Analytical modeling and performance evaluation of an adjustable permanent magnet linear eddy current brake with mechanical magnetic adjuster

Li, Zhao; Wang, Jialei; Li, Yan; Yang, Hui; Wang, Dazhi

doi:10.1063/5.0152032

A novel mechanical flux-adjusting permanent magnet linear eddy current brake with high performance and its nonlinear analytical model are proposed. The innovative point lies in attaching an additional mechanical magnetic adjuster directly above the interior permanent magnet bar, which can achieve flexible adjustment of the air-gap flux and improve the generation capacity of the braking force and its regulation range. Considering the importance of the analytic model in practical design, the average air-gap flux density and eddy current density are estimated based on the equivalent magnetic circuit method and Ampere’s law, where the critical flux-weakening effect and leakage flux effect are calculated quantitatively. Then a practical theoretical model of braking force is derived based on the energy conversion and verified by the 3D finite-element method. Through analysis of the magnitude and regulation range of the braking force in a given range of speed, the advantages of the device and the electromagnetic force model are proved separately. Moreover, design suggestions for selecting the key geometrical parameters are further given, which offer useful information for pre-design and optimization.

I. INTRODUCTION

As a simple and practical electromagnetic device, the eddy current brake (ECB) has a clear mechanism, which follows Faraday’s law and the Lorentz force formula. Eddy current braking takes full advantage of the electromagnetic damping generated between the magnetic field and eddy current field to brake a moving object, and the kinetic energy will be eventually lost in the form of heat. Currently, such non-adhesive braking technology is widely used in many engineering fields, such as braking,¹ vibration suppression,² and transmission.^3,4 Especially, the advantages of ECB are more obvious in the linear motor test system.^5–7

As research moves along, the development of ECB presents the trend of diversification. According to the different excitation modes, ECB can be classified into three kinds: electric excitation ECB,⁸ permanent magnet ECB,⁹ and hybrid excitation ECB.¹⁰ By comparison, electric excitation ECB has adjustable braking force but requires external power supply accompanied by the disadvantages of large volume and large loss, permanent magnet ECB employing permanent magnet material technology has been developed successfully from the viewpoint of energy conservation and environmental protection and requires no other power supply but has some difficulty in field adjustment and dynamic brake control, and hybrid excitation ECB has the advantages of the aforementioned ECBs but further raises the complexity of the structure and the risk of instability.

Furthermore, ECB can be classified into linear ECB, axial ECB, radial ECB, and axial-radial ECB depending on the configuration. Functionally, linear ECB can provide the braking force required, while the rotating one can produce the braking torque. This paper focuses on linear ECB. To improve the performance of linear ECB, researchers have carried out a lot of work. A comparative study has shown that magnetization patterns have an important effect on the force–speed characteristics of permanent magnet linear ECB.¹¹ The dynamic characteristics of linear electric excitation ECB in a maglev train were analyzed.¹² Novel parallel hybrid excitation linear ECB was analyzed and designed.¹³ Dual-sided hybrid excitation linear ECB was studied to suppress the vibration of a linear motor.¹⁴ To widen the speed range, novel cage-secondary PM linear ECB was presented.¹⁵ One contribution of this paper is that a novel permanent magnet linear ECB (NPMLECB) is proposed, as shown in Fig. 1, and it can be seen that the NPMLECB has the potential of flexible adjustability and large braking force with a wide speed range.

FIG. 1.

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Structure of the proposed NPMLECB: (a) total structure; (b) y-z cross section; (c) y-x cross section.

The quantitative calculation of the key performance of ECB, which can be achieved by numerical or analytical methods, is quite necessary and meaningful at the initial design stage.^16–18 The finite element method (FEM) is one of the most common numerical methods in the field of theoretical research and enterprise. The corresponding software is powerful and can be used to analyze various electromagnetic devices with complex topologies. However, it requires a high hardware configuration and long calculation time, especially the 3D FEM. Hence, most of the time, the FEM is employed to evaluate the accuracy of the theoretical model without experiments, which we have also done in this paper. In comparison, due to the flexibility, intuition and relative accuracy, the analytical method gets more attention in theoretical research. By solving Maxwell’s equations, analytical models (AMs) of the magnetic field, eddy current, and torque are established for axial ECB^19,20 and radial ECB.^21,22 This method still requires a great deal of programming to solve a series of partial differential equations (PDEs), especially with complex boundary conditions²³ and 3D analysis.²⁴ Another analytic method is to construct a magnetic equivalent circuit (MEC) of the field system. Some modified MECs are developed and adopted in rotating ECB devices. The key is how to accurately incorporate eddy current effects into the MEC model. In addition, some hybrid approaches have also been proposed to overcome the drawbacks of a single approach, for example, a combination of MECs and Maxwell’s equations for hybrid excitation linear ECB²⁵ and interior permanent magnet axial ECB,²⁶ a combination of the 3D-FEM and Maxwell’s equations for radial ECB,²⁷ and a combination of subdomain technology and MECs for linear cylindrical ECB.²⁵ In light of this, another contribution of this paper is that the braking force model of the NPMLECB is proposed based on the modified MEC, which can reflect the regulation process of the magnetic field and braking force.

In this paper, a novel hybrid magnetic path permanent magnet linear eddy current brake is presented. Based on the analysis of the design feature and working principle of the NPMLECB, the magnetic flux paths of the system are given qualitatively. To further simplify the analysis process of the device performance, a no-load and on-load magnetic field model is constructed in sequence by using the equivalent magnetic circuit and electromagnetic theory. Then the expression of the braking force is derived and verified by using the 3D FEM, and parametric analysis is further executed in the end.

II. DESIGN FEATURE AND WORKING PRINCIPLE

A. Design feature

As shown in Fig. 1, the proposed NPMLECB mainly consists of a primary and secondary mechanical magnetic adjuster (MMA). The secondary MMA comprises a low resistivity conductor plate and a high permeability iron core. The primary MMA comprises two permanent magnet plates (PMPs) and one permanent magnet bar (PMB). Every permanent magnet plate is composed of a series of permanent magnets and adjacent iron poles, which are mounted on the iron core. Moreover, the permanent magnets located on different PMPs are vertically magnetized, and the direction of magnetization is opposite, while the PMB is transversely magnetized. The MMA is made of ferromagnetic material with high permeability, which is free to move along the transversal direction.

B. Working principle

According to Faraday’s law of electromagnetic induction, eddy currents will be caused because of the relative movement between the primary and the secondary conductor plate. The interactions between the induced magnetic field and the permanent magnetic field will produce the braking force. The existing MMA serves as an air-gap flux adjuster. By shifting the MMA under a given lateral displacement, the flux produced by the PMB passing through the iron poles can be controlled quantitatively, as well as the air-gap field. Therefore, the braking force and braking distance of the system are kept under effective control without complex power electronic devices. Furthermore, the smaller the area covered in the PMB by the MMA, the smaller the flux-weakening effect, and the greater the air-gap magnetic flux. The characteristics and advantages of the proposed NPMLECB are summarized as follows:

Compared with the hybrid excitation linear eddy current brake, no additional power supply is required; as a result, potential energy savings are achieved.
Combined excitation of multiple permanent magnets improves the amplitude of the braking force and the braking force density.
The mechanical magnetic modulation mode, where no complex control is needed, can further improve the stability and safety of the braking system.

III. THEORETICAL MODEL

A. Magnetic flux paths

In order to facilitate the analysis, the magnetic flux paths of the PMP and PMB can be described independently. The magnetic flux paths of the PMP should be divided into two parts: first, it will pass through permanent magnet S (N-pole) → air gap → conductor plate → iron core → air gap → iron pole A → iron core A → permanent magnet S (S-pole); second, it will pass through permanent magnet S (N-pole) → air gap → conductor plate → iron core → air gap → permanent magnet N → iron pole B → PMB → iron core A → permanent magnet S (S-pole). Compared to the PMP, there are different magnetic flux paths for PMB, which will pass through PMB (N-pole) → iron core B → permanent magnet N and iron pole B → air gap → conductor plate → iron core → air gap → permanent magnet S and iron pole A → iron core A → PMB (S-pole). These flux paths will contribute to the construction of equivalent magnetic circuits.

B. No-load magnetic field without magnetic leakage consideration

Due to inconsistencies in the excitation region, it is not a good choice to establish the model of static magnetic field by solving Maxwell’s equation directly. A reasonable MEC model can be employed to solve these complicated magnetic field problems. According to the superposition principle, the effective value of the air-gap magnetic field is the sum of the magnetic fields when the PMP and PMB act alone. On the basis of the analysis of magnetic flux paths, the MEC model of the proposed NPMLECB is shown in Figs. 2(a) and 2(b). However, to reduce computational complexity, the MEC model within PMB and PMP excitation is constructed and employed, as shown in Fig. 2(c).

FIG. 2.

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MEC model of the proposed NPMLECB: (a) PMP excitation; (b) PMB excitation; (c) PMB and PMP excitation together.

The magnetomotive force (MMF) and reluctance of the PMP and PMB are given as follows

F_{pmp} = H_{c} h_{p}, F_{pmb} = H_{c} W_{b},

(1)

R_{pmp} = \frac{h_{p}}{μ_{0} μ_{r} W_{p} L_{p}}, R_{pmb} = \frac{W_{b}}{μ_{0} μ_{r} h_{b} L_{b} / p},

(2)

where, H_c, h_p, W_p, L_p, and p are the coercive force, height, width, length, and pole-pairs of the PMP, respectively; W_b and L_b are the width and length of the PMB, respectively; μ₀ is the permeability of vacuum; and μ_r is the relative permeability of the permanent magnet.

Because of the consistency in permeability, the reluctance of the conductor plate together with the inner air-gap can be calculated as follows:

R_{eg} = \frac{g_{i} + h_{c}}{μ_{0} W_{p} L_{p}},

(3)

where g_i and h_c denote the inner air-gap length and conductor plate thickness, respectively.

The correction coefficient of reluctance under the action of a PMB is approximately expressed as

γ = (2 - α_{p}) / α_{p},

(4)

where α_p is the pole-arc coefficient of the PMP.

Because the permeability of the iron region is very large, the corresponding reluctance can be negligible. In this paper, the reluctance of the iron core and iron pole regions is not considered further. During the modeling process of no-load magnetic field, the key point is that we need to quantitatively calculate the flux-weakening effect, which corresponds to the adjustable reluctance R_ad in the MEC model. According to the working mechanism of the NPMLECB, the value of R_ad is closely related to the relative position between the MMA and PMB and the MMA width in the z-direction. Thus, three cases are divided and analyzed as follows:

Case 1: PMB is partially covered with MMA, as shown in Fig. 3. When the MMA width is sufficiently large, the weakening magnetic flux path is demonstrated in Fig. 3(a), while Fig. 3(b) shows the weakening magnetic flux path when the MMA width is not big enough. R_ad is calculated as follows:

R_{a d} = \{\begin{gathered} 2 R_{a 1} + R_{a 2} + R_{a 3}, W_{m} - l_{d} \geq h_{b} & & l_{d} < W_{b}, \\ R_{a 1} + R_{a 2} + R_{a 3} + (R_{a 5} + R_{a 6}) ‖ R_{a 4}, W_{m} - l_{d} < h_{b} & & l_{d} < W_{b}, \end{gathered}

(5)

where W_m is the width of MMA and l_d is the distance between the leading edge of the MMA and the trailing edge of the PMB. R_a1, R_a2, R_a3, R_a4, R_a5, and R_a6 denote the reluctance in different parts of the air, and they can be expressed by

\{\begin{gathered} R_{a 1} = g_{o} / (μ_{0} h_{b} L_{b} / p), \\ R_{a 2} = 1 / (μ_{0} L_{b} / p), \\ R_{a 3} = (W_{b} - l_{d}) / (μ_{0} h_{m} L_{b} / p), \\ R_{a 4} = g_{o} / [μ_{0} (W_{m} - l_{d}) L_{b} / p], \\ R_{a 5} = 1 / (μ_{0} L_{b} / p), \\ R_{a 6} = g_{o} / [μ_{0} (h_{b} - W_{m} + l_{d}) L_{b} / p] . \end{gathered}

(6)

FIG. 3.

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PMB is partially covered with MMA: (a) MMA width is sufficiently large; (b) MMA width is not large enough.

Case 2: PMB is fully covered with MMA, as shown in Fig. 4. When the MMA width is sufficiently large, the weakening magnetic flux path is demonstrated in Fig. 4(a), while Fig. 4(b) shows the weakening magnetic flux path when the MMA width is not big enough. R_ad is calculated as follows:

R_{a d} = \{\begin{gathered} (R_{b 1} + R_{b 3}) ‖ R_{b 2} + R_{b 4}, W_{m} - l_{d} \geq h_{b} & & l_{d} > W_{b}, \\ (R_{b 1} + R_{b 3}) ‖ R_{b 2} + (R_{b 5} + R_{b 7}) ‖ R_{b 6}, W_{m} - l_{d} ⟨h_{b} & & l_{d}⟩ W_{b}, \end{gathered}

(7)

where R_b1, R_b2, R_b3, R_b4, R_b5, R_b6, and R_b7 can be expressed by

\{\begin{gathered} R_{b 1} = g_{o} / [μ_{0} (h_{b} - l_{d} + W_{b}) L_{b} / p], \\ R_{b 2} = g_{o} / [μ_{0} (l_{d} - W_{b}) L_{b} / p], \\ R_{b 3} = 1 / (μ_{0} L_{b} / p), \\ R_{b 4} = g_{o} / (μ_{0} h_{b} L_{b} / p), \\ R_{b 5} = 1 / (μ_{0} L_{b} / p), \\ R_{b 6} = g_{o} / [μ_{0} (W_{m} - l_{d}) L_{b} / p], \\ R_{b 7} = g_{o} / [μ_{0} (h_{b} - W_{m} + l_{d}) L_{b} / p] . \end{gathered}

(8)

FIG. 4.

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PMB is fully covered with MMA: (a) MMA width is sufficiently large; (b) MMA width is not large enough.

Case 3: To prevent the MMA from moving inefficiently over a wide area in the z-direction, the maximum value of l_d is preset, and the location where the centerline of the MMA and PMB coincide is the suitable candidate. As shown in Fig. 5, there are two different flux paths, and R_ad is calculated as follows:

R_{a d} = \{\begin{gathered} 2 [(R_{b 1} + R_{b 3}) ‖ R_{b 2}], W_{m} - l_{d} = {(W_{m} - W_{b}) / 2}_{b} < h_{b}, \\ 2 R_{b 8}, W_{m} - l_{d} = {(W_{m} - W_{b}) / 2}_{b} \geq h_{b}, \end{gathered}

(9)

where R_b8 can be expressed by

R_{b 8} = g_{o} / (μ_{0} h_{b} L_{b} / p) .

(10)

FIG. 5.

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PMB is fully covered with MMA when l_d is at maximum: (a) MMA width is not big enough; (b) MMA width is big enough.

By applying the Kirchhoff voltage law to the MEC model shown in Fig. 2(c), the solution equations of flux are obtained as follows:

[\begin{matrix} 4 R_{e g} + 2 R_{i p} + 2 R_{pmp} & - (2 R_{pmp} + 2 R_{e g}) & 0 & 0 \\ - (2 R_{pmp} + 2 R_{e g}) & 4 R_{e g} + 4 R_{pmp} + R_{pmb} & - (2 R_{pmp} + 2 R_{e g}) & - R_{pmp} \\ 0 & 0 & 4 R_{e g} + 2 R_{i p} + 2 R_{pmp} & - (2 R_{e g} + 2 R_{pmp}) \\ 0 & - R_{pmb} & 0 & R_{pmb} + R_{a d} \end{matrix}] [\begin{matrix} φ_{1} \\ φ_{2} \\ φ_{3} \\ φ_{4} \end{matrix}] = [\begin{matrix} - F_{pmp} \\ 2 F_{pmp} - F_{pmb} \\ - F_{pmp} \\ F_{pmb} \end{matrix}] .

(11)

The no-load air-gap flux densities associated with the regulation parameter l_d in a one pole pitch can be calculated by

B_{s - p m p} = \{\begin{gathered} \frac{φ_{2} - φ_{1}}{α_{p} L_{p} W_{p}}, - L_{p} / 2 \leq x \leq L_{p} / 2, \\ 0, e l s e, \end{gathered}

(12)

B_{s - i p} = \{\begin{gathered} \frac{φ_{1}}{α_{p} L_{p} (W_{p} + h_{p})}, - L_{p} / 2 \leq x \leq L_{p} / 2, \\ 0, e l s e . \end{gathered}

(13)

C. No-load magnetic field with magnetic leakage consideration

The magnetic leakage effect is not considered in –; therefore, there may be some errors in the model of the no-load air-gap magnetic field. In order to further improve the model accuracy, the magnetic leakage effect should be introduced into the MEC model of the proposed NPMLECB, as shown in Fig. 6. Herein, the leakage reluctance between the PMP and the iron pole (R_mi), the leakage reluctance inside the magnetic pole (R_lm), and the leakage reluctance between the magnetic poles (R_mm) are taken into consideration,

[\begin{matrix} 4 R_{e g} & - 2 R_{e g} & 0 & 0 & R_{m i} & 0 & 0 & 0 & 0 \\ 0 & 4 R_{e g} & - 2 R_{e g} & 0 & 0 & 0 & R_{m m} & 0 & 0 \\ 0 & - 2 R_{e g} & 4 R_{e g} & 0 & 0 & 0 & 0 & 0 & R_{m i} \\ 0 & 0 & 0 & R_{44} & 0 & 0 & R_{pmb} & 0 & 0 \\ R_{m i} & 0 & 0 & 0 & R_{55} & 2 R_{pmp} & 0 & 0 & 0 \\ R_{m i} & 0 & 0 & 0 & 2 R_{pmp} & R_{66} & - R_{l m} & 0 & 0 \\ 0 & R_{m m} & 0 & R_{pmb} & 0 & R_{l m} & R_{77} & R_{l m} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & R_{l m} & R_{88} & 2 R_{pmp} \\ 0 & 0 & R_{m i} & 0 & 0 & 0 & 0 & 2 R_{pmp} & R_{99} \end{matrix}] [\begin{matrix} φ_{1} \\ φ_{2} \\ φ_{3} \\ \begin{gathered} φ_{4} \\ φ_{5} \\ φ_{6} \\ φ_{7} \\ φ_{8} \\ φ_{9} \end{gathered} \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \\ \begin{gathered} F_{pmb} \\ F_{pmp} \\ F_{pmp} \\ F_{pmb} \\ F_{pmp} \\ F_{pmp} \end{gathered} \end{matrix}],

(14)

where

\{\begin{gathered} R_{44} = R_{pmb} + R_{a d}, \\ R_{55} = R_{99} = R_{m i} + 2 R_{i p} + 2 R_{pmp}, \\ R_{66} = R_{88} = 2 R_{pmp} + R_{l m}, \\ R_{77} = 2 R_{l m} + R_{m m} + R_{pmb} . \end{gathered}

(15)

FIG. 6.

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MEC model of the proposed NPMLECB considering magnetic leakage.

The flux tubes of leakage are depicted in Fig. 7. Therefore, R_mi, R_lm, and R_mm are computed approximately as

R_{l m} = 1/ \int_{0}^{g_{i} + h_{c}} \frac{μ_{0} W_{p}}{h_{p} + 2 π x} d x = \frac{2 π}{μ_{0} W_{p} \ln [1 + 2 π (g_{i} + h_{c}) / h_{p}]},

(16)

\begin{align} R_{m i} & /= 1 \int_{0}^{g_{i}} \frac{μ_{0} W_{p}}{π x + L_{p} (1 / α_{p} - 1)} d x \\ = \frac{π}{μ_{0} W_{p} \ln {1 + π g_{i} / [L_{p} (1 / α_{p} - 1)]}}, \end{align}

(17)

R_{m m} /= 1 \int_{0}^{g_{i}} \frac{μ_{0} W_{p}}{π x + X_{d}} d x = \frac{π}{μ_{0} W_{p} \ln [1 + π g_{i} / X_{d}]},

(18)

where

X_{d} = \sqrt{{[L_{p} (1 / α_{p} - 1)]}^{2} + {(W_{p} + W_{b})}^{2}} .

(19)

By solving , the fluxes φ₁ and φ₂ in and are recalculated based on the MEC model of the proposed NPMLECB, considering magnetic leakage.

FIG. 7.

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Flux tubes of (a) leakage between the PMP and the iron pole, (b) leakage inside the magnetic pole, and (c) leakage between the magnetic poles.

D. On-load magnetic field

When working with load, similar to the other eddy current devices, the eddy currents generated on the conductor plate will have a serious demagnetization effect, which distorts the air-gap magnetic field. Based on Faraday’s law of electromagnetic induction, the average induced eddy currents can be expressed by

J_{a v} = (J_{pmp} + J_{i p}) / 2 = σ v (B_{epmp} - B_{eip}) / 2,

(20)

where v is the speed, σ is the conductor conductivity, and B_epmp and B_eip are calculated by

\{\begin{gathered} B_{epmp} = B_{s - p m p} + B_{c p - p m p}, \\ B_{eip} = B_{s - i p} + B_{c p - i p}, \end{gathered}

(21)

where B_cp-pmp and B_cp-ip denote the induced magnetic flux density below the permanent magnet pole and the iron pole, respectively.

According to Ampere’s law, the eddy currents below the permanent magnet pole and the iron pole can be approximately calculated by

\{\begin{gathered} \oint_{C_{1}} H d l = h_{c} \int_{x_{1}}^{x_{2}} J_{pmp} d x, \\ \oint_{C_{2}} H d l = h_{c} \int_{x_{3}}^{x_{4}} J_{i p} d x, \end{gathered}

(22)

where C₁ and C₂ denote the integration path of the induced field below the permanent magnet pole and the iron pole, respectively, as shown in Fig. 8. For simplicity, the MMF drop across iron materials is ignored, so the different flux path lengths C₁ and C₂ can be given by

\{\begin{gathered} C_{1} = 2 (g_{i} + h_{c} + h_{p}), \\ C_{2} = 2 (g_{i} + h_{c}) . \end{gathered}

(23)

FIG. 8.

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Magnetic flux path of the induced field.

The induced magnetic flux densities below the permanent magnet pole and the iron pole can be expressed as

B_{c p - p m p} = \{\begin{gathered} B_{c p - l} = k_{1} e^{\frac{μ_{0}}{2 C_{1}} h_{c} σ v (1 + C_{1} / C_{2}) x}, - L_{p} / 2 α_{p} \leq x < - L_{p} / 2, \\ B_{c p - m} = k_{2} e^{\frac{μ_{0}}{2 C_{1}} h_{c} σ v (1 + C_{1} / C_{2}) x} - (B_{s - p m p} - B_{s - i p}) / (1 + C_{1} / C_{2}), - L_{p} / 2 \leq x \leq L_{p} / 2, \\ B_{c p - r} = k_{3} e^{\frac{μ_{0}}{2 C_{1}} h_{c} σ v (1 + C_{1} / C_{2}) x}, L_{p} / 2 < x \leq L_{p} / 2 α_{p}, \end{gathered}

(24)

B_{c p - i p} = - C_{1} B_{c p - p m p} / C_{2},

(25)

where the undetermined constants k₁, k₂, and k₃ are obtained using the continuity conditions of magnetic field and eddy currents as follows:

\{\begin{gathered} B_{c p - l} (- L_{p} / 2) = B_{c p - m} (- L_{p} / 2), \\ B_{c p - m} (L_{p} / 2) = B_{c p - r} (L_{p} / 2), \\ B_{c p - m} (x_{0}) = 0, \\ \int_{0}^{h_{c}} \int_{- L_{p} / 2 α_{p}}^{x_{0}} J_{a v} d x d y = \int_{0}^{h_{c}} \int_{x_{0}}^{L_{p} / 2 α_{p}} J_{a v} d x d y . \end{gathered}

(26)

Furthermore, the induced eddy currents can be expressed as

J_{a v} = \{\begin{gathered} k_{1} (1 + C_{1} / C_{2}) e^{\frac{μ_{0}}{2 C_{1}} h_{c} σ v (1 + C_{1} / C_{2}) x}, - L_{p} / 2 α_{p} \leq x < - L_{p} / 2, \\ k_{2} (1 + C_{1} / C_{2}) e^{\frac{μ_{0}}{2 C_{1}} h_{c} σ v (1 + C_{1} / C_{2}) x}, - L_{p} / 2 \leq x \leq L_{p} / 2, \\ k_{3} (1 + C_{1} / C_{2}) e^{\frac{μ_{0}}{2 C_{1}} h_{c} σ v (1 + C_{1} / C_{2}) x}, L_{p} / 2 < x \leq L_{p} / 2 α_{p}, \end{gathered}

(27)

where the unknown constant coefficients are calculated as

\{\begin{gathered} k_{1} = m_{1} e^{- m_{0} x_{0}} - m_{1} e^{m_{0} L_{p} / 2}, \\ k_{2} = m_{1} e^{- m_{0} x_{0}}, \\ k_{3} = m_{1} e^{- m_{0} x_{0}} - m_{1} e^{- m_{0} L_{p} / 2}, \\ x_{0} = - \ln \cosh (m_{0} L_{p} / 2 - m_{0} L_{p} / 2 α_{p}) / \cosh (m_{0} L_{p} / 2 α_{p}) / m_{0}, \end{gathered}

(28)

where

\{\begin{gathered} m_{0} = \frac{μ_{0}}{2 C_{1}} h_{c} σ v (1 + C_{1} / C_{2}), \\ m_{1} = \frac{B_{s - p m p} - B_{s - i p}}{1 + C_{1} / C_{2}} . \end{gathered}

(29)

E. Formulation of braking force

The developed braking force is associated with the total ohmic losses dissipated in the conductor plate and can be expressed as

F = \frac{P}{v} = \frac{2 p h_{c} W_{p} \int_{- L_{p} / 2 α_{p}}^{L_{p} / 2 α_{p}} {|J_{a v} (x)|}^{2} d x}{σ v} .

(30)

As illustrated in our previous research,²⁵ the actual paths of induced eddy currents have a great impact on the accuracy of the analytical model. The fundamental reason is that only the z-direction component is considered in the eddy current model. Within the region of the conductor plate overhung in the z-direction, the eddy currents form a series of closed loops. In order to consider the end effect and improve the accuracy of the braking force model, a 3D correction factor²³ is employed as follows:

k_{3 - D} = 1 - \frac{\tanh [π W_{p} / (2 l_{p} / α_{p})] / [π W_{p} / (2 l_{p} / α_{p})]}{1 + \tanh [π W_{p} / (2 l_{p} / α_{p})] \tanh [2 π L_{o} / (2 l_{p} / α_{p})]},

(31)

where L_o is the overhung length of the conductor plate.

The braking force expression with 3D effects can be finally calculated as

F_{3 - D} = k_{3 - D} F .

(32)

IV. VALIDATION AND ANALYSIS

Due to the accuracy of the finite element method, it can be employed to evaluate the validity of the analytical model (AM) before the prototype goes into production and testing. In this paper, the air-gap magnetic field and braking force characteristics are estimated by the analytical method and verified using the FEM results. The major modeling parameters are tabulated in Table I.

TABLE I.

Main design parameters.

Parameters	Value	Parameters	Value
Height of the PMP and iron pole, h_p	10 mm	Pole pairs, p	6
Length of the PMP and iron pole, L_p	30 mm	Pole-arc coefficient, α_p	0.75
Width of the PMP and iron pole, W_p	30 mm	Conductor height, h_c	3 mm
Inner air-gap length, g_i	1 mm	Height of MMA, h_m	20 mm
Height of the iron core, h_i	20 mm	Width of the MMA, W_m	30 mm
Width of PMB, W_b	15 mm	Height of the PMB, h_b	20 mm
Outer air-gap length, g_o	1 mm	Conductor conductivity, σ	57 MS/m
PM coercivity, H_c	−890 kA/m	Remanence, B_r	1.23 T
Length of the PMB, L_b	480 mm	Overhung length, l_o	5 mm

Parameters	Value	Parameters	Value
Height of the PMP and iron pole, h_p	10 mm	Pole pairs, p	6
Length of the PMP and iron pole, L_p	30 mm	Pole-arc coefficient, α_p	0.75
Width of the PMP and iron pole, W_p	30 mm	Conductor height, h_c	3 mm
Inner air-gap length, g_i	1 mm	Height of MMA, h_m	20 mm
Height of the iron core, h_i	20 mm	Width of the MMA, W_m	30 mm
Width of PMB, W_b	15 mm	Height of the PMB, h_b	20 mm
Outer air-gap length, g_o	1 mm	Conductor conductivity, σ	57 MS/m
PM coercivity, H_c	−890 kA/m	Remanence, B_r	1.23 T
Length of the PMB, L_b	480 mm	Overhung length, l_o	5 mm

The mesh of the proposed NPMLECB is shown in Fig. 9, which includes 5 151 699 triangular elements. An example of 3D FEM results for the magnetic field is presented in Fig. 10, and the whole simulation process will take more than 8 h. It further shows the necessity of constructing a simple and reliable mechanism model.

FIG. 9.

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Mesh of the 3D finite element model.

FIG. 10.

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3D finite element results for the magnetic field.

A. Magnetic field characteristic

Figure 11 shows the amplitude variation in B_s-pmp and B_s-ip with the change in l_d from 0 to 20 mm when the NPMLECB is in a no-load condition. It can be observed that the results obtained by the AM and FEM are consistent, which shows the validity of the field model. As l_d reaches 20 mm, B_s-pmp increases to 0.55 T, while B_s-ip decreases to 0.17 T, and their rates of change are 22% and −72%, respectively, compared to those when l_d = 0. More pertinently, B_s-pmp and B_s-ip embrace different characteristics: (1) The rangeability of B_s-pmp is less than the air-gap magnetic density on the surface of the iron pole, and this is because the reluctance of the PMP is much larger than that of the iron pole in the flux path excited by the PMB. (2) With the increase in l_d, B_s-pmp increases smoothly, and the growth gradually decreases; however, B_s-ip decreases gradually, and the damping is gradually reduced; this is mainly due to the fact that the direction of the MMF excited by the PMB is opposite to the direction of the MMF excited by the PMP in the flux loop, the magnetic flux of PMB excitation is short-circuited by the MMA, and the effect of the MMF of PMB excitation on the magnetic density of the air gap is weakened.

FIG. 11.

View large Download slide

Amplitude comparison of air-gap flux density at the no-load condition.

Figures 12(a) and 12(b) show the comparison of the air-gap magnetic density curves on the surface of the PMP obtained from the AM and FEM under different on-load cases. The first observation is that AM and FEM results are in good agreement, which suggests the validity of the on-load field model. As shown in Fig. 12(a), at the fixed MMA location with l_d = 10 mm, for different speeds of v = 1 and 6 m/s, the air-gap magnetic density curves present different trends. On the whole, the higher the velocity, the lower the average density, and the more distorted it is at the same time. As shown in Fig. 11(b), for the same speed of v = 5 m/s, when the MMA is regulated from l_d = 0 to 20 mm, the air-gap magnetic density curves present similar trends with different amplitudes. The results show that the MMA has a fairly good regulation function for the magnetic field.

FIG. 12.

View large Download slide

Comparison of the air-gap flux density: (a) at l_d = 10 mm with v = 1 and 6 m/s; (b) at v = 1 m/s with l_d = 0 and 20 mm.

B. Braking force characteristic

The braking force vs speed with different MMA locations for the proposed NPMLECB is given in Fig. 13. It can be seen that there is a high degree of consistency between the AM and FEM, which indicates that the mathematical model for braking force containing structural parameters is highly reliable and can be employed in further analysis and optimization. The deviation is mainly from the simplification of the magnetic field and eddy current, which are essentially three-dimensional distribution. In addition, AM-1 represents the analytical result, where the magnetic leakage is not considered, and there are large errors in the prediction results of AM-1. Compared with the FEM, the maximum deviation of the AM and AM-1 is 4.5% and 10.2%, respectively, and it can be inferred that the error mainly comes from the deviation of the magnetic field.

FIG. 13.

View large Download slide

Braking force-speed curves with different l_d.

With the increase in l_d from 0 to 20 mm, the braking force decreases significantly at a fixed speed. Taking v = 5 m/s as an example, the braking force decreases from 2404 to 1165 N, so the variant rate is about 51.5%, which presents a wide braking space of the proposed topology. In addition, for different locations of the MMA, the critical speeds corresponding to the maximum braking force are fairly close (about v = 5 m/s); therefore, there is relatively stable braking performance for the proposed NPMLECB.

C. Parametric analysis

As mentioned earlier, the proposed braking force model is reliable and has higher prediction accuracy. Therefore, the analytical model is employed to further explore the influence of structural parameters on the features. According to previous research,²⁸ the inner air-gap, external air-gap, conductor plate thickness, and conductor material are selected to be discussed in this paper. In addition, the adjustment range of the braking force is defined as follows:

T_{range} = \frac{T_{\max} - T_{\min}}{T_{\max}} \times 100 %,

(33)

where T_max is the maximum braking force and T_min is the maximum braking force.

Figure 14 shows the braking force characteristic with different inner air-gap lengths. As illustrated in Fig. 14(a), at the maximum braking force state (l_d = 0 mm), with the increase in the inner air-gap length from 1 to 5 mm, the braking force diminishes gradually at the same speed. In addition, the results show that the critical speeds corresponding to the maximum braking force have changed and the turning points have moved backward. Figure 14(b) shows the regulation range of the braking force for different inner air-gap lengths. It can be observed that the regulation range increases gradually along with the inner air-gap length but the overall change is modest. Through the above-mentioned analysis, we can speculate that the smaller inner air-gap length can improve the overall performance of the system. Although the braking force can be adjusted by changing the inner air-gap length, compared with the proposed strategy, the operation is complex, and the axial force has a negative effect on the regulating mechanism.

FIG. 14.

View large Download slide

Braking force characteristics with different inner air-gap lengths: (a) braking force surface at l_d = 0 mm; (b) regulation range.

Figure 15 shows the braking force characteristic with different external air-gap lengths. As illustrated in Fig. 15(a), at the maximum braking force state (l_d = 0 mm), with the increase in the external air-gap length from 1 to 5 mm, the braking force characteristic surface is significantly different from the surface shown in Fig. 14(a). First, the braking force increases gradually along with the external air-gap length at the same speed; second, the change in the maximum braking force for each inner air-gap length is not obvious, and the turning speed remains basically at v = 5 m/s. Figure 15(b) shows the regulation range of the braking force for each external air-gap length. It can be observed that the regulation range decreases significantly along with the inner air-gap length, and for the given case, the regulation range is estimated to range from 56.5% to 42.5%. Through the above-mentioned analysis, we can speculate that the external air-gap length should be chosen cautiously.

FIG. 15.

View large Download slide

Braking force characteristics with different external air-gap lengths: (a) braking force surface at l_d = 0 mm; (b) regulation range.

Figure 16 shows the braking force characteristic with different conductor thicknesses. As illustrated in Fig. 15(a), at the maximum braking force state (l_d = 0 mm), with the increase in the conductor thickness from 1 to 9 mm, the braking force characteristic surface looks more complicated. First, the braking force increases first and then decreases along with the conductor thickness at the same lower speed, but it continuously decreases at the same higher speed; second, the maximum braking force for each conductor thickness is gradually decreased, from 2694 to 1379 N. Figure 16(b) shows the regulation range of the braking force for different conductor thicknesses. It can be observed that the regulation range increases slightly along with the conductor thickness, and for the given case, the regulation range is estimated to range from 51.2% to 54.1%. Through the above-mentioned analysis, we think that 3 mm is a suitable choice for the conductor thickness in the balance of the braking force and regulation range.

FIG. 16.

View large Download slide

Braking force characteristics with conductor thickness: (a) braking force surface at l_d = 0 mm; (b) regulation range.

Figure 17 shows the braking force characteristic with different conductor materials, such as zinc, aluminum, and copper. As illustrated in Fig. 17(a), at the maximum braking force state (l_d = 0 mm), with the increase in the conductivity of the conductor material, the critical speed decreases, and the peak braking force is likely to be the same. In addition, the higher the conductivity of the conductor material, the better the braking performance in the field of lower speed. However, in the field of high speed, the advantage of the NPMLECB using copper is lost. Figure 17(b) shows the regulation range of the braking force for conductor materials. It can be observed that the regulation range hardly changes with the conductor materials because they have similar permeability. Accordingly, the conductor material should be chosen according to the speed region.

FIG. 17.

View large Download slide

Braking force characteristics with conductor material: (a) braking force curve at l_d = 0 mm; (b) regulation range.

To make sure the proposed model is reliable and stable in parameter design and optimization, a metric, normalized root mean square deviation (NRMSD), is employed to evaluate the deviation between analytical (ANA) and numerical (FEM) results,

N R M S D = \frac{\sqrt{\frac{1}{v_{e}} \int_{0}^{v_{e}} {[F_{FEM} (v) - F_{AM} (v)]}^{2}}}{\max [F_{FEM} (v)] - \min [F_{FEM} (v)]} \times 100 % .

(34)

For the inner air-gap length, external air-gap length, conductor plate thickness, and conductor material, the NRMSD does not exceed 0.1% in each case. By comparison, the NRMSD from the analytical model without consideration of magnetic leakage is relatively large, which is more than 0.5%. This means that no extreme dimension case that corresponds to the bad metric exists. Therefore, the proposed analytical model can be used for follow-up research.

V. CONCLUSION

In this paper, a novel field-control permanent magnet linear eddy current brake, named the NPMLECB, is presented. Based on the weak magnetic regulation and hybrid magnetic circuit technology, the NPMLECB can have great potential in amplitude and regulation flexibility of the braking force. To avoid excessive computation, especially the time-consuming solving of complicated PDEs, the practical nonlinear mechanism models for the air-gap magnetic field, eddy current, and braking force are developed and validated using the 3D FEM. The main conclusions of this study are summarized as follows:

The composite magnetic circuit structure of the permanent magnet, iron pole, and permanent magnet bar can produce sufficiently high magnetic flux density, which avoids introducing other types of energy sources, and this will save energy effectively. The unique mechanical magnetic adjuster can be adjusted in real time, which generates the required braking force as soon as possible, and this purely mechanical transmission will improve the reliability of the system.
The proposed NPMLECB has a wide regulation range for braking force. With the increase in l_d from 0 to 20 mm, the rates of change for B_s-pmp and B_s-ip are 22% and −72%, respectively, the braking force decreases form 2404 to 1165 N at the turning speed, and the regulation range of braking force is more than 50%. Furthermore, the turning speed remains relatively constant (about v = 5 m/s) at any operation condition, which shows stable braking performance.
To improve the accuracy of the nonlinear analytical model, the flux-weakening effect and leakage flux effect are calculated quantitatively. To this end, the models of adjustable reluctance, which correspond to different locations of the MMA, are developed. Compared with 3D FEM results, it demonstrates that nonlinear analytical models of the air-gap magnetic density and braking force can provide high prediction accuracy.
The sensitivity analysis for the key device geometries, for example, the inner air-gap, external air-gap, conductor plate thickness, and conductor material, is put into action, which is conducted to demonstrate the versatility of the nonlinear analytical model. Among them, the NRMSD does not exceed 0.1% in any case, which proves the sufficient precision of the nonlinear theoretical model. Thus, this provides a tool for the pre-design and optimization of the NPMLECB.

ACKNOWLEDGMENTS

This work was supported in part by the Key Projects of Science and Technology of Henan Province, Grant No. 232102320210, in part by the Young Backbone Teacher Foundation of Zhongyuan University of Technology, Grant No. 2020XQG06, and in part by the Project of Discipline Strength Improvement Plan, Grant No. SD202208.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Zhao Li: Conceptualization (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Jialei Wang: Data curation (equal); Formal analysis (equal); Investigation (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal). Yan Li: Data curation (equal); Formal analysis (equal); Investigation (equal); Supervision (equal); Writing – review & editing (equal). Hui Yang: Formal analysis (equal); Investigation (equal); Project administration (equal); Validation (equal). Dazhi Wang: Conceptualization (equal); Investigation (equal); Methodology (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Analytical modeling and performance evaluation of an adjustable permanent magnet linear eddy current brake with mechanical magnetic adjuster

I. INTRODUCTION

II. DESIGN FEATURE AND WORKING PRINCIPLE

A. Design feature

B. Working principle

III. THEORETICAL MODEL

A. Magnetic flux paths

B. No-load magnetic field without magnetic leakage consideration

C. No-load magnetic field with magnetic leakage consideration

D. On-load magnetic field

E. Formulation of braking force

IV. VALIDATION AND ANALYSIS

A. Magnetic field characteristic

B. Braking force characteristic

C. Parametric analysis

V. CONCLUSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Analytical modeling and performance evaluation of an adjustable permanent magnet linear eddy current brake with mechanical magnetic adjuster Open Access

I. INTRODUCTION

II. DESIGN FEATURE AND WORKING PRINCIPLE

A. Design feature

B. Working principle

III. THEORETICAL MODEL

A. Magnetic flux paths

B. No-load magnetic field without magnetic leakage consideration

C. No-load magnetic field with magnetic leakage consideration

D. On-load magnetic field

E. Formulation of braking force

IV. VALIDATION AND ANALYSIS

A. Magnetic field characteristic

B. Braking force characteristic

C. Parametric analysis

V. CONCLUSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Analytical modeling and performance evaluation of an adjustable permanent magnet linear eddy current brake with mechanical magnetic adjuster