Seediscussions,stats,andauthorprofilesforthispublicationat:http://www.researchgate.net/publication/224598419

Measurementofparametersforinteriorpermanentmagnetmotors

CONFERENCEPAPER·AUGUST2009

DOI:10.1109/PES.2009.5275584·Source:IEEEXplore

CITATION

1

DOWNLOADS

8

VIEWS

56

4AUTHORS,INCLUDING:

Rahman

UniversityofOttawa

226PUBLICATIONS2,778CITATIONS

SEEPROFILE

P.Zhou

ANSYS

49PUBLICATIONS709CITATIONS

SEEPROFILE

D.Lin

43PUBLICATIONS461CITATIONS

SEEPROFILE

Availablefrom:P.Zhou

Retrievedon:14July2015

1

Abstract—In this paper, a brief review of methods for

measurement of reactance parameters of interior permanent magnet (IPM) synchronous motors is presented. An update of new approaches is also given for extracting saturated parameters Ld, Lq and permanent-magnet flux linkage λmd with arbitrary load conditions for IPM motors. The new procedure starts with the parameter extraction in actual abc reference frame based on transient finite element analysis (FEA) solutions in time-domain. Then the parameters in abc reference frame are transformed to d-q axis reference frame via Park transformation. As a result, this approach is able to provide better accuracy and allow easy implementation because the parameters are derived from a realistic physical representation and the effects of the slot leakage and phase-spread harmonic fields are included.

Index Terms—d-q axis parameters, flux linkage, interior permanent-magnet synchronous motors, transient finite element analysis, measurement of machine parameters.

I. INTRODUCTION

NTERIOR permanent-magnet synchronous motors (IPMSM) are widely used in various electrical devices due to its high torque-to-current and power-to-weight ratios, and its high

efficiency. The motor generates the torque not only by magnets, but also by the reluctance difference between the d and q axes. To take geometric complexity, high local magnetic saturation, induced eddy currents, dynamic core loss, magnet orientations and mechanical stress into account, the transient finite element analysis (FEA) coupled with the driven circuit can reach a desirable accuracy with high computation efficiency [1-23]. However, the extraction of d-q axis parameters is highly desirable because not only d-q axis parameters provide good insight of the feasibility of a design and the possibility of conducting complex system level simulation based on the equivalent circuit model, but also they are fundamental parameters to many vector control algorithms in the d-q reference frame for fast and accurate response with quick recovery from a disturbance [22].

Clearly, the effectiveness of the two-axis model on the accuracy of simulation results and the robustness of a controller will definitely rest on how closely the d-q axis parameters can represent a true operating condition. Due to the saturation in the rotor and stator cores, the d-q axis quantities and the excitation voltage E0 are no longer independent of each other. For example, the q-axis armature reaction may have a considerable effect on the saturation in the rotor magnetic bridge region. This situation will also have an influence on the reluctance of d-axis path and therefore cause

a change in magnet operating point, hence the field. However, the conventional method for obtaining the saturated synchronous parameters does not consider such cross magnetic effects, that is, the induced EMF E0 is determined by the permanent magnets only, the d-axis inductance Ld is determined from the magnets and d-axis current, and the q-axis inductance Lq is determined from the q-axis current only. To this end, some efforts have been made in both simulation and experiment aspects. The significance of such cross magnetic effects has been clearly mentioned in the early work [1-7], and a procedure was proposed to determine the saturated parameters Ld, Lq and E0 with the cross effects considered at various load conditions. Two classical methods including load angle measurement are briefly presented.[20,23] However, the procedure may not flexible enough for general use since the numerical error due to the arbitrary choice of a small displacement from operating point may cause some uncertainty for determining both Ld and E0 at the same time. In addition, the introduction of the equivalent reluctivity associated for the use of time-phasor may not always assure good accuracy during the simulation of starting performance.

In this paper, the saturated parameters Ld, Lq and permanent-magnet induced flux linkage λmd at arbitrary load conditions are extracted based on the frozen method, instead of the small-displacement method. The procedure starts with the parameter extraction in abc reference frame from the transient FEA solution in the time domain, without using the equivalent BH curve associated with the frequency domain. Then the derived parameters are transformed to d-q axis reference frame via Park transformation. Thus, the extracted d-q parameters have taken into account the effects of the saturation, slot leakage and phase-spread harmonic fields. As a result, this approach is able to provide better accuracy.

II. STEADY STATE MODELS

It is well kwon that an IPM motor is basically a salient pole synchronous machine. Unlike in conventional salient pole ac synchronous machine, the saliency is created inside the rotor by magnets arrangement in an IPM Motors having a smooth air gap. The classical analysis of salient pole synchronous machine provides the established expression of developed power Pd in a 3-phase 2-pole balanced synchronous motor as [24-25]:

δ2sinX2X

)X(X3Vsin

X

E3VP

qd

qd2

p

d

0pd

−+= δ (1)

Measurement of Parameters for Interior Permanent Magnet Motors

M. A Rahman, P. Zhou, D. Lin and M. F. Rahman,

I

978-1-4244-4241-6/09/$25.00 ©2009 IEEE

2

Where, Vp is the per phase stator supply voltage, Eo is the per phase excitation voltage , Xd and Xq are the direct and quadrature (d-q) axis reactances, respectively and δ. is the angle between the Vp and Eo voltage phasors. Using the three-phase voltages in abc system are transformed into the corresponding voltages in dq0 reference frame , the expression of developed torque can be easily obtained by dividing in equation (1) by synchronous angular speed and rearranging it for an IPM motor as:

[ ]qimdqididLqLpdT λ+−= )(2/3 (2)

Where P is number of poles and λmd is the permanent-magnet flux linkage. Ld and Lq are the d-q axis inductances, respectively. id and iq are the d-q axis components of the IPM motor stator current, respectively. The first term of equation (2) is the reluctance torque and the 2nd term is the electric torque like that in any PM dc conventional motor.

III. TRANSIENT FE ANALYSIS MODELS

In 2D FEM analysis, the magnetic vector potential A is normally used since the magnetic vector potential and the current are reduced to one component. The field equation can be expressed in matrix form as

[ ] [ ] gvDiHAdt

dQAS bw =+++ ][][ (3)

where A is the column matrix of the nodal value of the z component of A; iw is the terminal current flowing into each winding, which is unknown when the winding is excited by a voltage source; vb is a column matrix of unknown induced voltage differences between the far and near ends of passive solid conductors and accounts for eddy currents that are induced inside the solid conductors; and [S], [Q], [H] and [D] are coefficient matrices. The excitation g on the right side includes the prescribed current density and the contribution from permanent magnets.

To reduce computation costs for 3D problems, instead of treating a vector in the whole domain, we use just a magnetic scalar potential Ω in the whole domain and a current vector potential T in the conducting region. To be more general, the magnetic field H can be further split into different components:

kkc i HTHHH s ∑+Ω∇+++= (4)

where Hs is the source field due to known total currents or known current densities in either solid conductors or stranded conductors, Hc is the excitation due to permanent magnets, Hk is a source field representing 1A current flowing in voltage-driven winding k, and ik is the unknown current to be determined. Applying Ampere’s law, Faraday’s law and Gauss’s law for the solenoidality of the flux density yields the differential equations as

( )

( )cs

kk

dt

d

idt

d

HH

HTT

μμ

μμμσ

+−=

+Ω∇++⎟⎠

⎞⎜⎝

⎛ ×∇×∇ ∑1

(5)

( ) ( )cskki HHHT μμμμμ +−∇=+Ω∇+∇ •• ∑ (6)

with μ being the permeability. In non-conducting region, (3) is no longer applicable. The problem region is then discretized into tetrahedral elements. The scalar field associated with Ω is represented by node-based shape functions, and the vector field components T, Hs and Hk are represented by edge-based shape functions. Applying the Galerkin’s approach, the discretized field equations are given in matrix form as [16]

giHdt

dB

dt

dQ

dt

dS w =+Ω+⎟

⎠

⎞⎜⎝

⎛ + ][][][][ T (7)

where T and Ω are column vectors of the edge values of the current vector potential and of the nodal values of magnetic scalar potential, respectively; iw is a column vector of currents in stranded windings or in solid conductors associated with a voltage driven source, and the matrices [S], [Q], [B], [H] and column vector g are derived from the standard Garlerkin’s method. In ferromagnetic materials, the matrices [Q], [B] and [H] are nonlinear. Normally, these matrices are not constant from time step to time step because of the non-linearity and the position change of moving objects.

Because the currents iw in equations (3) and (5) are unknown, the field equations must be coupled with the circuit equations and solved simultaneously. A simple electrical circuit associated with a voltage source is:

[ ] sww ueidt

dLiR =++ ][ (8)

where [R] and [L] are the resistance and inductance matrices of the windings, and us and e are the column vectors of applied voltage sources and induced back EMF across the terminals of the windings, respectively. When mechanical transients are involved, the equation of motion is included.

IV. DETERMINATION OF D-Q PARAMETER S

As mentioned above, the first step is to extract the parameters in abc reference frame. It is clear that these parameters will vary with load conditions, and thus, under steady state, it is dsirable to extract these parameters in terms of different torque angles. This means that we need find a way to setup a transient FEA problem for a specfic load condition which is assiciated with a particular torque angle δ. This modeling request can be easily realized by the following scheme: rotate rotor to the initial position θ0 at which the d-axis oppositely aligns with the phase-A axis as shown in Fig. 3. At this initial position, phase-A will have the negative maximum flux linkage produced by permanent magnets, which corresponds to zero crossing from nagative to positive of the induced voltage in phase-A. Consequently, the different load conditions can be modelled by appling three phase voltages in terms of the torque angle as:

⎪⎩

⎪⎨

⎧

−+=−+=+=

)3/4sin(2

)3/2sin(2

)sin(2

1

1

1

πδωπδω

δω

tVv

tVv

tVv

c

b

a

(9)

3

where ω is the angular frequency, δ is the torque angle. After solving transient FEA under a specific load condition

at each time step, the FEA system’s coefficient matrix is frozen, which is equal to freezing the permeability of each element. Then the field is resolved with the frozen coefficient matrix together with different right-hand sides corresponding to the excitation of 1A current in turn in each winding with the exclusion of magnet and all other winding excitations. As a result, it is the calculated winding flux linkages that provide the self and mutual inductances for each winding. Similarly, by enabling the excitation of permanent magnet alone with zero currents in all windings, we can obtain the permanent magnet flux linkage in each winding. It is clear that, from implementation point of view, the key issue here is how to compute flux linkage which provides the link between the magnetic field domain and the electric circuit.

V. MEASUREMENT OF PARAMETERS

Figure 1 shows the circuit configuration to measure the input power of a 3-phase IPM synchronous machine. It is evident that for a fixed supply voltage Vp the expression of developed power in equation (1) has four unknown quantities, namely, Eo , Xd , Xq and δ.. Three of these unknown parameters can be determined, by measuring the input power for a known value of torque angleδ. Figure 2 shows a schematic of determining IPM motor parameters by torque angle measuring method [3, 7, 9]..

FIG. 1: CIRCUIT CONFIGURATION TO MEASURE THE INPUT

POWER OF A 3-PHASE IPM MACHINE

FIG. 2: SCHEMATIC FOR DETERMINING IPM MOTOR

PARAMETERS BY MEASURING TORQUE ANGLE.

VI. APPLICATION EXAMPLE

As an application example, the proposed FE approach is applied to simulate the performance of Toyota Prius IPM motor [17] at the synchronous speed, whose specifications are given in Table I. As discussed before, the first step is to derive the saturated parameters in abc reference frame under different load conditions. To this end, total 16 transient solutions are applied for the torque angle range from 0° to 150° with an interval of torque angle of 10°.

TABLE I. PRIMARY SPECIFICATIONS OF THE PRIUS IPM MOTOR

Description Value Rated Output Power 50 kW Rated Line-to-line Voltage 245 V Synchronous Speed 3600 rpm Number of Poles 8 Outside Diameter of Stator 269 mm Number of Stator Slots 48 Axial Length 84 mm

For each transient FE solution at a specific torque angle

associated with the specified three phase voltages. The simulation period is set to be 0.6945ms which is just enough to cover the phase-spread of 60 electrical degrees, or 15 mechanical degrees. Accordingly, the time step is chosen to be 0.0463ms so that the rotor would rotate one mechanical degree for each time step with a total of 16 time steps. After FEA at each time step, the inductance matrix and the flux linage vector are extracted from the field solutions. Thereafter, the inductance matrix and the magnet flux linkage vector in abc reference frame will be further transformed to d-q reference frame. Figure 3 shows the straight magnets orientation with the initial rotor position, where d-axis oppositely aligns with phase-A axis, for using. finite element method to calculate the air gap flux linkage. Pertinent results will be presented at the panel session for discussion.

4

FIG. 3. INITIAL ROTOR POSITION WHERE D-AXIS OPPOSITELY

ALIGNS WITH PHASE-A AXIS

VII. CONCLUSIONS

This paper presents the classical methods to measure IPM machine parameters. A new finite element approach to extract the saturated inductance parameters Ld, Lq and permanent-magnet induced flux linkage λmd at arbitrary load conditions for IPM synchronous motors is also presented. The procedure starts with the parameter extraction in natural abc reference frame based on the transient FEA solution in the time domain. Then the derived parameters are further transformed to d-q axis reference frame. This approach is able to provide better accuracy because the extracted d-q parameters have taken into account the effects of the slot leakage and phase-spread harmonic fields, eliminated the uncertainty due to the arbitrary choice of a small angle displacement and avoided introducing the equivalent BH curve in the frequency domain.

REFERENCES

[1] V. B. Honsinger, “The Fields and Parameters of Interior AC Permanent Magnet Machines”, IEEE Transactions on Power Apparatus and Systems, Vol. 101, No. 4, 1982, pp. 867-876. [2]. B..J. Chalmers, S.A. Hamid and G.D. Baines, “ Parameters and Performances of a High Field Permanent Magnet Synchronous Motor for Variable Frequency Applications;, Proc. IEE, Vol. 132, Part B, No. 3, 1985, pp. 117-124. [3]. S.F.Gorman, C. Chen and J.J. Cathey, “Determination of Permanent Magnet Synchronous Motor Parameters’, IEEE Transactions on Energy Conversion, Vol. 3, No. 3, 1988, pp.674-681. [4]. D. Pavlik, V.K. Garg, J.R. and J.Weiss, “ A Finite Element Technique for Calculating the Magnet Sizes and inductances of Permanent Magnet Machines”, IEEE Transactions on Energy Conversion, Vol. 3, No. 1, 1988, pp.116-122. [5]. P. Mellor, F. Chaaban and K. Binns, “ Estimation of Parameters and Performance of a Rare Earth Permanent Magnet Motors Avoiding Measurement of Load Angle”, Proc. IEE, Vol. 138, Part B, No.11, 1991, pp. 322-330.

[6]. M.A. Rahman and Ping Zhou, “Determination of Saturated Parameters of PM Motors Using Loading Magnetic Fields”, IEEE Transactions on Magnetics, Vol.27, No. 5, 1991, pp.3947-3950. [7]. M.A. Rahman and Ping Zhou, “An Accurate Determination of Permanent Motor Parameters of PM Motors by Digital Torque Angle Measurement”, Journal of Applied Physics, Vol.76, No. 10, 1994, pp. 6868-6870. [8]. M.A. Rahman and Ping Zhou, “ Analysis of Brushless Permanent Magnet Motors”, IEEE Transactions on Industrial Electronics, Vol. 43, No. 2, April 1996, pp. 256-267. [9]. J.F. Gieras, E. Santini and M.Wing, “Calculation of Synchronous Reactances of Small Permanent Magnet Alternating Current Motors: Comparison of Analytical Approach and Finite Element Method with Measurements”, IEEE Transactions on Magnetics, Vol.34, No. 5, 1998, pp. 3712-3720. [10.] B. Stumberger, B. Kreca and B. Hribernik, “Determination of parameters of synchronous motor with permanent magnets from measurement of load conditions”, IEEE Trans. on Energy Conversion, Vol. 14, No. 4, December. 1999, pp. 1413-1416,. [11] G. Kang, J .Hong, G. Kim and J. Park,“ Improved Parameter Modeling of Interior Permanent Magnet Motor based on Finite Element Analysis ”, IEEE Transactions on Magnetics, Vol.36, No. 4, 2000, pp. 1867-1870. [12]. H. Nee, L. Leferve, Thelin and J. Soulard, “ Determination of d-q axis reactances of Permanent Magnet Synchronous Motors without Measurement of Rotor Position”, ”, IEEE Transactions on Industry Applications, Vol. 36, No.5, 2000, pp. 1330-1335. [13]. E. C. Lovelace, T.M. Jahns and J.H. Lang, “ A Saturating Lump Parameter Model for an Interior PM Synchronous Motor”, IEEE Transactions on Industry Applications, Vol. 38, No.3, 2002, pp.645-650. [14] B. Stumberger, G. Stumberger, D. Dolinar, A. Hamler and M. Triep, “ Evaluation of Saturation and Cross Magnetization Effects in Interior Permanent Magnet Synchronous Motor”, ”, IEEE Transactions on Industry Applications, Vol. 39, No.5, 2003, pp.1264-1271. [15] P. Zhou, W. N. Fu, D. Lin, S. Stanton and Z. J. Cendes, “Numerical modeling of magnetic devices,” IEEE Trans. on Magnetics, vol. 40, No. 4, July 2004, pp. 1803-1809,. [16]. D. Lin, P. Zhou, W.N. Fu, Z. Badics and Z. J. Cendes, “A dynamic core loss model for soft ferromagnetic and power Ferrite materials in transient finite element analysis,” IEEE Trans. on Magnetics, vol. 40, No.2 March 2004, pp. 1318-1321, [17] J. S. Hsu, C. W. Ayers, C. L. Coomer, R. H. Wiles, S. L. Campbell, K. T. Lowe, R. T. Michelhaugh, “Report on Toyota/Prius Motor Torque Capability, Torque Property, No-Load Back EMF, and Mechanical Losses”, Oak Ridge National Laboratory, Oak Ridge Institute for Science and Education, ORNL/TM-2004/185. [18]. Y.S. Chen, Z.Q. Zhu and D. Howe, „ Calculation of d- and q- axis inductances of PM brushless ac Machines Accounting for Skew”, IEEE Transactions on Magnetics, Vol. 41, No.10, 2005, pp. 3940-3942. [19]. P. Zhou, D. Lin, W. N. Fu, B. Ionescu, and Z. J. Cendes, “A general cosimulation approach for coupled field-circuit problems,” IEEE Trans. on Magnetics, vol. 42, No. 4, April. 2006, pp. 1051-1054,. [20]. R. Dutta and M.F. Rahman, “ A Comparative Analysis of Two Test Methods of Measuring D-and Q-axis inductances of Interior Permanent Magnet Machine”, IEEE Transactions on Magnetics, Vol.42, No. 11, 2006, pp. 3712-3718. [21]. P. Zhou, Z. Badics, D. Lin and Z.J. Cendes, “Nonlinear T-Ω formulation including motion for multiply connected 3-D problems”, IEEE Trans. on Magnetics, Vol. 44, No. 6, June 2008., pp. 718-721, [22]. M.A. Rahman, R.M. Milasi, C. Lucas, B.N. Araabi and T.S. Radwan, “Implementation of emotional controller for interior permanent-magnet synchronous motor drive,” IEEE Transaction on Industry Applications, Vol. 44, No. 5, pp. 256-267, April. 2008. [23]. K.J. Meessen, P. Thelin, J. Soulard and E.A. Lomonova, “Inductance Calculations of Permanent Magnet Synchronous Machines Including Flux Change and Cross-Saturations”, IEEE Transactions on Magnetics, Vol.44, No. 10, 2008, pp. 2324-2331. [24] IEEE Standard 115-1995, “ IEEE Guide: Test Procedure for Synchronous Machine, Part 1- Acceptance and Performance Testing and Part 2- Test Procedures and Parameters Determination for Dynamic Testing. [25] IEEE Standard 115A-1987, “IEEE Standard Procedure for Obtaining Synchronous Machine Parameters by Standstill Frequency Response Testing.