Kuwait J. Sci. Eng. 39 (2B) pp. 149-164, 2012
Calculation of ®eld intensity inside the plunger of tubular linear reluctance motor using three dimensional FEM
Department of Electrical Engg.
Iran University of Science and Technology, Tehran, Iran
This paper reports a study of magnetic ®eld intensity in the plunger of tubular linear reluctance motors (TLRMs) with open type magnetic circuits. In this paper, magnetic
®eld intensity is described with regard to the eddy current e€ect in the iron core of tubular linear reluctance motors. In addition, magnetic ®eld intensity is calculated in the motor operating frequency. The proposed method is able to calculate magnetic ®eld intensity in any radius of the plunger. Electromagnetic ®nite-element analysis is used for motor simulation and magnetic ®eld intensity calculation when the plunger is completely inside the coil. Simulation results of magnetic ®eld intensity calculation using 3-D FEM with the coil current excitation is compared to theoretical results. The comparison yields a good agreement.
Keywords: Tubular linear reluctance motors; magnetic ®eld intensity; FEM analyses.
Using electric circuit theory, linear motors under both ac and dc supplies have been greatly studied. It must be mentioned here that linear dc motors have lower eciency than ac ones. However, they are widely used in many applications.
Linear motors have severaltypes which di€er in construction. TLRM is a linear motor that can operate in di€erent modes of operation such as self-oscillating, switched-oscillating and accelerator.
A TLRM consists of two major parts i.e. a moving part, also known as a plunger and a stator. A stator is a winding which is fed by a dc source and produces the magnetic ®eld. A TLRM in its simplest form is illustrated in
Figure 1.

150 A. Mosallanejad and A.Shoulaie
Fig. 1. Tubular linear reluctance motor with open magnetic circuit
A TLRM is a linear motor where the operation is based on the tendency of its movable part to move to a position with higher inductance or, in other words, lower reluctance. This is exactly how force and velocity is produced in this motor.
As the plunger moves into the coil, its ferromagnetic material reduces the reluctance and therefore magnetic force is developed due to the change in motor reluctance. The ferromagnetic plunger has a greater magnetic permeability than the air it replaces. As a result, the magnetic ¯ux can be formed more easily when the plunger is placed in the center of the coil. At this point, the reluctance is at its minimum for a given ¯ux level; it is also the position of the least energy.
When displaced from the centered position, magnetic forces will always act to restore the plunger to its previous position. TLRM consists of a series of coils activated sequentially to pull the plunger along the coil. Note that the plunger is only pulled, and is never pushed. This is a disadvantage of TLRMs when compared to other synchronous accelerators that can push and pull by choosing the relative polarity of the armature and stator windings.
TLRMs have been investigated in various types of magnetic circuits with analyzing magnetic ®eld, calculating integral parameters of the ®eld and determining static characteristics of the motor (Tomczuk & Sobol, 2005).
Moreover, the performance of the motor under both ac and dc supplies is studied (Mendrela, 1999).
Design rules of a reluctance accelerator are described in (Bresie et al., 1991).
This paper also discusses the accelerator control methods and their predicted performances. TLRM di€erent modes of operation are discussed completely
(Mendrela & Pudlowski, 1992). This paper also discusses the design and modeling of TLRM in self-oscillating mode. Equivalent circuit parameters of
TLRM are obtained using experimental method (Corda & Jamil, 2010). Also, a permanent magnet linear oscillating accelerator is analyzed theoretically and validated via experimental results (Zhu & cher, 2009). The coupled ®eld-circuit

Calculation of Field Intensity inside the Plunger of Tubular Linear Reluctance Motor .....
151 model of the three-stage reluctance accelerator is presented (Waindok & Mazur,
2009). Another work has studied the design features of variable air gap, cylindrical and variable reluctance actuators (Gibson et al., 2008).
In analyzing TLRM, obtaining inductance pro®le and motor output characteristics are very pro®table. For obtaining motor inductance pro®le, it is necessary to determine minimum inductance (motor inductance when the plunger is out of the coil) and maximum inductance (motor inductance when the plunger is fully inside the coil). Methods used for calculating inductance of air core solenoid that are practical for TLRM minimum inductance calculation are presented
(Haghmaram et al., 2006; Tomczuk & Sobol, 2005; Noval & Smith, 2003).
However, there is no method represented for maximum inductance calculation.
According to Tomczuk et al. (2005), ``The dynamic inductance of the motors was not calculated on the basis of the magnetic ®eld analysis. They often were obtained with using measured inductance only''. One of the methods used for inductance calculation is based on energy method in which magnetic ®eld intensity must be calculated. This paper calculated the magnetic ®eld intensity in TLRM iron core when the plunger is fully inside the coil. Since in TLRMs iron core is integrated, considerable eddy current is produced in it. Hence, this paper calculates magnetic
®eld intensity with regard to the e€ect of eddy current in iron core. As the frequency of the current applied to motor in¯uences magnetic penetration depth and ®eld intensity, motor operation frequency must be speci®ed. This is achieved by the means of half-wave Fourier series expansion.
Finite element method is used for motor simulation and ®eld intensity calculation when the plunger is fully inside the coil. For evaluating proposed method, simulation and calculation results are compared. The comparison yields a great agreement.
When TLRM is connected to a power supply since the plunger is outside the coil, current increases and pulls the plunger into the coil. Consequently, the coil inductance increases and limits the current. The supply must be switched o€ when the plunger is completely inside the coil. Therefore, the pulse duration of
TLRMs is very short. TLRM is supplied with a pure DC voltage; however, according to its short pulse duration, the current contains harmonic components. In this paper, to precisely determine motor current harmonic contents, half-wave Fourier series expansion method is used (Hernandez et al.,
2004; Neuber et al., 2003). In this approach, the main waveform is approximated through components with speci®c amplitudes and frequencies.
Figure 2 shows main waveform and its approximated waveform using half-wave
Fourier series expansion.

152 A. Mosallanejad and A.Shoulaie
Fig. 2. Main waveform and its approximated waveform
The approximated waveform will have more accuracy compared to the main waveform if more harmonic contents are considered. Figure 3 illustrates each harmonic content and its amplitude.
As depicted in Figure 3, many di€erent harmonic contents exist and their e€ect on the motor performance should be considered. Taking into account each harmonic content and its impact, is tedious work. Consequently, we have utilized average frequency approach.
Fig. 3. Harmonic components with their amplitude in motor current
According to each harmonic content frequency and amplitude, average frequency is calculated as follows:

Calculation of Field Intensity inside the Plunger of Tubular Linear Reluctance Motor .....
153 where
A i
= Amplitude of i th component f i
= Frequency of i th component
F mean
ˆ iˆ1
P
A i
2
A i
2 iˆ0 f i
1†
Magnetic ®eld strength H is the amount of magnetizing force. It is inversely proportional to the length of the ¯ux path and directly proportional to the current passing through it. Magnetic ®eld strength is a vector quantity whose magnitude is the strength of a magnetic ®eld at a point in the direction of the magnetic ®eld at that point. Flux density B, the amount of magnetism induced in a body, is a function of the magnetizing force H.
As mentioned before, TLRM consists of a solenoid, operating as a stator, and an integrated iron core called plunger. When the plunger is out of the coil, motor operates alike an air-coiled winding.
If the solenoid has a ®nite length l and consists of N closely wound turns of a
®lament that carries the current I, then the ®eld at the points well within the solenoid is given by:
H ˆ
NI l
2†
Equation (2) calculates ®eld intensity inside the coil when the plunger is out of the coil. However, when the plunger enters the coil, all motor parameters change in addition to ®eld intensity. Therefore, it is necessary to calculate the magnetic ®eld intensity inside the iron core when the plunger is completely inside in the coil.
Usually in electrical motors the iron core is laminated, while in TLRMs, it is integrated. Therefore, a high eddy current is produced inside the core. Figure 4.
shows a TLRM cylindrical core (plunger) without the coil.

154 A. Mosallanejad and A.Shoulaie
Fig. 4. Schematic outline of the iron core
In Figure 4, R is the core radius and l c is the iron core length. The current i …t† in the winding ¯ows in the ' direction of the cylindrical coordinate system shown in Fig. 4. With ¯owing current in the coil, in one dimensional analysis, the ¯ux line and the magnetic ®eld intensity vector in the core has only the component along thezaxis which depends only on the rcoordinate along the core radius and time, t, and the eddy current density vector has the '-directed component J
'
r ; t† only. The Ampere's circuital law is applied to a rectangular line consisting of a path located inside the iron core and the air gap (as indicated by the dashed line in Fig. 4. Therefore, we can write:
H c
†l c
‡ H a
†l a
† ‡ l c
Z R
J r; t r
3† where H c
† and H through the air gap.
a
† are the magnetic ®eld intensities in the iron core and in its related air gap, respectively. Also, l a is the average length of ¯ux path
( a
Assuming that the ¯uxes in the air and the core are equalto each other
ˆ c
†, we can write:
H a
ˆ rc
H c
A c
A a
4†
In TLRMs, the ¯ux path consists of two parts i.e. air gap and iron core. In
(3), core length (l c
† is speci®ed but the average length of ¯ux path through the air gap should be determined. The following formula is proposed to calculate l a
: l a
ˆ o
N i…t†
B c
:
A a
A c
ÿ l c rc
:
A a
A c
5† where rc is the relative magnetic permeability of the iron core. Substituting (5) and (4) into (3) yields:

Calculation of Field Intensity inside the Plunger of Tubular Linear Reluctance Motor .....
c
H c
B c
† ‡ l c
Z R
J r; t r
In left side of (6), N :i can be rewritten in the form follows:
N :i…t† ˆ H:l
155
6†
7† and
H ˆ
B c e where e is the equivalent magnetic permeability de®ned by: e
ˆ o rc
1 ‡ n … rc
ÿ 1†
8†
9† where n 
D 2 c l 2 c
ln
2l c
D c
ÿ 1† …10†
If the inner diameter of winding is equalto iron core diameter, the above equation has a very high accuracy, otherwise its accuracy will be decreased. In
TLRMs, (9) is not usable since there is signi®cant air gap between the plunger and the coil; therefore in this paper, equivalent magnetic permeability ( e measured by experimentalresults.
† is
Substituting (8) and (7) into (6), equation (6) can be rewritten in the following form: c
H c e
† ‡ l c
Z R
J r; t r
11†
From (11), we can write:
@H c
@r
ˆ l c l
: e c
:
@
@r
2
Z R
4 J r; t 5 ˆ ÿ l l c
: r
3 e c
J r; t
Faraday's law for the problem under consideration can be written as
12†

156 A. Mosallanejad and A.Shoulaie
@E
'
@r
ˆ ÿ
@B c
@ t
Applying Ohm's law and using (12) and (13), we can write:
@ 2 H c
@ r 2
ˆ l l c
: e
:
@H c
@
13†
14†
Equation (14) can be rewritten into the following equation for sinusoidal steady state conditions of an angular frequency !.
@ 2 ^ c
†
@r 2
ˆ K 2 :
@H c r
@ r
15†
Where
K ˆ s  j ! l c c l e
ˆ r l
 l c
1 ‡ j t
16† and t
ˆ r  c e f
17†
In (17) f is the average frequency of the coil current that is calculated from
(1). The di€erentialequation (15) has a generalsolution given by:
H c
18†
Applying the boundary conditions to (18) yields:
H…r† ˆ H
BO cosh…Kr† cosh…KR†
19† where H
BO is the magnetic ®eld intensity phasor at the boundary of the core .
In order to calculateH follows:
BO using the di€erential form of the Maxwell-Faraday law, ®rst we should calculate the eddy-current density phasor ^J …r† in the core as

Calculation of Field Intensity inside the Plunger of Tubular Linear Reluctance Motor .....
^J
'
H c
r††
'
ˆ ÿH
BO r l
 c l
1 ‡ j t sinh… q l
 c cosh… q l l
 c l
1‡j t
1‡j t r†
R†
157
20†
Substituting (20) into (11), and solving (11), the following relation is achieved:
H
BO
ˆ e c
NI l
21†
As previously mentioned, H
BO is the magnetic ®eld intensity phasor at the boundary of the core and it is the phasor of the sinusoidalcurrent ¯owing through the coil. Substituting (21) into (19), magnetic ®eld intensity inside the iron core is calculated as follows: e c
NI l cosh…Kr† cosh…KR†
22†
The basic geometricaldata of the prototype TLRM which was built in laboratory is given in Table 1.
Symbol
N
D ci
D co
D po l c l w r rc d
Table 1. Motor parameters
Quantity
Number of turn
Coilinner diameter(mm)
Coilouter diameter(mm)
Plunger outer diameter(mm)
Length of plunger(mm)
Length of winding(mm)
Relative magnetic permeability resistance coil(†
Conductor diameter(mm)
Value
691
20
45
16
200
200
380
0.98
0.0016
Regarding the motor operating frequency, i.e. 27Hz, and iron core small radius, it was expected that the magnetic ®eld intensity at the center of the iron core not to be zero. Figure 5 represents the magnetic ®eld intensity inside the iron core calculated from (22).

158 A. Mosallanejad and A.Shoulaie
Fig. 5. Magnet ®eld intensity inside the plunger
In Figure 5, magnetic ®eld intensity at the center of the iron core is at its minimum of 632 At/m while it reaches its maximum of 1402 At/m at the surface of iron core. In other words, getting near to the boundary layers of the iron core, the ®eld intensity increases up to its maximum. Here, it is essential to compare the calculation results of magnetic ®eld intensity with the simulation results using 3-D FEM.
Because of plunger axial symmetry, the three dimension FEM analysis is performed for a quarter of plunger area which is shown in Figure 6.
Fig. 6. The plunger model used in three dimension FEM
In order to evaluate the proposed method, a TLRM is modeled using 3-D
FEM with the coil current excitation when the plunger is completely inside the

Calculation of Field Intensity inside the Plunger of Tubular Linear Reluctance Motor .....
159 coil. The results obtained from FEM analysis are compared with the ones obtained from the proposed method.
The ®eld intensity in the iron core of TLRM when the plunger is completely inside the coilis shown in Figure 7.
Fig. 7. Magnetic ®eld intensity simulation results using 3-D FEM
Figure 7 shows ®eld intensity in di€erent parts of the winding and the iron core. In magnetic circuits with air gap, the larger part of energy is stored in the air gap. Therefore, ®eld intensity in the air gap is higher than the one in the iron core. As the air gap in TLRMs is considerably long, it is expected that the ®eld intensity in the iron core be considerably low.
Figure 7 show the ®eld intensity inside and outside the iron core is depicted with di€erent colors which indicate di€erent magnetic ®eld intensities. For calculating the exact magnetic ®eld intensity inside the coil, one can implement core ¯ux density. Figures 8 and 9 show magnetic ¯ux density in di€erent parts of the TLRM from two di€erent views.

160 A. Mosallanejad and A.Shoulaie
Fig. 8. Magnetic ¯ux density calculation using 3-D FEM (front views)
Fig. 9. Magnetic ¯ux density calculation using 3-D FEM (behind views)
As mentioned, highest magnetic ¯ux density in TLRM exists in its iron core.
However, ¯ux density in outer layer is higher than the one in the center of the iron core and therefore the maximum ¯ux density shown in Figure 9 is relevant to the outer layer of the iron core. According to Figure 9, magnetic ¯ux density in iron core surface is equal to 0.67 T. With respect to this value, it can be easily concluded that magnetic ®eld intensity is equal to 1403 At/m. It must be mentioned that the core relative magnetic permeability was considered equal to
380. Comparing the results obtained from FEM analysis and those achieved

Calculation of Field Intensity inside the Plunger of Tubular Linear Reluctance Motor .....
161 from our proposed method which is shown in Figure 5, the proposed method is veri®ed for calculating ®eld intensity inside the iron core.
It should be mentioned that in (22) the value of equivalent magnetic permeability ( e
† is calculated as follow e
ˆ o re
23†
In (23) re is the value of relative equivalent magnetic permeability that is achieved by the curve shown in Figure 10.
Fig.10. Experimental measured relative equivalent magnetic permeability according to position of plunger
In Figure 10, relative equivalent magnetic permeability that was measured from experimentalresults is shown for allplunger positions. If the coilis without the plunger, relative equivalent magnetic permeability is equal to that of the air, but with the plunger entering the coil it increases and reaches its maximum when the plunger is completely inside the coil. The maximum relative equivalent permeability is 67.7 according to Figure10.
The exact value of ¯ux density and magnetic energy stored in TLRMs can be obtained by specifying precise value of ®eld intensity inside the plunger.
Furthermore, accurate ®eld intensity calculation facilitates motor behavior prediction. Field intensity in the center of iron core is not zero as a reason of low frequency of motor current. Calculation results indicate that the minimum ®eld intensity exists in the center of the iron coil. Getting near to the boundary layers

162 A. Mosallanejad and A.Shoulaie
of the iron core, the magnetic ®eld intensity increases up to its maximum. The most important characteristic of the proposed method is its ability to calculate magnetic ®eld intensity in any radius of plunger.
e is the most important parameter for calculating magnetic ®eld intensity which was measured from experimental results. Since there is an air gap between the plunger and the coil in
TLRMs, equation (9) is not usable. Therefore, in this paper, equivalent magnetic permeability ( e
† is measured by the experimentalresults.
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Submitted : 14/3/2011
Revised : 25/9/2011
Accepted : 4/1/2012

164 Calculation of Field Intensity inside the Plunger of Tubular Linear Reluctance Motor .....
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