Efficiency Optimization of PMSM Based Drive System

Transcription

1 1 IEEE 7th Intenational Powe Electonics and Motion Contol Confeence - ECCE Asia June -, 1, Habin, China Efficiency Optimization of PMSM Based Die System Waleed Hassan and Bingsen Wang Depatment of Electical and Compute Engineeing Michigan State Uniesity 1 Engineeing Building East Lansing, MI s: hassanwa@msu.edu; bingsen@eg.msu.edu Abstact The poposed wok in this pape is a combination of analytical and numeical methods that ae able to calculate hamonic losses of pemanent magnet synchonous machine PMSM at any opeating point with less execution time and adequate accuacy. Both fundamental and hamonic losses of machine in conjunction with inete losses ae modeled such that inete-moto efficiency can be tested fo diffeent conditions. Numeical simulation of field oiented contol hae been conducted and esults include cuent, modulation index, powe facto, switching fequency and all aiables in die system. Fumoe, Maximum efficiency contol stategy has been chosen such that die system can wok at most efficient condition at all opeation points. The poposed wok is pesented in context of die system that is based pemanent magnet synchonous machine PMSM. Howee, poposed method can be extended to induction moto die system o any moto use PWM in contol of inete output oltage. I. INTRODUCTION Loss modeling and contol of high pefomance moto die system has attacted significant eseach attention [1] []. The educed system losses not only can be tanslated to highe efficiency and loweed opeating cost, but also can educe mal stess on aious components o sub-systems and in etun will impoe eliability of oeall system. The eduction of life-cycle cost due to impoed eliability is of geat pactical significance in many of cost-sensitie applications of electic die systems. The efficiency and powe density of pemanent magnet synchonous machines PMSMs hae been studied by seeal authos [], [] []. It has been shown that PMSM can achiee significantly highe efficiencies than induction motos when used in adjustable dies. Analytical method fo estimating fundamental losses of suface mounted PMSM has been poposed in []. Seeal eseaches hae utilized electical model of PMSM that includes a paallel esistance that accounts fo coe losses in high pefomance applications [], []. It can be concluded that both coppe and coe losses need to be accounted fo in analysis and contol of high pefomance moto dies. The coe losses ae due to both fundamental and hamonics excitations. The existing optimization methods of die system is focused on fundamental losses of electic machines. The inete losses ae often not counted in optimization pocess although modeling of inete loesses is aleady known [7]. A study on hamonics losses of tansfomes has been epoted []. This pape poposes an integatie appoach to optimize efficiency of die that including fundamental losses and losses due to switching of inete. The est of pape is oganized as following. Section II poides analytical model of PMSM hamonic losses based on amatue eaction field poduced by theephase cuent of stato winding. Double Fouie seies fo natually sampled sinusoidal PWM SPWM has been utilized fo ealuation and analysis. Section III pesents analyticalnumeical method of loss calculation fo thee-phase inete contolled by SPWM. The optimal-efficiency contol stategy is implemented in section IV. The esults and conclusion ae pesented in section V. II. HARMONICS LOSS MODELING OF PMSM The coe losses in machine consist of two components, i.e., hysteesis and eddy cuent losses. Both types of coe loss ae due to time aiation of flux density in coe. The complete expession fo coe losses in [W m ] is: P cd = K e nω s B p,n K h nω s B β p,n 1 whee K e is eddy loss coefficient; K h is hysteesis loss coefficient; B p,n is peak flux density of nth-ode; and ω s is fundamental angula fequency of applied oltage [9]. A. Hamonic Flux Density In this section analytical model fo suface mounted PMSM moto hamonic loss is deeloped based on wok published in whee amatue eaction field poduced by stato winding is pedicted. The amatue eaction field poduced by a thee-phase stato winding is: B φ α,, t = B p sinnω t α θ n whee ω = P p ω m is electical oto angula in [ad s 1 ]; amplitude of flux density B p due to nth /11/$.@1 IEEE

2 ode amatue eaction cuent is detemined by: W 1 B p,n = µ I n πδ K sok dp F B. Eddy Cuent Loss n Once maximum amplitude of flux density has been detemined, which esults fom amatue space ecto cuent, pocedue of finding accuate model fo coe loss is shown in following appoximate expession. P ed = K e nω Bp,n n Substituting into and defining olume in which loss is ealuated, expession fo eddy coe loss as will be eached as P e = K em nω I n whee ρ i is mass density of steel coe mateial in [kg m ]; V is steel coe olume in [m ]; and i poduct gies weight of coe mateial in [kg]. K em is defined as: K em = ρ i V K e µ W πδ C. Hysteesis Loss 1 K sok dp F Following same pocedue as last section, an expession fo hysteesis loss can be eached as pesented in. P h = K hm nω I n whee K hm is defined as: W 1 K hm = ρ i V K h µ πδ K sok dp F The final expession fo coe loss in [W] is gien by: P c = P e P h = K em nω I n K hm nω I n 7 Close examination of K em and K hm shows that y ae constant fo a gien machine design since y solely depend on machine dimensions. D. SPWM Hamonics Analysis If losses associated with fundamental is excluded fom 7, coe hamonic loss can be expessed as: P c,h = P e,h P h,h = K em nω In K hm n= n= nω I n It is clea that moto hamonics loss can can be fully ealuated if amplitudes of cuent hamonics detemined. Fo thee-phase inete with two-leel phase legs, a balanced set of thee-phase line-line output oltages is obtained if efeences ae displaced by 1. Unde se conditions, line to line oltages ae gien fo natually sampled PWM by [1] ll = V dc π Vdc m=1 M cosω o t π/ J mπ n M m nπ sin sin nπ m cos mω c t nω o t π/ n= n The majo significant side-band hamonics ae at fequency of f h = f s ± f, f s ± f, f s ± f. The suface mounted PMSM can be analyzed as a single phase cicuit that consists of seies impedance and back emf since eactances on d-axis and q-axis ae identical. At hamonic fequencies,,9 stato esistance and back emf ae neglected so that moto modeled as synchonous inductance only. The moto phase cuent can be detemined by phase oltage and synchonous inductance of stato winding. With assumption of sta connection of PMSM, phase cuent is gien by: 9 I = ll/ ωl s 1 Plugging equation 9 in equation1 yields hamonics of stato cuent. Once stato hamonic cuent is detemined, hamonic losses ae eadily to be calculated. III. LOSS OF VOLTAGE SOURCE INVERTER The majoity of hamonic losses of moto, which is caused by PWM caie fequency, can be minimized by inceasing inete switching fequency. Noneless, inete switching loss will concuently incease, as detailed in subsequent section. Theefoe, fo achieing global optimization of system, one needs to define alue of optimal fequency that minimizes moto hamonic loss and inete losses toge. The aim of this section is to poide analytical model fo calculation of powe losses in IGBT-based oltage souce inete used in PMSM die system. The inete paametes of semiconducto switches in inete hae been extacted fom data sheets of FAIRCHILD, RURG and HTGTNA modules. A numbe of diffeent methods hae been poposed to estimate losses of oltage souce inetes. The fist is based on complete numeical simulation of cicuit by specific simulation pogams with integated o paallel-unning losses calculations. The second is to calculate electical behaiou of cicuit based on analytical behaiou model [7]. Fo accuate estimation of losses duing tansient time and all opeating conditions, a hybid model has been deeloped. In this model, opeating conditions that ae esulted fom numeical simulation of moto die system ae utilized in analytical model. The losses in a powe-switching deice consist of conduction losses, switching losses and off-state blocking loss. 1

3 A. Switching Losses of VSI The equation fo switching loss P ls of a VSI with sinusoidal ac line cuent and with IGBT switching deices is gien by: P ls = π f s E on,i E off,i E off,d Vdc V IL I 11 whee f s is VSI switching fequency; V dc is dc link oltage; I L is peak alue of ac line cuent that is assumed sinusoidal; E on,i and E off,i ae tun-on and tunoff enegies of IGBT, espectiely; E off,d is tun-off enegy of powe diode due to eese ecoey cuent. B. Conduction Losses of VSI In contast to switching losses, conduction losses ae diectly depending on modulation function. Fo caie based PWM method, conduction losses of IGBT and diode can be expessed as: P lc,i = V CE, π CE, π P lc,d = V F, π I L CE, π 1 M π I L π M 1 M π π M cos φ cos φ cos φ cos φ 1 1 whee M is modulation index and φ is displacement angle between fundamental of modulation function and load cuent. The total inete losses at diffeent opeation points will be analyzed in next subsection. C. Total Losses of VSI The total losses of VSI hae been ealuated based on paametes that ae listed in Table I. TABLE I PARAMETERS OF THE SEMICONDUCTOR DEVICES OF THE VSI. Powe Deices IGBT Diode Paametes I ef A V ef V E on,i mj. E off,i mj.9 V CE, V 1. CE, mω 1 F, mω V F, V 1. E off,d mj.7 A plot of switching loss fo a ange of switching fequencies is shown in Fig. 1. It is clea that switching loss is popotional to switching fequency and maintains constant fo a gien load cuent and switching fequency. The conduction loss depends on load displacement angle, modulation index and load cuent. Fig. illustates aiations of conduction losses with load displacement angle unde diffeent loading conditions. The plot of conduction loss equation shows that loss is maximized when displacement angle is zeo and deceases as displacement angle inceases o deceases fom zeo. In field oiented contol of aiable die wods, wods, inceasing inceasing moto moto will will incease incease inete inete losses losses since since cuent cuent and and modulation index will be highe. modulation index will be highe. Switching Powe Losses W Switching Powe Losses W 7 i L = 1 A 7 i i 1 L = 9 A i i 9 L = A i L = A Switching Fequency. Hz.. 1. Switching Fequency Hz 1 Figue Fig. 1..1: Switching Switchinglosses lossesofof VSI VSIat atdiffeent diffeentload loadcuents. Figue.1: Switching losses of VSI at diffeent load cuents. VSI Conduction Powe Losses W VSI Conduction Powe Losses W MI=1, i L =1 A MI=1, i L =1 A MI=., i L =9. A MI=., i L =9. A MI=., i L = A MI=., i L = A Load Displacement Angle degee Fig.. Switching Load losses Displacement of VSI at diffeent Angle degee modulation indices. Figue.: Conduction losses of VSI at diffeent modulation indices. system, Figue.: foconduction a gien lossescommand of VSI at diffeent contolle modulation willindices. assign oltage popotional to that by means of modulation action. Howee, inceasing moto equies highe line to line oltage and causes moto to daw highe cuent fo constant load toque. In o wods, inceasing moto will incease inete 1 losses since cuent and modulation index will be highe. 1 IV. LOSS MINIMIZATION CONTROL STRATEGY The combined coppe losses and ion fundamental losses ae minimized though contol stategy deeloped in [] The contollable losses can be minimized by pope choice of amatue cuent ecto. It is a common pactice to set cuent i d =. As a esult, amatue cuent ecto that is in phase with back EMF is applied. In addition, ieesible demagnetization of pemanent magnets can be aoided. The ecent deelopment of pemanent magnets has bought mateials with high coeciity and high esidual magnetism. Theefoe, seeal contol methods hae been poposed to impoe pefomance of PM moto dies. In such contol methods, d-axis component of amatue cuent is actiely contolled accoding to opeating and load conditions. In next subsection, basic equations of moto model needed to deelop contol algoithm ae pesented. A. Modeling Fundamental Losses of PMSM The steady state d, q model of PMSM is shown in Fig.. The fundamental ion loss consists of hysteesis loss and eddy cuent loss. Heein, se two loss components ae lumped 19

4 i ds ds R s od i od i cd L d L qω i oq ω i qs Σ PI Σ - i ds - Σ ω - PI PI INV.PARK TRANSFORMATION qs ds d, q α, β sα sβ SPWM V dc -PHASE INVERTER θ i qs qs whee R s oq i oq i cq L q L d ω i od ω λ af of output cuents and oto, which is mamatically epesented by: I od = I ds I cd ; I cd = I oq = I qs I cq od L d p L q ω = i od Lq oq L d ω L q p i oq ω λ af. ω I oq ; I cq = 1 λ af L d I od ω 1 The amatue cuent i a, teminal oltage a and toque T e ae expessed as: i a = i d i q; a = d q 1 whee p denotes dt d. Reaanging of equation. leads to equations. and.: a = R s i d ω L q i oq R s i q ω λ af L d i od T E = P [λ af I oq L d L q I od I oq ] 17 The coppe loss P Cu and ion loss P F e ae detemined by: P Cu = [ ] R s i od Lqioq i oq λ af L d i od ω P F e = [ Lq i oq λ af L d i od ] 1 B. Condition fo Minimized Losses The total electic powe loss P E = P Cu P F e can be expessed as a function of i od, T e and ω. Unde steady state, electical loss P E is function of i od. Neeless, fo suface mounted PMSM L d = L q. Hence, simple expession fo P E can yield. By diffeentiation of P E with espect to i od, Loss Minimization Algoithm dθ dt i qs i ds d, q α, β i sα i sβ α, β a, b PARK CLARKE TRANSFORMATION TRANSFORMATION Figue.: Dynamic equialent cicuit of PMSM that includes coe esistance. Fig.. Steady state equialent cicuit of PMSM including coe loss esistance. Figue.: Field oientation contol diagam incldues loss minimization algoithm.. : ds i Fig.. Field oiented contol diagam including loss minimization algoithm. od Rc R s od = R s.1 qs R into a single quantity, which c i oq is epesented oq by coe loss esistance. As shown in Fig. Rc, oltage equations of condition fo i od that minimizes contollable losses od = R ds Rs i PMSM in steady-state ae expessed od. s as: esults has been found. [ ] [ ] Rc oq = ds Iod R qs R s i oq. = R s R [ ] c R s od I od = λ af R s ωl d s qs I oq R s R oq c ωl d R 19 s [ ] [ ] [ ] [ ] 1 In equations. od and., output Lq oltages ω ae functions of stato output cuents = Iod The ole of loss minimization algoithm in field and oltages. The oq dynamic pat L d of ω cicuit indicatesithat oq output λ af oltages ae functions oiented contol system is shown in Fig.. The simulation esults will be pesented in next section. V. SIMULATION RESULTS AND DISCUSSION i s a i s b PMSM The machine paametes of suface mounted PMSM that hae been used in simulation ae listed in Table II. The field oiented contol system includes loss minimization algoithm implemented in MATLAB/SIMULINK. The Moto hamonic losses of machine ae calculated by MATLAB block using aiables that is obtained fom SIMULINK fo aious opeation conditions. The conete losses ae diectly calculated by use of SIMULINK system with paametes that ae geneated by FOC subsystem. The esults and discussion ae pesented in thee subsections; FOC unde loss minimization contol stategy LMCS, VSI-moto fundamental losses, and VSI-moto hamonic losses. TABLE II MACHINE PARAMETERS. Paamete Value Units Numbe of pole pais Stato esistance. Ω Coe loss esistance Ω Rated pm Rated toque Nm Roto flux linkages.7 Wb Rated peak oltage pe phase 1 V Roto inetia J m.179 kgm Viscous fiction coefficient B m 9. 1 Nms L d.1 H L q.1 H A. FOC with Loss Minimization Contol The dynamic esponses of cuents, and toque at ated conditions ae shown in Fig. and Fig., espectiely. The system eaches steady state with smooth dynamic 1

5 isa 1 id = contol stategy [1]. Since suface mounted PMSM has no saliency Ld = Lq, e is no eluctance toque. Howee, influence of deceased flux linkage and applied oltage is still effectie. ids Speed pm ids A Speed* Speed 1 i ds Toque N.m iqs A iqs i qs Te ing fequency and moto phase cuent. Hence, if moto is contolled unde constant 1 T switching fequency, l switching losses ae popotional to moto phase cuent. Howee,.1 1. moto cuent inceases/deceases fo any incease/decease in load toque o/and efeence Time sec.1.. Time sec.... As a esult, switching loss inceases/deceases fo any inceases/deceases in load toque o/and efeence. Altenatiely, conduction loss gien in Equations.9 Fig.. of phase phasecuents, cuents,i ids and i of suface mounted Figue 7.1: Dynamic Dynamic esponse esponse of d and iq ofqssuface mounted PMSM. Figue 7.: Toque dynamicesponse esponseof of suface suface mounted PMSM. Fig...1 Toque andand dynamic PMSM. and is popotional to moto phase cuent, modulation index andmounted displacement PMSM VSI and PMSM Fundamental Losses at i = 9 Efficiency % 7. at i d gien by loss minimization algoithm d 9 As pesented in Chapte, losses of VSI ae diided into two types; conduction losses esponse in appoximate.1 second fo system paametes listed in Table II. The cuent and contolles ae de- and switching7losses. Equation. shows that switching loss is popotional to switch signed by a symmetic optimum appoach which is addessed 9 in []. Fo calculation of coe loss esistance, it is equied that moto uns at ated and toque of sinusoidal 1 1 powe supply such that hamonic losses ae not geneated. Speed pm The fundamental ion loss PF e is calculated by subtacting mechanical and coppe losses fom total output powe. The FiguePMSM 7.: PMSM efficiency esus. Fig. 7. efficiency esus. ion loss esistance, Rc, can be obtained fom fundamental coe losses as in 1, whee measued Rc epesents angle. The modulation index is contolled by load and since any incease in load o ion loss esistance at ated output powe. Pactically B. VSI and PMSM Fundamental Losses Rc depends on opeating conditions. Howee, it is assumed equies highe cuent which is supplied ia highe oltage. Theefoe modulation Asindexpesented in Section III, losses of VSI ae diided constant in this wok. inceases as well to supply equied oltage. Regading displacement angle, its into effect conduction losses and switching losses. Since switchon conduction loss is discussed in Chapte. Figue 7. a illustates elationship ing loss is popotional to switching fequency and moto Fig. 7 shows efficiency of suface mounted PMSM e phase cuent, switching loss inceases/deceases fo any sus fo two contol stategies: zeo id contol stategy and LMC stategy. It is clea that moto efficiency inceases/deceases in load toque o/and efeence. In unde maximum efficiency contol stategy is highe than addition, conduction loss is popotional to moto phase efficiency unde zeo id contol stategy. Howee, diffe- cuent, modulation index and displacement facto. The ence in efficiency is not as emakable fo suface mounted modulation index is contolled by load and since any PMSM as fo inteio PMSM due to following easons. incease in load o equies highe cuent which is In inteio PMSM The negatie d-axis cuent poduces a supplied ia highe oltage. Theefoe modulation index positie eluctance toque due to saliency Ld < Lq, inceases as well to supply equied oltage. Regading and accodingly stato cuent and coppe loss PCu is displacement angle, its effect on conduction loss is discussed smalle compaed with id = contol. Moeoe, in both in Section III. types of PMSMs negatie d-axis cuent educes flux Fig. a illustates elationship between moto fundalinkage. Consequently, oltage V, and ion loss PF e, mental loss and id unde loss minimization contol stategy. ae smalle as compaed with id = contol stategy []. The cue is conex with global minimum loss point at Since suface mounted PMSM has no saliency Ld = Lq, id =. A fo gien paamete in Table II. Fig. b e is no eluctance toque. Howee, influence of shows elationship between inete losses and id. The deceased flux linkage and applied oltage is still effectie. esult shows that minimum loss point occus at id = 11

6 PMSM Fundamental Losses W VSI Losses W Combined Losses W i d efeence A a 11 x x1 x1 x:- y:.9 i d efeence A b x:- m=1 n= y:7.9 1 P h,h K hm = ω m,n I m,n ω ω m=1 n= i d efeence A c The calculated alues ae K em =.79 1 and K Figue 7.: Fig. Moto-inete. Moto-inete fundamental fundamental losses losses esus esus i hm =.1. The hamonic losses in [W] fo any opeating d ati dated ; ated : a PMSM a fundamental losses; of PMSM b VSI and losses; VSI. c Combined fundamental losses of PMSM and VSI. condition is: PMSM fundamental losses; b VSI losses; c Combined fundamental losses P c,h =.79 1 ω m,n I m,n ω ω 1. This esult can be explained as following: ms alue of phase cuent is i d i q. When i d =, it is clea that phase cuent has minimum alue. Howee, it was explained that most inete loss is contolled by cuent. When phase cuent is minimum inete loss will be minimum as well. Fig. c shows elationship between combined losses of moto and inete esus i d. The elationship shows that combined minimum loss alue of inete and moto occus at a point of i d that lies between zeo i d and optimal i d that is geneated by loss minimization algoithm. Fo all conditions of loads and s, it can be obseed that inete minimum loss alue occus at zeo i d, and i d geneated by loss minimization algoithm is always negatie fo same conditions. Moeoe, combined minimum loss alue will stictly lie in egion between zeo i d and i d geneated by minimum loss algoithm. C. VSI and PMSM hamonic losses The moto hamonic losses at any opeating point can be calculated as following. Fist The machine opeates at ated and toque off sinusoidal powe supply. The input powe is n measued and coppe losses and output powe ae calculated, which gies moto efficiency and fundamental coe losses. The same pocedue is epeated except that moto is supplied with SPWM-aiable die contolle unde same toque and conditions. The coe losses that ae calculated with SPWM epesent summation of fundamental and hamonic losses at ated output powe. Subtacting sinusoidal-coe losses fom SPWMcoe losses yields moto hamonics loss at ated toque and. At T e = N m and N = pm calculated hamonics loss is 9. W. Howee, moto efficiency is educed fom 9. to 9 % when switching sinusoidal supply with SPWM-aiable die contolle. In PMSM hysteics losses ae ey small and can be neglected. In this wok hysteics losses alue is assumed to be % of total hamonic losses. : P e,h =.9P c,h ; P h,h =.P c,h The next step is to calculate coefficients K em and K hm K em =.1 P e,h m=1 n= m=1 n= ω m,n I m,n ω ω ω m,n I m,n ω ω The moto hamonics loss, VSI loss and total loss esus switching fequency ae shown in Fig. 9. The esults show that PMSM hamonics loss deceases as switching fequency inceases. Neeless, VSI losses incease as switching fequency incease. The summation of aboe two losses show that cue is conex and has global minimum loss alue at switching fequency of 19 khz. Howee specific optimal switching fequency will depend on paticula set of design paametes of VSI. In o wods, when paametes of VSI such that loss of VSI highe than moto hamonics loss at same ange of switching fequency, cue will be no moe conex and almost will be linea. Fig. 1 shows elationship between PMSM hamonic loss and modulation index. It can be seen that up to M =.7 1

7 PMSM Hamonic Losses W x 1 Switching Fequency Hz specific optimal switching fequency will depend on paticula set of design paametes of VSI. In o wods, when paametes of VSI such that loss of VSI highe than moto hamonics loss at same ange of switching fequency, cue will be no moe conex and almost will be linea. PMSM Hamonics Loss W Modulation Index X=.7 Y=11. VSI Losses W Combined Losses W Switching Fequency Hz x x:1.9e y:1. PMSM and VSI esus switching fequency at ated. hamonics loss inceases as modulation index inceases. As modulation index passes M =.7, hamonics loss begins to decease to locally minimum loss alue at unity modulation index. Because loss is minimum nea unity modulation index, it is suitable to use aiable dc link such that modulation index is kept nea unity at all opeating conditions. VI. CONCLUSIONS The main esults of wok can be summaized in following aspects: The moto loss that is caused by fundamental component of phase cuents can be minimized by using maximum efficiency contol stategy with assumption that coe loss esistance and coppe loss esistance ae constant unde all opeating conditions. The combined moto-inete fundamental losses ae minimized unde maximum efficiency contol stategy in egion in which i d < i d <. Fig. 1. Figue 7.7: PMSM hamonics loss esus modulation index. PMSM hamonics loss esus modulation index. Figue 7. shows combined loss of moto hamonics loss and VSI loss esus switching fequency at diffeent s and ated toque. The esult shows that total losses incease PMSM hamonics loss can be minimized by inceasing conete switching fequency with assumption that SPWM inete is used without any hamonics injection technique. as moto inceases. Also, at low s up to % of ated VSI loss is dominant. When inceases, moto hamonic losses become dominant such that loss cue becomes conex. Figue 7.7 shows elationship between PMSM hamonic Fo alossgien and modulation ange index. of It canswitching be seen that up to fequencies M =.7 hamonics in loss which inceases moto as modulation can index wok inceases. withas acceptable modulation indexpefomance, passes M =.7, choice of minimal switching fequency would minimize conete loss. Fo a gien ange of switching fequencies in which moto can delie acceptable pefomance, choice of optimum switching fequency that will minimize moto-inete losses follows paametic optimization. The optimal loss point is affected by system paametes. Hence, changing of alues of VSI paametes would ay optimal point of moto-inete losses. REFERENCES [1] D. S. Kischen, D. W. Nootny, and T. A. Lipo, On-line efficiency x 1 optimization of a aiable fequency induction moto die, Industy Switching Fequency Hz Applications, IEEE Tansactions on, ol. IA-1, no., pp. 1 1, 19. Fig. 9. Hamonic losses of PMSM, VSI losses and combined hamonic [] R. Colby and D. Nootny, Efficient opeation of pm synchonous losses of PMSM and VSI esus switching fequency at ated. motos, IEEE Tansactions on Industy Applications, ol. IA-, no., Figue 7.: Hamonic losses of PMSM, VSI losses and combined hamonic losses pp. of 1 1, No./Dec [] L. Xu, X. Xu, T. Lipo, and D. Nootny, Vecto contol of a synchonous eluctance moto including satuation and ion loss, IEEE Tansactions on Industy Applications, ol. 7, no., pp. 97 9, Sep./Oct [] S. Moimoto, Y. Tong, Y. Takeda, and T. Hiasa, Loss minimization contol of pemanent magnet synchonous moto dies, IEEE TRAN. on Industial Electonics, ol. 1, no., pp , Oct 199. [] G. Slemon and X. Liu, Coe losses in pemanent magnet motos, IEEE Tansactions on Magnetics, ol., no., pp. 1 1, sep 199. [] R. Schife and T. Lipo, Coe loss in buied magnet pemanent magnet synchonous motos, Enegy Conesion, IEEE Tansactions on, ol., no., pp. 79, jun 199. [7] M. Biehoff and F. Fuchs, Semiconducto losses in oltage souce and cuent souce igbt conetes based on analytical deiation, se. in Poc. Powe Electonics Specialists confeence, ol.,, pp.. [] C. Mi, G. R. Slemon, and R. Bonet, Modeling of ion losses of pemanent-magnet synchonous motos, IEEE Tans. Ind. Appl., ol. 9, no., pp. 7 7, May. [9] Z. Zhu and D. Howe, Instantaneous magnetic field distibution in bushless pemanent magnet dc motos. ii. amatue-eaction field, IEEE Tansactions on Magnetics, ol. 9, no. 1, pp. 1 1, Jan [1] D. G. Holmes and T. A. Lipo, Pulse Width Modulation fo Powe Conetes. John Wiley & Sons,. 1