IPMSM Inductances Calculation Using FEA

Transcription

1 X International Symposium on Inustrial Electronics INDEL 24, Banja Luka, November 68, 24 IPMSM Inuctances Calculation Using FEA Dejan G. Jerkan, Marko A. Gecić an Darko P. Marčetić Department for Power, Electronic an Telecommunication Engineering Faculty of Technical Sciences, University of Novi Sa Novi Sa, Serbia Abstract Accurate etermination of interior permanent magnet synchronous machine s (IPMSM inuctances is very important issue, especially in areas of high-performance rives an systems. This paper presents the metho for calculation of the irect an uarature inuctances of permanent magnet synchronous machine using finite element analysis (FEA, where the calculation of these parameters is base on the etermination of flux linkages. Two types of IPMSMs are investigate, with tangentially an raially magnetize permanent magnets. The results of the calculate inuctances are presente by iagrams an they are iscusse an compare with those obtaine by measurements. Keywors Finite element analysis, irect an uarature axis inuctances, magnetic flux ensity, interior permanent magnet synchronous machine NOMENCLATURE v,v stator - an - axis voltages i, i stator - an - axis currents i a,b,c stator phase currents R s stator phase resistance m permanent magnet flux L,L stator - an - axis self inuctances s stator -axis flux s stator -axis flux ω actual rotor angular spee m el electromagnetic torue m m loa torue J motor inertia p number of pole pairs saliency ratio (L /L J s current ensity vector A z z component of the magnetic vector potential µ permeability of material I. INTRODUCTION The interior permanent magnet synchronous motors (IPMSM have many avantages, such as high power ensity an possibility for spee regulation in wie range of spees [], [7], [9]. IPMSMs are wiely use in high-performance rives such as inustrial robots an high-performance machine tools because of their avantages on high-torue with aitional reluctant component. In recent years, the magnetic an thermal capabilities of the PM have been enhance by employing permanent magnets with high coercitivity []. IPMSMs are use in more an more applications because of their small volume, very goo efficiency, lower moment of inertia, rotor without heat problem, etc. [2]. Because of emans of high-performance rives it is very important to calculate as accurate as possible the values of the parameters of the IPMSM. Of the most important significance are the irect- an the uarature- axis inuctances, as they are etermining corresponing synchronous reactances []. Also, they are the most important parameters when steay state an ynamic moels of IPMSM are evelope [4]. Unlike surface PM motors, which have the same value of inuctance in irect an uarature axes an where all the torue is prouce by the magnet flux, interior permanent magnet motors have ifferent irect an uarature inuctances which results in an aitional torue component calle reluctance torue [5]. The conventional methos of testing for etermination of synchronous machine parameters are often inappropriate in the case of permanent magnet machines, because magnetic fiel prouce by the permanent magnets cannot be cancele uring measurements, an its fiel affects the total level of saturation of iron uring experiments. On the other han, finite element metho provies great opportunities for accurate numerical analysis of IPMSM, because using FEA algorithms allows calculations of funamental fiel uantities (such as flux linkages an store magnetic energy, an also the fiels prouce by PMs can be cancele easily [6]. Stanar experiments are reconstructing parameters of the machine base on uantities which can be measure through electrical connection of the machine, without knowlege of fiel istribution insie of the machine. During the last two ecaes the finite element metho prove to be the most appropriate numerical metho in terms of moeling, flexibility an accuracy to solve the nonlinear Poisson s euation governing the magnetic fiel who's concerne a principal element in calculation of machine parameters [6]. 4

2 II. EQUIVALENT CIRCUIT AND BASIC EQUATIONS Fig. shows the - an -axis euivalent circuits of IPMSM in which magnetic losses ue to variable magnetic fiel in stator core are neglecte. Base on Fig. the mathematical euations of the euivalent axis steay state moel of IPMSM in the rotor reference frame are given with: v v i = Rs i + ωl ωl i + i ωψ m The electromagnetic torue of the IPMSM has two components: funamental magnetic torue (which is proportional to the prouct of the magnet flux an -axis stator current, an the reluctance torue (which is epenent on the saliency ratio an to the prouct of -axis stator current components. It is essential to etermine IPMSM inuctances to preict reluctance torue as an aitional torue component. Base on Fig. torue can be expresse as: ( Ψ i + ( ρ L i i T = p m 2 Figure. - an -axis euivalent circuits of IPMSM a -axis euivalent circuit, b -axis euivalent circuit III. FINITE ELEMENT ANALYSIS OF IPMSM The finite element analysis is use in many areas of technical sciences, such as magnetics, electrostatic problems, heat transfer, flui ynamics etc. All FEA algorithms are base on solution of fiel euations over omain of interest using ivision of that omain with small segments of simple geometric shapes, calle finite elements (in 2-D FEA the most common are triangular shapes, in orer to reconstruct the fiel of the entire omain. Values of fiel variables insie any of the elements are represente using 2-D interpolation functions [6]. These functions are efine on each element using the values of the calculate variable in each noe. ( (2 Knowing the value of variable of interest in every noe of the region, combine with the usage of interpolation functions allows complete efinition of the behavior of the variable fiel on each element. The precision of the metho epens not only on the imensions of elements an their number but also on the type of the interpolation function. As for the numerical metho, the FEA algorithm converges to the exact solution provie to increase the number of subivisions of the solution omain an to ensure continuity of the interpolation function of its first erivatives along the borers of ajacent elements [6]. FEA algorithms use in the problems of electrical machines analysis are base on the evaluation of the magnetic vector potential A. As alreay mentione, 2-D FEA algorithm will be use in this research, an the starting euation for FEA is given by: δ δa z δ δa ( ( ( ( z = δx µ δx δy µ δy Euation ( is use for magneto-static solutions, so there is no time varying of currents or motion of rotor. Aeuate usage of FEA algorithms emans efinition of proper bounary conitions, which etermine how lines of magnetic potential vector pass through areas that separate regions with ifferent magnetic properties. Analyze machines have small area of 2-D cross-section (please see Table for machine s imensions, so only one bounary conition nees to be efine for this type of FEA, so-calle Dirichlet bounary conition, an the most common use of it is to efine A z = along outer stator surface. This means that with this bounary conition we force the magnetic fiel to stay insie bounary efine by stator outer iameter. For larger geometries it is useful to efine aitional bounary conitions which allow usage of only one slice of machine s geometry, an the with of that slice is efine by pole with, which means that for large machines with large number of pole pairs these bounary conitions can spee-up the simulations consierably [6]. FEA algorithm solves euation ( for every noe of the mesh create by ivision of region of interest with finite elements using some of well-known numerical algorithms, such as Newton-Raphson s, for instance. After fining values of A z in all noes of meshe region, calculating flux linkages of specific phase winings can be one easily. The flux linkage Ψ of the j-th wining when the i-th wining is ij J s supplie with current can be expresse by [7]: Ψij = p Az AzS N jl j S Ω j + Ω j Ω j (4 where N j is the number of turns of j-th wining, l j is the length of the stator core an S j is the cross-section of the coil ( 5

3 region. + Ω j an Ω j represent the positive an negative trace of the wining in (x,y plane. The flux vector in - omain can be forme as follows: 2 2 Ψ = Ψs + jψs = ( Ψa + aψb + a Ψc (5 2 Ψ s = ( Ψa Ψb Ψc 2 2 (6 Ψ s = ( Ψb Ψc (7 IV. INDUCTANCE CALCULATION Accurate inuctance calculation of PMSM is a relevant topic, since the inuctances etermine large part of the electrical machine behavior []. The inuctances estimation in the - axis is crucial not only for etermination of the torue an flux weakening capability but also for esigning control systems in orer to optimize the efficiency, power factor, etc. []. In this paper two types of IPMSM geometries will be analyze. Both types have ientical stator (three-phase fourpole concentrate wining, an the rotors are chosen in such a way to represent two most significant types of PM orientation use in IPMSMs. First type, here calle IPMSM-T (Figure 2a is with tangentially magnetize PMs, an the secon one, here calle IPMSM-R (Figure 2b is with raially magnetize PMs. Figure an 4 respectively show PMs air-gap flux istribution for the two mentione types of motors. It is well known that in the type IPMSM-R uarature inuctance L is greater than irect inuctance L, because there is more iron along axis (Figure 2b. However, such conclusion is not so straightforwar for the type IPMSM-T, because PMs flux path is between two neighboring magnets, which means through iron (which increases the inuctance, but the flux path is much longer (which ecreases the inuctances. The first FEA simulation will be use to calculate L. The magnet flux is turne off by setting the magnet remanence Br to zero. The current vector must be aligne with the axis. Variation of inuctance as a function of current amplitue is of interest, so series of magneto-static simulations with ifferent levels of current excitation will be conucte in orer to obtain these relationships. For example, if the magnitue of stator current is chosen to be A, then to align the current axis with phase a axis the instantaneous phase currents have to be efine as: ia i a = A, ib = ic = 5A 2 =. (8 The an components of the current vector are then efine as: i = A, i = A (9 Figures 5 an 7 respectively show the fiel solution for this case, for both types of IPMSM. The flux linkages of phases a, b an c are then calculate using euation (4. The flux vector is constructe an it s irect an uarature components are calculate using expressions (6 an (7. It is important to state that the 2-D FEA simulation neglects the leakage flux in the en region, because it cannot be inclue without usage of -D FEA algorithm [8], [9]. Analyze machines have concentrate wining, which means that they have very short en connections, especially in comparison with istribute winings. Because of that, one may expect very small en turn leakage inuctances. However, their influence can be inclue by aing some analytical expressions [7] on calculate inuctances, or by comparing the 2-D FEA results with measurements, because measure inuctances inclue those effects. In this paper en connections leakage inuctances have been neglecte. The inuctances L are calculate in a similar manner, only this time the current vector nees to be aligne with the axis. The phase currents are then given as: i a = A, ib = ic = A ( 2 The an components of the current vector are then efine as: i = A, i = A ( Figures 6 an 8 show the fiel solution for this case, for both types of IPMSM. After calculating the flux components, the inuctances for one turn per coil are given by: L Ψ s =, i L Ψs = (2 i b Figure 2. Two types of IPMSM: a IPMSM with tangentially magnetize PMs (IPMSM-T, b IPMSM with raially magnetize PMs (IPMSM-R 6

4 B[T] angle [eg] Figure. Flux istribution of permanent magnets in air gap of IPMSM-T.5 Figure 7. Distribution of flux lines in IPMSM-R when there is only axis current in stator winings B[T] angle [eg] Figure 4. Flux istribution of permanent magnets in air gap of IPMSM-R Figure 8. Distribution of flux lines in IPMSM-R when there is only axis current in stator winings Figure 5. Distribution of flux lines in IPMSM-T when there is only axis current in stator winings Figure 6. Distribution of flux lines in IPMSM-T when there is only axis current in stator winings V. RESULTS OF INDUCTANCE CALCULATIONS Rate parameters for IPMSM-T machine are given in Table, an IPMSM-R machine was generate for the sake of comparison with IPMSM-T configuration. As mentione above, both machines have the same stator. Results of calculations escribe in previous chapter are shown in Figures 9 (for inuctance L an (for inuctance L. It is interesting to notice that for both types of IPMSMs uarature inuctance L is approximately 5% greater than irect inuctance L, an that inuctance L is more influence by saturation. Results are showing goo match with experimental results (existent only for type IPMSM-T, but there are some ifference between simulations an experiments (both L an L calculate with FEA are approx. % larger from those obtaine with experiments. This can be explaine by the fact that PMs cannot be exclue from experiments without estroying the rotor, an their fiel is ae on time-varying fiel prouce by excitation uring experiments, which has influence on saturation level uring measurement. Also, for the purpose of FEA simulations authors i not have precise ata for steel use for stator stack (relative permeability as a function of fiel strength. 7

5 L[H] L[H]..2 IPMSM-T FEA IPMSM-R FEA IPMSM-T measure Irms[A] Figure 9. Results of FEA simulations an measurements for irect axis inuctance L.5. IPMSM-T FEA IPMSM-R FEA IPMSM-T measure Irms[A] Figure. Results of FEA simulations an measurments for uarature axis inuctance L TABLE I. IPMSM-T rate ata IPMSM-T NAMEPLATE DATA Number of stator slots 2 Number of poles 8 Stator core outsie imension [mm] 2 Stator core stack thikness [mm] 42 Stator core insie iameter [mm] 6.6 Air gap with [mm]. Wining specification Resistance (2 phase 2 C 7.5Ω Direct an uarature inuctances [mh] Rate spee [rpm] 5 Rate torue [Nm]. VI. CONCLUSION Values φ.65 84T, Y,Al L=2.8 L=. In this paper, the metho for calculation of inuctances for two types of IPMSMs was presente. The metho is base on the FEA calculation of flux linkages. It was shown that the uarature inuctance L is greater than irect inuctance L for both types of motors. Further research using FEA software combine with measurements may be useful for etermination of en connection leakage inuctance influence, an also to investigate saturation of axis flux path because of the presence of permanent magnets. Results for type IPMSM-T are showing goo match with the results from the manufacturer of the machine. Presente metho can be useful for etail analysis of ifferent constructions of IPMSM with possibility to separate influence of excitation from influence of permanent magnets fiel, which cannot be one easily uring measurements. ACKNOWLEDGMENT This work was supporte by the Ministry of Eucation, Science an Technological Development of the Republic of Serbia uner project III424. VII. REFERENCES F. Shiferl, T.A. Lipo, Power Capability of Salient Pole Permanent Magnet Synchronous Motors in Variable Spee Drive Applications, IEEE Trans. In. Appl., vol. 26, pp. 5 2, Jan./Feb. 99. Y. Li, X.P. Yan, The perspective an status of PMSM electrical servo system, Micromotors Servo Techniue, vol. 4, pp. -, 2. M. Pastorelli, J. Bottomley, P. Giangrane an C. Geraa, Sensorless control of PM motor rives A technology status review, IEEE Work. Elec. Mach. Design Contr. Diag. (WEMDCD, vol. 48, no. 4, pp , March -2, 2. K. Jang-Mok an S. Seung-Ki, Spee control of interior permanent magnet synchronous motor rive for the flux weakening operation, IEEE Trans. In. Appl., vol., no., pp. 4 48, January/February 997. D. Zarko, A Systematic Approach to Optimize Design of Permanent Magnet Motors With Reuce Torue Pulsations, P.D. thesis, University of Winsconsin-Maison, USA, 24. P. Silvester, Finite Elements for Electrical Engineers, Cambrige University Press, 99. [7] J. Cros an P. Viarouge, Synthesis of High Performance PM Motors With Concentrate Winings, IEEE Trans. Energy. Conv., vol. 7, no. 2, pp , Jun 22. [8] T. Cox, F. Eastham, an J. Proverbs, En Turn Leakage Reactance of Concentrate Moular Wining Stators, IEEE Transactions on Magnetics, vol. 44, no., pp , November 28. [9] A. M. EL-Refaie, Fractional - Slot Concentrate-Winings Synchronous Permanent Magnet Machines: Opportunities an Challenges, IEEE Trans.In. El., vol. 57, no., January 2. 8