CHAPTER 2 DERIVATION OF STATE EQUATIONS AND PARAMETER DETERMINATION OF AN IPM MACHINE. 2.1 Derivation of Machine Equations

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1 1 CHAPTER DERIVATION OF STATE EQUATIONS AND PARAMETER DETERMINATION OF AN IPM MACHINE 1 Deivation of Machine Equations A moel of a phase PM machine is shown in Figue 1 Both the abc an the q axes ae shown in the figue The magnetizing cuent, ue to the pesence of the magnet, is epesente in Figue 1 by the cuent souce i f The fictitious cuent souce along with the associate fiel wining m ) will be use in the eivation to follow to obtain an expession fo the flux which is equivalent to the flux ceate by the pesence of the magnet It is assume in the following eivation that the oto spee of the machine is constant Thus, no cuent flows in the ampe winings containe within the oto an thei pesence may be ignoe It is also assume that the machine is balance The stato voltage equations of the IPM ae [1] va = a + ia vb = b +ib vc = c + ic 1 1 1, 1) whee v a, v b, v c ae the a,b,c stato teminal voltages, a, b, c ae the flux linkages in the abc plane, i a, i b,i c ae the abc stato cuents, an 1 is the stato esistance of each phase

2 Figue 1 Moel of a bushless PM synchonous machine 14

3 15 of the stato wining Thus, the assumption has been mae that the esistances fo each phase of the stato ae the same The flux linkages expesse in tems of the stato an tems of the cuent ae a b = c aa ba ca ab bb cb ac bc cc ia ib + ic af bf cf i f ) The inuctances given in Equation ) will be kept in thei symbolic fom, tansfome into thei equivalent q axis countepats, an then be efine in tems of the physical constuction of machine Due to the salient natue of the oto, the inuctances of Equation ) ae functions of the position of the oto an assuming that the oto is spinning) ae functions of time This means that the inuctance paametes ae constantly changing - making the analysis of the machine vey ifficult in its pesent fom A tansfomation commonly efee to as Pak s tansfomation allows the equations escibing the machine to be tansfome into a efeence fame whee the inuctances ae not functions of time The efeence fame chosen to accomplish this goal is epenent upon the type of machine being looke at Fo a synchonous machine, this efeence fame is that of the oto In othe wos, the stato voltages, cuents, an inuctances will be pojecte onto the oto sie of the machine an, thus, be otating at the same spee with which the oto is spinning With this accomplishe, the inuctances no longe vay with the position of the oto

4 The tansfomation of a phase balance voltage, cuent, o flux linkage fom the 16 hqo=t ) habcs, ) abc to the q efeence plane may be expesse as [] T )= ) sin ) 1 π - ) π sin - ) 1 π + ) π, sin + ) 1 whee an enotes the efeence fame chosen Fo the oto efeence fame, = Expessing the q an o tems iniviually an substituting = gives π π hq = [ ha )+hb - )+hc + )] π π h = [ ha sin )+hb sin - )+hc sin + )] 1 ho= [ ha + hb + hc ], 4) whee, fo balance conitions, h = an the o tem will, fom this point fowa, be ignoe So, using Equation 4), the flux linkages may be expesse in the q plane as

5 17 Taking the eivative of q in Equation 5) with espect to time gives Compaing the fist backete expession of Equation 6) with the expession fo in Equation 5), gives Reaanging the voltage equation of 1) in tems of the ate of change of flux linkage gives Substituting Equation 8) into the secon backete expession of 6) gives )] + )+ - )+ [ = )] + )+ - )+ [ = c b a c b a q π π π π sin sin sin cos cos cos 5) )] + )+ - )+ [ + ] ) + + ) - + ) [ = - c b a c b a q π π π π cos cos cos sin sin sin 6) ] ) + + ) - + ) = [ - c b a π π sin sin sin 7) i - = v i - v = i - = v 1 c c c 1 b b b 1 a a a 8)

6 a b π c π )+ - )+ + )= π π [ va )+ vb - )+ vc + )] - π π [ ia 1 )+ib 1 - )+ic 1 + )] = v - i 1 9) 18 Substituting the esults of Equations 7) an 9) into 6) gives q = - + v - i 1 1) etting p=, an ω =, then p 1 q=- ω +v-i 11) Theefoe, the q-axis voltage can be witten as v = p q+i 1+ ω 1) Similaly, taking the eivative of the -axis flux linkage of Equation 5) gives

7 19 π π = [ a ) + b - ) + b + ) ] 1) a b π c π + [ sin )+ sin - )+ sin + )] π π q = [ a cos ) + b - ) + c + ) ] 14) Compaing the fist backete expession in Equation 1) with the equation fo q axis flux linkage in Equation 5) gives the esult Substituting the esults of Equation 8) into the secon backete expession of Equation 1) gives a b π c π sin )+ sin - )+ sin + )= π π [ va sin )+ vb sin - )+ vc sin + )] - π π [ia 1 sin )+ib 1 sin - )+ic 1 sin + )] 15) = v Substituting the esults of Equations 14) an 15) into Equation 1) gives s - i s 1 = q + v s - i s 1 16)

8 Afte eaanging Equation 16) an substituting Equation 1) into it, the final esult fo the axis voltage is v s =p +i 1-q ω s 17) In oe to obtain the iniviual self an mutual inuctance tems given in Equation )), the paametes may be efine in tems of the physical constuction of the machine, ie numbe of winings, physical imensions, etc, an then the equations obtaine may be tansfome into the q efeence fame Altenatively, as was one in this thesis, the flux linkage equations may be tansfome iectly into the q oto efeence fame equals ) an the q an axis inuctances may be efine iectly in that fame of efeence Thus, q as =T ) bs=t ) o cs aa ba ca ab bb cb ac i -1 bct ) ) is+t ) cc io af bf cf a b c aq bq cq i i i f q, 18) whee T ) aa ba ca ab bb cb ac bc cc -1 T ) ) = ls + mq ls + m, ls

9 1 an T ) af bf cf = m m mq It is wothwhile to eiteate that the fiel cuent i f is fictitious an has been use to epesent the flux linkage fom the magnet souce Fom this point foth the tem e will be use instea of the pouct tem i f * m Since the oto spee is assume constant, i an i q ae equal to zeo The flux linkage equations in the q axis can be expesse iniviually as = is + e q = q i, 19) whee = ls + q = ls + m mq, an m an mq ae the an q axis mutual inuctances an ls is the leakage in the stato

10 The an q axis mutual inuctances ae given as [1] mq m = C = C q m m, ) whee m is the inuctance of a machine with a unifom ai gap an no magnets This inuctance is etemine fom the flux linking with N 1 C w effective tuns, an is given as N 1 C w Di -8 m = 17 µ o m1 ) 1 H, 1) P g whee all the tems given in Equation 1) ae in mete, kilogam, secon MKS) units µ o = pemeability of fee space m 1 = numbe of phases of the machine N i = numbe of seies tuns pe phase C w = a wining facto which is a pouct of the istibution an pitch factos D i = stato inne iamete P = numbe of poles = coe length g = effective ai gap length C q an C ae factos which account fo the pesence of the magnets an ae, fo an inteio pemanent magnet, given as

11 C ρπ C q = ρ - sin π 8 π ) sin ρπ ρ = ρ + sin - π Rg 1+ Rm π ρ, ) whee ρ = the pole ac R g = eluctance of the ai gap R m = eluctance of the magnet The open cicuit magnet flux e fo an IPM machine is given as [1] 444 π Di )! -8 e= N i C w B f * 1, ) π P whee B! f is the amplitue of the funamental flux ensity ceate by an iniviual magnet In summay, the voltage an flux equations neee to analyze a pemanent magnet une the state assumptions ae v v s = p +i = p s +i s s+ s ω s - ω, 4) whee s = s i = + e i, s

12 4 an the subscipt s on the q an axis stato tems has been ae, s has eplace 1 as the symbol use to epesent the stato esistance, an the subscipt has been oppe fom ω Une steay-state, the eivatives of the state vaiables of Equation 4) ae zeo so, at steay state the voltage equations may be witten as V = I s+ s ω 5) V s = I s s - ω A -q axis schematic iagam epesenting the equations given in 5) is shown in Figue Figue Schematic iagam epesenting the steay state q an axis voltage equations of a PM machine

13 Detemination of Paametes of the IPM 5 The single most impotant etemination to be mae in the analysis of an electical machine is to fin what the paametes of the machine ae Due to the effects of satuation at heavy loa an emagnetization of the magnet at light loas, the paametes of the machine change significantly as the loa pesente to it changes Thus, the paametes namely the mutual inuctances in the an q axes an the flux linkage ceate by the magnet) must be functions of the opeating conitions Geneally, when effects ue to satuation ae inclue in an analysis, the paametes ae mae functions of eithe the total mutual flux mm o the total stato cuent Is The ecision was mae to make the paametes functions of stato peak cuent because, when the paametes wee plotte as a function of flux, thee wee egions in which two possible values of a paticula paamete coul be obtaine fo a single value of flux which coul be poblematic when solving equations) The paametes ae foun by using the equations given in Equation 5) an epeate below fo the sake of continuity V = V s s I = + ω s I s s I - ω s + ω e I 6)

14 6 Figue Schematic iagam of c test use to etemine stato esistance The stato esistive value s was foun by applying a c voltage acoss two teminals of the stato an measuing both the voltage an the cuent which flowe though the teminals see Figue ) The stato esistance fo a single phase is given as V c s = 7) I c The voltages an cuents in Equation 6) wee foun by vaying a thee phase balance esistive loa fom a high value to a low value an ecoing the teminal voltage an the cuent output fom the geneato The measuements mae wee the line to line voltages an the phase cuents In oe to convet the voltages an cuents into thei q components, the toque angle δ was neee The powe facto of the machine is also neee in paamete etemination, but, since the geneato was feeing a esistive loa, the cuent out of the geneato was in phase with the voltage at the teminals, so the powe facto was

15 7 unity The toque angle was foun by measuing the iffeence in angle of the voltages of a seach coil locate acoss phase a of the stato an the teminal voltage appeaing at the stato teminal of phase a This metho was not an ieal way to measue the toque angle because the oscilloscope use to measue the angle between the two voltages gave a vaying eaout even though the loa an spee of the geneato wee constant An aveage of the numbes was taken an use as the toque angle It woul have been much easie an pobably moe accuate) to have a commecially available toque angle measuing evice; howeve, no such evice was available Nevetheless, the stong cooboation between measue an peicte esults suggests that the metho use was an acceptable means of obtaining the toque angle Once the stato voltages, cuents, an toque angle ae known, the q voltages an V =V s Cos δ ) V s = -V s Sin δ ) I = I s Cos γ ) I s = - I s Sin γ ), 8) cuents can be foun by the following elations: whee V s is the peak line to neutal voltage, I s is the peak stato cuent, an γ is the sum of the toque angle δ an the powe facto angle Since the powe facto is unity since the geneato is feeing a puely esistive loa), then γ is equal to δ With the q voltages an cuents an the stato esistance known, the inuctance in the q axis can easily be foun an is given as

16 8 V s - s = - I ω I s 9) The inuctance in the axis an the magnet flux linkage ae not as easy to fin as the q axis inuctance The two tems ae containe in the same equation an ae, in a sense, couple togethe One metho of fining the magnet flux involves unning a no loa test on the PM machine fo a ange of fequencies an measuing the teminal voltage of the machine an the voltage acoss the teminals of the seach coil An empiical elationship between the seach coil voltage an the magnet voltage an thus the magnet flux linkage) can be evelope since, at a no loa conition, the teminal voltage of the machine is equal to the magnet voltage Figue 4 shows a plot of the ms voltage of the magnet vs the ai gap voltage fo both the seies connection high voltage) an the paallel connection low voltage) of the stato wining of the PM machine The low voltage connection was not use in any of the expeiments epote in this thesis except fo the one just escibe) The main eason fo this is that, since the machine is being opeate in geneato moe, one woul nomally like a high teminal voltage an the low voltage connection is, as one woul expect, one half of the teminal voltage of the high voltage connection fo any paticula opeating fequency This bings up one othe point of inteest, which is that one woul ieally like to have a geneato having a magnet voltage geate than 1 volts ms when opeating at 6

17 E o h v = * V a g o E o l v = 8 9 * V a g E o V ) A i g a p v o l t a g e m V ) Figue 4 Measue line to neutal ms teminal geneato voltage vs ai gap voltage fo no loa conition fo machine connecte in high an low voltage stato connections

18 Hetz since most loas wee esigne to opeate at o nea that paticula voltage Although not shown explicitly on Figue 4, the machine is opeating at 6 Hetz when the ai gap voltage is appoximately 5 mv This opeating point coespons to a magnet voltage of about 88 volts line to neutal ms which is cetainly low if one wante to ive, fo example an inuction moto with it As will be seen in subsequent chaptes, shunt capacitos place at the teminals of the PM machine boosts the teminal voltage, but a bette metho woul be to esign a PM machine having a magnet voltage of aoun 1 volts when opeating at 6 Hetz Incientally, the lowe voltage of this IPM machine is not an inication of poo esign; athe, it is an inication that it was esigne to be use as a moto The empiical elationship between the ai gap voltage Vag) an the magnet voltage on a pe phase ms basis E o ) was foun to be E = 168 * V o ) ag With this elationship establishe, the magnet voltage an thus the magnet flux) coul be appoximate une loa conitions by measuing the ai gap voltage at each opeating conition While this metho is not entiely accuate since, une loa conitions, the voltage acoss the seach coil is also affecte by the cuent flowing in the mutual inuctances of the an q winings of the stato, the appoximation seems easonable an, absent the use of finite element analysis, thee is little othe option but to use this metho if the axis inuctance an the magnet flux tems ae to both emain as functions of the opeating conitions

19 foun by 1 Afte the magnet flux tem has been etemine by e = E o / ω), then s can be s V = - s I ω I s - ω e 1) The plots of, s, an e ae given in Figues 5-8 The empiical elationships of the paametes as a function of peak stato cuent I s ae given as 1 4 ln ) = -1 Is + 9 Is - Is Is ln ) = -11Is + 51Is -1 Is Is +155 s e = Is -41Is + 8 Is ) In a conventional synchonous machine with a fiel wining, the axis inuctance is lage than the q axis inuctance; howeve, compaing the magnitues of an s in Figues 5 an 6, it can be seen that is lage This phenomenon, calle invese saliency, is cause by the magnet epth appeaing as basically an ai gap in the -axis As the machine is loae, it can also be seen that the magnet voltage E o fist inceases an then eceases The initial incease in magnet voltage is ue to the fact that, at vey light loas, the bige becomes highly satuate an much of the magnet flux flows though it an oes not contibute towas a useful ai gap voltage It was mentione ealie in this section that the eason the paametes wee mae functions of the peak stato cuent an not the total flux linkage was that the when the

20 5 + E x p e i m e n t - F i t q s [ H ] I s p e a k [ A ] Figue 5 Measue values of q axis inuctance vs peak stato cuent

21 1 1 + E x p - F i t s [ H ] I s p e a k [ A ] Figue 6 Measue axis inuctance vs peak stato cuent

22 4 1 M a g n e t i c F l u x [ W b ] E x p - F i t I s p e a k [ A ] Figue 7 Measue magnetic flux vs peak stato cuent

23 5 5 q s H ) M u t u a l F l u x [ W b ] Figue 8 Measue q axis inuctance vs peak mutual flux

24 6 paametes wee plotte as a function of the total mutual flux linkage mm, thee wee egions in which two possible solutions exist An example of this is shown in Figue 8 whee the q axis inuctance is plotte vs the mutual flux linkage It can be seen fom the figue why etemining fom a given value of mm woul be ifficult Fo example, if mm was given as 8 Wb, then coul eithe be 1 o 15 H The same poblem was pesent fo the paametes s an e as well

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