IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006 1303

Determination of Parameters Considering MagneticNonlinearity in an Interior Permanent Magnet

Synchronous MotorJi-Young Lee1, Sang-Ho Lee1, Geun-Ho Lee1, Jung-Pyo Hong1, and Jin Hur2

Department of Electrical Engineering, Changwon National University, Changwon 641-773, KoreaPrecision Machinery Research Center, Korea Electronics Technology Institute (KETI), Puchon 420-140, Korea

This paper presents a method to calculate motor parameters considering magnetic nonlinearity in interior permanent magnet syn-chronous motors (IPMSM). To analyze motor characteristics by using equivalent circuits, inductance and iron loss resistance, which areaffected critically magnetic saturation, are obtained by using static finite-element analysis. For a fabricated IPMSM, input current andmotor characteristics such as torque and efficiency are calculated, and the proposed method is verified by the comparison of calculatedand experimental results.

Index Terms—Equivalent circuits, IPMSM, iron loss resistance, phase angle of armature current.

I. INTRODUCTION

THESE days, interior permanent magnet synchronous mo-tors (IPMSM) are very attractive for application to the sys-

tems required high power density such as hybrid vehicles andcompressors. As those systems become more compact, the elec-tric motors are getting highly saturated. It is, therefore, natu-rally required to consider the magnetic nonlinearity when themachines are designed or analyzed.

Although there are several papers for the parameter determi-nation in equivalent circuits considering iron loss [1]–[3], theparameters are obtained from measurement, experimental equa-tions, or electric linear equations. For example, even though theauthors in [3] have used finite-element analysis (FEA) to calcu-late flux, the iron loss resistance and inductance are calculatedby an experimental equations and voltage equations.

This paper presents an analytical method to calculate motorparameters considering magnetic nonlinearity in IPMSM whenthe characteristics are estimated.

The motor characteristics are evaluated by using equiv-alent circuits, and some parameters, consisting the circuitand affected magnetic saturation, are obtained by static FEA.Permanent magnet flux, iron loss resistance, and d- and q-axisinductances are the examples for the parameters. The otherparameters can be calculated by common equations [1].

Since the method about calculation of permanent magnet fluxis very fundamental in FEA [4], only the computation methodsabout inductance and iron loss resistance are presented in thispaper. It is evident that each method is not very new. However,it is new challenge that the critical parameters of equivalent cir-cuit are calculated directly by FEA, and the characteristics areestimated from the circuit. The proposed method is verified bythe comparison of calculated and experimental results.

Digital Object Identifier 10.1109/TMAG.2006.871951

Fig. 1. Equivalent circuits of IPMSM. (a) d-axis equivalent circuits. (b) q-axisequivalent circuits.

II. EQUIVALENT CIRCUIT

Equivalent circuits for IPMSM based on a synchronous d-qreference frame including iron losses [2], [5] are presented inFig. 1. The mathematical model of the equivalent circuits isgiven as follow equations. Iron loss is considered by equiva-lent resistance , and the d- and q-axis voltages and effectivetorque equations are given by (1), (2), and (3), respectively

(1)

(2)

(3)

0018-9464/$20.00 © 2006 IEEE

1304 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006

Fig. 2. Procedure of iron loss calculation.

where and are d- and q-axis component of armature cur-rent, and are d- and q-axis component of iron loss current,

and are d- and q-axis component of terminal voltage,is armature winding resistance per phase, is iron loss resis-tance, is flux linkage of permanent magnet per phase (rms),

and are d- and q-axis armature self inductance, and ispole pair.

III. PARAMETER CALCULATION METHODS

There are four parameters to be calculated in order to use thecircuit model of Fig. 1, which are as follows [1]:

1) flux linkage of permanent magnet per phase, ;2) armature winding resistance per phase, ;3) iron-loss resistance, ;4) d- and q-axis armature self inductances, and .This paper, however, only deals with the two parameters, iron

loss resistance and inductances, which are affected critically bynonlinearity of magnetic material.

A. Equivalent Iron Loss Resistance,

Fig. 2 shows the procedure of iron loss calculation using ironloss data of magnetic material. Another group of these authorshas given the detail explanation about the flowchart in [6]. Aftercalculating total iron loss, , the iron loss resistance canbe calculated by (4)

(4)

where is terminal voltage at no load and base speed.

B. Inductances, and

Fig. 3 shows the flowchart for the inductance calculation. Thed- and q-axis inductances are obtained directly by using staticFEA according to current and phase angle of armature current,

Fig. 3. Procedure of inductance calculation.

Fig. 4. Configuration of IPMSM with experimental devices.

TABLE ISPECIFICATIONS OF IPMSM

which is defined as an angle between back-EMF (q-axis) andcurrent in vector diagram. This calculation lasts for interestingcurrent and current angle steps.

The detail process of calculating inductance by FEA is listedas follows.

— Step 1: The permeability of each element is obtained andsaved at operating point for the variation of current andphase angle of armature current by nonlinear analysis.

— Step 2: Linear analysis is performed with the permeabilityobtained in Step 1 after setting the magneto-motive forceof PM to zero. For the input value, the phase angle ofarmature current is fixed at 90 in the d-axis inductance

LEE et al.: DETERMINATION OF PARAMETERS CONSIDERING MAGNETIC NONLINEARITY 1305

Fig. 5. Calculated inductance for current and phase angle of armature currentat operating temperature 120 C.

calculation, and at 0 in the q-axis inductance calculation.The magnitude of current is 1 for simple calculation.

— Step 3: The phase inductance is calculated by the effec-tive inductance equation which is defined as twice the en-ergy stored in the magnetic filed divided by the squareof the current in the device winding [7], [8]. The d- andq-axis inductances are times of phase inductance bythe theory of Park’s transformation for considering theidentical power flow. Therefore, the inductance is as (5).When the magnitude of input current is unit value as men-tioned in Step 2, the equation for inductance computationis simplified as shown in Fig. 3

(5)

IV. ANALYSIS RESULTS AND DISCUSSION

The proposed method is applied to the fabricated IPMSM fora compressor. Fig. 4 shows a cross section of the IPMSM alongwith d- and q-axis and its configuration with experimental de-vices. The specifications are in Table I.

Fig. 5 shows the calculated d-and q-axis inductance for cur-rent and phase angle of armature current in operating tempera-ture. The operating temperature is considered by the demagne-tizing flux density of PM. Even though the magnetomotive force(MMF) of PM is reduced because of high temperature, the mag-netic circuit is highly saturated by both PM and armature MMF,and the saturation effect is different depending on phase angleof armature current.

Fig. 6 shows the calculated flux density variation accordingto rotor movement and load condition at stator and rotor. Thisresult indicates that iron losses are generated at both the statorand the rotor. In comparison with the result in [3], iron loss isgenerated not only rotor surface and bridge, but also rotor yokebecause of flux path variation due to saturation as high as in therotor bridge.

Fig. 7 shows the iron loss data of laminated core S18 ac-cording to flux density and frequency, and the calculated ironloss depending on speed and load condition. The total iron lossincreases due to the increment of speed and armature currentbecause the flux density in the motor increases according to in-

Fig. 6. Magnetic flux density variation at no load and rated load condition.(a) Stator yoke. (b) Tooth center. (c) Rotor yoke.

creasing of total MMF, and the frequency of flux density varia-tion increases according to motor speed.

Figs. 8 and 9 present the comparison of calculated and exper-imental results to shows the validity of the proposed method.Fig. 8 is a comparison of current for the variation of load andphase angle of armature current, and Fig. 9 is a comparison ofspeed/torque characteristic for the values, which measured onlyfor the speed range keeping constant torque. Between compar-ison values, the error is small enough to be ignored for overallcharacteristic estimation.

V. CONCLUSION

Parameter computation methods are introduced to estimatethe characteristics of IPMSM by using the equivalent circuit

1306 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006

Fig. 7. Iron loss data and calculated results. (a) Iron loss data of S18.(b) Calculated iron loss.

Fig. 8. Comparison of calculated and experimental current.

with the consideration of saturation. Among several parame-ters, which need to be calculated for characteristic analysis, d-and q-axis inductances and iron loss resistance are computedby static FEA. The d- and q-axis inductances are obtained asthe functions of the d- and q-axis currents. The iron loss re-sistance is obtained considering load condition and the motor

Fig. 9. Comparison of calculated and experimental characteristics. (suffix,TEST means experimental data).

speed. The input current and characteristics such as torque andthe efficiency are calculated by the proposed method, and theresults agree well with the experimental results. It can be con-sidered that the proposed method is useful for the design of themotors.

ACKNOWLEDGMENT

This work was supported by MOCIE through the IERCprogram.

REFERENCES

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Manuscript received June 25, 2005 (e-mail: jyecad@korea.com).