Solid State Phenomena Vol. 144 (2009) pp 53-58
Online available since 2008/Sep/26 at www.scientific.net
© (2009) Trans Tech Publications, Switzerland
doi:10.4028/www.scientific.net/SSP.144.53
Modeling of Magnetic Attraction Force of Electromagnetic Module in a
Relative Base – Air-gap – Absolute Base System
Tomasz Huścio1,a, Krzysztof Falkowski1,b
1
Mechanical Department, Technical University of Bialystok, ul. Wiejska 45C, 15-351 Białystok
a
tomekh@pb.edu.pl, bfalkowski@ pb.edu.pl
Keywords: relative base - air-gap - absolute base system, magnetic attraction forces.
Abstract. In this paper a formula of estimation of magnetic attraction force in the relative base –
air-gap – absolute base system is presented. The attraction force of the relative base (forcer) to the
ferromagnetic absolute base (stator) is a result of the attraction of permanent magnets, which are the
components of the electromagnetic modules. The physical model and mathematical description of
the particular electromagnetic module are presented.
Introduction
The relative base - air-gap - absolute base system (planar stepping motor) is the main element of the
precise coordinate positioning systems of the specialized devices (manipulators, machine tools,
measuring machines, assembly operations in microelectronics and instrument engineering, surface
mounting elements and units assembly, laser technological complexes [6]) (Fig. 1).
1- planar aerostatic two-coordinate relative base - the forcer with gas-film
lubrication (aerostatic lubrication); 2- absolute base - the immovable stator
(platen); 3- motion control card; 4- air filter and regulator
Fig. 1. Basic elements of the relative base - air-gap - absolute base system
In the paper a model of the magnetic attraction force, which is produced by permanent magnet
being part of the planar two-coordinate relative base is presented. The model enables us to estimate
the value of the attraction force Fp of the relative base (force) towards the absolute base (stator).
The attraction force Fp is generated by the permanent magnets, which are placed into the
electromagnetic modules. Basing on these dependences we worked out a module of the computer
program. The module will be used for calculations connected with the flow of the magnetic flux
through the electromagnetic path.
Object of research
The basic elements of the planar aerostatic two-coordinate relative base are shown in Fig. 2. The
planar aerostatic two-coordinate relative base consists of the following elements: aluminum frame;
electromagnetic modules: I, II; pneumatic system: supply conduit, built-in pneumatic system,
bearing chambers. The principal element of the pneumatic system is the built-in pneumatic system
with nozzles and bearing chambers, which provide the air into the air-gap between the absolute base
and the relative base. The air cushion acting as an air layer, which pressure is greater than
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Mechatronic Systems and Materials II
atmospheric pressure, produces aerodynamic lift Fn. By means of the aerodynamic lift the relative
base is suspended at height h above the absolute base. The use of the air-gap completely eliminates
the physical contact of the working surfaces, and the application of the aerostatic lubrication
reduces friction and wear of the sliding surfaces.
The working surface of the relative base contains two groups of orthogonal electromagnetic
modules. The electromagnetic modules (I) are responsible for movement of the relative base along
X-axis, and the modules (II) are responsible for movement of the relative base along Y-axis.
The planar stepping motor converts control pulse signals directly into a sequence of linear shifts.
The direction of shifts made by the forcer is related to the sequence of passing impulses and the
speed of the forcer depends on frequency of the current. The value of the shift of the forcer in
regard to the stator depends on the current pulses. The operation of the planar stepping motor is not
possible without an appropriate control system. The control system ensures the desirable
performance of the step motor.
Fig. 2. The basic components of the planar aerostatic two-coordinate relative base
Figure 3 shows the forces of the relative base - air-gap - absolute base system. The aerodynamic
lift Fn (reactive force of air cushion) counterbalances the loads resulting from gravity (weight of the
relative base Pp + weight of the object mounted on relative base - payload Pr) and magnetic
attraction forces Fp. The attraction force of the relative base towards the ferromagnetic absolute
base is caused by the permanent magnets, which are the part of the electromagnetic modules.
Solid State Phenomena Vol. 144
55
Fig. 3. Forces of the relative base - air-gap - absolute base system
Figure 4 shows elementary electromagnetic modules. The electromagnetic module consists of
two magnetic cores and the permanent magnet, which is sandwiched between the cores. The
magnetic cores are joined by means of a yoke (Fig. 4c, 4d). The winding is wound on each core.
The working surface of electromagnetic modules has a spaced toothed structure with a definite
phase shift from pole to pole.
Fig. 4. Cross sections of the elementary electromagnetic modules
Physical model
The electromagnetic module of the planar stepping motor consists of two magnetic circuits: the first
magnetic circuit, in which the magnetic flux is generated by the permanent magnet, and the second
one, in which the magnetic flux is generated by the motor coil. The magnetic flux generated by the
motor coil produces force Fc. The force Fc makes the relative base (forcer) move. The magnetic flux
generated by the permanent magnet produces magnetic attraction force Fp.
To estimate the value of the attraction force Fp between the relative base (force) and the absolute
base (stator) the following simplifying assumptions are made:
- the permanent magnet is characterized by the linear demagnetization curve (this is correct for
modern permanent magnets),
- magnetic reluctance of the iron is considerably smaller then magnetic reluctance of the air gap,
and is not taken into consideration,
- because of the facts that there is an air gap between the absolute base and relative base and the
small values of the magnetic reluctance of the iron, the magnetic circuit can be treated like linear.
Figure 5 shows the cross section of a relative base - air-gap - absolute base system.
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Mechatronic Systems and Materials II
Fig. 5. Electromagnetic module
Mathematical model of the magnetic circuit with the permanent magnet
The magnetic circuit with permanent magnet (Fig. 5) consists of the two air gaps ho high,
ferromagnetic magnetic core l long and the permanent magnet.
The magnetic circuit with permanent magnet in accordance with the Ampere’s law is described
by the equation:
Bm Am = B p A p
(1)
where: Bm – magnetic induction at working point of the permanent magnet, Bp – magnetic induction
in an air gap, Am – cross-sectional area of the permanent magnet, Ap – effective cross-sectional area
of the magnetic core.
Sum of the voltages for any closed loop is equal to zero in accordance with Kirchoff's law for
magnetic circuit:
∑U
i
=0
(2)
i
where: i – number of element of magnetic circuit; Ui – magnetic potential of i element.
Kirchoff’s law for the magnetic circuit of the electromagnetic module has the form:
U mr + U mh = 0
U m = U mr + U mh
(3)
where: Umr – magnetic potential of magnet, Umh – magnetic potential in the air gap.
The magnetic potential is equal:
U µ = H µ lµ
(3a)
where: Hµ - magnetic field, lµ - length of an element.
We consider Eq. 3a in Eq. 3:
H m l m + 2 H p ho = 0
(4)
where: Hm – magnetic field at the point of work of magnet, Hp – magnetic field in the air gap ho, lm –
length of permanent magnet.
Solid State Phenomena Vol. 144
57
The magnetic field in the air gap is equal:
Hp =
− H m ⋅ lm
2ho
(5)
We can estimate the magnetic field using the equation:
Hp =
Bp
(6)
µo
H
] - magnetic permeability of vacuum.
m
If we introduce Eq. 6 into Eq. 5, we obtain a flux density in the air gap. The flux density is
generated by the permanent magnet:
where: µ 0 = 4π ⋅ 10 −7 [
Bp = −
− µ o ⋅ lm
⋅ Hm
2ho
(7)
If we insert Eq. 7 into Eq. 1, we obtain the flux density generated by the permanent magnet at the
working point:
Bm = −
− µ o ⋅ l m Ap
⋅
⋅ Hm
2ho
Am
(8)
We assume that a demagnetization curve of the permanent magnet is linear. The demagnetization
curve is estimated by the function (Fig. 6):
Bm =
Br
H m + Br
Hc
(8a)
The flux density Bm and the magnetic field Hm at the working point of the permanent magnet can
be computed using Eq. 8 and Eq. 8a:

− µo ⋅ lm Ap
 Bm = − 2h ⋅ A ⋅ H m

o
m

 B = Br ⋅ H + B
m
m
r

Hc
(8b)
Fig. 6. The demagnetization curve
If we insert magnetic field Hm into Eq. 8, we can estimate the flux density in the air gap:
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Mechatronic Systems and Materials II
− µ o ⋅ lm
⋅ Hm
(9)
2ho
The value of the magnetic force generated by an elementary electromagnetic module equals:
Bp = −
Fe =
Ap
2µ o
⋅ Bp
2
(10)
The attractive force Fp between the relative base and the absolute base equals:
Fp = ne ⋅ Fe
(11)
where: ne- number of the elementary electromagnetic modules that belong to the driving module (I)
and (II).
Conclusions
Modeling of the magnetic attraction force of the electromagnetic module of the relative base – airgap – absolute base system consists of the following stages:
1. Creation of the physical model;
2. Creation of the mathematical model;
3. Research of the mathematical model (analytical-numerical solution of the equations of the
model, computer simulation);
4. Verification of the results;
5. Optimization of the system from the point of view of the defined assumptions and set
requirements;
6. Visualization of the results, computer animation.
On the base of the obtained relationships we developed a module of the computer program
SPAP. The computer program SPAP will be used for computer-aided design (design and
optimization of a structure) of the planar aerostatic two-coordinate relative base with
electromagnetic drive. Computer program SPAP consists of the two modules: SPAPAERO – used for
calculations connected with airflow in the relative base - air-gap - absolute base system, SPAPMAG –
used for calculations connected with flow of the magnetic flux in the electromagnetic drive. The
SPAPAERO has been already worked out [1].
References
[1] T. Huścio: Komputerowo wspomagana symulacja płaskich aerostatycznych podpór
współrzędnościowych Program SPAPAERO, International Scientific-Technical Conference.
Hydraulic and Pneumatics ‘2005. Problems and development tendencies in the beginning
decade of the 21st Century (2005), p. 145.
[2] Z. Gosiewski, K. Falkowski: Wielofunkcyjne łoŜyska magnetyczne, Wydawnictwa Naukowe
Instytutu Lotnictwa (2003).
[3] A dissertation by Tiejun Hu: Design and control of a 6-degree-of-freedom levitated positioner
with high precision. Submitted to the office of Graduate Studies of Texas A&M University.
May 2005.
[4] Boldea I., Syed A. Nasar: Linear electric actuators and generators. Cambridge University
Press, 1997.
[5] Hamers M.R.: Actuation Principles of Permanent Magnet Synchronous Planar Motors. A
Literature Survey. Philips Applied Technologies Department Mechatronics Technologies.
Acoustics and Control Group. Eindhoven, December 2005.
[6] Information on http:// www.ruchserwomotor.com
Mechatronic Systems and Materials II
10.4028/www.scientific.net/SSP.144
Modeling of Magnetic Attraction Force of Electromagnetic Module in a Relative Base – Air-Gap –
Absolute Base System
10.4028/www.scientific.net/SSP.144.53
DOI References
[4] Boldea I., Syed A. Nasar: Linear electric actuators and generators. Cambridge University Press, 1997.
doi:10.1017/CBO9780511529641