Study of Eddy Current Brake Based on Motion of
Permanent Magnet in the Nonmagnetic Metal Tube
Authors:
Yuxuan Xia,
Songming Yan,
Haining Tan
Adviser:
Mr. Wei Song
117 Xi’an Road, Heping District, Tianjin, P. R. China
ABSTRACT
Eddy current will be induced in a non-magnetic metal tube when the permanent
magnet moves in the non-magnetic metal tube, hindering the movement of the
permanent magnet. In this paper, this physical phenomenon is researched. Equivalent
current model is employed to analyze the magnetic field around the permanent
magnet. According to the basic laws of electromagnetic fields, eddy current
distribution in the metal tube is obtained and electromagnetic force on the permanent
magnet is modeled. With the help of Finite Element (FE) method, permanent magnet's
surrounding magnetic field, eddy current distribution in metal tube and eddy current
loss distribution are calculated, obtaining the changing law of permanent magnet's
magnetic field in the non-magnetic metal tube. Besides, a discrete calculation method
is proposed to complete numeric calculation of the electromagnetic force.
Furthermore, experimental platform is established and experiments demonstrating the
relation between velocity of permanent magnet and electromagnetic force on it are
designed. The experiments' results show influences of metal resistivity, the size of
tube and permanent magnet and the initial position of the permanent magnet on the
electromagnetic force and validate the correctness of theoretic analysis.
Key words: Electromagnetic Induction, Eddy Current Effect, Electromagnetic
Braking, Finite Element Method
i
TABLE OF CONTENTS
ABSTRACT ............................................................................................................................................ I
NOMENCLATURE ............................................................................................................................ III
.
INTRODUCTION.........................................................................................................................1
.
ANALYSIS AND MODELING ...................................................................................................2
2.1 ANALYSIS OF PERMANENT MAGNET MAGNETIC FIELD ....................................................................2
2.1.1 The magnetization intensity of permanent magnet.................................................................2
2.1.2 The equivalent current model of permanent magnet..............................................................3
2.1.3 The magnetic induction intensity of permanent magnet.........................................................4
2.2 ANALYSIS OF EDDY CURRENT IN METAL TUBE ................................................................................6
2.3 MODELING OF ELECTROMAGNETIC FORCE ......................................................................................8
.
ELECTROMAGNETIC CALCULATIONS............................................................................10
3.1 CALCULATION OF MAGNETIC FIELD ..............................................................................................10
3.2 DISCRETE CALCULATION METHOD OF ELECTROMAGNETIC FORCE ................................................12
3.3 MAGNETIC CALCULATION OF MAGNETS WITH DIFFERENT SIZES ...................................................13
3.4 CALCULATION OF EDDY CURRENT DISTRIBUTION .........................................................................15
V.
EXPERIMENTS AND RESULTS.............................................................................................18
4.1 DESCRIPTION OF EXPERIMENTAL SYSTEM AND FORCE ANALYSIS ..................................................18
4.2 RELATION BETWEEN VELOCITY AND ELECTROMAGNETIC FORCE ..................................................21
4.3 EXPERIMENT OF CHANGING THE RESISTIVITY ...............................................................................22
4.4 EXPERIMENT OF CHANGING THE MAGNETIC FIELD DISTRIBUTION .................................................25
4.5 EXPERIMENT OF CHANGING THE INITIAL POSITION .......................................................................26
V.
CONCLUSIONS AND PROSPECTS .......................................................................................29
ACKNOWLEDGEMENT ...................................................................................................................31
REFERENCE .......................................................................................................................................31
ii
NOMENCLATURE
A
B
Bm
Br
Bz
Bθ
Din
Dout
Fe
I
J
Irotary
Jm
Jsm
H
M
P
R
T
a
d
fc
g
h
m
r
v
β
μ0
ρ
Magnetic vector potential
Magnetic induction intensity
Remanent magnetic flux density
Radial component of magnetic induction intensity
Vertical component of magnetic induction intensity
Tangential component of magnetic induction intensity
Inner diameter of metal tube
Outer diameter of metal tube
Electromagnetic force
Current
Current density
Rotary inertia
Volume current density
Surface current density
Magnetic field intensity
Magnetization intensity
Loss density
Radius of pulley
Force on rope
Acceleration
Diameter of magnet
Friction
Acceleration of gravity
Height of magnet
Mass
Radius vector
Velocity
Angular acceleration
Permeability of vacuum
Resistivity
iii
. Introduction
The phenomenon that the falling time is largely greater than that of the free fall in
the air time when permanent magnet plummet in the nonmagnetic metal tube
(Hereinafter referred to as metal tube), which is often used in the qualitative
demonstration of Lenz's law. Through the theoretical analysis of electromagnetism,
the induced electromotive force and eddy current forming eddy magnetic field are
produced in the metal tube when permanent magnet move in the metal tube to prevent
the movement of permanent magnet[1-2]. The current induced due to variation of
magnetic field passing through a conduct is called eddy current. Eddy current will
result in electromagnetic power loss and heating of the conduct. Meanwhile, eddy
current will also produce electromagnetic force applied to the permanent magnet or
electricity conduct [1, 3, 4].
Eddy current effect is often applied in the electromagnetic mechanism. By the
heat effect of eddy current, high frequency current is injected into the smelting device
in the metal smelting process, which produces large eddy current in the smelting
metals and the heat generated by the vortex can make the metal melt quickly [5, 6].
Meanwhile, eddy current effect also can cause some negative effects. Eddy heat needs
to consume additional energy, reducing the efficiency of the electromagnetic
mechanism, and eddy heat may shorten the service life of insulating material, and
then reduce the electromagnetic properties of materials [7, 8]. Hence, research on the
influencing factors of the eddy current effect has a certain practical significance for
the rational utilization of the eddy current effect. The eddy current braking devices
based on the principle of eddy current effect producing electromagnetic force, with
low mechanical wear and tear, fast response time, high braking stability and simple
daily maintenance, can be used as auxiliary brake devices, and have a wide
application prospect in traction elevators, electric cars, rail transit and aerospace
industry.
This paper studies the physical phenomenon that when the permanent magnet
moves in nonmagnetic metal tube, it will be subject to electromagnetic force braking.
Firstly, the magnetic field of permanent magnet and the eddy current of nonmagnetic
metal tube are analyzed, and the mathematical model of electromagnetic force is
established. Then we explore the main factors which influence the electromagnetic
force. Secondly, the finite element method is used to carry out the numerical
calculation of electromagnetic force, and eddy current distribution of nonmagnetic
metal tube is described. Finally, the experiment system is designed to verify the
correctness of theoretical analysis result. In this paper, the conclusion can provide
certain reference for reasonable use of eddy current magnetic field to make buffer
braking effectively.
1
. Analysis and Modeling
In order to study the influencing factors of the electromagnetic force that
permanent magnets endure in nonmagnetic metal tube, first, this chapter analyses the
magnetic field of permanent magnet and describe the spatial distribution of permanent
magnet magnetic field, and then analyses the eddy current produced through the
movement of permanent magnet in the metal tube. Finally, we establish the model of
the electromagnetic force of permanent magnet caused by the permanent magnetic
field interacting with eddy current magnetic field.
The established model bases on the following basic assumptions:
1) The permanent magnet is uniform magnetized along the axial direction.
2) Place the metal tube vertically, and the direction of magnetic pole of permanent
magnet coincides with the symmetry axis of the tube.
3) The permanent magnet falls down vertically along the axial line of the metal tube
without the translation and deflection in the horizontal direction.
4) The metal tube is nonmagnetic, and the distribution of permanent magnet
magnetic field in the metal tube is the same as that of vacuum environment.
5) The relative permeability of permanent magnet equals 1.
2.1 Analysis of permanent magnet magnetic field
Permanent magnet can generate constant magnetic field in its surrounding space.
The metal tube induces eddy current by the movement of magnetic field, when the
permanent magnet moves in the metal tube. Therefore, the analysis of the magnetic
field of permanent magnet is the basis of study on the phenomenon of eddy current
brake.
2.1.1 The magnetization intensity of permanent magnet
By electromagnetic theory, the relationship between the magnetic induction
intensity and magnetic field intensity in a vacuum is written as
B  0 H
(2-1)
Where B is magnetic induction intensity, H is magnetic field intensity, μ0 is the
permeability of vacuum, and in magnetic materials we have
B  0 M   0  r H
(2-2)
Where M is magnetization intensity, which is the vector sum of magnetic moment of
each magnetic domain per unit volume in the magnetic material and can be used to
describe the magnetization of magnetic material, B contains two components, and one
is μ0μrH the same as the component in the air, the other is μ0M produced by the
magnetization of the magnetic material. Magnetic induction intensity of magnetic
2
materials is greatly enhanced after magnetization by the external magnetic field.
Considering the relative magnetic permeability μr = 1, when the magnetic materials is
uniform magnetized, the vector equation (2-2) can be written as algebraic sum
B  0 M  0 H .
(2-3)
2.1.2 The equivalent current model of permanent magnet
In this section, we calculate the magnetic field of permanent magnet by using the
equivalent current model, and the magnetic field is regarded as the one generated by
the distribution of current element with the volume current density Jm. According to
Maxwell's equations, we can derive the differential equations of magnetic vector
potential by introducing magnetic vector potential. By the equivalent current model,
the permanent magnet will be equivalent to a number of unit current with volume
current density Jm and volume dV. Magnetic vector potential A can be expressed as
0
1
M   dV
V
R
0   M
M

   dV

4π V R
R
0   M
0 M  n
dV 
dS

4π V R
4π  S R
A
4π 
(2-4)
Where S is the surface surrounding permanent magnet, R is the distance from source
point to field point, n is the unit normal vector on the outer surface of permanent
magnet. Equivalent volume current density is written as
Jm    M
(2-5)
The expression of equivalent surface current density is
J sm  M  n
(2-6)
Substituting (2-5), (2-6) into (2-4), we have
A
0
4π 
V

J
Jm
dV  0  sm dS
S
R
4π
R
(2-7)
According to ampere molecular current hypothesis, any point of magnetic field in
the outer space is motivated by all molecular currents in neat rows in permanent
magnet. For the cylindrical permanent magnet used in the experiment as an example,
the permanent magnet magnetizes evenly along the axial direction, hence, the
magnetization density M is constant and the equivalent volume current density Jm is
zero. For the equivalent current model of the permanent magnet, broadly only the
surface current density Jsm works. As a result, (2-7) can be simplified as
3
A
0
4π 
S
J sm
dS
R
(2-8)
2.1.3 The magnetic induction intensity of permanent magnet
The magnetic induction intensity of permanent magnet can be expressed as
B   A 
0

4π 
S
J sm
dS .
R
(2-9)
According to (2-6), considering that the permanent magnet is uniform magnetized
along the axial direction, the equivalent surface current on the two upper and lower
surface of the permanent magnet is zero. The density of the equivalent surface current
on the cylindrical surface of permanent magnet is the magnetization amplitude M and
its direction is along the tangent direction on the horizontal cross-section. Therefore,
analyze the circular line current field at first, and then accumulate, so that we can
obtain the permanent magnet magnetic induction intensity.
As shown in Fig. 2.1, we regard the center of the cylindrical permanent magnet as
origin, take the permanent magnet N-pole direction as the z axis positive direction,
and establish a right-handed three-dimensional coordinate system. First analyze the
thin layer circle l between z axis coordinates (z，z+dz), then calculate the magnetic
field produced. The line current existing in the circle of ring is expressed as
I = J sm dz  Mdz
(2-10)
Its direction is tangent direction of the ring, according to Biot-Savart's law
dB 
0 Idl  r
4π
r3
(2-11)
Fig. 2.1 The right-handed three-dimensional coordinate of the cylindrical permanent magnet
4
According to the field superposition principle, the sum of dB vectors produced by all
the current elements in a field point is the magnetic induction intensity the whole
section of line current produce. Take any field point P, the magnetic induction
intensity dBl produced by the circular line current I at P is expressed as
0 Idl  r
4π 
dBl 
r3
(2-12)
Substituting (2-10) into (2-12), we obtain
dBl 
0
Mdl  r
dz
l
r3
4π 
(2-13)
According to the field superposition principle, compute the vector integral of dBl
produced by circular current loops, then derive the magnetic induction intensity at P
produced by the whole cylindrical surface current
B=
h /2
 h /2
dBl 
0
h /2
Mdl  r
dz
l
r3
4π  
 h /2
(2-14)
According to (2-14), the magnetization density M, height h and bottom radius r all
have influences on the magnetic induction intensity.
The following analysis process clarifies that in the circular line current magnetic
field, the radial component and vertical component of magnetic induction intensity
only exist on the horizontal cross section, but tangential component doesn’t exist.
As shown in Fig. 2.1, in the circular line current of XOY plane we take two
current elements P1(rcosθ, rsinθ, 0) and P2(rcosθ, -rsinθ, 0), which are symmetrical
relative to x axis, and the corresponding element current vectors are respectively
(-Msinθ，Mcosθ，0) and (Msinθ，Mcosθ，0). Take a point P3(x1, 0, z1), and then the
magnetic induction intensity produced respectively by P1 and P2 at P3 are derived as
dB1 
0 Idl  r
r3
   Msin , Mcos , 0   ( x1  rcos , rsin , z1 )
 0
dl
2
2
4π
 x1  rcos    rsin   z12

4π
(2-15)
0 ( z1Mcos , z1Msin , rM  x1M cos  )
dl
2
2
4π
 x1  rcos    rsin   z12
dB2 
0 Idl  r
r3
  Msin , Mcos , 0   ( x1  rcos , rsin , z1 )
 0
dl
2
2
4π
 x1  rcos    rsin   z12

4π
(2-16)
0 ( z1Mcos ,  z1Msin , rM  x1M cos  )
dl
2
2
4π
 x1  rcos    rsin   z12
The synthesis of magnetic induction intensity produced by two symmetrical current
elements at P3 is written as
5
dB1  dB2 
 0 ( z1Mcos , 0, rM  x1M cos  )
dl
2π  x1  rcos 2   rsin 2  z12
(2-17)
By (2-17), we know that the resultant magnetic field at point P produced by any two
current elements symmetrical relative to x axis in a horizontal plane don’t have
component on the y axis. Assuming that two current elements are on the x axis,
located at point (r，0，0) and (-r，0，0), the corresponding current element vectors are
respectively (0，M，0) and (0，M，0), and the current magnetic induction intensity at
P3 is expressed as
dB 
0 Idl  r
r3
  0,  M , 0   ( x1  r , 0, z1 )
 0
dl
4π
x12  r 2  z12

4π
0 ( Mz1 , 0,  M ( x1  r ))
x12  r 2  z12
4π
(2-18)
dl
By (2-18), the magnetic field produced by the current elements doesn’t have
component on the y axis.
In conclusion, according to the field superposition principle and the symmetry of
the ring line current, the radial component Br and vertical component Bz of magnetic
induction intensity exist on the horizontal section, without tangential component.
Therefore, the magnetic induction intensity generated by the cylindrical surface
current of the permanent magnet also doesn’t have tangential component.
2.2 Analysis of eddy current in metal tube
According to the spatial distribution of magnetic fields analyzed in section 2.1,
this section will analyze the eddy current in the metal tube induced by the motion of
permanent magnet, and obtain the eddy current distributions and influencing factors.
When the permanent magnet moves in the metal tube, the moving magnetic field
caused by the motion of permanent magnet induces a current in the metal tube.
According to Faraday’s law of electromagnetic induction, we have
 
dm
d
   B  dS
dt
dt S
(2-19)
Where Φm is the magnetic flux through the metal tube. Assuming the reference frame
of permanent magnet is stationary, the magnetic field generated by the permanent
magnet doesn’t change with the time, and the metal tube moves relative to the
permanent magnet. The motional electromotive force (EMF) induced in the closed
loop of metal tube is expressed as
    v  B   dl .
l
6
(2-20)
By (2-16) ~ (2-18), the magnetic induction intensity produced by the permanent
magnet only has the radial component Br and vertical component Bz. The magnetic
induction intensity B at point P can be expressed as (Brcosθ, Brsinθ, Bz), where θ is
the angle between the projection of OP onto the plane XOY and x axis. Taking the
vertical upward as positive direction, when the permanent magnet falls in the metal
tube vertically, the speed of the metal tube relative to magnetic field only has a
vertical component, that is (0，0，v). Therefore, the motional induction electromotive
force is written as
    v  B   dl
l
   0, 0, v    Br cos , Br sin , Bz    dl
l
(2-21)
   vBr sin , vBr cos , 0   dl
l
By (2-21), the induction electromotive force in the metal tube only has to do with
the radial component Br of magnetic induction intensity, while vertical component Bz
has no influence on it. Therefore, the vertical component Bz is not considered when
we analyze the eddy current field. With the symmetry of the magnetic field and metal
tube, the analysis of electromagnetic induction will be carried out in the vertical cross
section through center axis of the metal tube as shown in Fig. 2.2(a).
Metal tube
Permanent
magnet
Metal tube
Vertical cross
section
dS
y
Permanent
magnet
Br
P(x,y)
O
(a)
x
(b)
Fig. 2.2 The vertical cross section through center axis of the metal tube
As shown in Fig. 2.2(b), a two-dimensional Cartesian coordinate system in the
cross section is established. The direction of Br in the cross section is given, and the
direction of eddy current in the metal tube can be determined by the right-hand rule.
For a random point P(x, z), its area is dS, and its thickness is dl. Therefore，the
induction electromotive force at P with the Br(x, z) is derived as
7
d =Br ( x, z )vdl
(2-22)
The direction of the induction electromotive force is perpendicular to the plane XOZ,
and the resistance is expressed as
dR =
dl
dS
(2-23)
So the current density is
J ( x, z )=
1 d Br ( x, z )v

.
dS dR

(2-24)
In conclusion, in any vertical cross section of the metal tube, the direction of the
current density is perpendicular to the cross section, and its value has to do with the
speed v of permanent magnet, the resistivity ρ of metal tube and the radial component
Br(x, z). Extending the two-dimensional analysis results to three-dimensional area, we
can obtain that the induced current density J in the tube only has tangential
component Jθ in any horizontal plane, and there is no vertical component Jz. Hence,
the direction of eddy current is along with the tangent direction of the metal tube.
2.3 Modeling of electromagnetic force
When the permanent magnet moves along the axial direction of metal tube, the
eddy current is induced in the metal tube. The eddy current magnetic field interacts
with the magnetic field of permanent magnet, so the electromagnetic force will be
generated hindering the relative motion. By Newton's third law, the forces on the
metal tube and the permanent magnet are action and reaction forces. Therefore, by
analyzing the force on the metal tube, electromagnetic force on the permanent magnet
can also obtained.
According to the previous conclusions, the magnetic induction intensity of
permanent magnet in the metal tube only has the radial component Br and vertical
component Bz, and we will analysis the force on the metal tube from the two
components. As shown in Fig. 2.2 (b), with the Ampere's law, the force generated by
the Bz that the current element J(x,z)dSdl at the point P(x, z) endures is expressed as
dFBz  J ( x, z )dSdl  Bz ( x, z )
(2-25)
Because the direction of dl is perpendicular to the cross section in Fig. 2.2, the
resultant force generated by Bz that the metal tube endures is derived as
8
FBz   dFBz
  J ( x, z )dSdl  Bz ( x, z )
(2-26)
   dl  Bz ( x, z )J ( x, z )dS
Sa l
Where Sa is half the area of cross section.
2π
 dl  Bz ( x, z)   r (cos  ,sin  , 0)  (0, 0, Bz )d  0
0
(2-27)
l
With (2-27), the resultant force generated by the Bz that the metal tube endures is
zero. Similarly, the force generated by the Br is expressed as
dFBr  J ( x, z )dSdl  Br ( x, z )
(2-28)
The resultant force generated by Br can be calculated by
FBr   dFBr
  J ( x, z )dSdl  Br ( x, z )
(2-29)
   dl  Br ( x, z )J ( x, z )dS
Sa l
As shown in Fig. 2.2(b), the left-hand rule can be used respectively in the four
quadrants, and the force each current element endures from the Br is directed along
the negative z axis. Hence, the resultant force generated by the Br is also directed
along the negative z axis. The equation (2-30) shows the resultant electromagnetic
force on the metal tube.
F    dl  Br ( x, z )J ( x, z )dS
Sa l
   dl  Br ( x, z )
Br ( x, z )v
Sa l
 k 
Sa
2πrBr2 ( x, z )v


dS
(2-30)
dS
Where k is the unit vector of z axis. By (2-30), electromagnetic force on the metal
tube has a reverse direction relative to the velocity v, so it is a braking force. Its value
is related to the Br, the relative velocity v, resistivity ρ and the area Sa.
By Newton's third law, electromagnetic force on the permanent magnet is
directed along the positive z axis, and its value equals the value of F.
This chapter analyzes the magnetic field of permanent magnet and the eddy
current field of metal tube, and then establishes the mathematical model of
electromagnetic force. In experiments, we will mainly explore the influence of
velocity v, resistivity ρ, and the area Sa on the motion of permanent magnet.
9
. Electromagnetic Calculations
By the analysis and modeling in the chapter 2, when the permanent magnet falls
in the metal tube, the direction of electromagnetic force on it is upward, the value is
F  
2πrBr2 ( x, z )v

Sa

2πv

dS
(3-1)
 rB ( x, z)dS
2
r
Sa
According to (3-1), if we want to calculate the electromagnetic force, we need to
calculate the surface integral ∫∫SarBr2(x,z)dS. Because of the higher complexity of the
analytic calculation for this surface integral, this chapter with the aid of finite element
method (FEM) to calculate permanent magnet magnetic field distribution and analyze
eddy current distribution. Besides, the discretization method is proposed for
calculating the electromagnetic force in this chapter.
3.1 Calculation of magnetic field
Because the material of the tube is nonmagnetic, when the permanent magnet
falls through the metal tube, magnetic field produced by permanent magnet is not
affected by the tube. So magnetic field distribution in the metal tube can be obtained
by calculating the static magnetic field of permanent magnet in vacuum.
With the electromagnetic field finite element analysis (FEA) software ANSOFT
MAXWELL, three-dimensional calculation model is set up. The permanent magnet
material is set to be Nd-Fe-B, which residual flux density Bm= 1.4T. The permanent
magnet shape is set to be cylinder, which bottom diameter d = 20mm and height h =
30mm. The cylinder center is the origin and the central axis of the permanent magnet
is z axis (the positive direction is the direction of N pole). As shown in Fig. 3.1(a), the
three-dimensional model is meshed by tetrahedron grid. The magnetic field
distribution of permanent magnet is shown in Fig. 3.1 (b).
(a) Mesh plots
(b) The magnetic field distribution
Fig. 3.1 The mesh generation and magnetic field distribution
10
As shown in Fig. 3.1(b), the static magnetic field distribution of permanent
magnet is symmetrical and there is only radial component and vertical component but
no tangential component. This conclusion is consistent with the analysis result in
Section 2.1. Due to the characteristic of the permanent magnet’s magnetic field, only
with analysis of the magnetic field on a vertical cross section containing permanent
magnet central axis, precise calculation results of the distribution of permanent
magnet magnetic field in the whole space can be obtained. In the above established
model of permanent magnet, 10 adjacent vertical straight line with even interval of 1
mm with distance of 12mm ~ 16mm from the axis of permanent magnet are chosen in
XOZ plane, and the static magnetic field of every line are calculated respectively. As
shown in Fig. 3.2, five lines located on the right wall of tube are marked with l1, l2, l3
and l4, l5 from left to right, respectively. The distances from the central axis of the
permanent magnet to the five lines are 12 mm, 13 mm, 14 mm, 15 mm, 16 mm
respectively. Finally, after calculation of magnetic field, results are shown in Fig.3.2.
Fig 3.2 Magnetic field distribution on vertical section
The calculation results shown in Fig. 3.2 indicate that the distribution of the
permanent magnet’s magnetic field is symmetrical along each line. Besides, there are
only vertical component Bz existed on the horizontal central line and radial
component Br=0. In the upper half of the permanent magnet, the magnetic induction
intensity B is directing at the outside of the permanent magnet, that is to say, Br>0,
while in the lower half, B is directing at the inside of the permanent magnet, that is to
say, Br<0. In the horizontal planes nearby the top and bottom surfaces, the direction of
the vertical component of B changes and its magnitude reach peak values.
In order to analyze quantitatively of components of magnetic field of permanent
magnet in the space, obtain the distribution trend of magnetic field in the metal tube,
the tangential component Bθ, radial component Br and vertical component Bz on l1，l2，
l3，l4，l5 are calculated respectively. Calculation results are shown in Fig. 3.3, and the
magnetic induction density B on the five lines l1~l5 are shown in Fig. 3.3(a) ~ (e)
respectively. The horizontal axis shows distance between each point on the line and
midpoint of the line.
11
Fig 3.3 Calculation results of Bθ, Br, and Bz on five lines
According to the calculation results of each component of magnetic induction
density shown in Fig. 3.3, some conclusions can be obtained as follows
1) Magnetic field distribution’s law is basically the same in each figure and the
tangential component Bθ is always zero.
2) The radial component Br is an odd function. Its direction changes in the midpoint of
each line and its amplitude reaches maximum on the positions of upper and lower
surfaces of the cylindrical permanent magnet.
3) The vertical component Bz is an even function and its direction changes on the
upper and lower surfaces of the cylindrical permanent magnet.
3.2 Discrete calculation method of electromagnetic force
As shown in Fig. 3.3, each component of magnetic induction intensity on the five
adjacent lines l1~l5 can be calculated. Because the space between the five adjacent
lines is narrow, the B on each line whose distance from line l1 is less than Δh/2 on the
vertical plane can be considered as equal to the B on the line l1. Therefore, as shown
in Fig. 3.2, plane Sa could be divided into five regions Sa1~Sa5, and then the surface
integral in (3-1) can be rewritten as
5
Z   rBr2 ( x, z)dS    rBr2i ( z)dS
i 1 Sai
Sa

Din h /2

Din
4 Din ( i 1) h /2
xdx Br12 ( z)dz  
z

xdx  Br2i ( z)dz 
i 2 Din ( i 1) h /2
z
Dout
 xdx B
Dout h /2
z
2
r5
( z)dz
(3-2)
1
h
ih
1
h2

( Din h 
)1   ( Din h 
)i  ( Dout h 
)5
2
4
2
2
4
i 2
2
2
4
Where Din and Dout denote inner and outer diameters of the metal tube respectively,
Λi(i=1~5) respectively denote calculation values of ∫Bri2(z)dz (i=1~5) which are
12
obtained with the help of Finite Element software. Then, the force on permanent
magnet can be simplified as
F
2πv

Z
(3-3)
If the inner and outer diameters of the metal tube vary, calculation of
electromagnetic force on permanent magnet can be achieved only by choosing lines of
different numbers and positions to analyze with variation of Sa in the surface integral.
3.3 Magnetic calculation of magnets with different sizes
To figure out how magnetic field distribution of permanent magnet changes when
its size varies, three-dimensional models of cylinder permanent magnets with four
different sets of dimensions are established and their magnetic field are calculated.
Four sets of dimensions are listed, as shown in Tab. 3.1.
Table 3.1 Different dimensions of permanent magnet for FEA
Test groups
Diameter d (mm)
Height h (mm)
a
20
20
b
20
20
c
15
15
d
15
15
Four magnetic fields of the models are calculated. As shown in Fig. 3.4, Four
permanent magnets’ three-dimensional magnetic field distribution is obtained.
Fig. 3.4 The field distributions of permanent magnets with different dimensions
From Fig. 3.4, it can be seen that the surrounding magnetic field distribution of
permanent magnet changes with variation of its dimensions. Besides, the static
13
magnetic field distribution of the permanent magnet is symmetrical and there are only
axial and vertical components but no tangential components.
To analyze components in every direction of the magnetic field of permanent
magnet quantitatively, the analysis is carried out in the vertical cross section through
center axis of permanent magnet by the method proposed in Section 3.1. After
calculation, distributions of magnetic field of the permanent magnets in the vertical
cross section are obtained as shown in Fig. 3.5.
Fig. 3.5 The distributions of magnetic fields on vertical cross section
Fig. 3.6 The components of magnetic induction density with different dimensions
14
Magnetic induction density’s components in every direction on the five lines
equally spaced in the vertical cross section are shown in Fig. 3.6. As shown in Fig.
3.6(a) ~ 3.6(d), different columns correspond to the permanent magnets with different
dimensions and each figure. From top to bottom respectively corresponds to different
lines in the vertical cross section.
According to magnetic field component calculation results in Fig 3.5 and Fig 3.6,
it can be concluded that:
1) For permanent magnet with different dimensions, the magnetic induction density
magnitude on each line is different, but their distribution rules are consistent
basically.
2) The magnitude of magnetic induction density’s component in every direction
increases with growth of diameter d of the permanent magnet.
3) When the height of the permanent magnet h varies, in vertical direction
distribution of magnetic induction density’s component in every direction changes
but its magnitude stays the same approximately.
3.4 Calculation of eddy current distribution
According to the theoretical analysis results of eddy current field in Section 2.2,
when the permanent magnet moves through the metal tube, induced current density J
in the metal tube only has tangential component Jθ in each vertical cross section. In
order to directly observe eddy current density distribution, validate theoretical
analysis results and obtain eddy current loss distribution, in this section, eddy current
distribution is obtained using FE method. Parameters needed in FE calculation are
listed as shown in Tab. 3.2
Table 3.2 Parameters for FE eddy current calculation
Items
Value
Magnet height h
30mm
Magnet diameter d
20mm
Remanent magnetic density Bm
1.4T
-6
0.01810 Ω/m
Resistivity of metal tube ρ
Inner diameter of metal tube Din
26mm
Outer diameter of metal tube Dout
32mm
Length of metal tube sm
500mm
Magnet velocity v
0.0941m/s
The three-dimensional model of the permanent magnet and metal tube is shown
in Fig 3.7(a), and the magnetic field produced in the metal tube when the permanent
magnet drops at a constant speed is calculated by transient field solver. As shown in
3.7 (b),eight vertical lines evenly distributed along a cylindrical surface which have a
distance of 14.5mm from the center axis of the tube are selected to analyze the eddy
current density.
15
(a) Model of eddy current calculation
(b) Eddy current distribution
Fig. 3.7 Model and distribution of eddy current calculation
It can be seen from Fig 3.7 (b), the eddy current distribution is symmetrical
relative to the horizontal center plane of the metal tube, and the eddy current is
directed along the tangent line. In upper half of the permanent magnet, eddy current
density is clockwise while in lower half anticlockwise.
According to the calculation results of eddy current field, eddy current loss
density in two ring surfaces which respectively have a distance of 13mm and 16mm
from the center axis is further calculated and the results are shown in Fig 3.8.
Fig 3.8 Loss density distribution in two ring surfaces
The equation to calculated eddy current loss density is expressed as
Peddy-current   J 2
16
(3-4)
Substituting (2-24) into (3-4), (3-5), we have
Peddy-current   J 2  ( Br v)2 
(3-5)
It can be seen from Fig 3.8, there is a positive correlation between eddy current
loss and the square of the component of magnetic field in radial direction and the
eddy current distribution is symmetrical relative to the horizontal central plane of the
metal tube. Besides, the eddy current loss density evidently decreases with distance
between permanent magnet and metal tube increasing.
According to the energy conservation law, under the electromagnetic force
permanent magnet endures caused by eddy current in the metal tube, the essence of
phenomenon of permanent magnet eddy current brake in metal tube is the process in
which mechanical energy of permanent magnet is transformed into internal energy of
the metal tube. And the transformation process mainly occurs between the permanent
magnet and its surrounding metal tube. Besides, the metal tube should have proper
thickness and it is no use choosing too thick metal tube to improve braking force.
17
V. Experiments and Results
In order to verify the correctness of the theoretical analysis, and further analyze
the motion of the permanent magnet in the metal tube. In this chapter, an experimental
system was established and an experimental force analysis was carried out. The
electromagnetic force expression calculated in the chapter 2 is written as
F  k 
2πrBr 2 ( x, z )v

Sa
dS
(4-1)
By (4-1) and Newton's first law, the speed can be derived as
v
F 
2π  rBr 2 ( x, z )dS
.
(4-2)
Sa
The equation (4-2) and (4-1) are equivalent, Four groups of experiments are
designed respectively to verify the correctness of above theories
1) Ensuring that the size and material of permanent magnet and metal tube are
changeless, analysis the relationship between the electromagnetic force and
velocity of permanent magnet.
2) Ensuring that the size of permanent magnet and metal tube are changeless, change
the metal resistivity by changing the metal material.
3) Ensuring that the metal material is changeless, change the magnetic field
distribution in the metal tube by changing the height of the permanent magnet or
the size of the metal tube.
4) Ensuring that the size and material of the metal tube are changeless, change the
initial velocity of the permanent magnet when it begin to fall into the metal tube
by changing the initial position of falling permanent magnet.
Based on the finite element method mentioned in the third chapter, we made the
theoretical calculation of ∫∫SarBr2(x,z)dS with finite element software, and then
compared the experimental results with the theoretical values.
4.1 Description of experimental system and force analysis
Based on the experimental system shown in Fig. 4.1, we conduct research on the
braking phenomenon of the permanent magnet in the metal tube. Experimental
operating system mainly includes the coils fixed on iron stand, permanent magnet,
vertical mental tube, photoelectric encoder, and pulleys. To ensure the initial space
position of permanent magnet beginning to fall, we place a plastic tube with smooth
inner wall between the coils and the metal tube.
18
Fig. 4.1 Experimental system
The permanent magnet is connected with the two upper and lower pulleys by the
light cotton thread. As shown in Fig. 4.2, by adjusting the output current of DC power
supply, make the permanent magnet suspended in a fixed position by the
electromagnetic force many times in the experiments. Disconnecting the DC power
supply, the permanent magnet begins to fall by gravity, and at the same time the light
cotton thread makes the pulleys rotate.
Fig. 4.2 The permanent magnet suspended in energized coils
The facility for measurement of velocity is composed by the photoelectric rotary
encoder coaxially connected with the upper pulley, DSP development board and
oscilloscope DLM2024. Encoder output the pulse signal based on the shaft speed, the
digital signal processor TMS320F28335 on the development board transfers the pulse
19
signal to the speed digital signal, and then the 12-bits DAC chip DAC7724 on the
development board converts the processed digital signal into the voltage signal
recorded by the oscilloscope. With the pulley radius of 0.02653m, we can achieve the
permanent magnet velocity linearly corresponding to the voltage displayed on the
oscilloscope. The update frequency of permanent magnet velocity in the experiments
reaches 40 kHz.
By adjusting the output current of DC power supply, make the lower surface of
permanent magnet aligned over the lower plane of energized coils, and the initial
position of the permanent magnet relative to the metal tube can be changed by
adjusting the vertical position of energized coils. Ignoring the shaft friction of encoder,
without slipping between the light cotton thread and pulleys, the force analysis of
experimental device is shown in Fig. 4.3.
Fig 4.3 Force analysis of movement device
In Fig. 4.3, fc1 and fc2 are respectively the force of static friction acting on the two
upper and lower pulleys by the thread, Fe is the electromagnetic force acting on the
permanent magnet, mg is the gravitational force acting on the permanent magnet.
Assuming that the acceleration of permanent magnet is a, and Irotary, β, R denote
respectively the moment of inertia, angular acceleration and radius of the pulleys,
according to the kinematics equation, we have
ma  mg  Fe  T2  T1

2  I rotary   f c1  f c2  R

a   R
20
(4-3)
Ignoring the mass of the light cotton thread, we have
T1  T3  f c1

T3  T2  f c2
(4-4)
2 I rotary 

mg  Fe   m 
a
R2 

(4-5)
By (4-4) and (4-3), we have
4.2 Relation between velocity and electromagnetic force
By (4-5), when the 2Irotary/R2 and the mass of permanent magnet m are acquired,
the electromagnetic force Fe can be obtained by measuring the acceleration a.
In order to measure the value of 2Irotary/R2 in (4-5), removing the metal tube in
experimental device, the permanent magnet doesn’t endure the electromagnetic force,
and the equation (4-5) can be written as
2I


mg   m  rotary
a
2
R 

(4-6)
By calculating the acceleration of the falling permanent magnet a, combined with
the measured mass of permanent magnet m, the value of 2Irotary/R can be calculated.
The velocity curve of permanent magnet measured in the experiment is shown in Fig.
4.4.
Fig 4.4 v-t curve without metal tube
As shown in Fig. 4.4, we can acquire the acceleration a = 5.461 m/s2 through the
curve fitting method. By (5-7), we can obtain the calculated value of 2Irotary/R2 equals
0.0523 kg with m = 0.06583 kg and g = 9.8011 m/s2.
21
In order to validate the relation between the velocity and electromagnetic force on
the permanent magnet, under the condition that the materials and dimensions of
permanent magnet and metal tube have no changes, velocity of the permanent magnet
is measured surrounding a fixed position the metal tube. The experimental results are
shown in Fig. 4.5.
Fig 4.5 The velocity curve of permanent magnet surrounding a fixed position
Fitting the speed line recorded in the experiment, we can obtain relation between
speed and time in form of a polynomial.
v  9095t 3  179.1t 2  3.325t  0.4376.
(4-7)
By taking the derivative of t in the polynomial in (4-7) we can get the
acceleration a of the permanent magnet. According to (4-5), we can calculate the
electromagnetic force Fe. From Fe/v curve in Fig 4.5, it can be seen that Fe/v keeps a
constant whatever the velocity is, verifying the linear relation between speed and
electromagnetic force of the permanent magnet surrounding a fixed position.
4.3 Experiment of changing the resistivity
In order to validate the influence of resistivity on falling velocity of the
permanent magnet and the electromagnetic force on it, we make the permanent
magnet falls through T2 copper tube, H62 brass tube and 6061 aluminum alloy tube
respectively. As shown in Fig. 4.6, the outer diameters of the metal tubes are all
32mm, inner diameters are all 26mm and lengths are 50mm. The distance from the
initial position of the center of the permanent magnet to the inlet of metal tube Δs
keeps 0.1m. Referring to related books we know the resistivity of three kinds of metal
tubes ρT2=0.01810-6Ω/m，ρH62=0.07110-6Ω/m，ρ6061=0.0410-6Ω/m, and the velocity
curves measured are shown in Fig. 4.7.
22
Fig 4.6 The metal tube with three kinds of materials
Fig 4.7 The v-t curve with different metal resistivity
23
According to the experimental results shown in Fig. 4.7, the whole movement
process can be divided into five parts:
① Free falling process.
② Deceleration by electromagnetic force.
③ Uniform motion of permanent magnet in the metal tube.
④ Acceleration with the electromagnetic force reduced at the end of the tube.
⑤ Free falling process.
In the three kinds of metal tubes, the acceleration in process ① and ⑤ are all
the same, so it shows that at this process there exists no electromagnetic force. In the
process ②, ③, ④ permanent magnet is under the effect of electromagnetic force.
With the different resistivity of metal tube, the velocities during the processes of
uniform motion and deceleration both change evidently.
Through the analysis of uniform motion process, we can obtain the velocities of
the permanent magnet in different tubes are vT2=0.0869m/s, vH62=0.3284m/s,
v6061=0.1825m/s respectively. When permanent magnet moves at a constant speed,
according to (4-5), it can be known that Fe=mg. Substituting it into equation (3-3), and
calculating Z with the discrete calculation method proposed in Section 3.2, theoretical
value of velocity at uniform motion process can be obtained as
v
mg
.
2πZ
(4-8)
As listed in Tab. 4.1, theoretical values contrast with experimental results, and
percentages of relative errors are calculated.
Table 4.1 The contrast between calculated and measured velocity
with different metal resistivity
T2 copper
Calculated velocity
vc (m/s)
0.0941
Measured velocity
ve (m/s)
0.0869
Relative error
(vc- ve)/ vc (%)
7.65
H62 brass
0.3864
0.3284
15.01
6061 aluminum alloy
0.2176
0.1825
16.13
Materials of tube
It can be seen from the Tab. 4.1, theoretical calculation values of permanent
magnet falling velocity are almost the same with the experimental results. Due to the
friction between pulleys and shafts in the experiment and the fact that the permanent
magnet can not be guaranteed to move along the central axis precisely which results
the friction between the permanent magnet and metal tube, the experimental
measurements are all slightly smaller than the theoretical values.
24
4.4 Experiment of changing the magnetic field distribution
The theoretical analysis shows that when the diameter of the metal tube or the
dimension of permanent magnet varies, magnetic field distribution of the metal tube
changes, resulting in the changes of electromagnetic force and the motion velocity. In
order to validate the influences of the magnetic field distribution in the metal tube on
the motion process, metal tubes of different dimensions and permanent magnet of
different heights are chosen for experiments. The different diameters of tube and
dimensions of permanent magnets used in each experiment are listed in Tab. 4.2, and
the material of metal tube is all T2 copper.
Table 4.2 Four combinations of permanent magnets and metal tubes for experiments
Inner diameter
Outer diameter
Height of PMs
Diameter of PMs
Din(mm)
Dout(mm)
h(mm)
d(mm)
a
26
32
30
20
b
28
32
30
20
c
26
32
50
20
d
28
32
50
20
Test group
In the experiment, the distance between the permanent magnet center and the tube
inlet is Δs=0.1m. Velocity curves measured are shown in Fig. 4.8.
Fig 4.8 v-t curve of four combinations of permanent magnets and metal tubes
25
According to the experiment results shown in Fig. 4.8, we can get the velocities
of permanent magnets in the uniform velocity process are respectively va = 0.0869m/s,
vb = 0.1634 m/s, vc = 0.1211 m/s, and vd = 0.2231 m/s. The masses of the permanent
magnets are m1 = 0.0658 kg, m2 = 0.1102 kg respectively. Similarly, theoretical
values of velocity in uniform speed process are calculated by using the discrete
calculation method proposed in Section 3.2. The theoretical values and experimental
results are contrasted in Tab. 4.3 and the percentages of relative errors are calculated.
Table 4.3 The contrast between calculated and measured velocity with different dimensions
Measured velocity
relative error
Calculated velocity
Test groups
ve (m/s)
(vc- ve)/ vc (%)
vc (m/s)
a
0.0941
0.0869
7.65
b
0.1645
0.1634
0.67
c
0.1403
0.1211
13.68
d
0.2311
0.2231
3.46
According to the results shown in Tab. 4.3, theoretical calculation values of the
velocity in uniform speed process basically coincide with the measured values in the
experiment. The reason that the calculated velocities are slightly higher than the
measured ones is existence of friction between the permanent magnet and the metal
tube in the motion process. It can be found the relative error with the inner diameter
being 26mm is larger. This is because the smaller the inner diameter is, the more
easily friction between permanent magnet and metal tube occurs in the process.
4.5 Experiment of changing the initial position
Observing the measured velocities in the Section 4.3 and Section 4.4, it can be
found that in permanent magnet falling process, acceleration will change considerably
at a certain position, which shows that permanent magnet is enduring a strong
electromagnetic force at this moment.
Once the dimensions and materials of the metal tube and the permanent magnet
are determined, velocity of permanent magnet in uniform speed process is a constant
value, and the magnitude of the electromagnetic force is determined by the velocity
and position of the permanent magnet. To study the influence of different initial
positions of falling permanent magnet on motion state, the distances from permanent
magnet center to metal tube inlet are respectively set to 0.05m, 0.1m and 0.15m,
which is realized by adjusting the position of the energized coil, and in the experiment
the tube is a fixed-specification T2 copper tube whose inner diameter is 26mm and
out diameter is 32mm. The velocity of permanent magnet is measured and the
displacement of permanent magnet is calculated by using the integral method.
Velocity-time (v-t) curve and displacement-time (s-t) curve of the permanent magnet
are obtained as shown in Fig. 4.9.
26
Fig 4.9 v-t curves and s-t curves with different initial positions of the permanent magnet
27
As shown in Fig. 4.9, the coordinate value of displacement highlighted by red
horizontal line is the distance between the initial positions of permanent magnet
center and the inlet of the tube. As shown in Fig. 3.3, nearby the top and bottom
surfaces of the permanent magnet, the amplitude of radial component Br is obviously
higher than other positions. Hence, the permanent magnet starts to endure a strong
electromagnetic force at a certain position nearby the inlet of the metal tube.
The displacement curves in Fig. 4.9 show that although the position where the
permanent magnet begins to fall is different, after the displacements of 0.027m,
0.077m and 0.127m respectively, the motion of permanent magnet changes from
acceleration to deceleration. And at this time, distances from permanent magnet
center to inlet of metal tube are all 0.023m. So the experimental phenomenon
coincides with theoretical analysis well.
In addition, in the three experiments where the initial position of the permanent
magnet is varied, time used in the deceleration process is 0.21s, 0.12s, and 0.26s
respectively, and the initial velocities of permanent magnet in the deceleration process
are 0.547m/s, 0.920m/s, and 1.205m/s. Velocities of permanent magnet in the uniform
process are 0.0865 m/s, 0.0876 m/s, and 0.0867 m/s. The initial velocity of permanent
magnet in Fig. 4.9(b) is between those in Fig. 4.9(a) and Fig. 4.9(c), but the
deceleration time in Fig. 4.9(b) is below those in Fig. 4.9(a) and Fig. 4.9(c)
significantly.
To sum up, for the velocity curves shown in Fig. 4.9(a) and Fig. 4.9(b), the
deceleration time is mainly affected by electromagnetic force, so the deceleration time
of permanent magnet shown in the Fig. 4.9(b) is shorter than that in Fig. 4.9(a). And
for the velocity curves shown in Fig. 4.9(b) and Fig. 4.9(c), the deceleration time is
mainly affected by the initial speed in the deceleration process, so the deceleration
time of permanent magnet shown in the Fig. 4.9(b) is shorter than that in Fig. 4.9(c).
28
V. Conclusions and prospects
To research on permanent magnet’s braking phenomenon when it moves in a
nonmagnetic metal tube, theoretical analysis and modeling, numerical calculation and
experimental validation in the thesis are concluded as follows.
1) Magnetic field distribution of the permanent magnet is analyzed and expression
for magnetic induction density is derived. Besides, eddy current distribution law is
studied and eddy current density in a vertical cross section of the metal tube is
calculated. Based on the analysis results of permanent magnet’s magnetic field
and metal tube’s eddy current, electromagnetic force on the permanent magnet is
modeled.
2) Magnetic field of permanent magnet, eddy current of metal tube and eddy current
loss are observed directly with the help of finite element software, and the
distribution law of magnetic induction density is obtained spatially. Conclusions
and equations coincide with each other well.
3) With the help of finite element software, a discrete calculation method is proposed
to simplify the surface integral in expression of electromagnetic force. Theoretical
values obtained by the discrete calculation method coincide with experimental
results basically.
4) Experimental platform is established and four groups of experiments are designed.
Movement processes of permanent magnet falling through a metal tube under
different conditions are observed and the results validate the correctness of the
theoretical analysis.
Permanent magnet
Metal tube
Fig 5.1 Schematic diagram of elevator dropping accident prevention device
The results of research on permanent magnet’s braking phenomenon when it falls
through a nonmagnetic metal tube could be expected to be applied to practical
occasion where braking is needed. Taking the commonly used elevator in the daily
life as an example, as shown in Fig 5.1, a nonmagnetic metal tube could be installed
29
at the bottom of the elevator shaft, and a permanent magnet could be installed at the
bottom of the elevator cabin. Then when the cabin is suddenly dropping, a serious
accident could be avoided due to the buffering capability of the system containing a
permanent magnet and a metal tube. In a word, the system could be expected to serve
as an auxiliary braking device.
In addition, as shown in Fig. 5.2, according to the result of this paper, a
permanent magnet eddy current brake is also invented during this research process.
The novel brake device produce braking torque by the electromagnetic force
generated by reciprocating motion of permanent magnet in a seamless nonmagnetic
metal tube. This novel eddy current brake can have advantages for compact structure,
long service life, energy conservation and environmental protection.
Fig 5.2 A kind of permanent magnet eddy current brake device
(Applying for Chinese invention patent, No.201410675449.8)
30
Acknowledgement
We would like to show sincere gratitude to our physics teacher Mr. Wei Song for
his daily instruction and assistance. The improvement of our research ability is closely
connected to his patient cultivation and it plays a key role in our research. We also
would like to express gratitude to our extramural academic advisers Mr. Xinmin Li
and Mr. Siyu Yang from Tianjin University. They spare no effort to care about our
research, provide us some experimental operation training and experimental
equipments, such as the digital oscilloscope.
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