Iron Losses Sohail Thesis KLAR 230408

8/3/2019 Iron Losses Sohail Thesis KLAR 230408

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Doc. Title: Revision Page

Effect of harmonics on iron losses 01 2/70



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Summary:

This thesis investigates the effect of non-sinusoidal flux waveforms on iron losses through

Epstein frame measurements.

The report includes a survey of empiric loss calculation methods. First, it is investigated how to

find the loss coefficients for calculations of losses with sinusoidal wave forms. For this purpose,

both new and previously obtained measurement data is used. Different methods to find the

coefficients are studied and their short comings are pointed out. It is found that the curve fitting

method using a series of measurements, ±4% agreement between empiric and experimental

values is possible for wide range of frequencies (e.g. up to 400 Hz). This can be compared to

±30% difference for the constant coefficient method utilizing experimental data for only two test

points.

Consequently, an experimental investigation is made on the iron loss in presence of harmonics.

One or two harmonics are superimposed on the fundamental flux wave. The effects of both the

harmonic amplitudes as well as the phase angles on iron losses are studied experimentally. It is

observed that the phase angle between fundamental and harmonic waves is important for low

order harmonics (e.g. 5th /7th) but has minor effect on higher order harmonics.

Further, a time domain analytical expression for calculation of iron losses with distorted

waveform is recommended and it is found that the recommended expression gives ±10%

agreement with the experimental results. However, it is shown that the impact of the phaseangle is not covered using the said expression. The results also show that iron loss coefficients

found using measurements with sinusoidal wave forms can be used for distorted waveforms as

well.



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TABLE OF CONTENTS

1 INTRODUCTION....................................................................................................................................6 1.1 PURPOSE..........................................................................................................................................6 1.2 SCOPE..............................................................................................................................................6 1.3 DEFINITIONS .....................................................................................................................................6 1.4 STRUCTURE ......................................................................................................................................7

2 LITERATURE STUDY ON IRON LOSS CALCULATION.....................................................................8

2.1 CALCULATION OF IRON LOSS COMPONENTS ........................................................................................8 2.2 METHOD OF REDUCING IRON LOSSES ...............................................................................................10 2.3 EXPERIMENTAL SETUP FOR MEASUREMENT OF IRON LOSS................................................................11

2.3.1 Epstein frame.........................................................................................................................11 2.3.2 Toroid Tester..........................................................................................................................13 2.3.3 Single Sheet Tester (SST).....................................................................................................14

2.4 METHODS TO PREDICT IRON LOSSES WITH SINUSOIDAL WAVEFORM ...................................................14 2.4.1 Introduction ............................................................................................................................14 2.4.2 Determination of Loss Cofficients..........................................................................................15 2.4.3 Conclusion .............................................................................................................................20

2.5 EVALUATION OF METHODS FOR LOSS PREDICTION WITH NON SINUSOIDAL WAVEFORMS........................20

3 MEASURMENT OF IRON LOSSES WITH NON SINUSOIDAL WAVE FORMS ...............................24

3.1 AUTHENTICITY OF MEASUREMENTS..................................................................................................24 3.1.1 Accuracy ................................................................................................................................24 3.1.2 Repeatability ..........................................................................................................................25

3.2 BLOCK DIAGRAM OF EXPERIMENTAL SETUP.......................................................................................26

4 EFFECT OF A SINGLE FLUX HARMONIC ON IRON LOSSES........................................................27

4.1 FUNDAMENTAL FREQUENCY 50 HZ...................................................................................................27 4.1.1 5th Harmonic..........................................................................................................................27 4.1.2 7th Harmonic..........................................................................................................................31 4.1.3 11th Harmonic........................................................................................................................33 4.1.4 13th Harmonic........................................................................................................................35 4.1.5 Conclusion .............................................................................................................................35

4.2 FUNDAMENTAL FREQUENCY 30 HZ...................................................................................................37 4.2.1 5th Harmonic..........................................................................................................................37 4.2.2 7th Harmonic..........................................................................................................................39 4.2.3 11th Harmonic........................................................................................................................41 4.2.4 13th Harmonic........................................................................................................................43 4.2.5 Conclusion .............................................................................................................................44

4.3 FUNDAMENTAL FREQUENCY 70 HZ...................................................................................................44 4.3.1 5th Harmonic..........................................................................................................................44 4.3.2 7th Harmonic..........................................................................................................................46 4.3.3 11th Harmonic........................................................................................................................47 4.3.4 13th Harmonic........................................................................................................................48

4.4 FUNDAMENTAL FREQUENCY 5 HZ.....................................................................................................49

5 EFFECT OF TWO FLUX HARMONICS ON IRON LOSS...................................................................50

5.1 FUNDAMENTAL FREQUENCY (50 HZ) ................................................................................................50 5.1.1 Effect of the 5th and 7th harmonic.........................................................................................50 5.1.2 Effect of the 5th and 13th harmonic.......................................................................................53

5.2 FUNDAMENTAL FREQUENCY 30 HZ...................................................................................................54 5.2.1 Effect of the 5th and 7th harmonic.........................................................................................54

5.2.2 Effect of the 5th and 13th harmonic.......................................................................................56 5.3 FUNDAMENTAL FREQUENCY (70 HZ) ................................................................................................57 5.3.1 Effect of the 5th and 7th harmonic.........................................................................................57



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5.3.2 Effect of the 5th and 13th harmonic.......................................................................................59 5.4 CONCLUSION ..................................................................................................................................60

6 CONCLUSIONS...................................................................................................................................62

7 FUTURE WORK ..................................................................................................................................64

8 REFERENCES.....................................................................................................................................65

9 ENCLOSURES ....................................................................................................................................67

9.1 APPENDIX – 1..................................................................................................................................67 9.2 APPENDIX – 2..................................................................................................................................68 9.3 APPENDIX – 3..................................................................................................................................69 9.4 APPENDIX – 4..................................................................................................................................70



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1 INTRODUCTION

1.1 Purpose

Nowadays, variable speed drives are becoming more and more popular. Variable speed is

achieved by use of an inverter. Inverter fed drives experience more iron loss as compared to

directly connected ones. The increase in iron loss is a result of the harmonics introduced by the

inverter supply and can effect both the lifetime of a machine and efficiency as well. Therefore, it

becomes important to have a fair idea about the effect of iron loss. The suggested thesis work

deals with quantifying the iron loss due to harmonics. The effect of different harmonic amplitude

and effect of phase difference is also studied.

1.2 Scope

This report aims to quantify the increase in iron loss due to introducntion of different time

harmonics in the fundamental alternating flux wave. The effect of amplitude and phase angle

due to presence of one as well two harmonics are studied. The results described in this report

are based upon the experiments made both at Surahammar Bruks AB and KTH. It is to be

noted that this report does not describe the iron loss associated with the rotational flux density.

1.3 Definitions

P hys Hystereses Iron loss component

P classical Classical iron losss component

P excess Excess iron loss compenent

P total Total iron loss

n Hystereses loss exponent

Kh Hystereses loss coefficient

Ka Excess loss coefficient

Ke Classical loss coefficient

f Frequency

Bmax Peak flux density

l Length of test specimen strip

ma Active mass of test specimen

d Thickness

Electrical Conductivity

Ch Hystereses loss correction factor

N1 Total number of primary winding turns



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N2 Total number of secondary winding turns

pw Power measured by wattmeter

W Specific iron loss in W/kg

1.4 Structure

This report has the following structure:

Chapter 2 makes an overview of some previous work that were done in order to understand and

predict iron loss with sinusoidal as well as non sinusoial wave forms.

Chapter 3 explains the experimental setup used for measurments for iron loss. Accuracy andrepeatability of the setup is also discussed.

Chapter 4 presents the results obtained from measurments in presence of one harmonic at

different fundamental frequencies. Observation and conclusions are also made on bases of

these measurments.

Chapter 5 investigates the effect of the presence of two harmonics on iron loss.

Chapter 6 presents the conclusion of this thesis work and relates the theory with results made

on basis of the Epstein frame measurments during the thesis work.

Chapter 7 includes some recommendations for the future works.

Chapter 8 Specifies a list of references for source material and further reading.

Chapter 9 Encloses some supplementary information.



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2 LITERATURE STUDY ON IRON LOSS CALCULATION

In order to calculate iron loss in electrical machines, it is essential to have the information of

shape or variation of the flux density. Flux density variation can be alternating or rotational [1].

Alternating flux density can further divide in to sinusoidal or non sinusoidal (due to presence of

different harmonics). Depending on the magnitude of the harmonic; non sinusoidal flux density

may or may not cause minor loops in the BH loop. On the other hand, rotational distribution of

the flux density can also be subdivided in to circular and elliptical (in the case of orthogonal flux

density components sinusoidally varying in time with different amplitudes). This can yet be

distorted (almost any shape), as happens in the case when alternating distribution in one of the

directions is non- sinusoidal. In the case of the alternating flux, there is a quite well known

theory, which is generally accepted and essentially based on the works of Bertotti, Fiorillo and

Novikov [1]. As for the rotational loss, the situation is not so clear and the problem of measuringand calculating the rotational loss has not yet been completely resolved [1]. In the following part;

iron loss calculation due to alternating flux density (sinusoidal and non-sinusoidal) will be

discussed.

2.1 Calculation of Iron loss components

2.1.1 Iron loss calculation with sinusoidal wave forms

Traditionally iron loss had been divided up in to two components, hysteresis loss “Phys” and eddycurrent or classical loss “Pclassical”. Therefore iron loss was expressed by (2.1). A brief description

of each part is given below.

classicalhysTotal PPP += 2.1

i) Hysteresis loss

The energy required to move the magnetic domain walls in the core magnetization is calledhysteresis loss. The hysteresis loss can be calculated using the empirical Steinmetz formula

B f k Pn

hhys max

=2.2

Where “K h ” and “n” are coefficients depending upon the magnetic material used, “f” is thefrequency and “B max ” is the peak flux density.

ii) Classical loss

Classical losses are caused by circulating currents in the core induced by flux variation. It canbe calculated as

6

2max

222 B f d

P classical

σ π =

2.3

The equation 2.3 is valid only when flux penetrates in material completely; In other words, thelamination thickness must be smaller than the skin depth.



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f

µπ

ρ <d 2.4

Where “ ” is the electrical conductivity, “d” is the thickness and “ ” is the resistivity of thelamination.

However, equation (2.1) has some limitations as some papers describes that it is onlyapplicable under the assumption that the maximum magnetic flux density of 1.0 T is notexceeded and the Hysteresis loop is under the static situation [17]. When P hys and P classical areadded together, the sum is always less than the measured total loss, the difference beingreferred to as the anomalous or excess loss. The excess loss accounts for 20 % or more of thetotal loss in electrical steels. In other materials it can comprise 90 % of the loss [3]. The excess

loss can be described as below.

iii) Excess loss

These losses are caused by parasitic micro-currents with high frequency that circulate around ofthe wall domain in the move of material magnetization. It can be calculated as

f BK p aexcess

5,15,1

max= 2.5

Whereoa GSV K σ = 2.6

“” is conductivity of the material, “G” and “Vo” are constants which appear to be material andmagnetization condition dependent, and “S” is the cross sectional area of the material.Therefore, now total iron loss can be represented by (2.6), which is a modification of (2.1).

excessclassicalhysTotaPPPP ++= 2.7

2.1.2 Iron Loss calculation with non-sinusoidal wave forms

i) Hysteresis loss

With introduction of harmonics, minor loops are the only thing that influences the hysteresis

loss. Minor loops appear due to the occurrence of flux reversal in the flux density wave formwhich is due to the presence of different harmonics components. The flux density reversaldepends on the magnitude as well as the phase angle of the harmonics [18]. There are someother factors that effects the magnitude of minor loops i.e. the order of harmonic components(higher order harmonic component gives more flux reversal in number and magnitude), totalpeak flux density (fundamental plus harmonic) and location of the minor loops (loops nearsaturation will cause larger loops and more losses). If the flux density waveform causes minorloops, the actual hysteresis loop area is required for the Hysteresis loss per cycle; the equation(2.2) is no more valid. Lavers [18] suggest a method, as given below

B f k C Pn

hhhys max=

2.8

Where “Ch” is the Hysteresis loss correction factor, which is



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=+=

N

i

ih db B

k C

1max1 2.9

Where “k” varies between 0.6 and 0.7 and dbi, i = 1,2,…, n, represents the flux reversals of theflux density waveforms (the change in the flux density during the excursion around a minorloop). If the minor loops do not occur, the hysteresis loss does not depend on the flux densitywaveform and is only related to the peak value of flux density. In that case equation (2.2) is stillvalid.

ii) Classical loss

The expression for calculation of iron loss for non-sinusoidal flux can be represented as

∞

=

=

0

22

222

6 nnclassical Bn

f d P

σ π 2.10

where “Bn” are the harmonic flux densities.

iii) Excess loss

Similarly, the expression for under non-sinusoidal flux can be represented as

dt dt

dB

T p

T

excess =

0

5,11

2.11

2.2 Method of reducing Iron Losses

Following are some methods used to reduce iron losses:

a) Lamination: The core is built up by thin lamination sheets piled on each other and insulated

from each other. This has an effect of reducing eddy currents.

b) Alloying: Iron is a good conductor and it is found that the addition of alloying elements

increase the electrical resistivity of iron which could help in reducing eddy current intensity. This

is normally done by adding silicon contents. A drawback of this method is that the introduction

of silicon makes iron brittle and difficult to roll and form.

c) Purification (and annealing): In a soft magnetic material the domains need to be able to alter

their disposition rapidly and easily according to overall magnetization of the metal. Things that

can hold up the easy movement of domain walls damage this intention. Any non-metallic

inclusions in the metal impede domain activity, so great efforts are made to purify electrical

steels. Other magnetic in-homogeneties spoil domain activity, such as stressed regions and

dislocations of the crystal lattice. Careful heat treatment (annealing) can remove most of these.



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d) Grain Size: Since the boundaries between adjacent crystals amount to magnetic in-

homogeneities, the less grain boundaries there are the better. As a consequence, efforts are

made to end up with grains as large as can be managed as this facilities domain boundary

motion.

2.3 Experimental Setup for Measurement of Iron Loss

2.3.1 Epstein frame

The Epstein frame is the most popular equipment used in industry to measure specific iron loss.

The detailed operating instructions are described in the international standard IEC 404-2.

The industrial standard frame is usually a 28 cm x 28 cm frame with four coils each having 700turns both on the primary and the secondary windings [9]. Each side have a primary winding

(magnetising winding) on the outer side and a secondary winding (voltage winding) on the inner

side as shown in Fig. 2.1.

Fig. 2.1 Epstein frame

The investigated steel samples (strips) should be 28 cm long (±2.05 cm) and 3 cm wide and

must be of multiple of 4, with a recommended minimum number of 12 strips. Strips cut across

the rolling direction are loaded on the opposite sides of the frame, while those cut along the

rolling direction are loaded on the opposite sides. The strips are loaded in the Epstein frame

making double-lapped joints as shown in Fig. 2.2.



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Fig. 2.2 Test specimen arrangements in the Epstein frame (double-lapped joint)

The total iron-loss of the test specimen, is given by

( )i

mc R

U P

N

N P

2

2

2

111,1

−=

2.12

where N1 and N2 are the total number of turns of the primary and secondary winding, “P m” is

the power measured by the wattmeter, “Ri” is the total resistance of the instruments in the

secondary circuit and 2U is the average value of the rectified voltage induced in the secondary

winding.

The measured specific total loss, W in W/kg is obtained by dividing “Pc” by the weight of the

active mass of the test specimen. Mass of the part of specimen where flux lines are existing is

called the active mass.

m

c

a

c

ml

lP

m

PW

4==

2.13

Where “l” is the length of a test specimen strip, “lm” is the conventional effective magnetic pathlength and m is the total mass of the test specimen.



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Fig. 2.3 Circuit used to calculate specific loss in the Epstein frame

Some of the shortcomings of this method are that flux density is not uniformly distributed due to

leakage flux around the joints [9]. The corners have been found to cause errors. The magnetic

length (94 cm) is estimated and is not an accurate value. The 3 cm strips width is not wide

enough for cutting stresses not to propagate to the centre of the strips and influence the loss

results. Therefore, the material under test must be annealed to relieve stress before testing,

especially for grain-oriented steel. The preparation and loading of the strips on to the frame is

time consuming [9]. The magnetic flux conditions in the Epstein frame correspond to those

found in power transformers where good correspondence between calculation and

measurements is obtained. However, such correspondence is not observed for inductionmotors, where the error from the Epstein test results can be larger than 50% [11].

The standard Epstein test does not include the effect of the flux harmonics that exist in induction

motors, since sinusoidal excitation at the fundamental frequency is specified [11]. To perform

core loss measurements in electrical steel samples under non-sinusoidal excitation, various test

benches are built based on the Epstein frame [11], [12] and [13].

2.3.2 Toroid Tester

The toroid has primary and secondary windings with excitation applied to the primary and the

induced voltage measured on the secondary. Toroid geometry is more similar to the geometry

of a stator in an electrical machine [14], hence some people prefers toroids over Epstein frames.One of the problems with toroids is the non uniform distribution of flux density.

Fig. 2.4 Geometrical characteristic of the core assembled using two concentric rings



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A disadvantage of this tester is that the toroid must be properly wound, which is time consuming

compared to the Epstein frame. The toroidal tester takes longer to prepare and set up for

testing.

2.3.3 Single Sheet Tester (SST)

In some of the research papers it is mentioned that due to a considerable easier sample

preparation and substantial saving of material, the single sheet tester (SST) with yokes is

increasingly replacing the Epstein frame [15] and in the future, SST may become the preferred

method [9]. In another paper [16], the single sheet method is stated to be the most precise and

economic one for magnetization characteristics measurement of the magnetic steel sheets, as

compared to Epstein frame and ring specimen method. A small description for the measurement

principle of SST is given as below, see Fig. 2.5.

Fig. 2.5 Measurement principle of SST

The significant difference between the single sheet method and the conventional method is that

in SST, an H coil is used for the measurements of magnetic field strength [16]. While a separate

B coil is used to measure flux density.

A major drawback of this tester is that it requires calibration with either an Epstein frame or atoroid tester. Moreover, the international standard recommends a double yoke tester which is

heavy, costly and large. Further some pneumatic suspension may be required to place the yoke

on the magnetic sheet to avoid damaging the sheet [9].

2.4 Methods to predict Iron Losses with sinusoidal waveform

2.4.1 Introduction

The modelling of power losses in Ferro-magnetic material has been continuously under study.

As described in section 2.1, Jordan defined the Hysteresis and eddy-current components on



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which the analysis of electrical machines is still based [4]. Bertotti proposed a model including

third term excess or anomalous loss [5]; mathematical formulation is given as below.

PPPP excessclasshysTotal++=

2.14

f BK B f d

B f K P a

n

hTotal

5,15,1

max

2

max

222

max737,8

6++=

σ π

2.15

The most difficult part in finding a correct prediction of iron loss is the determination of loss

coefficients. Different researchers have proposed different methods for loss coefficient

calculation. A few of those are explained in the following section.

2.4.2 Determination of Loss Cofficients

Constant coefficient Method

One approach of calculating loss coefficients is calculating the classical loss component by the

formula stated above (2.21) and assuming n = 2. Using experimental values of total specific iron

losses measured from the Epstein frame it is possible to determine the unknown coefficients

(Kh & Ka). Hence two unknown coefficients (Kh and Ka) can be found by two known values of

specific losses at two different values of flux density or frequency. Figure 2.6 shows the results

obtained using the said method. It is observed that using this approach, the difference between

calculated and experimental values may become high. For example, coefficient determined at

50 and 60 Hz at 1.5 T fails to give good agreement at 50 or 60 Hz at other values of flux

densities (e.g. difference of 14% is observed in case of 1.0 T).

-10

-5

0

5

10

15

20

25

30

35

0 50 100 150 200 250 300 350 400 450

Frequency ( HZ ) ( Kh,Ka calculated at 50, 60 Hz at 1,5 T )

D

i f f e r e n c e ( % )

1.5 T

1.4 T

1.3 T

1.2 T

1.1 T

1.0 T

0.9 T

0.8 T0.7 T

Fig. 2.6 Difference B/W measured and calculated values (Ref. Measured values)

Iron loss prediction using curve fitting methods

There are several approaches to find the loss coefficients which are based on curve fitting. Oneof these methods is described by ( Fig. 2.7). In this method, a graph of the specific loss



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(measured from the Epstein frame) versus frequency at a given value of (B or specific loss

versus B at a given frequency) is plotted. Similarly, a plot of specific losses calculated by using

(2.21) is drawn by assuming some values of kh and ka. Then the relative differences of both

values are observed. The values of kh and ka that gives low relative difference are selected.

0

10

20

30

4050

60

70

80

90

100

0 100 200 300 400 500

Frequency

L o s s e s ( w / k g )

Experimental Values

Analytical Values

Fig. 2.7 Curve fitting Method

There are many other models that claim to give good agreement between calculated andmeasured loss values e.g. [4], [5], [6], [7]. In the following section, the approach discussed in [4]

is studied and implemented for the M400-50A material, together with some suggested changes.

In the first step of the procedure, in order to identify the values of the coefficient, (2.14) is

divided by the frequency, resulting in

( )2

f c f ba f

W ++=

2.16

Where Bk a h= 5.1

Bk b a= 2

Bk c e= 2.17

Using (2.16) and (2.17), values of a, b and c can be calculated using quadratic fitting as shownin Fig. 2.8 .



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y = 0,0004x2

+ 0,0013x + 0,0416

0

0,05

0,1

0,15

0,2

0,25

0 5 10 15 20 25

Square root of frequency [Hz]

S p

e c i f i c L o s s p e r F r e q u e n c y [ W / K g / H z ]

0,1 T

0,2 T

0,4 T

0,5 T

0,6 T

0,8 T

1,0 T

1,1 T

1,2 T

1,3 T

1,4 T

1,5 T

Poly. (1,5 T)

Fig. 2.8 Ratio of core loss and frequency w/f, as function of Sqrt (f) according to (2) for 812737 F Steel

During trials, it is observed that a use of five sample points, represented by measurements at

the same induction and different frequencies, is beneficial in improving the overall stability of the

numerical procedure [4],[9]. In this case, measurements at one low frequency of 30 Hz, one

intermediate frequencies of 50, 100 and 250 Hz and one high frequency of 400 Hz are used.

The derivation and use of single ka and ke, as a polynomial function of induction for the entirefrequency range introduces very large errors and hence it is recommended to split the data and

perform fitting separately on three frequency ranges identified as low (up to 400 Hz), medium

(400 to 1000 Hz) and high [9]. The values of the fitting residual for (2.16) were very close to

unity i.e. r2, indicating a very good approximation. From (2.16) and (2.17), the eddy-current

coefficient ke and the excess loss coefficient ka are calculated. These coefficients are

independent of frequency [4]. Third order polynomials were employed for curve fitting of ke and

ka as shown in Fig. 2.9 and Fig. 2.10. Then, r2 of 0,94 and 0,92 are obtained for ka and ke.

3

3

2

210 Bk Bk Bk k k eeeee +++=

2.18

3

3

2

210 Bk Bk Bk k k aaaaa +++=

2.19



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y = -0,0002x3

+ 0,0006x2

- 0,0004x + 0,0001

0,00E+00

2,00E-05

4,00E-05

6,00E-05

8,00E-05

1,00E-04

1,20E-04

1,40E-04

1,60E-04

1,80E-04

2,00E-04

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6

Induction [T]

C o e f f i c i e n t K e [ W / K g / H z ^ 2 / T ^ 2 ]

Fig. 2.9 Variation of eddy-current loss component coefficient ke with magnetic induction; ke is independent of

frequency

y = 0,0002x3

- 0,0029x2

+ 0,0035x + 0,0007

R2

= 0,9431

0,00E+00

2,00E-04

4,00E-04

6,00E-04

8,00E-04

1,00E-03

1,20E-03

1,40E-03

1,60E-03

1,80E-03

2,00E-03

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6

Induction [T]

C o e f f i c i e n t K a

Fig. 2.10 Variation of excess (anomalous) loss component coefficient ka with magnetic induction; ka is

independent of frequency

The coefficient “z” represents the ratio of hysteresis loss and frequency, which is calculated

from (2.16) by substituting the values of b and c from (2.17) and making use of (2.18) and

(2.19). In order to calculate kh and n the following equation was used.

Bnk z h logloglog += 2.20



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The plot of logz against induction at a set frequency indicates two intervals of different variation

types, which can be approximately set to induction ranges of 0,1 to 0,3 and 0,4 to 1,5 T as

shown in Fig. 2.11.

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6

Induction [ T ]

L o g z [ W / K g / H z ]

Fig. 2.11 Logarithm of ratio of hysteresis loss and frequency for M400 (822158)

Calculation of core losses using the method described above shows good agreement betweenexperimental and calculated values. Maximum difference is observed at low values of flux

density (e.g. difference of -4.16% is observed at 0.2 T) as shown in Fig. 2.12

-5

-4

-3

-2

-1

0

1

2

3

4

30 50 100 250 400

Frequency (Hz)

D i f f e r e n c e ( % )

0,1 T

0,2 T

0,3 T

0,4 T

0,5 T

0,6 T

0,7 T

0,8 T

0,9 T

1,0 T

1,1 T

1,2 T

1,3 T

1,4 T

1,5 T

Fig. 2.12 Difference B/W measured and calculated values (Ref. Measured values)



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2.4.3 Conclusion

Depending on the experimental data available and accuracy required, someone can select anyof the method described above. For example if very little experiment data is available, one

would have no choice except to adopt the constant coefficient method. On the other hand,

accuracy of the result has to be compromised a bit. Similarly, curve fitting methods needs a lot

of experimental data but results in very good accuracy.

2.5 Evaluation of methods for loss prediction with non sinusoidal waveforms

Iron loss per cycle can mainly be divided in to two parts [18], [23]. One is frequency

independent (hysteresis loss or static loss) and the other is varying with frequency (“eddy +

excess loss” or dynamic losses). One way of finding hysteresis loss is to excite a test specimenat very low frequency and measure the iron loss. The iron loss at such a low frequency will

represent Hysteresis loss as the eddy currents induced in the test specimen will be negligibly

small [24]. However when higher frequencies are involved the effect of the frequency dependent

part will make the BH loop broader. Keeping above mentioned facts in mind one can say that

when higher frequencies are involved there is no or a small increase in the hysteresis loss

component. On the other hand, the effect on dynamic loss components will be much higher.

Therefore, in the following discussion when comparing experimental measured iron losses with

calculated iron losses from analytical expressions, we will assume that the effect of harmonics

on Hysteresis loss component is small. The relative difference between measured and

calculated iron losses also support the simplification that if we ignore the effect of harmonic on

hysteresis loss still an acceptable amount of accuracy can be achieved.

In literature, quite a few methods to predict iron loss with non-sinusoidal flux wave forms can be

found. These methods are based on the prior information about the iron loss with sinusoidal

waveforms. Most of the researchers have proposed different methods for loss prediction with

and without minor loops [1], [9], [19], [20]. The method explained in [1] is already discussed in

section 2.1. Whereas [18] has proposed a method comprising of different imperial multiplying

factors for hysteresis and eddy loss components without reveling any details about these.

Marubbini [22] has proposed a method that accounts both for sinusoidal and non-sinusoidal

wave forms. It can be expressed as

+ += dt

dt dB

T K dt

dt dB

T K PP aehTotal

5.12

112.21

Equation (2.21) is similar to the sum of equations (2.2, 2.3, and 2.4). The big difference is that

equation 2.21 is in time domain, in order to account for the presence of harmonics in the flux

wave forms. As a consequence, Ke must be divided by 22 and ka by (1.41) 1.5 in case of a

pure sinusoidal waveform.

For the sake of simplicity and because less knowledge of material science is needed, equation

2.21 is preferred to calculate the iron losses, and later on compared with experimental results.

Phys, Ka and Ke can be calculated using any method explained in the early part of this chapter.

However, the detail of another method used by [23] that is widely used and gives relative good

results is explained below.



Master Thesis



In this method, information about the total iron loss at different frequencies and at a particular

flux density is required. In order to explain the method, an example to calculate the individual

loss components at 1.5 T is given. Total iron losses (WT) for 1.5 T at three different frequencies

(30, 50, 100 Hz) are measured using the Epstein frame method. At each frequency the total iron

loss (WT) is subtracted from the eddy current loss (We) and scaled by dividing by frequency;

(WT- - We) / f, whereas eddy current loss is calculated using equation 2.6. Now a graph is drawn

between scaled losses and the square root of frequency, as shown in Fig. 2.13 .

Fig. 2.13 Loss separation

The interception with the y-axis gives hysteresis loss and the excess loss can be calculated by

subtracting eddy current and hysteresis from total loss. Ka is now calculated from excess loss

while Ke is from eddy current loss component.

Once hysteresis loss, Ka and Ke is known equation 2.27 can be used to calculate the iron loss

with distorted waveforms. The Fig. 2.14 and Fig. 2.15 show the difference between calculated

and measured iron losses.

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 0.052

0.054

0.056

0.06

0.062

0.064

Square Root of Frequency

y = 0.0023*x + 0.041

Data 1 Linear

(W total - W eddy) / f



Master Thesis



Fig. 2.14 Comparison of measured and calculated values (Ref. Measured Values)


It is observed that in case of 20% 5 th and 7th harmonic the calculated losses are less then the

measured losses. This is due to that we assumed that hysteresis losses will not increase with

introduction of harmonics. This difference can be acceptable and it can conclude that equation

2.27 can be used to predict iron loss in machine with fair amount of accuracy in the

circumstances where many researchers agree that any methods which results in ±10%

accuracy is good enough [22].

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 -8

-6

-4

-2

0

2

4

6

Peak Flux density (T)

10% 5th

10% 7th 10% 11th 10% 13th

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-15

-10

-5

0

5

10


20% 5th 20% 7th 20% 11th 20% 13th

Difference (%)

Difference (%)



Master Thesis



Similarly if we compare the calculated and measured losses in case of two flux harmonics we

also came to the same result that still equation 2.27 can be used to predict the iron loss with fair

amount of accuracy (See Fig. 2.16 and Fig. 2.17).



0 0.2 0.4 0.6 0.8 1 1.2 1.4 -2

0

2

4

6

8

10


10% 5th + 10% 7th

0 0.2 0.4 0.6 0.8 1 1.2 1.4 -10

-5

0

5

10

15


10% 5th + 10% 13th

Difference (%)

Difference (%)



Master Thesis



3 MEASURMENT OF IRON LOSSES WITH NON SINUSOIDAL WAVE FORMS

The following section describes accuracy and repeatability of apparatus used for measuring iron

loss. The measurements were performed at KTH, Stockholm. The measurements from the said

setup are compared with the measurements obtained from apparatus at Cogent, Surahammar.

The apparatus at cogent is believed to be accurate and according to international standards.

In the end of this section, a brief description about the experimental setup used for

measurements is given.

3.1 Authenticity of Measurements

3.1.1 Accuracy

Iron losses at different frequencies were compared with the losses measured from apparatus atCogent. Fig. 3.1 shows the difference between the two measurements (Ref. measurements at

cogent).

-10

-5

0

5

10

15

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6

B (T)

D i f f e r e n c e

( % )

50 Hz

30 Hz

Fig. 3.1 Comparison b/w cogent and KTH measurements (Ref. Cogent)

Following important observation are made from Fig. 3.1.

• Nearly same results are obtained for 30 and 50 Hz.

• Maximum difference is observed at low and high value of induction.

• Difference is positive (losses measured with KTH apparatus are higher) at low value of

peak flux density (<0.3 T).

•

Difference is negative (losses measured with KTH apparatus are lower) at high value ofpeak flux density (>1.2 T).



Master Thesis



• Good agreement of measurements is obtained at flux densities between 0.3 T to 1.2 T.

Therefore we can conclude that the apparatus used during the thesis work is very accurate from0.3T to 1.4T. A small difference at very high flux density will make no difference in comparative

study.

3.1.2 Repeatability

Repeatability is another parameter that indicates the credibility of measurements. Repeatability

of apparatus was also checked at different frequencies and the results found are shown in Fig.

3.2 and Fig. 3.3.

-6

-4

-2

0

2

4

6

8

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6

B (T)


Experiment No. OneExperiment No. Two

Fig. 3.2 Repeatability at 30 Hz

-3

-2

-1

0

1

2

3

4

5

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6

B (T)


Experiment No. One

Experiment No. Two

Fig. 3.3 Repeatability at 50 Hz



Master Thesis



Following important conclusions are made from Fig. 3.2 and Fig. 3.3.

• Repeatability at 30 Hz for lower values of induction (< 0.4T) has the maximum difference

of 7%, while for higher value of peak flux density it is less then 2%. This even improves

with increasing values of peak flux density.

• At 50 Hz for lower values of induction (< 0.4T) has the maximum difference of 2.3%,

while for higher value of induction it is less then 1%.

• Repeatability graph at 70 Hz (See annexure-4) shows good agreements between

experiments with maximum difference of 2.2 % at 3 T.

• Repeatability graph at 5 Hz (See annexure-4) also shows good agreements between

experiments with maximum difference of 1.2 %.

3.2 Block diagram of Experimental setup

Fig. 3.4 Block diagram of apparatus

A block diagram of the experimental setup is shown in

Fig. 3.4. A computer is used as a signal generator, controller as well as data recorder. Theprogramming is made in Simulink. The compiled program is then downloaded to a dSpace

system. Real time control of different parameters such as the amplitude of the fundamental,

harmonics as well as the phase angle is possible via a graphical interface. Digital to analogue

converters supplies the amplifier with input signals. The signal is amplified by an amplifier and

fed to the Epstein frame.

A flux-meter and an ampere-meter are used to record flux and current through the Epstein

frame. The flux values are used in a feed-back loop so that it is possible to specify the flux

wave-form (i.e. the B-field) instead of the current wave-form (i.e. the H-field). Iron loss is

calculated by integrating the area of BH loop.

Simulink

dSPASE

DACAmplifie

Computer

ADC

A

Flux Meter

Epstein frame



Master Thesis



4 EFFECT OF A SINGLE FLUX HARMONIC ON IRON LOSSES

To understand the effect of harmonic on iron loss, a quantitative study of the 5 th, 7th, 11th and

13th harmonic with different fundamental frequencies is made. Fundamental frequencies of 5,

30, 50 and 70Hz are selected, as a wide range of the machines operates in these frequencies.

Different graphs are shown to indicate the percentage increase of iron loss in presence of each

harmonic. Effect of the harmonic amplitude and phase angle is discussed. Reasons for increase

in loss iron with respect to pure sinusoidal wave (non harmonics) as well as the effect of change

in phase angle are also explained.

4.1 Fundamental Frequency 50 Hz

4.1.1 5th Harmonic

4.1.1.1 Effect of harmonic amplitude

In order to analyze effect of the 5th harmonic on iron losses, different percentage of the 5th

harmonic were introduced on the fundamental frequency (50 Hz). Fig. 4.1 and Fig. 4.2 show the

trends observed.

Fig. 4.1 Specific loss due to introduction of the 5th harmonic

0 0.5 1 1.5 0

0.5

1

1.5

2

3

3.5

4

Fundamental peak flux density B (T)

Sin With 20% 5th harmonic With 15% 5th Harmonic With 10% 5th harmonic

Specific Loss (w/kg)



Master Thesis



0

10

20

30

40

50

60

70

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



20 % Harmonic

15 % Harmonic

10 % Harmonic

Fig. 4.2 Effect on Iron loss due to introduction of the 5th harmonic (Ref. Sin)

Following observations are made from Fig. 4.1 and Fig. 4.2

• Increase of loss is observed with increase of magnitude of 5th harmonic.

• Gradual increase in iron loss up to 1.2T.

• At higher values of peak flux density there is less effect of introduction of the 5th

harmonic as the core is already saturated and that results in a sharp decrease (

see figure 4.2 ) at higher values of peak flux densities (>1,2 T).

• A maximum Increase of 58% in case of 20% of the 5th harmonic, 37% in case of

15% of the 5th harmonic and 20% in case of 10% of the 5th harmonic is

observed.

4.1.1.2 Effect on the BH loop

Fig. 4.3 shows minor loops existence due to the presence of the 5th harmonic at 1.1 T.

It can be seen that minor loops are generated with introduction of 20% 5 th harmonics as shown

in figure 4.3; however area of minor loop is not so large and no minor loop generation is

observed in case of 10% and 15% of the 5th harmonic.



Master Thesis



Fig. 4.3 Effect of 5th harmonic on BH curve

4.1.1.3 Effect of phase angle

The effect of phase angle on iron loss was studied at different phase angles (30, 60, 90 and 120

degrees). Fig. 4.4 shows the effect of different phase angles for 20% of the 5th harmonic.

It can be seen that the effect on iron loss increases with increasing phase difference and asmall phase difference has little effect on iron losses, such as in the case of 30 degrees and 60

degrees the effect is very small. On the other hand, high phase angle difference plays an

important role. In fact a decrease in losses is observed for flux densities less than 1.1 T. This is

due to the fact that an introduction of the 5th harmonic with increasing phase shift causes a

reduction of total flux density. A typical example of a phase shift of 90 degrees with 20% of the

5th harmonic is shown in Fig. 4.5. It is further observed that the differences above 1.1 T first

decreases and eventually ends up with increase in the iron loss that can be contributed to the

following factors:

• The core gets more saturated in case of the phase angle of zero degrees due to

a relatively high peak flux density.• The occurrence of minor loops near to saturation results in an increase of the

minor loop area itself, as seen in Fig. 4.6.

Similar kinds of trend of iron losses are observed with 15% and 10% of the 5th harmonic.

-200 -100 0 100 200 300 400 500 -1.5

-1

-0.5

0

0.5

1

1.5

H(A/m)

B (T) Minor

Loops



Master Thesis



-10

-8

-6

-4

-2

0

2

4

6

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6

B (T)


30 degree

60 degree

90 degree

120 degree

Fig. 4.4 Effect of Phase angle with 20 % of the 5th harmonic ( Ref. Zero Phase Shift)

Fig. 4.5 Effect of Phase Angle difference on the peak flux density

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 -1.5

-1

-0.5

0

0.5

1

1.5

Time (Sec)

B (T)

90 Degree phase shift Zero Degree phase Shift



Master Thesis



Fig. 4.6 Effect on minor loop in saturation

4.1.2 7th Harmonic

4.1.2.1 Effect of harmonic Amplitude

There is a sharp decrease at higher values of induction (>1.2 T) and the increase in losses ishigher as compared to the 5th harmonic. For example, 20% of the 5th harmonic causes

maximum 58% difference (Fig. 4.2) as compared to 20% of the 7th which causes maximum

85%.

0

10

20

30

40

50

60

70

80

90

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



20%

15%

10%

Fig. 4.7 Effect on Iron loss due to introduction of the 7th harmonic (Ref. 50 Hz)

-1500 -1000 -500 0 500 1000 1500

-1.5

-1

-0.5

0

0.5

1

1.5

H (A/m)

B (T)



Master Thesis



The same trend is observed in the case of the 7th harmonic as in the case of the 5th harmonic:

4.1.2.2 Effect on the BH Curve

It is seen from the Fig. 4.8 that the number of minor loops appearing in the BH curve has

increased compared to the introduction of the 5th harmonic which results in an increase of iron

loss and the area covered by minor loops has become larger as compared to the 5th harmonic.

Further, maximum six numbers of minor loops are observed.

Fig. 4.8 Effect of 20% of the 7th harmonic on the BH curve

-400 -300 -200 -100 0 100 200 300 400 -1.5

-1

-0.5

0

0.5

1

1.5

B (T)

H (A/m)

Minor

Loops



Master Thesis



4.1.3 11th Harmonic


It is seen from Fig. 4.9 that the same amplitude of the 11th harmonic causes more iron loss as

compared to the loss caused by lower order harmonics and 20% of the 11th harmonic causes

maximum 168% increase in losses while 15% of the 11th harmonic causes maximum about

95% increase.

0

20

40

60

80

100

120

140

160

180

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



20%

15%

10%

Fig. 4.9 Effect on Iron loss due to introduction of 11th Harmonic (Ref. 50 Hz)

4.1.3.2 Effect on the BH curve

It is seen from the Fig. 4.10 that the Increase in number as well as areas of minor loops are

observed, therefore an increase in iron loss is expected and maximum ten number minor loops

are observed.



Master Thesis



Fig. 4.10 Effect of 20% of the 11th harmonic on the BH curve


In general the effect of phase angle on 11th harmonic is very small in fact it can be called

neglect able (see Fig. 4.11). The reasons are that a change in phase angle for higher order

harmonics has not much effect on the minor loop area. Further, the total peak flux density also

does not experience any large change.

-2

-1

0

1

2

3

4

5

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6

B (T)


45 degree

90 degree

135 degree

Fig. 4.11 Effect of phase angle on 10% of the 11th harmonic (Ref. zero phase shift)

-200 -150 -100 -50 0 50 100 150 200 -1.5

-1

-0.5

0

0.5

1

B (T)

H (A/m)



Master Thesis



4.1.4 13th Harmonic


It is observed in Fig. 4.14 that the increase in iron loss due to an introduction of 20% of the 13th

harmonics is 222% and the effect of 13th order harmonic is higher as compare to lower order

harmonics in terms of in iron losses.

0

50

100

150

200

250

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



20%

15%

10%

Fig. 4.12 Effect on Iron loss due to introduction of the 13th harmonic (Ref. 50 Hz)

4.1.4.2 Effect on the BH Loop

Twelve numbers of minor loops are observed in all cases (10%, 15% and 20% 13th harmonic).

4.1.5 Conclusion

The over all effect of harmonic on 50 Hz can be summarized as below:

• At a fixed harmonic order, an in iron losses is observed with increasing harmonic

amplitude; this is due to an increase in area of the BH curve itself as well as an

introduction of minor loops if any.

• For a given amplitude of the harmonic, increase in the order of the harmonic

results in an increase of iron losses.

•

Gradual increase in iron loss is observed with increasing peak flux density. Thereason to this can be found if we review the theory of iron loss components



Master Thesis



variations with induction in case of sinusoidal wave forms. Fig. 4.13 shows the

variation of individual iron loss components with flux density. It shows that the

hysteresis loss component decreases with increasing flux density while eddy

current and excess loss components increase with increasing in flux density.

Therefore, it can be concluded that with the introduction of harmonics eddy

current and excess losses increase, while the effect of hysteresis loss is small.

• At higher values of peak flux density, the increase in losses tends to decrease as

the iron gets more and more saturated.

• In case of low order harmonic phase angle effect iron loss to some extent but this

effect seems vanishing with increasing harmonic order.

• The number of minor loops appearing in BH curve depends on the following:

i. Relative amplitude of harmonic with fundamental signal e.g. In case of 20% 5th

harmonic total 4 minor loop are observed while no minor loop is observed in case

of 10% for the same harmonic order, as shown in Figure 1 and 2 (see annexure-

2 ).

ii. Harmonic order also plays an important role in generation of minor loops e.g. 4

no. of minor loops are generated in case of 20% 5th harmonic while number of

minor loops increases to 6 in case of 20% 7th harmonic as shown in Figure 1 and

2 (see annexure-3). This is due to the fact that minor loop appear if the dB/dt for

the harmonic is negative and bigger than for the fundamental.iii. With increasing harmonic order minor loops are more likely to appear even at

relatively low amplitude of harmonic (e.g. 15% 5th harmonic do not generate

minor loop while 15% 7th harmonic results in to generation of minor loops)

Fig. 4.13 Iron loss compenents versus flux density

0.5 1 1.5

10

20

30

40

50

60

70


Hysteresis Eddy current

Excess

Percentage of total Iron loss



Master Thesis




4.2.1 5th Harmonic


The general trend of increase in losses is the same as it was in the case of 50 Hz

(fundamental). As it is seen in Fig. 4.14 that the gradual increase in loss up to 1.2 T in case of

higher harmonic amplitude while the Increase remains nearly same for lower harmonic

amplitude. Further, as core is forced to saturation above 1.2 T that causes reduction in

percentage increase. The increase of 52 % is observed in case of 20% of the 5th harmonic that

was 56% in case of 50Hz fundamental with 5th

harmonic.

0

10

20

30

40

50

60

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6


D i f f e r e

n c e ( % )

20 % 5th harmonic

15 % 5th harmonic

10 % 5th harmonic

Fig. 4.14 Effect of the 5th harmonic on iron loss (Ref. 30 Hz,Fundamental)


Same no of minor loops are observed as it was in case of 50 Hz that indicates the number of

minor loops appearing depends on the relative amplitude of harmonic with the fundamental

signal not on the frequency of the fundamental signal itself.



Master Thesis



Fig. 4.15 Effect of the 5th harmonic on the BH curve with 30Hz fundamental


As shown in Fig. 4.16, the effect of phase angle on iron loss is low at low value of phase angle

difference but at higher values of the peak flux density different trend is observed due to thereasons already explained in section 4.1.1.3. Similar trend is observed in case of 15 and 10% of

the 5th harmonic.

-7

-6

-5

-4

-3

-2

-1

0

1

2

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



45 degree

90 degree

Fig. 4.16 Effect of phase angle on 20 % of the 5th harmonic (Ref. Zero Phase Shift)

-200 -150 -100 -50 0 50 100 150 200 -1.5

-1

-0.5

0

0.5

1

H (A/m)

B (T)



Master Thesis



4.2.2 7th Harmonic


As suggested by the Fig. 4.17, same trend is observed as it was in the case of 50Hz with 7th

harmonic. However percentage increase is little less as compare to 50Hz with 7th harmonic. For

example maximum increase in case of 30 Hz with 20 % of the 7th harmonic is 72% while in

case of 50 Hz it is 86%.

0

10

20

30

40

50

60

70

80

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



20% 7th harmonic

15 % 7th harmonic10 % 7th harmonic

Fig. 4.17 Effect of the 7th harmonic on iron loss (Ref. 30 Hz, Fundamental)


The effect of introduction of the 7th harmonic on the BH loop is same as it was in case of 50 Hz

fundamental frequency (with 7th harmonic).


In general increase in the iron loss is caused with introduction of phase angle shift however the

increase is small (about 5%). The increase is due to the fact that the total peak flux density is

increase with introduction of the 7th harmonic (See figure Fig. 4.19) which is quite opposite to

the 5th harmonic case. Further, introduction of the phase angle does not affect the location of

minor loop, hence the trend of increase in loss in saturation is not observed as it was evident in

case of 5th harmonic (At higher value of peak flux density).



Master Thesis



-5

0

5

10

15

20

25

30

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



20% 7th Harmonic

15% 7th Harmonic

10% 7th Harmonic

Fig. 4.18 Effect of 90 degree phase Shift (REF Zero degree)

Fig. 4.19 Effect of the phase angle difference on peak flux density

0.005 0.01 0.015 0.02 0.025 0.03

-1

-0.5

0.5

1

Time (Sec)

B (T)

90 Degree phase shift

Zero Degree phase Shift



Master Thesis



Fig. 4.20 Effect of the phase angle difference on minor loops

s

4.2.3 11th Harmonic


The increase in loss with introduction of 10% of the 11th harmonic is nearly same as it was in

case of 50 Hz fundamental. Maximum increase in case of 30 Hz (10% of the 11th harmonic) is

38 % while it is 41% in case of 50 Hz fundamental (10 % of the 11th harmonic).

-800 -600 -400 -200 0 200 400 600 800 -1.5

-1

-0.5

0.5

1

1.5

H(A/m)

B (T)

90 Degree phase shift In Phase



Master Thesis



0

5

10

15

20

25

30

35

40

45

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



10 % 11th harmonic

5% 11th harmonic

Fig. 4.21 Increase in Iron loss due to introduction of the 11th Harmonic (Ref. Sin)


Same effect of 11th harmonic on the BH loop is observed as it was in case of 50 Hz.


The effect of phase angle on iron loss is very little as shown in Fig. 4.22. Maximum of 1.9%

increase is observed.



Master Thesis



-5

0

5

10

15

20

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



90 Phase Shift

Fig. 4.22 Effect of phase shift on the iron loss (Ref. Zero Phase Shift)

4.2.4 13th Harmonic


Maximum Increase in the iron loss with introduction of 10 % of the 13th harmonic is 37 % while

in case of 50 Hz it was above 50%. Therefore we can say that at higher harmonic order effect of

the harmonic in 30 Hz fundamental frequency is much less than 50 Hz fundamental.

0

5

10

15

20

25

30

35

40

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



10 % 13th harmonic

5% 13th harmonic

Fig. 4.23 Increase in iron loss due to the introduction of the 13th harmonic (Ref. Sin)



Master Thesis



4.2.5 Conclusion

We can conclude that the increase in loss due to introduction of different harmonic with 30Hzfundamental is slightly less than 50Hz fundamental. However this gap seems to be increasing at

the higher order harmonic.


4.3.1 5th Harmonic


The increase in iron loss due to introduction of the 5th harmonic with fundamental 70Hz isnearly same as it was in the case of 50Hz or 30Hz fundamental e.g. the increase of about 55%

in iron loss is observed in all the three cases.

0

10

20

30

40

50

60

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



20 % Harmonic

15 % Harmonic

10 % Harmonic

Fig. 4.24 Effect on iron loss due to introduction of the 5th Harmonic (Ref. Sinusoidal)

4.3.1.2 Effect on the BH loop

Four numbers of minor loops are observed at 20% of the 5th harmonic. Further, the flux

reversals are observed in case of 15% and 10% of the 5th harmonic that it is not enough to

cause minor loops however effects the shape of the BH loop.


Small decrease in loss in observed up to 1.2 T peak flux density for different percentage of the

5th harmonic and change of phase angle will create minor loop in case of 15% of the 5th



Master Thesis



harmonic near the saturation which were not there in case of harmonic being in phase with

fundamental (See figure 4.25). However the minor loop is so small that it has little effect on iron

loss. Further, with the phase shift of 90 degree, In case of 20% of the 5th harmonic area of

existing minor loop will increase as minor loop will shift towards saturation. Saturation will cause

increase in loss at higher flux density.

-10

-5

0

5

10

15

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6


D i f f e r e n c e ( %

)

20%

15%

10%

Fig. 4.25 Effect of the phase angle on iron loss (Ref. Phase Shift Zero)

Fig. 4.26 Creation of minor loops with the phase angle shift

-1000 -800 -600 -400 -200 0 200 400 600 800 1000 -1.5

-1

-0.5

0

0.5

1

1.5

H (A/m)

B (T)

90 degree phase shift In phase



Master Thesis



4.3.2 7th Harmonic

4.3.2.1 Effect of 7th harmonic amplitude

Maximum 90% increase of the iron loss is observed in the case of 20% of the 7th harmonic.

Similarly 50% increase is observed in case of 15% of the 7th harmonic and 20% in case of 10%

of the 7th harmonic is observed (see Fig. 4.27).

0

10

20

30

40

50

60

70

80

90

100

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



20 % Harmonic

15 % Harmonic

10 % Harmonic

Fig. 4.27 Effect on the iron loss due to introduction of the 7th Harmonic (Ref. Sinusoidal)


Maximum 5.2% of increase in iron loss is observed due to phase shift of 90 degrees (see Fig.

4.28).



Master Thesis



-5

0

5

10

15

20

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



20%

15%

10%

Fig. 4.28 Effect of the phase angle on iron loss (Ref. Phase shift Zero)

4.3.3 11th Harmonic


Maximum increase in loss of 177% is observed with introduction of 20% of the11th harmonic

while increase in case of 15% and 10% of the 11th harmonic is about 100% and 40%

respectively.

0

20

40

60

80

100

120

140

160

180

200

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



)

20 % Harmonic

15 % Harmonic

10 % Harmonic

Fig. 4.29 Effect on iron loss due to introduction of 11th Harmonic (Ref. Sinusoidal)



Master Thesis




Maximum increase of about 4.5% is observed with the phase shift of 90 degrees.

-10

-5

0

5

10

15

20

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



20%

15%

10%

Fig. 4.30 Effect of the phase angle on iron loss (Ref. Phase Shift Zero)

4.3.4 13th Harmonic


Maximum increase of 56% in the iron loss is observed in case of 10% of the 13th harmonic

while the maximum increase was 16% in case of 5% of the 13th harmonic.



Master Thesis



0

10

20

30

40

50

60

70

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



10 % Harmonic

5 % Harmonic

Fig. 4.31 Effect on iron loss due to introduction of the 13th harmonic (Ref. Sinusoidal)


The effect of different harmonic with the fundamental frequency 5 Hz is also studied andfollowing results can be concluded

• The increase in the iron loss due to different harmonic with the fundamental 5 Hz

is nearly same as it was for other fundamental frequencies (i.e. 50 Hz, 70 Hz and

30 Hz).

• Effect of the phase angle on iron loss is more on low order harmonic while phase

angle has less effect in case of higher order harmonics.



Master Thesis



5 EFFECT OF TWO FLUX HARMONICS ON IRON LOSS

In electrical machines several harmonics are present simultaneously. These harmonics may or

may not be in phase with the fundamental frequency. As in section 4 it was observed that the

phase angle is important for the iron losses generated by lower order harmonics while the effect

is very minor for losses generated by higher order harmonics. In the following section, the effect

of two flux harmonics on iron loss is studied. The combination of (5 th + 7th) and (5th + 13th)

harmonic is selected because these are often dominant in electrical machines. The effect of the

phase angle is studied when:

• Two low order harmonic (5th and 7th ) are present

• One low order (5th) and one high order (13th) harmonic is present.

One should know that in the following section when the effect of phase angle is reported in caseof the 5th and 7th harmonics, the 5th harmonic is in phase with the fundamental frequency while

the 7th harmonic is having a 90 degrees phase shift.

Similarly in case of the 5th and 13th harmonics, the 5th harmonic is in phase with the fundamental

frequency while the 13th harmonic is having a 90 degrees phase shift.

However, the cases with the 5th harmonic having a phase shift of 90 degrees and the other

harmonic in phase with the fundamental frequency is also studied but this leads toward similar

kind of trends and conclusions.

5.1 Fundamental Frequency (50 Hz)

5.1.1 Effect of the 5th and 7th harmonic

5.1.1.1 Effect of the harmonic amplitude

Let us take the expample of a wave form that contains two harmonics, having amplitude of 10%

of the 5th and 7th harminics (both in phase with the fundamental frequency). It is observed that

the maximum iron loss induced by such wave form is about 23%. Further, gradual Increase in

iron loss with increasing flux density untill saturation comes in to effect, as it is the case with

single harmonic.



Master Thesis



0

5

10

15

20

25

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



10% 5th and 10% 7th harmonic

Fig. 5.1 Effect on iron loss due to introduction of the 5th and 7th harmonic (Ref. Sin)


Six minor loops are created in case 10% of the 5th and 10% of the 7th harmonics are introduced.

This suggests that the number of created minor loops are totally dependent on the wave form of

the flux density as shown in Fig. 5.3. However in this particular case the minor loops appearingat peak is so small that it is not visible in Fig. 5.2.

Fig. 5.2 Effect of 10% of the 5th and 10% of the 7th harmonic on BH curve

-150 -100 -50 0 50 100

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

H (A/m)

B (T)



Master Thesis



Fig. 5.3 Flux density as function of time


It is seen in Fig. 5.4 that the increase in iron loss up to 1 T is observed in case of introduction of

the phase angle difference of 90 degrees. It is because of the increase in peak flux density. It is

also observed, the introduction of the phase angle causes increase in peak flux density it is

obvious that saturation will be reached early and the difference will tends to decrease after the

saturation reaches. Further, the percentage increase in the iron loss due to the phase shift is

nearly equal to the percentage increase in the peak flux density.

0

2

4

6

8

10

12

14

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



90 degree phase shift

Fig. 5.4 Effect of the phase angle (Ref. Phase angle zero)

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 -1.5

-1

-0.5

0

0.5

1

1.5

Time (Sec)

B (T) Flux reversal

causing minor

Loops



Master Thesis



Fig. 5.5 Flux density as function of time



Maximum increase of 75% in iron loss is observed and the saturation effect at high peak flux

density is also evident here, as it is in almost every case.

0

10

20

30

40

50

60

70

80

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



)


Fig. 5.6 Effect on the iron loss due to introduction of the 5th and 13th harmonic (Ref. Sinusoidal)

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 -1.5

-1

-0.5

0

0.5

1

1.5

Time (Sec)

B (T)

90 Degree phase shift Zero Degree phase Shift



Master Thesis




Six numbers of the minor loops are seen following the flux reversals in flux density wave form.


The effect of phase angle on iron loss is very little. In fact, it can be neglected. This confirms the

conclusion that at higher order harmonics phase angle effect on iron loss is very small.

-10

-8

-6

-4

-2

0

2

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6




Fig. 5.7 Effect of phase angle on the 5th and 13th harmonic (Ref. Phase angle zero)




It is observed in the figure below that the increase in iron loss is slightly less as compare to the

increase with the fundamental frequency of 50 Hz and the maximum increase of about 20% is

seen in case of the fundamental frequency of 30 Hz while it is 23% in case of 50 Hz.



Master Thesis



0

5

10

15

20

25

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



10 % both harmonic

Fig. 5.8 Increase in iron loss due to introduction of the 5th and 7th harmonic (Ref. Sinusoidal)


The effect on BH loop is same as it is in case of the 50 Hz. i.e. same numbers of the minor

loops are observed in both the cases.


On average 14% increase is observed with the introduction 90 degrees phase shift until

saturation effects losses at 1.2 T.



Master Thesis



0

2

4

6

8

10

12

14

16

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



90 degree phase shift

Fig. 5.9 Effect of phase angle on the 5th and 7th harmonic (Ref. Zero degree phase shift)



Maximum 65% increase in the iron loss is observed that is slightly lower then in case of the 50

Hz fundamental frequency.

0

10

20

30

40

50

60

70

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



10 % 13th harmonic

Fig. 5.10 Increase in the iron loss due to introduction of the 5th and 13th harmonic (Ref. Sinusoidal)



Master Thesis




Similar kind of effect on the BH loop is observed as it is in case of 50Hz.


In general decrease in iron loss is observed with introduction of phase shift of 90 degrees.

Following figure suggests that maximum decrease of 4% is observed.

-10

-8

-6

-4

-2

0

2

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



90 degree Phase shift

Fig. 5.11 Effect of the phase angle on iron loss (Ref. Zero degree phase shift)




Maximum increase due to introduction of 10% of the 5 th and 7th harmonics with 70Hz

fundamental is about 27% that is slightly larger than same harmonics with 50Hz fundamental.



Master Thesis



0

5

10

15

20

25

30

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



10% 5th, 10% 7th (In Phase)

Fig. 5.12 Effect on iron loss due to introduction of the 5th and 7th harmonic (Ref. Sinusoidal)


Effect on the BH loop is same as it was in case of the 50 Hz fundamental frequency.


The maximum increase in iron loss due to phase shift of 90 degree is about 10% that is about

the same as it is in case of 50Hz with same harmonics.



Master Thesis



0

2

4

6

8

10

12

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



90 degree Phase Shift

Fig. 5.13 Effect of the phase angle on iron loss (Ref. Phase angle Zero)



Maximum increase with introduction of 10% of the 5th and 7th harmonics is about 77%. It is

further observed that the gradual increase in loss in observed with increasing the peak flux

density, the trend that is observed in almost every case.

0

10

20

30

40

50

60

70

80

90

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



10% 5th, 10% 13th (In Phase)

Fig. 5.14 Effect on iron loss due to introduction of the 5th and 13th harmonic (Ref. Sinusoidal)



Master Thesis




Generally small decrease in observed due to phase shift of 90 degrees phase shift. This smalldecrease is due to decrease in the peak flux density as already stated.

-10

-8

-6

-4

-2

0

2

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6



)

90 degrees phase shift

Fig. 5.15 Effect of the phase angle on iron loss (Ref. phase shift zero degree)

5.4 Conclusion

The overall effect of two flux harmonics is the same for all three fundamental frequencies (i.e.

30Hz, 50Hz and 70Hz). However it is noted that the increase in iron loss is slightly increasing

with increase in fundamental frequency (e.g. increase in loss in case of 50 Hz fundamental

frequency is slightly more than 30Hz while it is slightly less than 70Hz) as shown in figure 5.15.

This is due to the fact that with increasing fundamental frequency, eddy current and excess

losses becomes more and more effective or in other words; the percentage of eddy current and

excess losses in total iron loss increases with increasing fundamental frequency.In case of low order harmonic a change of phase angle may result in either a decrease or an

increase in iron loss. There may be two reasons for such a behavior.

• The main reason is “change in phase angle results into a change in total peak

flux density (fundamental + harmonic)”.

• The area of the minor loops can also be affected with a phase shift. However,

most of the time it has a less important role to play regarding the iron loss.

In case of higher order harmonics both of the factors mentioned above do not exist, hence a

change in phase angle does not change the iron loss.



Master Thesis



0

5

10

15

20

25

30

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6

Fundamental peak flux density B(T)


30 Hz

50 Hz

70 Hz

Fig. 5.16 Increase in iron loss due to 10% of the 5th and 7th harmonic at different fundamental frequencies (Ref.

fundamental frequency)



Master Thesis



6 CONCLUSIONS

Some methods for prediction of iron loss coefficients are discussed in chapter 2, these methods

can basically be divided into two main categories. One is called the constant coefficient method

and the other is referred to as the curve fitting method. It is found that the constant coefficient

method is not feasible to use for a wide frequency range although it requires fewer amounts of

experimental data. On the other hand, different curve fitting methods are found in literature.

These methods usually give fairly accurate information about the total iron loss but if correct

information about individual loss components is required one has to be careful in selecting

among these different curve fitting methods. In such a case, methods in section 2.4.2.2 are not

recommended at all. The first method in section 2.4.2.2 involves a lot of guessing while the

second method described in the same section shows slightly too high hysteresis loss.

Therefore, it is recommended that the method described in section 2.5 should be followed.Table 6.1 shows loss coefficients for M400 at 50 Hz using the method given in section 2.5 with

the assumption of n = 2.

Table 6.1 Loss coefficient for M400 (813727F) at 50 Hz

Ke = 0.1437 * 10-3 [W/kg/Hz2 /T2]

Kh [W/kg/Hz/T2] Ka [W/kg/Hz1.5 /T1.5]

0,1 T - -

0,2 T - -

0,3 T0.0306 0.000344

0,4 T0.0285 0.000324

0,5 T0.0245 0.000515

0,6 T0.0218 0.000638

0,7 T0.0199 0.000725

0,8 T0.0186 0.000790

0,9 T

0.0173 0.000869

1,0 T0.0165 0.000930

1,1 T0.0158 0.001016

1,2 T0.0156 0.001058

1,3 T0.0159 0.001102

1,4 T0.0167 0.001167

1,5 T0.0180 0.001226



Master Thesis



In general similar kind of trends is observed in the presence of one or two harmonics. It is found

that an increase in peak flux density, generation of minor loops and involvements of high

frequencies are the main reason of increase in losses with distorted waveforms. Table 6.2

describes the maximum increase of iron loss with an introduction of 20% of different order

harmonics at 50 Hz fundamental frequency.

Table 6.2 Maximum increase in iron loss due to an introduction of 20% of different orders of harmonics at 50 Hz

fundamental frequency.

Harmonic order 5th 7th 11th 13th

Maximum

increase58% 85% 168% 222%

It is important to note that the eddy current and excess losses increase with the introduction of

harmonics while hysteresis loss is less affected. The effect of harmonics on increase in iron loss

is observed with increasing harmonic amplitude as well as harmonic order. Further, it is found

that the higher the fundamental frequency the higher will be the increase in iron loss, as seen in

Table 6.3.

Table 6.3 Percentage increase in iron loss with the introduction of 10% of the 5th and 7th harmonics at different

fundamental frequencies

0.8 T 0.9 T 1.0 T 1.1 T 1.2 T

30 Hz 18 20 17 20 17

50 Hz 22 21 22 22 23

70 Hz 25 25 27 28 28

Section 2.5 discusses different methods to predict iron loss with distorted flux waveforms.

Following the time domain expression (6.1) to predict iron loss in such conditions isrecommended.

+

+= dt

dt

dB

T K dt

dt

dB

T K PP

aehTotal

5.12

11 6.1

It is found that no additional information is needed. However, the correct prediction of loss

coefficients is very important. Use of the time domain equation accounts for change in phase

angle in the resultant waveform. However, the change in iron loss due to a phase angle shift is

even bigger in reality. Therefore, loss coefficients need to be changed accordingly. It was found

that the percentage change (increase/decrease) in iron loss due to a phase angle shift is nearly

equal to the percentage change in peak flux density (e.g. in case of which 20% of the 5 th

harmonic 90 degrees phase shift results in average decrease of 3% in iron loss that is nearly

equal to the percentage decrease in peak flux density).



Master Thesis



7 FUTURE WORK

For continuation of this work the following aspects should be considered:

• The impact of rotational iron losses.

• Defining an imperial coefficient to account for phase angle effects, or some other

method could be followed.

• Other iron materials, having different grades and thickness should be studied.

• Implementation of the recommended expression in suitable software and

calculation of iron loss in the complete motor.



Master Thesis



8 REFERENCES

[1] Julus Saitz “Calculation of Iron losses in Electrical Machines”,PhD dissertation at

Helsinki University of Technology. March 1997.

[2] Chandur Sadarangani “Electrical Machines”Department of Electric Power

Engineering, Royal Institute of Technology, Stockholm, August 2000

[3] Anthony J. Moses “Loss Prediction in Electrical steel Laminations and Motor

Core”, Cardiff university, Cardiff school of Engineering, Cardiff/ U.K.

[4] Dan M. lonel “On the variation with flux and frequency of the core loss coefficients

in Electrical Machines”, IEEE transactions on Industry Applications vol. 42 no. 3,

pp658-667, May/june 2006

[5] G. Bertotti, “General properties of power losses in soft ferromagnetic materials”,

IEEE transaction on magnetics, vol.24, no. 1, pp621-630, jan 1988.

[6] E. Della Torre, “Magnetic Hysterisis”, Piscataway, NJ: IEE, 200. 1998.

[7] H. Domeki “Investigation of benchmark model for estimating iron loss in rotating

machine” IEEE transaction on magnetics, Vol 40, no 2 pp794-797, Mar 2004”

[8] Mircea Popescu “A best fit model of power losses in cold rolled moter lamination

steel operating in a wide range of frequency and magnetization”, IEEE trans on

magnetics. Vol.43, no. 4, pp 1753-1756 April 2007.

[9] Lotten Tsakani “Core losses in Motor laminations exposed to high frequency or

nonsinusoidal excitation”, IEEE transaction on Industrial applications. Vol40, no.

5, pp -1325-1332 sep-oct 2004.

[10] M. S. Lancarotte “Predection of magnetic losses under sinusoidal or

nonsinusoidal induction by analysis of magnetization rate”, IEEE transaction on

Energy conversion. Vol 16, pp. 174-179.

[11] Andre G. Torres “A generalized Epstein test method for the computation of core

losses in induction motors” IEEE transaction on industrial electronics, pp 1150-

1155, 2002.

[12] Lotten T. Mthombeni “Lamination core loss measurements in machines operating

with PWM or non-sinusoidal excitation”,IEEE Electric machines and drivesconference. Pp-742-748, 2003.

[13] Aldo Boglietti “About the possibility of defining a standard method for iron loss

measurment in soft magnetic material with inverter supply”, IEEE transaction on

industry applications vol. 33, no. 5, sept. 1997.

[14] A. Boglietti “The annealing influence onto the magnetic and energetic properited

in soft magnetic material agter punching process”, IEEE Electric machines and

drives conference , pp 503- 508, 2003.

[15] T. Nakata “Numerical analysis and experimental study of the error of magnetic

field strength measurements with single sheet testers”, IEEE transaction onmagnetics, Volume 22, pp 400 – 402, Sep 1986.



Master Thesis



[16] Liu Shuo “Study of single sheet tester for A.C magnetization characteristics

measurment”, IEEE transaction on Electrical machines and systems, Volume

1, pp 361 – 364, 18-20 Aug. 2001.

[17] Yicheng Chen “An improved formula for lamination core loss calculations in

machines operating with high frequency and high flux density excitation”, IEEE

transaction on industry applications, pp 759-766, 2002.

[18] J. D Lavers “A simple method of estimating the minor loop Hystereses loss in thin

laminatios”, IEEE transactions on magnetics, vol. MAG-14, No. 5, September

1978.

[19] Edoardo Barbisio “Predicting loss in magnetic steels under arbitrary indiction

waveform and with minor Hystereses loops”, IEEE transactions on magnetic,

vol40, no 4, July 2004.

[20] A. Boglietti “Iron loss prediction with PWM supply:an overview of proposed

methods from and engineering applocation point of view”, IEEE transactions on

magnetics , 2007.

[21] M. Amar “A general formula for prediction of iron losses under nonsinusoidal

voltage waveform”, IEEE transactions on magnetic, vol 31, no 5, September 1995.

[22] Marubbini J. “Low voltage high current PM traction motor design using recent core

loss results”, IEEE transaction on industry applications,pp 1560-1566, 2007

[23] A. Broddefalk “Dependence of the power losses of a non-oriented 3% Si-steel on

frequency and guage”, Journal of Magnetism and Materials 304 (2006) e586-e588.

[24] P. C. Sen “Principles of electric machines and power electronics”, Second edition,

Queens university, Kingston, Ontario, Canada.



Master Thesis



9 ENCLOSURES

9.1 Appendix – 1

-200 -150 -100 -50 0 50 100 150

-1

-0.5

0

0.5

1

B (T)

H(A/m)

50 Hz 30 Hz

Fig. 1 BH loop with the 5th harmonics at different frequencies



Master Thesis



9.2 Appendix – 2

-150 -100 -50 0 50 100 150

-1

-0.5

0.5

1

H (A/m)

B (T)

-150 -100 -50 0 50 100 150 200

-1

-0.5

0

0.5

1

H (A/m)

B (T)

Fig. 1 (4 No. minor loops are generated with introduction of 20% of the 5thharmonic

Fig. 2 (no minor loop generated with introduction of 10% of the 5th harmonic)



Master Thesis



9.3 Appendix – 3

-200 -100 0 100 200 300

-1

-0.5

0

0.5

1

H (A/m)

B (T)

-250 -200 -150 -100 -50 0 50 100 150 200 250 -1.5

-1

-0.5

0

0.5

1

1.5

B (T)

H (A/m)

Fig. 1 (4 No. minor loops generated with introduction of 20% of the 5th harmonic)

Fig. 2 6 No. minor loops generated with introduction of 20% of the 7th harmonic



Master Thesis



9.4 Appendix – 4

Repeatability at 70 Hz

-2,5

-2

-1,5

-1

-0,5

0

0,5

1

1,5

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6

B (T)


Experiment No. One

Experiment No. Two

Repeatability at 5 Hz

-1,5

-1

-0,5

0

0,5

1

1,5

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6

B (T)


Experiment No. One

Experiment No. Two

Documents

Iron Losses Sohail Thesis KLAR 230408