Modelling and experimental research on the equivalent magnetic circuit network of hybrid magnetic couplers considering the magnetic leakage effect
2019; Institution of Engineering and Technology; Volume: 13; Issue: 9 Linguagem: Inglês
10.1049/iet-epa.2018.5808
ISSN1751-8679
AutoresShuang Wang, Yongcun Guo, Deyong Li, Chang Su,
Tópico(s)Electrical Contact Performance and Analysis
ResumoIET Electric Power ApplicationsVolume 13, Issue 9 p. 1413-1421 Research ArticleFree Access Modelling and experimental research on the equivalent magnetic circuit network of hybrid magnetic couplers considering the magnetic leakage effect Shuang Wang, Shuang Wang Anhui Intelligent Mine Technology and Equipment Engineering Laboratory, Anhui University of Science and Technology, Huainan, Anhui, 232001 People's Republic of China School of Mechanical Engineering, Anhui University of Science and Technology, Huainan, Anhui, 232001 People's Republic of ChinaSearch for more papers by this authorYongcun Guo, Corresponding Author Yongcun Guo guoyc1965@126.com Anhui Intelligent Mine Technology and Equipment Engineering Laboratory, Anhui University of Science and Technology, Huainan, Anhui, 232001 People's Republic of China School of Mechanical Engineering, Anhui University of Science and Technology, Huainan, Anhui, 232001 People's Republic of ChinaSearch for more papers by this authorDe-yong Li, De-yong Li School of Mechanical Engineering, Anhui University of Science and Technology, Huainan, Anhui, 232001 People's Republic of ChinaSearch for more papers by this authorChang Su, Chang Su School of Mechanical Engineering, Anhui University of Science and Technology, Huainan, Anhui, 232001 People's Republic of ChinaSearch for more papers by this author Shuang Wang, Shuang Wang Anhui Intelligent Mine Technology and Equipment Engineering Laboratory, Anhui University of Science and Technology, Huainan, Anhui, 232001 People's Republic of China School of Mechanical Engineering, Anhui University of Science and Technology, Huainan, Anhui, 232001 People's Republic of ChinaSearch for more papers by this authorYongcun Guo, Corresponding Author Yongcun Guo guoyc1965@126.com Anhui Intelligent Mine Technology and Equipment Engineering Laboratory, Anhui University of Science and Technology, Huainan, Anhui, 232001 People's Republic of China School of Mechanical Engineering, Anhui University of Science and Technology, Huainan, Anhui, 232001 People's Republic of ChinaSearch for more papers by this authorDe-yong Li, De-yong Li School of Mechanical Engineering, Anhui University of Science and Technology, Huainan, Anhui, 232001 People's Republic of ChinaSearch for more papers by this authorChang Su, Chang Su School of Mechanical Engineering, Anhui University of Science and Technology, Huainan, Anhui, 232001 People's Republic of ChinaSearch for more papers by this author First published: 28 June 2019 https://doi.org/10.1049/iet-epa.2018.5808Citations: 3AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this work, the massive modelling and computation costs required by the calculation of the magnetic leakage coefficient were avoided by using the three-dimensional finite element method (3D-FEM) in the initial design and optimisation of hybrid magnetic couplers (HMCs). The equivalent magnetic circuit network model of HMCs was established, and the equivalent reluctance of the model was calculated to obtain the analytic expression of the magnetic leakage coefficient of HMCs. A set of 1:2 prototypes was designed and manufactured. Simulation analysis and experimental tests were conducted to verify the correctness of the calculation of magnetic leakage. The calculated and experimental values of the magnetic leakage coefficient and the 3D-FEM were in good agreement. The output torque value of the HMC was analysed and calculated by considering that the air-gap leakage was closer to the test value and was more accurate than the value calculated without considering the air-gap leakage effect. This study provided a theoretical reference for the design and investigation of HMCs. 1 Introduction Couplers based on magnetic drive technology are new types of speed control equipment [1]. Magnetic couplers have attracted widespread attention at home and abroad because of their light load start, overload protection, isolation vibration, and other characteristics [2]. The three-dimensional finite element method (3D-FEM) can be used to analyse and design magnetic couplers [3, 4]. Nevertheless, 3D-FEM requires long modelling and computation times. These requirements are detrimental to the initial design and optimisation of magnetic couplers [5]. The magnetic circuit method is used for the initial design and optimisation of magnetic coupler motors; then, 3D-FEM can be adopted for performance verification and confirmation [6]. The magnetic leakage coefficient must be determined in the performance calculation and magnetic coupler design based on magnetic circuit analysis regardless of the principal structure of the motor [7]. Kim et al. [8] established a model of cylinder magnetic couplers through 3D-FEM and tested its linear coupling characteristic. Wang et al. established a 2D Cartesian coordinate system on a rotating conductor by improving the 3D magnetic field modelling of the axial disc-type magnetic coupler. They regarded the permanent magnet region as a travelling wave and solved the multilayer boundary problem [9]. Budhia et al. pointed out that existence of the axial magnetic force in the disc structure of the single disc-type magnetic coupler complicates the achievement of the complete balance of the axial force. They stated that disc wiping and damage equipment damage might occur. They proposed the use of 3D-FEM for model establishment and experimentally proved that axial force can be balanced by the symmetry of its structure [10]; Mohammadi and Mirsalim optimised the design of the single/double disc magnetic eddy current coupler and proposed an analytic model based on the traditional Faraday's law of electromagnetic induction and 3D-FEM. Their model could easily handle complex geometrical shapes and material properties [11]. Ge Yanjun et al. analysed and computed the magnetic leakage coefficient of a permanent asynchronous magnetic coupler by combining the integral method and the regional division method given the various defects of the magnetic field method, 3D-FEM, and magnetic line closing circuit method in calculating the magnetic leakage coefficient of permanent asynchronous magnetic couplers [12, 13]. Yang Chaojun et al. [14] used a set of magnet modulated asynchronous magnetic coupler with 14 antipodes and 21 magnet regulating pole pieces to establish a model of the permanent magnetic field and the modulation magnetic field in the air gap of magnetic induction-type asynchronous magnetic coupler and obtained the static and transient 3D air-gap magnetic field distribution and periodicity through 3D-FEM. Existing magnetic couplers can be roughly classified as radial cylindrical magnetic couplers [15] and axial disc-type magnetic couplers [16]. The electromagnetic torque of the cylindrical magnetic coupler is adjusted by adjusting the engagement area. This coupler has a singular speed control method. Under the same power conditions, the cylindrical magnetic coupler has lighter weight, smaller size, smaller rotary inertia, and higher efficiency than the disc magnetic coupler. In view of the above shortcomings, a hybrid magnetic coupler (HMC) that combines the traditional cylinder-type magnetic coupler and the double disc-type magnetic coupler was proposed. This coupler could increase the area of the induced magnetic field and improve transmission efficiency via magnetisation from the axial and radial directions to reduce the start-up impact vibration of the mechanical equipment and to provide protection against regular overload. Excessively large magnetic leakage is the main shortcoming of HMCs. In addition, its special structure complicates the calculation of magnetic leakage. Thus, the minimisation of magnetic leakage to improve the performance of HMCs has become a hot issue. The accurate and rapid calculation of the magnetic leakage of HMC components is particularly important for reducing magnetic leakage. In this work, the equivalent magnetic circuit network of HMCs is first analysed. Then, the magnetic leakage coefficient of the magnetic coupler is calculated. Finally, the 3D-FEM and prototype experimental method are utilised to verify the accuracy of the equivalent magnetic circuit and the correctness of the calculated air-gap magnetic leakage coefficient. 2 No-load equivalent magnetic circuit model of HMCs The structure of the HMC is shown in Fig. 1. The HMC is composed of a double disc-type magnetic coupler and a concentric axial magnetic coupler. It has a cylindrical symmetry. The active rotor comprises an axial copper conductor disc and a copper conductor ring. The upper slot of the active rotor is embedded with the copper conductor. The inner and outer of the conductor is wrapped with a thin ring copper layer to form a closed induction current loop. The permanent magnet rotor (slotted aluminium disc) is furnished with permanent magnets with axial magnetisation and closely spaced alternating arrangement. The active rotor of the HMC is not in contact with the driven rotor (permanent magnet rotor) to avoid vibration interference and reduce the loss of transmission parts. The active rotor and the driven rotor transmit electromagnetic torque through air-gap magnetic field interaction, and the electromagnetic torque increases under the same size and dimension conditions because the HMC can simultaneously achieve shaft radial air-gap magnetic induction. Fig. 1Open in figure viewerPowerPoint HMC (a) Sectional view of the HMC, (b) One-quarter hybrid magnetic field path indication The working principle of the HMC is illustrated in Fig. 1a. The outer box rotates when the input shaft rotates, and then the copper ring and two axial copper rotors rotate with the outer box. An eddy current is generated between the axial copper rotor and axial permanent magnet rotor because of electromagnetic induction. It drives the axial permanent magnet rotor to rotate and then drives the output shaft to rotate. The eddy current is generated because of the existence of an electromagnetic induction engagement area between the copper ring and radial permanent magnet rotor and drives the rotation of the output shaft. The axial permanent magnet rotor and radial permanent magnet rotor simultaneously generate an eddy current on the copper conductor to increase the area opposite to the permanent magnet. This effect improves transmission capacity. The magnetic field path of the HMC is illustrated in Fig. 1b, which shows the axial main magnetic circuit 1, radial main magnetic circuit 2, and pole-to-pole magnetic leakage circuit of the permanent magnet and the hybrid magnetic leakage circuit. In Fig. 2, Φr is the virtual intrinsic magnetic flux per pole of the permanent magnet (Wb), Φm is the total equivalent magnetic flux supplied outward of the permanent magnet per pole (Wb), Φm1 is the equivalent magnetic flux supplied outward of the axial permanent magnet per pole (Wb), and Φm2 is the equivalent magnetic flux supplied outward of the radial permanent magnet per pole (Wb). Given that the lengths of polarisation direction of the axial and radial permanent magnets are equal t, Φm1 = Φm2. Φδ is the air-gap equivalent main flux of the permanent magnet per pole (Wb), Rδ is the air-gap reluctance of the permanent magnet per pole (H−1) and Rσ is the reluctance of the hybrid magnetic leakage circuit (H−1). Given that the same permanent magnet is used in the axial and radial directions in this work, the reluctance of the axial permanent magnet and the radial permanent magnet is the same, Rm is the reluctance of the permanent magnet itself per pole (H−1), Rp is the magnetic leakage reluctance of the permanent magnet edge to the rotor (H−1), Rmm is the reluctance of the pole-to-pole magnetic leakage circuit of the permanent magnet (H−1) and Rmr is the reluctance of the magnetic leakage circuit of circumference direction of the permanent magnet (H−1). Fig. 2Open in figure viewerPowerPoint No-load equivalent magnetic circuit network of the HMC The air-gap equivalent main flux per pole Φδ and the total equivalent magnetic flux per pole supplied outward of the permanent magnet Φm can be obtained in reference to the no-load equivalent network diagram shown in Fig. 2, i.e. (1) (2) (3) where α = Rm/Rσ, β = Rm/Rmm, γ = Rm/Rmr and η = Rm/Rp. From (1) and (2), the air-gap leakage coefficient σ0 is obtained as (4) 3 Analysis and calculation of the reluctance of different components 3.1 Permanent magnet reluctance Rm and air-gap reluctance Rδ Equation (4) shows that the magnetic leakage coefficient of the HMC depends on the size of the reluctance of each component. The permanent magnet reluctance and air-gap reluctance can be expressed as (5) (6) where h is the length of the permanent magnet along the polarisation direction (mm), Spm is the opposite area of the permanent magnet (mm2); μ0 = 4π × 10−7 is the air permeability (H/m); μr is the relative permeability of the permanent magnet (H/m); δ1 is the axial air-gap length (mm); and δ2 is the radial air-gap length (mm). 3.2 Reluctance of inner rim magnetic circuit Rpi and reluctance of outer rim magnetic circuit Rpo Per pole of the permanent magnet The magnetic leakage equivalent magnetic circuit path of the inner and outer rims of the permanent magnet is shown in Fig. 3.The magnetic conductance of the inner rim magnetic leakage circuit of the axial permanent magnet and the magnetic conductance of inner rim magnetic leakage circuit of the radial permanent magnet can be expressed as Fig. 3Open in figure viewerPowerPoint Calculation sketch of reluctance of rim magnetic circuit of the permanent magnet (a) 3D cutaway view of axial inner rim magnetic circuit and outer rim magnetic circuit, (b) 2D calculation chart of axial inner rim magnetic circuit and outer rim magnetic circuit, (c) 3D cutaway view of radial inner rim magnetic circuit and outer rim magnetic circuit, (d) 2D calculation chart of radial inner rim magnetic circuit and outer rim magnetic circuit The magnetic conductance of the inner rim magnetic leakage circuit of the axial permanent magnet Λpri is (7) The magnetic conductance of inner rim magnetic leakage circuit of the radial permanent magnet Λpai is expressed as (8) where li is the distance between the inner rim of the permanent magnet and the circumference of the centre hole of the aluminium disc (mm); lo is the distance between the outer rim of the permanent magnet and the circumference of the centre hole of the aluminium disc (mm); p1 is the number of the pole pairs of the axial permanent magnet; and p2 is the number of the pole pairs of the radial permanent magnet. Similarly, the magnetic conductance of the outer rim magnetic leakage circuit of the axial permanent magnet and the magnetic conductance of the outer rim magnetic leakage circuit of the radial permanent magnet can be expressed as shown below The magnetic conductance of outer rim magnetic leakage circuit of the axial permanent magnet Λpro is (9) The magnetic conductance of the outer rim magnetic leakage circuit of the radial permanent magnet Λpao is (10) Therefore, the reluctance of the inner rim magnetic leakage circuit Rpi and the reluctance of the outer rim magnetic leakage circuit Rpo of the axial and radial permanent magnets can be obtained through the reciprocal valuation of (9) and (10), respectively, i.e. (11) (12) The reluctance of the magnetic leakage circuit of the rim of the permanent magnet is expressed as (13) 3.3 Reluctance of the pole-to-pole magnetic leakage circuit of the permanent magnet Rmm The pole-to-pole magnetic leakage path of the permanent magnet is presented in Fig. 4. The pole-to-pole leakage magnetic conductance of axial adjacent permanent magnets and radial adjacent permanent magnets are derived through the integration of the magnetic leakage area. Fig. 4Open in figure viewerPowerPoint Calculation sketch of reluctance of pole-to-pole magnetic leakage circuit of the permanent magnet (a) 3D calculation chart of axial pole-to-pole leakage magnetic circuit, (b) 2D calculation chart of axial pole-to-pole leakage magnetic circuit, (c) 3D cutaway view of radial pole-to-pole leakage magnetic circuit The pole-to-pole leakage magnetic conductance of axial permanent magnets Λmm1 is (14) The pole-to-pole leakage magnetic conductance of radial permanent magnets Λmm2 is expressed as (15) Thus, the reluctance of the pole-to-pole magnetic leakage circuit Rmm of axial and radial permanent magnets can be obtained from the reciprocal valuation of (14) and (15), respectively, i.e. (16) 3.4 Reluctance of the circumferential magnetic leakage circuit of the permanent magnet Rmr The circumferential magnetic leakage path of the permanent magnet is shown in Fig. 5. The circumferential leakage magnetic conductances of axial adjacent permanent magnets and radial adjacent permanent magnets are derived through the integration of the magnetic leakage area. Fig. 5Open in figure viewerPowerPoint Calculation sketch of the reluctance of the circumferential magnetic leakage circuit of permanent magnet (a) 3D calculation chart of the axial circumferential magnetic leakage circuit, (b) 3D cutaway view of radial circumferential magnetic leakage circuit The circumferential leakage magnetic conductance of the axial permanent magnet Λmr1 is as follows: (17) The circumferential leakage magnetic conductance of the radial permanent magnet Λmr2 is expressed as (18) Thus, the reluctance of the circumferential magnetic leakage circuit of the permanent magnet Rmr of axial and radial permanent magnets can be obtained from the reciprocal valuation of (17) and (18), respectively, i.e. (19) 3.5 Reluctance of hybrid magnetic leakage circuit Rσ The hybrid magnetic leakage path is illustrated in Fig. 6. The hybrid leakage magnetic conductance Λσ is obtained by the integration of the magnetic leakage area (20) where lh is the hybrid air gap, i.e. the distance between the axial permanent magnet rotor and radial permanent magnet rotor (mm). Fig. 6Open in figure viewerPowerPoint Calculation sketch of the reluctance of the hybrid magnetic leakage circuit Similarly, the reluctance of the hybrid magnetic leakage circuit Rσ can be obtained from the reciprocal valuation of magnetic conductance represented by (20). 4 Computational formula for output torque Let r1 and r2 be the inner and outer diameters (mm) of the axial copper disc, respectively and r3and r4 be the inner and outer diameters (mm) of the radial copper ring, respectively. The axial copper disc is considered to be composed of countless copper bars passing through the centre of the circle with the length of (r2–r1), and the radial copper ring is assumed to be composed of countless copper bars passing through the centre of the circle with the length of (r4-r3), as shown in Fig. 7. Fig. 7Open in figure viewerPowerPoint Equivalent structure diagram of the copper conductor (a) Axial copper plate, (b) Radial copper ring The induced electromotive force generated on the dl section of the copper bar is (21) where Bg is the magnetic flux density in the air gap (T), and ωs is the slip angular velocity of the axial copper disc (or radial copper ring) relative to the permanent magnet disc and is expressed as ωs = ω1–ω2. In this expression, ω1 and ω2 are the angular velocity of the rotation of the copper conductor (copper ring) and the axial permanent magnet disc (or the radial permanent magnet disc), respectively. Let s be the slip ratio, which has the following equation (22) The following equation can be obtained in reference to a previous study [17] (23) where T1a is the torque transmitted by the axial permanent magnet per pole (N/m); T1r is the torque transmitted by the radial permanent magnet per pole (N/m); σc is the electrical conductivity of copper (S/m); Δ is the skin depth of the copper conductor (mm); Npa is the number of axial pole pairs; Npr is the number of radial pole pairs, and kR is the correction coefficient of the resistance at different rotational speeds, the variation range of which is 0.6–4.6 [18]. As can be seen from analytic expression (23), the transmission torque of the HMC is related to air-gap lengths and rotational speed differences in the axial and radial directions. Small air gaps are associated with large air-gap permeability and large torque. However, the value of the transmitted torque is maximised when the air gap is constant. The transmitted torque increases first and then decreases as the rotational speed difference increases. Therefore, the total transmitted torque T of the HMC is (24) 5 3D-FEM verification and experimental verification 3D-FEM and the experimental method are used to verify the correctness of the results of magnetic leakage analysis. 5.1 Simulation The 3D magnetic field simulation software Ansoft is used to establish the model in accordance with the parameters of the experimental prototype of the HMC shown in Table 1. In this model, the solution type is defined as transient electromagnetic field simulation. The material of the permanent magnet is NdFe35, the yoke material of the permanent magnet is steel_1010, the solution time is set as 0.3 s, the step is 0.001 s, and the outer rotor input speed is 450 r/min. Table 1. Dimensional parameters of the HMC Parameters Value axial magnet pole pairs 4 axial magnet rotor outer diameter (mm) 200 axial magnet rotor thickness (mm) 25.4 axial copper conductor outer diameter (mm) 200 axial copper conductor thickness (mm) 8 axial yoke iron outer diameter (mm) 200 axial yoke iron thickness (mm) 10 radial magnet pole pairs 5 radial magnet rotor thickness (mm) 25.4 radial magnet rotor inner diameter (mm) 200 permanent magnet size (mm) 50.8 × 25.4 × 12.7 Given that mesh generation directly affects the accuracy of the results of finite element simulation, a selective generation method is adopted to improve mesh generation quality. That is, mesh generation should be dense for components that require high solution accuracy. These components include the copper conductor, yoke, permanent magnet, and the air gap. The number of mesh units is 35,000 and the number of nodes is 17,971. Mesh generation should be sparse for other components that require low solution accuracy to shorten simulation time and improve simulation quality [19]. The dimensional parameters of the HMC are shown in Table 1. The cloud picture of the magnetic flux density distribution of the HMC when the axial, radial, and hybrid air gaps have lengths of 5, 30, and 4 mm, respectively, is shown in Fig. 8. Fig. 8Open in figure viewerPowerPoint Simulation diagram (a) Magnetic density distribution nephogram of HMC, (b) Currents density distribution of radial copper, (c) Currents density distribution of axial copper The magnetic leakage coefficients computed through 3D-FEM and analytical calculation are shown in Table 2, which shows that the 3D-FEM and analytical calculation results are in good agreement. Table 2. Comparison of 3D-FEM results with analytical calculation results Axial air gap (mm) Radial air gap (mm) Hybrid air gap (mm) Analytic calculation 3D-FEM calculation Error% 3 2 20 1.236 1.1378 7.9 3 4 30 1.285 1.1401 9.7 4 2 20 1.237 1.1542 7.0 4 4 30 1.287 1.1487 10.0 5 2 20 1.238 1.1228 9.3 5 4 30 1.288 1.1927 7.4 Table 3 shows the values of the four magnetic leakage parameters α, β, γ, and η calculated with different axial, radial, and hybrid air gaps. The magnitudes of β, γ, η, and α are not in a uniform order. This result indicates that the four kinds of magnetic leakage carry different weights in the total air-gap magnetic leakage. The hybrid magnetic leakage has the highest weight. Thus, the remaining three kinds of magnetic leakage are ignored in the calculation. Table 3. Comparison of calculation results for α, β, γ, and η Axial air gap (mm) Radial air gap (mm) Hybrid air gap (mm) α β γ η 3 2 20 0.36 0.014 0.00000041 0.00000033 3 4 30 0.43 0.022 0.00000016 0.00000012 4 2 20 0.36 0.016 0.00000096 0.00000072 4 4 30 0.43 0.025 0.0000008 0.00000062 5 2 20 0.36 0.017 0.000001 0.0000008 5 4 30 0.43 0.027 0.0000013 0.0000012 3D-FEM is used to simulate the magnetic flux density distribution of the axial magnetic field and the radial magnetic field and the loss caused by the magnetic leakage, as shown in Fig. 9. Analysing the cloud picture reveals that the magnetic density distributed at the axial aluminium disc and the radial aluminium disc is low, whereas magnetic permeability is large. These results indicate that the aluminium disc has a good magnetism-isolating effect of and that the magnetic leakage reluctance of the inner and outer rims of the permanent magnet and the circumferential magnetic leakage reluctance of the permanent magnet is small. These results are in agreement with the results presented in Tables 2 and 3 and the analytical calculations. Fig. 9Open in figure viewerPowerPoint Simulation result diagram (a) Magnetic density distribution nephogram of radial permanent magnetic, (b) Magnetic density distribution nephogram of axial permanent magnetic, (c) Magnetic leakage loss diagram of radial permanent magnetic, (d) Magnetic leakage loss diagram of axial permanent magnetic 5.2 Experimental verification The 1:2 prototype experimental device of the HMC is shown in Fig. 10. It is mainly composed of a YE2-90S-4 three-phase asynchronous AC motor (with a rated speed of 1400 r/min), frequency converter (with a frequency range of 10–50 Hz), YH-502 dynamic torque sensor (with a range capacity of 0–500 Nm and precision of 0.5%, Beijing Yuhang Instrument Technology Co., Ltd), an elastic coupler, UX-52 digital display governor, HMC (1:2 prototype), YE2-80L-4-type load motor, WT-10A digital display gauss metre (with a range of 0–2000 mT), and MS6208B noncontact digital display tachometer. Fig. 10Open in figure viewerPowerPoint Experimental prototype As shown in the diagram of the experimental prototype, the input motor adopts the frequency inverter to control rotational speed, the load motor adopts the governor to control rotational speed and the dynamic torque sensor can monitor the output torque and the rotational speed of the experimental prototype in real time over the measurement range of 0–500 Nm and 0–6000 r/min. In the experiment, the output shaft of the experimental prototype is connected to one end of the torque sensor by using an elastic coupler, and the other end of the torque sensor is connected to the load motor to rotate synchronously with the load motor. The rotational speed of the load motor is rated at 450 r/min through the governor to facilitate data analysis. The rotational speed of the input motor is controlled by the frequency converter to provide the rotational speed of 50–450 r/min, and the input rotational speed can be measured by using the tachometer. Given that the torque sensor and the prototype of the HMC rotate synchronously, the output torque of the experimental prototype is obtained as the display reading of the torque sensor. First, the load motor starts to enter the smooth operation phase under non-loaded conditions. Then, the input motor starts. The rotational speed of the input motor gradually changes, and the torque sensor is used to read and record the output torque and rotational speed of the experimental prototype. The noncontact tachometer is adopted to test the stable output rotational speed of the experimental prototype after each change in the rotational speed of the input motor. Equation (28) is used to calculate the output torque of the HMC in consideration of the magnetic leakage effect. The output torque of the HMC is calculated without considering the magnetic leakage effect in reference to a previous work [20]. The contrast relationship curve is obtained and presented in Fig. 11. Fig. 11Open in figure viewerPowerPoint Comparative curve (a) Axial air gap is different, (b) Slip ratio is different Fig. 11 shows the comparison of the output torque curves constructed by using values calculated without considering the magnetic leakage effect, the values calculated by considering the magnetic leakage effect and the experimental values. As shown in Fig. 11a, as the axial air gap gradually decreases, the difference between the value calculated without considering the magnetic leakage effect and the experimental value increases, and the error between the calculated value and experimental result is ∼13.8%. The error between the values calculated by considering the magnetic leakage effect and the experimental value decreases and is ∼5.6%. As shown in Fig. 11b, as the slip ratio increases, the difference between the calculated value obtained without considering the magnetic leakage effect and the experimental value increases, and the error between the calculated and experimental values is ∼19.6%. The error between the values calculated by considering the magnetic leakage effect and the experimental value decreases and is ∼7.5%. In summary, the actual situation can be simulated well through the equivalent network modelling of the HMC that considers the magnetic leakage effect. The model can rapidly and accurately analyse the magnetic leakage of the complex magnetic coupler by using the simple magnetic circuit formula. The calculation of the magnetic leakage coefficient through traditional 3D-FEM requires massive modelling and calculation time resources. The study provides a theoretical reference for the design and investigation of HMCs. 6 Conclusions Large magnetic leakage is a major shortcoming of HMCs. In addition, the special structures of HMCs complicate the calculation of magnetic leakage. Therefore, accurately calculating the magnetic leakage of each component of magnetic couplers is highly important. This work provided the following contributions and obtained the following conclusions: (i) The equivalent magnetic circuit network model of HMCs that considers the leakage effect is established, and the calculation formula of the magnetic leakage coefficient of the magnetic coupler is obtained. (ii) 3D-FEM is used to simulate the magnetic leakage coefficient of the HMC. The error between the magnetic leakage coefficient provided by this method and the calculated magnetic leakage coefficient is only 8.7%. (iii) The self-developed prototype is adopted in an experiment, and the output torque values of the HMC calculated with and without accounting for the magnetic leakage effect are compared with the experimental results. The error between the values calculated by considering the magnetic leakage effect and experimental values is smaller than that between the values calculated without considering the magnetic leakage effect and experimental values. These results validate the modelling method. (iv) The equivalent network modelling of HMCs accounting for the magnetic leakage effect is simpler and faster than the traditional 3D-FEM model. The research results can provide a theoretical basis for the design and optimisation of HMCs. 7 Acknowledgments This research work was supported by the National Natural Science Foundation of China (Grant No. 51874004), the Key Project of Youth Natural Fund of Anhui University of Technology (Grant No. QN 2018116), and the Major Science and Technology Projects in Anhui Province (Grant No. 1908085QE227). 8 References 1Merdzan, M., Jumayev, S., Borisavljevic, A., et al: 'Electrical and magnetic model coupling of permanent magnet machines based on the harmonic analysis', IEEE Trans. Magn., 2015, 51, (11), pp. 1– 4 2Dutt, A., Varshney, S.K., Mahapatra, S.: 'Design of tunable couplers using magnetic fluid filled three-core optical fibers', IEEE Photonics Technol. Lett., 2012, 24, (3), pp. 164– 166 3Gok, K.: 'Development of three-dimensional finite element model to calculate the turning processing parameters in turning operations', Measurement, 2015, 75, pp. 57– 68 4Wen, T., Ou, W.X., Liu, Q., et al: 'Predication and analysis of positioning status of large-scale billets on forging dies using multi-body dynamics simulation', Int. J. Adv. Manuf. Technol., 2015, 80, (1), pp. 447– 453 5Fetzer, J., Kurz, S., Lehner, G., et al: 'Application of BEM-FEM coupling and the vector preisach model for the calculation of 3D magnetic fields in media with hysteresis', IEEE Trans. Magn., 2000, 36, (4), pp. 1258– 1262 6Christen, R., Bergamini, A., Motavalli, M.: 'Three-dimensional localization of defects in stay cables using magnetic flux leakage methods', J. Nondestruct. Eval., 2003, 22, (3), pp. 93– 101 7Wang, J., Zhu, J.: 'A simple method for performance prediction of permanent magnet eddy current couplings using a new magnetic equivalent circuit model', IEEE Trans. Ind. Electron., 2018, 65, (3), pp. 2487– 2495 8Kim, J.M., Choi, J.Y., Koo, M.M., et al: 'Characteristic analysis of tubular-type permanent-magnet linear magnetic coupling based on analytical magnetic field calculations', IEEE Trans. Appl. Supercond., 2016, 26, (4), pp. 1– 5 9Wang, J., Lin, H., Fang, S., et al: 'A general analytical model of permanent magnet eddy current couplings', IEEE Trans. Magn., 2014, 50, (1), pp. 1– 9 10Budhia, M., Boys, J.T., Covic, G., et al: 'Development of a single-sided flux magnetic coupler for electric vehicle IPT charging systems', IEEE Trans. Ind. Electron., 2013, 60, (1), pp. 318– 328 11Mohammadi, S., Mirsalim, M.: 'Design optimization of double-sided permanent-magnet radial-flux eddy-current couplers', Electr. Power Syst. Res., 2014, 108, (3), pp. 282– 292 12Mohammadi, S., Mirsalim, M., Vaez-Zadeh, S.: 'Nonlinear modeling of eddy-current couplers', IEEE Trans. Energy Convers., 2014, 29, (1), pp. 224– 231 13Yanjun, G.E., Yunzhuo, S.H.I., Feng, J.I.A., et al: 'Leakage coefficient computation of permanent magnetic asynchronous coupling', Mach. Des. Manuf., 2013, 1, (7), pp. 67– 70 14Chaojun, Y., Lingying, K., Tao, Z., et al: 'Research on 3D air-gap magnetic field of field modulated asynchronous magnetic couplings', J. Mech. Eng., 2016, 52, (8), pp. 8– 15 15Mohammadi, S., Mirsalim, M., Vaez-Zadeh, S., et al: 'Analytical modeling and analysis of axial-flux interior permanent-magnet couplers', IEEE Trans. Ind. Electron., 2014, 61, (11), pp. 5940– 5947 16Mohammadi, S., Mirsalim, M.: 'Double-sided permanent-magnet radial-flux eddy current couplers: three-dimensional analytical modelling, static and transient study, and sensitivity analysis', IET Electr. Power Appl., 2013, 7, (9), pp. 665– 679 17Menezes, L.R.A.X.D., Ajayi, A., Christopoulos, C., et al: 'Efficient computation of stochastic electromagnetic problems using unscented transforms', IET Sci. Meas. Technol., 2008, 2, (2), pp. 88– 95 18Zhang, B., Wan, Y., Li, Y., et al: ' Optimized design research on adjustable-speed permanent magnet coupling'. IEEE Int. Conf. on Industrial Technology 2013, Cape Town, South Africa, February 25–27, 2013 19Jafarboland, M., Sargazi, M.M.: 'Analytical modelling of the effect of pole offset on the output parameters of BLDC motor', IET Electr. Power Appl., 2018, 12, (5), pp. 666– 676 20Shuang, W., Yongcun, G., Pengyu, W., et al: 'Design and experimental research on hybrid magnetic coupler', J. Xi'an Jiaotong Univ., 2017, 51, (7), pp. 115– 123 Citing Literature Volume13, Issue9September 2019Pages 1413-1421 FiguresReferencesRelatedInformation
Referência(s)