Cryogenics 117 (2021) 103328

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Cryogenics
journal homepage: www.elsevier.com/locate/cryogenics

Frozen image analysis of a superconducting magnetic levitation system

consisting of multi-surface superconductor and Halbach array permanent
magnet configuration
Ahmet Cansiz a, *, Ahmet F. Reisoglu a, Kemal Ozturk b, Murat Abdioglu c
a
Istanbul Technical University, Electrical Engineering Department, Istanbul, Turkey
b
Karadeniz Technical University, Department of Physics, Trabzon, Turkey
c
Bayburt University, Department of Mathematics and Science Education, Bayburt, Turkey

A R T I C L E I N F O A B S T R A C T

Keywords: Levitation strength provided by high temperature superconductors are limited for device applications. Although
Superconducting magnetic levitation superconducting material properties are continuously improving, there is still strong necessity of efficient design
Frozen image model mechanisms for the superconducting magnetic levitation systems. Studies in the last decades have shown that
Permanent magnet guideway
combining multi-surface superconductor and permanent magnet components in optimum configurations has
improved the levitation forces. In this respect, Halbach arraying permanent magnets interacting with multi-
surface superconductors has become one of the most utilized methods. This paper investigates frozen image
modeling of the levitation and guidance forces on a particular levitation system, which consists of a permanent
magnet guideway and high temperature superconductor car body. The levitation enhancement is investigated for
three configurations according to force interactions between the guideway and car body. These configurations
are based on the use of single permanent magnet-single superconductor, Halbach array permanent magnets-
single superconductor and Halbach array permanent magnets-multi-surface superconductors. The vertical and
guidance forces for the present configurations were calculated in terms of field cooling and zero field cooling
conditions by using frozen image model with magnetic dipole approximation. The predicted force calculations
are analyzed in terms of vertical and lateral traverses of the car body respect to guideway for particular mea­
surement distances. The force analysis provided by frozen image model qualitatively agree with the previously
obtained experimental data.

1. Introduction bulk. Manufacturing techniques for bulk shape HTS formed by Yttrium
Barium Copper Oxide (YBCO) has been improved so far have lead
Levitation enhancement has significant importance in the de­ numerous applications, such as low loss cables for electrical power
velopments of superconducting magnetic levitation (SML) systems for transmission [1], electromagnets for the superconducting magnetic en­
linear transportation and bearings. Levitation and guidance forces are ergy storage (SMES) [2,3], superconducting fault current limiters (SFCL)
the major parameters for optimizing the magnetic levitation (maglev) for power reliability [4], magnetic gears (MG) for electromechanical
systems, and they strongly depend on the superconducting and magnetic energy conversion [5,6], transformers [7] motors-generators [8], fly­
material properties. Bulk shape high temperature superconductors wheels for energy storage systems [9], bearings for industrial applica­
(HTS) and permanent magnets (PM) are the most suitable materials for tions [10–14]. In addition, the SML technology has particularly opened
maglev applications as they provide levitation without active control. up new aspects on transport applications, such as superconducting
However, the hysteresis effect in the superconductor materials causes magnetic levitation trains [15–18].
levitation drift in the levitated body, this in turn forces to draw the Designing SML system requires the effective use of superconducting
attention of alternative design considerations for device applications. and magnetic components. Bulk YBCO is the best superconducting ma­
The industrial use of HTSs is generally in the form of cable, wire and terial for the levitation purpose that operates at liquid nitrogen (LN)

* Corresponding author.
E-mail address: acansiz@itu.edu.tr (A. Cansiz).

https://doi.org/10.1016/j.cryogenics.2021.103328
Received 7 January 2021; Received in revised form 13 April 2021; Accepted 7 June 2021
Available online 2 July 2021
0011-2275/© 2021 Elsevier Ltd. All rights reserved.
A. Cansiz et al. Cryogenics 117 (2021) 103328

temperature. NdFeB is the most suitable magnetic material to couple

with bulk superconductor for effective levitation and guidance forces in
SML. Beside constructing the cryogenic mechanism, the important
strategy in manufacturing SML system relies on the optimization of the
levitation and guidance forces. Since the magnetic force density in the
HTS-PM systems is limited, one of the most efficient way of optimizing
the force strength is to choose appropriate number of superconducting
and magnetic components.
Various methods have been developed to contribute to the in­
vestigations of superconducting levitation studies [11]. Since the su­
perconductors are highly non-linear materials, a particular method for
modeling the levitation characteristics requires certain simplifications
and assumptions. The frozen image model (FIM) and its implementation
by using magnetic dipole approximation is one of the frequently applied
methods for this purpose [19]. According to present literature, analyzing
SML system with FIM so far only included single HTS-PM(s) arrange­
ments [20,21]. In this paper, multi-surface HTS-PM is introduced in FIM
for the first time to analyze the magnetic force capacity of the SML
system and compared with the experimental measurements given in
[22].
This paper is organized in terms of the frozen image model analysis
of a SML system regarding the levitation and guidance forces existed in
three different configurations (Con). The configuration of the forces in
SML system is introduced together with the concept of frozen image
model in Section 2, and the frozen image model agreements with pre­
viously obtained experimental results were evaluated in Section 3. The
conclusion of the study is given at the end.

2. Configuration of the forces in SML system

The frozen image model analysis performed for SML vehicle system
is given in Fig. 1(a) [22]. The vehicle system consisting of a car body and
a guideway is schematically drawn in Fig. 1(b). As shown in Fig. 1(b),
Con-1 (indicated with 1) consists of a single HTS on the car body, which
is levitated by a single PMG. In Con-2 (indicated with 2), a single HTS on
the car body is levitated by Halbach arrayed PMG. In Con-3 (indicated
with 3), the car body is also levitated by Halbach arrayed PMG but this
time there are three bulk HTS on the car body (one on the top and two on
the sides). The thick red arrows represent magnetization directions of
PMs. The detailed frozen image representations of the car body levitated
by the HTS and permanent magnet guideway (PMG) interaction in terms
of the three different configurations is shown in Fig. 2. According to
frozen image representations of PM-HTS interactions, Fig. 2(a) shows
Con-1, Fig. 2(b) shows Con-2, and Fig. 2(c) shows Con-3. In the image Fig. 1. (a) Picture of the experimental setup for force measurement of SML
representations given in Fig. 2, the black arrows indicate magnetic system [22]. (b) Schematic demonstration of the SML vehicle. (1) Con-1: Single
moment of PMs, the blue arrows indicate frozen images and the red PM facing with single HTS (2) Con-2: Halbach array PMs facing with single HTS
arrows indicate diamagnetic images for corresponding configuration. and (3) Con-3: Halbach array PMs facing with three HTSs.
Note that since the superconducting region is assumed to be semi-
infinite planes the shaded areas in Fig. 2 indicate the superconducting Halbach array consist of PMs with the magnetization directions both in
regions. Following is the description of FIM and its implementation on vertical and lateral.
the force calculation based on the dipole moment approximation. The accuracy of FIM not only depends on the configuration of the
FIM predicts the forces between the HTS and PM without taking levitation system but also selecting appropriate assumptions related to
hysteresis into account [19]. However, this method is very quick and the calculation parameters. Important of these parameters are the
effective for determining the qualitative behavior of the forces in the interaction distance, dimensions and magnetic properties of the com­
levitation system. In the present design, FIM is used to predict the forces ponents. For example, if the interaction distance is higher than the di­
between the HTS and PMs for the arrangements given in Con-1, -2 and mensions of the components (HTS and PMG in this case) the size of the
-3. The forces are predicted in the cases of the vertical force versus components are not taken into account in the model. In this case, the
vertical distance z (or displacement) for cooling height (h), the vertical model can be implemented by using dipole approximation, where the
force versus lateral displacement for cooling height (h) and measure­ PMs are assumed to be point like dipole moments. On the other hand,
ment height (z), and last, the lateral (guidance) force versus lateral when the dimensions of the components are higher than the interaction
displacement for cooling and measurement heights. Since it is analyti­ distance, the model can only provide realistic results via Amperian
cally formulated, FIM allows all kinds of force interactions in respective current approximation [11], where the dimensions become the most
directions and particular cooling conditions. The calculations for the important parameters. Due to the complexity of Con-2 and -3, the dipole
sophisticated systems require advance computing with limited geome­ approach is preferred in this study.
tries. For instance, FIM with the consideration of Amperian current The interaction force between two magnetic dipole moments of m1
approximation [11] cannot be easily applied on Con-2 and -3, because and m2 with a distance of r apart from each other is derived from the

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A. Cansiz et al. Cryogenics 117 (2021) 103328

Table 1
Parameters of configurations: PM: magnetic moment of permanent magnet, Dia:
diamagnetic image, Fro: frozen image, rPM-d: position vector between PM and its
diamagnetic image, rPM-f: position vector between PM and its frozen image.
PM Dia Fro rPM-d rPM-f

1 m11 = m12d = m12f = rd = (2 h-2z)az rf = (2 h-z)az + rar

maz -maz maz

2 m21 = m21d = m21f = rd = (2 h-2z)az rf = (2 h-z)az + rar

mar mar -mar
m22 = m22d = m22f =
maz -maz maz
m23 = m23d = m23f =
-mar -mar mar

3 m31 = m31dL = m31fL = rd31L = (2 h-2d rf31L = (2 h-2d + r)

mar -mar mar + 2r)ar ar + zaz
m31dU = m31fU = rd31U = (2 h-2z) rf31U = (2 h-z)az +
mar -mar az rar
m31dR = m31fR = rd31L = (2 h + rf31R = (2 h + 2d-
-mar mar 2d-2r)ar r)ar + zaz

m32 = m32dL = m32fL = rd32L = (2 h + rf32L = (2 h + r)ar

maz maz -maz 2r)ar + zaz
m32dU = m32fU = rd32U = (2 h-2z) rf32U = (2 h-z)az +
-maz maz az rar
m32dR = m32fR = rd32L = (2 h-2r) rf32R = (2 h-r)ar +
maz -maz ar zaz

m33 = m33dL = m33fL = rd33L = (2 h + rf33L = (2 h + 2d

-mar mar -mar 2d + 2r)ar + r)ar + zaz
m33dU = m33fU = rd33U = (2 h-2z) rf33U = (2 h-z)az +
-mar mar az rar
m33dR = m33fR = rd33L = (2 h-2d- rf33R = (2 h-2d-r)
mar -mar 2r)ar ar + zaz

configuration number, while second subscripts indicate which PM is in

which set. The third and fourth columns represent the diamagnetic (dia)
and frozen (fro) image dipole moments, respectively. The fifth and sixth
columns represent the position vectors between the PMs and their
diamagnetic and frozen image dipole moments, respectively. The
magnitude of dipole moments (m) of the PM and its images are assumed
to be the same.
The magnetic potential can be obtained from Eq. (1) for the PM-HTS
system given in Con-1 as;
[ ]
μ0 m2 1 2r2 − 4(2h − z)2
Fig. 2. Frozen image model demonstration of the SML vehicle. (a) Con-1: U1 = + (2)
4π 4(h − z)3 [(2h − z)2 + r2 ]5/2
Single PM facing with single HTS (b) Con-2: Halbach array PMs facing with
single HTS and (c) Con-3: Halbach array PMs facing with three HTSs. Consider next, the second configuration (Con-2) where the car body
(with single HTS) is levitated by Halbach arrayed three PMs guideway,
following magnetic potential [23], as shown in Fig. 2(b). Halbach arrayed PMs couple with the diamagnetic
[ ] (red color) and frozen (blue) images in the HTS on the top. In this
μ m1 ⋅m2 3(m1 ⋅r)(m2 ⋅r)
U= 0 − (1) configuration, the dipole moments of the PMs from left to right are given
4π r3 r5
as m21, m22 and m23. The dipole moment of the diamagnetic and frozen
images of PMs in Halbach array and associated position vectors are
where μ0 is permeability of free space. The magnetic interaction po­
constituted accordingly in Table 1. Eventually, the cumulative magnetic
tential given in Eq. (1) is used to implement FIM on the SML system
potential obtained for the PMs represented by the dipoles of m21, m22
under investigation for three different configurations given in Fig. 2(a),
and m23 with their images appeared in the single HTS are obtained from
2(b) and 2(c).
Eq. (1) as;
Now consider the simplest case of Con-1 shown in Fig. 2(a), where a
[ ]
single PM is interacting with a single HTS. r is the center to center dis­ μ0 m2 1 10r2 − 8(2h − z)2
tance between the PM (represented by a dipole moment) and its images U2 = + (3)
4π 4(h − z)3 [(2h − z)2 + r2 ]5/2
in the bulk HTS (represented by an image dipole moment). Con-1 con­
sists of only a single PM guideway and a single HTS on the car body. In In Con-3, the car body consists of three HTSs levitated by Halbach
this configuration dipole moment of the PM is given as m11. The dipole arrayed three PMs, as shown in Fig. 2(c). From this figure, it is easy to
moment of diamagnetic and frozen images of the PM, and the position visualize Con-3 such that Halbach arrayed PMs couple with the
vectors for the PM-image couple for diamagnetic and frozen cases are diamagnetic (red color) and frozen (blue) images in the HTS on the car
given in Table 1. Note that h is the field cooling (FC) height, z is the body. In Con-3, the dipole moments of the PMs are exactly the same as in
vertical and r is the lateral component of the position vector. In Table 1, Con-2. From left to right, the dipoles are given as m31, m32 and m33.
first column indicates the configuration number. First subscripts in Similar to Con-1 and -2, the dipole moment of the diamagnetic and
dipole moments of the PMs (given in the second column) indicate the frozen images of PMs in Halbach array and associated position vectors

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A. Cansiz et al. Cryogenics 117 (2021) 103328

are constituted accordingly. The position vectors are tabulated in vectors between the PM and its diamagnetic or frozen images are exactly
Table 1: First, the position vectors for the PM on the left (m31) with the same for any of the PM in Halbach array. It was determined from the
diamagnetic and frozen images appear in the HTSs, which are located on calculations that this assumption simplifies the interaction forces, which
the left, top and right side of the car body. Second, the position vectors in turn reduce the number of equations without affecting the magnitude
for the PM in the middle (m32) with the diamagnetic and frozen images of forces.
appear in the HTSs located on the left, top and right side of the car body.
And last, the position vectors for the PM on the right (m33) with the 3. Frozen image analysis of the SML system
diamagnetic and frozen images appear in the HTSs located on the left,
top and right side of the car body. The cumulative magnetic potential Even if FIM is well explained in previous studies [19], it must be
obtained for the PMs, represented by the dipoles of m31, m32 and m33 emphasized here that applying this model varies very much in terms of a
with their images appeared in the HTSs on the left, top and right are particular configuration of interest and associated interaction compo­
obtained from Eq. (1) as; nents. For instance, implementing FC and ZFC conditions for deter­
mining the force between PMs and HTSs is an important concern from
μ0 m 2
U3 = [A1 + A2 + A3 + A4 + A5 + A6 ] (4) the point of interaction. Considering the simplest case of Con-1, which
4π
only takes into account the force between single PM and bulk HTS, the
where ZFC distance is much higher than the dimensions of the components. In
[ ] this case, there are two images of the PM in the HTS: one is the frozen
A1 =
1 4
+
1
+
1
(5) image and the other is the diamagnetic image. Therefore, the experi­
8 (h − z)3 (h + r)3 (h − r)3 mental measurement for vertical and guidance forces can be exactly
[ ] realized by FIM in Con-1 for the FC and ZFC conditions.
A2 =
1 1
+
1
+
1
−
1
(6) Con-2 is more complicated than 1 from the modeling point of view. A
4 (h − d + r)3 (d + 2h − r)3 (d + h + r)3 (d − h + r)3 single HTS is interacting with Halbach arrayed PMG. Each one of the PM
in the array has their own frozen and diamagnetic images. It is assumed
2z2 − 4(2h − 2d + r)2 that every dipole moment from the array only interacts with their own
A3 = (7)
[(2h − 2d + r)2 + z2 ]5/2 images. This means that the center to center distance (d) between the
dipoles in Halbach array is not taken into account in Con-2. Therefore, as
4z2 − 4(2d + 2h − r)2 2z2 − 4(2d + 2h + r)2 2z2 − 4(2d − 2h + r)2 well as in the case of Con-1, the experimental measurements can also be
A4 = 2
+ + properly modeled by FIM in Con-2. The vertical and guidance force
[(2d + 2h − r) + z2 ]5/2 [(2d + 2h + r) + 2
z2 ]5/2 [(2d − 2h + r)2 + z2 ]5/2
expressions for Con-1 and -2 are obtained from the derivatives of U1 (Eq.
(8)
(2)) and U2 (Eq. (3)) magnetic potentials respect to spatial directions (z
4r2 − 2(2h − z)2 2r2 − 4(2h − z)2 4r2 − 2(2h − z)2 and r), respectively.
A5 = 2
+ + (9) Although it is possible to measure the forces both in FC and ZFC
[(2h − z) + r2 ]5/2 [(2h − z) + 2
r2 ]5/2 [(2h − z)2 + r2 ]5/2
conditions, from the point of FIM it is not practical to derive the mag­
netic potential of the system for Con-3 in case of ZFC. This is due to
4z2 − 2(2h + r)2 4z2 − 2(2h − r)2
A6 = 2
+ (10) geometry of Con-3 in which the size of the bulk HTS along the vertical
[(2h + r) + z2 ]5/2 [(2h − r)2 + z2 ]5/2 direction is finite (superconductors positioned on the sides of car body
The force expressions for Con-1, -2 and -3 are obtained by taking the cannot be assumed semi-infinite during the vertical traverse). For
derivatives of the magnetic potentials given in Eqs. (2), (3) and (4), in instance, as the PM moves along the vertical direction, existence of the
respect to vertical (z) and lateral (r) directions, respectively. The deri­ HTS on the sides of the car body prevents FIM to consider ZFC distance,
vation of the vertical and guidance forces for particular configuration is which is practically an infinite number. Therefore, as it was also
straightforward with the symbolic Matlab toolbox. The equations for the explained in Section 2 in describing the property of cooling distance (h),
forces are not given in the text because they occupy too much space. The a ZFC distance cannot be defined properly. In order to overcome this
force expressions obtained from symbolic toolbox are directly imple­ obstacle for implementing FIM in ZFC conditions for Con-3, the FC
mented according to particular configuration and cooling procedure. height is taken into consideration separately when using the derivative
Note that in the above formulation regarding Eq. (4), the term d is 5 mm of U3 (Eq. (4)). In this case, the forces are calculated separately for Con-
and it is the distance between dipole moments of Halbach arrayed PMs 3. First, the vertical force is calculated with the consideration of the
and only matters for Con-3. interaction between Halbach array on the PMG and single HTS on the
It is also important to point out here that the definition of FC height car body only (exactly the case of Con-2), and second the vertical force is
(h) in Con-3 is different from Con-1 and -2. When a single or multiple calculated with the consideration of the interaction between Halbach
number of PM is levitated above a single bulk HTS, h in the formulations array on the PMG and the HTS located on the sides of the car body only.
of Con-1 and -2 is directly implemented in frozen image model: For the Eventually, the two force results are summed to form vertical force as a
case of FC procedure, h has a finite value, for the case of zero field function of particular vertical distance from the bottom of the car body
cooling (ZFC), h is taken as infinite. The situation is different in Con-3; to top surface of the PMG.
beside the HTS on the top, there are additional two HTSs on the left Magnetic levitation and guidance (lateral) force measurements on
and right sides of the PM guideway. In this case, implementing FIM the car body given in Fig. 1(a) is performed with the three axes mea­
requires different task when the HTS included car body approaches to­ surement system [22]. The force measurement system basically collects
wards the PM guideway. Since the HTS regions are assumed to be semi- the data during the corresponding vertical and lateral movements for the
infinite planes in terms of FIM, it is not possible to define a ZFC pro­ three different PMG, represented by Con-1, -2 and -3. The PMs are
cedure exactly same as is in the experimental procedure. The PMs on the NdFeB with the dimension of 40x30x30 mm and their surface magnetic
guideway produces frozen and diamagnetic images at the same time on flux densities are 0.53 T. Bulk shape HTSs (YBCO) are installed on the
the multi surfaces (one on the top and two on the sides). car body have the dimensions of 65x34x14 mm. The detailed experi­
In above potential formulation regarding Con-2 and -3, we assume mental setup and material properties of the YBCO is given elsewhere
one to one correlation between the PM and its own images. This means [21,22]. The vertical levitation force versus vertical distance between
that a particular PM is not interacting with the images of other PMs on HTSs and the PMGs are measured in h = 75, 25 and 5 mm at liquid
the guideway. Therefore, for Con-2 shown in Table 1, the position nitrogen temperature. Here, h = 75 mm is actually accounted as ZFC
procedure. Note that the experimental setup does not allow the force

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A. Cansiz et al. Cryogenics 117 (2021) 103328

measurement for the vertical distances less than FC distance (h) of 5 mm. the design in Con-3. At this point, the model only provides qualitative
For the case of lateral displacement, the levitated car body above the analysis for the SML system.
PMG is free to move up to 20 mm from the center point for Con-1 and -2, Fig. 4 shows the magnitude of the vertical force versus vertical dis­
while it is only free to move 5 mm for Con-3. For complete comparison of tance between the top surface of the PMG and the bottom of the car body
guidance forces for all configurations, however, the lateral displacement (HTS) for the case of h = 25 mm predicted by FIM. The inset figure shows
is limited to 4 mm, which leaves enough clearance between the cage of the force measurements, which are performed for all of the configura­
the PMG and car body. tions [22]. Similar to the results given for h = 5 mm, the experimental
To start the FIM analysis, as a first step the vertical force on the car setup allows to measure the forces between the PMG and car body for
body is calculated by FIM and then the predicted calculation is evaluated the distances ranging from 5 mm to ZFC conditions. There is no
by previously obtained experimental data [22] shown in the inset figure. noticeable attractive force for the case of Con-1 and -2 for distances
Fig. 3 shows FIM prediction of the vertical force versus vertical distance higher than h = 25 mm both in theoretical predictions and experimental
between the top surface of the PMG and HTS for the case of h = 5 mm, measurement (inset figure). For Con-3, a substantial amount of attrac­
where the inset figure shows the corresponding experiment. Note that in tive force (maximum − 35 N and minimum − 25 N) still exists between
Fig. 3, black straight line is for Con-1 (1_T), red long dashed line is for the PMG and HTS even at the distances of up to 40 mm. This is due to the
Con-2 (2_T) and blue dashed line is for Con-3 (3_T). Also in the inset effect of the HTSs placed on the side walls of the car body. This behavior
figure, black line with circle is for Con-1 (1_E), red line with squares is is also predicted by FIM at h = 25 mm with less accuracy compared to
for Con-2 (2_E) and blue line with triangles is for Con-3 (3_E). Moreover, the case in h = 5 mm; the maximum attractive force − 25 N is obtained at
arrows 1 and 2 in the inset represent first ascent and first descent, around 22 mm in vertical distance, which is due to the effect of addi­
respectively. tional superconductor (as again assumed semi-infinite superconducting
The examination of experimental force data seen in the inset of Fig. 3 surfaces) located on the sides of car body. Moreover, the side effect from
indicates large hysteresis in first ascent and descent for all configura­ the HTS constructed on the car body can also be examined for the dis­
tions. For example, Con-1 has the maximum (first ascent) and minimum tance between 5 mm and cooling distance of 25 mm. As shown in the
(first descent) attractive forces of − 24 N and − 13 N at the vertical dis­ inset of Fig. 4, for the vertical distance of 5 mm, the maximum forces are
tance (z) of 12 mm, respectively. Note that the first descent (2) is fol­ achieved as 80, 140 and 206 N for Con-1, -2 and -3, respectively. Again,
lowed right after the first ascent (1) in the inset figure. Con-2 has the the overall magnitude of the force is the highest for Con-3. Note that the
maximum and minimum attractive forces of − 55 N and − 18 N at z = 16 maximum force in Con-3 is only a little higher than that of Con-2. This is
mm, respectively. Con-3 has the maximum and minimum attractive actually valid for all of FC conditions because of the fact that the addi­
forces of − 122 N and − 66 N at z = 18 mm, respectively. The magnitude tional two bulk HTSs installed in the sideways of the car body do not
of theoretically predicted vertical forces for all configurations qualita­ contribute to the vertical force as much as they contribute to the guid­
tively agree with experimentally measured vertical forces. For instance, ance force. In other words, to gain more vertical force, Con-2 is more
in the case of Con-1, the maximum vertical force is obtained as − 25 N at convenient, while on the other hand to gain both in the vertical and
z = 7 mm. In the case of Con-2, the maximum force is obtained as − 50 N guidance forces, Con-3 must be preferred over the other two
at z = 9 mm. And in the case of Con-3, the maximum attractive force is configurations.
obtained as − 140 N at z = 11 mm. Fig. 5 shows the vertical force versus vertical distance for ZFC case
The ability of FIM in the vertical force described above can be (FC = h = 75 mm) for all configurations predicted by FIM. The inset
examined with a closer look at the data in Fig. 3. While the maximum figure shows the experimentally measured values of the forces. For ZFC
value of the vertical force is predicted at 12 mm for Con-3, this force case, the force is not only the highest but also tend to stay high as the car
value was measured at 17 mm in the corresponding experiment (inset body approaches the PMG, which is observed in both theoretical
figure). The presence of a 5 mm shift on the data is attributed to the calculation and experimental measurement. Unlike in the other config­
quantitative inadequacy of the dipole approach. It is known from pre­ urations, the vertical force in Con-3 tends to keep high values as the car
vious studies that this problem is reduced if the same situation was to be body moves away from the PMG. As can be seen from inset figure the
modeled with the Amperian currents approach in a more advanced and repulsive force is sustained up to the vertical distance of 50 mm, while
comprehensive way [11,17]. However, within the framework of the this trend is up to 25 mm seen in theoretical prediction, however. As in
assumptions addressed in this study, the Amperian currents approach is the cases of FC conditions, this is again due to the additional bulk HTS on
not practical, beside the difficulty in implementation, especially due to the sides of the car body, which is more effective in ZFC condition. In
every way, Con-3 has an advantage over the other configurations in ZFC
procedure. In fact, in the ZFC procedure, Con-3 substantially contributes
0 not only to the guidance force but also to the vertical force that is not the
FC = h = 5 mm
1_T
2_T 200
3_T FC= h = 25 mm
Vertical Force (N)

-50 Configuration
150 1_T 200
40
2 2_T FC= h = 25 mm
FC=h=5mm
Vertical Force (N)

3_T 150
Vertical Force (N)

Vertical Force (N)

0 Configuration
100 100
1_E
2_E
-40 3_E
-100 Configuration
50

1
-80 1_E
2_E
50 0

3_E -50
-120 0.0 10 20 30 40 50 60 70
Vertical Distance (mm)
0.0 10 20 30 40 50 0
Vertical Distance (mm)
-150
0.0 10 20 30 40 50
Vertical Distance (mm) -50
0.0 10 20 30 40 50
Vertical Distance (mm)
Fig. 3. FIM prediction of vertical force between the PMG and car body versus
vertical distance for all of the configurations in h = 5 mm. (Inset: Corresponding Fig. 4. FIM prediction of vertical force between the PMG and car body versus
experimental results [22], arrows 1 and 2 represent first ascent and first vertical distance for all of the configurations in h = 25 mm. (Inset: Corre­
descent, respectively). sponding experimental results [22].)

5
A. Cansiz et al. Cryogenics 117 (2021) 103328

200 other hand, for the case of guidance force given in Fig. 6, the force is
FC= h = 75 mm (ZFC) double itself from Con-1 to -2, while it is quadruple from Con-2 to -3.
Configuration
1_T 200 This situation shows that Con-3 seems to be not significantly important
150 2_T FC= h = 75 mm in vertical force case as much as in the guidance force. Actually, this is
Vertical Force (N)

3_T 150 Configuration

not the case for the vertical force when a lateral traverse is concerned at

Vertical Force (N)

1_E
100
2_E particular cooling height and measurement height, which can be
100 3_E
analyzed in terms of FIM results in the following.
50
In order to explore the ability of FIM for analyzing SML system
50 0 further, the vertical forces versus lateral displacement is calculated. The
0.0 10 20 30 40 50 60 70 calculation is performed according to the various vertical traverses for
Vertical Distance (mm)

0.0 10 20 30 40 50
Vertical Distance (mm)

Fig. 5. FIM prediction of vertical force between the PMG and car body versus
vertical distance for all of the configurations in ZFC case of CH = 75 mm. (Inset:
Corresponding experimental results [22].)

case in FC.
Con-3 inherently contains stronger guidance force for both FC and
ZFC conditions. In order to compare the configurations, the guidance
force is measured for the FC conditions. As can be expected, although
Con-3 is suitable to obtain a guidance force in both cooling procedures,
Con-1 and -2 only provide guidance force for the case of FC. The guid­
ance force on the car body versus the displacement in the lateral di­
rection is measured for all of the configurations for h = 5 mm. The FC
distance from the PMG is also established as 5 mm away from all of the
HTS surfaces. Fig. 6 shows the guidance force versus lateral displace­
ment from the FC position. Since the agreement between the theoretical
prediction and experimental [22] in guidance force is better than the
vertical force case, the prediction from FIM and experimental results are
plotted in same graph. The restoring force tends to increase as the lateral
displacement is increased from the center point (cooling position). Since
there are additional HTSs on the sides of the car body, the guidance force
is maximized in Con-3 by a substantial amount. For example, from the
experimental data the maximum guidance force at 4 mm (ultimate
lateral displacement available) from the cooling position is well below
20 N for Con-1 and -2, while it is more than 60 N for Con-3.
As can be inferred from the experimental data, the forces between
HTSs and PMs tend to increase when the number of PM and HTS com­
ponents are increased. However, the major concern of this study is to
demonstrate how these forces are increased in terms of particular con­
ditions with the experimental verification. As shown in Fig. 3 for
instance, when the FC procedure of h = 5 mm is selected, the vertical
force increases by doubling itself starting from Con-1 through -3. On the

40
FC = h = 5 mm Configuration
30 1_T
2_T
20
3_T
Guidance Force (N)

10 1_E
2_E
0 3_E

-10

-20

-30

-40
-2.0 -1.5 -1.0 -0.50 0.0 0.50 1.0 1.5 2.0
Lateral Displacement (mm)

Fig. 6. FIM prediction and experimental results of the guidance force versus Fig. 7. FIM prediction of the vertical forces versus lateral displacement for z =
lateral displacement for all of the configurations for h = 5 mm. 4, 5 and 6 mm for h = 5 mm for (a) Con-1, (b) Con-2 and (c) Con-3.

6
A. Cansiz et al. Cryogenics 117 (2021) 103328

particular cooling height. As shown in Fig. 7(a), (b) and (c), the vertical editing. Ahmet F. Reisoglu: Software. Kemal Ozturk: Funding acqui­
force is calculated at the measurement heights of 4, 5 and 6 mm for FC sition, Investigation. Murat Abdioglu: Investigation.
height of 5 mm for Con-1, -2 and -3, respectively. Comparing all of the
configurations, as in the case of vertical and guidance forces shown in Declaration of Competing Interest
Figs. 3, 4, 5 and 6, the vertical force is again much higher in Con-3
during the lateral traverse when the car body is displaced upward and The authors declare that they have no known competing financial
downward. This results indicate that the car body has higher restoring interests or personal relationships that could have appeared to influence
force along the vertical direction during the lateral traverse, which is the work reported in this paper.
clearly contributed by the HTSs on the sides of the car body. These
theoretical results however, cannot be prone to valid without experi­ Acknowledgements
mental verifications except providing qualitative analysis, which is
planned for future task. This work was supported by the Scientific and Technological
Research Council of Turkey (TUBITAK – Turkey), with project no.
4. Conclusions 118F426.

In this study, we investigated the multi-use of superconducting and References

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CRediT authorship contribution statement

Ahmet Cansiz: Conceptualization, Methodology, Writing – review &