Summary 7. ELECTROMAGNETIC JOINT. ROTATING MAGNETIC FIELD. SPACE-PHASOR THEORY... 2
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1 uay 7. ELECTROMAGETIC JOIT. ROTATIG MAGETIC FIELD. PACE-PHAOR THEORY ELECTROMAGETIC JOIT UMER OF POLE DITRIUTED WIDIG TORQUE EXPREIO PACE PHAOR THREE-PHAE MACHIE AD THE ROTATIG MAGETIC FIELD PACE PHAOR ALGERA pace pha they appled t an electcal achne... 1
2 7. Electanetc jnt. Rtatn anetc feld. pace-pha they 7.1 Electanetc jnt Cnde an tpc tatn achne, wth ne t wndn. Fue 7-1 Itpc t pweed achne cncentated wndn In de t calculate the anetc feld H (and the flux denty, we need t apply the Apee' Law, that t pef the Lne nteal aln a cled path,.e. ne an the flux lne f Fue 7-1. Due t the efactn law, the tanent f the anle β between the nal t the epaatn uface and the dectn f the utput flux fle equal t the at between the peeablte f the tw ateal (feanetc ateal and a. ut μfe=, the flux lne pependcula t the epaatn uface, (tβ=μ/μfe. Futhee, the anetc feld H nde the feanetc ateal ay be cndeed equal t ze (μfe=. Thu, the lne nteal f H aln a flux lne ha nly the cntbutn due t the tw eent f flux lne nde the aap (I the anettve fce due t the t cuent: H dl H C / Fue 7- Qualtatve tend f the anettve fce aln the a ap cncentated wndn In the a, the flux denty =μh. Thu, the apltude f the flux denty cntant (f half ped:
3 ( ( Fue 7- Qualtatve tend f the flux denty aln the a ap cncentated wndn The ft hanc f a quae wave ha an apltude f 4/π f the apltude f the quae wave. c( 4 ( 1 Cnde the Fue 7-4. Fue 7-4 Itpc t pweed achne cncentated wndn The flux lnkae ψ (flux lnked wth the t cl, due t the t cuent l l ld nd b 4 4 ( 1 Un the defntn f the nductance: l L 4 l ( μ / 4/π μ /
4 Cnde, nw, the tat wndn, wth a enec echancal ptn θ f the t epect the tat. Fue 7-5 Itpc t pweed achne cncentated wndn The flux lnkae wth the tat wndn, due t the t cuent : l l ld c 4 n n 4 ( 1 L l L c 4 In a la anne l L 4 7. ube f ple Cnde the tuctue f Fue 7-6. θ
5 Fue 7-6 Itpc t pweed achne, fu ple cncentated wndn Yu fnd tw th' and tw uth'. Th achne ha fu ple. If yu walk aln the aap, yu fnd a ped f the anetc quantte equal t an half f the echancal ped (π. Th ean that the fequency f the electcal quantte tw te the fequency f the echancal ne. Wth np ple pa, the at np. In anthe way, the echancal peed een n the electcal wld np te the echancal peed een n the echancal wld. ω=npω 7. Dtbuted wndn Cnde nw a achne wth a t wth dtbuted wndn ntead f a cncentated ne. 1 b(θ θ 1 θ 1 Fue 7-7: Flux denty wavef aln the a-ap n a dtbuted wndn achne, due t tat cuent 1 The dtbutn f tun n the lt pduce an effect (wndn fact that eay t analyze. Each cl pduce a feld epeented by a pace pha. The vect u f thee feld pvde the eultn feld (ee Fue 7-8
6 ψ ψ e -jε k j k e ψ e jε ψ Fue 7-8: Wndn fact Yu need t ntduce a ceffcent kw: wndn fact (kw <=1 k w 4 l 7.4 Tque expen Wth bth tat and t cuent The tque expen L L L L T e W, cnt b(
7 b( -θ Applyn the upeptn Pncple: The eney denty 1/ H. b b b c c k k w w 4 4 Wth μfe=, thee n eney ted n the feanetc ateal. Theefe, the eney ted nly n the a-ap. l Te W c b HdV dv V V ld d c d c c The ft te de nt depend n the echancal ptn, t patal devatve ze. d The ecnd te ha an aveae value whch de nt depend n the echancal ptn and a c[(+θ] whe nteal between and π equal t. The thd ne : c c c c c c The nteal between and π f the ft pat equal t. Theefe the ecnd pat the nly dffeent f ze, T l e c l d c eatve ean an attactve tque: k ψψ n(θ (Electanetc Jnt 7.5 pace pha n l If we cnde nly the ft hanc and nelectn the hhet hanc, the tuatn n the a ap ay be epeented by a vect whe dectn aln the th pla ax (Fue 7-9.
8 Fue 7-9: Mf epeentatn by ean f a pace pha The pace pha (whe apltude a functn f te allw the knwlede f the value f the cepndn quantty at any lcatn wthn the a-ap and at each ntant. In fact, n de t knw the value f the apltude at a ven pnt f the a ap, t uffcent t pject the pha nt the equed dectn. It appea that M (, t M ( t c( (Fue 7-1. M(,t M(t Fue 7-1: pace pha applcatn All electcal quantte (vltae and cuent and anetc quantte (f and fluxe can be epeented by pace pha, allwn f eay ntepetatn f electanetc phenena. 7.6 Thee-phae achne and the tatn anetc feld Cnde thee wndn, equal each the but wth a dplaceent f 1. Lk at the Fue 7-11.
9 1 1 ' ' 1 Fue 7-11: Thee-phae achne uppe t upply the achne by ean f a yetcal thee-phae pwe upply. At teadytate, the cuent wll aue the wavef f Fue 7-1: they have the ae apltude, ae fequency and a dplaceent f 1 1(t (t (t t1 t π/ω t Fue 7-1: Thee-phae cuent At te t=, the cuent 1 aue t axu value Iax whle ==Iax c(1 =-.5 Iax. Applyn the upeptn pncple, the ttal flux ae the u f the thee fluxe due t the thee cuent. Thu, the ttal flux ha the dectn f 1 and an apltude equal t / f the flux ψ1 (ee Fue 7-1.
10 1 ψ ψ1 1 ' ψ ' ψtt 1 Fue 7-1: Ttal flux at t= At t=t1, the cuent aue t axu value Iax whle 1==-.5 Iax. (Fue Thu, the ttal flux ha the dectn f and an apltude equal t the pevu ne. ψtt ψ 1 ψ1 ψ 1 ' ' 1 Fue 7-14: Ttal flux at t=t1 At t=t, the cuent aue t axu value Iax whle 1==-.5 Iax. (Fue Thu, the ttal flux ha the dectn f and an apltude equal t the pevu ne. It ean that, at teady tate, wth a yetcal pwe upply, the ttal flux aue a cntant value and t vn at a cntant peed: f a tw ple achne, the peed f the tatn flux Ω equal t the anula fequency ω f the electcal quantte (f np ple pa, t eult ω=np Ω. Th effect called "tatn feld" (cap anetc tante and the letne f the electechancal cnven n ac achne.
11 1 ψ1 1 ' ψ ' ψ ψtt 1 Fue 7-15: Ttal flux at t=t 7.7 pace pha aleba Cnde the efeence fae,β f Fue 7-16 ( ha the ae dectn f the anetc ax 1 β 1 β 1 ' ' 1 Fue 7-16: feence fae, fxed wth the tat wndn uppe t have thee cuent nt the wndn wth a ttal effect epeented by the pace pha I. The expen f the pace pha f the cuent a a functn f the tw cuent and β the fllwn: j ( Cnveely, knwn the pace pha, the value f wndn cuent can be calculated ply pjectn the pace pha aln the dectn f the cepndn anetc ax; f exaple: R( whee R defned a the pjectn f the pha nt the anetc ax f phae "". The ttal effect acheved by un thee wndn, dplaced by 1 electcal deee n pace, and thee-phae cuent, ay be ealy calculated by pjectn. The fula a fllw: ( 1
12 j whee e On the the hand, t pble t calculate, knwn the pace pha and aun that the u f the phae cuent ze, the value f a phae cuent un the fula: 1 R1( whee R1 defned a the pjectn f the pha nt the anetc ax f phae "1". Tadtnally (f th cue, hweve, the pace pha defned a fllw: ( 1 R1( 1 bette: 1 ( ( ( a t antan the ae expen f pwe and eney bth n the phae quantte and n the pace pha ( ean "eal pat f". The pace pha, lke all vect, cpletely defned by tw vaable: the apltude and the auent ( phae anle by t cpnent wth epect t tw thnal axe (efeence fae. q β d Fue 7-17: Dffeent feence fae The axe f a enec efeence fae ae cnly aked wth "d" and "q" (whee d the eal ax, whle q epeent the anay ax. If the efeence fae fxed wth the tat (wth the eal ax alned wth 1, the tw axe ae called and β (a we aw. Gven β a the pha wth epect t a efeence fae "β", the cepndn pha n a efeence fae "dq" dplaced by an anle wth epect "β " : dq j e The pha alway the ae, but t "een" f a dffeent vewpnt. The peatn f the devatve f a pha lead t the fllwn elatnhp: dq j d d e d j j d e j e j dt dt dt dt j pace pha they appled t an electcal achne uppe t upply the achne by ean f a yetcal thee-phae pwe upply. At teadytate, the cuent wll aue the wavef f Fue 7-1. e
13 Applyn the pace Pha fula [ ( 1 ] the cuent pace pha aue jt jt jt the value Iaxe Iaxe Ie. If the ae fula appled t the vltae, the vltae pace pha ha a cntant apltude, equal t the value f the lne-lne vltae and t vn at a cntant peed ω. The actve pwe f the thee-phae ccut tadtnally calculated by: P=VphIphc(φ P=qt(VllIphc(φ, whee Vph and Iph epeent the phae vltae and phae cuent, φ the dplaceent anle between the and Vll the lne t lne vltae. Un the pace pha epeentatn yu have: P=VIc(φ (wthut any quae t f, whee V and I epeent the vltae and cuent pace pha.
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