XXIV Symposium
Electromagnetic Phenomena in Nonlinear Circuits
June 28 - July 1, 2016 Helsinki, FINLAND
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CALCULATION OF EDDY CURRENT AND HYSTERESIS LOSSES DURING
TRANSIENT STATES IN LAMINATED MAGNETIC CIRCUITS
Marek Gołębiowski
Rzeszow University of Technology, The Faculty of Electrical and Computer Engineering,
ul. W. Pola 2, 35-959 Rzeszow, Poland, e-mail: yegolebi@prz.edu.pl
Abstract - The article presents a complex magnetic permeability
for the presumed angular frequency in a laminated magnetic
circuit. On this basis, the synthesis of a magnetic permeability as a
function of the Laplace variable ‘s’ was presented. After
transformation of the variable 's' to a variable 'z' of the Z
transformation it was possible to conduct discrete time calculation
of transient states of magnetic circuits including the eddy current
losses. An Iterative process is developed to take the saturation of
the magnetic circuit in these calculations into account. To take into
account the hysteresis losses, the scalar model of magnetic
hysteresis by Juhani Tellinen is implemented. The new method is
validated by calculations of a two-coil transformer.
I. INTRODUCTION
The magnetic circuits of electrical machines is stacked
from the magnetic sheets of iron. The magnetic properties of
the electrical steels are described in DIN EN 10106 and DIN
EN 10207. The metal sheets have a thickness of 0.25 to 1
mm. They are insulated from each other by a layer of silicate
or water-soluble paints.
The methods for calculating the eddy current loss in the
iron, presented in literature are not accurate. In order to
calculate the losses in laminated magnetic systems, usually
equivalent sheet metal conductivity is calculated. The
calculations are then implemented by finite element method
[1]. However, this requires "a priori" assumptions of
waveforms to calculate the equivalent conductivity of metal
sheets. Adjusting an equivalent magnetic circuit of the
transformer [2], and then determining its parameters based on
a frequency response, might be encumbered with an error and
difficulties. Used in [3, 4] method for determining losses of
metal sheets is based on induction values given at the
beginning and calculated at the end of the finite element
method time step. For accurate calculation of the losses
including transients, more than one previous time step has to
be regarded. This article proposes the use of equivalent metal
sheet magnetic permeability (Fig. 2). It allows the implicit
calculation of transients in laminated magnetic systems.
II. COMPLEX MAGNETIC PERMEABILITY IN
MONOHARMONIC CALCULATIONS
The phenomenon of eddy current loss and displacement of
the flux in the sheets of the magnetic core (cf. Fig. 1), in te
monoharmonic calculations, can be taken in account by
introducing an equivalent complex magnetic permeability
𝜇̂ (𝜔). While the imaginary part of the magnetic permeability
corresponds to the active power, the real part represents the
reactive power. The equivalent magnetic permeability can be
derived on the basis of fig. 1 [5].
Fig.1. The magnetic field By in transformer metal sheet induced by a current
density J
The complex magnetic permeability was obtained:
𝜇̂ =
𝜇
𝑝∙𝑑
2
∙ tanh �
𝑝∙𝑑
2
� , 𝑝 = �𝑗𝜔𝜇𝛾
(1)
III. THE SYNTHESIS OF THE EQUIVALENT
MAGNETIC PERMEABILITY OF LAMINATED
SYSTEMS 𝜇̂
The equivalent magnetic permeability 𝜇̂ from the formula
(1) can be written in the Laplace domain (𝑠 = 𝑗𝜔)
𝜇̂ =
where:
𝑑
1
(2)
𝑘
𝑘
𝑠 � 1 𝑐𝑜𝑡ℎ� 2 √𝑠��
𝑘1
√𝑠
𝑑 2
𝛾
𝑘1 = � ; 𝑘2 = � � ∙ 𝛾
2 𝜇
2
(3)
Formula (2) is synthesized in the form of impedance.
𝐶0 =
𝑘2
𝑘1
2
= 𝜇 ; 𝑅𝑛 =
1
C0
s
2
1
C0 s
R1
2𝑘2
𝑛2 ∙𝜋2
=
1 𝑑2 𝛾
𝑛2 2𝜋2
R2
1
C0
s
2
; 𝑛 = 1,2, … ∞
(4)
1
C0
s
2
Rn
Fig. 2. The impedance derived from (2)
IV. SIMULATION OF A 2-COIL TRANSFORMER
INCLUDING EDDY CURRENT LOSSES
The input current and the magnetic induction in the iron
core of a transformer (cf. Fig. 3) are calculated.
Parameters of the transformer are presented in table 1.
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45
B(H)
Sp
B
Ls1
Rs1
i1
zw1
2
1
Ls 2
i2
1 1
3
0.5
Rs2
zw2
B, T
e(t )
d
l
0
3
-0.5
Fig. 3. The diagram of 2-coil transformer with a single laminated magnetic
core
-1
-1.5
TABLE I PARAMETERS OF THE TRANSFORMER
𝑆𝑝 [m²]
0.001
𝑑𝑙 [m]
0.5
𝑍𝑤1
10
𝐿𝑠1[µH]
25
𝑍𝑤2
10
𝐿𝑠2[µH]
25
𝑅𝑠1[mΩ]
79
𝑅𝑠2[mΩ]
∞
where: Sp - cross section core area, dl - length of the magnetic
core, zw1, zw2 - number of turns in the windings.
The magnetomotive force is given by:
𝜃 = 𝑖1 ∙ 𝑧𝑤1 + 𝑖2 ∙ 𝑧𝑤2 − 𝐼ℎ 𝑑𝑙
(5)
where 𝐼ℎ is a hysteresis component of a current of iron
losses [8]. The equations describing the transformer can be
presented in the Laplace transform domain ′𝑠′:
�
�𝐿𝑠1 +
𝑅𝑠1
�𝐿𝑠2 +
� ∙ 𝐼1 (𝑠) + �𝑆𝑝 ∙ 𝑧𝑤1 � ∙ 𝜇̂ (𝑠) ∙
𝑠
𝑅𝑠2
𝑠
𝜃(𝑠)
� ∙ 𝐼2 (𝑠) + �𝑆𝑝 ∙ 𝑧𝑤2 � ∙ 𝜇̂ (𝑠) ∙
=
𝑑𝑙
𝜃(𝑠)
𝑑𝑙
𝐸(𝑠)
𝑠
=0
(6)
Where 𝜇̂ (𝑠) is the impedance of the circuit synthesised in
chapter III. For the calculations, an exemplary non-linear
characteristics of magnetization is adapted:
𝐻 = 𝑎1 𝐵 + 𝑎3 𝐵3 + 𝑎9 𝐵9
(7)
where: 𝑎1 = 398; 𝑎3 = 30; 𝑎9 = 55.
The resulting waveforms of current 𝑖1 (𝑡), supply voltage
𝑒(𝑡) and magnetic induction 𝐵(𝑡) is shown in Fig. 4.
i1(t), e(t), 100*B(t)
150
100
100*B(t)
i(t)
e(t)
50
0
-50
-100
-150
0
5
10
time, s
15
20
x 10
-4
Fig. 4. The waveforms of current 𝑖1 (𝑡), assumed value of supply voltage
𝑒(𝑡) and the magnetic induction 𝐵(𝑡) in the magnetic core (multiplied
by 100)
In Fig. 5 the difference between the curve 1 and 2 for the
assumed value of the induction B defines the hysteresis loss.
V. CONCLUSIONS
Used to date equivalent coefficient of the electrical
conductivity for laminated magnetic systems required the
"a priori” assumption of the magnetic induction waveform.
2
2
1
3
-2000
-1000
0
H, A/m
1000
2000
Fig. 5. The dependence of the magnetic induction 𝐵 on the magnetic field
strength 𝐻 at frequency 𝑓 = 1500 𝐻𝑧; 1 - curve taking into account the eddy
current and hysteresis losses; 2 - curve taking into account only the eddy
current loss; 3 - the initial magnetization curve (7)
Calculated magnetic induction could not correspond to the
actual distribution of induction in such magnetic circuits.
It is important to consider the losses in metal sheets directly
in implicit calculation of transients. This possibility is
provided by presented method based on the synthesis of the
equivalent magnetic permeability 𝜇̂ (𝑠) according to the
scheme shown in Fig. 2. The nonlinear magnetization
characteristics of iron and hysteresis loss can also be
considered in the presented method.
The calculated metal sheets losses were compared with the
results the results presented in [6,7].
Good conformance of results depending on the angular
frequency was attained. It was found that a thickness of metal
sheets significantly impacts the loss. This is an important
parameter of the equivalent magnetic permeability.
REFERENCES
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Using
Frequency
Response
Analysis,
IEEE
TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 1, JANUARY
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Proceedings of EPNC 2016, June 28 - July 1, 2016 Helsinki, FINLAND