Computation of periodic magnetic field in ferromagnetic conductive medium and supply current harmonics by using harmonic balance finite element method
The harmonic balance finite element method for two dimensional periodic magnetic field in a conductive ferromagnetic medium is formulated. To convert of partial differential equation system into the system of nonlinear algebraic equations the weak formulation of Galerkin method is used. Structural s...
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Інститут електродинаміки НАН України
2017
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| Cite this: | Computation of periodic magnetic field in ferromagnetic conductive medium and supply current harmonics by using harmonic balance finite element method / I.S. Petukhov // Технічна електродинаміка. — 2017. — № 5. — С. 18-22. — Бібліогр.: 14 назв. — англ. |
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| author | Petukhov, I.S. |
| author_facet | Petukhov, I.S. |
| citation_txt | Computation of periodic magnetic field in ferromagnetic conductive medium and supply current harmonics by using harmonic balance finite element method / I.S. Petukhov // Технічна електродинаміка. — 2017. — № 5. — С. 18-22. — Бібліогр.: 14 назв. — англ. |
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| container_title | Технічна електродинаміка |
| description | The harmonic balance finite element method for two dimensional periodic magnetic field in a conductive ferromagnetic medium is formulated. To convert of partial differential equation system into the system of nonlinear algebraic equations the weak formulation of Galerkin method is used. Structural steel and silicon steel with different magnetic properties and various electrical conductivity are studied. It is described how to take into account the nonlinear properties of medium in mathematical model. The spectrum of supply current and flux density on the surface and inside the medium were computed provided the sinusoidal voltage fed. The dependencies of the amplitudes of high harmonics versus the steel properties and amplitude of the first harmonic of magnetic flux density at the surface of medium are presented.
Представлено метод гармонічного балансу сумісно з методом скінченних елементів для двовимірного магнітного поля в електропровідному феромагнітному середовищі. Для перетворення диференціального рівняння в часткових похідних до системи нелінійних алгебраїчних рівнянь використовувався метод Гальоркіна в слабкому формулюванні. Досліджувалися конструкційна та легована сталь із різними магнітними властивостями та різною електропровідністю. Описано врахування нелінійних властивостей у математичній моделі. Розраховано спектри струму живлення та магнітної індукції на поверхні та всередині середовища за умов живлення синусоїдальною напругою. Представлено залежності амплітуд вищих гармонік від властивостей сталі та амплітуди першої гармоніки магнітної індукції на поверхні середовища.
Сформулирован метод гармонического баланса совместно с методом конечных элементов для двумерного магнитного поля в электропроводящей ферромагнитной среде. Для преобразования дифференциального уравнения в частных производных в систему нелинейных алгебраических уравнений использовался метод Галеркина в слабой формулировке. Исследовались конструкционная и легированная сталь с различными магнитными свойствами и различной электропроводностью. Описан учет нелинейных свойств в математической модели. Рассчитаны спектры тока питания и магнитной индукции на поверхности и внутри среды при условиях питания синусоидальным напряжением. Представлены зависимости амплитуд высших гармоник от свойств стали и амплитуды первой гармоники магнитной индукции на поверхности среды.
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| first_indexed | 2025-12-07T16:12:44Z |
| format | Article |
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18 ISSN 1607-7970. Техн. електродинаміка. 2017. № 5
COMPUTATION OF PERIODIC MAGNETIC FIELD IN FERROMAGNETIC
CONDUCTIVE MEDIUM AND SUPPLY CURRENT HARMONICS BY USING
HARMONIC BALANCE FINITE ELEMENT METHOD
I.S. Petukhov,
Institute of Electrodynamics of National Academy of Sciences of Ukraine,
pr. Peremohy, 56, Кyiv, 03057, Ukraine. E-mail: igor_petu@mail.ru
The harmonic balance finite element method for two dimensional periodic magnetic field in a conductive ferromagnetic
medium is formulated. To convert of partial differential equation system into the system of nonlinear algebraic equa-
tions the weak formulation of Galerkin method is used. Structural steel and silicon steel with different magnetic proper-
ties and various electrical conductivity are studied. It is described how to take into account the nonlinear properties of
medium in mathematical model. The spectrum of supply current and flux density on the surface and inside the medium
were computed provided the sinusoidal voltage fed. The dependencies of the amplitudes of high harmonics versus the
steel properties and amplitude of the first harmonic of magnetic flux density at the surface of medium are presented.
References 14, figures 5.
Key words: harmonic balance method, finite element method, ferromagnetic medium, magnetic field spectrum, current
spectrum.
Introduction. Magnetic field simulation in ferromagnetic medium requires taking into account mag-
netic saturation and hysteresis phenomena. These phenomena cause occurrence of high harmonics in mag-
netic field, flux density and supply current. If it is necessary to study a periodic process of an electromag-
netic device the simulation in the frequency domain is preferable to numerical integration in time-domain.
Currently, the finite element method (FEM) mainly is used for modeling of magnetic field. In periodic proc-
ess complex variable representing the only main harmonic is used. And in such case no spectra can be found.
In [7,8,9,13] it was shown that harmonic balance finite element method (HBFEM) is the most effective
method under in this situation. Unlike the single-harmonic simulation, which uses magnetization curve for
effective values, HBFEM uses real one and gives the instantaneous values of magnetic field and eddy cur-
rents in a period. Therefore this method allows to increase the accuracy of flux density and iron losses de-
termining. The situation is rather complicated by the presence of eddy currents but HBFEM can be effec-
tively used in this case too [12].
Apart from the computation of iron losses another important problem is evaluation of current supply
spectrum due to nonlinear properties of medium. But it is not possible to calculate the current spectrum in
voltage fed devices by using mono-harmonic approach. Computation of a transient is not preferable due to
some known reasons [7]. HBFEM gives the spectrum directly [3,14] and for that reason it is most suitable
method which allows to find the stationary periodic solution. Aim of the article is to study influence of iron
saturation and eddy current to spectrum of supply current by using HBFEM.
The simulation was carried out with GE2D package which supports the HBFEM for two-
dimensional low frequency electromagnetic problems. The package was developed in the Institute of elec-
trodynamics of the NAS of Ukraine and used first order triangle finite elements [6]. This package was used
earlier for computation of particle tracing in stationary magnetic field [1] and parameters of permanent mag-
net electrical machines [5].
Mathematic model. The model of nonlinear magnetic field in conductive immovable medium is
based on partial differential equation in terms of vector magnetic potential A
( ) / tγ ϕ γ∇× ∇ × = − ∇ − ∂ ∂ +f A A J , (1)
where f (•) is nonlinear vector function representing dependence of field strength components (Hx, Hy) vs
magnetic flux density components (Bx, By) , γ is an electric conductivity, φ is a scalar electric potential, t is a
time, J is an external exciting current density. The above-mentioned nonlinear function for isotropic medium
is determined with the relationship
( , ) ( ) ( , )x y x yH H H B B B B= , (2)
where H(B) is an expression of the normal magnetization curve, || B || is the Euclidean norm of the magnetic
flux density vector.
Consider the case of a two-dimensional magnetic field which has the only spatial component A = Az,
provided that there are no external sources of electric potential φ and external current J inside the medium.
Taking into account above-mentioned, the equation (1) can be rewritten in the following form
© Petukhov I.S., 2017
ISSN 1607-7970. Техн. електродинаміка. 2017. № 5 19
0,, =
∂
∂
γ+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
∂
∂
∂
∂
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
∂
∂
∂
∂
t
A
y
A
x
AH
yy
A
x
AH
x xy
, (3)
where x and y are the axes of Cartesian coordinate system.
To study the influence of nonlinearity to wave form of the eddy currents and supply current respec-
tively without influence of any geometry features it is enough to consider subdomain of rectangle, shown in
Fig. 1. The subdomain is cut from the cross section of slab which is infinite in x and z directions. The alter-
nating magnetic field is excited by the thin layer of current density Jw located on the top and bottom surfaces
of slab (Fig. 1). Thanks to symmetry relative to the horizontal axis the only one half of cross section may be
examined. If the magnetic field is excited by a voltage source then the magnetic flux Φ along x - direction
inside the slab may be represented according to Stoke's theorem as
dldS ∫∫ ==Φ
lS
x AB . (4)
Consequently the voltage ΔU applied to selected part of winding of width b is given by expression
( )
b
l du w A t dx
b dt
Δ = Δ ∫ , (5)
where Δw is a number of turns, l is a reference length. Due to that the vector potential does not depend on x
coordinate, the relationship between voltage and am-
plitude sine vector potential AΓ on the slab surface
may be found from (5) as
)sin(2)( tAlwtu m ωΔω=Δ , (6)
where ω is an angular frequency. Thus the boundary
condition on the slab surface (Fig. 1) is
)2/()sin( lwtUA m ΔωωΔ=Γ , (7)
where ΔUm is an amplitude of sine supply voltage.
According to harmonic balance method the
approximate solution of vector potential Ã(x,y,t) and
other variables are represented by trigonometric
polynomials like this
[ ]
1 1
( , , ) cos ( ) sin ( ) ( , )
gnn
c i s i i
i g
A x y t A t A t N x yν νν ω ν ω
= =
= +∑ ∑% , (8)
where n − number of nodes of mesh; ng − number of harmonic base function; / 2gν = ⎡ ⎤⎢ ⎥ ; Ni is node base
function of element. As mentioned above, triangular first order finite elements have been used. In accordance
with HBFEM we should find all node variables, namely, flux density, field strength and current as truncated
Fourier series containing the same number of harmonics like (8). Thus, discrete analog of derivatives for de-
termination of flux density components in (3) is
( ) ( ) ( ) ( )
3 3
1 11 1
; ,
g gj jn n
j j
x j h h y j h h
h j j h j j
A N NAB A B A
y y x x
∂ ∂ ∂∂α ξ α α ξ α
∂ ∂ ∂ ∂= = = =
= = = − = −∑ ∑ ∑ ∑
% %
% % (9)
where j1 – j3 is number of element nodes, ξg(α) is a trigonometric base function: cos(νωt) or sin(νωt) in (8).
Application of Galerkin method in weak formulation gives integral relationship
( ) ( )
( )
0 0
0
2 2
2 0 ,
i i
y x g i g
i g
N NH H d d H N d d
x y
A N d d
π π
τ
π
∂ ∂ ξ α α ξ α α
π ∂ ∂ π
∂γ ω ξ α α
π ∂ α
Ω Γ
Ω
⎡ ⎤
− − Ω + Ω +⎢ ⎥
⎣ ⎦
+ Ω =
∫ ∫ ∫ ∫
∫ ∫
% % %
%
(10)
where Hτ is unknown surface field strength. Assuming that the magnetic field outside the infinite slab is zero
(like outside a solenoid) one can say that the surface field strength Hτ is equal to surface current density of
excitation winding Jw. As the result, the algebraic equation system of FEM includes two unknown vectors.
The first one is a vector of harmonic amplitudes of potential AΩg, in nodes, which do not belong to surface of
slab. The second one is amplitudes of harmonics of supply current density Jg, in nodes belonging to the sur-
face (top boundary of the selected rectangle on Fig. 1). Let the period be divided into L equal intervals. Then
A
A
20 ISSN 1607-7970. Техн. електродинаміка. 2017. № 5
for a node with the index i (i = 1 … n) belonging to the element e and for a harmonic base function with the
index h (h = 1 … ng) of a contribution Re
ih to the residual vector R gives
( )
( ) ( ) ( ) } ( )
1 2 3
1
, 1
, , 1
1
1 ,
e
L
e i i
ih y x h s
sе
L
e
j h h s j h s i e h h s
j j j j s
N N
R H H
L x y
A N N d S J d
L
∂ ∂
ξ α
∂ ∂
γ ω ν ξ α ξ α ξ α
=
±
= =Γ
⎧ ⎡ ⎤⎪= − − +⎨ ⎢ ⎥
⎣ ⎦⎪⎩
⎡ ⎤+ + Γ⎣ ⎦
∑∫
∑ ∑∫
% %
m
(11)
where j1, j2, j3 are nodes of element e , Se is the square of element, (αs = 2π (s - 1)/L; s = 1 … L) are the nodes
in time domain; Гe is the part of the boundary belonging to element e. The major step in the formulation of a
nonlinear system of equation is to get expressions for Jacobi matrix obtaining from (11). This way, by using
of through indexes k = ng (i - 1) + h and n = ng (j - 1) + g , we have contribution of finite element e to k, n-cell
of global matrix
( )
( ) ( ) ) ( )
1
2
1
1
1 ,
e
L ye i x i
kn h s
s n ne
L
e
h s i j e h s
s
H N H N
J
L A x A y
N N d S d
L
∂ ∂ ∂ ∂
ξ α
∂ ∂ ∂ ∂
γ ω ν ξ α ξ α
=
=Γ
⎛ ⎡ ⎤
= − − +⎜ ⎢ ⎥⎜ ⎣ ⎦⎝
⎡ ⎤+ + Γ⎣ ⎦
∑∫
∑∫m
(12)
where Hx, Hy are nodal instantaneous values of field strength. Derivatives in the first term of (12) may be
written as
;yy y y yx x x x x
n y n x n n y n x n
HH H B BB H H H B
A B A B A A B A B A
∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
= + = + , (13)
in turn, the derivatives from field strength components ∂Hη / ∂Bη, (η = x, y) are defined from the vector func-
tion (2) but derivatives ∂Bη / ∂An in accordance to (9) are determined by
( ) ( )
y
N
A
B
x
N
A
B j
sh
n
xj
sh
n
y
∂
∂
αξ
∂
∂
∂
∂
αξ
∂
∂
=−= ; . (14)
In order to study the influence of magnetic properties of steel monotone Frohlich function for ap-
proximation of relative magnetic permeability was used
( ) ]1/[1 m
smx BB++= μμ , (15)
where μmx is initial magnetic permeability, Bs is magnetic flux density, corresponding to the inflection point
(half saturation), m is a factor of function steepness.
Two steel compositions were studied, namely, low silicon (structural steel, μmx = 1000; Bs =1,44; m
= 6,0; γ =7,14·106 S·m-1) and silicon steel (μmx =5000; Bs =1,15 T; m=10, γ=2,3·106 S·m-1). The correspon-
dence between approximating dependencies and catalog data is shown in Fig. 2.
If indexing of all nodes starts from the nodes
belonging to top edge of domain, the system equation
can be represented in general matrix form as
[1]
0
gZ
H G
Γ
Ω
⎧ ⎫⎡ ⎤ ⎧ ⎫⎪ ⎪ =⎨ ⎬ ⎨ ⎬⎢ ⎥
⎪ ⎪⎣ ⎦ ⎩ ⎭⎩ ⎭
J A
A
, (16)
where boundary values of AΓ are defined from (7).
For solving the nonlinear system (16) the modi-
fied Newton’s method [10] which includes original op-
timization algorithm and gives quick and reliable con-
vergence [4,6] is used in GE2D package. The reliability
of convergence is provided by the monotone approxima-
tion of magnetic permeability too. To increase the effi-
ciency of computation the algorithm used the reception
proposed in [11], comprising converting of expression
for magnetic permeability (15) to dependence of specific
conductance on magnetic flux density squared ν(B2).
Results and discussion. The dependence of supply current spectrum versus value of applied voltage
has been studied. As more suitable criterion of process intensity the first harmonic amplitude of flux density in
the surface can be adopted. It is caused by the fact that voltage or flux depends on the size of slab, but the flux
ISSN 1607-7970. Техн. електродинаміка. 2017. № 5 21
density does not. Skin effect
leads to non sinusoidal wave-
form of flux density both on
the surface and inside the
metal. This fact, provided that
the amplitude of main flux
density harmonic B1m equals
to 1,8 T, is confirmed by time
dependencies shown in Fig. 3
and Fig. 4 for flux density and
field strength respectively in
ten slices along the depth of
metal. The modeling was car-
ried out taking into account three first odd harmonics (1, 3, 5). As preliminary numerical experiments have
proved, the maximum harmonic number of 5 is enough to ensure the accuracy of both losses and amplitudes
calculation [4]. The current waveform has therefore some peaks (Fig. 4). On the contrary, the waveforms of
flux density are more flat than sinusoid. It is necessary to note, that in spite of non sinusoidal flux density char-
acter the flux and the voltage are sinusoidal as stated in problem formulation. The reason of this is phase varia-
tion of magnetic field along the depth.
The solution of system (16) gives spectra of supply current density Jg. Curves in Fig. 5 display a re-
lationship between amplitude of main harmonic of the flux density and relative amplitudes of 3rd and 5th
current harmonics (with respect to the amplitude of its first harmonic), where μ represents the maximum (ini-
tial) magnetic permeability, contained in formula (15). For comparison in Fig. 5 curves corresponding to non
conductive medium are represented, marked with γ = 0. This case corresponds to laminated magnetic core
which has been studied in some publications [5,7].
As one can see, provided that the amplitude of fundamental harmonic of flux density is up to 2 T, the
amplitude of its third harmonic reaches 8 ... 18%. And the higher specific resistance of steel or the more
sharp bending of the magnetization curve (steepness of magnetic permeability curve (15) at II IIB =Bs) is,
the more significant is contribution of high harmonics. Thus the maximum part of high harmonics, more then
50%, is observed in laminated magnetic core. The experimental data for amplitude of high harmonics in
magnetization current arising in laminated core of real transformer is described in [2]. For third harmonic
under condition that the amplitude of flux density Bm corres-ponds to 1; 1,4; 2 T, corresponding relative ampli-
tude of third harmonic reaches 21; 27,5 and 69 % respectively. The results presented above have no good cor-
relation with respect to obtained curves. It can be justified by the fact that ideal infinite object was consid-
ered. To uphold the obtained result it may be noted, that represented investigation was carried out not taking
into account resistance of winding, flux leakage, external circuit parameters and other factors taking place in
real transformers or inductors. With respect to the charts presented in Fig. 5 one may observe that the growth
of amplitudes of current harmonics decreases with increasing of flux density. This feature may be explained
by the increasing of linear part of magnetization curve in satu-
ration region as magnetic flux density grows.
Conclusion. The spectrum of higher harmonics of sup-
ply current generated by nonlinear medium is significantly in-
fluenced by eddy currents therein. Laminated core causes
maximal amplitude of the harmonics contained in supply cur-
rent. Conversely, solid magnetic core causes lower amplitude of
the high harmonics.
In addition in a case of solid magnetic core waveform
of flux density on the surface is considerably flattened and sig-
nificantly differs from a sinusoid. This fact should be taken
into account when designing the devices indicating flux den-
sity in the surface of solid ferromagnet.
To obtain a high accuracy computation of current and
flux density amplitudes, losses etc. it is enough to take into
account the first five time harmonics of the field. If only three
22 ISSN 1607-7970. Техн. електродинаміка. 2017. № 5
first odd harmonics exist the size of FEM matrix increases only three times compared to mono-harmonic
simulation.
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dynamika. – No 1. – 1999. – Pp. 64–68. (Rus)
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3. Cherepin V.T., Olykhovskiy V.L., Petukhov I.S. Optimizing the deflecting magnetic system of the mass spectrometer type of
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УДК 621.3.01
РОЗРАХУНОК ПЕРІОДИЧНОГО МАГНІТНОГО ПОЛЯ У ФЕРОМАГНІТНОМУ ЕЛЕКТРО-ПРОВІДНОМУ СЕ-
РЕДОВИЩІ ТА ГАРМОНІК У СТРУМІ ЖИВЛЕННЯ МЕТОДОМ ГАРМОНІЧНОГО БАЛАНСУ СУМІСНО З МЕ-
ТОДОМ СКІНЧЕННИХ ЕЛЕМЕНТІВ
І.С. Пєтухов, докт.техн.наук
Інститут електродинаміки НАН України,
пр. Перемоги, 56, Київ, 03057, Україна. E-mail: igor_petu@mail.ru
Представлено метод гармонічного балансу сумісно з методом скінченних елементів для двовимірного магнітного поля в елек-
тропровідному феромагнітному середовищі. Для перетворення диференціального рівняння в часткових похідних до системи
нелінійних алгебраїчних рівнянь використовувався метод Гальоркіна в слабкому формулюванні. Досліджувалися конструкційна
та легована сталь із різними магнітними властивостями та різною електропровідністю. Описано врахування нелінійних вла-
стивостей у математичній моделі. Розраховано спектри струму живлення та магнітної індукції на поверхні та всередині
середовища за умов живлення синусоїдальною напругою. Представлено залежності амплітуд вищих гармонік від властивос-
тей сталі та амплітуди першої гармоніки магнітної індукції на поверхні середовища. Бібл. 14, рис. 5.
Ключові слова: метод гармонічного балансу, метод скінченних елементів, феромагнітне середовище, спектр магнітного
поля, спектр струму.
УДК 621.3.01
РАСЧЕТ ПЕРИОДИЧЕСКОГО МАГНИТНОГО ПОЛЯ В ФЕРРОМАГНИТНОЙ ПРОВОДЯЩЕЙ
СРЕДЕ И ГАРМОНИК ПИТАЮЩЕГО ТОКА МЕТОДОМ ГАРМОНИЧЕСКОГО БАЛАНСА
СОВМЕСТНО С МЕТОДОМ КОНЕЧНЫХ ЭЛЕМЕНТОВ
И.С. Петухов, докт.техн.наук
Институт электродинамики Национальной академии наук Украины,
пр. Победы, 56, Киев, 03057, Украина. E-mail: igor_petu@mail.ru
Сформулирован метод гармонического баланса совместно с методом конечных элементов для двумерного магнитного поля в
электропроводящей ферромагнитной среде. Для преобразования дифференциального уравнения в частных производных в сис-
тему нелинейных алгебраических уравнений использовался метод Галеркина в слабой формулировке. Исследовались конструк-
ционная и легированная сталь с различными магнитными свойствами и различной электропроводностью. Описан учет нели-
нейных свойств в математической модели. Рассчитаны спектры тока питания и магнитной индукции на поверхности и
внутри среды при условиях питания синусоидальным напряжением. Представлены зависимости амплитуд высших гармоник
от свойств стали и амплитуды первой гармоники магнитной индукции на поверхности среды. Библ. 14, рис. 5.
Ключевые слова: метод гармонического баланса, метод конечных элементов, ферромагнитная среда, спектр магнитного
поля, спектр токов.
Надійшла 03.02.2017
Остаточний варіант 24.04.2017
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| id | nasplib_isofts_kiev_ua-123456789-158939 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-7970 |
| language | English |
| last_indexed | 2025-12-07T16:12:44Z |
| publishDate | 2017 |
| publisher | Інститут електродинаміки НАН України |
| record_format | dspace |
| spelling | Petukhov, I.S. 2019-09-18T20:04:55Z 2019-09-18T20:04:55Z 2017 Computation of periodic magnetic field in ferromagnetic conductive medium and supply current harmonics by using harmonic balance finite element method / I.S. Petukhov // Технічна електродинаміка. — 2017. — № 5. — С. 18-22. — Бібліогр.: 14 назв. — англ. 1607-7970 DOI: https://doi.org/10.15407/techned2017.05.018 https://nasplib.isofts.kiev.ua/handle/123456789/158939 621.3.01 The harmonic balance finite element method for two dimensional periodic magnetic field in a conductive ferromagnetic medium is formulated. To convert of partial differential equation system into the system of nonlinear algebraic equations the weak formulation of Galerkin method is used. Structural steel and silicon steel with different magnetic properties and various electrical conductivity are studied. It is described how to take into account the nonlinear properties of medium in mathematical model. The spectrum of supply current and flux density on the surface and inside the medium were computed provided the sinusoidal voltage fed. The dependencies of the amplitudes of high harmonics versus the steel properties and amplitude of the first harmonic of magnetic flux density at the surface of medium are presented. Представлено метод гармонічного балансу сумісно з методом скінченних елементів для двовимірного магнітного поля в електропровідному феромагнітному середовищі. Для перетворення диференціального рівняння в часткових похідних до системи нелінійних алгебраїчних рівнянь використовувався метод Гальоркіна в слабкому формулюванні. Досліджувалися конструкційна та легована сталь із різними магнітними властивостями та різною електропровідністю. Описано врахування нелінійних властивостей у математичній моделі. Розраховано спектри струму живлення та магнітної індукції на поверхні та всередині середовища за умов живлення синусоїдальною напругою. Представлено залежності амплітуд вищих гармонік від властивостей сталі та амплітуди першої гармоніки магнітної індукції на поверхні середовища. Сформулирован метод гармонического баланса совместно с методом конечных элементов для двумерного магнитного поля в электропроводящей ферромагнитной среде. Для преобразования дифференциального уравнения в частных производных в систему нелинейных алгебраических уравнений использовался метод Галеркина в слабой формулировке. Исследовались конструкционная и легированная сталь с различными магнитными свойствами и различной электропроводностью. Описан учет нелинейных свойств в математической модели. Рассчитаны спектры тока питания и магнитной индукции на поверхности и внутри среды при условиях питания синусоидальным напряжением. Представлены зависимости амплитуд высших гармоник от свойств стали и амплитуды первой гармоники магнитной индукции на поверхности среды. en Інститут електродинаміки НАН України Технічна електродинаміка Теоретична електротехніка та електрофізика Computation of periodic magnetic field in ferromagnetic conductive medium and supply current harmonics by using harmonic balance finite element method Розрахунок періодичного магнітного поля у феромагнітному електро-провідному середовищі та гармонік у струмі живлення методом гармонічного балансу сумісно з методом скінченних елементів Расчет периодического магнитного поля в ферромагнитной проводящей среде и гармоник питающего тока методом гармонического баланса совместно с методом конечных элементов Article published earlier |
| spellingShingle | Computation of periodic magnetic field in ferromagnetic conductive medium and supply current harmonics by using harmonic balance finite element method Petukhov, I.S. Теоретична електротехніка та електрофізика |
| title | Computation of periodic magnetic field in ferromagnetic conductive medium and supply current harmonics by using harmonic balance finite element method |
| title_alt | Розрахунок періодичного магнітного поля у феромагнітному електро-провідному середовищі та гармонік у струмі живлення методом гармонічного балансу сумісно з методом скінченних елементів Расчет периодического магнитного поля в ферромагнитной проводящей среде и гармоник питающего тока методом гармонического баланса совместно с методом конечных элементов |
| title_full | Computation of periodic magnetic field in ferromagnetic conductive medium and supply current harmonics by using harmonic balance finite element method |
| title_fullStr | Computation of periodic magnetic field in ferromagnetic conductive medium and supply current harmonics by using harmonic balance finite element method |
| title_full_unstemmed | Computation of periodic magnetic field in ferromagnetic conductive medium and supply current harmonics by using harmonic balance finite element method |
| title_short | Computation of periodic magnetic field in ferromagnetic conductive medium and supply current harmonics by using harmonic balance finite element method |
| title_sort | computation of periodic magnetic field in ferromagnetic conductive medium and supply current harmonics by using harmonic balance finite element method |
| topic | Теоретична електротехніка та електрофізика |
| topic_facet | Теоретична електротехніка та електрофізика |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/158939 |
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