ISSN 1843-6188
Scientific Bulletin of the Electrical Engineering Faculty – Year 11 No. 2 (16)
PERFORMANCE ANALYSIS OF A SOLENOIDAL ELECTROMAGNET
D. CAZACU, C. STANESCU
University of Pitesti, no. 1, Targul din Vale, Pitesti, 110040
E-mail: [email protected]
Abstract. In this paper a dc solenoidal electromagnet was
considered. The magnetic forces were measured and the
static characteristic was obtained. Also the forces were
computed with finite element method using the Maxwell
stress tensor method and compared. Good agreement was
noticed among measured, computed and manufacturer
characteristics.
Magnetic forces fv, within a certain volume vΣ can be
reduced to a system of fictitious stresses acting on the
surface Σ that bounds the considered volume.
Equivalent condition is that the resultant force must be
the same:
F = ∫ fν dV = ∫ Tn dA
Keywords: electromagnet, finite element method, Maxwell
stress tensor method
1.
VΣ
where Tn is magnetic Maxwell stress tensor.
The volume magnetic force density f v
magnetostatic field has de following expression:
INTRODUCTION
Computing the magnetic forces is one of the most
important issues within designing of the electromagnetic
devices such as: actuators, sensors and electric motors.
Electromagnetic actuators (EMA) became widespread
used because of their rugged construction, low cost,
simple structure, relative high density force and ease of
control. Solenoidal electromagnets (with plunger) are
used in electromagnetic valve actuation systems, fuel
injection actuation, exhaust gas recirculation systems,
refrigerators, and washing machines.
Electromagnets, and the variety of devices in which they may
be mounted, have been studied intensively to achieve high
quality standards. However, electromagnets are complex
electromechanical systems, whose behavior is governed by
an interaction of magnetism, electricity and mechanics. As a
consequence, analysis and design of such systems, and the
calculation
of
their
parameters,
is
inherently
multidisciplinary, demanding interdisciplinary knowledge.
Forces can be derived from field solutions obtained from
numerical computation, using finite element or finite
difference methods.
Different techniques for force techniques are available in
numerical modeling: the Maxwell stress tensor method,
the virtual work method and the equivalent sources
method [1]-[5].
In this paper experimental and numerical static
characteristics of a solenoidal electromagnet ITS-LS
2924 are obtained and compared.
Also the static characteristics determined numerically and
experimental were compared with the manufacturer results.
2.
(1)
Σ
in
a
1
1
dµ 

f v = J × B − H 2 grad µ + grad  H 2τ
 (2)
2
2
dτ 

where:
- The first term J × B is the volume magnetic force
density exerted, in a magnetostatic field B , on
media in electrokinetic regime.
1
- The second term − H 2 grad µ represents the
2
volume magnetic force density exerted on magnetic
heterogeneous media.
1
dµ 

- The last term
grad  H 2τ
 it is called the
2
dτ 

volume density of the magnetostriction force.
In a linear, homogenous, isotropic and non-compressible
ferromagnetic material, the force is concentrated on the
surface and the surface force density is given by:
1
f S = (υo − ν ) Bn2 − ( µ0 − µ ) H t2  n
(3)
2
where µ=1/v is the magnetic permeability of the material
and n is the outer normal vector.
This expression can be derived from the second term of
the volume force density (2). On the surface of the
material we can write:
B2
1
fV = − ( H t2 + n2 ) grad µ =
2
µ
(4)
1
= − ( H t2 grad µ − Bn2 gradν )
2
Since only the normal variation of µ is different from
zero, surface force density expression (3) is obvious.
THEORETICAL ASPECTS
To calculate the force exerted on the plunger, the
Maxwell stresses or the virtual work method can be used
[6] - [12].
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Scientific Bulletin of the Electrical Engineering Faculty – Year 11 No. 2 (16)
ISSN 1843-6188
The virtual work method for computing force and
torques is derived from magnetic energy Wm or
coenergy Wm’ changes against space displacement.
The expressions of magnetic forces are:
 ∂Wm 
F = −

 ∂δ Φ= ct
(5)
 ∂Wm 
F =

 ∂δ i = ct
(6)
Figure 1. The dc solenoidal electromagnet ITS-LS 2924
where Wm is magnetic field energy and δ is air gap of the
electromagnet. Assuming a constant magnetic flux, the
expression of force is:
d Λδ
1
F = (U mm )2
2
dδ
3.2 Experimental measurements
A laboratory device for determining the static
characteristic of the plunger electromagnet in described
in Figure 2.
(7)
where: Umm is magnetomotive force of the coil and
Λδ
is air gap permeance.
In the case of high magnetic permeability (ferromagnetic
material), a simplified expression for the force can be
obtained:
1
N 2 I 2 A0
.
F = − µ0 ⋅
2
δ2
(8)
From relation (8), some observations can be drawn:
- electromagnetic force is directly proportional to cross
section of the plunger;
- force is reverse proportional to the square air gap
length, and it tends to reduce the air gap;
- being an attraction force between two magnetic parts,
the mobile and fixed ones, a steady state is obtained
only for a minimum value of magnetic energy.
Figure 2. Measurement equipement to determine the static
characteristic of plunger electromagnet: 1. countertops 2.
fixed table 3. fixer 4. support colon 5. feed device 6.
dynamometer table 7. dynamometer 8. fixing device of the
comparator clock 9. comparator clock 10. controller 11.
connection terminals
3. EXPERIMENT
3.1 Studied device
In order to measure the air gap we choose a comparator
clock Ultra Prazision, which has the measurement range
0 to 30 mm.
In order to measure the force a precision dynamometer
with strain mark placed on a mobile plate was chosen.
We measured pull forces.
The scale can be set on 0 in any position by pressing
ON/TARE button, avoiding to introduce errors in
measurement, such as adding to the magnetic force the
weight of the plunger.
The experimental characteristic is shown in Figure 3.
A dc solenoidal electromagnet (with plunger), produced
by INTERTEC Components GmbH, was studied. It is
presented in Figure 1.
It has the following characteristics: power supply 12 V
dc, number of turns N = 288 and the connection current
I=0.33 A.
In this electromagnet with limiter type, the air gap is
between two truncated cone surfaces, and the force
exerted is pulling.
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ISSN 1843-6188
Scientific Bulletin of the Electrical Engineering Faculty – Year 11 No. 2 (16)
8
Caracteristica B(H)
7
6
1600
F [N ]
5
1400
4
1200
3
1000
2
800
1
0
600
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
400
δ
200
Figure 3. The experimental characteristic
0
0
200
400
600
800
1000
1200
1400
3.3 Numerical models
Figure 5. Characteristic B(H) for the ferromagnetic
material
The electromagnet can be designed using analytical
magnetic circuits technique but improved accuracy is
obtained by using finite element analysis.
2D finite element simulations were performed using
Comsol Multiphysics ver.3.4 software program,
Force calculations are effected by the approximated
nature of the finite element method.
In the Maxwell stress method for the force calculation,
the stress distribution arises from the meshes used to get
the field solution.
Copyright© by COMSOL AB. The AC/DC module,
magnetostatics sub-module was used.
The program solves a Poisson equation in order to find
the magnetic potential:
∆A = − µ J
The total force is found by setting up a closed surface
around the domain of interest and integrating the stresses
over the whole surface.
(3)
In order to find the static characteristic, the forces
exerted on the plunger were obtained for different
positions of it.
In order to obtain a good accuracy adaptive finite
element mesh was used.
In Figure 6 the finite element initial mesh of 6358 of
Lagrange elements is presented .
The air gap values varied between 1 mm up to 20 mm.
For this purpose a parameterized analysis were
performed using Parameterized Geometry facility from
Comsol.
The boundary conditions on the magnetic field domain is
magnetic insulation, Az = 0. The geometry is shown in
Figure 4.
Then the problem was solved for three increased levels
of degrees of freedom 12771, 46405 and 134581
respectively.
Figure 6. Finite element mesh
Figure 4. Geometry of the numerical model
The solution time was 10.968 s on a Dual Core computer
with Intel processor at 2 GHz and 2GB of RAM.
Materials for the plunger, case, and the limiter is steel
with the magnetic characteristic B(H) presented in Fig.5.
The current density value is J = 346728 A/m2.
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Scientific Bulletin of the Electrical Engineering Faculty – Year 11 No. 2 (16)
ISSN 1843-6188
The distribution of the magnetic flux lines and of the
magnetic potential vector are presented in Fig.7 and 8.
Figure 10.Distribution of the magnetic flux density B
Figure 7. Distribution of the magnetic flux lines
9
Experimental values
Numerical values
8
7
F[N]
6
5
4
3
2
1
0
0
2
4
6
8
10
12
14
16
18
20
delta[mm]
Figure 11. Comparative experimental and numerical static
characteristics
Figure 8. Map distribution of magnetic potential vector
The intensity of the magnetic field H distribution for
different positions of the plunger are described in Fig.9.
For the next range of the airgap, between 2 to 8 mm, the
numerical values are beyond the experimental values.
Then they become higher to the end of the of the range,
between 8 and 20 mm.We think that for a laboratory
device the meadured values has a resonable accuracy.
In Figure 12 the magnetic flux density variation along
longitudinal contour of the plunger, from the top to the
bottom, for the minimum air gap.
Figure 9. Distribution of intensity of the magnetic field H
The magnetic flux density B distribution for different
positions of the plunger are described in Fig.10.
For each position of the plunger the magnetic force was
computed.
Figure 12. Magnetic flux density variation along
longitudinal section of the plunger
In Figure 11 the numerical and experimental characteristics
are represented.
Along the straight part of the plunger the magnetic flux
increases to a certain almost constant level. Then in the
tilted area it decreases and increases again in the air gap.
There is a good agreement between them.
It can be noticed that at the beginning, for a small values
of the airgap, about 1 mm, the numerical force is bigger
then the experimental values.
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ISSN 1843-6188
Scientific Bulletin of the Electrical Engineering Faculty – Year 11 No. 2 (16)
Figure 13. The experimental and numerical characteristics,
compared with the manufacturer characteristic
In Figure 13 the experimental and numerical
characteristics are compared with the manufacturer
characteristic.
It can be seen that the initial measured and numerical
force values, corresponding to an air gap of 1mm, are
higher then that of the manufacturer.
The difference is about 25% for the measured force and
about 33% for the numeric force. After 2 mm the
differences decreases significantly and the curves are in
good agreement.
In Figure 14 is represented the stroke force values versus
the air gap, δ, for different values of the supply current.
They have similar shapes and variations.
The curves indicate that the magnetic forces generated in
a static regime is strongly influenced by variations in
electric current, a mean effect, with no coupling with
other design parameter.
The shape of the curves is in good agreement with the
theoretical curves, where the force is inverse
proportional with the squared air gap.
There are certain local differences to the normal
variation, such as increases of the force between 5-6 mm
and 7 to 10 mm. They might be produced by mesh.
Figure 15. Magnetic force values versus the supply current
values, for different values of air gap δ
In Figure 15 the variation of the magnetic force versus the
supply current and the air gap is presented .Obviously
the maximum value is obtained for the maximum current
and for the minimum air gap.
Computed Force
10
8
6
4
2
0
100
90
20
15
80
10
70
Ampere-Turns[A]
5
60
0
delta[mm]
Figure 15. Variation of the magnetic force versus the
supply current and the air gap
4. CONCLUSIONS
In this paper certain measurements and numerical
simulations were performed in order to obtain the
magnetic force that act on the plunger and the static
characteristic of the solenoidal electromagnet. Good
agreement was noticed among measured, computed and
manufacturer characteristics. The dynamic of the
electromagnet, using the co-simulation between finite
element analysis and a Simulink model, will be the
subject of a future work.
Figure 14. Magnetic force values versus the air gap δ,
for different values of the current
5. REFERENCES
In Figure 15 are represented the stroke force values
versus the supply currents values for different values of
the air gap, δ.
It can be noticed that for the first four values of the air
gap (7, 6, 5 and 4 mm) the rate of force increase is
almost the same and it is small, about 0.5 N. From 4 mm
to 3 mm the increase is almost double. Then for 2 and 1
mm the increase is much faster.
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Scientific Bulletin of the Electrical Engineering Faculty – Year 11 No. 2 (16)
[2] A. Nicolaide, Electromagnetics. General Theory of
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