新型三相磁通切换型双凸极永磁电机电感特性分析_英文_

2007年11月电工技术学报Vol.22 No.11 第22卷第11期TRANSACTIONS OF CHINA ELECTROTECHNICAL SOCIETY Nov. 2007 Inductance Characteristics of 3-PhaseFlux-Switching Permanent Magnet Machine WithDoubly-Salient StructureHua Wei Cheng Ming（Southeast University Nanjing 210096 China）Abstract In this paper, the inductance characteristics of a novel 3-phase 12/10 pole (12-stator- tooth/10-rotor-pole) flux-switching permanent magnet (FSPM) machine with doubly-salient structure is investigated based on finite element (FE) analysis. Firstly, the topology and operation principle of the machine are presented. Secondly, the apparent (static) and incremental (dynamic) inductances are defined. Then, the conventional calculation method of transforming the 3-phase inductances in stator frame into dq-axes rotor frame is performed, which consumes lots of time. To avoid it, a newly simplified and fast method is proposed to obtain the d-axis and q-axis inductances directly and accurately at only two special rotor positions. The proposed method is validated by the comparison of the predicted results in two ways. The experimental measurement verifies the predicted results.Keywords: Flux-switching, doubly-salient, permanent magnet (PM) machine, inductance, finite element (FE) analysis1 IntroductionConventional permanent magnet (PM) brushless machines usually have magnets on the rotor. However, brushless machines having magnets on the stator[1], namely doubly-salient permanent magnet (DSPM) machines[2-4], flux-reversal permanent magnet (FRPM) machines[5], and flux-switching permanent magnet (FSPM) machines[6-9], have recently been the subject of considerable research. Since inductances are essential parameters for the dynamic performance predictions of PM brushless machines, which significantly affect the output torque/power and the field-weakening capability, it is necessary to predict inductance accurately for the control system design by taking saturation-effect into account due to the inductances gradually reduce with the increase of armature current. For those novel machines, both the self-inductance inductance and mutual-inductance are mostly calculated by the conventional method step by step for each phase based on finite element (FE) analysis[3,5,6], which consumes much time and a large quantity of data need to be dealt. In this paper, in addition to the conventional way of transforming the inductance in stator frame into rotor frame, a newly simplified and fast method is proposed which can directly calculate d-axis and q-axis inductances at two special rotor positions avoiding the transforming procedure, significantly saving time and workload. The proposed method is validated by the comparison of the predicted results from two methods and the experimental results with acceptable accuracy.2 Topology and operation principleFig.1 shows the cross-sections of a 3-phase, 12/10-pole FSPM machine. It can be found that the rotor of the machine is similar to that of a switched reluctance (SR) motor. In addition, the concentrated windings, also same to SR motors, are employed, which leads to low copper consumption and low copper loss due to short end-windings. In the FSPM machine the concentrated coil is wound around the two adjacent teeth with a piece of magnet in the middle. Compared to SR motors, the main difference lies in the configuration of magnets in the stator, containing 12 segments of “U”-shape magnetic cores, between which 12 pieces of magnets are inset pre-magnetized circumferentially in alternative opposite directions. Unlike the conventional PM machinesThis project supported by National Natural Science Foundation of China (No.50377004), SRFDP (20050286020) and the Foundation for Excellent Doctoral Dissertation of Southeast University.Received November 13, 2006; received in revised form March 5, 200722电 工 技 术 学 报 2007年11月having magnets in the rotor, the placement of both magnets and windings in the stator is favorable for cooling and is desirable for the aerospace and EV applications where the ambient temperature of the machine may be high. The operation principle of FSPM machine can be described in Fig.2 Obviously, both the value and polarity of the phase PM flux-linkage vary versus rotor position [7]. Due to the PM flux-linkage and consequently the back-emf are essentially sinusoidal [6-7], it makes the FSPM machine an excellent candidate for brushless AC drive operation. The detailed analysis of the structure of the FSPM machine is presented in Ref.[7]. Tab.1 gives the corresponding topology dimensions and parameters ofthe analyzed 3-phase FSPM machine.Fig.1 Topologies of FSPM machineFig.2 Operation principle of FSPM machine Tab.1 Topology and parameters of FSPM motorRated speed n r /(r ·min −1) 1500 DC-link voltage U dc /V440Phase number m 3Stator outer diameter D so /mm 128Stator inner diameter D si /mm 70.4Airgap length g /mm 0.35Stack length l s /mm 75 Rotor inner diameter D ri /mm22Stator tooth number P s12Rotor pole number P r 10Rated RMS current I m /A 3.8 Current density J s /(A ·mm −2)5Winding turns per coil n coil 70Peak phase PM flux-linkageψm /Wb0.174Phase winding turns N phase 280 Phase resistance R ph /Ω 1.4263 Definition of InductanceIn generally, inductance can be characterized as the property of a circuit element by which energy is capable of being stored in a magnetic field [10]. The terminal voltage u of a general inductor can be expressed asi d d d d d d d d i iu Ri Ri Ri L t i t t ψψ=+=+=+ （1）L i =d ψ /d i （2）where R —— Resistance of the inductorψ —— Flux-linkage L i —— InductanceIn the absence of ferro-magnetic material, ψ is directly proportional to current i for all values and the inductance can be simply expressed asL a =ψ / i （3）Eq.(3) means that inductances are independent of the current and depended only on the topology of the inductor. In order to distinguish the different definitions of inductance, Eq. (3) is referred to as the apparent or static inductance, whilst Eq.(2) is referred to as the incremental or dynamic inductance [10]. Fig.3 shows the apparent and incremental inductances for a typical ψ-i curve. Obviously, the apparent inductance equals or very close to the incremental one when the electromagnetic system does not include a magnetic material or the magnetic material is not saturated. However, both the apparent and incremental inductances will gradually decrease as the material becomes saturated, and consequently, the incremental inductance will be less significantly than the corresponding apparent inductance.Fig.3 ψ-i curve4 Apparent inductance4.1 Inductance calculation in stator frame 4.1.1 Unsaturated conditionAccording to Eq.(3), the self- and mutual- inductance of the FSPM machine is performed based第22卷第11期花 为等 新型三相磁通切换型双凸极永磁电机电感特性分析 23on two steps to consider the saturation-effect.Firstly, the unsaturated inductance is investigated assuming the PMs being absent, dealt as free space. Whilst, a DC armature current density of J s =5A/mm 2 is injected into each phase winding respectively, corresponding to the rated current of 3.8A. In Fig.1a there are totally four coils contributing to one phase, e.g. coil A 1-A 4 for phase A. Due to the topology symmetry, coils A 1 and A 3 are identical from the viewpoint of magnetic circuit, and so do coils A 2 and A 4. Suppose only one turn wound on the corresponding tooth belonging to each coil, i.e. the winding turns of one coil n coil =1. Thus, for coil A 12a1a1a1a1coila1as slot s coils slot sL n i J A k n J A k ψψΦΛ==== （4）where L a1——The self-inductance of the coil A 1ψa1——Flux-linkage of the coil A 1 Φa1——Flux of the coil A 1Λa1——Self-permeance of the coil A 1 i a ——The injected DC armature current J s ——Current densityA slot ——The half area of one slotk s ——The slot packing or filling factorBased on finite element analysis (FEA), the armature reaction field with the rated current of phase A is shown in Fig.4a. Then, with the flux calculated, the unsaturated inductance can be obtained by Eq.(4). Fig.5a shows the corresponding self-inductances per turn of coil A 1 and coil A 2 as well as their sum under unsaturated field, i.e. neglecting the existence of magnets. Due to the asymmetry of magnetic circuit for coil A 1 and A 2 as the rotor rotates in a period of 36°m ech[7], there is a periodical difference between L a1 and L a2 as shown. Hence, if the four coils are connected serially to compose one phase, e.g. phase A, the relationship between phase inductance L a and coil inductances (L a1～L a4) are expressed as followsL a =L a1+L a2+L a3+L a4=2(L a1+L a2) （5）（a ）I m =3.8A （b ）Open-circuit (PM)Fig.4 Magnetic fielddistributions24 电工技术学报 2007年11月Fig.5 Apparent inductances of FSPM machineObviously, the combined phase inductance is more sinusoidal than the individual coil one, which indicates further that the FSPM machine is more suitable for AC operation. For mutual-inductances, similarly, Fig.5b shows the results of all mutual-inductances of three phase windings. Apparently, M ab is identical to M ba, for example. It is noted that the mean value of mutual-inductance is almost half of that of self-inductance due to the special machine topology. The relationship between the self-inductance and the mutual-inductance is shown in Fig.6, Fig.5c shows the unsaturated phase inductance including the self- and mutual-inductance, respectively. Fig.5d is the unsaturated 3-phase self-inductances, indicating the perfect symmetry between them with elec60D phase shift. Hence, the apparent inductances are obtained without taking the saturation-effect into account.Fig.6 Relationship of self- and mutual-inductances4.1.2 Saturated conditionSecondly, to investigate the saturation-effect on the inductance characteristics, the existence of magnets is considered, i.e. both the PM excitation and the armature excitation take effect simultaneously. In this case, the open-circuit (PM) field should be pre-analyzed to obtain the constant PM flux-linkage ψm. Fig.4b gives the magnetic field distribution excited solely by magnets. Hence, saturated inductance can be calculated based on the combined field both excited by the armature currents and the magnets. Hence, Eq.(4) can be re-written as follows2a1m a1m a1m a1coila s slot s coil s slot sL ni J A k n J A kψψψψΦΦ−−−===（6）where ψm——The flux-linkageΦm——Flux per coil produced by magnetsDue to the armature reaction may strengthen or weaken the PM magnetic field, so two cases are considered here by injecting the current density of 5A/mm2 and −5A/mm2 respectively into one phase windings. respectively. Fig.5e and Fig.5f shows the comparison results of self- and mutual-inductance under three loadings, i.e., only armature field, pro- magnetized field and de-magnetized field, respectively. Obviously, due to the saturation effect, both the self- and mutual-inductance are significantly reduced compared to the unsaturated results. However, in both the saturated fields, the average value of inductances are almost the same, i.e. the inductances under PM strengthening action is similar to that under PM weakening action due to bipolar PM flux-linkage, which is different from the unipolar PM flux-linkage in DSPM machine[3]. Moreover, the waveforms of第22卷第11期花 为等 新型三相磁通切换型双凸极永磁电机电感特性分析 25saturated inductances including self- and mutual- inductances are quite different from those on the unsaturated condition with the reversal rotor position of peak and bottom values. It should be noted that this difference will play an important role in the relative values of d-axis inductance L d and q-axis inductance L q as presented in the followings.4.1.3 Inductance calculation in rotor frameUp to now, the apparent inductance in stator frame reference has been obtained. However, since the 3-phase FSPM machine is proposed as a synchronous motor with the sinusoidal inductances, the machine can be analyzed and controlled in the rotor frame, i.e. the classic dq-axes model shown in Fig.7. Considering this, due to L =ψ/i , through the famous Park-transform, the L d and L q can be obtained as followsL (d,q,0)=P ψ(a,b,c)i (a,b,c)−1P −1=P L (a,b,c)P −1 （7）whereo o o o cos cos(120)cos(120)2/3sin sin(120)sin(120)1/21/21/2θθθθθθ⎡⎤−+⎢⎥=−−−−+⎢⎥⎢⎥⎢⎥⎣⎦P （8）ψ(a,b,c)=a b c ψψψ⎡⎤⎢⎥⎢⎥⎢⎥⎣⎦（9） i (a,b,c)=a b c i i i ⎡⎤⎢⎥⎢⎥⎢⎥⎣⎦（10）Fig.7 Definition of dq-axes of FSPM machineAccording to Eq.(7), the self- and mutual-inductance can be transformed into L d and L q . Fig.5g shows the transformed unsaturated L d and L q based on the unsaturated L (a,b,c). It can be seen that the variation of L d and L q in terms of the rotor position is so small that they can be regarded as constants, and the other components including L 0, L dq , L d0, L q0 are almost zero, satisfying exactly the request of the dq-axes model. Similarly, the L d and L q in rotor frame are also affected by the armature field as that does on self- and mutual-inductance in stator frame. Fig.5h shows the saturated L d and L q based on the Park-transform. It is interesting that the relativevalues between L d and L q are reversely changed, i.e., under unsaturated condition: L d ＞L q ; under saturated conditions: L q ＞L d . This unique characteristic should be paid more attention in the design and control of the machine since the relative values between L d and L q affect the control algorithm and the flux-weakening capability of the machine significantly [10]. The small value of L d will reduce the flux-weakening capability of the FSPM machine with the fixed PM flux-linkage ψm , and rated current I a , as shown ink fw =L d I a /ψm （11）where k fw ——The factor to reflect the flux-weakeningcapability of the machineFor the FSPM machines adopting ferrite material, the flux-weakening capability can reach infinite speed in theory [6] due to lower open-circuit flux-linkage ψm . However, in the case of NdFeB material, the merit will discount due to higher saturated ψm and smaller L d with the same rated current I a .To verify the other inductance components transformed into dq-axis frame are almost zero, Fig.5i shows the L 0, L dq , L d0, and L q0 per turn under three different loadings. It can be seen all the waveforms vary nearly zero. Compared with the value of L d and L q , they can be neglected. Further, to analyze the pulsation of the transformed L d and L q , a ripple factor called k rip is defined as followsk rip_d =max[(L d_max −L d_ave ),(L d_ave −L d_min )]/L d_ave （12）k rip_q =max[(L q_max −L q_ave ),(L q_ave −L q_min )]/L q_ave（13） where L d_max , L d_min , L d_ave ——The maximum, minimumand average of the trans- formed d-axis inductanceFor L q , the equations can also be adopted. Tab.2 lists the calculated results. It can be seen, the maximum error between the average and the offset is less than 5%, indicating the dq-axes inductances of FSPM motor can be expressed as constant values. Besides, the average values of other components are negligible compared with L d and L q .Tab.2 Inductance components in dq-axis frameUnit: µHL d_avek rip_d (%)L q_avek rip_q(%)L 0_ave L dq_ave L d0_ave L q0_aveUnsaturated5.0810.859 4.165 1.822 0.093 0.001Pro-magnetized 2.488 4.249 3.133 1.945 0.093 0 −0.0390.013De-magnetized 2.559 4.151 3.138 1.939 0.091 00.040−0.013Note: All the results in the table are based on the first method, i.e., Park-transform.26电 工 技 术 学 报 2007年11月4.2 The simplified calculation: two-position method The d- and q-axis synchronous inductance L d and L q can be deduced from the calculated self- and mutual-inductance L (a,b,c) as discussed above. However, both L (a,b,c)and L d and L q are dependent on the working point of both hard and soft magnetic materials when saturation is considered. As a fact, it will cost too much effort and computing time to calculate self- and mutual-inductances in the stator frame because the finite element (FE) analysis has to be repeated over a whole rotor pole pitch step by step to get the complete waveforms of L (a,b,c) before transforming them into L d and L q . Thus, a simplified and fast approach to avoiding the time-consuming procedure is proposed based on the dq-axes theory and the equations in the following sections, in which the d- and q-axis inductances can be directly obtained according to only two special rotor positions, so it is called two-position method.When the rotor position θr =0, i.e. the d-axis lags the center of magnet of coil A 1 by 9°, shown in Fig.7. According to Eq. (8), if the machine is supplied with the DC currents as I a =I , I b =−I /2, and I c =−I /2, then only the d-axis current exists, i.e. I d =I and I q =0. Thus, the corresponding d-axis flux-linkage d ψ can be calculatedd a b c 211()322ψψψψ=−− （14）Considering the defined (3) of apparent inductanced m d dL I ψψ−= （15）where ψm ——d-axis PM flux-linkage transformedinto rotor frame, equal to the peak value of the PM flux-linkage per phase in stator frameSimilarly, keeping the supplied armature currents unchanged, i.e. I a =I , I b =I c =−I /2, rotating the rotormech 9D clockwise, i.e. θr =−elec 90D(the rotor pole number is 10), then only the q-axis current exists, i.e. I d =0, and I q =I . Hence, the q-axis inductance can be calculated in the same way, which is only caused by I qL q =ψq / I q （16）By Eq.(15) and Eq.(16), the d- and q-axis inductance can be obtained only by these two special positions avoiding the repeated calculations of inductance in the stator frame step by step. Tab.3 compares the corresponding predicted results underunsaturated condition (only armature field), as well as saturated conditions (pro-magnetized field and de-magnetized field), respectively by the conventional method and the proposed two-position method. It is obviously that the predicted L d and L q are in good agreements by two methods under three conditions, effectively validating the two-position method.Tab.3 Apparent inductances predictionUnit: µHL da(abc-dq)L da (θr =0°)L qa (abc-dq)L qa (θr =−9°mech )Unsaturated 5.081 5.071 4.1654.167Pro-magnetized 2.488 2.345 3.133 3.128 De-magnetized2.559 2.6713.138 3.128Note: L da and L qa means the apparent d- and q-axis inductance, respectively.5 Incremental inductanceIn order to further accurately predict the dynamic behaviour and performance of electromagnetic systems with saturation, a more accurate knowledge of the value of the incremental inductance (L i =d ψ/d i ), rather than the apparent inductance (L a =ψ/i ) is required because the inductance voltage terms d ψ/d t encountered in such models can be readily expressed as (d Ψ/d i )×(d i /d t ) when saturation is an important factor [10].Since iron saturation has to be considered, the d- and q-axis synchronous inductances can be defined in terms of the apparent and incremental inductances respectively as followsd di d q qi q d d d d L I L I ψψ⎧=⎪⎪⎨⎪=⎪⎩（17） d m dad qqa q L I L I ψψψ−⎧=⎪⎪⎨⎪=⎪⎩（18） where L di , L qi ——The incremental dq-axes inductances,respectivelyL da ,L qa ——The apparent dq-axes inductancesrelative to the open-circuit working point, respectivelyBoth apparent and incremental inductances need to be used in order to account for saturation and to第22卷第11期花 为等 新型三相磁通切换型双凸极永磁电机电感特性分析 27predict the dynamic behavior and performance accurately in the dq-axes system.To calculate the incremental inductance, on the base of the results above, which are obtained from a DC armature currents density J s =5A/mm 2 field, simultaneously to keep the PMs work points almost the same, a new incremental DC armature field with a current density of 6A/mm 2 is applied. Thus, Eq.(17) can be modified asd d d2d1di d dd2d1q q q2q1qi q q q2q1d d d d L I I I I L I I I I ψψψψψψψψ∆−=≈=∆−∆−=≈=∆− （19）The incremental inductance in d-q frame can beobtained by Eq.(19) adopting two-position method. Besides, to compare with the results based on two-position, the incremental inductance in stator frame is also calculated as21212121k k kk k k j j kj k k L I I M I I ψψψψ−=−−=− （20）Here, k , j denotes either of (a,b,c). After all incremental inductances in stator frame are calculated, they are transformed into the incremental inductances in rotor frame as done in the section of apparent inductance. Tab.4 compares the results from the two methods, showing a good agreement between them. It can also be found that the incremental inductances are close or slightly lower than the corresponding apparent values, in consistent with the theory analysis in section III.Tab.4 Incremental inductances predictionUnit: µHL di (abc-dq)L di (θr =0o )L qi (abc-dq) L qi (θ=−9o mech )Unsaturated 5.027 5.019 4.123 4.115 Pro-magnetized 2.410 2.109 3.138 3.106 De-magnetized 2.5292.5483.1503.062Note: L di and L qi means the incremental d- and q-axis inductance, respectively.6 Experiment validationThe prototype of the analyzed 3-phase FSPM motor is shown in Fig.8a. To verify the predicted inductance based on FE analysis, the experimental results of the self-inductance of phase A is comparedwith the predicted one shown in Fig.8b. It can be seenthe good agreement between the two waveforms is obtained, validating accuracy of the proposed methodto calculate the inductances of FSPM motor.（a ）Prototype of the 3-phase FSPM motor（b ）Comparison of the predicted and measured L aaFig.8 Experiment validation of phase self-inductance7 ConclusionThis paper investigates the inductance characteristics of a 3-phase FSPM machine based on FE analysis. The apparent and incremental inductances are defined firstly. The conventional calculation method of transforming the inductance in stator frame into rotor frame is performed. Since the conventional method cost too much computation time, a newly simplified and fast method is proposed to directly calculate the d- and q-axis inductances based on two special rotor positions. The comparison between the results obtained by conventional and proposed methods verifies the effectiveness and accuracy of the proposed two-position method. Experimental results on the prototype verify the predictions. The proposed method lays a foundation for the further performance analysis, design and control of the FSPM motor.Acknowledgments : Hua Wei thanks the University of Sheffield for providing a one-year visiting studentship.Reference[1] Rauch S E, Johnson L J. Design principles of28 电工技术学报 2007年11月flux-switching alternators[J]. AIEE Trans., 1955, 74III: 1261-1268.[2] Liao Y, Liang F, Lipo T A. A novel permanent magnetmachine with doubly saliency structure[J]. IEEE Trans. Industry Applications, 1995, 3 (5): 1069-1078.[3] Cheng M, Chau K T, Chan C C. Static characteristicsof a new doubly salient permanent magnet machine[J].IEEE Trans. Energy Conversion, 2001, 16 (1): 20-25.[4] Cheng M, Chau K T, ChanC C, et al. Control andoperation of a new 8/6-pole doubly salient permanentmagnet motor drive[J]. IEEE Trans. on Industry Applications, 2003, 39 (5): 1363-1371.[5] Deodhar R P, Andersson S, Boldea I, et al. Theflux-reversal machine: a new blushless doubly-salientpermanent-magnet machine[J]. IEEE Trans. IndustryApplications, 1997, 33 (4): 925-934.[6] Hoang E, Ben Ahmed A H, Lucidarme J. Switchingflux permanent magnet polyphased machines[C]. 7thEuropean Conf. Power Electronic and Applications,Trondheim, Norway, 1997, 3: 903-908.[7] Hua W, Zhu Z Q, Cheng M, et al. Comparison offlux-switching and doubly-salient permanent magnetbrushless machines[C]. 8th International Conf. on Electrical Machines and System, Nanjing, China, 2005. 1: 165-170. [8] 花为, 程明, Zhu Z Q, 等. 新型磁通切换型双凸极永磁电机的静态特性研究[J]. 中国电机工程学报,2006, 26 (13): 129-134.Hua Wei, Cheng Ming, Zhu Z Q, et al. Study on static characteristics of novel flux-switching doubly- salientPM machine[J]. Proceedings of the CSEE, 2006, 26(13): 129-134.[9] 花为, 程明, Zhu Z Q, 等. 新型两相磁通切换型双凸极永磁电机的静态特性研究(英文)[J]. 电工技术学报, 2006, 21(6): 70-77.Hua Wei, Cheng Ming, Zhu Z Q, et al. Study on static characteristics of a novel two-phase flux-switching doubly-salient permanent magnet machine[J]. Transactionsof CES, 2006, 21 (6): 70-77.[10] Chen Y S. Motor topologies and control strategies forpermanent magnet brushless AC drives[D]. PhD Thesis, UK: the University of Sheffield, 1999.Brief notesHua Wei，male, born in 1978, Ph D, his areas of interests include novel permanent magnet machines design, analysis and control.Cheng Ming，male, born in 1960, Ph D, professor, his teaching and research interests include electrical machines, motor drives and power electronics.新型三相磁通切换型双凸极永磁电机电感特性分析花为程明（东南大学电气工程学院南京 210096）摘要本文基于有限元法研究了一种新型三相12/10极切换磁通型双凸极永磁电机的电感特性。