International Journal of Applied Electromagnetics and Mechanics 33 (2010) 243–249
DOI 10.3233/JAE-2010-1119
IOS Press
243
Attractive and repulsive forces of
ferromagnetic materials in time-varying
electromagnetic fields
Ronggang Cao∗ , Zhang Fang, Jun Zou and Jiansheng Yuan
State Key Lab of Power Systems, Tsinghua University, Beijing, 100084, P.R. China
Abstract. Electromagnetic forces of ferromagnetic materials consist of attractive and repulsive components, which can be
introduced from the Lorentz force formula and the Kelvin force formula. In harmonic fields, the attractive and the repulsive
forces both have DC and AC components and they cancel out with each other to reduce the total average force. The contributions
or influences of the frequency, magnetic permeability and electric conductivity of the material to attractive and repulsive forces
in time-varying fields are analyzed and simulated by the numerical experiments using FEM methods. Results show that the
total time average electromagnetic forces can be reduced greatly, even to zero in certain states, which is useful to control the
vibration problem.
Keywords: Electromagnetic forces, ferromagnetic materials, time-varying electromagnetic fields
1. Introduction
In time-varying electromagnetic fields, electromagnetic forces of ferromagnetic materials can be
classified to two components, deduced from Lorentz and Kelvin force [1,2], i.e. the repulsive and
attractive forces which respectively reflect the eddy current and the magnetization phenomenon. In
harmonic fields, two force components both have DC and AC components. Precisely speaking, Lorentz
force describes the force caused by the charge, including static charges and dynamic charges, i.e. the
current, and it can be expressed in the form F = ρu E + J × µ0 H . And especially for the conduct
problems, the former term is much less than the latter one. Kelvin force describes the force caused by the
polarization and the magnetization, and it can be expressed as F = (P · ∇)E + µ0 M · ∇H . Especially
for the ferromagnetic materials, the last formula’s first term can be cancelled.
The Lorentz and the Kelvin force constitute the total force of an object in the electromagnetic fields.
However, there are many other definitions for the force classification and calculation methods, such as
the Lorentz-Kelvin force density expression [2], the Korteweg-Helmholtz force density expression [1–3],
the magnetic charge method [4,9], the magnetizing currents method [5,9], the maxwell tensor method [1,
6], and the virtual work method [6,7,10]. The Lorentz-Kelvin force density formula can be got by adding
the Lorentz force density and the Kelvin force density. The Korteweg-Helmholtz force density formula
can be deduced from the Lorentz-Kelvin force density. Compared with the Lorentz-Kelvin force density
1
∗
Project 50677028 supported by NSFC.
Corresponding author. Tel.: +86 10 62791807; E-mail: [email protected].
1383-5416/10/$27.50  2010 – IOS Press and the authors. All rights reserved
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R.G. Cao et al. / Attractive and repulsive forces of ferromagnetic materials in time-varying electromagnetic fields
formula or the Korteweg-Helmholtz force density formula, the maxwell tensor and virtual work methods
are usually used to calculate the total force of an object. For the rigid and linear materials, all of these
methods give the same results. However, it would be different for the nonlinear materials [8].
In harmonic electromagnetic fields, the forces contain AC and DC components. For the DC component,
the first term in the Korteweg-Helmholtz formula describes the repulsive force, which is related to the
eddy current, and the second term describes the attractive force, which is related to the magnetization.
The DC components of the attractive and the repulsive forces cancel out with each other. And the total
time average electromagnetic forces can be reduced greatly, even to zero in certain states.
2. Analytical formulas of electromagnetic forces for ferromagnetic materials
Basically, there are two kinds of expressions for the total force, which are the Lorentz-Kelvin (LK)
and the Korteweg-Helmholtz (KH) force density formulas as follows:
F LK = J × µ0 H + µ0 M · ∇H
F KH = J × B −
1
H · H∇µ
2
(1)
(2)
Equation (2) can be derived from Eq. (1) for the rigid materials [2]. The first term of Eq. (2), denoted
by FJ , reflects the repulsive force in response to the eddy currents, and the second one (denoted by FM )
reflects the attractive force in response to the magnetization.
In harmonic fields, the current density and the magnetic flux density could be expressed as follows:
√
(3)
J (t) = 2J sin (ωt + ϕJ ) · aJ
√
B (t) = 2B sin (ωt + ϕB ) · aB
(4)
Thus the repulsive and the attraction force can be expressed as:
F J = BJ cos (ϕB − ϕJ ) sin (θ) · aJ×B − BJ cos (2ωt + ϕB + ϕJ ) sin (θ) · aJ×B
= F JDC + F JAC
1
1
F M = − H 2 ∇µ + H 2 cos (2ωt + 2ϕH ) ∇µ
2
2
= F M DC + F M AC
(5)
(6)
In Eq. (5), aJ × aB = sin(θ) · aJ×B . From Eqs (5) and (6), we can see that the repulsive and the
attractive forces both own the AC and DC components. From Eq. (5), we have
|F JDC | = |BJ sin (θ) · cos (φB − φJ )|
(7)
|F JAC | = |BJ sin (θ)|
(8)
So that,
|F JDC | 6 |F JAC |
(9)
R.G. Cao et al. / Attractive and repulsive forces of ferromagnetic materials in time-varying electromagnetic fields
245
Fig. 1. Axisymmetric model of a solenoid coil and a ferromagnetic sheet.
The direction of the DC component of F J is repulsive, and its magnitude is not more than the AC
component. Thus, the waveform of the total force can be positive or negative in the time domain.
According to Eq. (6),
1 2 |F M DC | = |F M AC | = H ∇µ
(10)
2
The direction of the DC component of F M has the attractive force’s direction, and its magnitude is equal
to the AC component. Thus, the waveform of the total force always shows the attractive force’s direction.
For the ferromagnetic materials, the DC and AC components of repulsive and attractive forces can
be cancelled each other. In the proper condition, the total forces can be reduced to zero. From Eqs (5)
and (6), it shows that the DC components of F J (denoted by F J DC ) and F M (denoted by F M DC ) can
cancel out, while the AC components are depending on the phases.
3. Numerical simulations of the forces for ferromagnetic materials
To investigate the cancellation effect of the forces, numerical simulations are conducted for an axisymmetric model that is a solenoid coil with a ferromagnetic sheet, as shown in Fig. 1. The solenoid is
5cm high, with 4 cm interior radius and 15 cm exterior radius. The cylindrical sheet is 0.5 cm thick, with
15 cm radius. The sheet’s surface facing to the solenoid is 1 cm far away from the solenoid end surface.
It is supposed that the solenoid has the strand currents and all the materials work in the linear states.
Then the simulation results in different situations are given below. There are two popular methods to
calculate the total force, which are the tensor method and the virtual work method [3,4]. The magnetic
field of the model is obtained by the finite element software ANSOFT MAXWELL SV, and the force on
the sheet is also provided by the software, which employs the virtual work method. The finite element
mesh and the magnetic lines of force in a situation are given in Fig. 2.
3.1. Only with the repulsive or the attractive force components
To simulate the repulsive force, the relative permeability is set to 1, because the attractive forces caused
by the magnetization must be zero. And to simulate the attractive force, the conductivity is set to zero,
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R.G. Cao et al. / Attractive and repulsive forces of ferromagnetic materials in time-varying electromagnetic fields
Table 1
Total average force with different relative permeability (3.8 × 107 S/m, 500 Hz)
Relative permeability
200
400
600
(646)
800
Total average force(N)
−54
−22
−3
0
10
(a)
(b)
Fig. 2. (a) Finite element mesh (local region). Fig.2. (b) Magnetic lines of force.
because the repulsive forces caused by the eddy currents must be zero.
Suppose the ampere-turn of the solenoid is 2 × 104 A. The results of repulsive forces (relative permeability is 1) are as follows:
–
–
–
–
For 3.8 × 107 S/m and 50 Hz, the F JDC of the sheet is −141N, and |F JAC | is 158N.
For 3.8 × 107 S/m and 500 Hz, F JDC is −177N and |F JAC | is 178N.
For 50 Hz and 1 × 107 S/m, F JDC is −38N and |F JAC | is 82N.
For 50 Hz and 3 × 107 S/m, F JDC is −130N and |F JAC | is 150N.
It is easy to understand that the higher frequency or conductivity will bring higher DC or AC components
of the forces, and from the results we can see that the magnitudes of the DC and AC components will
get closer each other.
The results of attractive forces (electric conductivity equals 0) are as follows:
– For 50 Hz and 1000 (relative permeability), F M DC is 134N and |F M AC | is 134N.
– Other results show that F M DC and |F M AC | do not change with the frequency.
The calculation results show that the magnitude of the DC and AC components are always the same.
3.2. With both the repulsive and the attractive force components
In order to get the values of permeability, frequency and conductivity, which make the total force zero,
we can fix one of them, and adjust the others. Table 1 shows the results.
The result show that the attractive and repulsive magnetic forces can be cancelled each other and the
total average is zero with the proper frequency, permeability and conductivity. For the certain model, the
R.G. Cao et al. / Attractive and repulsive forces of ferromagnetic materials in time-varying electromagnetic fields
247
Table 2
The aluminium sheet’s total average force
with frequency (µr = 1, σ = 3.8 × 107 S/m)
Frequency(Hz)
5
100
1000
Total average force(N)
−7
−166
−178
Table 3
The copper sheet’s total average force with
frequency (µr = 1, σ = 5.8 × 107 S/)
Frequency(Hz)
5
100
1000
Total average force(N)
−15
−172
−180
Table 4
The steel sheet’s total average force with
frequency (µr = 4000, σ = 1.03 × 107 S/m)
Frequency (Hz)
5
100
1000
Total average force (N)
33
30
22
null force can also occur with 3.8 × 107 S/m, 5 kHz, and 6400, or 5.8 × 107 S/m, 500 Hz and 1070, and
so on.
3.3. Total average force on the sheets made of practical materials
Actually, the permeability and the conductivity of the certain material can not be changed arbitrarily.
Therefore, copper, steel and aluminium sheets are analyzed following. The model structure is the same
as above, and the ampere-turn of the solenoid is 2 × 104 A. Tables 2, 3 show the results for copper and
aluminium. The results show that the total average forces are always repulsive and the force of the copper
sheet is larger than the aluminium sheet’s.
For the steel sheet, the ampere-turn of the solenoid is set to 1 × 104A, and the results are given in
Table 4. It shows that when the frequency is low, the total average force is attractive, and when the
frequency is getting higher, the repulsive component becomes larger. Thus, the average total force could
be zero in a certain state.
3.4. Two layers sheet with different materials
If the sheet is made up of two different material sheets, it is easy and practical to make the average
total force of the sheets be zero. It is supposed that the structure of the model is similar to the above one.
While, it is still possible to get the average total force to be zero. Supposing the solenoid is 20 cm high.
The interior radius is 6 cm and the exterior radius is 8 cm. The cylindrical sheet is made up of aluminium
and steel, and its radius is 8 cm. The aluminium sheet is nearer than the steel sheet to the solenoid. The
aluminium sheet’s surface facing the solenoid is 1 cm far way the end surface of the solenoid.
The ampere-turn of the solenoid is 1 × 104 A and the aluminium sheet is 0.5 cm thick and keeps
unchanged in the calculation. The different configuration of the steel sheet’s thickness can influence the
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R.G. Cao et al. / Attractive and repulsive forces of ferromagnetic materials in time-varying electromagnetic fields
Table 5
The sheets with aluminum and steel layers (50Hz)
The radius of the aluminum sheet(cm)
5
(8)
10
15
Total average force(N)
20
0
−14
−38
average total force. When the frequency is 18 Hz, the calculations show that the average total force is
zero with the thickness of the steel sheet is 0.43 cm.
3.5. Other sheet configurations
Besides of changing the thickness of the sheet, it is possible to change the shape of the sheet.
Supposing the two layers sheet is made up of aluminium and steel, and both sheets’ thickness is 0.5cm.
The aluminium sheet is nearer than the steel one to the solenoid, and the upper surface of the aluminium
sheet is 1cm far away the lower surface of the solenoid. The radius of the steel sheet is 15cm, and the
radium of the aluminium sheet is changed. The ampere-turn of the solenoid is 1 × 104A, and the results
are shown in Table 5.
Although the total average force of the two layers sheet can be fixed to zero, the force of the each layer
sheet can be very large. If the total forces of two layers are both have the direction to the inner of the
sheet, the fixation of the sheets would be convenient.
4. Conclusions
Electromagnetic forces of ferromagnetic materials own attractive and repulsive force components,
which are respectively related to the magnetization and the eddy currents. The analysis and numerical
simulations show that the attractive and the repulsive force components can cancel out, and the total
average force can decrease to zero. Because the relative permeability and the conductivity of the certain
material can not be changed arbitrarily, practically, the sheet can be made up of several layers with
different materials, which may get the state to make the total average force of the sheet to zero. It is
useful to deal with the problem of the vibration of the materials in time-varying electromagnetic fields.
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