Chapter 3 AC Machinery Fundamentals 1
The Vltage Induced in a Rtating Lp e v B ind v = velcity f the cnductr B = Magnetic Flux Density vectr l = Length f the Cnductr Figure 3-1 A simple rtating lp in a unifrm magnetic field. (a) Frnt view; (b) view f cil 2
1. Segment ab. v is tangential t the path f rtatin B pints t the right (v x B) pints int the page r in directin f Hence, e v B is int the page ind 2. Segment bc. (v x B) is perpendicular t l, hence e ab =0 3
3. Segment cd v is tangential t the path f rtatin B pints t the right (v x B) pints ut f the page r in directin f Hence, e vblsin( ) is ut f the page ab cd Nte that 180 ab cd 4. Segment da. (v x B) is perpendicular t, hence e ad =0. 4
Ttal Vltage induced e e e e e v bsin( ) v bsin( ) ind ba cb dc ad ab cd 2 v bsin( ) 2 v bsin( ) ab If lp rtates at a cnstant angular velcity, = t v r, r is the radius frm the axis f rtatin t the lp edge eind = 2rlBsin(t) = BAωsin(t), A = 2rl is the lp area 5
Trque Induced in a Current-Carrying Lp F i xb F = Magnetic frce n a segment f a current lp i = Current in the lp B = magnetic Flux Density = Vectr length f the segment its directin is defined t be in the directin f the current flw. Figure 3-4 A current-carrying lp in a unifrm magnetic field. (a) Frnt view; (b) view f cil 6
The trque n the segment will then be given by rxf rf sin( ) Where is the angle between r and F 1. Segment ab. i is int the page. B pints t the right xb pints dwn hence the induced frce is dwn ( F)( r sin ) ri Bsin clckwise ab ab ab Fab i B 7
2. Segment bc. i is in the plane f the page. B pints t the right xb pints int the page hence the induced frce F i B int the page ab ( F)( rsin ) 0, since 0 bc bc bc 8
3. Segment cd. i is ut f the page. B pints t the right xb pints up hence the induced frce F i B up ab ( F)( r sin ) ri Bsin clckwise cd cd cd 9
4. Segment da. i is in the plane f the page. B pints t the right Fda xb pints ut f the page hence the induced frce i B is ut f the page ( F)( rsin ) 0, since 0 da da da 10
Ttal induced trque ind = ab + bc + cd + da = rilbsinab + rilbsincd = 2rilBsin, since ab = cd = 11
The Rtating Magnetic Field An alternate frm fr the trque induced in the lp is B lp i Flux density f the lp current G ind lp S G is a factr depending n the gemetry f the lp. A Area f the lp. 2ri B sin KB B ind lp S sin AG B B 12
The generated trque depends n: The strength f the rtr (lp) magnetic field The strength f the applied magnetic field f statr The angle between the tw fields The generated trque tends t align B lp and B S If magnetic field f the statr rtates, then the rtr field (and hence rtr lp) will fllw. 13
Hw can statr magnetic field be made t rtate? Three cils spaced 120 electrical degrees apart. A balanced 3-phase currents are applied t them. i ( t) I sin( t) A aa ' i ( t) I sin( t 120 ) A bb ' i ( t) I sin( t 240 ) A cc ' M M M The field intensity prduced by these currents are H ( t) H sin( t) 0 A.turns/m aa ' H ( t) H sin( t 120 ) 120 A.turns/m bb ' H ( t) H sin( t 240 ) 240 A.turns/m cc ' M M M 14
The flux densities resulting frm these magnetic fields are give by: B ( t) B sin( t) 0 T aa ' B ( t) B sin( t 120 )120 T bb ' B ( t) B sin( t 240 ) 240 T cc ' M M M Where B H M M Figure 3-8 (a) A simple three-phase statr. Currents are assumed psitive if they flw int unprimed end f the cil. (b) The field intensity Haa (t) prduced by a current flwing in cil aa 15
At t 0 Baa' 0, B B sin( 120 )120, B B sin( 240 ) 240 bb ' M cc '' M B B B B 1.5BM 90 net aa ' bb ' cc ' 16
At t 90 1 1 B B 0 B 120 B 240 net M M M 2 2 B 1.5B 0 net M 17
The resulting magnetic field has mved in CCW. The magnitude f the magnetic field remains cnstant. At any time t, It can be shwn the resultant magnetic field has a frm given by: 1.5 sin 1.5 cs B B t x B t y net M M The magnitude f the field is a cnstant 1.5B M. The angular psitin f the field,, is 90 t The psitin changes at an angular velcity. 18
The Vltage Induced by the Rtating Magnetic Field Assumptins: Air gap flux density is radial. Flux density is distributed sinusidally in tangential directin. Figure 3-13 (a) A cylindrical rtr with sinusidally varying air-pap flux density. (b) The mmf r field intensity as a functin f angle in the air gap. 19
A sinusidal air-gap flux density may be achieved by arranging the number f cnductrs in clsely spaced slts in a sinusidal manner given by n N Cs( ) C C Figure 3-14 (a) An ac machine with a distributed statr winding t prduce a sinusidally varying airgap flux density. The mmf resulting frm the winding, cmpared t an ideal distributin. 20
Althugh nt strictly crrect, we will assume a sinusidal distributin in the fllwing discussins. Then the rtating magnetic field f statr is assumed t have a frm M B B Cs t m Where is measured frm the directin f the peak flux density, as shwn next. 21
Figure 3-15 (a) A rtating magnetic inside a statinary statr cil. Detail f cil. (b) The flux densities and velcities n the sides f the cil. (c) The flux density distributin in the air gap. 22
The Relatinship between Electrical Frequency and Mechanical Speed (Textbk Pages 165-167) se = (p/2)sm ωse = (p/2)sm fse = (p/2) fsm nsm = 60 fsm fsm = nsm / 60 fse = (p/2) (nsm / 60) = (p nsm)/120 23
The statr rtating field induces vltages in three-phase winding having N c turns per phase as, e ( t) N sint aa ' e ( t) N sin( t 120 ) bb ' e ( t) N sin( t 240 ) cc ' C C C Where 2r BM The rms vltage induced in any given phase f the statr is given by E 2N f A C 24
Induced Trque in an AC Machine Fr a simple lp, the generated trque was fund t be KB B ind Lp S sin If B R represents the rtr flux density and B S represents the rtating magnetic field f statr, as shwn in figure 3-19, then ind R S ind KB B sin KB xb KB x( B B ) KB xb Hence, KB B ind R net R S R net R R net sin where is the angle between B andb is called trque angle. R S and 25
Figure 3-19 A simplified synchrnus machine shwing its rtr and statr magnetic fields. 26
The Pwer-Flw Diagram Fr an AC generatr, mechanical pwer is input t the machine, and then stray, mechanical, and cre lsses are subtracted. The remaining pwer is what can be cnverted t electrical frm and is called cnverted pwer (labeled P CONV ). Other lsses are cpper (I 2 R) lsses in statr and rtr. Fr an AC mtr, the electrical pwer is input t the machine, and then the statr and rtr cpper lsses are subtracted. The remaining pwer is P CONV. The pwer-flw diagrams fr AC machines are shwn in Figure 3-21. 27
Figure 3-21 (a) The pwer-flw diagram f a three-phase AC generatr. (b) The pwer flw diagram f a three-phase AC mtr. 28