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 �̂�(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:

�̂� = 𝜇𝑝∙𝑑2∙ tanh �𝑝∙𝑑

2� , 𝑝 = �𝑗𝜔𝜇𝛾 (1)

III. THE SYNTHESIS OF THE EQUIVALENT

MAGNETIC PERMEABILITY OF LAMINATED SYSTEMS �̂�

The equivalent magnetic permeability �̂� from the formula

(1) can be written in the Laplace domain (𝑠 = 𝑗𝜔)

�̂� = 1

𝑠 �𝑘1√𝑠

𝑐𝑜𝑡ℎ�𝑘2𝑘1√𝑠��

(2)

where:

𝑘1 = 𝑑2 �

𝛾𝜇

; 𝑘2 = �𝑑2�2∙ 𝛾 (3)

Formula (2) is synthesized in the form of impedance.

𝐶0 = 𝑘2𝑘12

= 𝜇 ; 𝑅𝑛 = 2𝑘2𝑛2∙𝜋2

= 1𝑛2

𝑑2𝛾2𝜋2

; 𝑛 = 1,2, … ∞ (4)

R1 R2 Rn

sC0

1 sC2

10 sC

2

10sC

2

10

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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XXIV Symposium Electromagnetic Phenomena in Nonlinear Circuits June 28 - July 1, 2016 Helsinki, FINLAND

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B dl1i 1Rs 1Ls 2i 2Ls

2Rs)(te 1zw 2zw

Sp

Fig. 3. The diagram of 2-coil transformer with a single laminated magnetic core

TABLE I PARAMETERS OF THE TRANSFORMER

𝑆𝑝 [m²] 𝑑𝑙 [m] 𝑍𝑤1 𝑍𝑤2 𝐿𝑠1[µH] 𝐿𝑠2[µH] 𝑅𝑠1[mΩ] 𝑅𝑠2[mΩ]

0.001 0.5 10 10 25 25 79 ∞ 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 − 𝐼ℎ𝑑𝑙 (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

𝑠� ∙ 𝐼1(𝑠) + �𝑆𝑝 ∙ 𝑧𝑤1� ∙ �̂�(𝑠) ∙ 𝜃(𝑠)

𝑑𝑙= 𝐸(𝑠)

𝑠

�𝐿𝑠2 + 𝑅𝑠2𝑠� ∙ 𝐼2(𝑠) + �𝑆𝑝 ∙ 𝑧𝑤2� ∙ �̂�(𝑠) ∙ 𝜃(𝑠)

𝑑𝑙= 0

� (6)

Where �̂�(𝑠) is the impedance of the circuit synthesised in

chapter III. For the calculations, an exemplary non-linear characteristics of magnetization is adapted:

𝐻 = 𝑎1𝐵 + 𝑎3𝐵3 + 𝑎9𝐵9 (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.

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.

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 �̂�(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

[1] Jian Wang , Heyun Lin , Yunkai Huang , Xikai Sun, A New Formulation

of Anisotropic Equivalent Conductivity in Laminations, IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 5, MAY 2011.

[2] Shintemirov A., Tang W. H., Wu Q. H., Transformer Core Parameter Identification Using Frequency Response Analysis, IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 1, JANUARY 2010.

[3] Emad Dlala, Comparison of Models for Estimating Magnetic Core Losses in Electrical Machines Using the Finite-Element Method, IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 2, FEBRUARY 2009.

[4] Anouar Belahcen, Antero Arkkio, Comprehensive Dynamic Loss Model of Electrical Steel Applied to FE Simulation of Electrical Machines, IEEE TRANSACTIONS ON MAGNETICS, VOL. 44, NO. 6, JUNE 2008.

[5] Gołębiowski L., Mazur D., Comparison of calculations methods of dissipation inductance in DC- supplied multiple-winding autotransformers, s. 55-56, 2010, 21st Symposium of Electromagnetic Phenomena in Nonlinear Circuits, EPNC, 29.06 -2.07.2010, Dortmund and Essen, Germany.

[6] Azarewicz St., Węgliński B., Parameters of Chosen Generator Sheets at Elevated Frequency of Remagnetization, Zeszyty Problemowe – Maszyny Elektryczne Nr 80/2008.

[7] Zakrzewski K., Overloss coefficient in magnetic laminations during PWM supply voltage, ARCHIVES OF ELECTRICAL ENGINEERING VOL. 59(3-4), pp. 169-176 (2010).

[8] Tellinen J. : A Simple Scalar Model for Magnetic Hysteresis, IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 4, JULY 1998.

0 5 10 15 20x 10-4

-150

-100

-50

0

50

100

150

i1(t), e(t), 100*B(t)

time, s

i(t) e(t)

100*B(t)

-2000 -1000 0 1000 2000-1.5

-1

-0.5

0

0.5

1

B(H)

H, A/m

B, T 1

2

31

2

3

12

3

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Proceedings of EPNC 2016, June 28 - July 1, 2016 Helsinki, FINLAND