Iec tr 62383 2006

TECHNICAL REPORT IEC TR 62383 First edition 2006 01 Determination of magnetic loss under magnetic polarization waveforms including higher harmonic components – Measurement, modelling and calculation m[.]

General

The measurement method that incorporates higher harmonics can also be effectively utilized with the Epstein frame or a ring core as a magnetic circuit However, when using the Epstein frame, it is important to consider the specific path length characteristics, which remain uncertain in the higher frequency range.

LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.

The proposed test apparatus is based on the magnetic circuit of a double U-yoke SST It can be considered to consist of the following parts.

Yokes, windings and test specimen

Yokes are designed in a U shape and constructed from insulated sheets of electrical steel or nickel iron alloy The manufacturing processes for yokes adhere to specific guidelines.

According to Annex A of IEC 60404-3, while the dimensions of the yokes and specimens are flexible, the effective magnetic path length (\$l_{eff}\$) must match the inside width specified in the standard if the yoke size is reduced It is essential for the initial permeability of the yoke to remain relatively constant across frequencies, particularly up to the maximum higher harmonic frequency being measured For windings and test specimens, reference to IEC 60404-3 is necessary, especially for ring specimens.

At higher frequencies, capacitance and dielectric effects pose significant challenges To minimize dielectric loss, it is essential to carefully manage the winding space and select appropriate dielectric constants for the formers and wire insulation.

It is essential to continuously monitor the temperature of the test specimen In high-frequency measurements, the increase in temperature significantly impacts results, necessitating measures to minimize this effect.

Power amplifier

A power amplifier must possess low output impedance and a frequency bandwidth that exceeds the highest harmonic frequency to be measured Additionally, the output voltage should be sufficiently high to effectively magnetize the specimen across the entire higher harmonic frequency range For further details, please refer to the relevant documentation.

Waveform synthesizer

An arbitrary waveform can be generated through computer programming, ensuring that its frequency is synchronized with the digitizer frequency The frequency uncertainty of the waveform synthesizer must be better than 0.01%, allowing for the output of arbitrary waveforms created from synthesized digital wave data Additionally, the relative frequency uncertainty should remain below 0.01%.

Digitiser

To digitize the secondary induced voltage \$U_2(t)\$ and the voltage \$U_s(t)\$ across the non-inductive precision resistor \$R_s\$ connected in series with the primary winding for magnetizing current determination, a 2-channel digitizer is essential Both channels must be sampled and digitized simultaneously, with the resulting data subsequently recorded in memory.

To ensure accurate representation of the power integral, the ratio of the sampling frequency (\$f_s\$) to the magnetizing frequency (\$f_m\$) must be an integer, adhering to the Nyquist condition This allows the power integral to be replaced by a corresponding sum, which accurately reflects the integral up to the \$n\$th harmonic, where \$2n\$ represents the number of samples per fundamental period Maintaining the Nyquist condition requires synchronization of the sampling frequency and magnetizing frequency to a common fundamental clock, ensuring a fixed integer ratio.

To accurately scan the hysteresis loop, the sampling frequency $ f_s $ must exceed twice the bandwidth of the B- and H-signals, as expressed by the equation $ f_s = 2 m_s n f $, where $ n $ represents the highest harmonic to be measured.

Commercial hardware components often lack synchronization, resulting in a non-integer ratio of \$f_s / f_m\$ To minimize the deviation of the true period length from the nearest multiple of sampled measurement intervals, the sampling frequency must be significantly increased, such as using 1,024 samples per period.

Adhering to the Nyquist condition is crucial for accurately capturing higher frequencies To prevent the influence of non-existent low-frequency harmonics in the measurement signal, it is essential to implement a low-pass antialiasing filter This filter should restrict the system bandwidth to less than half of the sampling frequency (\$f_s / 2\$).

For optimal amplitude resolution, a minimum of 12-bit resolution is essential to minimize digitalization errors, especially in non-oriented materials with high silicon content Additionally, it is crucial that the two voltage channels transmit signals with minimal phase shift to ensure that total uncertainty remains unaffected.

When measuring magnetic loss under conditions of magnetic polarization with significant higher harmonic components, it is crucial that the digital sampling conditions for these higher harmonics meet the specified requirements This is particularly important when the amplitude of the higher harmonics is sufficient to generate minor loops.

Control of secondary voltage

The waveform of the secondary voltage should be controlled to have the required components This control can be achieved by feedback techniques using digital or analog means

The deviation should be below 1 % for each harmonic component.

Peak reading apparatus

To accurately measure the peak value of magnetic polarization, it is essential to utilize a Miller type analog integrator along with a peak reader This setup must have a frequency bandwidth that exceeds the highest harmonic frequency, denoted as \$f_h\$, that is intended for measurement.

The peak reader should be able to repeat peak readings at an appropriate time rate

The uncertainty of the peak reading apparatus should be better than 0,2 %

An average voltmeter is unsuitable for measuring the peak value of magnetic polarization, as the secondary induced voltage can exhibit multiple zero crossings within a single period.

Air flux compensation

To compensate for air flux, a mutual inductor can be utilized The primary winding of the mutual inductor is connected in series with the primary winding of the test apparatus, while the secondary winding is connected in series opposition to the secondary winding of the test apparatus.

To ensure accurate measurements, the mutual inductance value must be adjusted so that, when an alternating current flows through the primary windings without the specimen in the apparatus, the voltage between the non-common terminals of the secondary windings does not exceed 0.1% of the voltage measured across the secondary winding of the test apparatus alone.

The measuring system can be constructed using the components which are described in

Clause 2 The block diagram of the circuit is shown in Figure 1

Figure 1 – Block diagram of the measuring system for the measurement of magnetic loss of electrical steel sheets under magnetic polarization waveforms which include higher harmonic components

Generation of the magnetic polarization waveform including higher

The time dependent magnetic polarization including higher harmonics can be described by ¦= + + + +

( ω φ (2) where j is a non-negative integer;

N corresponds to the highest harmonic frequency f h ; ω1 is the fundamental angular frequency(ω 1 =2ʌf 1 );

J is the amplitude of the (2j+1) th harmonic at the angular frequency ω h =(2j+1)ω 1 ;

The synthesized reference voltage U (t ) is as follows; ¦ = + + + + +

N 2 is the number of turns of secondary winding;

A is the cross sectional area of specimen

The procedure of setting the desired magnetic polarization follows the following steps

The relative amplitudes of fundamental and higher harmonic waves are established, followed by the measurement of the resulting peak value of magnetic polarization Subsequently, the synthesizer's gain is adjusted to achieve the desired amplitude of magnetic polarization.

Determination of peak value of magnetic polarization

To measure the peak value of magnetic polarization $ \hat{J} $, a Miller type analogue integrator and a peak reader should be utilized The relationship between the peak value of magnetic polarization and the output voltage $ \hat{U}_J $ of the peak reader is established.

Uˆ J = N2 ˆ (4) where RC is the time constant of the Miller integrator.

Determination of the magnetic polarization

The instantaneous value of the magnetic polarization J (i ) at the time t = i/nf of the specimen can be calculated from the secondary induced voltage U 2 (t) using the following numeric equation: ¦ =

( (5) where i is an integer; n is the number of sampling points per period; f is the magnetizing frequency;

J 0 is the integration constant such that ¦ = n i i

The peak value of the magnetic polarization J (t ) is identical to the maximum value of J (i )

Determination of magnetic field strength

The instantaneous value of the magnetic field strength H (i ) at the time t = i/nf can be calculated from the digitised value of the voltage U s (i ) across the non-inductive precision resistor R s :

= (7) where l eff is the effective magnetic path length;

N 1 is the number of turns of primary winding.

Determination of the magnetic loss

The magnetic loss $ P_c $ can be determined using data from the 2-channel digitizer, specifically the secondary induced voltage $ U_2(t) $ and the voltage $ U_s(t) $ across the non-inductive precision resistor $ R_s $, which is connected in series with the primary winding.

P c is the specific total loss, in watt per kilogram; i is an integer; n is the number of sampling points per period;

N 1 is the number of turns of primary winding;

N 2 is the number of turns of secondary winding;

A is the cross sectional area of specimen; ρ m is density of the test specimen in kilogram per cubic meter.

Plotting the a.c hysteresis loop including the higher harmonics

The a.c hysteresis loop can be plotted using the magnetic polarization from equation (5) and the magnetic field strength from equation (7)

Magnetic loss measurement of non-oriented electrical steel sheets

Figure 2 illustrates the results obtained from a non-oriented electrical steel specimen at a fundamental frequency of 60 Hz and a maximum magnetic polarization of \$\hat{J} = 1.5 \, T\$, under varying harmonic amplitude conditions Specifically, Figure 2a presents the magnetic polarization curves.

Figure 2b the magnetic field strength curves, and Figure 2c the a.c hysteresis loops The higher harmonic frequency is f h = 23f 1 and the higher harmonic amplitudesJˆ 23 amount to

Figure 3 illustrates the total loss as a function of higher harmonic frequency $ f_h $ and higher harmonic polarization $ \hat{J}_h $ for non-oriented electrical steel sheets at $ \hat{J} = 1.5 \, T $ The total loss for a 0.5 mm thick specimen significantly exceeds that of a 0.35 mm thick specimen, primarily due to the eddy current effect.

Magnetic loss measurement under stator tooth waveform conditions

To determine the a.c hysteresis loop of an induction motor's stator-tooth, measuring the magnetic polarization using search coil windings is feasible, but assessing magnetic field strength within the motor poses challenges A viable method involves amplifying the induced voltage from the B-coil and connecting it to the measuring system outlined in Clause 4 This allows for the measurement of the a.c magnetic properties of a specimen made from the same material as the stator-tooth It is essential to adjust the amplifier's gain to satisfy equation (9), ensuring that the magnetic polarization of the sample in the yoke aligns with that of the stator tooth.

The equation \$U = (A)(9)\$ relates the cross-sectional areas of the sample (\$A_s\$), the number of B-coil windings (\$N_s\$), and the voltage applied to the single sheet tester (\$U_s\$) Additionally, it incorporates the cross-sectional area of the stator tooth (\$A_m\$), the number of B-coil windings (\$N_m\$), and the voltage induced from the system (\$U_m\$).

B-coil of the induction motor, respectively

To measure the a.c hysteresis loop of an induction motor's stator tooth, we utilized a 10 cm × 10 cm double U-yoke SST along with a 3-phase, 3.75 kW induction motor featuring 4 poles, 36 stator teeth, and 44 rotor teeth Two B-coils were wound, one aligned with the rolling direction and the other perpendicular to it, to facilitate the measurement process.

In the setup illustrated in Figure 4, we observe 10 turns on a single sheet, which allows us to measure magnetic induction B instead of magnetic polarization J When the magnetic field strength is relatively low, the air flux becomes negligible compared to the magnetic flux density of the core, enabling us to approximate magnetic polarization J through the voltage induced in the B-coil Additionally, an a.c dynamometer can be employed to load the induction motor effectively.

1.5 Fundamental frequency f 1 = 60 Hz Higher harmonic frequency f h = 23f 1

M a gn et ic P o la ri z a ti on ( T )

M a g net ic F ield St reng th ( A /m )

The higher harmonic amplitude was 0 %, 2 %, 5 %, and 10 % of Jˆ 1 respectively

The dependency of higher harmonic polarization components on magnetic polarization $ J(t) $ and magnetic field strength $ H(t) $ is illustrated in Figure 2 This analysis focuses on the a.c hysteresis loops of non-oriented electrical steel at a fundamental magnetizing frequency of $ f_1 = 60 $ Hz, with a maximum magnetic polarization of $ \hat{J} = 1.5 $ T, and considers higher harmonic frequencies where $ f_h \neq f_1 $.

M a g n e tic F ie ld S tre n g th (A /m )

H ig h e r h a rm o n ic p o la riz a tio n c o m p o n e n t(% )

Figure 3 – Specific total loss depending on the higher harmonic frequency and higher harmonic polarization for the non-oriented electrical steel sheet at J ˆ = 1,5 T

Hig her har m pol (J h /J 1 ) higher harmonic frequency ( f h =n f 1 , f 1 ` Hz)

Figure 4 – B -coil winding positions of stator tooth of a 3,75 kW induction motor to measure the a.c hysteresis of the stator tooth depending on the load

Figure 5 illustrates the a.c hysteresis loops for a rotor without skew, with the B-coil wound coaxially in the rolling direction of non-oriented electrical steel In Figure 5a, the a.c hysteresis loop is presented under sinusoidal magnetic polarization, while Figure 5b depicts the a.c hysteresis loop of the stator-tooth under no load Both cases exhibit the same maximum magnetic polarization, denoted as \$\hat{J}\$, yet they differ in the specific total loss.

Figures 5a and 5b indicate an increase of over 10%, while Figures 5c and 5d illustrate the a.c hysteresis loops of the stator-tooth at 40% and 80% of full load The specific total loss values rose by more than 30% and 50%, respectively Figure 6 demonstrates the relationship between specific total loss and the load of the induction motor, revealing that specific total loss is also influenced by the direction of magnetization due to the macroscopic magnetic anisotropy of non-oriented electrical steel The analysis includes a) sinusoidal polarization waveform (standard method), b) no load, c) 1.5 kW load, and d) 3 kW load.

Figure 5 – AC hysteresis loop of the stator teeth of a 3,75 kW induction motor measured in single sheet tester

This document is licensed to MECON Limited for internal use at the Ranchi and Bangalore locations, and it has been supplied by the Book Supply Bureau The notation "Ƴ" represents the direction parallel to the rolling direction of non-oriented electrical steel when the rotor is not skewed, while "Ɣ" denotes the direction perpendicular to the rolling direction under the same conditions.

Figure 6 – Specific total loss of the stator tooth depending on the load

7 Prediction of magnetic loss including higher harmonic polarization

General

The magnetization process of magnetic materials exhibits complex non-linear and hysteresis behavior that is challenging to physically describe Currently, a hysteresis loop cannot be represented as a mathematically analytical function To effectively predict magnetic loss during this process, a suitable model is essential This report will introduce several typical methods for predicting total magnetic loss.

Energy loss separation [14]

The physically based separation of specific total loss P ( f )at a given frequency f is expressed as the sum of the hysteresis loss P h , classical eddy current loss P cl ( f ), and excess loss

The equation $ P = h + cl + exc $ (10) defines the total energy loss, where $ P h ( f ) = W h f $ and $ W h $ represents the hysteresis energy loss per cycle Generally, $ W h $ is treated as a frequency-independent quantity, except in certain special cases [15] Consequently, equation (10) can be reformulated to express the energy loss per cycle as $ W = P / f $.

There are no limitations, in principle, as to the kind of polarization waveform The fundamental aim is the prediction of W h , W cl ( f ) , and W exc ( f )

T o ta l loss at st at or t oot h (W /k g)

7.2.2 Energy losses with arbitrary flux waveform and no minor loops

The instantaneous classical eddy current loss per unit volume P cl ( t ) can be defined for a magnetic lamination of conductivity σ and thickness d as

This relationship is only valid when the flux penetrates the material completely

The classical energy loss per cycle T = 1/f is then obtained as ³ ³ =

= T cl T cl dt dt t dJ dt d t P f

( σ (13) and it can be calculated exactly for all waveforms of J (t )

In the absence of minor loops, hysteresis loss remains unaffected by the polarization waveform The instantaneous excess loss, denoted as \$P_{exc}(t)\$, is expressed in its most general form by equation (14): \$\frac{dP_{exc}}{dt} = V \cdot n \cdot \frac{dJ}{dt}\$.

= σ (14) where n o is the number of simultaneously active magnetic objects in the limit f ặ 0;

V o is a parameter defining the statistics of the magnetic objects;

S is the cross-sectional area of the lamination

A magnetic object (MO) is defined as a collection of neighboring walls that interact strongly enough to be considered a single entity The dynamic behavior of an individual MO is characterized by a damping coefficient $ G $, which is approximately independent of the MO's internal details, with a theoretical value of $ G = 0.1356 $ It is important to note that $ V_o $, which accounts for the excess loss due to the material structure, is a function of $ \hat{J} $ In many cases, the value of $ n_o $ is sufficiently small, ensuring practical consistency in applications.

(15) and the excess energy loss becomes ³ ³ = ⋅

= T exc T exc dt dt t GSV dJ dt t P f

( σ (16) which is easily specialized to the desired J (t ) waveform

By combination of equation (11), equation (13) and equation (16), energy loss per cycle becomes as follows:

= h cl exc h T T dt dt t GSV dJ dt dt t d dJ W f W f W W f

In order to predict the total loss for a given J ( t ) , the following quantities are required: conductivity σ, cross-sectional area S and the pre-emptive determination of W h and V o only

Let us consider the typical experiment in a grain-oriented Fe-Si lamination reported in

Figure 7, where the behaviour of the quantity

The equation $ W_{dif} = -cl = h + exc $ (18) is measured under sinusoidal flux and is derived from its optimal straight fitting line as a function of $ f^{1/2} $ It is evident that the values of $ W_h $ and $ V_o $ can be determined from the intercept and slope of the theoretical line, respectively.

Figure 8 shows the frequency behaviour of the magnetic energy loss experimentally found in non-oriented Fe-(3 wt %)Si lamination under sinusoidal polarization at peak magnetization

The fitting lines were predicted using the method described earlier, focusing on the loss separation equation (18) and the direct calculation of \$W_{cl}\$ The representation of \$W_{dif} = W - W_{cl} = W_h + W_{exc}\$ as a function of \$f^{1/2}\$ reveals a deviation from this dependence at low frequencies Consequently, the best fit of \$W_{dif}\$ is illustrated in Figure 8, allowing for the determination of the parameters \$W_h\$ and \$V_o\$.

Figure 7 – Examples of experimental dependence of the quantity W dif =W −W cl =W h +W exc on the square root of frequency in grain-oriented Fe-Si laminations (thickness 0,29 mm)

The experimental quantity W dif =W−W cl =W h +W exc is plotted as a function of f 1 / 2 This results in the dash-dot straight fitting line, by which the two parameters W h 1 and V o are determined

Figure 8 – Energy loss per cycle W and its analysis in a non-oriented Fe-(3wt %)Si lamination energy loss with arbitrary flux waveform and minor loops

For the total loss at a given frequency f in the presence of minor loops, equation (17) should be modified as follows:

= h M h m exc M exc m T dt dt t dJ

The quasi-static loss contribution, denoted as \$W_{h, M} + W_{h, m}\$, is determined by the sum of the quasi-static major loop area, where the subscripts M and m refer to the major and minor loops, respectively.

The total energy loss is derived from the summation of the areas $ W_{h,m,i} $ of the $ 2n $ minor loops, each exhibiting a polarization swing of $ \pm J_{m,i} $ The excess energy loss linked to the major loop is denoted as $ W_{exc,M} $, while $ W_{exc,m} $ represents the excess energy loss associated with the collection of $ 2n $ minor loops.

For the calculation of W h , M + W exc , M , only the total loss value W M under sinusoidal polarization at two different frequencies and peak amplitude J ˆ needs to be known

W di f = (W - W cl) (mJ/kg) f 1/2 (Hz 1/2 )

LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU. dt dt t

The equation $ M - = + = 1 + \sigma_o ( ) \cdot ³ \| ( )^{3/2} $ under the specified waveform $ J(t) $ at the target frequency $ f $ simplifies the analysis of loss versus frequency behavior, as shown in Figure 8 Here, $ W_{h1, M} $ represents the linearly extrapolated major loop hysteresis loss term.

It is noticed that the time integration of ( )| 3 / 2

| dt t dJ is extended only over the portion ¦ T M of the period T which is not occupied by the minor loops

To determine W h , m , the simplified Preisach model [14] was used and excess energy loss associated with the ensemble of minor loops ¦ =

W , 2 1 , , (21) was calculated using the following equation: dt dt t

= σ (22) where the integration is made over the associated i-th time interval of duration T m , i

Equation (19) is utilized to predict energy loss in the scenario depicted in Figure 9, under the conditions of J ˆ= 1.4 T The analysis considers minor loops with a consistent peak-to-peak amplitude of 2 J m and a variable number of loops, specifically 2n = 2, 4, …, 12, while maintaining a constant product value of 2n⋅∆J m = 1.2 T.

The parameter $ V_o (J^\hat) $, which is associated with the statistical characteristics of the magnetization process and the distribution of local coercive fields, increases as $ J^\hat $ rises This trend is exemplified in the current non-oriented Fe-Si lamination.

Figure 10 By introducing the results provided by equation (19), we arrive at the prediction shown in Figure 11, which compares well with the experimental data

NO Fe-(3wt%)Si f = 1 Hz

The minor loops exhibit a peak-to-peak amplitude of 2J m The theoretical loops were reconstructed using a Preisach model, which relied solely on the experimental major hysteresis loop as input, along with appropriate simplifying assumptions.

Figure 9 – Examples of composite experimental (solid lines) and reconstructed (dashed lines) d.c hysteresis loops at peak magnetization J ˆ = 1,4 T in non-oriented

Fe-(3 wt %) Si laminations (thickness 0,34 mm) generated by the J(t) waveforms

Figure 10 – Experimental dependence of the statistical parameter of the magnetization process V o on the peak magnetization value in the tested non-oriented Fe-Si laminations

NO Fe-(3wt%)Si f = 50 Hz

Figure 11 – Loss evolution with the number of minor loops in a non-oriented Fe-Si lamination, subjected to controlled constant magnetization rate dJ dt ( t ) = 4 f ⋅ ( J ˆ + 2 J m ) , with J ˆ = 1,4 T and 2nJ m = 1,2 T

Neural network method [17]

A neural network consists of interconnected nodes that mimic the functionality of animal neurons Its processing capability is determined by the strengths of the connections between these units, known as weights, which are adjusted through a learning process based on training patterns.

Figure 12 – Artificial neuron (also termed as unit or nodes)

Neural networks are structured in layers, including an input layer, one or more hidden layers, and an output layer Each layer is interconnected, allowing output signals to be transmitted from one layer to the next In fully connected networks, every node in a layer is connected to all nodes in the preceding layer Information is introduced into the network via the input layer and subsequently processed in the following layers.

Figure 13 – Neural network design topology

The study utilized a commercial neural network package to predict loss in 0.5 mm thick non-oriented electrical steel with silicon content between 0.2% and 4% The predictions were made under various excitation waveforms, including sinusoidal, square, and PWM, at a fundamental frequency of 50 Hz Input data was collected across a flux density range of 0.1 T to 1.5 T, with PWM conditions set at a 50 Hz fundamental frequency, a frequency ratio of 12, and a modulation index varying from 0.5 to 1.0.

In total, four input nodes were used, as shown in Table 1, the peak flux density varying from

The study examines materials with a magnetic field strength ranging from 0.1 to 1.5 T, where the density varies between 7.85 and 7.60 (10³ kg/m³) The silicon content is analyzed from 0.2% to 4%, and the form factor (f.f) ranges from 1.11 for a sine wave to 1.63 for a single-phase unipolar PWM waveform at a modulation index of 0.5 and a frequency ratio of 12 Notably, the fundamental frequency is fixed at 50 Hz, thus not included as an input parameter.

Input layer nodes 4 Hidden layer-1 nodes 4 Hidden layer-2 nodes 3 Output layer nodes 1 Transfer function Sigmoid

The effectiveness of back propagation neural networks is significantly influenced by the number of training cases utilized Generally, a larger dataset leads to a more accurate model In this study, a total of 450 records were employed for training purposes.

The software allows for the incorporation of test sets during training, which are essential for evaluating network overtraining and model integrity The standard deviation between the training output and the target for the test set was approximately 5.5%, indicating a strong performance, especially given the inclusion of various waveforms like PWM Performance metrics for the trained neural network are detailed in Table 2 and Table 3, highlighting results for data points both within and outside the training set.

Table 2 – Error of the specific total loss recalled from the trained neural network compared with the measured values at 1,6 T (point not used during the training)

Table 3 – Error of the specific total loss recalled from the trained neural network compared with the measured values at 1,5 T (point used during the training)

Modified superposition formula [20]

When the harmonic order exceeds 9, the impact of the phase angle on magnetic loss diminishes The traditional superposition principle applied in analyzing magnetic loss is as follows:

J 1 is magnetic polarization of fundamental frequency;

J h is magnetic polarization of higher harmonic frequency f h =nf 1 ; f 1 is the fundamental frequency; n is an integer

The magnetic losses, denoted as P_c and P_c(J_h, nf 0), were measured from a normally demagnetized state Although this method has been previously applied, it fails to predict magnetic loss using the superposition principle when the polarization is high or when its direction differs from the rolling direction.

The analysis of the a.c hysteresis loops, depicted in Figure 15, reveals the behavior of H(t) and J(t) when only one period of higher harmonic polarization is considered It is evident that the superposition principle fails in the high polarization region and in directions that differ from the rolling direction Additionally, the minor loops generated in both the zero polarization and saturation polarization regions exhibit significant differences, which are also influenced by the measurement direction.

Figure 14 – Waveforms of dt t dJ ( )

, H (t ) and J (t ) when higher harmonic polarization is included

Figure 16 illustrates that the magnetic loss $ P_c $ is influenced by the position of the a.c minor loop of the higher harmonic in relation to the phase of the fundamental wave loop and its amplitude At low higher harmonic amplitudes, $ P_c $ remains unchanged regardless of the minor loop's position However, as the higher harmonic amplitude increases, $ P_c $ rises, becoming dependent on both the harmonic amplitude and the minor loop's position Specifically, when the peak magnetic polarization $ \hat{J} $ is 1 T, $ P_c $ does not vary with the minor loop's position In contrast, at a peak magnetic polarization of 1.5 T, a significant change in $ P_c $ occurs with the minor loop's position as the harmonic polarization amplitude increases This suggests that when the minor loop is situated in high polarization regions of the major hysteresis loop, irreversible magnetization rotation occurs, while in the zero polarization region, irreversible domain wall movement may significantly contribute to the increase in magnetic loss $ P_c $.

The article discusses the polarization regions in magnetization, specifically highlighting the zero polarization region and the saturation polarization region It emphasizes the differences in magnetization when aligned with the rolling direction versus when it is perpendicular to the rolling direction This information is crucial for understanding the magnetic properties of materials in various applications.

The generation of two symmetrical alternating current (a.c.) minor loops is observed in both the zero polarization region and the saturation polarization region of the fundamental hysteresis loop This analysis focuses on magnetization occurring in the rolling direction as well as perpendicular to it.

Figure 16 – Specific total loss P c of the combined waves, with harmonic frequency 23 f 1, depending on the position of a.c minor loop at maximum magnetic polarization of

This effect was also measured when the higher harmonic frequency was low, i.e f h

Tiêu đề	Determination of Magnetic Loss Under Magnetic Polarization Waveforms Including Higher Harmonic Components – Measurement, Modelling and Calculation Methods
Trường học	Unknown University
Chuyên ngành	Electrical Engineering
Thể loại	Technical report
Năm xuất bản	2006
Thành phố	Geneva

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