a study for characteristics of variable reluctance ... · sensor for ev motor 9 ;10. the winding of...

A study for characteristics of variablereluctance resolver considering end slotleakage inductance and eccentricity

Zhi-LongWang1, Ki-Chan Kim*2

1,2Department of Electrical Engineering,Hanbat National University,

Daejeon, 305-719, South [email protected], [email protected],

Corresponding author*

Phone:+82-042-821-1090

February 4, 2018

Abstract

Background/Objectives: As a position sensor, vari-able reluctance (VR) resolver is widely applied to manyfields as well as electric vehicles (EV) and hybrid electricvehicles (HEV).

Methods/Statistical analysis: In this paper, a methodfor deriving the transformer ratio of the VR resolver is pro-posed by using Finite Element Method (FEM) simulations.In addition, the generalization of the inductance correctionfactor is studied according to the design parameter of theVR resolver such as the stack length. Moreover, the charac-teristics of the VR resolver when some abnormal problemsoccur such as eccentricity of the rotor are studied.

Findings: As a result, 2D FEM simulation is not con-sidering the end slot leakage inductance. Therefore, 3DFEM simulation is conducted in order to derive the endslot leakage inductance. But 3D FEM simulation takes alot of analysis time. Therefore, the correction factors of

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International Journal of Pure and Applied MathematicsVolume 118 No. 19 2018, 1791-1804ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu

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2D FEM simulation by the VR resolver design variablessuch as stack length were derived. When applying the valueof the correction factor using the 3D simulation, the errorof transformation ratio between the experimental data andsimulation data was about 3 %. By applying these correc-tion factor derivation methods, the value of the correctionfactor corresponding to the stack length of the VR resolverwas linearly derived. In addition, when eccentricity of therotor occurs, the output waveform of VR resolver is unbal-anced due to the unbalance of the magnetic field. Therefore,the unbalance phenomenon causes a large rotor angle error.By the results of the derivation, we can judge the cause ofthe error when the angle error occurs due to eccentricity.

Improvements/Applications: VR resolver is widelyapplied to EV fields because it is not greatly affected bythe external environment.In the future, we plan to studythe characteristics of VR resolver considering leakage in-ductance at the same time as eccentricity occurs.

Key Words: Eccentricity, end slot leakage inductance,FEM, position sensor, variable reluctance resolver.

1 Introduction

Recently, the automotive industry is gradually changing from en-gine car that use gasoline as the main power source to HEV andEV that use motor as the main driving source due to environmen-tal pollution and energy depletion 1,2. The motor used as a drivingsource of EV and HEV needs to be precisely controlled under avariety of variables. Therefore, it is important to accurately knowthe position information of rotor in order to control the motor 3.The position sensor of rotor is needed to grasp rotor informationof motor. The encoder and the resolver are widely used positionsensor of motor 4. Compared to the encoder, resolver is not affectedby the external environment such as temperature and vibration 5,6.In addition, unlike the encoder, resolver can grasp the position in-formation of rotor in the static state of motor 7. Therefore, rotorposition sensor of traction motor that has more vibration and dis-turbance is more suitable for use resolver than use encoder 8.

The VR resolver has a simple structure and a structure thinnerthan other resolvers. Therefore, it is mainly used as a position

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sensor for EV motor 9,10. The winding of the stator are woundexcitation coil, COS and SIN coil 11. The rotor of VR resolver isa pole structure in which the reluctance changes depending on theposition without a winding 12.

In this paper, a method for deriving the transformer ratio of theVR resolver is proposed by using the 2D and 3D FEM simulations.In addition, the generalization of the inductance correction factor isstudied according to the design parameter of the VR resolver suchas the stack length. Moreover, the influence of the transforma-tion ratio when some abnormal problems such as axial-gap of therotor. The transformer ratio of the VR resolver should be accu-rately derived in order to accurately grasp the position informationof the rotor. Also, the transformer ratio has an influence on theback-EMF generated at the output winding. Therefore, accuratelyderiving the transformer ratio of the VR resolver by simulation isvery important for accurate control of the motor.

2 Basic principle of VR resolver andRo-

tor angle derivation method

2.1 Basic principle of VR resolver

The basic principle of the VR resolver is shownin figure 1. Oneprimary winding, called exciting winding, and secondary winding ofsin and cos, called output winding, are placed in the stator insteadwithout winding in the rotor.

Fig. 1. Basic principle of VR resolver

If an exciting signal is applied at exciting winding, the value ofreluctance is changed when the position of the rotor changes. The

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secondary winding of output is placed by difference of 90 electricaldegrees.

Esin = K × E0 × sinωt× sin θ (1)

Ecos = K × E0 × sinωt× cos θ (2)

where Eo is the maximum of the exciting input voltage,ω(ω =2πf) is the frequency of exciting input voltage,θ is the angle of rotorrotation, ESIN is the induced electromotive force of sin secondarywinding, ECOS is the induced electromotive force of cos secondarywinding, K is the transformer ratio of VR resolver. The magneticflux passing through the air-gap during the unit time is changedby change of the reluctance when the rotor rotates. Accordinglythe induced electromotive force waveform from the secondary out-put windings can be represented as sin, cos function by using therotation angle as shown in equation (1) (2).

2.2 Rotor angle derivation method by simula-tion

In this paper, the rotor angle derivation of the resolver uses thearctangent method. The sampling process of the output waveformsof the sin winding and the cos winding is shown in figure 2(a).Through these sampling processes, a sinusoidal waveform can bederived as shown in figure 2(b). Using these sampled waveforms,the harmonics of the resolver are analyzed. As shown in figure 2(b),the sinusoidal waveform derived from the sampling is converted intoarctangent. Consequently, the converted arctangent curve showsthe angle of the rotor according to time.

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Fig. 2. Figure 2.Processes of the rotor angle derivation (a)Extract the max value sin and cos output waveform within a cycle

(b) Derive the position information by arctangent

3 Study for characteristics of VR re-

solver considering end slot leakage in-

ductance

3.1 Specifications and experimental results ofbase model

A FEM 2D analysis model of the VR resolver is shown in figure3(a). Model as shown in the figure is configured largely to thestator and the rotor. Input and output windings are wound atteeth in the stator. As shown in figure 3(b), x-axis and y-axis showthe teeth number and turns per tooth respectively. As shown infigure, the sin winding and the cos winding are arranged with aphase difference of 90 degrees.

Fig. 3. Analysis Model of VR resolver (a) FEM 2D analysismodel (b) The distribution of winding

The model applied to analysis is 16 slots and the saliency ofrotor is 4X. The details of specification of the analysis model areshown in Table 1.

Table 1.The specification of analysis model

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The experimental results are shown in table 2. The resistanceand the impedance of the input windings are 24 Ω and 110 Ω,respectively in table 2. Transformer ratio is also known to be 0.286(±10%).

Table 2.The experimental results of analysis model

3.2 FEM analysis of base model

The transformer ratio should be firstly analyzed in order to derivethe position information of the rotor. The error rate of the positioninformation of the rotor is accurately analyzed by correctly derivingthe transformer ratio.

K =Emax

E0

(3)

Z =√R2 + (2πfL)2 (4)

whereEmax is the maximum of induced electromotive force ofsecondary winding, Z is the impedance of input winding, R is theresistance of input winding, L is the inductance of the input wind-ing.

In 2D FEM simulation, maximum value of the output waveformof sin winding is 3 V as shown in figure 4(a), and the effectivevalue of the exciting input voltage is 5 Vrms. The transformer

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ratio is calculated as 0.424 by equation (3). The simulation resultis different from the experimental data as shown in table 2. Inthe FEM simulation, the value of impedance should be confirmed.Therefore, the impedance of the input winding can be obtained byusing equation (4). The value of winding resistance and frequencyare known by table 1 and 2. The inductance of winding can beconfirmed by the results of the simulation analysis as 1.17 mH infigure 4(b). In the 2D analysis, the value of winding impedance is77.33 Ω less than 110Ω, because the 2D analysis does not accountfor the end slot leakage inductance. In this case, the simulationmust proceed through the 3D FEM simulation.

Fig. 4. Results of 2D simulation (a) Output waveform of sin andcoswinding (b) Inductance of input winding

3D simulation is analyzed after modeling, as shown in figure5. 3D analysis can consider end slot leakage inductance and thetotal inductance of the input winding is shown as 1.727mH. Bycomparing the inductance of the 2D and 3D analysis, the end slotleakage inductance can be calculated.

As a result, in equation (4), the impedance of the input windingis derived as 111.13 Ω which is the similar results with the measureddata. The slot leakage inductance as a correction factor is appliedto the input winding then the Maxwell 2D simulation was analyzedagain.

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Fig. 5. FEM 3D analysis model

The output waveform that adds a correction factor is shown infigure 6. Maximum voltage of SIN winding is 2.09 V and the trans-former ratio is 2.09V / 7.071V = 0.295. The error of the transfor-mation ratio of the actual measurement result and the simulationresult is about 3%. Therefore, deriving the transformation ratio ofthe resolver is very important to accurately ascertain the positionof the rotor.

Fig. 6. 2D FEM output waveform by adding a correction factor

3.3 Deduction of correction factoraccording tostack length

The correction factor is derived using 3D FEM simulation due toend slot leakage inductance. The transformer ratio is substantiallythe same as the actual experimental result when the derived correc-tion factor is used for 2D FEM simulation. However, 3D simulationtakes a lot of analysis time. Therefore, the generalization study ofthe correction factor is performed to reduce the analysis time ac-cording to the design variables of VR resolver. The stack lengthis chosen as the design variables of VR resolver. The correctionfactor is derived according to stack length through the method inthe section above. Table III shows inductance of the 2D and 3Dsimulation according to stack length. The inductance increases asthe stack length increases as shown in table 3.

Table 3.The inductance according to stack length

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The correction factor of the inductance according to stack lengthis shown in figure 7. It can be seen that the correction factordecreases proportionally as the stack length increases. The trans-former ratio can be derived according to stack length of VR resolverthrough 2D FEM simulation without 3D FEM simulation from theresults of figure 7.

Fig. 7. The correction factor according to stack length

4 Study for characteristics of VR re-

solver considering eccentricity

In this paper, eccentricity is defined as the nonuniform air gap thatexists between the stator and rotor of VR resolver. The left figuresin figure 8 show three models for comparing the characteristics ofthe VR resolver according to eccentricity of the rotor. In figure 8(a) (b) (c), the base model, 0.3 mm eccentric model and 0.6 mmeccentric model are shown respectively. The right figures in figure8 show the output waveform of three model derived using FEManalysis. As can be seen in figure 8, the VR resolver has an un-balanced waveform due to the eccentricity of the rotor. This isbecause the magnetic field of the VR resolver is unbalanced due tothe eccentricity of the rotor. Since the reluctance changes irregu-larly according to the rotation angle of the rotor, the magnetic fluxpassing through the magnetic path also irregularly changed.

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Fig. 8. Comparison of output waveform of VR resolver accordingto rotor eccentricity (a) Base model (b) 0.3 mm eccentric model

(c) 0.6 mm eccentric model

The output sampling waveform for the above three models areshown in figure 9. As shown in figure 9, the output is unbalancedwhen the eccentric models are compared with base model. In figure9 (b), the angle information of the sampled waveform over time us-ing the arctangent method are derived. It was confirmed that theerror of the angle information largely occurs according to eccentric-ity.

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Fig. 9. Comparison of angle calculation of VR resolver accordingto rotor eccentricity (a) Comparison of output waveform sampling

(b) Comparison of angle derivation

5 Conclusion

In this paper, firstly, study on the method of deriving the transfor-mation ratio of the VR resolver was conducted. 2D FEM simulationis not considering the end slot leakage inductance. Therefore, 3DFEM simulation is conducted in order to derive the end slot leakageinductance. But 3D FEM simulation takes a lot of analysis time.Therefore, the correction factors of 2D FEM simulation by the VRresolver design variables such as stack length were derived. As aresult, when applying the value of the correction factor using the3D simulation, the error of transformation ratio between the ex-perimental data and simulation data was about 3 %. By applyingthese correction factor derivation methods, the value of the correc-tion factor corresponding to the stack length of the VR resolver waslinearly derived.

In addition, when eccentricity of the rotor occurs, the outputwaveform of VR resolver is unbalanced due to the unbalance ofthe magnetic field. Therefore, the unbalance phenomenon causesa large rotor angle error. By the results of the derivation, we canjudge the cause of the error when the angle error occurs due toeccentricity.

AcknowledgmentThis material is based upon work supported by the Ministry of

Trade, Industry & Energy(MOTIE, Korea) under Industrial Tech-nology Innovation Program. No.10063006, ‘Development of 2kW/kg,100kW, IPMSM Electric Vehicle Drive System Based on High Ef-ficiency Cooling System’.

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