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Citation: Steckiewicz, A. Efficient
Transfer of the Medium Frequency
Magnetic Field Using Anisotropic
Metamaterials. Energies 2023,16, 334.
https://doi.org/10.3390/en16010334
Academic Editor: Nunzio Salerno
Received: 30 November 2022
Revised: 14 December 2022
Accepted: 23 December 2022
Published: 28 December 2022
Copyright: © 2022 by the author.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
energies
Article
Efficient Transfer of the Medium Frequency Magnetic Field
Using Anisotropic Metamaterials
Adam Steckiewicz
Department of Electrical Engineering, Power Electronics and Power Engineering, Faculty of Electrical
Engineering, Bialystok University of Technology, Wiejska 45D Str., 15-351 Bialystok, Poland;
a.steckiewicz@pb.edu.pl
Abstract:
This paper introduces a novel waveguide intended for the spatial transfer of alternating
magnetic fields. Instead of ferromagnetic material, an air core was proposed, while the cladding
was realized using anisotropic metamaterial, built of the resonators and a paramagnetic compos-
ite. Since prior works regarding magnetic field transfer concentrated on static or high frequency
fields, the proposed device complements the range of medium frequencies (several to hundreds of
kilohertz). The three-dimensional model of the 50 cm long and 20 cm wide rectangular structure
with metamaterial cladding was made in COMSOL and computed using the finite element method.
Multi-turn inductors were considered and homogenized by the current sheet approximation, while an
optimization solver was used to identify an optimal design of the waveguide. The analysis was made
with respect to different resonators and permeability of the paramagnetic material. Additionally, the
frequency response of the structure was determined. On these bases, the dependencies of the mean
energy density and magnetic field intensity at the output of the waveguide were characterized. It
was shown that discussed structure was able to provide an efficient transfer of the magnetic field
between two ports. Thus, this device can be used to extend the distance of the wireless power transfer,
especially between devices isolated by a thick barrier (e.g., wall), in which the meta-structure may
be embodied.
Keywords: energy transfer; magnetic fields; magnetic metamaterials; numerical analysis
1. Introduction
The progress in magnetic materials and metastructures had risen sharply in the last
decade [
1
,
2
]. Due to the development of novel devices such as magnetic cloaks [
3
,
4
],
concentrators [
5
,
6
], enhanced sensors [
7
] and artificial magnetic wormholes [
8
], new possi-
bilities appeared. Nowadays one can improve the measurement range of magnetic field
probes [
9
], simultaneously ensure the effective shielding and ‘hide’ objects [
10
], absorb
power nearly perfectly in the energy harvesting systems [
11
] and minimize the influence
of a device under test and an environment on each other [
12
]. Many extraordinary meta-
material properties were achieved through their ability to exhibit diamagnetic behavior
and also due to anisotropy of the effective permeability and/or permittivity [
13
]. The well-
thought usage of these attributes can lead to more efficient solutions and improvements of
existing devices.
Since metamaterials are known for the static [
14
] and transient fields [
15
], the scope
of interest and applications had expanded significantly. The most promising is the us-
age of metastructures in the wireless power transfer (WPT) to improve effectiveness [
16
],
reduce the negative impact of a misalignment [
17
] and increase the distance of energy
transfer [
18
]. For these purposes, one-, two- or three-dimensional metalenses were used.
They are a combination of the resonators, i.e., planar inductors with a parasitic or lumped
capacitance, resonating at a specific frequency [
19
]. The goal was to design the metamateri-
als with a near-zero effective permeability (if they were located behind the transmitting
Energies 2023,16, 334. https://doi.org/10.3390/en16010334 https://www.mdpi.com/journal/energies
Energies 2023,16, 334 2 of 17
and receiving coils) or negative permeability (for the metamaterial located between the
coils) [
17
,
18
]. In WPT studies, the enhancement of power transfer distance and efficiency
are the two most important objectives. Therefore, it was proposed and studied to utilize
the array of resonators [
20
] oriented horizontally [
21
,
22
] or perpendicularly [
23
] to the
direction of an energy transfer. The idea may be found in transportation, where coupled
resonators, placed below the surface, can transfer the power from a single source to several
receivers distributed along the road [
24
]. As the result, the actual distance between the
transmitter and receiver can be significantly extended. However, the exploration of similar
metastructures operating in fast-varying fields and involving a range of frequencies from
several to hundreds of kilohertz is still less expanded than the microwave or static magnetic
field metamaterials.
In the low-frequency fields, the arrays of resonators cannot operate effectively, since
the extremely high inductances and capacitances are necessary to achieve the resonance
state. Therefore, in order to transfer the magnetic field through space, magnetic hoses were
proposed [
8
,
25
,
26
]. A ferromagnetic core acted as a low-reluctance “path” for magnetic
flux, whereas a diamagnetic shell (superconductor) preserved the magnetic flux from
propagating outside the ferromagnetic interior. Although this composition was highly
effective for the static magnetic field, the alternating field (e.g., operating in the kilohertz or
megahertz band) will cause the eddy current inside the core, which leads to a power loss
and attenuation. The superconductor was able to provide near-zero magnetic permeability,
yet it required thermal insulation and a cooling system.
An alternative approach is to use the anisotropic metamaterial as a cladding of the
waveguide, since meta-surfaces can obtain anisotropy of the effective properties with either
negative or positive-negative values in, e.g., terahertz range [
27
], as well as simultaneously
enhance the field intensity [
28
]. Both these properties are crucial in the design of the
potential waveguide. In the GHz range, the dielectric resonator antenna, having a high gain
and wide bandwidth, was synthesized using multi-layer stacks and changed into a cylinder
possessing anisotropic permittivity [
29
]. The ferrite-filled rectangular waveguides were
proposed as leaky-wave antennas [
30
], where two slots on the broad wall helped to improve
the gain. This solution has led to further applications, such as integrated filters [
31
] proving
that anisotropic structures can ensure several applications related to the transmission and
conditioning of an electromagnetic (EM) signal.
In this paper, the magnetic hose for the medium frequency field was introduced.
Compared to the high-frequency solutions, the presented waveguide has been intended for
a medium frequency range. The ferrite filler was replaced by the air to eliminate the losses.
In dielectric stacks, both the perpendicular and parallel components of permittivity can be
negative [
27
], while it is important to note that the introduced waveguide ensures only a
positive-negative combination of these components. What is more, the dielectric materials
can be efficiently used when the propagation of the EM wave is considered, whereas in
the magnetic fields the permittivity has a less contribution to the field distribution than
the permeability. Hence, in the proposed solution, a dielectric shell can only serve as a
frame mechanically supporting the structure, having a negligible impact on the effective
properties. To compensate for the lack of a permittivity medium that controls the EM field
propagation, the paramagnetic composite was used instead. Consequently, the idea of
routing the static magnetic field (with the use of an anisotropic cladding) and the concept
of the magnetically coupled array of resonators, had been combined into the anisotropic
metamaterial, acting as the hose for transporting the magnetic field. The considered
structure opens the possibility of the effective transfer of the alternating magnetic fields.
In any application, increasing the distance of the magnetic field transfer is crucial, such
as WPT through thick walls [
32
], magnetic resonance imaging [
33
,
34
] or in the distant
magneto-resistive sensors [
35
], the hose operating in kilohertz range may become the
solution. The article is organized as follows: in Section 2, the structure of the magnetic hose
with metamaterial and the numerical model are shown; in Section 3, the parameters and
results of numerical simulation are discussed, and the optimized constructions and the
Energies 2023,16, 334 3 of 17
magnetic field distributions are presented; in Section 4, the most important conclusions
are listed.
2. Materials and Methods
In this section, an overview of the introduced waveguide is presented. The structure is
a combination of the magnetic hose for a static magnetic field and a microwave waveguide
with a metamaterial cladding. Hence, the term “magnetic meta-hose” (MMH) was used in
this study to distinguish the considered device from the two abovementioned. Additionally,
the three-dimensional numerical model, used for the analysis of MMH, is characterized.
2.1. Structure of the Magnetic Meta-Hose
The design of the magnetic meta-hose was based on the single-layer hose [
25
], origi-
nally proposed and successfully tested for the static magnetic field. The non-static fields
(i.e., electromagnetic waves) can be transferred by the classic waveguide. However, the
transverse dimensions of the guide must be greater than the size of the wave. For ex-
ample, an electromagnetic wave with a frequency of 100 kHz needs a waveguide with
(approximately) 300 m wide input to propagate inside. To overcome some disadvantages
of hoses and waveguides, the indirect structure can be used. An air was proposed instead
of the ferromagnetic core and the superconducting shell was replaced by the anisotropic
metamaterial cladding (Figure 1a). To ensure a similar effect to the one observed in the
magnetic hose, the cladding was made of stacked resonators and the paramagnetic compos-
ite. As the result, resonators were able to mimic the effect of a near-zero permeability, while
the paramagnetic hybrid composite introduced a higher permeability (
µm
) than the core
(
µcore
= 1) to compensate for the absence of a low-reluctance path. It is important that the
hybrid composite is characterized by the low value of the imaginary part of the permeabil-
ity, resulting from a high surface resistivity [
36
], and thus the electromagnetic loss is vastly
reduced in the kilohertz range.
Energies 2022, 15, x FOR PEER REVIEW 3 of 17
shown; in Section 3, the parameters and results of numerical simulation are discussed,
and the optimized constructions and the magnetic field distributions are presented; in
Section 4, the most important conclusions are listed.
2. Materials and Methods
In this section, an overview of the introduced waveguide is presented. The structure
is a combination of the magnetic hose for a static magnetic field and a microwave
waveguide with a metamaterial cladding. Hence, the term “magnetic meta-hose” (MMH)
was used in this study to distinguish the considered device from the two abovemen-
tioned. Additionally, the three-dimensional numerical model, used for the analysis of
MMH, is characterized.
2.1. Structure of the Magnetic Meta-Hose
The design of the magnetic meta-hose was based on the single-layer hose [25],
originally proposed and successfully tested for the static magnetic field. The non-static
fields (i.e., electromagnetic waves) can be transferred by the classic waveguide. However,
the transverse dimensions of the guide must be greater than the size of the wave. For
example, an electromagnetic wave with a frequency of 100 kHz needs a waveguide with
(approximately) 300 m wide input to propagate inside. To overcome some disadvantages
of hoses and waveguides, the indirect structure can be used. An air was proposed instead
of the ferromagnetic core and the superconducting shell was replaced by the anisotropic
metamaterial cladding (Figure 1a). To ensure a similar effect to the one observed in the
magnetic hose, the cladding was made of stacked resonators and the paramagnetic
composite. As the result, resonators were able to mimic the effect of a near-zero permea-
bility, while the paramagnetic hybrid composite introduced a higher permeability (µ
m
)
than the core (µ
core
= 1) to compensate for the absence of a low-reluctance path. It is im-
portant that the hybrid composite is characterized by the low value of the imaginary part
of the permeability, resulting from a high surface resistivity [36], and thus the electro-
magnetic loss is vastly reduced in the kilohertz range.
(a) (b)
Figure 1. Structure of the medium frequency magnetic meta-hose: (a) the source of the magnetic
field should have been placed near the input port, while the resulting transfer of the magnetic field
is observed at an output port. (b) A single resonator.
Figure 1.
Structure of the medium frequency magnetic meta-hose: (
a
) the source of the magnetic
field should have been placed near the input port, while the resulting transfer of the magnetic field is
observed at an output port. (b) A single resonator.
Energies 2023,16, 334 4 of 17
The inner shell of MMH was made of a non-conductive (and non-magnetic) material
and operated only as a frame, at which the resonators and the composite were mounted.
The length of the hose was d
x
, the width was d
y
and the height was d
z
. Throughout the
research, it was assumed that d
x
= 2.5d
y
and d
y
=d
z
. The composite and resonators were
stacked alternately, where the thickness of the composite layer was d
m
. The number of
layers can be changed, although fewer layers are expected to decrease the efficiency of
MMH, whereas more layers will lead to the extensive construction of the side walls.
Resonators were proposed as the square planar coils with outer size 2r
o
and inner
resection 2r
i
(Figure 1b) and were wound using ultra-thin wires with a diameter of d
w
and an insulation thickness of d
i
(the number of turns was n). This approach may help to
ensure as high inductance as possible since many turns can be used at the limited surface
of the planar inductor. Furthermore, to achieve a specific resonant frequency, the identical
capacitors were connected in series with each coil. Their capacitance was adjusted during
an optimization process.
The lumped parameters of planar inductors can be expressed by analytical formulae;
however, they are valid for an isotropic and non-magnetic environment. A self-inductance
of the planar coil is [37]
L=c1µrµ0davg n2
2hlnc2
v+c3v+c4v2i, (1)
where c
1
,c
2
,c
3
,c
4
are coefficients specific to the different shapes of inductor, e.g., a square,
circular, octagonal, etc. [37]. A mean diameter is
davg =ro+ri, (2)
while a fill factor is defined as
v=ro−ri
ro+ri
, (3)
and an inner resection is expressed as
ri=ro−ndw−(n−1)di. (4)
In Equation (1), the relative permeability (
µr
) of the surrounding area occurs. However,
its value for the mixed type of environment (i.e., the air and layers of paramagnetic
composite), is non-linearly dependent on an amount of paramagnetic material with a
permeability of
µm
; hence, 1 <
µr
<
µm
. To identify
µr
or Lfor the different amounts of
paramagnetic material, extra experiments or numerical calculations would be required.
Nevertheless, Equations (1)–(4), even for
µr
= 1 and the imposed operating frequency (f
r
) of
the external magnetic field, can still be useful to estimate an initial capacitance as
C0=1
(2πfr)2L, (5)
which was used as a starting point for the optimization process.
2.2. Metamaterial Properties
The anisotropic metamaterial is characterized by the complex effective permeability
tensor (if the non-diagonal elements are omitted) [13]
µeff =
µx0 0
0µy0
0 0 µz
, (6)
where at least one component
µx
,
µy
or
µz
has a value that is different from the two others.
In the case of a typical approach without the additional paramagnetic or ferromagnetic
Energies 2023,16, 334 5 of 17
material, according to magnetic field components in Figure 2, the meta-cell simply pos-
sesses
µx
=
µy
= 1 and some complex permeability
µz6=
1. The value of
µz
component is
the result of an adjustment of the geometrical parameters of the inductor and the series
capacitance [38].
Energies 2022, 15, x FOR PEER REVIEW 5 of 17
sesses µ
x
= µ
y
= 1 and some complex permeability µ
z
≠ 1. The value of µ
z
component is the
result of an adjustment of the geometrical parameters of the inductor and the series ca-
pacitance [38].
Figure 2. An exemplary cell of the considered anisotropic metamaterial: the meta-cell acts as an
anisotropic, homogenous medium characterized by the complex effective permeability tensor µ
eff
.
By an application of the paramagnetic composite and limiting the thickness (z-axis)
of the meta-cell to the thickness of the composite (d
m
), the effective permeability is
00
00
00
m
m
z
μ
μ
μ
=
eff
μ
(7)
where µ
z
also depends on µ
m
. Still, it is possible to achieve the near-zero or negative real
part of µ
z
. To estimate its value, numerical computations need to be conducted since
available analytical formulae are limited to cases where µ
m
= 1 [38,39].
2.3. Simulation Model
The three-dimensional (3D) numerical model of MMH was created in COMSOL
Multiphysics software, as shown in Figure 3. During simulations, only a quarter of the
hose was represented due to symmetry of the system. The Magnetic Insulation boundary
condition [40]
0
0×=nA
(8)
was assigned to an external xz surface, and the Perfect Magnetic Conductor boundary
condition [40]
0
0×=nH
(9)
was used on xy boundary of the model. Where n
0
was the normal vector to the xy surface,
A was the magnetic vector potential and H was the magnetic field. The Magnetic Insula-
tion was also applied on the remaining external surfaces to imitate an expansive space
outside the model, where the magnetic field tends to zero (isotropic, dielectric back-
ground). The infinitely long and negligibly thin conductor with the current I, located at a
distance d
I
= r
o
from an input port, was the source of the external magnetic field. The ports
were interpreted as surfaces at the beginning and the end of MMH. On these surfaces,
magnetic field and magnetic energy were then examined.
The multi-turn planar inductors were modeled using the current sheet approximation
approach [41–43], known in the COMSOL software as the Multi-turn coil boundary con-
dition [44]. Many turns (nearly 200) and thin wires (d
w
<< r
o
) must be imitated, leading to
an undesirably high number of degrees of freedom. To reduce the memory demand of a
computational unit, each coil was homogenized to four ‘sheets’, characterized by pa-
rameters such as the number of turns (n), an electrical conductivity (σ
w
), a winding cross
section (a
w
), and the total diameter of the wire (d
i
+ d
w
). Additionally, a lumped capacitor
(C) was connected in series.
Figure 2.
An exemplary cell of the considered anisotropic metamaterial: the meta-cell acts as an
anisotropic, homogenous medium characterized by the complex effective permeability tensor µeff.
By an application of the paramagnetic composite and limiting the thickness (z-axis) of
the meta-cell to the thickness of the composite (dm), the effective permeability is
µeff =
µm0 0
0µm0
0 0 µz
(7)
where
µz
also depends on
µm
. Still, it is possible to achieve the near-zero or negative
real part of
µz
. To estimate its value, numerical computations need to be conducted since
available analytical formulae are limited to cases where µm=1[38,39].
2.3. Simulation Model
The three-dimensional (3D) numerical model of MMH was created in COMSOL
Multiphysics software, as shown in Figure 3. During simulations, only a quarter of the
hose was represented due to symmetry of the system. The Magnetic Insulation boundary
condition [40]
n0×A=0 (8)
was assigned to an external xz surface, and the Perfect Magnetic Conductor boundary condi-
tion [40]
n0×H=0 (9)
was used on xy boundary of the model. where
n0
was the normal vector to the xy surface,
A
was the magnetic vector potential and
H
was the magnetic field. The Magnetic Insulation
was also applied on the remaining external surfaces to imitate an expansive space outside
the model, where the magnetic field tends to zero (isotropic, dielectric background). The
infinitely long and negligibly thin conductor with the current I, located at a distance
d
I
=r
o
from an input port, was the source of the external magnetic field. The ports were
interpreted as surfaces at the beginning and the end of MMH. On these surfaces, magnetic
field and magnetic energy were then examined.
The multi-turn planar inductors were modeled using the current sheet approximation
approach [
41
–
43
], known in the COMSOL software as the Multi-turn coil boundary con-
dition [
44
]. Many turns (nearly 200) and thin wires (d
w
<< r
o
) must be imitated, leading
to an undesirably high number of degrees of freedom. To reduce the memory demand
of a computational unit, each coil was homogenized to four ‘sheets’, characterized by
parameters such as the number of turns (n), an electrical conductivity (
σw
), a winding cross
section (a
w
), and the total diameter of the wire (d
i
+d
w
). Additionally, a lumped capacitor
(C) was connected in series.
Energies 2023,16, 334 6 of 17
Energies 2022, 15, x FOR PEER REVIEW 6 of 17
Figure 3. Numerical model of the magnetic meta-hose with a line conductor as the source of the
magnetic field. The model was analyzed as a quarter of the original structure, and symmetry con-
ditions were applied on the external xz and xy surfaces.
The system was analyzed in the frequency domain with angular frequency ω, with
respect to the magnetic vector potential (A). The external current density J
e
= f(I) acted as
the magnetic field source. The Helmholtz equation was used to find A and the magnetic
field (H) [45]
1j
e
ωσ
μ
∇× ∇× − =
AAJ
, (10)
1
μ
=∇×HA
, (11)
Within the domains characterized by the known magnetic permeability (µ) and the
electrical conductivity (σ). These parameters, the homogenized inductors, Equations (8)–
(11) and the line current source I = 1 A were sufficient to describe the electromagnetic
phenomena in the model and find the distribution of magnetic field (H). The system was
solved numerically using the finite element method (FEM) [41,46], where the number of
degrees of freedom resulting from a finite element mesh was around 500,000.
3. Results and Discussion
The magnetic meta-hose was analyzed for frequencies from 10 kHz to 1 MHz. The
parameters of resonators and a nonconductive shell were presented in Table 1. To make
the study more general, the fill factor (ν) was used instead of the number of turns (n) to
show the influence of the inductor on the magnetic field transfer. The second variable
was the relative magnetic permeability (µ
m
) of a composite. Two operating frequencies
were considered: f
r
= 25 kHz and f
r
= 100 kHz, to show the dependability and effectiveness
of the system on the frequency of an external magnetic field.
Since the magnetic field was considered in the study, the related quantities were
calculated. To show the increase in the magnetic energy at the output port, the gain of
energy transfer was calculated as
Figure 3.
Numerical model of the magnetic meta-hose with a line conductor as the source of the
magnetic field. The model was analyzed as a quarter of the original structure, and symmetry
conditions were applied on the external xz and xy surfaces.
The system was analyzed in the frequency domain with angular frequency
ω
, with
respect to the magnetic vector potential (
A
). The external current density
Je
= f(I) acted as
the magnetic field source. The Helmholtz equation was used to find
A
and the magnetic
field (H) [45]
∇ × 1
µ∇ × A−jωσA=Je, (10)
H=1
µ∇ × A, (11)
Within the domains characterized by the known magnetic permeability (
µ
) and the electri-
cal conductivity (
σ
). These parameters, the homogenized inductors, Equations (8)–(11) and the
line current source I= 1 A were sufficient to describe the electromagnetic phenomena in the
model and find the distribution of magnetic field (
H
). The system was solved numerically
using the finite element method (FEM) [
41
,
46
], where the number of degrees of freedom
resulting from a finite element mesh was around 500,000.
3. Results and Discussion
The magnetic meta-hose was analyzed for frequencies from 10 kHz to 1 MHz. The
parameters of resonators and a nonconductive shell were presented in Table 1. To make
the study more general, the fill factor (
ν
) was used instead of the number of turns (n) to
show the influence of the inductor on the magnetic field transfer. The second variable was
the relative magnetic permeability (
µm
) of a composite. Two operating frequencies were
considered: f
r
= 25 kHz and f
r
= 100 kHz, to show the dependability and effectiveness of
the system on the frequency of an external magnetic field.
Energies 2023,16, 334 7 of 17
Table 1. Parameters used during simulations.
Length dx50 cm
Width dy20 cm
Height dz20 cm
Permeability of background and coils µ04π·10−7H/m
Electrical conductivity of background
and paramagnetic composite σ00 S/m
Electrical conductivity of wires σw5.8 ×107S/m
Wire diameter dw0.2 mm
Wire insulation thickness di0.02 mm
Composite thickness dm2 mm
Coil diameter 2ro10 cm
Shape coefficients
c11.46
c21.9
c30.18
c40.13
Nelder-Mead reflection coeffcient α1
Nelder-Mead expansion coeffcient γ2
Nelder-Mead contraction coeffcient ρ10.25
Nelder-Mead shrink coeffcient ρ20.25
Optimality tolerance ε0.01
Maximum number of iterations Max_it 20
Since the magnetic field was considered in the study, the related quantities were
calculated. To show the increase in the magnetic energy at the output port, the gain of
energy transfer was calculated as
η=
s
Sout
µ
Hν,µm
2ds
s
Sout
µ|H0|2ds , (12)
while the ability to preserve the high value of the magnetic field was defined as a ratio
between the mean magnetic field at the output port and the input port
k=
s
Sout
Hν,µm
ds
s
Sin
Hν,µm
ds . (13)
where S
in
is the area of the input port, S
out
is the area of the output port, |
H0|
is the
magnetic field norm at the output port without MMH and |
Hν,µm|
is the magnetic field
norm at the output port with MMH, characterized by the fill factor
ν
and the composite
with relative permeability µm. Hence, ηand kwere calculated with respect to νand µm.
The capacitor establishes a resonant frequency of the metamaterial; thus, a precise
adjustment of the capacitance was necessary. To automatize this process, the optimization
algorithm (Nelder-Mead method) was utilized. The algorithm was conjugated with the
numerical model and its task was to maximize the one-dimensional objective function
η= f(C). Therefore, the following procedure was realized:
(1)
Start from ν= 0 and µm= 1.
(2)
Estimate the inductance (L) using Equations (1)–(4).
(3)
Estimate the initial capacitance (C0) using Equation (5).
(4)
Run the simulation and optimization algorithm.
(5)
Adjust the capacitance during the optimization process to find the maximum η.
(6)
Stop and save the magnetic field distribution after identification of max(η).
(7)
Increase ν. If ν> 0.9 then increase µmand set ν= 0.
(8)
Execute points (2)–(7).
Energies 2023,16, 334 8 of 17
As the result, at a certain point (
ν
,
µm
) only optimized values of Cand
η
were saved,
while kwas calculated. The exemplary values of optimized capacitances were listed in
Tables S1 and S2 in the Supplementary Material.
3.1. Optimum Cases for Different Parameters of Metamaterial
After performing the optimization process, the maximum values of magnetic energy
gain, i.e., max(
η
(
ν
,
µm
)) and corresponding magnetic field ratio k(
ν
,
µm
) were saved. Then,
from the entire set of results, the best cases were found. On these bases it was possible to
identify a structure of a coil (
ν
) and properties of composite (
µm
) which ensured the most
efficient MMH for fixed geometrical parameters of the hose.
At the operating frequency f
r
= 25 kHz, the maximum energy gain
ηbest
= 1260.5 was
found for
ν
= 0.15 and
µm
= 150, as shown in Figure 4a. Compared to the case without
any hose, the energy density increased more than three orders of magnitude. For
µm
= 1
and different inductors (dashed red line in Figure 4a), the best energy gain was
η
= 26.86
(
ν
= 0.29). While for different permeabilities (without resonators), it was less than
η
= 1.7
(
µm
= 150). The differences in
ηbest
shows that it is unlikely to achieve the high energy gain
using only resonators or paramagnetic/ferromagnetic material. However, by combining
these two approaches, at some specific point, the gain can be elevated significantly. A
similar situation was found with the magnetic field ratio k, where the best case kbest = 0.39
was identified for
ν
= 0.45 and
µm
= 26, and an area of high values of kin Figure 4b was
much wider than in Figure 4a. This suggests that the magnetic field transfer “from input to
output” of MMH can be achieved with many combinations of
ν
and
µm
. Nevertheless, the
utilization of resonators (k= 0.32) or the composite material (k= 0.01) gave poorer results
than their combination.
Energies 2022, 15, x FOR PEER REVIEW 8 of 17
(8) Execute points (2)–(7).
As the result, at a certain point (ν, µ
m
) only optimized values of C and η were saved,
while k was calculated. The exemplary values of optimized capacitances were listed in
Tables S1 and S2 in the Supplementary Material.
3.1. Optimum Cases for Different Parameters of Metamaterial
After performing the optimization process, the maximum values of magnetic energy
gain, i.e., max(η(ν, µ
m
)) and corresponding magnetic field ratio k(ν, µ
m
) were saved. Then,
from the entire set of results, the best cases were found. On these bases it was possible to
identify a structure of a coil (ν) and properties of composite (µ
m
) which ensured the most
efficient MMH for fixed geometrical parameters of the hose.
At the operating frequency f
r
= 25 kHz, the maximum energy gain η
best
= 1260.5 was
found for ν = 0.15 and µ
m
= 150, as shown in Figure 4a. Compared to the case without any
hose, the energy density increased more than three orders of magnitude. For µ
m
= 1 and
different inductors (dashed red line in Figure 4a), the best energy gain was η = 26.86 (ν =
0.29). While for different permeabilities (without resonators), it was less than η = 1.7 (µ
m
=
150). The differences in η
best
shows that it is unlikely to achieve the high energy gain using
only resonators or paramagnetic/ferromagnetic material. However, by combining these
two approaches, at some specific point, the gain can be elevated significantly. A similar
situation was found with the magnetic field ratio k, where the best case k
best
= 0.39 was
identified for ν = 0.45 and µ
m
= 26, and an area of high values of k in Figure 4b was much
wider than in Figure 4a. This suggests that the magnetic field transfer “from input to
output” of MMH can be achieved with many combinations of ν and µ
m
. Nevertheless, the
utilization of resonators (k = 0.32) or the composite material (k = 0.01) gave poorer results
than their combination.
(a) (b)
Figure 4. The results of maximizing objective function η = f(C) for different fill factors ν and per-
meabilities µ
m
at f
r
= 25 kHz: (a) energy transfer gain η and (b) magnetic field ratio k.
For f
r
= 100 kHz, the gain and ratio increased, as shown in Figure 5. The best case η
best
= 8466 (nearly 7 times higher than at f
r
= 25 kHz) was found for ν = 0.15 and µ
m
= 124. At
this time the highest assumed value of µ
m
(i.e., 150) was not required to maximize the
gain. While the fill factor ν for 25 kHz and 100 kHz, respectively, was 0.15 which (in this
model) suggests that a large number of turns was not necessary to maximize a magnetic
coupling. It is worth noting that more turns will increase the inductance and resistance,
and hence after some ν, the quality factor Q = ωL/R starts decreasing, limiting the effec-
Figure 4.
The results of maximizing objective function
η
= f(C) for different fill factors
ν
and perme-
abilities µmat fr= 25 kHz: (a) energy transfer gain ηand (b) magnetic field ratio k.
For f
r
= 100 kHz, the gain and ratio increased, as shown in Figure 5. The best case
ηbest
= 8466 (nearly 7 times higher than at f
r
= 25 kHz) was found for
ν
= 0.15 and
µm
= 124.
At this time the highest assumed value of
µm
(i.e., 150) was not required to maximize the
gain. While the fill factor
ν
for 25 kHz and 100 kHz, respectively, was 0.15 which (in this
model) suggests that a large number of turns was not necessary to maximize a magnetic
coupling. It is worth noting that more turns will increase the inductance and resistance, and
hence after some
ν
,the quality factor Q=
ω
L/Rstarts decreasing, limiting the effectiveness
of the magnetic energy transfer. What is more, the highest ratio k
best
= 1.15 was found again
for
ν
= 0.45 and
µm
= 26 and shows a possibility of not only transferring the magnetic field
Energies 2023,16, 334 9 of 17
with no attenuation but also an ability to enhance it at the end of the hose. Comparing to
f
r
= 25 kHz the area of high values of k(yellow parts) had changed. Here, the number of
turns of inductors may have a more substantial impact on transferring the magnetic field
from the input to the output of MMH than the permeability of the composite material.
Energies 2022, 15, x FOR PEER REVIEW 9 of 17
tiveness of the magnetic energy transfer. What is more, the highest ratio k
best
= 1.15 was
found again for ν = 0.45 and µ
m
= 26 and shows a possibility of not only transferring the
magnetic field with no attenuation but also an ability to enhance it at the end of the hose.
Comparing to f
r
= 25 kHz the area of high values of k (yellow parts) had changed. Here,
the number of turns of inductors may have a more substantial impact on transferring the
magnetic field from the input to the output of MMH than the permeability of the com-
posite material.
(a) (b)
Figure 5. The results of maximizing objective function η = f(C) for different fill factors ν and per-
meabilities µ
m
at f
r
= 100 kHz: (a) energy transfer gain η and (b) magnetic field ratio k.
3.2. Magnetic Field Distribution
To show the effect of the magnetic field transfer, the distribution of |H|, in the best
cases, was shown in this section. In the beginning, the cases without any hose and only
with paramagnetic cladding were calculated. Additionally, the power flow was shown.
For the line current source in Figure 6, that is, an infinitely thin conductor, the
magnetic field distribution has a typical shape. The magnetic field around the source has
the highest values, while with growing distance, the magnetic flux tends to zero. The
power flow was directed outside the conductor; hence, the field propagates to a distant
space. In this case, the mean magnetic field norm (≈0.46 A/m) and the energy density
(≈2.9 nJ/m) at the distance r
o
+ d
x
(the output of MMH) were negligible, and at the distance
r
o
(the input of MMH), their values were 7 A/m and 361 nJ/m, respectively. The situation
changed after applying the composite material with µ
m
= 150, as shown in Figure 7.
However, at the output, the mean magnetic field was reduced (0.44 A/m) while the
magnetic energy increased (4.9 nJ/m). As the result, the paramagnetic material was not
able to achieve the task of transferring the magnetic field. The only visible effect on the
magnetic field distribution was the change in the curvature of isolines at the input port of
the hose. The magnetic flux had been ‘drawn’ into the hose, but the effective distance of
the transfer was less than r
o
.
Identical values and distributions were observed for both 25 and 100 kHz, due to the
lack of conductive material in the system and no dispersion of permeability.
Figure 5.
The results of maximizing objective function
η
= f(C) for different fill factors
ν
and perme-
abilities µmat fr= 100 kHz: (a) energy transfer gain ηand (b) magnetic field ratio k.
3.2. Magnetic Field Distribution
To show the effect of the magnetic field transfer, the distribution of |
H|
, in the best
cases, was shown in this section. In the beginning, the cases without any hose and only
with paramagnetic cladding were calculated. Additionally, the power flow was shown.
For the line current source in Figure 6, that is, an infinitely thin conductor, the magnetic
field distribution has a typical shape. The magnetic field around the source has the highest
values, while with growing distance, the magnetic flux tends to zero. The power flow was
directed outside the conductor; hence, the field propagates to a distant space. In this case,
the mean magnetic field norm (
≈
0.46 A/m) and the energy density (
≈
2.9 nJ/m) at the
distance r
o
+d
x
(the output of MMH) were negligible, and at the distance r
o
(the input
of MMH), their values were 7 A/m and 361 nJ/m, respectively. The situation changed
after applying the composite material with
µm
= 150, as shown in Figure 7. However, at
the output, the mean magnetic field was reduced (0.44 A/m) while the magnetic energy
increased (4.9 nJ/m). As the result, the paramagnetic material was not able to achieve
the task of transferring the magnetic field. The only visible effect on the magnetic field
distribution was the change in the curvature of isolines at the input port of the hose. The
magnetic flux had been ‘drawn’ into the hose, but the effective distance of the transfer was
less than ro.
Identical values and distributions were observed for both 25 and 100 kHz, due to the
lack of conductive material in the system and no dispersion of permeability.
Comparing Figure 7with Figure 8, the differences in the magnetic field distribution
are clearly visible. At f
r
= 25 kHz, the magnetic flux was ’pulled’ in the MMH which can be
noticed as an orange area (1
÷
4 A/m) inside the hose. The highest magnetic field in the
hose was found in the resonators and the composite material. The Poyting vectors showed
that the power traveled across the shell and then returned to an air core. Additionally,
a kind of the power circulation can also be observed near the conductor (in Figure 6the
vectors were directed from the conductor to the outside space). The meta-hose visibly
changed the magnetic field distribution and the power flow. This helped to achieve the
increase in the magnetic field norm at the output (1.37 A/m) and magnetic field energy
Energies 2023,16, 334 10 of 17
density (992 nJ/m). Although the magnetic flux had not reached the value initially found
at the input port (7 A/m), the magnetic energy gain was significant.
Energies 2022, 15, x FOR PEER REVIEW 10 of 17
Figure 6. Magnetic field norm (color) and normalized power flow (black arrows) for the model
without the magnetic meta-hose.
Figure 7. Magnetic field norm (color) and normalized power flow (black arrows) for the model
with the paramagnetic cladding (the highest considered permeability, µ
m
= 150) without resonators.
Comparing Figure 7 with Figure 8, the differences in the magnetic field distribution
are clearly visible. At f
r
= 25 kHz, the magnetic flux was ’pulled’ in the MMH which can
be noticed as an orange area (1 ÷ 4 A/m) inside the hose. The highest magnetic field in the
hose was found in the resonators and the composite material. The Poyting vectors
showed that the power traveled across the shell and then returned to an air core. Addi-
tionally, a kind of the power circulation can also be observed near the conductor (in Fig-
Figure 6.
Magnetic field norm (color) and normalized power flow (black arrows) for the model
without the magnetic meta-hose.
Energies 2022, 15, x FOR PEER REVIEW 10 of 17
Figure 6. Magnetic field norm (color) and normalized power flow (black arrows) for the model
without the magnetic meta-hose.
Figure 7. Magnetic field norm (color) and normalized power flow (black arrows) for the model
with the paramagnetic cladding (the highest considered permeability, µ
m
= 150) without resonators.
Comparing Figure 7 with Figure 8, the differences in the magnetic field distribution
are clearly visible. At f
r
= 25 kHz, the magnetic flux was ’pulled’ in the MMH which can
be noticed as an orange area (1 ÷ 4 A/m) inside the hose. The highest magnetic field in the
hose was found in the resonators and the composite material. The Poyting vectors
showed that the power traveled across the shell and then returned to an air core. Addi-
tionally, a kind of the power circulation can also be observed near the conductor (in Fig-
Figure 7.
Magnetic field norm (color) and normalized power flow (black arrows) for the model with
the paramagnetic cladding (the highest considered permeability, µm= 150) without resonators.
Energies 2023,16, 334 11 of 17
Energies 2022, 15, x FOR PEER REVIEW 11 of 17
ure 6 the vectors were directed from the conductor to the outside space). The meta-hose
visibly changed the magnetic field distribution and the power flow. This helped to
achieve the increase in the magnetic field norm at the output (1.37 A/m) and magnetic
field energy density (992 nJ/m). Although the magnetic flux had not reached the value
initially found at the input port (7 A/m), the magnetic energy gain was significant.
Figure 8. Magnetic field norm (color) and normalized power flow (black arrows) for the best model
with the magnetic meta-hose (ν = 0.15, µ
m
= 150) at f
r
= 25 kHz.
Further observations regarding the spatial distribution of the magnetic field come
from Figures 8–10. A higher magnetic field can be observed around the centrally located
resonators than the field around resonators closer to the source. This was probably due to
the induced current inside the coils, which also was the highest in central inductors (e.g.,
for the waveguide in Figure 10, the current of the central coil was 3.95 mA, while the
current of the coil at the input port was 1.59 mA). Furthermore, the ‘reversed’ power flow
in the center of the hose appeared. Numerical calculations indicated that the power den-
sity inside MMH, mostly near the input port, was greater than in the surrounding space.
This confirmed the effect of ‘pulling’ the field and energy into the hose. The changed di-
rection of the Poyting vector arises, most probably, from the metamaterial cladding. In
these examples, the interior of the magnetic hose acted as another metamaterial medium
(with a negative refractive index), changing the direction of the x-component of the
Poyting vector.
Figure 8.
Magnetic field norm (color) and normalized power flow (black arrows) for the best model
with the magnetic meta-hose (ν= 0.15, µm= 150) at fr= 25 kHz.
Further observations regarding the spatial distribution of the magnetic field come
from Figures 8–10. A higher magnetic field can be observed around the centrally located
resonators than the field around resonators closer to the source. This was probably due
to the induced current inside the coils, which also was the highest in central inductors
(e.g., for the waveguide in Figure 10, the current of the central coil was 3.95 mA, while
the current of the coil at the input port was 1.59 mA). Furthermore, the ‘reversed’ power
flow in the center of the hose appeared. Numerical calculations indicated that the power
density inside MMH, mostly near the input port, was greater than in the surrounding
space. This confirmed the effect of ‘pulling’ the field and energy into the hose. The
changed direction of the Poyting vector arises, most probably, from the metamaterial
cladding. In these examples, the interior of the magnetic hose acted as another metamaterial
medium (with a negative refractive index), changing the direction of the x-component of the
Poyting vector.
At f
r
= 100 kHz, as shown in Figure 9, the enhancement of the mean magnetic field at
the output was more effective (
≈
5.7 A/m), and the magnetic energy reached the highest
gain (3.64
µ
J/m). It can also be noticed that the field was nearly uniform across MMH, with
values of 4
÷
5 A/m, and transferred to the end of the hose. The power flow was similar to
those in Figure 8. The last example in Figure 10 shows the magnetic field distribution for
the case with the maximum k. The mean magnetic field at the output port reached nearly
8 A/m. While the magnetic flux was transferred effectively inside the air core, the field
of centrally located resonators was almost as high as the one near the line conductor. The
effect can be observed better in the 3D projection in Figure 10a.
Energies 2023,16, 334 12 of 17
Energies 2022, 15, x FOR PEER REVIEW 12 of 17
Figure 9. Magnetic field norm (color) and normalized power flow (black arrows) for the best model
with the magnetic meta-hose (ν = 0.15, µ
m
= 124) at f
r
= 100 kHz.
(a) (b)
Figure 10. Magnetic field norm (color) and normalized power flow (black arrows) for the model
with the highest k = 1.15, i.e., the magnetic meta-hose (ν = 0.45, µ
m
= 26) at f
r
= 100 kHz: (a) 3D view
on the system with xz and xy cut planes and (b) 2D view on xy cut plane.
At f
r
= 100 kHz, as shown in Figure 9, the enhancement of the mean magnetic field at
the output was more effective (≈5.7 A/m), and the magnetic energy reached the highest
gain (3.64 µJ/m). It can also be noticed that the field was nearly uniform across MMH,
with values of 4 ÷ 5 A/m, and transferred to the end of the hose. The power flow was
similar to those in Figure 8. The last example in Figure 10 shows the magnetic field dis-
tribution for the case with the maximum k. The mean magnetic field at the output port
reached nearly 8 A/m. While the magnetic flux was transferred effectively inside the air
core, the field of centrally located resonators was almost as high as the one near the line
conductor. The effect can be observed better in the 3D projection in Figure 10a.
Figure 9.
Magnetic field norm (color) and normalized power flow (black arrows) for the best model
with the magnetic meta-hose (ν= 0.15, µm= 124) at fr= 100 kHz.
Energies 2022, 15, x FOR PEER REVIEW 12 of 17
Figure 9. Magnetic field norm (color) and normalized power flow (black arrows) for the best model
with the magnetic meta-hose (ν = 0.15, µ
m
= 124) at f
r
= 100 kHz.
(a) (b)
Figure 10. Magnetic field norm (color) and normalized power flow (black arrows) for the model
with the highest k = 1.15, i.e., the magnetic meta-hose (ν = 0.45, µ
m
= 26) at f
r
= 100 kHz: (a) 3D view
on the system with xz and xy cut planes and (b) 2D view on xy cut plane.
At f
r
= 100 kHz, as shown in Figure 9, the enhancement of the mean magnetic field at
the output was more effective (≈5.7 A/m), and the magnetic energy reached the highest
gain (3.64 µJ/m). It can also be noticed that the field was nearly uniform across MMH,
with values of 4 ÷ 5 A/m, and transferred to the end of the hose. The power flow was
similar to those in Figure 8. The last example in Figure 10 shows the magnetic field dis-
tribution for the case with the maximum k. The mean magnetic field at the output port
reached nearly 8 A/m. While the magnetic flux was transferred effectively inside the air
core, the field of centrally located resonators was almost as high as the one near the line
conductor. The effect can be observed better in the 3D projection in Figure 10a.
Figure 10.
Magnetic field norm (color) and normalized power flow (black arrows) for the model with
the highest k = 1.15, i.e., the magnetic meta-hose (
ν
= 0.45,
µm
= 26) at f
r
= 100 kHz: (
a
) 3D view on
the system with xz and xy cut planes and (b) 2D view on xy cut plane.
3.3. Frequency Response of MMH
The frequency responses of the magnetic meta-hose were calculated to study their
ability to enhance the magnetic energy transfer at the medium frequency range. Only
the best cases from Section 3.1 were analyzed. As shown in Figure 11, the hose with the
highest gain at f
r
= 25 kHz (
η
= 392.11) achieved a nearly identical value at f= 112.2 kHz
(η= 318.24), while at f= 84.14 kHz, the gain was even higher (η= 417.03).
Energies 2023,16, 334 13 of 17
Energies 2022, 15, x FOR PEER REVIEW 13 of 17
3.3. Frequency Response of MMH
The frequency responses of the magnetic meta-hose were calculated to study their
ability to enhance the magnetic energy transfer at the medium frequency range. Only the
best cases from Section 3.1 were analyzed. As shown in Figure 11, the hose with the
highest gain at fr = 25 kHz (η = 392.11) achieved a nearly identical value at f = 112.2 kHz (η
= 318.24), while at f = 84.14 kHz, the gain was even higher (η = 417.03).
Figure 11. Frequency response of the magnetic meta-hose with the highest energy transfer gain η (ν
= 0.15, µm = 150) identified at fr = 25 kHz.
This example shows that MMH acts as a band-pass filter, able to transfer the mag-
netic energy for the designed frequency (e.g., 25 kHz) and simultaneously in the higher
frequencies range. It was also confirmed by the magnetic field ratio k, which reached the
highest values (for the abovementioned frequencies) with the peak k = 0.71 at f = 84.14
kHz (more than four times higher than k = 0.172 at f = 25 kHz). In Figure 11 and Figure 12,
it can be seen that obtaining the optimal gain at some imposed frequency will not exclude
the possibility of the effective magnetic field transfer at the other (higher) frequencies.
Figure 12. Frequency response of the magnetic meta-hose with the highest magnetic field ratio k (ν
= 0.45, µm = 26) identified at fr = 25 kHz.
The results were much more promising for the operating frequency fr = 100 kHz,
shown in Figure 13. The gain η was nearly 8466.06 (f = 100 kHz) and 594.75 (f = 316.23
kHz) as well as 1107.29 (f = 421.7 kHz), which was at least an order of magnitude more
0.0
0.3
0.6
0.9
1
10
100
1000
10 100 1000
kη
Frequency (kHz)
η
k
0.0
0.3
0.6
0.9
1
10
100
1000
10 100 1000
kη
Frequency (kHz)
η
k
Figure 11.
Frequency response of the magnetic meta-hose with the highest energy transfer gain
η
(ν= 0.15, µm= 150) identified at fr= 25 kHz.
This example shows that MMH acts as a band-pass filter, able to transfer the magnetic
energy for the designed frequency (e.g., 25 kHz) and simultaneously in the higher frequen-
cies range. It was also confirmed by the magnetic field ratio k, which reached the highest
values (for the abovementioned frequencies) with the peak k= 0.71 at f= 84.14 kHz (more
than four times higher than k= 0.172 at f= 25 kHz). In Figures 11 and 12, it can be seen
that obtaining the optimal gain at some imposed frequency will not exclude the possibility
of the effective magnetic field transfer at the other (higher) frequencies.
Energies 2022, 15, x FOR PEER REVIEW 13 of 17
3.3. Frequency Response of MMH
The frequency responses of the magnetic meta-hose were calculated to study their
ability to enhance the magnetic energy transfer at the medium frequency range. Only the
best cases from Section 3.1 were analyzed. As shown in Figure 11, the hose with the
highest gain at fr = 25 kHz (η = 392.11) achieved a nearly identical value at f = 112.2 kHz (η
= 318.24), while at f = 84.14 kHz, the gain was even higher (η = 417.03).
Figure 11. Frequency response of the magnetic meta-hose with the highest energy transfer gain η (ν
= 0.15, µm = 150) identified at fr = 25 kHz.
This example shows that MMH acts as a band-pass filter, able to transfer the mag-
netic energy for the designed frequency (e.g., 25 kHz) and simultaneously in the higher
frequencies range. It was also confirmed by the magnetic field ratio k, which reached the
highest values (for the abovementioned frequencies) with the peak k = 0.71 at f = 84.14
kHz (more than four times higher than k = 0.172 at f = 25 kHz). In Figure 11 and Figure 12,
it can be seen that obtaining the optimal gain at some imposed frequency will not exclude
the possibility of the effective magnetic field transfer at the other (higher) frequencies.
Figure 12. Frequency response of the magnetic meta-hose with the highest magnetic field ratio k (ν
= 0.45, µm = 26) identified at fr = 25 kHz.
The results were much more promising for the operating frequency fr = 100 kHz,
shown in Figure 13. The gain η was nearly 8466.06 (f = 100 kHz) and 594.75 (f = 316.23
kHz) as well as 1107.29 (f = 421.7 kHz), which was at least an order of magnitude more
0.0
0.3
0.6
0.9
1
10
100
1000
10 100 1000
kη
Frequency (kHz)
η
k
0.0
0.3
0.6
0.9
1
10
100
1000
10 100 1000
kη
Frequency (kHz)
η
k
Figure 12.
Frequency response of the magnetic meta-hose with the highest magnetic field ratio k
(ν= 0.45, µm= 26) identified at fr= 25 kHz.
The results were much more promising for the operating frequency f
r
= 100 kHz, shown
in Figure 13. The gain
η
was nearly 8466.06 (f= 100 kHz) and 594.75 (f= 316.23 kHz) as well
as 1107.29 (f= 421.7 kHz), which was at least an order of magnitude more than the values
observed for f
r
= 25 kHz. The high values of
η
were also accompanied by the elevated
values of k. The imposed frequency f
r
= 100 kHz allowed us to achieve k= 0.715 and, similar
to the previous cases, two additional peaks were found where kwas more significant. For
f= 316.23 kHz the magnetic field ratio exceeded 1, which means the magnetic field was
amplified at the output (compared to the magnetic field at the input).
Energies 2023,16, 334 14 of 17
Energies 2022, 15, x FOR PEER REVIEW 14 of 17
than the values observed for fr = 25 kHz. The high values of η were also accompanied by
the elevated values of k. The imposed frequency fr = 100 kHz allowed us to achieve k =
0.715 and, similar to the previous cases, two additional peaks were found where k was
more significant. For f = 316.23 kHz the magnetic field ratio exceeded 1, which means the
magnetic field was amplified at the output (compared to the magnetic field at the input).
Figure 13. Frequency response of the magnetic meta-hose with the highest gain of energy transfer η
(ν = 0.15, µm = 124) identified at fr = 100 kHz.
In Figure 14, the frequency response of MMH with the highest recorded k at fr = 100
kHz was presented. The same observations can be made in Figure 12. In the considered
frequency range, the highest gain and ratio were found at the imposed frequency of 100
kHz, and two subsequent peaks were nearly combined. By analyzing Figures 10–13 one
may conclude that magnetic meta-hose was able to act as a spatial band-pass filter for the
transient magnetic field. The passed frequencies can be adjusted, for instance, by chang-
ing the parameters of the inductors and the capacitor.
Figure 14. Frequency response of the magnetic meta-hose with the highest magnetic field ratio k (ν
= 0.45, µm = 26) identified at fr = 100 kHz.
The analyzed cases indicated that MMH led to greater values of energy density gain
and magnetic field ratios with higher operating frequencies. However, for the frequen-
cies over 1 MHz some negative effects, such as the eddy currents, dispersion, and a
magnetic loss in a composite material will be observed, leading to a reduced ability of
MMH to transfer the magnetic field and/or electromagnetic waves effectively. The prac-
0.0
0.3
0.6
0.9
1.2
1
10
100
1000
10000
10 100 1000
kη
Frequency (kHz)
η
k
10,000
0.0
0.3
0.6
0.9
1.2
1
10
100
1000
10000
10 100 1000
kη
Frequency (kHz)
η
k
10,000
Figure 13.
Frequency response of the magnetic meta-hose with the highest gain of energy transfer
η
(ν= 0.15, µm= 124) identified at fr= 100 kHz.
In Figure 14, the frequency response of MMH with the highest recorded kat f
r
= 100 kHz
was presented. The same observations can be made in Figure 12. In the considered
frequency range, the highest gain and ratio were found at the imposed frequency of
100 kHz, and two subsequent peaks were nearly combined. By analyzing Figures 10–13 one
may conclude that magnetic meta-hose was able to act as a spatial band-pass filter for the
transient magnetic field. The passed frequencies can be adjusted, for instance, by changing
the parameters of the inductors and the capacitor.
Energies 2022, 15, x FOR PEER REVIEW 14 of 17
than the values observed for fr = 25 kHz. The high values of η were also accompanied by
the elevated values of k. The imposed frequency fr = 100 kHz allowed us to achieve k =
0.715 and, similar to the previous cases, two additional peaks were found where k was
more significant. For f = 316.23 kHz the magnetic field ratio exceeded 1, which means the
magnetic field was amplified at the output (compared to the magnetic field at the input).
Figure 13. Frequency response of the magnetic meta-hose with the highest gain of energy transfer η
(ν = 0.15, µm = 124) identified at fr = 100 kHz.
In Figure 14, the frequency response of MMH with the highest recorded k at fr = 100
kHz was presented. The same observations can be made in Figure 12. In the considered
frequency range, the highest gain and ratio were found at the imposed frequency of 100
kHz, and two subsequent peaks were nearly combined. By analyzing Figures 10–13 one
may conclude that magnetic meta-hose was able to act as a spatial band-pass filter for the
transient magnetic field. The passed frequencies can be adjusted, for instance, by chang-
ing the parameters of the inductors and the capacitor.
Figure 14. Frequency response of the magnetic meta-hose with the highest magnetic field ratio k (ν
= 0.45, µm = 26) identified at fr = 100 kHz.
The analyzed cases indicated that MMH led to greater values of energy density gain
and magnetic field ratios with higher operating frequencies. However, for the frequen-
cies over 1 MHz some negative effects, such as the eddy currents, dispersion, and a
magnetic loss in a composite material will be observed, leading to a reduced ability of
MMH to transfer the magnetic field and/or electromagnetic waves effectively. The prac-
0.0
0.3
0.6
0.9
1.2
1
10
100
1000
10000
10 100 1000
kη
Frequency (kHz)
η
k
10,000
0.0
0.3
0.6
0.9
1.2
1
10
100
1000
10000
10 100 1000
kη
Frequency (kHz)
η
k
10,000
Figure 14.
Frequency response of the magnetic meta-hose with the highest magnetic field ratio k
(ν= 0.45, µm= 26) identified at fr= 100 kHz.
The analyzed cases indicated that MMH led to greater values of energy density gain
and magnetic field ratios with higher operating frequencies. However, for the frequencies
over 1 MHz some negative effects, such as the eddy currents, dispersion, and a magnetic loss
in a composite material will be observed, leading to a reduced ability of MMH to transfer
the magnetic field and/or electromagnetic waves effectively. The practical frequency range
of the proposed magnetic hose will probably be located between several dozen to several
hundred kilohertz. Furthermore, it was not a rule that optimum cases for
η
also provided
the highest k. Figures 11 and 13 revealed that the peak values of the magnetic field ratio
were found at higher frequencies than fr.
Energies 2023,16, 334 15 of 17
4. Conclusions
The paper concentrated on an analysis of the magnetic meta-hose intended for the
transfer of the medium frequency magnetic field. The structure of the device was proposed
as a hybrid between the magnetic hose for the static field and the classic waveguide.
The anisotropic metamaterial was introduced as a cladding of the waveguide to ensure
a propagation of the magnetic field along the specified path. Two layers of the planar
resonators and a paramagnetic composite were used to form the metamaterial. Due to the
layered construction of the hose, it was possible to reduce the thickness of the walls and
utilize an air core which combined, accordingly minimizing the mass of the device.
The numerical model of the meta-hose was created, and the magnetic field distribution
was calculated. The results showed that the proposed meta-structure was able to transfer
the magnetic field and its energy at the considered distance effectively. The magnetic
energy density was increased more than a thousand times at a lower frequency (25 kHz)
and more than eight thousand times at 100 kHz. For the second operating frequency, the
mean magnetic field norm at the output of the magnetic hose was not lower than at the
input, and hence the task of transferring the magnetic flux had been achieved. Furthermore,
the frequency characteristics demonstrated a band-pass behavior of the magnetic meta-hose
since only signals with the specific frequencies were transferred. This property is similar to
the one exhibited by optical fibers, where only some modes of the electromagnetic field, at
particular frequencies, can propagate across the waveguide.
Although the numerical results were promising, a comparison with the experimental
measurements is essential. The verification would require a prototype of the magnetic meta-
hose and to test the power transfer efficiency from some magnetic field source to a load
(e.g., a receiving coil). The advantage of the introduced structure is the set of commercially
available components (dielectric frames, thin wires and paramagnetic composites); hence,
the proposed magnetic hose can be manufactured with current technologies.
Supplementary Materials:
The following supporting information can be downloaded at: https://
www.mdpi.com/article/10.3390/en16010334/s1, Table S1: Optimized capacitances in (nF) resulting
from the Nelder-Mead algorithm used for the design of a magnetic meta-hose, intended to operate in
the external magnetic field at f
r
= 25 kHz; Table S2: Optimized capacitances in (nF) resulting from
the Nelder-Mead algorithm used for the design of a magnetic meta-hose, intended to operate in the
external magnetic field at fr= 100 kHz. Tables with exemplary optimized capacitances.
Funding:
This research was funded by the Ministry of Science and Higher Education in Poland at
the Bialystok University of Technology under research subsidy No. WZ/WE-IA/2/2020.
Data Availability Statement: Not applicable.
Acknowledgments: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
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