Analytical modeling of permanent magnet linear eddy current brake using magnetic equivalent circuit method

Abstract

In order to predict the performance of a permanent magnet linear braking (ECB) with cylindrical structure, an analytical model is established in this paper. The model is based on the magnetic equivalent circuit (MEC) method with the consideration of the often ignored magnetic saturation in pole shoe. The influence of induced eddy current is taken into account by introducing the magnetomotive force (MMF) in the model. To obtain braking force, a simple method for approximating the cross-section of the electric field with mean value method is proposed. The proposed model that can predict the braking force performance of ECB in a wide range of structural dimensions is obtained through referring to a small number of finite element models. A prototype test is carried out. The validity of the proposed method is verified through experiment and FEM. Results show that the braking force predicted by the presented method are in good agreement with experimental results and FEM results under different design parameters. The proposed model can accurately represent the variation trend of braking force and critical speed with design parameters.

Keywords

Eddy current brake analytical modeling magnetic equivalent circuit (MEC)permanent magnet (PM)saturation

1. Introduction

The eddy current brake (ECB), of which the braking force is independent of friction, has advantages of simple structure, high reliability, and convenient maintenance. At present, ECB has been widely applied in industrial fields, such as vibration suppression, vehicle suspension systems, high speed train braking systems and so on [1–5].

The magnetic field source of ECB can be produced by permanent magnets (PMs) as well as winding systems, according to which can ECB be divided into three types: electric excitation ECB: permanent magnet ECB, and hybrid excitation ECB [6–8]. Permanent magnetic linear ECB can generate large damping forces sufficient to brake strong impact loads, while with simpler structure than electric excitation and hybrid excitation linear ECB. The ECB studied in this paper is a permanent magnetic linear ECB with cylindrical structure, as is shown in Fig. 1. In contrast to the more commonly studied linear ECB with relatively independent primary and secondary, the cylindrical linear ECB has a more integrated structure. However, this type of model has not been studied much, even though it has good applicability in some brake devices with high impulse.

To analyze and evaluate the braking force performance of ECB, a reliable theoretical model is necessary. Finite element method (FEM), being able to provide not only accurate results, but also data visualization that can be very helpful for understanding and analyzing, are commonly used in ECB model analyzing. FEM can be applied to ECB of various structures, but its modeling and meshing as well as calculating process are time-consuming [9,10]. Therefore, FEM is less convenient to analytical method (ANA) for the dimension parameter analysis of ECB models.

The analytical model of ECB, with clear physical meaning and low computing resource requirements, plays an important role in the preliminary design of ECB. There are two possible approaches to establish an analytical model of ECB, one is based on Maxwell’s equations and adequate boundary conditions, and other is based on the magnetic equivalent circuit (MEC). The former approach is normally combined with field analysis and subdomain modeling techniques and can accurately describe slotting effects and eddy current reaction fields [11–13]. However, the derivation and programming of it are complicated, and its complexity is related to topology.

The MEC model is prevalently employed in the design and optimization of ECB, for its simplicity and effectiveness [14–18]. The effectivity of ECB MEC model is related to the consideration of impacts of induced eddy currents. In MEC models of previous studies, the influence of eddy currents can reasonably be omitted, which will inevitably cause error when eddy current effect is significant. In some recent reports, the influence of induced eddy current was taken into account to build more accurate MEC models [19,20]. Besides the eddy current, the accuracy of MEC model can also be affected by the magnetic saturation in pole shoes. The magnetic saturation phenomenon is usually neglected in previous MEC models for it makes the model nonlinear while causing little error. However, in the proposed ECB model, magnetic saturation can cause relatively large error of the reluctance of pole shoes, especially when magnetic field intensity is large. Therefore, to improve the accuracy of MEC model, the consideration of the magnetic saturation in pole shoes is required.

Fig. 1.

(a) PM linear ECB and (b) its partial schematic diagram.

In order to obtain accurate braking force performance, MEC model is always combined with subdomain model technology to obtain in previous studies. The precise description of slotting effect has great influence on the accuracy of this approach. However, for the proposed cylindrical linear ECB model, slots have a more complex figure than the linear ECB with conventional structure, making it difficult to describe and programming. And the calculation of the subdomain model considering slotting effect will be time-consuming [21–23].

Based on MEC method, an analytical model for predicting the performance of the proposed ECB will be established in this paper. The magnetic saturation in pole shoes is taken into account by presenting the relationship between air gap magnetic flux density and the relative permeability used when calculating the magnetic resistance of pole shoes. The impact of eddy current on MEC model is inherently taken into account by introducing the magnetomotive force (MMF) produced by the induced eddy current in cylinder conductor. This approach has been applied in previous ECB analyze and design studies [24–27]. A practical method is developed to obtain the eddy current and braking force by approximating the electric field cross section with regular distribution. The analytical model is verified by prototype test and FEM in this paper.

2. Theory

Figure 1(a) illustrates the ECB studied in this paper, which mainly consists of two parts: the primary part, with multiple sets of cylindrical PM and pole shoe alternately arranged on the shaft; and the secondary part, which is composed of conductor cylinder and back iron cylinder. PMs are axially magnetized, and adjacent PMs always have opposite magnetized directions. As shown in Fig. 1(b), d _m, d are the axial thickness of PMs and pole pitch; r ₀, r _pm, r _ip, r _i, r _b are radius of the shaft, external radius of PMs and iron pole shoe, internal radius of the inner cylinder conductor and back iron, respectively. h _b is the radial thickness of the back iron. In addition, d _s is the axial thickness of pole shoes, the radial thickness of air gap and cylinder conductor are denoted by h _a and h _i, the external radius of back iron is denoted by r _eb.

2.1. Assumptions

Since the ECB structure is cylindrical, as shown in Fig. 1, it can be simplified to a two-dimensional axisymmetric model. Magnetic saturation and MMF will be considered in the MEC model. To further simplify the difficulty of analysis, the following assumptions are made:

(1) There is no magnetic saturation phenomenon in iron back, and the relative permeability of iron back in MEC model is constant with the chosen material.

(2) PMs exhibit linear demagnetization characteristics and thus the relative recoil permeability of PM is a constant quantity.

(3) The permeability of shaft and conductor cylinder are assumed the same as Vacuum permeability.

2.2. Magnetic equivalent circuit model

Figure 2(a) shows the magnetic circuit distribution and magnetic flux flow direction of the eddy current brake in no-load state. The no-load state of ECB refers to the state in which the relative velocity of primary and secondary is zero, that is, the eddy current in conductor cylinder is zero. Analyze under no-load condition can rapidly indicate whether the performance and size are compatible with the envisioned application without involving comprehensive eddy current and braking force calculation. As exhibited in Fig. 2(a), the main flux path (black solid lines) passes through PMs, pole shoes, air gap, conductor cylinder, and back iron.

Fig. 2.

(a) Magnetic path and flux flow direction and (b) its equivalent magnetic circuit.

Each R _x in Fig. 2(b) refers to the reluctance of the corresponding part x of magnetic circuit in Fig. 2(a), R _n refers to the reluctance of magnetic circuit part that pass through air gap and conductor cylinder. The calculation formula for each unit of magnetic circuit is as follows.

The magnetomotive force generated by PM is: $\begin{eqnarray}F_{c}=H_{c}d_{m}\end{eqnarray}$ (1) where H _c is the permanent magnet coercivity. Specifically, under the assumption that there is no irreversible demagnetization in this model, the magnetic field constitutive relation of permanent can be considered simplified as: $\begin{eqnarray}B={\mu}_{0}{\mu}_{m}H+B_{r}.\end{eqnarray}$ (2)

So, it makes that: $\begin{eqnarray}H_{c}=\left|-\frac{B_{r}}{{\mu}_{0}{\mu}_{m}}\right|\end{eqnarray}$ (3) where, B _r and μ_m are remanence and relative magnetic permeability of permanent magnet.

Permanent magnets have a ring structure. The reluctance of permanent magnet is as (4), and the formula of R ₄ can be obtained similarly. The reluctance of part 1 is as (5). R ₃ can be obtained similarly. $\begin{eqnarray}\displaystyle R_{m} & = & \displaystyle \frac{d_{m}}{{\mu}_{0}{\mu}_{m}{\pi}(r_{pm}^{2}-r_{0}^{2})}\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle R_{1} & = & \displaystyle \frac{\ln ((l-r_{0})/r_{0})}{2{\mu}_{0}{\mu}_{i}{\pi}d_{s}}\end{eqnarray}$ (5) where, l is the length of the part in pole shoe where magnetic circuit flows to shaft.

Fig. 3.

The magnetic circuit of part 5 (a) main path, (b) fringe effect.

Fig. 4.

Magnetic circuit in air gap and conductive layer.

The magnetic permeance of part 5, the magnetic circuit in the shaft, consist of two types of magnetic flux tube, as shown in Fig. 3, considering fringe effect: $\begin{eqnarray}\displaystyle G_{51} & = & \displaystyle 2{\mu}_{0}\left[r_{0}\ln \left(1+\frac{d_{s}}{d_{m}}\right)-\frac{d_{s}}{{\pi}}\right]\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle G_{52} & = & \displaystyle {\mu}_{0}\frac{{\pi}}{6}\left[r_{0}({\pi}-{\theta})-d_{m}\ln \left(\frac{{\pi}\sin ({\theta}/2)}{{\theta}}\right)\right]\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle R_{5} & = & \displaystyle 1/(G_{51}+G_{52}).\end{eqnarray}$ (8)

Simplified magnetic circuits pass through air gap and conductive layer are illustrated in Fig. 4. The main flux that flows directly into secondary from lateral surface of pole shoes (red area) and the flux leakage that flows out from edges of pole shoes (blue area) are in parallel. It is approximated that the magnetic circuit of MFL are also directly connected to part 4, and the magnetic potential of this section is set as F _n. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle F_{n}=R_{n}{\Phi}_{{\delta}}=R_{mid}{\Phi}_{mid}=R_{sid}\left({\Phi}_{{\delta}}-\frac{{\Phi}_{mid}}{2}\right)\end{eqnarray}$ (9) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Phi}_{mid}={\Phi}_{{\delta}}\left(\frac{R_{sid}}{R_{mid}+R_{sid}/2}\right)\end{eqnarray}$ (10) where, R _mid can be derived by analogy from (5).

R = l∕(sμ₀μ_r) is used to calculate the leakage path R _sid reluctance, l is the average path, and s is obtained by dividing the volume of the leakage path by the length of the average path. The average lengths of R _sid (L _sid) is approximated to be 1.22((r _ip − r _pm)∕2 + h _a + h _i). The volumes of R _sid (V _sid) is as (11). And R _sid can be obtained. $\begin{eqnarray}\displaystyle V_{sid} & = & \displaystyle 2{\pi}(r_{b}-r_{pm})^{2}\frac{{\pi}r_{i}}{4}\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle R_{sid} & = & \displaystyle \frac{{L_{sid}}^{2}}{{\mu}_{0}V_{sid}}.\end{eqnarray}$ (12)

Integrate the magnetic circuits here: $\begin{eqnarray}F_{n}=\left(\frac{R_{sc}R_{sid}}{R_{sc}+R_{sid}/2}\right){\Phi}_{{\delta}}=R_{n}{\Phi}_{{\delta}}.\end{eqnarray}$ (13)

Instead of uniformly distributed in pole shoes, the magnetic flux density is greater as it is closer to the air gap. This results in, and makes it less accurate to calculate R ₂ integrally using a same relative permeability.

Fig. 5.

Magnetic circuits in pole shoe.

Approximate partition method is used to manage the magnetic saturation, and the right side of pole shoe is divided in 4 parts. Simplified magnetic circuits in pole shoe are illustrated in Fig. 5. Parts 21 to 24 have increasing magnetic flux density. Consider R ₂ as is composed of R ₂₁ ∼ R ₂₄ in the following proportion, which is roughly the proportion of the flux of this part to total flux that flow out of the pole shoe. $\begin{eqnarray}\displaystyle R_{2} & = & \displaystyle 0.25R_{21}+0.5R_{22}+0.75R_{23}+R_{24}\end{eqnarray}$ (14) $\begin{eqnarray}\displaystyle R_{21} & = & \displaystyle \frac{\ln ((l+0.25(r_{pm}-l))/l)}{2{\mu}_{0}{\mu}_{21}{\pi}d_{s}}\end{eqnarray}$ (15) $\begin{eqnarray}\displaystyle R_{22} & = & \displaystyle \frac{\ln ((l+0.5(r_{pm}-l))/(l+0.25(r_{pm}-l)))}{2{\mu}_{0}{\mu}_{22}{\pi}d_{s}}\end{eqnarray}$ (16) $\begin{eqnarray}\displaystyle R_{23} & = & \displaystyle \frac{\ln ((l+0.75(r_{pm}-l))/(l+0.5(r_{pm}-l)))}{2{\mu}_{0}{\mu}_{23}{\pi}d_{s}}\end{eqnarray}$ (17) $\begin{eqnarray}\displaystyle R_{24} & = & \displaystyle \frac{\ln ((r_{pm})/(l+0.75(r_{pm}-l)))}{2{\mu}_{0}{\mu}_{24}{\pi}d_{s}}\end{eqnarray}$ (18) where, u ₂₁, u ₂₂, u ₂₃ use fixed relative permeability, the value of u ₂₄ depends on the air gap flux density. In particular, the average flux density in segment 24 (B (u ₂₄)) depends on the flux density at the outer diameter of pole shoe (B _q). Their relationship is as Fig. 6, which is obtained by comparing the model trial calculation and the magnetic flux density calculation results of finite element models. And the permeability mentioned above is derived from the BH curve of iron.

Fig. 6.

Relationship between B _q and B(u ₂₄).

According to the equivalent magnetic circuit of the electromagnetic buffer and the Kirchhoff magnetic potential difference law, it can be get that: $\begin{eqnarray}\left\{\begin{array}{@{}l@{}}F_{AB}=F_{c}-R_{m}{\Phi}_{m}=(4R_{1}+R_{5}){\Phi}_{mm}=R_{mm}{\Phi}_{mm}\\ F_{AB}=(4R_{2}+2R_{n}+4R_{3}+R_{4}){\Phi}_{{\delta}}=R_{r}{\Phi}_{{\delta}}\\ {\Phi}_{m}={\Phi}_{mm}+{\Phi}_{{\delta}}.\end{array}\right.\end{eqnarray}$ (19)

Solving above equations, it is easy to get: $\begin{eqnarray}{\Phi}_{{\delta}}=\frac{F_{c}R_{mm}}{R_{r}R_{mm}+R_{m}R_{r}+R_{mm}R_{m}}.\end{eqnarray}$ (20)

The magnetic flux density of the air gap intermediate layer under no-load condition is: $\begin{eqnarray}B_{ag}=\frac{{\Phi}_{mid}}{{\pi}d_{s}(r_{ip}+r_{i})}.\end{eqnarray}$ (21)

In order to represent the magnetic saturation in pole shoes, which limits the extent to which the air gap magnetic flux can be improved through design, the calculation process of this MEC model is as follows. B(u ₂₄) should be first assumed to calculate the MEC model and obtain B _q and other required results, then B(u ₂₄) should be calculated again according to Fig. 6 and be substituted into the model for recalculation. After iterative calculation until B(u ₂₄) converges, MEC model results considering magnetic saturation can be obtained. The calculation flow chart is shown in Fig. 7.

Fig. 7.

Calculation flowchart for the proposed model.

The influence of whether to consider magnetic saturation on the accuracy of MEC model can be seen from Fig. 8. The blue line is the curve of air gap flux density changing with d _m when magnetic saturation is not considered in no-load state (B(u ₂₄) is unchanged), and the black line is the B _ag versus d _m curve when magnetic saturation is considered. It can be seen that if magnetic saturation is not taken into account, it will lead to errors in MEC model and the calculation of braking force.

Fig. 8.

Air gap magnetic flux density varies with d _m obtained by different models.

2.3. Braking force analysis

Since the ECB studied in this paper adopts a cylindrical structure, it is more concise and clear to use axisymmetric 2D model in finite element modeling. In FEM analysis, it can be seen that the electric field section distribution on secondary presents a relatively regular shape. And for models of different sizes, the deformation of electric field cross section shape with increasing velocity also have relatively consistent law. Inspired by the analysis of FEM results, this paper attempts to use the most basic formulas related to eddy currents and mean value method to approximately calculate the braking force of ECB.

Equation (22) is the basic formula for calculating the eddy current and the braking force generated by eddy current. The electric field is calculated by the cross product of magnetic induction intensity and speed, and the current density is obtained according to electric field intensity. The braking force generated by eddy current is achieved by dividing the heat loss power of eddy current by speed.

The distribution of electric field in secondary is highly consistent with the distribution of magnetic flux density. By calculating the MEC model proposed above, the magnetic induction intensity in secondary can be obtained, as well as the electric field strength and current density. It should be noted that the calculation results are actually average values. Using mean value method to calculate the cross-section of electric field, the eddy current and electromagnetic resistance generated can be obtained. $\begin{eqnarray}E=v\times B\quad J={\sigma}E\quad P=\int _{V}\frac{J^{2}}{{\sigma}}dv\quad F=\frac{P_{e}}{v}.\end{eqnarray}$ (22)

A series of finite element models are established. Under the condition that the structure being unchanged, three models are established for each dimension parameter. For each dimension parameter, build three models using the upper and lower design range of that parameter and its intermediate value. And for each finite element model, calculations are performed under the speed of 1–15 m/s. An analytical model is obtained based on the approximate electric field shape, the modification rule of the shape according to the dimension parameter change and eddy current formulas. Through analyzing a small number of finite element models, the proposed analytical model can be applicable in a large size range. The influence of eddy current on magnetic circuit model is integrated by introducing MMF. MMF is the current calculated by approximate electric field shape. The proposed model needs the reference of finite element model, but a relatively accurate prediction model can be obtained only by establishing a small number of FEMs. And the calculation time of this model is much less than exact analysis model and FEM.

Fig. 9.

Cross-section diagram of electric field distribution at different velocities.

Figure 9 are the electric field results obtained by FEMs under a certain size at increasing speeds. It can be seen that the electric field in conductor cylinder is concentrated at the position directly opposite to pole shoe, and the electric field in back iron presents a triangular shape. The reason why the electric field takes on this shape is that the magnetic field in secondary does not exactly follow the magnetic field in primary when it moves, but with a certain hysteresis, as shown in Fig. 10. In the meantime, the hysteresis deformation is actually caused by the influence of induced eddy current. In particular, the reason for the triangular cross-section of electric field in back iron is that the electric field is generated only by the radial component of magnetic field intensity perpendicular to the velocity.

As the speed increases, the deformation of magnetic field in secondary becomes increasingly serious, causing the shape of electric field to become longer and less concentrated. The variation of electric field follows similar rules at different sizes.

Fig. 10.

Magnetic field lines at different velocities.

At the same velocity, the cross-sectional area of electric field is also related to the size of structure. For example, it can be seen from Fig. 11 that the increase of the thickness of conductor cylinder reduces the electric field generated by ECB at work, which is the most noticeable. There are two reasons for that. First, the thicker the conductor cylinder, the greater its reluctance, which will lead to a smaller magnetic field shape in secondary. Second, the thicker the conductor cylinder, the greater the deformation of the magnetic field line in secondary.

Fig. 11.

Cross-section diagram of electric field distribution of models with different h _i.

Fig. 12.

Schematic diagram of electric field section zoning.

By analyzing the similarity of FEM results, the cross-section of electric field is divided into three parts, as shown in Fig. 12, so that the eddy current and braking force can be approximately calculated by multiplying the area by the average value of electric field. This division is based on the intensity, uniformity and position of electric field. As shown in Fig. 12, the three parts are the region 1 with the highest electric field intensity, the region 2 with the stronger electric field intensity in back iron, and the region 3 with the smaller electric field intensity in conductor cylinder. d ₁, d ₂, d ₃, and g ₂ represent the side lengths of each partition required to calculate the partition area.

Region 1 is the section where electric field is most concentrated, and the electric field intensity E ₁ at the precise middle layer of conductor cylinder is taken as the average value.

Region 2 is the section where electric field is concentrated in back iron. It can be seen from Fig. 10 that the magnetic field lines here are directly connected with the magnetic field lines at region 1. The magnetic flux density decreases slightly due to the increase of flux surface, but the electric field is relatively uniform. The mean electric field intensity E ₂ in region 2 is obtained by the cross product of the magnetic induction intensity at the radius (r ₀ + h _b∕4) in back iron and the velocity.

Region 3 is the electric field generated by the deformed and weakening magnetic field in conductor cylinder, and regions above and below region 1 are added together for calculation as an integral region. The electric field here is small at top and large at bottom and gradually decreases in intensity. Take E ₁ times a velocity dependent coefficient E _3' to get E ₃ as the average electric field of the third region.

The side lengths of the electric field cross-sectional shape required for calculation are functions of the structure parameters and velocity, as follows: $\begin{eqnarray}\displaystyle d_{1} & = & \displaystyle d_{s^{\prime }}k_{d1}\quad d_{2}=d_{s^{\prime }}k_{d2}+icd\quad g_{2}=h_{b}k_{g2}-icd\quad d_{3}=d_{m^{\prime }}k_{d3}\end{eqnarray}$ (23) $\begin{eqnarray}\displaystyle d_{m^{\prime }} & = & \displaystyle 1.25d_{m}+0.005\quad d_{s^{\prime }}=\frac{d_{m^{\prime }}}{r_{ms}}\quad r_{ms}=\frac{d_{m}}{d_{s}}.\end{eqnarray}$ (24)

k _d1, k _d2, k _d3 and k _g2 are the velocity correlation coefficients of each side length. In the original idea, the side lengths are obtained by multiplying these coefficients directly by structure parameters. However, through analyzing, it is found that the side lengths are also sensitive to d _m, d _s, h _i, and the ratio of the thickness of permanent magnet to the thickness of pole shoe (r _ms). Therefore, in the proposed method, d _m′ is used instead of d _m for calculation, and the relationship between them is shown in (24). Meanwhile, d _s′ is used instead of d _s for calculation, d _s′ is recalculated with d _m′ and the original r _ms. The formulas for d ₂ and g ₂ are similar with d ₁, but a decrement icd related to the thickness of inner cylinder is subtracted. After analyzing the established FEMs, the variation of each required coefficient with velocity are obtained, as shown in Fig. 13.

Fig. 13.

The coefficients and decrement needed.

According to (22) and the approximate electric field cross-section method, the braking force can be obtained: $\begin{eqnarray}\displaystyle E_{1} & = & \displaystyle \frac{{\Phi}_{mid}}{2{\pi}d_{s}\left(r_{i}+\displaystyle \frac{h_{i}}{2}\right)}v\quad E_{2}=\frac{{\Phi}_{mid}}{2{\pi}d_{s}\left(r_{b}+\displaystyle \frac{h_{b}}{4}\right)}v\quad E_{3}=E_{1}E_{3^{\prime }}\end{eqnarray}$ (25) $\begin{eqnarray}\displaystyle J_{1} & = & \displaystyle E_{1}{\sigma}_{i}\quad J_{2}=E_{2}{\sigma}_{b}\quad J_{3}=E_{3}{\sigma}_{i}\end{eqnarray}$ (26) $\begin{eqnarray}\displaystyle P_{1} & = & \displaystyle \frac{J_{1}^{2}}{{\sigma}_{i}}{\pi}d_{1}(r_{o}^{2}-r_{i}^{2})\quad P_{2}=\frac{J_{2}^{2}}{{\sigma}_{b}}\left(\frac{{\pi}d_{2}}{3}(r_{g}^{2}+r_{g}r_{o}+r_{o}^{2})-{\pi}d_{2}r_{o}^{2}\right)\quad P_{3}=\frac{J_{3}^{2}}{{\sigma}_{i}}{\pi}d_{3}(r_{o}^{2}-r_{i}^{2})\end{eqnarray}$ (27) $\begin{eqnarray}\displaystyle F & = & \displaystyle \frac{n_{ps}(P_{1}+P_{2}+P_{3})}{v}\end{eqnarray}$ (28) where, σ_i, σ_b are conductivity of conductor cylinder and back iron, J ₁, J ₂, J ₃ are the average current density of each region, n _ps is the number of pole shoe, r _g = r _b + g ₂.

According to (28), the eddy current and braking force can be approximated by the method of calculating the average electric field by partition together with MEC model results. At the same time, induced eddy currents weaken the magnetic field generated by permanent magnets. In order to consider its influence, the Magneto Motive Force (MMF) F _i generated by eddy current is added into the MEC model using a method similar to that of motor. The MMF of magnetic circuit model, corresponding to each pole shoe, is calculated by multiplication of the eddy current in secondary and a coefficient, as is shown in (29). As for the hysteretic deformation of magnetic circuit caused by relative motion, since the magnetic field lines are still relatively regular after deformation, it is approximately considered that other parts of the magnetic circuit model remain unchanged. $\begin{eqnarray}F_{i}=\left(J_{1}\cdot d_{i}h_{i}+J_{2}\cdot \frac{1}{2}d_{2}g_{2}+J_{3}\cdot d_{3}h_{i}\right)\cdot I\prime .\end{eqnarray}$ (29)

The eddy current position is corresponding to the permanent magnet and magnetic boot structure, but some offsets occur when the velocity or structure size changes, as shown in Fig. 14. As the electric field shift downward, the relationship between its induced magnetic field and the original field is more like overlap rather than intersection, which causes the induced magnetic field to weaken the original field more seriously. Thus, F _i is computed by multiplying the current calculated from electric field by a coefficient I′ about velocity and structure sizes. The influence of eddy current on the original magnetic field is analyzed by FEMs, and the variation rule of the corresponding coefficient I′ is obtained, as is shown in Fig. 15, the effect of structural size on I′ is ignored.

Fig. 14.

Schematic diagram of induced magnetic field of eddy currents at different speeds.

Fig. 15.

Coefficient I′ relative to velocity.

In order to obtain the magnetic field analysis model that internalizes eddy current, an iterative calculation method is used. Firstly, the radial flux density of conductor cylinder is calculated by no-load MEC model, and eddy current is obtained accordingly. Then, the eddy current is substituted into the model and recalculated, and iterative calculation is carried out until the eddy current converges, and braking force is obtained accordingly. The complete calculation flow chart is shown in Fig. 16.

Fig. 16.

Calculation flow chart of ECB analysis model.

3. Model verification and discussion

3.1. Test platform and 2-D FEM

To verify the proposed analytical model, a small size prototype is used for impact tests. At the same time, the corresponding finite element model is established for comparison and verification. In this study, it is expected to design an ECB with a reasonable size that can obtain more than 120 Kg braking force. However, the diameter and length of the prototype obtained through preliminary design are too large, and test equipment will also need special design and manufacture. Therefore, to preliminarily verify the correctness of FEM model and proposed model, a small prototype is made by reducing the overall size of the structure for testing. The small ECB prototype has a basic structure consistent with Fig. 1. The PM type is NdFeB, the material of cylinder conductor is aluminum alloy (6061). The shaft is made of stainless steel, the pole shoes and the back iron are made of ordinary carbon steel. The main specifications of the prototype are demonstrated in Table 1.

Table 1
Dimensions and material parameters of the prototype

Parameter Value Parameter Value

r ₀ 6 mm h _a 0.5 mm

r _pm 17.5 mm h _i 2 mm

r _ip 19.5 mm B _r 1.25 T

r _eb 29 mm σ_i 28 MS/m

d _m 10 mm σ_b 6.9 MS/m

d _s 5 mm n _pm 8

n _ps 9

Parameter	Value	Parameter	Value
r ₀	6 mm	h _a	0.5 mm
r _pm	17.5 mm	h _i	2 mm
r _ip	19.5 mm	B _r	1.25 T
r _eb	29 mm	σ_i	28 MS/m
d _m	10 mm	σ_b	6.9 MS/m
d _s	5 mm	n _pm	8
n _ps	9

Fig. 17.

Experimental platform configuration.

The experimental apparatus is illustrated in Fig. 17. A L-shaped plate installed on the slide rail is connected with the shaft. And the plate is hit by the air hammer to move the ECB prototype. The linear motion of the prototype is oriented by two guide rings placed beside pole shoes at both ends. Therefore, the measured value is the sum of braking force and friction force. And the friction force that comes from the mass of eddy current brake should be subtracted from the measured force value.

In order to verify the proposed method with more sizes at larger speed that are difficult to acquire by experiment, more FEMs are also carried out in this work. In the parameter analysis, the pitch number are set relatively large to minimize the edge effect. In this paper, single braking is studied without considering temperature change. All examples are assumed to be 20 °C.

3.2. Comparison with 2-D FEM and experimental results

Figure 18 shows the comparison between the braking force obtained by proposed method, FEM results, and experimental data. A good agreement between the FEM results and experimental data is observed. Still, it is also visible that the braking force calculated by the proposed method is greater than experimental data. That is due to the edge effect ignored in the proposed method is nonnegligible for the small pole pitch prototype. The analytical model can be more accurate with edge effect being consider in the proposed method.

Fig. 18.

Breaking force results comparison.

3.3. Braking force related to the different thickness of PM and the ratio of PM thickness to pole shoe thickness

Both the axial thickness of PM and the ratio of PM thickness to pole shoe thickness influence the braking force of PM linear ECB. It is obvious that the larger the PM, the greater the magnetic field and braking force will be. In addition, according to (23), (24) of the method proposed in this paper, it can be seen that the calculation of electric field cross section is sensitive to r _ms, ratio of PM thickness to pole shoe thickness. Therefore, to further investigate the validity of proposed method, the braking force associated with different thickness of PM under d _s and r _ms invariable are predicted respectively, using the proposed method and 2-D FEM model.

As can be seen from Fig. 19, the calculation results of braking force and critical velocity are in good agreement with FEM results in a large range. When d _s is constant, the increase in d _m leads to an increase in braking force but does not change the critical velocity. When r _ms remains unchanged (r _ms = 2.5), the simultaneous increase of d _m and d _s will not only increase braking force, but also decrease the critical velocity.

This indicates that the critical velocity is sensitive to d _s while insensitive to d _m. It can also be seen from figures that the braking force deviation will increase when d _m is large. This is because the magnetic field generated by the large permanent magnet is too strong, leading to a small deviation in the prediction of magnetic saturation in pole shoe and electric field in secondary. The proposed model accurately represents the variation trend of braking force and critical speed with design parameters. Although perfect accuracy is not achieved, the proposed approach is suitable for the preliminary optimization design of ECB.

Fig. 19.

Braking force varies with speed under different thickness of PM while (a) d _s = 8.5 mm and (b) r _ms = 2.5.

3.4. Braking force related to different thickness of conductor cylinder

The thickness of conductor cylinder is a sensitive parameter related to the critical speed. Increasing the thickness of conductor cylinder from a smaller value will significantly reduce the critical speed. As is shown in Fig. 20, braking force curves obtained by the proposed method and finite element method are in good agreement with each other at different conductor cylinder thicknesses. The variation trend of braking force and critical speed with design parameters predicted by the proposed model also match well with FEM prediction.

Fig. 20.

Braking force varies with speed under different thickness of conductor cylinder.

4. Conclusion

An analytical model for cylindrical linear ECB is presented, in which the MEC model considering eddy current and magnetic saturation is employed to calculate the magnetic field. And an approximate electric field cross section method, developed by eddy current formulas and FEM analysis, is employed to calculate the braking force.

By comparing the braking performance predicted by proposed method with the results of FEM and measurement, the accuracy of method is verified. This method can predict braking force in a wide speed range. Moreover, this method has the ability for parameter design of this ECB structure, for it can accurately represent the variation trend of braking force and critical speed with design parameters.

For the cylindrical PM linear ECB that has not been studied a lot, based on previous studies, the proposed method considers the influence of eddy current and magnetic saturation that improves the accuracy of MEC model. The proposed model needs the reference of FEM models, but the proposed analytical model applicable in a large size range can be obtained only by establishing a small number of FEMs. And the proposed braking force calculation method is simple, fast, and easy to use. Although the approximate method of electric field reduces the accuracy, it is sufficient for design application.

The accurate calculation of the proposed method is an interesting research direction in the future. In addition, considering edge effect and the effect of acceleration on braking force, the more targeted analysis model for braking curve design is also a meaningful research direction. From the perspective of energy conversion, continuous operation of eddy current brakes will inevitably cause temperature rise, thereby affecting braking performance. Coupling analysis of electromagnetic field and thermal field is an interesting topic for further research.

Footnotes

Acknowledgements

This research was financially supported by the “China National Postdoctoral Program for Innovative Talents” [Grant No. BX2021126], the “National Natural Science Foundation of China” [Grant No. 52105106] , the “Jiangsu Province Natural Science Foundation” [Grant No. BK20210342], the “Jiangsu Planned Projects for Postdoctoral Research Funds” [Grant No. 2021K008A], and the “Nanjing Municipal Human Resources and Social Security Bureau” [Grant No. MCA21121]. Besides, the authors wish to express their many thanks to the reviewers for their useful and constructive comments.

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