A Finite Element Method for Transient Analysis of Power Electronic Motor Drives Including Parasitic Capacitive Effect and External Circuit

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1 F m Mho for rs ss of or ro Moor rvs Ig rs pv ff r r W.. F S. L. Ho h Hog Kog oh Uvrs Hg Hom, Kooo, Hog Kog f@po..h bsr o mso (-) f m mho (FM) of rs romg f for mog por ro moor rvs s prs. h propos mho s o o h prs pv ff b op h rbrr o rs. h FM formo hh s h spm rr h ro of h mo s ph s. h spm rr ff h p of h soo om s rprs b opg h r of ps o h FM qos. B rog o os, h f s of h ssm qos hs smmr off mr. mho sg r hrg s o for h ompo of h p mr s so propos for rg h ompo m. h vop gorhm hs b pp o sm ssms h por sors hvg hgh frq /or hgh v/. Kors r, spm rr, romg f, f m mho, moor, prs p, por ro, rs. I. IROUIO os h s por ro rv r moors ( rvs) r rpg ro moors b sso vog m ppos. Wh hs rvs prov vb vopm oppors of r mhs, h so brg probms o srs rsrhrs. s fs shg rpv sp rsg or fg vogs of rvs hv hgh-frq os, hr prs pv ff s ff r vr sgf [-4]. hs for s o oform vog srbo mog os. h sso hgh v/ hgh / srsss r sors of prmr g so frs. If h moor s o o h vrr hrogh possb og fr bs, vog obg h moor rms, o s rf v phomo, so sros ff h prform of hs moors [5-7]. ovo mos of r mhs oprg r sso vog os por frq o prs osr hgh-frq ffs f h mhs r rg b sqr v vog os. s h shg frq rss, h f of -rr mg ors h s ffs o gs bom rsg sgf h mog of r mhs rvs. hrfor, h mog of rvs h h bov fors o o s hgh srb for h prpos of r mr smo. I [8] [9], r r mos r s o sm h rvg vs vog srbo r moors. h svg of hs mog s h h -rr ff or ro ors o b. o osr h mg f srbos, [] [], f ffr mhos r vop; [], rr sovr sg FM s propos o sm h mg f h frq om. h mos sb mho o s sh rs probms s o s h m sppg FM opg h r r o omp h mg fs h r ssm smos. Hovr sh mho s m m o mog o-frq probms bs h r f s o op h spm rr s gor. o ovrom hs mo, rfr [3] prss mho hh ss ps o h ff of h spm rr. Rfr [4] ss smr mho h h ommr sofr M. hr mhos o h spm rr h ro of h mo s ph. I hs ppr - FM s propos o h prs pv ff, -rr ff rbrr o r rs. h propos FM formo s h spm rr h ro of h mo s ph, rprss h spm rr ff h - p of h soo om b og ps mog oors. For h ompo of h p mr, mho sg r hrg s o s prs o r h ompg m. I hs b s o sm h rs probms hh h r sors hv hgh frq ompos. II. I H RIO OF SOLI OUORS h foog hpohss r ro: () h mg f s B hs ompos of o h - p. hrfor h mg vor po hs ompo h ro o, h s, ; s os og h s. h oomb gg s ssf. (3) h oo rr fos og h s. (4) h hrg s s.. h r F h r f b ompos o o orhogo vors: () For h ompo h ro: V () hr V s h r sr po of h oor h om of mg f. qo () ms h h hgg mg f h r sr po pro r f h ro. If s h vog /8/$5. 8 I hor s s m o: LUL KISK UIVRSI. oo o Mrh 9, 8:5:36 from I Xpor. Rsros pp.

2 rop b h o rms of h oor h h gh h ro, o hs: V (3) hr s h por ( or ) o rprs hr h forr or rr ph. Sbsg (3) o (), o hs: (4) S h ompos r g, o hs: ( V ) (5) I ms h h mg f hs o r ff o h r f o h - p. org o ρ ε, o hs: ε ( ε V ) ρ (6) O h os of h oors, h hrg s ρ. I oors, h r ro m τ ε/σ ( F/m / sms/mr oppr) s vr sm, h s ssm h ρ. h oor s hs q-po rgo h - p [5]. h r f qo o h - p s: ( ε V ) (7) B. h Mg F h r vor spm grs J ; h J grs, hh ms h hgg r f grs mg f h - p; grs J, J grs. I h - FM h f of s ggb. org o h M-mpr s : ( ν ) σ ε (8) ( ν ) ( ν ) (9) o hs: V V ( ν ) σ σ ε () hr s smpf s ; ν s h rv of h mr σ s h ov. hrfor, sbsg (3) o (), h o mg f qo h oor b r s: σ ε ( ν ) σ ε (). h rr Brh qo h o rr s h oor og h ro of h s s: J σ σ ε () h o rr h oor og h s s: σ ε σ σ ε ε (3) R hr h rss of h oor: R (4) σ σs hr S s h ross-so r of h oor. h b.m.f. of h oor s: σ (5). h rr h - h rr s h - p s: J σ (6) h rr fos o from h oor h - p: Q S S (7) S S hr S s h r r srf of h oor Q s h hrg. I -, qo (7) s smpf s: ε (8) Γ hs gro s og Γ, hh s h oor of h oor h - p. h s h rr fog o from h oor v pv ff. For smp, h ros f srb b (7) s frs sov; h h p mr s r. h rr fog o from h oor h - p b omp b: oors (9) h ros f h m rgh h ss (s So VII) o s o b omp o f hr s o moo. Ur sh oo, o h r r h hs ps, s of h r f, s o b op h h mg f r.. h qos of O oor o ovrom h probm h qo () s h so rvv of h rsp o m, o p h off mr of h s ssm qos smmr, h o s p h ssm qo. Mpg ε o h o ss of (4), o hs: ε ε ε () org o (4) (), o hs: σ ( ν ) σ ε () org o (3), o hs: σ ε () R h ssm qos h rgo of so oors r (), () (). F. h rop oors ro bor oos b s o r h soo om. Smr o h sr gs, h so oors sho so b grop oghr f hr r hor s s m o: LUL KISK UIVRSI. oo o Mrh 9, 8:5:36 from I Xpor. Rsros pp.

3 mg f qs ssf hr pro oo or -pro oo [6]. h bs roshps b o oor h oor grop r: ps S (3) (4) (5) R R (6) hr p s h smmr mpr hh s f s h ro of h org f ross-so r o h soo r; S s h o ross-so r of h rgo op b hs oor grop h soo om; s h vog ffr b h o rms of h so oor grop; s h o rr; s h o oor mbr; s h mbr of pr brhs R s h.. rss of h oor grop. Sbsg (5) o (), h f qo h so oor grop s: σ ( ) σ ε ν (7) Sbsg (5) o (), o hs: ε ε ε (8) Sbsg (3), (4), (5) (6) o (), h r brh qo of h so oor grop s: σ ε (9) pr p hrfor, h ssm qos h rgo of h so oors r (7), (8) (9). III. I H RIO OF SR WIIS. h Mg F qo oors h sm sr g r grop oghr. h mg f qo h rgo of h sr gs b prss s: ( ν ) (3) Sp hr s h g rr; S s h o ross-so r of h rgo op b hs o grop h soo om; s h o oor mbr of hs g s h mbr of pr brhs hs g. B. h r Brh qo h brh qo of h sr gs s: R S (3) hr s h vog o h o rms of h g; s h mo ph h s; R s h o.. rss of h g: R (3) p σs I h rgo of h sr gs, org o h brh qo (3) of h sr gs, o hs: () R hr s h b.m.f.: (34) S Sbsg () o (3), h f qo boms: σ σ ( ν ) (35) qo (34) so b r s σ (36) pr h brh qo (3) b r s: σ (37) pr p qos (35), (36) (37) r h bs qos for h sr gs. h o o s ro o p h off mr of h ssm qos smmr. IV. I H OHR RIOS I rgos of r ro, so rgos hvg prm mgs (M) r rr ss, h f ffso qo s: ( ) M M ν σ J s ν µ (38) hr h frs rm h rgh h s o ss rgos hvg r rr ss; h so rm h rgh h s o ss M; ν qs o h qv rv h M; µ s h mg prmb vm; M M r, rspv, h ompos of h mgo vor (mprs/mr) h M. V. H MRIX FORM OF FIL QUIOS. Smmr of F qos h bs qos rgos b smmr s: σ σ ( ν ) σ ε M M J s ν µ (mg f qo) (39) ε ε ε (r f qo) (4) σ (o qo) (4) pr σ ε pr p (brh qo) (4) hr s h rr h sr gs or h so oor grops s h vog ffr b h o rms of h sr gs or h so oor grops. h forh rm o h f s of h f qo (39), so h o qo (4) o s h sr g rgo. h so h hr rms h hor s s m o: LUL KISK UIVRSI. oo o Mrh 9, 8:5:36 from I Xpor. Rsros pp.

4 f s of (39), h r f qo (4), h so rm h brh qo (4) o s h so oor rgo. B. h Mr Form of F qos fr ppg sr r pror, h op f r brh qos h mg f rgo b r bo mr form: p (43) h offs sso h h gr of froms (ofs) of, j () (hr h sbsrp, j r h s of h os) r: j j j ν () j j σ (45) ε (46) 4 ε (47) 4 3 σ (48) ε (49) ) ( pr (5) 4 3 σ (5) M M J H s µ ν (5) hr s h shp fo. H ν h frs rm o h rgh h s of (5) o s o o-ro m bor oos. I s obsrv h,,,,,, 4 4, VI. OULI WIH XRL IRUI Boh h r qos of h gs h f rgo h r r b op h h mg f qos oghr sov smos.. h r r qo I m om p r prss s qv rss. h brh qo of h r r h m sp b smpf s: [ ]{ } { } { } (53) hr r h brh vog rr, rspv; s h mr of h o; s h om mr sso h vog sors, rr sors s s h soo from h prvos m sp for v pv ms. B. h op Ssm qos Usg h br r s mho o sr h m vrb mpg h m sp s o h o qo h brh qo, o obs h rrr forms of h h sp for h m sppg pross s foos: p (54) hr h sprsrp s h ppropr vs h h sp of h m sppg ompo. Mpg /p o h o ss of (53), o hs h brh qo of h r r: { } p p p (55) g h r r qo (55) o (54), o obs: p p p p (56) h brh vog om mr h brh rr om mr r: { } b (57) { } b (58) Usg h o mho, h roshp b h hor s s m o: LUL KISK UIVRSI. oo o Mrh 9, 8:5:36 from I Xpor. Rsros pp.

5 brh vog b h o vog s: { b} [ b ]{ } (59) hr b s h o-o-brh mr. h Krhhoff s rr b prss s, [ b]{ b } (6) h mr b b r s: [ b] [ ] (6) hr s h o-o-brh mr of h gs s h o-o-brh mr of h r r. h qo (59) b r s: { } (6) h Krhhoff s rr b prss s, [ ] (63) Sbsg h roshps (6) (63) o h ssm qos (56), h f gob qos r ob: p (64) 3 ( ) 4 4 p hr h off mr s smmr. VII. I MRIX OMUIO h r f qo o h - p s: ( ε V ) ρ (65) oor 3 oor 3 oor 3 Hr h hrg o s s o h ps. Bs of h roshp of Q V, h hrg o h obj s: Q 3 3 V Q 3 3 V Q V3 3 V (66) 3 V 3 3 V3 B fg mr: 3 3 (67) o hs: V 3 Q 3 Q V 3 Q 3Q V3 3 3 Q3 3 3 Q3 (68) If h hrg os r s s: { Q Q Q } { } 3 (69) o hs: V 3 (7) V 3 V fr h f s sov, V, V V 3 r o. h o hs: { } { V V V } 3 (7) 3 Smr, If h hrg os r s s: { Q Q Q } { } 3 (7) o hs: { } { V V V } 3 (73) 3 If h hrg os r s s: { Q Q Q } { } 3 (74) o hs: { } { V V V } 3 3 (75) 3 fr h mr s ob, h mr b ob b (67). Usg hs gorhm, h off mr of h FM qos b p h sm. O m rgh h s probm s o b sov. hrfor, h ompg m o r h p mr b r. 3 Fg.. h ps mog hr oors For smp, ssm h hr oors s sho Fg. s s mp. Us h ps r omp b sovg h f qos 3 ms h sg h vog pos s: (V, V, V 3 ) (,, ), (,, ) (,, ), rspv (hr V, V, V 3 r h vog pos o h oor, 3, rspv). Bs sh sg of h vog pos s s osrs (65), s o ov FM progrmmg. VIII. XML h vop FM s pp o sm h rspos of sp vog fr b o ssm. Fg. shos o s of o h 8 rs so. so oors r o s sho Fg. 3. h r-o-gro p r-o-r p r omp frs sg ros f sovr s pr-prossor. h p b o-j rs s ggb. h ff of h og fr b b h sor h o s h smo o s h mp msmh b h b h moor. Hr h b s hor s s m o: LUL KISK UIVRSI. oo o Mrh 9, 8:5:36 from I Xpor. Rsros pp.

6 mo s r h srs of s mp prmr sos s sho Fg. 4. h b r h o r r o oghr b h r r opg h romg f. -9 s -8 s -7 s Fg.. So so of form-o o s f pos v vs v v v3 v4 v5 v6 v7 v8 Fg. 3. o r Mg f rgo - o r b r Fg. 4. h o s o o h sp vog hrogh og b Fg. 5. o vog po srbo sp vog (smg h WM vfro; s V s Fg. 4 Fg. 5) hvg 5 s rs-m 8 V mg s pp o h rm of h o. h sm rs o vog pos r sho Fg. 5 h h sp vog s pp o h o rm hrogh b h gh of m. h mgs of h vog pos V, V,, V 8 h Fg. 5 b fo h Fg. 3. h vog obg phom h o rm b r obsrv from h vforms. of h soo om b opg h p r h FM qos. h p mr b r b sovg h r f qos rv b hrgs, so o s of qos h m rgh h ss s o b sov for h p mr ompo. h vop mho s s o sm h sp-fro vog pp o o o b fr b. I rvs h h vog rops o h oors r o form f h rs-m s vr shor. I so shos h h vog o h rm of h o m b mh hghr h h vog pp o h rm of h fr b. h mho so b pp o s romh qpms h h r sbj o ps sors or srg vs from hrsorm or r brrs. Rfrs [] H. Bo, ss of h mp of ps-h mo vrr vog vforms o o moors, I rs. I. pp., vo.3, pp , Mr./pr [] J. Mhor, L. g, rs ffs of WM rvs o o moors, I/I&S of. R., pp.-7, M 995. [3] R. J. Krm,. Lgg,. Shg,. Sbs, WM vrrs hr f o moor ovr-vogs, I of. R., pp.3-3, 997. [4] M. Mf, J. Sg, S. B,. Sbs, ff of srg vog rsm o h so of o vog mhs f b WM ovrrs, I IS of. R., pp.39-46, 997. [5]. rsso, rs ffs ppo of WM vrrs o o moors, I rs. I. pp., vo.8, pp.95-, Sp./O. 99. [6] R. Krm,. Lgg,. Sbs, Iro of rv moo b prmrs o moor rss, I IS of. ro., pp.43-5, 996. [7]. Sbs,. Lgg, R. J. Krm, b hrrss hr f o moor ovr-vogs, I of. R., pp.4-, 997. [8] L. bb,. vo Jo,.. j,. Sgh, H.. o, Vog srbo h gs of moor sbj o hgh v/ WM vogs, I S of. ro., pp , 995. [9] Hss,. Joos, Mog smo of rvg vs o moor rvs, I of. R., pp.8-34, 997. []. B. Hpo, Smo of b g rspos o spfro vog vs, I IS of. R., pp.8-86, 995. [] Y. g, ss of sp-fro vog srbo r so fr vrr f moor, I IS of. R., pp.59-56, 997. []. Srsh, H.. o,.. Rsr.. j, rg h rs ffs of WM vog vform o h sor gs of rom o o moors, I rs. o or ros, Vo.4, o., J. 999, pp.3-3. [3] J.F. hrpr, Y. Lfvr M. Ljo-M, F m Formo for h S of h Hgh Frq Bhvor of Wo ompos, I rs. Mg. Vo.3, o.3, M 996, pp.98-. [4] H.. o,. Srsh,. br, Smo of vog srss o h vrr f o moor gs spp hrogh fr b, I IS of. R., pp.43-5, 997. [5] R. Mr,.M. Rr K.J. Ovrsho, h ff of Hgh-Vog Lghg Impss o h or roprs of rsformrs, I rs. Mg. Vo.5, o.5, Sp. 989, pp [6] W.. F,. Zho,. L, S. So Z. J. s, Mog of So oors o-mso rs F-m ss Is ppo o r Mhs, I rs. Mg. Vo.4, o., Mrh 4, pp IX. OLUSIOS - FM for h romg f ompo of por ro rvs s, hh h spm rr h mo s ph. h propos mo rprss h spm rr ff o h p hor s s m o: LUL KISK UIVRSI. oo o Mrh 9, 8:5:36 from I Xpor. Rsros pp.