# Nonlinear Modeling of Transmission Performance for Permanent Magnet Eddy Current Coupler

Authors: Xiaowei Yang; Yongguang Liu; Liang Wang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Establishing the analytical model for accurately predicting the transmitted torque of the permanent magnet eddy current coupler (PMEC) with double conductor rotors is very important to study the nonlinear transmission performance of the PMEC. In this paper, based on magnetic equivalent circuit (MEC) approaches, considering the effects of permanent magnet (PM) end leakage, side leakage, and back iron saturation, the 3D magnetic field model of the PMEC is established. Based on the magnetic field model, combined with Faraday’s law and Ampere’s law, the 3D nonlinear model of the PMEC is established. The proposed model adopts the Jenei method to describe the inductance limited eddy current field. The theoretical data, 3D finite element method (FEM) data, and experimental data are compared. The research shows that the predicted value of the analytical model matches well with the torque obtained by 3D FEM and experiments in the range of 0-100% slip, with an error less than 5.3% observed. Finally, the influence of structural parameters on the transmission performance of the PMEC is studied by using the proposed model.


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## Body

## 1. Introduction

Large-scale rotating equipment is the important power equipment in many areas, such as electric industry, iron & steel industry, petrochemical industry, transportation, and so on. The power system is connected by coupling to transfer the power. The traditional coupling is very strict in alignment; otherwise it will cause serious vibration, even major accidents such as equipment damage [1, 2]. In order to solve the problem of alignment, researchers have successfully explored a nonphysical contact transmission technology. It is magnetically coupled technology [3–6]. The the permanent magnet eddy current coupler (PMEC) usually consists of permanent magnets (PMs) rotor and conductor rotor, as shown in Figure 1. The PMs rotor includes PMs and back iron, and the conductor rotor includes copper and back iron [7, 8]. The research of electromagnetic field is the core and foundation of the PMEC. At present, the research methods mainly include numerical methods and analytical methods [9–12]. The finite element method (FEM) is a typical numerical method [13, 14]. Li Z et al. study the magnetic field, eddy current, and torque of the PMEC by the 3D FEM. Compared with experiment, the validity of the 3D FEM was verified [3]. Hyeon-JaeShin et al. take the PMEC as the research object. The influence of structure parameters on the transmission of the PMEC is studied by 3D FEM. The accuracy of the FEM results was verified by experiments [4]. Dong K et al. establish a 1/6 3D FEM model of the PMEC and study the efficiency of the PMEC [15]. The FEM can give accurate prediction results in the study of electromagnetics. But the FEM has long calculation times and is less flexible; therefore, FEM is not applicable in the initial design stage of electromagnetic fields [16–19]. The magnetic equivalent circuit (MEC) method is a common analytical method. The MEC simplifies the geometry or material nonlinearity and can quickly determine the influence of key design parameters [20–24]. Mohammadi S et al. use the MEC to study the performance of the PMEC. Compared with experiment and FEM, the validity of the model was verified under 8% slip [25–27]. Wang J et al. study the transmission performance of the PMEC by the MEC, and accurate prediction torque was obtained at 10% slip [28, 29]. Based on MEC, the conductivity of copper is regarded as a constant value, and the inductance characteristics of copper are neglected. Current studies show that results according to this model are not always accurate. Further, when slip is higher than 10% the error model is unacceptable, such that present studies are limited to less than 10% slip [25–31]. In literature [32], according to the slip, the Russell-Norsworthy factor is continuously revised, and the theoretical value is basically consistent with the FEM value within 100% slip. Literature [33–35] adopts Maxwell’s equations, and Zheng D et al. adopt the accurate subdomain method [36–38], where torque characteristics are studied under 100% slip. Good results were obtained, especially at low slip speed. In addition, literature [39, 40] regarding the back iron with infinite or a set permeability has shown that the saturation effect is neglected in the back iron.Figure 1
Structural diagram of the PMEC.In this paper, the PMEC with double conductor rotor is taken as the research object. The PMEC is shown in Figure2. Compared with Figure 1, it consists of a set of the PMs rotor and two sets of the conductor rotors. Combining MEC, Faraday’s, and Ampere’s laws, a 3D magnetic field model is built that includes edge effect and iron saturation effect. The Jenei method is used to describe the inductance characteristics of copper, and the dynamic conductivity of copper rotor is proposed. The nonlinear torque model of the PMEC is established, which can predict the transmission performance of the PMEC from 0-100% slip. Compared with the 3D FEM and experiment, the validity of the proposed model is verified, and the influence of sensitive parameters on the transmission performance of the PMEC is researched.Figure 2
Structural diagram of the PMEC with double conductor rotor.


## 2. Magnetic Field Modeling

### 2.1. Problem Description and Idealization

The distribution of the PMEC on the r-z and r-θ planes is shown in Figure 3. The conductor rotor is a composite rotor composed of copper and iron, which combines the high conductivity of copper with the high permeability of iron. This enables the PMEC to produce efficient transmission torque. The N-S pole of the PMs is alternatively distributed. The PMs rotor, the copper rotor, and the back iron rotor are coaxial, constituting a closed magnetic loop. Each pair of magnetic poles produce the same electromagnetic characteristics and mechanical properties.Figure 3
The graph of the PMEC on r-z and r-θ planar.Determining the electromagnetic field of the PMEC requires measuring the air gap magnetic field, which consists of both a static magnetic field produced by the PMs and the eddy current magnetic field. The superposition of the two magnetic fields constitutes the dynamic effective air gap magnetic field, which is the basis of the research for understanding PMEC performance.In order to study the influence of structure and operation parameters on the transmission torque characteristics of the PMEC, idealization of the PMEC is as follows:( 1 ) There is no eddy current effect in PMs, and the PMs are uniformly magnetized in axial direction( 2 ) The influence of the magnetization of the back iron on the air gap magnetic field is not consideredAs shown in Figure4, a seven-layer field model is established to study the magnetic circuit of the PMEC. The PMs magnetic circuits are as follows: closed magnetic circuit between adjacent PMs passing through the copper rotor and the back iron rotor; closed magnetic circuit between adjacent PMs passing through copper rotor or air gap; and closed magnetic circuit formed by the PM. The closed magnetic circuit passing through the copper rotor and the back iron rotor is the effective magnetic circuit, and the rest is the magnetic flux leakage. The copper rotor cuts the magnetic line with the velocity V, and the eddy current is generated in the copper rotor. The induced eddy current field is coupled with the excitation magnetic field to form the magnetic power of transmission torque.Figure 4
Electromagnetic field distribution of seven-layer field model.


### 2.2. Static Magnetic Field Model

When the PMEC is static, the speed difference between the conductor rotor and the PMs rotor is zero. At this time, the PMs produce the air gap magnetic field. According to analysis of the MEC and magnetic flux, the PMs represent the current source, and the copper rotor, back iron, and air gap represent the reluctance in the magnetic circuit. Because of the symmetry of the magnetic circuit, only half the pole pairs need to be studied. The MEC diagram of the PMEC is constructed as shown in Figure5.Figure 5
MEC diagram of the basic unit.The expressions of the source magnetic flux and the internal reluctance of the PM are(1)φr=BrZSr(2)Rr=hrμ0μrSrin whichSr is the axial area of the PM, expressed as(3)Sr=α2r22-r12(4)α=πpnn+1The reluctance of the back iron rotor is as follows:(5)RF=1PFin whichPF is the permeance of the back iron rotor, expressed as (6)PF=∫r3r4μ0μbhFrπ/pdrThe relative permeability of air gap and the copper rotor is approximately 1, so the reluctance of air gap and the copper rotor is as follows:(7)Rac=Ra+Rc=g+hcμ0αr1+r2/2r2-r1Magnetic flux leakage includes the leakage of adjacent PMs and PM leakage. The leakage magnetic circuit of adjacent PMs is shown in Figure6.Figure 6
The leakage magnetic circuit of adjacent PMs.The reluctance of the leakage magnetic of adjacent PMs is(8)Ra2=1Pa2(9)Pa2=∫r1r2∫0xa2μ0πx+rα/ndxdrin whichxa2 is (10)xa2=min⁡αr1+r24,g+hcThe PM leakage magnetic field is composed of lateral, inner, and outer edge magnetic leakage. The PM leakage magnetic circuit is shown in Figure7.Figure 7
Self-leakage magnetic circuit of the PM.The reluctance of the lateral edge flux leakage is(11)Ra11=1Pa11(12)Pa11=∫0ra11μ0r2-r12πr+hrdrin whichra11 is (13)ra11=min⁡αr1+r24,αr1+r24n,g+hcThe reluctance of the inner edge magnetic leakage is(14)Ra12=1Pa12(15)Pa12=∫0g+hcμ0αr12πr+hrdrThe reluctance of the outer edge magnetic leakage is(16)Ra13=1Pa13(17)Pa13=∫0g+hcμ0αr22πr+hrdrThe reluctance of self-leakage can be expressed as(18)22Pa11+Pa12+Pa13=Ra1According to Figure5 and  Eq. (1)–(7), (8), (10), (11), (13), (14), (16), and (18), the effective magnetic flux φe is as follows:(19)φe=Ra1Ra2RrRrRa24Rac+RF+Ra1Ra2+4Rac+RF+Ra1Ra22Rac+RF/2φrThe effective magnetic field intensity is(20)Beθ=Be=φeSr;-α2≤θ≤α20-π2p<θ<-α2,α2<θ<π2pThe relative permeabilityμb of the back iron rotor varies with the magnetic field intensity. Therefore, μb is determined by the B-H curve of the back iron rotor, as exhibited in Figure 8.Figure 8
B-H curve of the back iron rotor.If the thickness of back iron is too thin, magnetic saturation will occur. This increases the reluctance. The air gap flux density decreases and the transmission ability of the PMEC is reduced. If the thickness of the back iron is too thick, it does not contribute to the increase of transmission torque. On the contrary, inertia increases.According to Figure5 and B-H curves,(21)φe/2hFr2-r1≤1.3According to Eq. (19), hF is solved.(22)hF≥0.017mTherefore,hF is 0.02m in this paper. The relative permeability μb is calculated in Figure 9.Figure 9
Flow chart of the relative permeabilityμb [27].According to the B-H curve of the back iron rotor,μb is calculated as follows:(23)μbk=Bkμ0Hk0.5μbk-10.5(see [27]). The relative permeability is judged as follows:(24)μbk-μbk-1μbk≤0.001(see [27]).


### 2.3. Dynamic Magnetic Field Model

In the operation of the PMEC, the dynamic eddy current generated on the conductor rotor is the alternating eddy current. Therefore, the inductance characteristics of the alternating eddy current should be considered. The eddy current consists of that from the copper rotor and back iron rotor. However, because the contribution to the overall eddy current from the back rotor is far less than that in the copper motor, its influence can be neglected during the rapid optimization design.Effective magnetic fields are formed by superposing the induced magnetic field with the excited magnetic field, producing a stable, dynamic magnetic field.(25)Bper,θ=Beθ+Bpr,θThe induced current density in the copper rotor is as follows:(26)J=σpev→r,θ×B→r,θ=σperωBper,θwhereσpe is effective conductivity of copper rotor, and calculation is shown in Section 2.4. According to Ampere’s law,(27)∮lHdl=∯sJds(28)∮lBpθμ0dl=∯sσperωBpeθds(29)Bpθμ0·2g+2hc+hr=σpeωr2hc∫θ1θ2Bpeθdθm is the intermediate variable.(30)m=μ0hcr2σpeω2g+2hc+hrAccording to Eq. (25)-(29),(31)Bpr,θ=m∫θ1θ2Bpr,θ+BeθdθThe differential of Eq. (31) is as follows:(32)dBpr,θdθ=mBpr,θ+mBeθThe general solution ofBp(r,θ) is derived as follows:(33)Bpr,θ=k1emθ-π2p<θ<-α2,r1<r<r2k2emθ-Be-α2≤θ≤α2,r1<r<r2k3emθα2<θ<π2p,r1<r<r2B p(r,θ) satisfies the following conditions.(a)
There is a pointθ0 in [-α/2, α/2] that satisfies the following.(34)Bpr,θ0=0(b)
Bp(r,θ) satisfies continuity in point -α/2 and α/2.(35)k1e-αm/2=k2e-αm/2-Be(36)k3eαm/2=k2eαm/2-Be(c)
The currents on both sides ofθ0 are equal.(37)∫-π/2pθ0Jhcrdθ=∫θ0π/2pJhcrdθAccording to Eq. (34)-(37), (38)k1=Becosh⁡m/2pπ-αpcosh⁡mπ/2p-eαm/2k2=Becosh⁡m/2pπ-αpcosh⁡mπ/2pk3=Becosh⁡m/2pπ-αpcosh⁡mπ/2p-e-αm/2θ0=-1mln⁡cosh⁡m/2pπ-αpcosh⁡mπ/2p(39)Bpr,θ=Becosh⁡m/2pπ-αpcosh⁡mπ/2p-eαm/2emθ-π2p<θ<-α2,r1<r<r2Becosh⁡m/2pπ-αpcosh⁡mπ/2pemθ-Be-α2≤θ≤α2,r1<r<r2Becosh⁡m/2pπ-αpcosh⁡mπ/2p-e-αm/2emθα2<θ<π2p,r1<r<r2


### 2.4. Effective Conductivity of the Copper Rotor

The eddy current distribution of the copper rotor is shown in Figure10. The Greenhouse method is the piecewise superposition algorithm, which can accurately calculate eddy current inductance. However, there are many superposition operations in this algorithm, which makes the computational efficiency low [41, 42]. In order to simplify the calculation of inductance, Jenei et al. improve the Greenhouse method [42]. The Jenei method is the global average algorithm. In contrast, the Jenei method can predict inductance accurately, but its operation efficiency is higher than the Greenhouse method.(40)l=l1+l2+l3+l44Figure 10
Path diagram of eddy current in the copper rotor.in whichli is the effective length of the i segment conductor.According to the Jenei method, inductance is calculated as follows:(41)Lc=μ0l2πln⁡lhc+rAVα/2-0.67in which rAV=r1+r2/2.The calculation for impedance is as follows:(42)Zc=Rzc+jXLc(43)RZC=2lσcαrAVhc(44)X=2πf=πpsN115(45)Zc=2lσcαrAVhc2+πpsN1Lc152(46)cos⁡φRZC=RZCZc(47)s=N1-N2N1Therefore, the equivalent conductivity is calculated as follows:(48)Zc=2lσpeαrAVhc(49)1σpe=1σc2+πpsN1LcαrAVhc30l2The relationship betweenZc and slip is shown in Figure 11. The inductance of copper increases rapidly and the equivalent conductivity decreases rapidly. As slip increases, the characteristic copper inductance also increases. However, because the characteristic resistance remains unchanged, the equivalent conductivity is formed by both the inductance and the resistance. As such, the reduction of the equivalent conductivity gradually slows down.Figure 11
Z c-slip curve diagram.


## 2.1. Problem Description and Idealization

The distribution of the PMEC on the r-z and r-θ planes is shown in Figure 3. The conductor rotor is a composite rotor composed of copper and iron, which combines the high conductivity of copper with the high permeability of iron. This enables the PMEC to produce efficient transmission torque. The N-S pole of the PMs is alternatively distributed. The PMs rotor, the copper rotor, and the back iron rotor are coaxial, constituting a closed magnetic loop. Each pair of magnetic poles produce the same electromagnetic characteristics and mechanical properties.Figure 3
The graph of the PMEC on r-z and r-θ planar.Determining the electromagnetic field of the PMEC requires measuring the air gap magnetic field, which consists of both a static magnetic field produced by the PMs and the eddy current magnetic field. The superposition of the two magnetic fields constitutes the dynamic effective air gap magnetic field, which is the basis of the research for understanding PMEC performance.In order to study the influence of structure and operation parameters on the transmission torque characteristics of the PMEC, idealization of the PMEC is as follows:( 1 ) There is no eddy current effect in PMs, and the PMs are uniformly magnetized in axial direction( 2 ) The influence of the magnetization of the back iron on the air gap magnetic field is not consideredAs shown in Figure4, a seven-layer field model is established to study the magnetic circuit of the PMEC. The PMs magnetic circuits are as follows: closed magnetic circuit between adjacent PMs passing through the copper rotor and the back iron rotor; closed magnetic circuit between adjacent PMs passing through copper rotor or air gap; and closed magnetic circuit formed by the PM. The closed magnetic circuit passing through the copper rotor and the back iron rotor is the effective magnetic circuit, and the rest is the magnetic flux leakage. The copper rotor cuts the magnetic line with the velocity V, and the eddy current is generated in the copper rotor. The induced eddy current field is coupled with the excitation magnetic field to form the magnetic power of transmission torque.Figure 4
Electromagnetic field distribution of seven-layer field model.


## 2.2. Static Magnetic Field Model

When the PMEC is static, the speed difference between the conductor rotor and the PMs rotor is zero. At this time, the PMs produce the air gap magnetic field. According to analysis of the MEC and magnetic flux, the PMs represent the current source, and the copper rotor, back iron, and air gap represent the reluctance in the magnetic circuit. Because of the symmetry of the magnetic circuit, only half the pole pairs need to be studied. The MEC diagram of the PMEC is constructed as shown in Figure5.Figure 5
MEC diagram of the basic unit.The expressions of the source magnetic flux and the internal reluctance of the PM are(1)φr=BrZSr(2)Rr=hrμ0μrSrin whichSr is the axial area of the PM, expressed as(3)Sr=α2r22-r12(4)α=πpnn+1The reluctance of the back iron rotor is as follows:(5)RF=1PFin whichPF is the permeance of the back iron rotor, expressed as (6)PF=∫r3r4μ0μbhFrπ/pdrThe relative permeability of air gap and the copper rotor is approximately 1, so the reluctance of air gap and the copper rotor is as follows:(7)Rac=Ra+Rc=g+hcμ0αr1+r2/2r2-r1Magnetic flux leakage includes the leakage of adjacent PMs and PM leakage. The leakage magnetic circuit of adjacent PMs is shown in Figure6.Figure 6
The leakage magnetic circuit of adjacent PMs.The reluctance of the leakage magnetic of adjacent PMs is(8)Ra2=1Pa2(9)Pa2=∫r1r2∫0xa2μ0πx+rα/ndxdrin whichxa2 is (10)xa2=min⁡αr1+r24,g+hcThe PM leakage magnetic field is composed of lateral, inner, and outer edge magnetic leakage. The PM leakage magnetic circuit is shown in Figure7.Figure 7
Self-leakage magnetic circuit of the PM.The reluctance of the lateral edge flux leakage is(11)Ra11=1Pa11(12)Pa11=∫0ra11μ0r2-r12πr+hrdrin whichra11 is (13)ra11=min⁡αr1+r24,αr1+r24n,g+hcThe reluctance of the inner edge magnetic leakage is(14)Ra12=1Pa12(15)Pa12=∫0g+hcμ0αr12πr+hrdrThe reluctance of the outer edge magnetic leakage is(16)Ra13=1Pa13(17)Pa13=∫0g+hcμ0αr22πr+hrdrThe reluctance of self-leakage can be expressed as(18)22Pa11+Pa12+Pa13=Ra1According to Figure5 and  Eq. (1)–(7), (8), (10), (11), (13), (14), (16), and (18), the effective magnetic flux φe is as follows:(19)φe=Ra1Ra2RrRrRa24Rac+RF+Ra1Ra2+4Rac+RF+Ra1Ra22Rac+RF/2φrThe effective magnetic field intensity is(20)Beθ=Be=φeSr;-α2≤θ≤α20-π2p<θ<-α2,α2<θ<π2pThe relative permeabilityμb of the back iron rotor varies with the magnetic field intensity. Therefore, μb is determined by the B-H curve of the back iron rotor, as exhibited in Figure 8.Figure 8
B-H curve of the back iron rotor.If the thickness of back iron is too thin, magnetic saturation will occur. This increases the reluctance. The air gap flux density decreases and the transmission ability of the PMEC is reduced. If the thickness of the back iron is too thick, it does not contribute to the increase of transmission torque. On the contrary, inertia increases.According to Figure5 and B-H curves,(21)φe/2hFr2-r1≤1.3According to Eq. (19), hF is solved.(22)hF≥0.017mTherefore,hF is 0.02m in this paper. The relative permeability μb is calculated in Figure 9.Figure 9
Flow chart of the relative permeabilityμb [27].According to the B-H curve of the back iron rotor,μb is calculated as follows:(23)μbk=Bkμ0Hk0.5μbk-10.5(see [27]). The relative permeability is judged as follows:(24)μbk-μbk-1μbk≤0.001(see [27]).


## 2.3. Dynamic Magnetic Field Model

In the operation of the PMEC, the dynamic eddy current generated on the conductor rotor is the alternating eddy current. Therefore, the inductance characteristics of the alternating eddy current should be considered. The eddy current consists of that from the copper rotor and back iron rotor. However, because the contribution to the overall eddy current from the back rotor is far less than that in the copper motor, its influence can be neglected during the rapid optimization design.Effective magnetic fields are formed by superposing the induced magnetic field with the excited magnetic field, producing a stable, dynamic magnetic field.(25)Bper,θ=Beθ+Bpr,θThe induced current density in the copper rotor is as follows:(26)J=σpev→r,θ×B→r,θ=σperωBper,θwhereσpe is effective conductivity of copper rotor, and calculation is shown in Section 2.4. According to Ampere’s law,(27)∮lHdl=∯sJds(28)∮lBpθμ0dl=∯sσperωBpeθds(29)Bpθμ0·2g+2hc+hr=σpeωr2hc∫θ1θ2Bpeθdθm is the intermediate variable.(30)m=μ0hcr2σpeω2g+2hc+hrAccording to Eq. (25)-(29),(31)Bpr,θ=m∫θ1θ2Bpr,θ+BeθdθThe differential of Eq. (31) is as follows:(32)dBpr,θdθ=mBpr,θ+mBeθThe general solution ofBp(r,θ) is derived as follows:(33)Bpr,θ=k1emθ-π2p<θ<-α2,r1<r<r2k2emθ-Be-α2≤θ≤α2,r1<r<r2k3emθα2<θ<π2p,r1<r<r2B p(r,θ) satisfies the following conditions.(a)
There is a pointθ0 in [-α/2, α/2] that satisfies the following.(34)Bpr,θ0=0(b)
Bp(r,θ) satisfies continuity in point -α/2 and α/2.(35)k1e-αm/2=k2e-αm/2-Be(36)k3eαm/2=k2eαm/2-Be(c)
The currents on both sides ofθ0 are equal.(37)∫-π/2pθ0Jhcrdθ=∫θ0π/2pJhcrdθAccording to Eq. (34)-(37), (38)k1=Becosh⁡m/2pπ-αpcosh⁡mπ/2p-eαm/2k2=Becosh⁡m/2pπ-αpcosh⁡mπ/2pk3=Becosh⁡m/2pπ-αpcosh⁡mπ/2p-e-αm/2θ0=-1mln⁡cosh⁡m/2pπ-αpcosh⁡mπ/2p(39)Bpr,θ=Becosh⁡m/2pπ-αpcosh⁡mπ/2p-eαm/2emθ-π2p<θ<-α2,r1<r<r2Becosh⁡m/2pπ-αpcosh⁡mπ/2pemθ-Be-α2≤θ≤α2,r1<r<r2Becosh⁡m/2pπ-αpcosh⁡mπ/2p-e-αm/2emθα2<θ<π2p,r1<r<r2


## 2.4. Effective Conductivity of the Copper Rotor

The eddy current distribution of the copper rotor is shown in Figure10. The Greenhouse method is the piecewise superposition algorithm, which can accurately calculate eddy current inductance. However, there are many superposition operations in this algorithm, which makes the computational efficiency low [41, 42]. In order to simplify the calculation of inductance, Jenei et al. improve the Greenhouse method [42]. The Jenei method is the global average algorithm. In contrast, the Jenei method can predict inductance accurately, but its operation efficiency is higher than the Greenhouse method.(40)l=l1+l2+l3+l44Figure 10
Path diagram of eddy current in the copper rotor.in whichli is the effective length of the i segment conductor.According to the Jenei method, inductance is calculated as follows:(41)Lc=μ0l2πln⁡lhc+rAVα/2-0.67in which rAV=r1+r2/2.The calculation for impedance is as follows:(42)Zc=Rzc+jXLc(43)RZC=2lσcαrAVhc(44)X=2πf=πpsN115(45)Zc=2lσcαrAVhc2+πpsN1Lc152(46)cos⁡φRZC=RZCZc(47)s=N1-N2N1Therefore, the equivalent conductivity is calculated as follows:(48)Zc=2lσpeαrAVhc(49)1σpe=1σc2+πpsN1LcαrAVhc30l2The relationship betweenZc and slip is shown in Figure 11. The inductance of copper increases rapidly and the equivalent conductivity decreases rapidly. As slip increases, the characteristic copper inductance also increases. However, because the characteristic resistance remains unchanged, the equivalent conductivity is formed by both the inductance and the resistance. As such, the reduction of the equivalent conductivity gradually slows down.Figure 11
Z c-slip curve diagram.


## 3. Torque Model

According to the analysis of the dynamic magnetic field, the magnetic coupling force provides traction for torque transmission. Therefore, the output torque is expressed as follows [28, 29]:(50)T=4pkscos⁡φRZC∫VrJ×BdVwhereks is Russell-Norsworthy factor [28], the correction factor of eddy boundary effect, as calculated below:(51)ks=1-tanh⁡pr2-r1/2rpr2-r1/2r1+tanh⁡pr2-r1/2rtanh⁡pr4-r3-r2+r1/2rThe 3D integral process of space electromagnetic field is complex, which is unfavorable for rapid optimization design. The validity of the average radius for solving the output torque has been proved in the literature [25–30]. Therefore, the output torque is as follows:(52)T=4pksrAV3r2-r1hcσpeωcos⁡φRZC∫-π/2pπ/2pBpe2rAV,θdθ


## 4. Model Verification

The structure of the PMEC is shown in Figure2. The PM adopts NdFeB. The conductor rotor is constructed with copper and steel 10. The main structural parameters of the PMEC are shown in Table 1.Table 1
The structural parameters of the PMEC.
Parameter value Parameter value Inner radius of the PMs [m] 0.1775 Inner radius of conductor rotor [m] 0.15  Outer radius of the PMs [m] 0.2625 Outer radius of conductor rotor [m] 0.3  Thickness of the PMs [m] 0.032 Thickness of the copper rotor [m] 0.009  Thickness of the PMs 10 Thickness of the back iron rotor [m] 0.02  Remanence of the PMs [T] 1.2 Conductivity of the copper rotor [20°C] [MS/m ] 57.1Torque value is related to the temperature, and the eddy current loss produces heat [29]. Therefore, the air-cooling fin is designed according to the heat power of the PMEC. The thermodynamic analysis of the PMEC is carried out by ANSYS. The structure of the air-cooling fin is optimized, so that the working temperature of the copper rotor is about 90°C. The simulation results are shown in Figure 12.Figure 12
Temperature distribution of the PMEC.


### 4.1. Study on the Distribution of Electromagnetic Field

In literature [3, 4], the validity of the FEM is verified by comparative study of FEM and experiment. Therefore, the proposed model is compared with the FEM. Brz is the magnetic flux density of the PM in the Z direction, and it is extracted along r direction as shown in Figure 13. The average value of Brz is taken as the input value of the proposed model.Figure 13
Distribution of magnetic flux density of the PM.In the static state, the speed difference between the conductor rotor and the PMs rotor is 0. The effective magnetic flux density distribution of the air gap is shown in Figure14. The magnetic flux density distribution is relatively uniform in [-α/2, α/2] and is almost zero in other intervals. The average error between the theoretical and FEM data is 3.4%.Figure 14
Air gap flux density distribution curve at 0 speed difference.In the operation state, there is a speed difference between the conductor rotor and the PMs rotor. The effective magnetic flux density distribution of the air gap is exhibited in Figure15 at a 50rpm difference in speed. The air gap magnetic induction intensity gradually increases in [-α/2, α/2]. The magnetic induction intensity is about -0.1T in [-π/2p, -α/2], and it is about 0.1T in [α/2, π/2p]. The deviations between the theoretical and FEM data are less than 3.7%.Figure 15
Air gap flux density distribution curve at 50rpm speed difference.The eddy current distribution at a 50 rpm speed difference is shown in Figure16. The position of the eddy current ring corresponds to that of the PMs. The induced magnetic field is produced by the eddy current field. By comparing the data from Figures 14 and 15, it is found that the induced magnetic field weakens the excitation magnetic field in the first half of the period and strengthens it in the latter half of the period. As the speed difference also increases, the effect of the induced magnetic field on the excitation magnetic field is strengthened.Figure 16
Eddy current distribution at 50rpm speed difference.


### 4.2. Study on the Influence of Slip on Transmission Performance

Based on the structural parameters in Table1, the PMEC was established, and an experimental platform was built as shown in Figure 17. The conductor rotor is symmetrically distributed on both sides of the PMs rotor. The power equipment adopts the gear motor. The torque was tested in the range of 5% slip. The effect of different slip on torque was studied by changing the thickness of air gap.Figure 17
Experimental platform.When the air gap is 0.004m, the variation of torque is as shown in Figure18. When the air gap is 0.002m, the variation of torque is shown in Figure 19. The predicted torque model is compared with the 3D FEM and the experiment in the range of 5% slip. The results show that difference between theoretical and experimental data is always below 4.5%. Theoretical torque data based only on the resistance characteristics of the copper rotor is compared to FEM and experimental data. This data shows that, at low slip, the model agrees well, whereas torque error increases as slip also increases. When the slip is 4.7%, the error reaches 40%. The reason is that, with the increase of slip, the inductance effect of the copper rotor is enhanced, which makes the theoretical error increase.Figure 18
The torque curve of the PMEC (g=0.004m).Figure 19
The torque curve of the PMEC (g=0.002m).The torque-slip curve when the air gap is 0.004m is shown in Figure20. The theoretical data are in good agreement with 3D FEM data at 0-100% slip where the deviations between the theoretical data and the FEM data are less than 5%, also showing that the validity of the proposed torque model is proved in the 100% slip. In the range of 0-8% slip, the torque rapidly increases with the increase of slip. In the range of 8-100% slip, the torque decreases rapidly in the initial stage before stabilizing.Figure 20
The torque curve of the PMEC (g=0.004m).Based on the analysis in Section4.1, the induced magnetic field affects the distribution of the excitation magnetic field. This influence is strengthened as slip increases. The cutting speed of the copper rotor is proportional to slip. In the initial stage of slip, the cutting speed of the copper rotor increases, and the induced potential increases rapidly in the copper rotor. Although the equivalent conductivity of the copper rotor decreases, the eddy current field strengthens as does the intensity of the induced magnetic field. At this time, the induced magnetic field has little influence on the intensity of the excitation magnetic field, so the coupling force between the induction magnetic field and the excitation magnetic field is enhanced. The torque transmission ability of the PMEC increases rapidly. As the induced magnetic field increases, the intensity of excitation magnetic field is continuously weakened, which is clearly observed when the slip exceeds 8%. As slip increases, the increase of induction potential is limited, and the equivalent conductivity decreases rapidly. This reduces the intensity of the induced magnetic field. Therefore, the magnetic coupling force decreases and the torque transmission ability drops rapidly. When the slip exceeds 20%, both the equivalent conductivity and torque transmission decrease slowly and eventually stabilize.


### 4.3. Study on the Influence of Structural Parameters on Transmission Performance

(A) Study on the Influence of Air Gap on Transmission Performance.The thickness of the air gap was changed, and the effect of air gap on transmission performance was studied. As shown in Figure 21, with the decrease of the air gap, the torque transmission ability of the PMEC increased, and the peak value of torque was greatly improved. As the air gap decreased, the leakage of the PM decreased, which made the intensity of the excitation magnetic field increase, so the torque transmission ability was enhanced. The deviations between the theoretical data and the FEM data were always below 5%.Figure 21
Transmission characteristic curves at different air gaps.(B) Study on the Influence of Pole Pairs on Transmission Performance.The number of pole pairs was changed, and the total volume of the PMs remained unchanged. The effect of pole pairs on transmission performance was studied. The results are shown in Table 2, showing that the best number of pole pairs is 10 or 12. When the number of pole pairs is less than 10, the transmission torque ability increases obviously. When the number is more than 12, the transmission torque ability decreases significantly. This is because the distance of the PM decreases with the increase of the pole pairs, which increases the flux leakage of the PM. Thus the transmission torque ability of the PMEC decreases.Table 2
Table of relation between the pole pairs and torque (g=0.004m).
s p Analytical 3D FEM 4% 6% 8% 4% 6% 8% 6 1899 2162 2159 1805 2072 2124  8 2229 2570 2589 2125 2468 2535  10 2346 2692 2702 2235 2600 2638  12 2355 2660 2639 2271 2595 2612  14 2327 2604 2562 2225 2549 2541  16 2283 2560 2521 2203 2470 2437(C) Study on the Influence of the Thickness of the Copper Rotor on Transmission Performance.The influence of the thickness of the copper rotor on transmission performance was studied as shown in Table 3. As the thickness of the copper rotor increased to 0.009 m, the transmission torque of the PMEC increased and eventually leveled off. When the thickness was higher than 0.01m, the transmission torque ability increased slightly. Therefore, when the thickness of the copper rotor is 0.009~0.01m, transmit torsion for the PMEC is optimized.Table 3
Table of relation between the thickness of the copper rotor and torque (g=0.004m).
s h c Analytical 3D FEM 4% 6% 4% 6% 0.006 2053 2577 1966 2559  0.007 2186 2660 2102 2662  0.008 2280 2689 2167 2660  0.009 2346 2692 2235 2600  0.01 2363 2621 2266 2528  0.011 2360 2538 2229 2443  0.012 2347 2462 2221 2344When the thickness is less than 0.009m, the thickness of the copper has little influence on the magnetic leakage of the PMs. Increasing the thickness is beneficial to enhancing the eddy current magnetic field and the magnetic coupling force. Therefore, the transmission torque ability is enhanced. When the thickness is greater than 0.01m, the leakage increases and the effective air gap magnetic field decreases; thus the transmission torque ability of the PMEC decreases.(D) Study on the Influence of Thickness of the PM on Transmission Performance.The thickness of permanent magnets was changed, and the remaining parameters are shown in Table 1. As shown in Table 4, the results show that the thickness of the PM is positively correlated with the transmission torque ability of the PMEC. The deviations between the theoretical data and the FEM data are always below 5.3%. The validity of the proposed model was therefore verified.Table 4
Table of relation between the thickness of the PM and torque (g=0.004m).
s h r Analytical 3D FEM 2% 4% 6% 2% 4% 6% 0.026 1306 2121 2421 1240 2030 2334  0.028 1353 2202 2521 1279 2096 2431  0.03 1396 2275 2610 1332 2158 2522  0.032 1435 2346 2692 1372 2235 2600  0.034 1470 2406 2772 1406 2310 2681  0.036 1503 2465 2845 1429 2354 2719T0 is the value of torque under unit thickness of the PMs, T0=T/1000hr, and T0 is used to measure the utilization rate of the PMs. As shown in Table 5, with the increase of the thickness of the PMs, the contribution of the PMs to the transmission torque ability of the PMEC decreases. The reason for this phenomenon is that as the thickness of the PMs increased, so does leakage, thereby decreasing the contribution of the PMs to the effective air gap magnetic field as well as transmission torque. Therefore, the thickness of the PMs is negatively correlated with the utilization ratio of the PMs.Table 5
Table of relation between the T0 and torque.
s h r Analytical 3D FEM 2% 4% 6% 2% 4% 6% 0.026 50.1 81.5 93.1 47.7 78.1 89.7  0.028 48.3 78.6 90.0 45.6 74.8 86.8  0.03 46.5 75.8 87.0 44.4 71.9 84.1  0.032 44.8 73.3 84.1 42.9 69.8 81.2  0.034 43.2 70.7 81.5 41.3 67.9 78.8  0.036 41.8 68.5 79.0 39.7 65.4 75.5(E) Comparative Study with the Double-Sided PM Eddy Current Coupler.The double-sided PM eddy current coupler is shown in Figure 22. The main structural parameters of the double-sided PM eddy current coupler are consistent with those in Table 1. By 3D FEM, the transmission torque ability of the PMEC studied in this study was compared with that of the double-sided PM eddy current coupler as shown in Figure 23. The peak torque of the double-sided PM eddy current coupler is increased by 1.6 times. The utilization ratio of the PMs decreases. The torque value increases by 2.6 times under 100% slip. The increase of torque value is greater under 100% slip. When the load is blocked, the slip between the motor and the load is 100%. The PMEC becomes the load of the motor. The torque value transmitted by the PMEC is the output torque value of the motor. Torque value under 100% slip should be as small as possible. Therefore, the PMEC studied in this paper has more advantages than the double-sided PM eddy current coupler.Figure 22
Structural diagram of the double-sided PM eddy current coupler.Figure 23
Comparison of transmission characteristics (g=0.004m).


## 4.1. Study on the Distribution of Electromagnetic Field

In literature [3, 4], the validity of the FEM is verified by comparative study of FEM and experiment. Therefore, the proposed model is compared with the FEM. Brz is the magnetic flux density of the PM in the Z direction, and it is extracted along r direction as shown in Figure 13. The average value of Brz is taken as the input value of the proposed model.Figure 13
Distribution of magnetic flux density of the PM.In the static state, the speed difference between the conductor rotor and the PMs rotor is 0. The effective magnetic flux density distribution of the air gap is shown in Figure14. The magnetic flux density distribution is relatively uniform in [-α/2, α/2] and is almost zero in other intervals. The average error between the theoretical and FEM data is 3.4%.Figure 14
Air gap flux density distribution curve at 0 speed difference.In the operation state, there is a speed difference between the conductor rotor and the PMs rotor. The effective magnetic flux density distribution of the air gap is exhibited in Figure15 at a 50rpm difference in speed. The air gap magnetic induction intensity gradually increases in [-α/2, α/2]. The magnetic induction intensity is about -0.1T in [-π/2p, -α/2], and it is about 0.1T in [α/2, π/2p]. The deviations between the theoretical and FEM data are less than 3.7%.Figure 15
Air gap flux density distribution curve at 50rpm speed difference.The eddy current distribution at a 50 rpm speed difference is shown in Figure16. The position of the eddy current ring corresponds to that of the PMs. The induced magnetic field is produced by the eddy current field. By comparing the data from Figures 14 and 15, it is found that the induced magnetic field weakens the excitation magnetic field in the first half of the period and strengthens it in the latter half of the period. As the speed difference also increases, the effect of the induced magnetic field on the excitation magnetic field is strengthened.Figure 16
Eddy current distribution at 50rpm speed difference.


## 4.2. Study on the Influence of Slip on Transmission Performance

Based on the structural parameters in Table1, the PMEC was established, and an experimental platform was built as shown in Figure 17. The conductor rotor is symmetrically distributed on both sides of the PMs rotor. The power equipment adopts the gear motor. The torque was tested in the range of 5% slip. The effect of different slip on torque was studied by changing the thickness of air gap.Figure 17
Experimental platform.When the air gap is 0.004m, the variation of torque is as shown in Figure18. When the air gap is 0.002m, the variation of torque is shown in Figure 19. The predicted torque model is compared with the 3D FEM and the experiment in the range of 5% slip. The results show that difference between theoretical and experimental data is always below 4.5%. Theoretical torque data based only on the resistance characteristics of the copper rotor is compared to FEM and experimental data. This data shows that, at low slip, the model agrees well, whereas torque error increases as slip also increases. When the slip is 4.7%, the error reaches 40%. The reason is that, with the increase of slip, the inductance effect of the copper rotor is enhanced, which makes the theoretical error increase.Figure 18
The torque curve of the PMEC (g=0.004m).Figure 19
The torque curve of the PMEC (g=0.002m).The torque-slip curve when the air gap is 0.004m is shown in Figure20. The theoretical data are in good agreement with 3D FEM data at 0-100% slip where the deviations between the theoretical data and the FEM data are less than 5%, also showing that the validity of the proposed torque model is proved in the 100% slip. In the range of 0-8% slip, the torque rapidly increases with the increase of slip. In the range of 8-100% slip, the torque decreases rapidly in the initial stage before stabilizing.Figure 20
The torque curve of the PMEC (g=0.004m).Based on the analysis in Section4.1, the induced magnetic field affects the distribution of the excitation magnetic field. This influence is strengthened as slip increases. The cutting speed of the copper rotor is proportional to slip. In the initial stage of slip, the cutting speed of the copper rotor increases, and the induced potential increases rapidly in the copper rotor. Although the equivalent conductivity of the copper rotor decreases, the eddy current field strengthens as does the intensity of the induced magnetic field. At this time, the induced magnetic field has little influence on the intensity of the excitation magnetic field, so the coupling force between the induction magnetic field and the excitation magnetic field is enhanced. The torque transmission ability of the PMEC increases rapidly. As the induced magnetic field increases, the intensity of excitation magnetic field is continuously weakened, which is clearly observed when the slip exceeds 8%. As slip increases, the increase of induction potential is limited, and the equivalent conductivity decreases rapidly. This reduces the intensity of the induced magnetic field. Therefore, the magnetic coupling force decreases and the torque transmission ability drops rapidly. When the slip exceeds 20%, both the equivalent conductivity and torque transmission decrease slowly and eventually stabilize.


## 4.3. Study on the Influence of Structural Parameters on Transmission Performance

(A) Study on the Influence of Air Gap on Transmission Performance.The thickness of the air gap was changed, and the effect of air gap on transmission performance was studied. As shown in Figure 21, with the decrease of the air gap, the torque transmission ability of the PMEC increased, and the peak value of torque was greatly improved. As the air gap decreased, the leakage of the PM decreased, which made the intensity of the excitation magnetic field increase, so the torque transmission ability was enhanced. The deviations between the theoretical data and the FEM data were always below 5%.Figure 21
Transmission characteristic curves at different air gaps.(B) Study on the Influence of Pole Pairs on Transmission Performance.The number of pole pairs was changed, and the total volume of the PMs remained unchanged. The effect of pole pairs on transmission performance was studied. The results are shown in Table 2, showing that the best number of pole pairs is 10 or 12. When the number of pole pairs is less than 10, the transmission torque ability increases obviously. When the number is more than 12, the transmission torque ability decreases significantly. This is because the distance of the PM decreases with the increase of the pole pairs, which increases the flux leakage of the PM. Thus the transmission torque ability of the PMEC decreases.Table 2
Table of relation between the pole pairs and torque (g=0.004m).
s p Analytical 3D FEM 4% 6% 8% 4% 6% 8% 6 1899 2162 2159 1805 2072 2124  8 2229 2570 2589 2125 2468 2535  10 2346 2692 2702 2235 2600 2638  12 2355 2660 2639 2271 2595 2612  14 2327 2604 2562 2225 2549 2541  16 2283 2560 2521 2203 2470 2437(C) Study on the Influence of the Thickness of the Copper Rotor on Transmission Performance.The influence of the thickness of the copper rotor on transmission performance was studied as shown in Table 3. As the thickness of the copper rotor increased to 0.009 m, the transmission torque of the PMEC increased and eventually leveled off. When the thickness was higher than 0.01m, the transmission torque ability increased slightly. Therefore, when the thickness of the copper rotor is 0.009~0.01m, transmit torsion for the PMEC is optimized.Table 3
Table of relation between the thickness of the copper rotor and torque (g=0.004m).
s h c Analytical 3D FEM 4% 6% 4% 6% 0.006 2053 2577 1966 2559  0.007 2186 2660 2102 2662  0.008 2280 2689 2167 2660  0.009 2346 2692 2235 2600  0.01 2363 2621 2266 2528  0.011 2360 2538 2229 2443  0.012 2347 2462 2221 2344When the thickness is less than 0.009m, the thickness of the copper has little influence on the magnetic leakage of the PMs. Increasing the thickness is beneficial to enhancing the eddy current magnetic field and the magnetic coupling force. Therefore, the transmission torque ability is enhanced. When the thickness is greater than 0.01m, the leakage increases and the effective air gap magnetic field decreases; thus the transmission torque ability of the PMEC decreases.(D) Study on the Influence of Thickness of the PM on Transmission Performance.The thickness of permanent magnets was changed, and the remaining parameters are shown in Table 1. As shown in Table 4, the results show that the thickness of the PM is positively correlated with the transmission torque ability of the PMEC. The deviations between the theoretical data and the FEM data are always below 5.3%. The validity of the proposed model was therefore verified.Table 4
Table of relation between the thickness of the PM and torque (g=0.004m).
s h r Analytical 3D FEM 2% 4% 6% 2% 4% 6% 0.026 1306 2121 2421 1240 2030 2334  0.028 1353 2202 2521 1279 2096 2431  0.03 1396 2275 2610 1332 2158 2522  0.032 1435 2346 2692 1372 2235 2600  0.034 1470 2406 2772 1406 2310 2681  0.036 1503 2465 2845 1429 2354 2719T0 is the value of torque under unit thickness of the PMs, T0=T/1000hr, and T0 is used to measure the utilization rate of the PMs. As shown in Table 5, with the increase of the thickness of the PMs, the contribution of the PMs to the transmission torque ability of the PMEC decreases. The reason for this phenomenon is that as the thickness of the PMs increased, so does leakage, thereby decreasing the contribution of the PMs to the effective air gap magnetic field as well as transmission torque. Therefore, the thickness of the PMs is negatively correlated with the utilization ratio of the PMs.Table 5
Table of relation between the T0 and torque.
s h r Analytical 3D FEM 2% 4% 6% 2% 4% 6% 0.026 50.1 81.5 93.1 47.7 78.1 89.7  0.028 48.3 78.6 90.0 45.6 74.8 86.8  0.03 46.5 75.8 87.0 44.4 71.9 84.1  0.032 44.8 73.3 84.1 42.9 69.8 81.2  0.034 43.2 70.7 81.5 41.3 67.9 78.8  0.036 41.8 68.5 79.0 39.7 65.4 75.5(E) Comparative Study with the Double-Sided PM Eddy Current Coupler.The double-sided PM eddy current coupler is shown in Figure 22. The main structural parameters of the double-sided PM eddy current coupler are consistent with those in Table 1. By 3D FEM, the transmission torque ability of the PMEC studied in this study was compared with that of the double-sided PM eddy current coupler as shown in Figure 23. The peak torque of the double-sided PM eddy current coupler is increased by 1.6 times. The utilization ratio of the PMs decreases. The torque value increases by 2.6 times under 100% slip. The increase of torque value is greater under 100% slip. When the load is blocked, the slip between the motor and the load is 100%. The PMEC becomes the load of the motor. The torque value transmitted by the PMEC is the output torque value of the motor. Torque value under 100% slip should be as small as possible. Therefore, the PMEC studied in this paper has more advantages than the double-sided PM eddy current coupler.Figure 22
Structural diagram of the double-sided PM eddy current coupler.Figure 23
Comparison of transmission characteristics (g=0.004m).


## 5. Conclusion

Based on MEC, combining Faraday’s law, Ampere’s law, and Jenei method, a simple and practical 3D nonlinear model is proposed to predict the magnetic field distribution and transmission characteristics of the PMEC. The theoretical data, 3D FEM data, and experiment data are comparatively studied. The validity of the model is verified in the range of 0~100% slip. The proposed theoretical model can quickly study the influence of structural parameters on the transmission performance of the PMEC.( 1 )The slip is less than 8%, the transmission torque ability is positively correlated with the slip, the slip is higher than 8%, and the transmission torque ability is negatively correlated with the slip.( 2 )The thickness of air gap and the PMs are the two most obvious parameters affecting the transmission torque ability of the PMEC. The thickness of air gap is negatively correlated and the thickness of the PMs is positively correlated.( 3 )The thickness of the PMs is negatively correlated with the utilization ratio of the PMs. Therefore, the thickness of the PMs should not be too thick.( 4 )When the number of pole pairs is greater than 12 pairs, with the increase of the number of pole pairs, the transmission torque ability decreases. The optimal number of magnetic poles is 10 or 12.( 5 ) The thickness of copper rotor is 0.009-0.01m. If the thickness of the copper rotor is too thick or too thin, the transmission torque of the PMEC will decrease.In addition, the PMEC of double conductor rotors is compared with the double-sided PM eddy current coupler, and the advantages of double conductor rotors structure are verified.



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*Source: 2098725-2019-05-02.xml*