CALCULATION OF MAGNETIC LOSSES AND ENERGY EFFICIENCY OF A MAGNETIC GEARBOX WITH A MODULATED MAGNETIC FIELD
In this article, a simplified mathematical model is used to calculate magnetic losses in the ferromagnetic elements of a magnetic gearbox to analyze the distribution in space and time of magnetic induction in its core. This takes into account the peculiarities of using such a magnetic gearbox (multi...
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| author | Grebenikov , V. Podoltsev , O. Tazhibaev, А. Arynov , N. Sakhno , O. |
| author_facet | Grebenikov , V. Podoltsev , O. Tazhibaev, А. Arynov , N. Sakhno , O. |
| author_institution_txt_mv | [
{
"author": "V. Grebenikov ",
"institution": "Dnipro University of Technology, Dnipro, Ukraine"
},
{
"author": "O. Podoltsev ",
"institution": "Institute of Transport Systems and Technologies of the National Academy of Sciences of Ukraine, Dnipro, Ukraine"
},
{
"author": "А. Tazhibaev",
"institution": "LLC \"UMAY R&D\", Astana, Republic of Kazakhstan"
},
{
"author": "N. Arynov ",
"institution": "LLC \"UMAY R&D\", Astana, Republic of Kazakhstan"
},
{
"author": "O. Sakhno ",
"institution": "LLC \"Energoavtomatyzatsiya\", Zaporizhzhia, Ukraine"
}
] |
| author_sort | Grebenikov , V. |
| baseUrl_str | https://ve.org.ua/index.php/journal/oai |
| collection | OJS |
| datestamp_date | 2026-07-18T06:32:21Z |
| description | In this article, a simplified mathematical model is used to calculate magnetic losses in the ferromagnetic elements of a magnetic gearbox to analyze the distribution in space and time of magnetic induction in its core. This takes into account the peculiarities of using such a magnetic gearbox (multiplier) in a lowpower wind power plant. It is shown that when considering only the first harmonic components in the distribution of the magnetomotive forces of both rotors and the specific magnetic conductivity of the air gap, the magnetic induction in the air gap contains six components that have different pole pitch, move with different angular velocities and in different directions. According to the results of the calculations, it is shown that when the modulator is used as a low-speed shaft connected to a wind turbine, in the steady-state mode, all components of magnetic induction in the moving coordinate system x of the modulator change with the same angular velocity Fe p 2 = , which depends on the number of pole pairs of the external rotor and the rotation speed of the modulator together with the wind turbine - Fe . Under these conditions, the magnetic losses in the ferromagnetic poles of the modulator are calculated. Based on these data and considering mechanical losses, an energy efficiency map of the magnetic gearbox is constructed, representing its efficiency's dependence on the modulator rotation speed and mechanical torque on its shaft. The constructed efficiency map of the magnetic gearbox is the basis for calculating the efficiency of the entire wind power plant containing such a gearbox. The calculation of the characteristics of the magnetic gear was carried out in the Simcenter Magnet software package. Bibl. 20, Fig. 5. |
| doi_str_mv | 10.36296/1819-8058.2025.1(80).92-99 |
| first_indexed | 2025-07-17T11:39:52Z |
| format | Article |
| fulltext |
92
Відновлювана енергетика. №1/2025 | Вітроенергетика
УДК 621.313.8 https://doi.org/10.36296/1819-8058.2025.1(80)92-99
CALCULATION OF MAGNETIC LOSSES AND ENERGY EFFICIENCY OF A MAGNETIC GEARBOX WITH A
MODULATED MAGNETIC FIELD
Received Nov. 10, 2024; accepted Mar. 14, 2025
Available online Apr. 01, 2025
Grebenikov V.1, Podoltsev O.2, TazhibaevА.3,
Arynov N.4, Sakhno O.5
Author for correspondence: Grebenikov Viktor,
e-mail: elm1153@gmail.com
In this article, a simplified mathematical model is
used to calculate magnetic losses in the ferromag-
netic elements of a magnetic gearbox to analyze the
distribution in space and time of magnetic induction
in its core. This takes into account the peculiarities of
using such a magnetic gearbox (multiplier) in a low-
power wind power plant. It is shown that when con-
sidering only the first harmonic components in the
distribution of the magnetomotive forces of both rotors and the specific magnetic conductivity of the air gap,
the magnetic induction in the air gap contains six components that have different pole pitch, move with differ-
ent angular velocities and in different directions.
According to the results of the calculations, it is shown that when the modulator is used as a low-speed shaft
connected to a wind turbine, in the steady-state mode, all components of magnetic induction in the moving coor-
dinate system x of the modulator change with the same angular velocity Fep 2= , which depends on the num-
ber of pole pairs of the external rotor and the rotation speed of the modulator together with the wind turbine -
Fe . Under these conditions, the magnetic losses in the ferromagnetic poles of the modulator are calculated.
Based on these data and considering mechanical losses, an energy efficiency map of the magnetic gearbox is
constructed, representing its efficiency's dependence on the modulator rotation speed and mechanical torque on
its shaft. The constructed efficiency map of the magnetic gearbox is the basis for calculating the efficiency of the
entire wind power plant containing such a gearbox. The calculation of the characteristics of the magnetic gear
was carried out in the Simcenter Magnet software package. Bibl. 20, Fig. 5.
Keywords: wind turbine, magnetic gearbox, magnetic losses, computer modeling, energy efficiency map of the
gearbox.
РОЗРАХУНОК МАГНІТНИХ ВТРАТ ТА ЕНЕРГЕТИЧНОЇ ЕФЕКТИВНОСТІ МАГНІТНОГО РЕДУКТОРА
З МОДУЛЬОВАНИМ МАГНІТНИМ ПОЛЕМ
Отримано 10 лист. 2024 р.; рекомендовано до публікації 14 бер. 2025 р.
Доступно онлайн 01 квіт. 2025 р
Гребеніков В. В.1, Подольцев О. Д.2,
Тажибаев А. А.3, Аринов Н. Н.4, Сахно О. А.5
Автор для кореспонденції: Гребеніков Віктор,
e-mail: elm1153@gmail.com
У роботі для розрахунку магнітних втрат у фе-
ромагнітних елементах магнітного редуктора
використовується спрощена математична мо-
дель для аналізу розподілу в просторі та часі
магнітної індукції в його активній зоні. При
цьому враховуються особливості використання
такого редуктора у вітроенергетичній уста-
новці малої потужності. Показано, що при
1 Dr. of Tech. Sciences, Senior Researcher
https://orcid.org/0000-0002-1114-1218
2 Dr. of Tech. Sciences, Senior Researcher
https://orcid.org/0000-0002-9029-9397
3 General Director
https://orcid.org/0009-0004-4395-3722
4 Leading Designer
5 PhD in Tech. Sciences
https://orcid.org/0000-0002-3283-3731
1, 2 Dnipro University of Technology, Dnipro, Ukraine
2 Institute of Transport Systems and Technologies of the
National Academy of Sciences of Ukraine, Dnipro,
Ukraine
1д-р техн. наук, ст. наук. співроб.
https://orcid.org/0000-0002-1114-1218
2 д-р техн. наук, ст. наук. співроб.
https://orcid.org/0000-0002-9029-9397
3 Генеральний директор
https://orcid.org/0009-0004-4395-3722
4 Провідний конструктор
5 канд. техн. наук
https://orcid.org/0000-0002-3283-3731
1, 2 Інститут електродинаміки НАН України, м. Київ,
Україна.
3, 4 ТОВ "UMAY R&D" м. Астана, Республіка Казахстан
5 ТОВ "Енергоавтоматизація", м. Запоріжжя, Україна
https://orcid.org/0000-0002-9029-9397
https://orcid.org/0000-0002-9029-9397
93
Відновлювана енергетика. №1/2025 | Вітроенергетика
розгляданні тільки перших гармонічних складових у розподілі магніторушійних сил обох роторів та
питомої магнітної провідності повітряного проміжку магнітна індукція в повітряному проміжку
містить шість складових, які мають різний полюсний крок, рухаються з різною кутовою швидкістю
та в різних напрямках.
За результатами проведених розрахунків показано, що при використанні модулятора як низькошвид-
кісного вала, що під’єднується до вітрової турбіни, в усталеному режимі всі складові магнітної ін-
дукції в рухомій системі координат х модулятора змінюються з однаковою кутовою швидкістю
Fep 2= , що залежить від кількості пар полюсів зовнішнього ротора 2p та швидкості обертання
модулятора разом з вітротурбіною – Fe . За цих умов проведено розрахунок магнітних втрат у фе-
ромагнітних полюсах модулятора. На основі цих даних та з урахуванням механічних втрат побудо-
вана мапа енергетичної ефективності магнітного редуктора, що представляє залежність значення
його ККД від швидкості обертання модулятора та механічного моменту на його валу. Побудована
мапа ефективності магнітного редуктора є основою для розрахунку ефективності всієї вітроенер-
гетичної установки, що містить такий редуктор. Розрахунок характеристик магнітного редуктора
виконувався в програмному комплексі Simcenter Magnet. Бібл. 20, рис. 5.
Ключові слова: вітрова турбіна, магнітний редуктор, магнітні втрати, комп’ютерне моделю-вання,
мапа енергетичної ефективності редуктора.
List of used designations and abbreviations
LR – low-speed rotor
HR – high-speed rotor
MG – magnetic gearbox
Introduction. Nowadays, the use of renewable energy
sources is becoming more widespread due to the increas-
ing obstacles to the use of energy obtained from the com-
bustion of natural substances. The use of wind energy is a
solution that helps to generate electricity in a rational way
[1]. Any wind power plant consists of two parts: mechanical
and electrical; the mechanical part includes a wind turbine
and a mechanical gearbox (a multiplier that increases the
speed of rotation of the electric generator shaft), and an
electric generator with a semiconductor converter and a
load makes up the electrical part. As is known [2], the pres-
ence of a mechanical multiplier containing the contacting
surfaces of two rotors rotating at different angular speeds
significantly complicates the maintenance of such systems
and prompts the search for other alternative circuitry solu-
tions to the structure of modern wind power plants.
Over the past decade, specialists in the field of electrical
machines have paid much attention to the creation and
study of magnetic gearboxes (MG) [3, 4]. The peculiarity of
their design is the absence of contacting surfaces, and the
transmission of mechanical power between two rotors is
carried out due to the contactless interaction between per-
manent magnets. At present, the best design is considered
to be the MG with a modulated magnetic field in the air
gap, which was proposed in [5]. To determine the feasibility
of using such MRs, for example, in wind power plants or
other devices, it is necessary to conduct preliminary studies
on the energy efficiency of such gearboxes. To do this, it is
necessary to develop a methodology for calculating losses
in the active magnetic elements of the MG at different val-
ues of the rotational speed of its rotors. It should be noted
that, unlike classical electric machines, the magnetic field in
the core of the MG, due to the presence of a modulator,
has a much more complex structure, which complicates the
calculation of magnetic losses.
The literature uses mathematical models based mainly on
the application of three main approaches: 1) analytical mod-
els based on the equivalent magnetic circuits [6-9], 2) analyt-
ical models based on the mathematical solution of the elec-
tromagnetic field equations in the magnetic core [10-12],
and 3) computer numerical models, usually using the finite
element method [13, 14]. To calculate the magnetic losses in
the active elements of the MG, it is necessary to calculate the
magnetic field in its magnetic elements as a function of time,
which depends on the angular rotation speed of both rotors,
as well as in the general case of the modulator. When calcu-
lating the magnetic field by the analytical method, as a rule,
a two-dimensional field problem is solved for stationary ro-
tors and a modulator, i.e., the magnetostatic approximation
is used. When using numerical methods, the calculation of a
nonstationary problem is potentially possible, but consider-
ing the different rotational speeds of the rotors requires sig-
nificant computer time. In this article, a simplified approach
is used to calculate magnetic losses in the MG based on the
results of calculating a one-dimensional magnetic field in an
air gap with a modulator, taking into account the movement
of all the main elements of the MG. This approach makes it
possible to determine the magnitude and nature of the time
variation of the magnetic field in all moving elements of the
MG and, based on these data, to calculate magnetic losses in
them, as well as the energy efficiency of the magnetic gear-
box.
Objective of the work
Development of a mathematical model for determining the
distribution of a one-dimensional unsteady magnetic field
in the core of the MG and, on this basis, calculation of mag-
netic losses in the main MG elements depending on differ-
ent angular speeds of their rotation, as well as construction
94
Відновлювана енергетика. №1/2025 | Вітроенергетика
of an energy efficiency map of the gearbox, taking into ac-
count the peculiarities of its use in low-power wind power
plants, see, for example, [15, 16].
1. Description of the magnetic gearbox design
The design of the magnetic gearbox studied in this paper is
shown schematically in Fig. 1. It consists of an inner low-
speed rotor 1 with permanent magnets and a magnetic flux
closure, an outer high-speed rotor 2 with permanent mag-
nets, and a magnetic flux modulator 3 consisting of blended
ferromagnetic poles. The inner and outer rotors have the
number of pole pairs 13,4 21 == pp and rotate with an
angular velocity 21, . The modulator has a pole number
of 17=FeN and can rotate with an angular velocity of
Fe . All three speeds are determined in a fixed coordinate
system associated with the MG body (Fig. 1). The outer di-
ameter of the magnetic gearbox is 101 mm, the diameter
of the inner rotor is 71 mm, the axial length of the active
part is 50 mm, and the air gaps are 1 mm. The outer rotor
has permanent magnets of radial magnetization, and the
inner rotor has tangential magnetization. Preliminary nu-
merical studies have shown that the tangential arrange-
ment of magnets on the inner rotor allows for a higher elec-
tromagnetic torque compared to the radial arrangement of
magnets.
The reduction ratio of the magnetic gearbox is equal to 𝐺 =
𝑝2/𝑝1 = 3,25 with a fixed modulator and 𝐺 = 𝑁𝐹𝑒/𝑝1 =
4,25 with a fixed outer rotor and a moving modulator. The
calculation of the magnetic field and electromagnetic mo-
ments acting on the HR (high-speed rotor), LR (low-speed
rotor) and modulator of the magnetic gearbox is performed
using the Simcenter Magnet software package. Fig. 1 also
shows a picture of the magnetic field of the MG, which
shows that there is steel saturation in some elements of the
MG. Fig.2 shows the dependence of the electromagnetic
torque on the rotation angle of the modulator with a
stationary high- and low-speed rotor. In this picture, we de-
note: Tin - torque on the inner high-speed rotor; T out -
torque on the outer low-speed rotor; Tm - torque on the
modulator.
According to the results of computer modeling of this MG,
using the numerical finite element method, it was deter-
mined that the maximum torque value on the modulator
shaft is 185 Nm.
2. Analytical calculation of the magnetic field in the core
of a magnetic gearbox
In the analytical calculation of the magnetic field in the MR
core, a simplified one-dimensional formulation of the prob-
lem is considered when it is assumed that the magnetic in-
duction vector in the air gap with the modulator has only
one radial component, which depends on the angular coor-
dinate ϕ and time t [17].
The magnetomotive forces (MMF) generated by the per-
manent magnets on rotor 1 and rotor 2 are periodic func-
tions of the angular coordinate ϕ and can be expanded into
a Fourier series. Leaving only the first harmonic in these se-
ries, we obtain the following expressions
𝐹1 = 𝐹1𝑚 𝑐𝑜𝑠[ 𝑝1(𝜙 − 𝜔1𝑡)],
𝐹2 = 𝐹2𝑚 𝑐𝑜𝑠[ 𝑝2(𝜙 − 𝜔2𝑡)], (1)
Where 21, is the mechanical speed of rotors 1 and 2,
respectively.
The specific magnetic conductivity of the air gap with the
located ferromagnetic poles of the modulator in a one-di-
mensional approximation is also a periodic function of the
angular coordinate
ϕ and can be expanded into a Fourier
series - Fig. 3. Leaving only the first two terms of this series,
we have
𝛬 = 𝛬0 + 𝛬𝑚 𝑐𝑜𝑠[ 𝑁𝐹𝑒(𝜙 − 𝜔𝐹𝑒𝑡)], (2)
where ωFe - is the angular speed of rotation of the modula-
tor (further in the paper, two options for connecting the
MG to the wind turbine are considered). For simplicity, ex-
pressions (1), (2) assume that all initial phases in the har-
monic functions are equal to zero.
Fig. 2. Dependence of the electromagnetic torque on
the modulator (Tm), outer rotor (Tout), and inner rotor
(Tin) on the modulator rotation angle
Fig. 1. Magnetic Gearbox
95
Відновлювана енергетика. №1/2025 | Вітроенергетика
The radial component of magnetic induction in the air gap
with the modulator can be calculated as
В = F1 + А2 = В11 + В12 + B13 + B21 + B22 + B23 (3)
where the induction components taking into account (1)
and (2) and considering that
𝑐𝑜𝑠 𝐴 ⋅ 𝑐𝑜𝑠 𝐵 = 0,5[𝑐𝑜𝑠( 𝐴 − 𝐵) + 𝑐𝑜𝑠( 𝐴 − 𝐵)],
can be determined by the following expressions:
𝐵11 = 𝛬0𝐹1𝑚 𝑐𝑜𝑠[ 𝑝1(𝜙 − 𝜔1𝑡)],
𝐵12 = 0,5𝛬𝑚𝐹1𝑚 𝑐𝑜𝑠[ (𝑁𝐹𝑒 − 𝑝1)(𝜙 −
𝑁𝐹𝑒𝜔𝐹𝑒−𝑝1𝜔1
(𝑁𝐹𝑒−𝑝1)
𝑡)],
𝐵13 = 0,5𝛬𝑚𝐹1𝑚 𝑐𝑜𝑠[ (𝑁𝐹𝑒 + 𝑝1)(𝜙 −
𝑁𝐹𝑒𝜔𝐹𝑒+𝑝1𝜔1
(𝑁𝐹𝑒+𝑝1)
𝑡)],
𝐵21 = 𝛬0𝐹2𝑚 𝑐𝑜𝑠[ 𝑝2(𝜙 − 𝜔2𝑡)], (4)
𝐵22 = 0,5𝛬𝑚𝐹2𝑚 𝑐𝑜𝑠[ (𝑁𝐹𝑒 − 𝑝2)(𝜙 −
𝑁𝐹𝑒𝜔𝐹𝑒−𝑝2𝜔2
(𝑁𝐹𝑒−𝑝2)
𝑡)],
𝐵23 = 0,5𝛬𝑚𝐹2𝑚 𝑐𝑜𝑠[ (𝑁𝐹𝑒 + 𝑝2)(𝜙 −
𝑁𝐹𝑒𝜔𝐹𝑒+𝑝2𝜔2
(𝑁𝐹𝑒+𝑝2)
𝑡)]
From the above expressions (4), it can be seen that when
taking into account only the first harmonics in the Fourier
series for periodic functions ,, 21 FF , six magnetic field
components are determined in the core, which have differ-
ent pole pitch, different angular velocity and move depend-
ing on the ratio of the above parameters both in the for-
ward and reverse directions.
Each of the components of magnetic induction in (3) corre-
sponds to its magnetic losses in the ferromagnetic ele-
ments of the MG, which will be calculated further. To sim-
plify such calculations, it is advisable to take into account
the relationship between the values of the angular veloci-
ties of all MG elements, considering the steady-state mode
of its operation. It should also be noted that the following
relation always holds for MG:
FeNpp =+ 21 . (5)
The magnetic gearbox studied in this paper is designed to
operate in a wind power plant. In such an installation, the
low-speed MG element is connected to the wind turbine,
and the high-speed element is connected to the shaft of an
electric generator. Two options for such a connection can
be realized.
Option 1. For the first possible option, the modulator is sta-
tionary, the outer LR 2 is connected to the wind turbine,
and the inner HR 1 is connected to the generator shaft.
Thus, for this option: Fe = 0 and 2 - is the adjusted
speed of the turbine. Under these conditions, from expres-
sion (4) for 22B , taking into account (5), we obtain
𝐵22 = 0,5𝛬𝑚𝐹2𝑚 𝑐𝑜𝑠[ 𝑝1(𝜙 −
−𝑝2𝜔2
𝑝1
𝑡)].
This shows that there is a magnetic field component in the
core, characterized by the number of pole pairs 1p and ro-
tating with an angular velocity
2
1
2
1
p
p
−=
and in the direction opposite to the outer LR - 2. The inter-
action of this field with the magnetomotive force (MMF)
1F of the inner HR 1 will generate a useful torque, which
will cause this rotor to rotate at a speed of 1 . The reduc-
tion ratio will be equal to 1221 p/p/G == = 13/4.
Note that we consider the steady-state operation of the
MG. The features of operation in transient modes are con-
sidered in [15, 16].
The components of magnetic induction (4) under the con-
dition Fe = 0, 2
1
2
1
p
p
−= and taking into account (5)
will be as follows:
𝐵11 = 𝛬0𝐹1𝑚 𝑐𝑜𝑠[ 𝑝1(𝜙 +
𝑝2
𝑝1
𝜔2𝑡)],
𝐵12 = 0,5𝛬𝑚𝐹1𝑚 𝑐𝑜𝑠[ 𝑝2(𝜙 − 𝜔2𝑡)],
𝐵13 = 0,5𝛬𝑚𝐹1𝑚 𝑐𝑜𝑠[ (𝑁𝐹𝑒 + 𝑝1)(𝜙 +
𝑝2𝜔2
(𝑁𝐹𝑒+𝑝1)
𝑡)],
𝐵21 = 𝛬0𝐹2𝑚 𝑐𝑜𝑠[ 𝑝2(𝜙 − 𝜔2𝑡)], (6)
𝐵22 = 0,5𝛬𝑚𝐹2𝑚 𝑐𝑜𝑠[ 𝑝1(𝜙 +
𝑝2𝜔2
𝑝1
𝑡)],
𝐵23 = 0,5𝛬𝑚𝐹2𝑚 𝑐𝑜𝑠[ (𝑁𝐹𝑒 + 𝑝2)(𝜙 −
𝑝2𝜔2
(𝑁𝐹𝑒 + 𝑝2)
𝑡)]
These expressions show that for this option, at any point on
the ferromagnetic pole of the modulator, for example, at
ϕ=0, all six components of the magnetic field change with
the same frequency -ω=p2ω2 - creating the corresponding
magnetic losses. Note that ω2 in this option is the rotational
speed of the wind turbine.
Option 2. For the second possible option of using the MG,
when the outer rotor 2 is stationary, the modulator is con-
nected to the wind turbine, and the inner HR 1 is connected
to the shaft of the electric generator. Given that ω2=0, and
ωFe is the set speed of the turbine, from expression (4) for
22B , taking into account (5), we obtain
𝐵22 = 0,5𝛬𝑚𝐹2𝑚 𝑐𝑜𝑠[ 𝑝1(𝜙 −
𝑁2𝜔𝐹𝑒
𝑝1
𝑡)].
This shows that there is a magnetic field component in the
core characterized by the number of pole pairs 1p and ro-
tating with an angular velocity
Fig. 3. Specific magnetic conductivity of the air gap
)(
0
m
96
Відновлювана енергетика. №1/2025 | Вітроенергетика
𝜔1 =
𝑁𝐹𝑒
𝑝1
𝜔𝐹𝑒
in the same direction as the modulator. The interaction of
this field with the magnetomotive force F1 of the HR 1 will
generate a useful torque, which will cause this rotor to ro-
tate at a speed of 1 . The reduction ratio will be equal to
𝐺 = 𝜔1/𝜔𝐹𝑒 = 𝑁𝐹𝑒/𝑝1 = 17/4.
The components of magnetic induction (4) under the con-
dition 𝜔2= 0, 𝜔1 =
𝑁𝐹𝑒
𝑝1
𝜔𝐹𝑒 and taking into account (5) will
be as follows:
𝐵11 = 𝛬0𝐹1𝑚 𝑐𝑜𝑠[ 𝑝1(𝜙 −
𝑁𝐹𝑒
𝑝1
𝜔𝐹𝑒𝑡)],
𝐵12 = 0,5𝛬𝑚𝐹1𝑚 𝑐𝑜𝑠( 𝑝2𝜙),
𝐵13 = 0,5𝛬𝑚𝐹1𝑚 𝑐𝑜𝑠[ (𝑁𝐹𝑒 + 𝑝1)(𝜙 −
2𝑁𝐹𝑒𝜔𝐹𝑒
(𝑁𝐹𝑒+𝑝1)
𝑡)],
𝐵21 = 𝛬0𝐹2𝑚 𝑐𝑜𝑠( 𝑝2𝜙), (7)
𝐵22 = 0,5𝛬𝑚𝐹2𝑚 𝑐𝑜𝑠[ 𝑝1(𝜙 −
𝑁𝐹𝑒𝜔𝐹𝑒
𝑝1
𝑡)],
𝐵23 = 0,5𝛬𝑚𝐹2𝑚 𝑐𝑜𝑠[ (𝑁𝐹𝑒 + 𝑝2)(𝜙 −
𝑁𝐹𝑒𝜔𝐹𝑒
(𝑁𝐹𝑒 + 𝑝2)
𝑡)]
From expressions (7), it is clear that for this option, at any
fixed point in the space to which the modulator moves,
there are six components of the magnetic field that vary
with different frequencies.
When calculating the losses in a modulator moving at the
speed Fe , it is necessary to calculate the magnetic field
in the coordinate system associated with this modulator.
To do this, we put ϕ = ωFet in expressions (7), and after sim-
ple transformations, we obtain:
𝐵11 = 𝛬0𝐹1𝑚 𝑐𝑜𝑠( − 𝑝2𝜔𝐹𝑒𝑡),
𝐵12 = 0,5𝛬𝑚𝐹1𝑚 𝑐𝑜𝑠( 𝑝2𝜔𝐹𝑒𝑡),
𝐵13 = 0,5𝛬𝑚𝐹1𝑚 𝑐𝑜𝑠( − 𝑝2𝜔𝐹𝑒𝑡),
𝐵21 = 𝛬0𝐹2𝑚 𝑐𝑜𝑠( 𝑝2𝜔𝐹𝑒𝑡), (8)
𝐵22 = 0,5𝛬𝑚𝐹2𝑚 𝑐𝑜𝑠( − 𝑝2𝜔𝐹𝑒𝑡),
𝐵23 = 0,5𝛬𝑚𝐹2𝑚 𝑐𝑜𝑠( 𝑝2𝜔𝐹𝑒𝑡)
An important result is evident from expressions (8): in the
moving coordinate system of the modulator, all six compo-
nents of magnetic induction change in time with the same
angular frequency Fep 2= , where Fe is the rotation
speed of both the modulator and the wind turbine.
Further, when calculating magnetic losses, the second con-
nection option of the MG is considered, in which its reduc-
tion factor has the highest of the two possible values.
3. Calculation of magnetic losses and energy efficiency of
a magnetic gearbox
In this article, magnetic losses are calculated only in the fer-
romagnetic elements of the modulator since the losses in
rotor elements 1 and 2 are much smaller. We consider an
option where the MG modulator is rigidly fixed to the shaft
of a wind turbine rotating at a given speed Fe
= (0÷40) rad/s. The LR 2 is stationary, i.e., 2 = 0, and the
HR 1 rotates at a synchronous speed 11 p/NFeFe = .
Under these conditions, the components of magnetic in-
duction in the modulator are calculated by expressions (8).
They change with the same angular frequency Fep 2=
, which depends on the modulator rotation speed (equal to
the wind turbine speed). Therefore, in the shuffled mag-
netic poles of the modulator, there is a resulting magnetic
field with a magnetic induction equal to the vector sum of
all components (8):
𝐵 = 0,5(𝛬0𝐹1𝑚 + 𝛬𝑚𝐹1𝑚 + 𝛬0𝐹2𝑚 +
𝛬𝑚𝐹2𝑚) 𝑐𝑜𝑠( 𝑝2𝜔𝐹𝑒𝑡) = 𝐵𝑚 𝑐𝑜𝑠( 𝑝2𝜔𝐹𝑒𝑡) (9)
Note that the initial phases of the six harmonic functions
(which were initially assumed to be zero in expressions (1)
and (2)) must be taken into account. The values of these
phases are difficult to determine, and they will depend on
the load of the MG, which greatly complicates further cal-
culations. For a simplified accounting of this factor, an em-
pirical coefficient of 0.5 was used in expression (9).
The Bertotti formula [18] is most often used to calculate
magnetic losses in the magnetic circuit of any electrical ma-
chine, according to which the total specific losses have
three components corresponding to hysteresis losses, eddy
currents, and additional losses:
𝑃𝑙𝑜𝑠𝑠 = 𝑘ℎ𝑦𝑠𝐵𝑚
𝛼 𝑓 + 𝑘𝑒𝑑𝑑𝑦𝐵𝑚
2 𝑓2 + 𝑘𝑒𝑥𝑐𝐵𝑚
1,5𝑓1,5 [W/kg],(10)
where Bm - is the amplitude value of magnetic induction, T,
f - is the frequency, Hz.
From this formula, the eddy current loss factor for charge
steel can be determined by the following expression [19]:
𝑘𝑒𝑑𝑑𝑦 =
𝜎𝐹𝑒𝜋2𝑑𝐹𝑒
6𝜌𝐹𝑒
, (11)
where Fed , Fe , Fe - are the thickness of the sheets, their
density, and specific electrical conductivity, respectively.
The steel used for the modulator's charge-coupled poles is
2212 steel with a thickness of =Fed 0.5 mm, for
which =Fe 7650 kg/m, =Fe 5∙106 (Ohm∙m)-1
With these data, using (11), we obtain the value =eddyk
2.7∙10 4. The values of the other three unknown coefficients
97
Відновлювана енергетика. №1/2025 | Вітроенергетика
in formula (10) were found by approximating the experi-
mental data on the dependence of the specific losses for
steel 2212 on the amplitude value of magnetic induction at
a frequency of 50 Hz - such data are given in [20, p. 130].
These data are shown in Fig. 5, where the rectangles indi-
cate the results of the calculation according to formula (10)
at certain values of the unknown coefficients:
=hysk 0.0176, =exck 0.0019, 2.
Further, the following data were used to calculate the mag-
netic losses in the modulator:
𝐹1𝑚 =6,8 А, 𝐹2𝑚 =7,3 кА, 𝛬0 =11,5∙10-5 H/м2,
=m 16,6∙10-5 H/м2, =2p 13.
The mass of all the modulator poles is =Fem 1.7 kg. Cal-
culated by formula (10), the total losses in the MG modula-
tor equal to FelossFe mPP = as a function of the modulator
rotation frequency ωFe are shown in Fig. 6. Here, it was
taken into account that the frequency f in formula (10) is
equal to )2/(2 Fepf = , and the calculated
value mB by formula (9) is equal to mB = 1.98 T.
4. Energy Efficiency of MG
As a characteristic of the energy efficiency of the MG, we
use the value of its efficiency, which is calculated as
Fe
FeFeFeFe
Fe
T
kkPT
T
)(
),(
2
11
2 ++−
= ,
FeG =1 , (12)
where T, Fe are the mechanical torque on the wind tur-
bine shaft and the angular speed of its rotation, the
Fig. 6. Efficiency map of a magnetic gearbox
1 5 10 15 20 25 30
Angular velocity Fe , rad/s
M
ec
h
an
ic
al
m
o
m
en
t
, N
∙m
Fig. 4. Specific steel losses
Fig. 5. Total losses in the modulator
,Fe rad/s
,FeP W
98
Відновлювана енергетика. №1/2025 | Вітроенергетика
dependence of magnetic losses in the MG )( FeFeP is
taken into account by the given Fig. 4, and the expressions
2
FeFek and
2
11k correspond to the mechanical losses
on the low- and high-speed MG shaft. When calculating the
efficiency, the following values of the coefficients were
chosen: =Fek 0.12 N∙m∙s, =1k 0.02 N∙m∙s.
The results of calculating the dependence of efficiency as a
function of mechanical torque on the wind turbine shaft T
and its rotation speed Fe using formula (12) are shown in
Fig. 6. In the articles, this dependence is called an efficiency
map for any electric machine, and in this case, it is the MG.
This figure shows that the minimum value of efficiency (
65%) is achieved in the region of the maximum possible
value of the input shaft speed of the MG, when all losses
are the largest, and the minimum possible value of the me-
chanical torque on its load, when the useful mechanical
power transmitted by the gearbox is small. The constructed
MG efficiency map is the basis for calculating the efficiency
of the entire wind power installation containing such a
magnetic gearbox.
Conclusions
In this paper, a simplified mathematical model for calculat-
ing the magnetic field in the core of a magnetic gearbox in-
tended for use in a low-power wind turbine is considered.
In this case, the option, where the wind turbine rotor shaft
is connected to the MP modulator and the high-speed shaft
of the electric generator is connected to the internal rotor
of the MG, is considered. It is shown that when considering
only the first harmonic components of the magnetomotive
force of both rotors and the specific magnetic conductivity
of the air gap, there are six components of magnetic induc-
tion in this gap that have different pole pitches, move at
different speeds and in different directions.
The calculation results show that when the modulator is
used as a low-speed shaft connected to a wind turbine, all
components of the magnetic field in the moving poles of
the modulator change with the same angular velocity
ω=p2ωFe, which depends on the number of pairs of poles of
the external p2 and the speed of rotation of the modulator
together with the wind turbine - ωFe. Under these condi-
tions, we calculated the magnetic losses in the ferromag-
netic poles of the modulator.
Based on these data and taking into account mechanical
losses, a map of the energy efficiency of the MG is con-
structed, which represents the dependence of its efficiency
on the modulator rotation speed and mechanical torque on
its shaft. It is shown that the minimum value of efficiency
( 65%) is achieved in the region of the maximum possible
value of the MG input shaft rotation speed and the mini-
mum possible value of the mechanical torque on its load.
The constructed map is the basis for calculating the effi-
ciency of the entire wind power plant containing such a
magnetic gearbox.
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|
| id | veorgua-article-510 |
| institution | Vidnovluvana energetika |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-19T01:15:08Z |
| publishDate | 2025 |
| publisher | Institute of Renewable Energy National Academy of Sciences of Ukraine |
| record_format | ojs |
| resource_txt_mv | veorgua/7c/5269ee589cfff61ef1c2f6a86f7f5e7c.pdf |
| spelling | veorgua-article-5102026-07-18T06:32:21Z CALCULATION OF MAGNETIC LOSSES AND ENERGY EFFICIENCY OF A MAGNETIC GEARBOX WITH A MODULATED MAGNETIC FIELD РОЗРАХУНОК МАГНІТНИХ ВТРАТ ТА ЕНЕРГЕТИЧНОЇ ЕФЕКТИВНОСТІ МАГНІТНОГО РЕДУКТОРА З МОДУЛЬОВАНИМ МАГНІТНИМ ПОЛЕМ Grebenikov , V. Podoltsev , O. Tazhibaev, А. Arynov , N. Sakhno , O. wind turbine, magnetic gearbox, magnetic losses, computer modeling, energy efficiency map of the gearbox. вітрова турбіна, магнітний редуктор, магнітні втрати, комп’ютерне моделю-вання, мапа енергетичної ефективності редуктора. In this article, a simplified mathematical model is used to calculate magnetic losses in the ferromagnetic elements of a magnetic gearbox to analyze the distribution in space and time of magnetic induction in its core. This takes into account the peculiarities of using such a magnetic gearbox (multiplier) in a lowpower wind power plant. It is shown that when considering only the first harmonic components in the distribution of the magnetomotive forces of both rotors and the specific magnetic conductivity of the air gap, the magnetic induction in the air gap contains six components that have different pole pitch, move with different angular velocities and in different directions. According to the results of the calculations, it is shown that when the modulator is used as a low-speed shaft connected to a wind turbine, in the steady-state mode, all components of magnetic induction in the moving coordinate system x of the modulator change with the same angular velocity Fe p 2 = , which depends on the number of pole pairs of the external rotor and the rotation speed of the modulator together with the wind turbine - Fe . Under these conditions, the magnetic losses in the ferromagnetic poles of the modulator are calculated. Based on these data and considering mechanical losses, an energy efficiency map of the magnetic gearbox is constructed, representing its efficiency's dependence on the modulator rotation speed and mechanical torque on its shaft. The constructed efficiency map of the magnetic gearbox is the basis for calculating the efficiency of the entire wind power plant containing such a gearbox. The calculation of the characteristics of the magnetic gear was carried out in the Simcenter Magnet software package. Bibl. 20, Fig. 5. У роботі для розрахунку магнітних втрат у феромагнітних елементах магнітного редуктора використовується спрощена математична модель для аналізу розподілу в просторі та часі магнітної індукції в його активній зоні. При цьому враховуються особливості використання такого редуктора у вітроенергетичній установці малої потужності. Показано, що при розгляданні тільки перших гармонічних складових у розподілі магніторушійних сил обох роторів та питомої магнітної провідності повітряного проміжку магнітна індукція в повітряному проміжку містить шість складових, які мають різний полюсний крок, рухаються з різною кутовою швидкістю та в різних напрямках. За результатами проведених розрахунків показано, що при використанні модулятора як низькошвид- кісного вала, що під’єднується до вітрової турбіни, в усталеному режимі всі складові магнітної ін- дукції в рухомій системі координат х модулятора змінюються з однаковою кутовою швидкістю Fe p 2 = , що залежить від кількості пар полюсів зовнішнього ротора 2p та швидкості обертання модулятора разом з вітротурбіною – Fe . За цих умов проведено розрахунок магнітних втрат у феромагнітних полюсах модулятора. На основі цих даних та з урахуванням механічних втрат побудо- вана мапа енергетичної ефективності магнітного редуктора, що представляє залежність значення його ККД від швидкості обертання модулятора та механічного моменту на його валу. Побудована мапа ефективності магнітного редуктора є основою для розрахунку ефективності всієї вітроенер- гетичної установки, що містить такий редуктор. Розрахунок характеристик магнітного редуктора виконувався в програмному комплексі Simcenter Magnet. Бібл. 20, рис. 5. Institute of Renewable Energy National Academy of Sciences of Ukraine 2025-03-31 Article Article application/pdf https://ve.org.ua/index.php/journal/article/view/510 10.36296/1819-8058.2025.1(80).92-99 Vidnovluvana energetika ; No. 1(80) (2025): Scientific and applied Journal renewable energy ; 92-99 Возобновляемая энергетика; ##issue.no## 1(80) (2025): Scientific and applied Journal renewable energy ; 92-99 Відновлювана енергетика; № 1(80) (2025): Науково-прикладний журнал Відновлювана енергетика; 92-99 2664-8172 1819-8058 10.36296/1819-8058.2025.1(80) en https://ve.org.ua/index.php/journal/article/view/510/419 Copyright (c) 2025 V. Grebenikov , O. Podoltsev , А. Tazhibaev, N. Arynov , O. Sakhno https://creativecommons.org/licenses/by-nc-nd/4.0 |
| spellingShingle | wind turbine magnetic gearbox magnetic losses computer modeling energy efficiency map of the gearbox. Grebenikov , V. Podoltsev , O. Tazhibaev, А. Arynov , N. Sakhno , O. CALCULATION OF MAGNETIC LOSSES AND ENERGY EFFICIENCY OF A MAGNETIC GEARBOX WITH A MODULATED MAGNETIC FIELD |
| title | CALCULATION OF MAGNETIC LOSSES AND ENERGY EFFICIENCY OF A MAGNETIC GEARBOX WITH A MODULATED MAGNETIC FIELD |
| title_alt | РОЗРАХУНОК МАГНІТНИХ ВТРАТ ТА ЕНЕРГЕТИЧНОЇ ЕФЕКТИВНОСТІ МАГНІТНОГО РЕДУКТОРА З МОДУЛЬОВАНИМ МАГНІТНИМ ПОЛЕМ |
| title_full | CALCULATION OF MAGNETIC LOSSES AND ENERGY EFFICIENCY OF A MAGNETIC GEARBOX WITH A MODULATED MAGNETIC FIELD |
| title_fullStr | CALCULATION OF MAGNETIC LOSSES AND ENERGY EFFICIENCY OF A MAGNETIC GEARBOX WITH A MODULATED MAGNETIC FIELD |
| title_full_unstemmed | CALCULATION OF MAGNETIC LOSSES AND ENERGY EFFICIENCY OF A MAGNETIC GEARBOX WITH A MODULATED MAGNETIC FIELD |
| title_short | CALCULATION OF MAGNETIC LOSSES AND ENERGY EFFICIENCY OF A MAGNETIC GEARBOX WITH A MODULATED MAGNETIC FIELD |
| title_sort | calculation of magnetic losses and energy efficiency of a magnetic gearbox with a modulated magnetic field |
| topic | wind turbine magnetic gearbox magnetic losses computer modeling energy efficiency map of the gearbox. |
| topic_facet | wind turbine magnetic gearbox magnetic losses computer modeling energy efficiency map of the gearbox. вітрова турбіна магнітний редуктор магнітні втрати комп’ютерне моделю-вання мапа енергетичної ефективності редуктора. |
| url | https://ve.org.ua/index.php/journal/article/view/510 |
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