The eddy current brake has the advantages of no frictional contact or hydraulic fluid, high structural reliability, etc. The existing linear eddy current brakes are mostly flat type. A cylindrical permanent magnet eddy current brake is proposed in this paper, whose air gap magnetic field is generated by a series of ring-shaped permanent magnets and guided by iron pole to conductor layers, and can achieve higher air gap magnetic flux density. This paper introduces its basic structure and working principle. In order to obtain the analytical model of magnetic circuit design, the equivalent magnetic circuit method is used to analyze and derive the magnetic circuit, and it is verified by the axisymmetric finite element model. To derive the braking force generated by the eddy current brake, the layer theory approach is applied. The influence of electromagnetic parameters on the force characteristic is obtained by finite element numerical calculation, which provides a theoretical basis for the optimal design of the eddy current brake

Guolai Yang

Yumeng Fan

English

Vibroengineering PROCEDIA

  ISSN PRINT 2345-0533, ISSN ONLINE 2538-8479, KAUNAS, LITHUANIA 111 
Design and analysis of magnetic circuit of permanent 
magnet eddy current brake 
Yumeng Fan1, Guolai Yang2 
School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, China 
1Corresponding author 
E-mail: 1fanyumengzoey@njust.edu.cn, 2yyanggl@njust.edu.cn 
Received 19 September 2019; accepted 26 September 2019 
DOI https://doi.org/10.21595/vp.2019.21039 
Copyright © 2019 Yumeng Fan, et al. This is an open access article distributed under the Creative Commons Attribution License, which 
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 
Abstract. The eddy current brake has the advantages of no frictional contact or hydraulic fluid, 
high structural reliability, etc. The existing linear eddy current brakes are mostly flat type. A 
cylindrical permanent magnet eddy current brake is proposed in this paper, whose air gap magnetic 
field is generated by a series of ring-shaped permanent magnets and guided by iron pole to 
conductor layers, and can achieve higher air gap magnetic flux density. This paper introduces its 
basic structure and working principle. In order to obtain the analytical model of magnetic circuit 
design, the equivalent magnetic circuit method is used to analyze and derive the magnetic circuit, 
and it is verified by the axisymmetric finite element model. To derive the braking force generated 
by the eddy current brake, the layer theory approach is applied. The influence of electromagnetic 
parameters on the force characteristic is obtained by finite element numerical calculation, which 
provides a theoretical basis for the optimal design of the eddy current brake. 
Keywords: eddy current brake, magnetic circuit design, analytical model, finite element analysis. 
1. Introduction 
Compared with the traditional brake, the eddy current brake has the advantages of simple 
working principle, high structural reliability and no friction [1]. It makes the eddy current brake 
being widely applied, such as high-speed train track braking system and electromagnetic aircraft 
ejection system braking [2-4]. In order to solve the problem of short-distance braking of a large 
impulse machinery, a new type of eddy current brake is designed. 
The fundamental mechanism of the eddy current brakes has been studied by many scholars [5], 
and some scholars have analyzed the performances of it with different structures [6]. Jia Qiang 
has designed a linear eddy current buffer by using Halbach array [7]. Kou Baoquan et al has 
designed two different kinds of magnetic circuit hybrid eddy current brake [1, 8], which can 
produce a greater braking force. 
For the braking force calculation of the eddy current brake, Romanian scholar Boldea et al. 
equivalent the primary excitation winding to an infinitely thin current layer, and then derived 
Expression [9] according to the layer theory. There are also many other different derivation 
methods for the braking force models of different structures of eddy current buffers [10, 11]. 
Most of the existing linear eddy current brakes are flat-plate structures or hybrid excitation 
circuits. This paper designs and analyzes a brake model with a cylindrical structure using only 
permanent magnet. It has the following advantages over other types of eddy current brakes. First 
of all, the structure is simple, the installation is convenient, and it is not easy to suffer loss and 
error. Secondly, a large braking force can be generated, which makes it possible to brake a 
high-speed moving object in a short time and a short distance. 
In order to carry out in-depth research, analytical mode were carried out for magnetic circuit 
design analysis, and finite element simulation and model analysis were also carried out for eddy 
current and braking force. 
DESIGN AND ANALYSIS OF MAGNETIC CIRCUIT OF PERMANENT MAGNET EDDY CURRENT BRAKE.  
YUMENG FAN, GUOLAI YANG 
112 VIBROENGINEERING PROCEDIA. OCTOBER 2019, VOLUME 28  
2. Cylindrical permanent magnet eddy current brake model 
Fig. 1(a) illustrates the partial three-dimensional structural schematic of the studied cylindrical 
linear EMB, which mainly consists of two parts: the primary part, which consists of a moving rod 
combined with a sequence of ring-shaped, axially magnetized permanent magnets separated by 
pure iron poles, and the secondary part, which consists of an outer tube and an inner tube. 
High-velocity relative movement between the primary and secondary parts will occur under the 
intensive impact load. Due to the electromagnetic induction, circumferential eddy currents are 
generated in the secondary. They are sequentially arranged along the iron pole interval, and is 
opposite to the adjacent eddy currents. At the same time, the circumferential eddy currents induce 
a new magnetic field interacting with the primary magnetic flux to generate braking force, which 
hinders the relative motion of the primary and secondary according to Lenz’s law. 
Fig. 1(b) exhibits the layout of the permanent magnets which are oriented with magnetic poles 
of the same polarity facing each other. The iron pole is located between the two permanent 
magnets to increase the amount of magnetic flux entering the secondary. Therefore, the decrease 
of the quiescent operation point caused by the flux leakage in the cylinder hole of permanent 
magnet and excessive air-gap reluctance is reduced. The thick bold arrows show the direction of 
field line. 
As shown in Fig. 1(b), 𝜏௠, 𝜏, 𝑟௜, 𝑟௢, 𝑟ௗ, 𝑅௜, 𝑅௢ are the thickness of the permanent magnet; the 
pole pitch; the internal radius of the inner cylinder; the external radius of the inner cylinder (the 
internal radius of the outer cylinder), the external radius of the EMB, the internal radius of the 
permanent magnet, the external radius of the permanent magnet, respectively. 
a) 
 
b) 
Fig. 1. a) Permanent magnet eddy current brake and (b) permanent magnet distribution map 
3. Analytical model of the eddy current brake 
3.1. Analytical model of no-load magnetic field 
The no-load state of the eddy current brake refers to a state in which the relative velocity of 
the primary and secondary is zero, that is, the eddy current in the conductor plate is zero. Analyze 
under no-load condition can rapidly provide verify whether the performance and size are 
compatible with the envisioned application without involving comprehensive eddy current 
formulation. Fig. 2(a) shows the magnetic circuit distribution and magnetic flux flow direction of 
the eddy current brake in the no-load state. This paper did not neglect the leakage flux, adding it 
into the equivalent magnetic circuit, which improve the accuracy of the analytical model. 
The calculation formula for each part of the magnetic circuit is as follows. 
The magnetomotive force generated by the permanent magnet is: 
𝐹௖ = 𝐻௖𝜏௠, (1)
DESIGN AND ANALYSIS OF MAGNETIC CIRCUIT OF PERMANENT MAGNET EDDY CURRENT BRAKE.  
YUMENG FAN, GUOLAI YANG 
 ISSN PRINT 2345-0533, ISSN ONLINE 2538-8479, KAUNAS, LITHUANIA 113 
where 𝐻௖ is the permanent magnet coercivity. 
The reluctance of the air gap is as follows, and the reluctance of the inner cylinder, the iron 
pole can be obtained similarly: 
𝑅ఋ =
ln(𝑟௜/𝑟ଵ)
2𝜇଴𝜋(𝜏 − 𝜏௠), (2)
where: 𝑟ଵ is the external radius of the iron pole. 
m
2
Φ
rΦ
δΦ
1 1
2 2
rΦ
r
2
Φ
2
δΦ
δΦ
33 4
6
5 5
A BC D
 
a) 
1
2
R
m
2
R
2Rδ
R
6
2
R
cR 3R
4
2
R
cF mmΦ
mΦ cΦ
δΦ
A B
C
5R
D
mm
2
R
 
b) 
Fig. 2. a) Magnetic path and flux flow direction and b) its equivalent magnetic circuit 
The magnetic circuit 1 and 4 and the permanent magnet have a ring structure. The reluctance 
of permanent magnet is as Eq. (3), and the reluctance of the 1 and 4 part can be obtained similarly: 
𝑅௠ =
𝜏௠
𝜇଴𝜇௠𝜋(𝑅௢ଶ − 𝑅௜ଶ)
, (3)
where: 𝜇௠ is the relative magnetic permeability of the permanent magnet.  
The magnetic permeance of the 6 part of the magnetic circuit can be expressed as: 
𝐺଺ = න න
𝜇଴𝑟
𝜋(𝑅௜ − 𝑟) 𝑑𝑟
ோ೔ିఛ೘
ோ೔ିఛଶ
𝑑𝜃
ଶగ
଴
. (4)
 
drr
 
Fig. 3. The magnetic circuit of the 6 part 
A B
r
 
Fig. 4. Leakage flux between iron poles 
DESIGN AND ANALYSIS OF MAGNETIC CIRCUIT OF PERMANENT MAGNET EDDY CURRENT BRAKE.  
YUMENG FAN, GUOLAI YANG 
114 VIBROENGINEERING PROCEDIA. OCTOBER 2019, VOLUME 28  
The integral path of the leakage reluctance at the inner and outer ends of the permanent magnet 
is shown in Fig. 4. Although the relative magnetic permeability of the iron pole is larger than that 
of the permanent magnet, the axial length of the permanent magnet is longer, thus the magnetically 
leakage cannot be ignored. 
Therefore, the expression of the leakage flux in the hole of the permanent magnet is: 
𝐺௠௠ଵ = න න
𝜇଴(𝑅௜ − 𝑟)
𝜋𝑟 𝑑𝑟
ఛ௠/ଶ
ఛ௠/ସ
𝑑𝜃
ଶగ
଴
. (5)
The outer leakage flux is divided into two parts: the part through conductor and part only in 
the air gap: 
𝐺௠௠ଶ = න න
𝜇଴(𝑅଴ + 𝑟)
𝜋𝑟
௥೔ିோబ
(௥೔ିோబ)/ଶ
𝑑𝑟𝑑𝜃
ଶగ
଴
, (6)
𝐺௠௠ଷ = න න
𝜇଴(𝑟଴ − 𝑅଴)/√2
𝜋(𝑟଴ − 𝑅଴) +
𝜇௦𝑟଴
𝜏௠ − 2𝑟 𝑑𝑟𝑑𝜃
௟బ
଴
,
ଶగ
଴
 (7)
𝑙଴ =
𝜏௠
2 − 2(𝑟௜ − 𝑅଴). (8)
The total leakage reluctance of the permanent magnet can be obtained by paralleling the 
three-part’s leakage reluctance: 
𝑅௠௠ =
𝑅௠௠ଵ𝑅௠௠ଶ𝑅௠௠ଷ
𝑅௠௠ଵ𝑅௠௠ଶ + 𝑅௠௠ଵ𝑅௠௠ଷ + 𝑅௠௠ଶ𝑅௠௠ଷ. (9)
According to the equivalent magnetic circuit of the electromagnetic buffer and the Kirchhoff 
magnetic potential difference law, we can get: 
𝐹௖ = 𝐹஺஻ +
𝑅௠
2 Φ௠,     𝐹஺஻ =
𝑅௠௠
2 Φ௠௠,     𝐹஺஻ = 𝐹஼஽ +
𝑅ଵ
2 Φ஼,    
𝐹஼஽ = ൬𝑅ହ +
𝑅଺
2 ൰ (Φ஼ − Φఋ),     𝐹஼஽ = ൬𝑅ଶ + 𝑅ଷ +
𝑅ସ
2 + 𝑅௖ + 𝑅ఋ൰ Φఋ,    Φ௠ = Φ௠௠ + Φ௖,
 (10)
make: 
൜𝑡ଵ = 𝑅௠ 2⁄ 𝑡ଶ = 𝑅௠௠ 2⁄ 𝑡ଷ = 𝑅ଵ/2,𝑡ସ = 𝑅ହ + 𝑅଺ 2⁄ 𝑡ହ = 𝑅ଶ + 𝑅ଷ + 𝑅ସ 2⁄ + 𝑅௖ + 𝑅ఋ, (11)
By solving the above equations, we get: 
𝛷ఋ =
𝑡ଶ𝑡ସ𝑡ହ𝐹஼
𝑡ହ(𝑡ସ + 𝑡ହ)(𝑡ଵ𝑡ଶ + 𝑡ଵ𝑡ଷ + 𝑡ଶ𝑡ଷ) + 𝑡ସ𝑡ହଶ(𝑡ଵ + 𝑡ଶ). (12)
Therefore, the gap magnetic density at the air gap place r under no-load condition is: 
𝐵ఋ =
𝑡ଶ𝑡ସ𝑡ହ𝐹஼
2𝜋𝑟(𝜏 − 𝜏௠)(𝑡ହ(𝑡ସ + 𝑡ହ)(𝑡ଵ𝑡ଶ + 𝑡ଵ𝑡ଷ + 𝑡ଶ𝑡ଷ) + 𝑡ସ𝑡ହଶ(𝑡ଵ + 𝑡ଶ)). (13)
The design parameters of the eddy current brake are shown in Table 1. 
DESIGN AND ANALYSIS OF MAGNETIC CIRCUIT OF PERMANENT MAGNET EDDY CURRENT BRAKE.  
YUMENG FAN, GUOLAI YANG 
 ISSN PRINT 2345-0533, ISSN ONLINE 2538-8479, KAUNAS, LITHUANIA 115 
Table 1. Design parameters of eddy current brake 
Parameters / mm Value Parameters / mm Value 
The internal radius of the permanent magnet 
𝑅௜ 20 
The external radius of the inner 
cylinder 𝑟଴ 64 
The external radius of the permanent 
magnet 𝑅଴ 61 
The external radius of the outer 
cylinder 𝑟ௗ 72 
The external radius of the iron pole 𝑟ଵ 62.5 The thickness of the pole pitch τ 30 
The internal radius of the inner  
cylinder 𝑟௜ 63 
The thickness of the permanent magnet 
𝜏௠  20 
3.2. Analysis model of braking force 
In this chapter, the layer method approach is used to derive the braking force of the inner tube 
conductor layer. The permanent magnet is equivalent to an infinitely thin current layer so that the 
magnetic motive force generated by the current layer is equal to the magnetic motive force 
generated by the actual winding and the permanent magnet. According to the electromagnetic field 
theory, the corresponding differential equations and their general solutions in different regions can 
be obtained. 
 
Fig. 5. Finite element and analysis results of air gap flux density 
For the air gap region, because the electrical conductivity of the air gap is zero, so the equation 
is as follows. And the region of the outer tube is similar to it: 
∂ଶ𝐴ଵ
∂𝑧ଶ +
∂ଶ𝐴ଵ
∂𝑟ଶ = 0. (14)
For the inner tube conductor layer, the eddy current is induced in the layer when the brake is 
working. And the differential equation in the inner tube layer can be derived: 
𝐽 = −𝜎௖𝑣
𝑑𝐴ଶ
𝑑𝑧 ,     ∇
ଶ𝐴ଶ = 𝜇଴𝜎௖𝑣
𝑑𝐴ଶ
𝑑𝑧 , (15)
where 𝜎௖ is the conductivity (S/m) of the inner cylinder, and 𝑣 is the relative velocity between the 
primary and secondary. 
The expression of the eddy current in the inner tube layer can be derived from the differential 
equations. And the expression of the eddy current loss and braking force can be obtained: 
𝐽௘ = 𝑣𝜎௖
𝑑𝐴ଶ
𝑑𝑧 ,      𝑃௘ =
𝜋𝑟ଵ𝑝
𝜎௖ න න 𝐽௘ ⋅ 𝐽௘
௘
଴
ఛ
ିఛ
𝑑𝑟𝑑𝑧,     𝐹 = 𝑃௘𝑣 , (16)
where 𝑝 is the polar logarithm; 𝐽௘ is the conjugate complex number of 𝐽௘. 
DESIGN AND ANALYSIS OF MAGNETIC CIRCUIT OF PERMANENT MAGNET EDDY CURRENT BRAKE.  
YUMENG FAN, GUOLAI YANG 
116 VIBROENGINEERING PROCEDIA. OCTOBER 2019, VOLUME 28  
4. Finite element analysis 
Using a set of design data as Table 1 to establish the finite element model, the magnetic flux 
density of the device at the speed of 5, 10 and 1 5m/s are as Fig. 6. It is observed that the reaction 
field of eddy currents tilts the field line coming into the secondary and differs in degree at different 
speed.  
Fig. 7 are force-speed characteristic curves obtained by using different air gap thickness and 
inner tube thickness, respectively. It can be seen from Fig. 7(a) that the thickness of the air gap 
has little effect on the braking force when the speed is small, but the larger the thickness of the air 
gap at a high speed, the smaller the braking force generated, because the air gap flux density is 
reduced. It can be seen from Fig. 7(b) that both the peak braking force and the critical speed 
increase as the thickness of the inner tube increases. At the same time, the braking force increases 
as the thickness of inner tube increases when the speed is less than 9 m/s, but the opposite when 
the speed is greater than 9 m/s. 
In addition, the calculation results show that the peak braking force of the permanent magnet 
eddy current brake can reach 1.5e5N, and the critical speed is about 8 m/s, which is larger than 
the ordinary eddy current brake. That is, at high speeds, the brakes can reach a sufficiently large 
braking force requirement. 
 
a) 
 
b) 
 
c) 
Fig. 6. Magnetic flux density (𝑇) at a) 5 m/s, b) 10 m/s, and c) 15 m/s 
 
a) 
 
b) 
Fig. 7. Force characteristic curves for different a) air gap thicknesses and b) inner tube thicknesses  
5. Conclusions 
In this paper, a new type of cylindrical permanent magnet eddy current brake is proposed, and 
its basic structure and working principle are introduced. In order to obtain the analytical model of 
the magnetic circuit, the equivalent circuit method and layer theory approach are used to analyze 
and derive the magnetic circuit. The model is also calculated by the axisymmetric finite element 
DESIGN AND ANALYSIS OF MAGNETIC CIRCUIT OF PERMANENT MAGNET EDDY CURRENT BRAKE.  
YUMENG FAN, GUOLAI YANG 
 ISSN PRINT 2345-0533, ISSN ONLINE 2538-8479, KAUNAS, LITHUANIA 117 
model, and the analytical model is verified through comparing. The influence of the thickness of 
the air gap and the thickness of the inner tube on the braking force characteristic curve is obtained 
using the finite element model, which provides a theoretical basis for the future optimal design of 
this eddy current brake. 
There are still errors of the analytical model proposed this paper with practical device, 
demagnetization effect of permanent magnet and armature reaction would be added into 
consideration in future study. 
References 
[1] Kou B., Jin Y., Zhang H., et al. Characteristic analysis of hybrid excitation linear electromagnetic 
dampers. Proceedings of the CSEE, Vol. 33, Issue 24, 2013, p. 143-151. 
[2] Wang Z., Hua X., Chen Z. Experimental study on vibration control of a model footbridge by a tiny 
eddy current tuned mass damper with permanent magnets. Journal of Vibration and Shock, Vol. 33, 
Issue 20, 2014, p. 129-132. 
[3] Yan Guobin Study of Hybrid Excitation Rail Eddy Current Brake System of High-Speed Train. 
Zhejiang University, 2010. 
[4] Xiao Yao, Wu Jun The analysis and design of permanent-magnet eddy current brake for EMALS. 
Small and Special Electrical Machines, Vol. 41, Issue 8, 2013. 
[5] Bae J. S., Hwang J. H., Park J. S., et al. Modeling and experiments on eddy current damping caused 
by a permanent magnet in a conductive tube. Journal of Mechanical Science and Technology, Vol. 23, 
Issue 11, 2009, p. 3024-3035. 
[6] Jang S. M., Lee S. H. Comparison of three types of permanent magnet linear eddy-current brakes 
according to magnetization pattern. IEEE Transactions on Magnetics, Vol. 39, Issue 5, 2003, 
p. 3004-3006. 
[7] Jia Q., Gao Y., Tong Y., et al. Design of linear eddy current buffer. Journal of Engineering Design, 
Vol. 18, Issue 3, 2011, p. 209-213. 
[8] Kou B., Jin Y., Zhang H., et al. Analysis and design of hybrid excitation linear eddy current brake. 
IEEE Transactions on Energy Conversion, Vol. 29, Issue 2, 2014, p. 496-506. 
[9] Boldea I., Babescu M. Multilayer theory of D.C. linear brakes with solid-iron secondary. Proceedings 
of the Institution of Electrical Engineers, Vol. 123, Issue 3, 1976, p. 220. 
[10] Yin Xiangrui Research on Long Stroke Secondary Moving Permanent Magnetic Linear Eddy Current 
Brake. Harbin Institute of Technology, 2017. 
[11] Zhao Xiaobo Study on Electromagnetic Properties and Braking Performance of Permanent Magnet 
Type Eddy Current Retarder. Nanjing Agriculture University, 2009. 


Design and analysis of magnetic circuit of permanent magnet eddy current brake

https://www.jvejournals.com/article/21039/pdf

Design and analysis of magnetic circuit of permanent magnet eddy current brake

Abstract

Similar works

Full text

Available Versions

Vibroengineering PROCEDIA