A STUDY ON DETERMINATION OF PARAMETERS FOR PERMANENT MAGNET SYNCHRONOUS MACHINE BY COMPARING LOAD TESTS AND FINITE ELEMENT

ANALYSIS

Author(s) David Carpenter, Sorin Deleanu, Cuong Nguyen, Michael Lau, Muhammad Mustaqur Rahman

NAIT (Northern Alberta Institute of Technology), Edmonton, Canada

ABSTRACT This paper presents a comparison between results, for two methods of obtaining the values of d-axis synchronous reactance Xsd and q-axis synchronous reactance Xsq, of the permanent magnet synchronous motor with radial magnets (RPMSM). The paper provides a background describing the benefits of using the RPMSM machine and how the synchronous reactance parameters are valuable in determining stability and control algorithms for the machine. The two methods described are a load test under constant frequency conditions and Finite Element Analysis (FEA). Results obtained from the load tests, including the internal load angle , have been used to calculate Xsd and Xsq and represent their values with respect to the armature current and internal angle . Comparative results have been extracted using FEA. Good agreement between results is illustrated. Future developments conclude the paper.

Index Terms Permanent Magnet Synchronous Motor (PMSM), Permanent magnets, synchronous reactance, radial permanent magnets synchronous motor (RPMSM), counter electromotive force (back EMF), saliency ratio, Finite Element Analysis (FEA), position transducer, Samarium Cobalt (SmCo), Normal Flux Density

1. SYMBOLS AND NOMENCLATURE fn - nominal frequency Isd , Isq - d-axis , respectively q-axis component of the stator current in the rotor reference frame Vsd, Vsq - d-axis , respectively q-axis component of the stator voltage in the rotor reference frame Ls-Stator self inductance Lsd, Lsq - stator synchronous inductance along the d, q axis RR

ngle)

s- stator resistance vs, is - space vector of the stator voltage and current i - electrical Angle between the phase voltage and the phase induced EMF (internal angle or load a - power Factor Angle (angle between vs and is) - torque Angle (the angle between is and the d-axis)

r- rotor angular frequency n - nominal angular frequency) Te - electromagnetic (developed) torque TL - load torque Ei - total Induced EMF per phase EPM - induced EMF due to the permanent magnets per phase Xsd, Xsq - direct (d-axis) and quadrature (q-axis) reactance Vs, Is- stator voltage, respectively current, steady state S1, P1- apparent , respectively real input power of PMSM cos input power factor of PMSM P1- input real power P2-output (shaft) power S1- input apparent power

2. INTRODUCTION

The PMSM has become popular in recent years because of its advantages over the wound rotor synchronous motor: These advantages include higher specific torque and shaft power, robustness, lower maintenance costs, better dynamic performances, as well as simplified converter configurations for variable frequency operation.

In comparison with the induction machine, a PMSM displays higher efficiency and power factor vales for the same load conditions. Consequently, a PMSM drive will require a variable frequency converter with lower ratings than an induction motor similar pole configuration and power rating.

Following significant advances in power electronic systems, control strategies and permanent magnet manufacturing process, PMSM are now available for a large range of applications from servo drives to large industrial AC drives. Also, for direct start across the line, (line-start PMSM), a squirrel-cage winding is included in the ferromagnetic circuit of the rotor as a damper winding.

To design control strategies for stable operation, it is essential to determine the PMSM parameters allows analyzing the operation stability of PMSM drives. This also allows for design of improved efficiency and torque-speed characteristics, through adaptive speed, current and/or torque regulators.

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The parameters required to be determined are the armature resistance , d-axis and respectively q-axis reactance . The latter two parameters are very dependent on the load conditions and the level of saturation of the magnetic circuit of the PMSM. A source of error can occur when determining the d-axis synchronous reactance if a constant value of the permanent magnets induced EMF is assumed.

SR sdX

sqX

sdX

In recent years, there have been a number of reports describing different approaches to PMSM parameter determination. Analytical methods of calculation following design steps are described in [6], [7] and [18] for common PMSM configurations. These including shaped high energy Neodymium-Iron-Boron and Samarium Cobalt permanent magnets. On-line identification methods, using the stator circuit response to auxiliary injected signal of imposed frequency are very suitable for servo drives, and detailed in [20], [21], and [22]. Another method of synchronous reactance determination, involving tests made with the rotor at standstill is described in papers [4], [10] and [23]. In this procedure, the saliency of the magnetic circuit appears when the self and mutual inductance is found by a trigonometric function of the rotor position, following measurements with the PMSM energized from a single-phase. Core loss determination, which cannot be separated through classical method from mechanical loss (friction and ventilation) is given in [3] and [9]. Load tests are considered in [1], [5] and [6] as the starting points for parameter identification when calculating synchronous reactance. In [5], authors propose a hybrid method of no-load and load tests where the internal angle is not directly measured. The method includes a simple algorithm for internal angle calculation. Yet, the authors pointed out the inaccuracy of the results in case that the tested PMSM has a heavy saturation along d-axis. Numerical methods, especially FEA can provide accurate values of synchronous reactance along d-axis, respectively q-axis and correlate with load conditions,[6], [16] ,and [17]. The two applied methods for parameter determination considered in this paper, load test and finite element analysis, show results with good correlation.

3. PMSM CONSTRUCTION

The investigation and comparison of the two methods has been carried out on a typical low cost PMSM with radial mounted SmCo permanent magnets, as shown in Figure. 1. The direction of magnetization is also radial. The stator winding has 36 slots; the rotor has four magnets and a damper winding with four sections of 9 slots each. Figure 1 presents the cross section areas of the rotor and stator, with all of the dimensions in mm. The active length of the PMSM is 130mm.

The RPMSM ratings are: Pn=5HP, p=4(Nr=1800RPM), Vn=230V, In=12A,3-phase

a)

b)

c)

d-axis

q-axis

Figure 1. Cross-sectional view of the RPMSM motor

a) rotor, b) stator, c) stator slot and tooth

4. LOAD TEST METHOD To determine the synchronous reactance along both d-axis and q-axis, direct load tests have been carried. The measurements system, as shown in Figure 2, provides (S1, P1, V1,I1, cos) and output (TL,P2, i ) parameters for PMSM. Stator resistance RRS was measured according to Std. IEEE-112 for each load point.

Figure 2. Experimental Circuit Used for No-Load And Load

Tests. Schematic Diagram

The internal angle was measured using a position encoder, capable to generate an index pulse voltage. The pulse was synchronized with one of the line-to-line voltages applied to the motor at no load and measured for different load conditions.

Using the phase voltage, current and load angle, PMSM equivalent circuit parameters were calculated using the phasor diagram shown in Figure 3 [1].Measurements have been performed in conditions of unity power factor or

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Figure 3. Phasor diagram for salient-pole PMSM (in rotor reference frame)

slightly lagging. The components can be expressed as the induced voltage along direct and quadrature axis, by decomposing the phasor diagram:

sinsinsinsin sssSsqiid IRVIXEE (1)

coscoscoscos sssSsdPMiiq IRVIXEEE (2)

The load and torque angles are related by:

090 (3) From Figure 4, it can be observed that there is cross

coupling of direct and quadrature magnetic axis, in which the q-axis current has an influence over the direct axis flux component, together with the flux due to the permanent magnets. Also note that the d-axis current has a contribution in the q-axis flux component. Following the work reported by Sturmberger et al. [1], it is possible to obtain Xsd through a variational algorithm. To smooth the data, Sturmberger et al. apply an orthogonal polynomial approximation for the curves representing the functions i(), Is() and Ei(). For a more accurate approach, the results presented here are based on a Legendre polynomial [18], because of the higher precision and improved convergence. When considering (k+1)=(k)+, then equation (2) becomes a linear system from where Xsd may be determined.

1cos11cos1 kkIXEkkE SsdPMii

Figure 4. Internal load angle dependency of torque angle for

different shaft loads and 230V, 250V line to line

kkIsdXEkkiE SPMi coscos

1cos1cos

cos1cos1

kkIkkIkkiEkkiE

sdXSS

ii

coscos

SiPM IsdXiEE

(4)

With =0.05 degrees, accurate results can be obtained

in terms of variable EPM and Xsq:

(5)

sinsin

sIiE

sqXi (6)

The above analysis has been applied to the motor

under test for two line voltages of 230V and 250V and the results of these parameters are shown in Figures 4 to 10. These values have then been used to determine the developed torque (calculated) for comparison with the shaft torque (measured) by use of equations (7) to (12). The final results show good correlation at supply voltages of 230V and 250V, as shown in Figure 11.

iPMsiisdiss ERXRVA sincos (7)

isdisqisiisqs XXRXVB sincos

(8)

isqisdisqiPM XXXED

(9)

2sisqisdiPM RXXEC

(10)

2sisqisd RXXE (11)

236.120)(

EDCBAftlbTe

(12)

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Figure 5. Stator current dependency of torque angle for different shaft loads and 230V, 250V line to line

Figure 6. Induced Voltage dependency of torque angle for

different shaft loads and 230V, 250V line to line

Figure 7. Induced Voltage dependency of internal load angle for different shaft loads and 230V, 250V line to line

at 250V

Figure 8. Isd, Isq dependency of internal load angle for

different shaft loads and 230V, 250V line to line

Figure 9. Permanent magnet induced voltage EMP dependency of internal load angle for different shaft loads

and 230V, 250V line to line

at 250V

Figure 10. Xsd, Xsq dependency of internal load angle for

different shaft loads and 230V, 250V line to line

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Figure 11. Electromagnetic torque Tc(i), (calculated from load tests) Te(i)(FEA) and load (shaft) torque TL(i)

(measured) as functions of the internal load angle for 230V respectively 250V line to line voltage

5. CONFIRMATION OF RESULTS USING FINITE

ELEMENT ANALYSIS

An alternative approach, based on FEA, has been used to provide further confirmation of the results. The OPERA-2d software [24] has been used to determine the values of Xsd and Xsq using a method described by Petkovska and Cvetkovski [16]. This method relies on replacing the magnets with demagnetized material. The FEA method is described elsewhere [24] and is based on solving Poissons equation. For synchronous operation, the rotor and the rotating field in the stator have no differential rotational speed. This implies that a magnetostatic solution can be used to determine flux crossing the air-gap. The advantage of using an FEA approach is that saturation of the appropriate parts of the magnetic circuit of the machine are automatically included in the analysis.

The finite element model used in the analysis is shown in Figure 12, for the d-axis analysis. The OPERA-2d software allows for easy parameterization of the model and this permits easy rotation of the rotor to model the q-axis also. The model has 15,610 elements and the mesh was refined using the built in error estimation. The rms error of the solution was determined at less than 9%. The solution is shown in Figure 13 by lines of flux represented by equi-potential lines of vector potential. The method requires that the flux crossing normal to the air-gap is integrated over one pole. Using the integration function within the software, it was possible to accurate determine this value. A plot of the air-gap flux is shown in Figure 14.

Table 1 provides a comparison between the values of inductance from the analytical method and the FEA approach. It should be noted that the direct load tests provide synchronous reactance while finite element analysis

yield the values are armature reaction reactance. In order to compare methods, it was necessary to make allowance for

Figure 12. FEA 2D model of PMSM under test

Figure 13. FEA solution showing flux lines in cross-section of motor

Figure 14. Graph of air-gap flux density for a single pole

Synchronous

Reactance Synchronous

reactance from direct load tests ()

Synchronous reactance

derived from FEA ()

Xsd=Xad+X1 8.35 8.87

Xsq=Xaq+X1 12.75 13.0

Table 1. Comparison of parameter results for both

methods (at nominal voltage and current)

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Polo SpaceResaltado

Polo SpaceResaltado

the armature leakage reactance. Using methods described elsewhere [10], the value for the motor under test is X1=2.48. The results show good agreement with the variances being attributed to variations in magnetic material been used in the 2D analysis. 6. CONCLUSIONS AND FUTURE DEVELOPMENTS

The relatively large variation between the electromagnetic (developed) and the shaft torque at 230V will be further investigated by developing a more accurate procedure meant to include the core losses and their separation from mechanical losses. It has been shown that the above approaches to determining the parameters of a PMSM have provided good agreement. It is expected that the variances between the two methods is due to magnetic material variations and end-winding effects that have not been accounted for in the FEA analysis.

In order to address these issues, it is proposed to perform a sensitivity analysis on the material characteristics of the motor in the FEA analysis and also to carry out 3D analysis so that end effects can be incorporated into the model.

7. REFERENCES [1] B. Sturmberger, B. Kreca,B. Hribernik. Determination of Parameters of Synchronous Motor with Permanent magnets from Measurement of Load Conditions. IEEE Trans. on Energy Convers., Vol. 14, No.4, p.1413 1416, December 1999. [2] U. Pahner, S. Van Haute, R. Belmans, F. Mameyer, B. Stumberger, D. Dolinar. Comparison of two methods to determine the d/q-axis lumped parameters of permanent magnet machines with respect to numerical optimisation. Proceedings of International conference on Electrical Machines (ICEM), Istanbul, Turkey, 1998, Vol.1, p. 352-357, September 2 4. [3] F. Fernandez-Bernal, A. Garcia-Cerrada, R. Fuare, Determination of Parameters in Interior-Permanent Magnet Synchronous Motors With Iron Losses Without Torque Measurement. IEEE Trans. on Ind. Appl., Vol. 37, No.5, p.1265 1272, September/October 2001. [4] M.E. Haque, M.F. Rahman. Dynamic Model and Parameter Measurement of Interior Permanent Magnet Synchronous Motor. Proceedings of the 2006 Australasian Universities Power Engineering Conference (AUPEC'06), T10, Melbourne, Victoria, Australia, 10 13 December 2006. [5] H-P. Nee, L. Lefevre, P. Thelin, J. Soulard. Determination of d and q Reactances of Permanent magnet Synchronous Motors Without Measurements of the Rotor Position. IEEE Trans. on Ind. Appl., Vol. 36, No.5, p.1330 1335, September/October 2000. [6] J.F. Gieras, M. Wing. Permanent Magnet Motor Technology. Design and Applications. Second Edition, Revised and Expanded. Marcel Dekker, ISBN 0-8247-0739-7, 2002. [7] S.A. Nasar, I. Boldea, L.E. Unnewehr. Permanent Magnet, Reluctance and self-Synchronous Motors, CRC Press, ISBN 0-8493-9313-2, 1993. [8] B.J. Chalmers, S.A. Hamed, G.D. Baines. Parameters and Performance of a high-field permanent magnet synchronous motor

for variable-frequency operation. IEE Proceedings, Vol. 132, Part B, No. 3, p. 117 - 124, May 1985. [9] M. Chunting., G.R. Slemon, R. Bonert. Modeling of Iron Losses of Permanent-Magnet Synchronous Motors. IEEE Trans. on Ind. Appl., Vol. 39, No.3, p.734 742, May/June 2003. [10] S.F. Gorman, C. Chen, J.J. Catey. Determination of Permanent Magnet Synchronous Motors Parameters for use in Brusless DC Motor Drive Analysis. IEEE Trans. on Energy Convers., Vol. 3, No.3, p.674 681, September 1988. [11] M.A. Rahman, P. Zhou, Analysis of Brushless Permanent Magnet Synchronous Motors. IEEE Trans. on Ind Eectron., Vol. 43, No.2, p.256 267, April 1996. [12] R. Krishnan. Electric Motor Drives. Modelling, Analysis, and Control. Pearson Education, Prentice Hall, ISBN 0-13-091014-7, 2001 [13] S. Wu, D.D. Reigoza. Y. Shibukawa, M. Leetma, R.D. Lorentz, Y. Li. Interior Permanent-Magnet Synchronous Motor Design for Improving Self-sensing Performance at Very Low Speed. IEEE Trans. on Ind. Appl., Vol. 45, No.6, p.1939 1946, November/December 2009. [14] F.J.T.E. Ferreira, M.V. Cistelecan. Simulating Multi-Connection, Three-Phase, Squirrel-Cage, Induction Motors by Means of Changing Per-Phase Equivalent Circuit Parameters, Proceedings of the 2008 International Conference on Electrical Machines, Paper ID 1314, 2008. [15] P.C. Krause. Analysis of Electrical Machinery, McGraw-Hill, 1986. [16] L. Petkovska, G. Cvetkovski. Steady State Performance Evaluation of a Permanent magnet Synchronous Motor Based on FEA Book of summaries of the 9th Spanish Portuguese Congress on Electrical Engineering 9th CHLIE'2005, pp. 235-242, 2005. [17] J. Kolehmainen, J. Ilkaheimo. Motors with Buried Magnets for Medium Speed Applications. IEEE Trans. on Energy Conversion, Vol. 23, No.1, p.86 90, March 2008. [18] Y. Huang, Y. Long. On orthogonal polynomial approximation with the dimensional expanding technique for precise time integration in transient analysis Communications in Nonlinear Science and Numerical Simulation, p.1584-1603, 2008. [19] J. Pyrhonen, T. Jokinen, V. Hrabovcova. Design of Rotating Electrical Machines, John Wiley & Sons, Ltd, 2008. [20] J. Huang, K.A. Corzine, M. Belkhayat, Online Synchronous Machine Paremeter Extraction From Small-Signal Injection Techniques. IEEE Trans. on Energy Convers., Vol. 24, No.1, p.43 51, March 2009.[21] J-H Jang, J-I Ha, M. Ohto, K. Ide, S-K Sul. Analysis of Permanent-Magnet Magnet Machine for Sensorless Control Based on High-Frequency Signal Injection.IEEE Trans. on Ind. Appl., Vol. 40, No.6, p.1595 1604, November/December 2004. [22] S Wu, D.D. Reigoza, Y. Shibukawa, M. Leetma, R.D. Lorentz, Y. Li. Interior Permanent-Magnet Synchronous Motor Design for Improving Self-sensing Performance at Very Low Speed. IEEE Trans. on Ind. Appl., Vol. 45, No.6, p.1939 1946, November/December 2009. [23] L. Sun, H. Li, B. Xu. Precise Determination of Permanent Magnet Synchronous Motor Parameters Based on Parameter Identification Technique ICEMS 2003, Electrical Machines and Systems, Vol. 1.p.34 36, 2003. [24] OPERA-2d Reference Manual. Cobham Technical Services, Version 13.014, December 2009.

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