvalent crcut s the thrd regn, the ar ga (δ). Fr the termnals f ths equvalent crcut, we use the bundary equatn fr the currents n (4) (usng (6) fr F m ): N, +, = F m, = ;, +,4 = F m, = N Here,, and,4 are zer. Further, we relace the subscrt by ra (rtr hase a) and the subscrt by sa (statr hase a) wth F m and leave ut the subscrt wth the currents n the equvalent crcut: N r N s = F m,ra = ra ; = F m,sa = sa (7) When we cmare the equatns fr λ () and (), we can see that we have t multly λ and λ by the number f turns f a wndng t fnd the flux lnkage f ths wndng. In ths way we get fr the cls crresndng wth ne le ar (all wndngs are cnnected n seres): λ rac = N r λ ; λ sac = N s λ (8) The equatns (7) and (8) crresnd wth tw deal transfrmers. S, we get the crcut n fgure 5. ra λ rac N r :1 Fgure 5 λ L σ L m L σ λ 1: N s sa λ sac An equvalent crcut fr the basc mdel T cmare ths fgure wth ts classcal frm, we nvestgate the man nductance. Usng (5) wth (19), we get L m = π 1 ; R mm = 1 r s 4 R mm µ 0 l rr r r rs (9) The exressn fr the man reluctance R mm may nt lk very famlar. Hwever when we wrk t ut fr the case that the ar ga s relatvely small, we wll fnd an usual exressn. Fr ths urse we use the relatn r s = r r + δ (40) where δ s the ar-ga length. When the ar ga s relatvely small, the exressn fr R mm n (9) becmes R mm = 1 µ 0 l δ δ = ( r s µ π 0 τ l ) ; δ r s (41) where τ = πr s / s the le tch. As we may see, R mm s twce the reluctance f the ar-ga ver ne le tch. The factr /π accunts fr averagng the snusdal dstrbutn f the flux densty. 7. - THE MACHINE MODEL In ths sectn we derve the usual machne-mdel equatns, usng the revus sectns. We start wth the cmutatn f the ar-ga flux. Next, the flux lnkages f the machne wndngs are evaluated. These flux lnkages are used n the vltage equatns, whch are transfrmed t the statr reference frame. In the revus sectn, n the statr, we nly used hase a. When we use all three statr hases, the surface current densty becmes (crresndng wth ()): K s (ϕ)=ẑ s sa sn ϕ + sb sn(ϕ π) + sc sn(ϕ 4 (4) π) The dfferent magnetc axes are dected n fgure. The rtr surface current densty s K r (ϕ)=ẑ r ra sn(ϕ θ)+ rb sn ( ϕ θ π) + rc sn ( ϕ θ 4 (4) π) A three-hase snusdally dstrbuted current densty may als be reresented by tw rthgnal snusdally dstrbuted current denstes. T btan these current denstes, we may use the well-knwn transfrmatns named after Clarke and Park: fsα = 1 1 1 f sβ 0 1 f sa 1 f sb f sc frα = cs θ cs(θ + π)cs(θ + 4π) f rβ sn θ sn(θ + π) sn(θ + 4π) f (44) ra f rb f rc When we aly these transfrmatns, the machne currents are reresented n the statr reference frame. Usng the transfrmed currents, the current denstes ((4) and (4)) may als be gven as K s (ϕ)= ( Ẑs sα sn ϕ sβ cs ϕ ) K r (ϕ)= ( Ẑr rα sn ϕ rβ cs ϕ ) (45) The sne wave n ths exressn crresnds wth the magnetc axs n the hrzntal (α) drectn, the csne wave wth the magnetc axs n the vertcal (β) drectn (see fgure ). In the sectns t 6, we used the surface current densty K m (ϕ)= m Ẑ m sn ϕ. If we cmare ths exressn fr K m (ϕ) wth (45), we can see that we have t aly a factr / fr the currents. Further, we may use all derved equatns. Hwever, we nw have tw magnetc axes (α and β). S the equatns fr the currents (magnetmtve frces) (7) becme α = N r rα β rβ ; α = N s sα β sβ Wth these exressns, we fnd fr the equvalent-crcut flux lnkages (see fgure 4): 47

SPEEDAM 004 June 16th - 18th, CAPRI (Italy) λα =L λ σ β λα =L λ σ β N s sα sβ N s rα rβ +L m +L m where we used n rs = N r /N s ( N s sα N s ) rα +n rs sβ rβ ( sα sβ +n rs rα rβ ) (46) We can fnd the flux lnked wth each f the statr and rtr wndngs by means f () wth (), where the angle α (the angle between the magnetc axs f the feld and the magnetc axs f the wndng) fllws frm fgure : λ sa 1 0 λ sb =N s 1 1 λ sc 1 1 λ ra cs θ λ rb =N r λ rc λα λ β sn θ cs(θ + π) sn(θ + π) cs(θ + 4 π) sn(θ + 4 π) λα λ β (47) In fact, we ntrduced the nverse transfrmatns f (44) n a natural way. When we add the statr and the rtr hase-wndng vltage equatns, whch have the frm u = R+ dt d λ,wehave a cmlete set f equatns fr the nductn machne. T get a mre ractcal (and usual) frm f the vltage equatns, we use the transfrmatns (44) fr the hasewndng vltage equatns: usα sα =R u s + d λsα sβ sβ dt λ sβ urα rα =R u r + d λrα 0 1 + ω rβ rβ dt λ m rβ 1 0 and fr the equatns fr the flux lnkages (47): λsα λα λrα λα = N λ s ; = N sβ λ β λ r rβ λ β Usng (46), these equatns may be elabrated t ( ) λsα sα sα rα =L λ sσ +L sβ sm +n sβ rs sβ rβ λrα λ rβ =L sσ n rs rα λrα λ rβ (48) (49) ( ) (50) sα rα +n rs L sm +n rβ rs sβ rβ where we ntrduced L sσ = Ns L σ ; L sm = Ns L m (51) The set f equatns (48) and (50) s the usual set f equatns fr nductn machnes. The leakage nductances fund n ths way nly ncrrate the leakage flux n the ar ga, whch s nrmally very small. Hwever, they can smly be enlarged t take nt accunt the ther cntrbutns t the leakage flux, whch are nt n the mdel here. 8. - EXAMPLES When we have fund the flux lnkages, we may use (10) wth (6) t cmute the vectr tental n the rtr,the ar ga and the statr. We can use ths vectr tental t draw (analytcally cmuted) feld lts n an easy way. As an examle, fgure 6 shws the bendng f the feld lnes n the ar ga crresndng t the trque fr nrmal mtr eratn. Fgure 7 makes the ar-ga leakage flux vsble fr the case f a hgh sl. In bth cases, the nner regn n fgure 1 () s left ut. In these (teachng) examles the ar-ga wdth s enlarged t make the nterestng henmena very clear. Fgure 6 Fgure 7 9. - REFERENCES Lw-sl eratn Hgh-sl eratn 1 Davd K. Cheng. Feld and Wave Electrmagnetcs. Addsn-Wesley, Readng, Massachusetts, nd edtn, 1989. B. Hague. The Prncles f Electrmagnetsm - Aled t Electrcal Machnes. Dver Publcatns, New Yrk, 196. H.A. Haus and J.R. Melcher. Electrmagnetc felds and energy. Prentce Hall, Englewd Clffs, New Jersey, 1989. 4 Jachm Hltz. On the satal ragatn f transent magnetc felds n ac machnes. IEEE Transactns n Industry Alcatns, (4):97 97, Jul/Aug 1996. 5 G.R. Slemn. Mdellng f nductance machnes fr electrc drves. IEEE Transactns n Industry Alcatns, 5(6):116 111, Nv/Dec 1989. 48