dq0 transformation applied to asymmetrically

fed electrical machines

By

Johan Christian Lamprecht

Submitted to the Department of Electrical Engineering in partial fulfillment of the

requirements for the Baccalaureus Technologiae in Electrical Engineering

at the

SUPERVISOR: E VOSS

NOVEMBER 2010

2

DECLARATION

I hereby submit this thesis in partial fulfillment of the requirements of the degree Baccalaureus Technologiae to the Department of Electrical Engineering at the Cape Peninsula University of Technology. I declare that this is my original work and that it has not been submitted in this or a similar form for a degree at any other tertiary institution.

Candidate ………………………………………………………………….. J.C. Lamprecht

3

ABSTRACT This paper deals with the analysis of asymmetrical faults of the supply to a reluctance synchronous machine. By applying the dq0 transformation the fundamental variables will be transformed from a stationary time dependent system (stator) to a rotating but time independent system (rotor). These stator quantities ir , iy, ib can be transformed by means of a Park’s transformation matrix into three individual rotor quantities id , iq, i0. The dq0 transformation is a fictitious mathematical transformation of the stator currents on to the rotor currents. Various case studies are dealt with to find a comparison regarding these dq0 currents.

Index terms- Reluctance synchronous machine, dq0, stator, rotor, and matrix

4

ACKNOWLEDGEMENTS I would like to express my gratitude to the following people: • God my Creator, Saviour, Provider and my Consoler for making this possible. • Mr. Egon Voss for his guidance and support during this research. • My family for their encouragement and support. • My classmates and friends for their help throughout the year.

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TABLE OF CONTENTS

DECLARATION ............................................................................................................................................................... 2

ABSTRACT ...................................................................................................................................................................... 3

ACKNOWLEDGEMENTS ............................................................................................................................................... 4

LIST OF FIGURES ........................................................................................................................................................... 7

LIST OF TABLES ............................................................................................................................................................. 8

LIST OF SYMBOLS ......................................................................................................................................................... 9

LIST OF ABBREVIATIONS .......................................................................................................................................... 10

I. Introduction ..................................................................................................................................................................... 11

II. Methodology .............................................................................................................................................................. 14

III. Results ........................................................................................................................................................................ 17

A. Symmetrical system .................................................................................................................................................... 17

TABLE I. ............................................................................................................................................................................. 19

B. Magnitude asymmetry in one phase ........................................................................................................................... 20

TABLE II. ............................................................................................................................................................................ 21

C. Magnitude asymmetry in all phases ........................................................................................................................... 22

TABLE III. .......................................................................................................................................................................... 23

D. Angle asymmetry........................................................................................................................................................ 24

TABLE IV. .......................................................................................................................................................................... 25

E. Angle and magnitude asymmetry in one phase .......................................................................................................... 26

TABLE V. ........................................................................................................................................................................... 27

F. Angle and magnitude asymmetry in all phases .......................................................................................................... 28

TABLE VI. .......................................................................................................................................................................... 29

IV. Analysis of Results ..................................................................................................................................................... 30

CASE A ........................................................................................................................................................................... 30

CASE B ........................................................................................................................................................................... 30

CASE C ........................................................................................................................................................................... 30

CASE D ........................................................................................................................................................................... 30

CASE E ........................................................................................................................................................................... 30

CASE F ............................................................................................................................................................................ 31

V. Conclusion .................................................................................................................................................................. 31

VI. References .................................................................................................................................................................. 32

6

7

LIST OF FIGURES Figure 1-1: Schematic of the cross-section of a traditional three-phase synchronous machine Figure 1-2: Equivalent d-axis circuit Figure 1-3: Equivalent q-axis circuit Figure 1-4: Equivalent 0-axis circuit Figure 2-1: Threé phase current system Figure 2-2: Phasor diagram for wt=0° Figure 2-3: Phasor diagram for wt=10° Figure 2-4: Space phasor diagram Figure 3-1: Phasor diagram at wt=0° and x0=0° Figure 3-2: Phasor diagram at wt=30° and x0=0° Figure 3-3: Phasor diagram at wt=0° and x0=30° Figure 3-4: Threé phase current system at x0=0° Figure 3-5: Threé phase current system at x0=30° Figure 3-6: Phasor diagram at wt = 30° and x0= 0° Figure 3-7: Threé phase current system at x0= 0° Figure 3-8: Phasor diagram at wt = 30° and x0= 0° Figure 3-9: Threé phase current system at x0= 0° Figure 3-10: Phasor diagram at wt = 0° and x0= 0° Figure 3-11: Threé phase current system at x0= 0° Figure 3-12: Phasor diagram at wt = 0° and x0= 0° Figure 3-13: Phasor diagram at wt =3 0° and x0= 0° Figure 3-14: Threé phase current system at x0= 0° Figure 3-15: Phasor diagram at wt = 0° and x0= 0° Figure 3-16: Phasor diagram at wt = 30° and x0= 0° Figure 3-17: Threé phase current system at x0= 0°

8

LIST OF TABLES Table I: Symmetrical system Table II: Magnitude asymmetry in one phase Table III: Magnitude asymmetry in all phases Table IV: Angle asymmetry Table V: Angle and magnitude asymmetry in one phase Table VI: Angle and magnitude asymmetry in all phases

9

LIST OF SYMBOLS Is Current space phasor id Direct current iq Quadrature current i0 Zero sequence current

10

LIST OF ABBREVIATIONS Dq0 Direct, quadrature, zero currents d-axis direct axis q-axis quadrature axis mmf magneto motive forces

11

I. INTRODUCTION As it is difficult to measure rotor currents the dq0 transformation is applied which was proposed by RH Park in 1929[1]. The dq0 reference frame is an “at the system frequency rotating” reference frame, which is commonly used in order to simplify the analysis of synchronous machine equations [2]. These axes (dq) guide the magnetic flux, produced by the current in the stator windings, and cause the rotor to move in the direction of the stator phase sequence which finally reaches the speed synchronous to that of the stator frequency. Consequently, this magnetic flux, that in effect “pulls” the rotor along the stator, produces the RSM’s torque.

As shown in Figure 1-1, two axes are defined in a synchronous machine, which are rotating together with the rotor of the machine: • the direct (d) axis • the quadrature (q) axis, located 90 electrical degrees ahead of the d-axis [1].

Figure 1-1: Schematic of the cross-section of a traditional three-phase synchronous machine

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Equivalent circuits of the dq0 axis are illustrated by Figures 1-2 to 1-4 [3].

Figure 1-2: Equivalent d-axis circuit

Figure 1-3: Equivalent q-axis circuit

Figure 1-3: Equivalent q-axis circuit

13

The d-axis represents the rotor magnet flux axis and the q-axis is in quadrature to the d-axis. The d-axis stator current component controls the air gap flux, while the q-axis current component is used for torque control. If id is positive or the phase advance angle is negative (phase lag), the armature or stator current produces an mmf around the air gap that tends to supplement the permanent magnet flux. If id is negative, the stator mmf is negative and causes demagnetization of the magnets. [4]. The dq0 reference frame is an “at the system frequency rotating” reference frame, which is commonly used in the analysis of synchronous machine equations [2]. By changing the frame of reference to one rotating synchronously at angular speed w, the projection of the r,y,b, vectors over the rotating dq0-axes will remain constant in time. Their value will depend on the angle between the rotating axis and the time reference. [5]. It is well known that the corresponding dq0 model of a synchronous machine is derived from the assumption that the machine windings are sinusoidally distributed. This implies that all higher space harmonics produced by the machine windings, except the fundamental one, are neglected. However, an internal fault in the stator windings will break the sinusoidally distributed characteristic of the windings. The faulted windings will produce stronger space harmonics in the air-gap magnetic field. Moreover, the symmetry between the machine windings will no longer be present. Therefore, the conventional dq0 model is not suited to analyze internal faults [6].

Figure 1-4: Equivalent 0-axis circuit

14

II. METHODOLOGY Every machine with asymmetry (either in the stator, the rotor or in both) offers two coórdinate systems. The three phase current system in Figure 2-1 can be described as: ir = i rmaxcos(wt), iy=i ymaxcos(wt-120°), ib=i bmaxcos(wt+120°) with ir being reference and the coordinate system fixed to the stator (namely to coil “R”)[7] .

The phasor diagrams (Figure 2-2 & 2-3) show a system which is fixed to the rotor. Figure 2-2 illustrates the situation for wt=0° el and Figure 2-3 illustrates the situation for wt=10° and it can be seen

that the current space phasor ( )byrs iiiI 2

3

2 αα ++= has turned through an angle of exactly 10°. (1)

Figure 2-1: Threé phase current system

Figure 2-2: Phasor diagram for wt=0° Figure 2-3: Phasor diagram for wt=10°

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Important to notice is that at wt=0°, Is= i r The space phasor Is can be represented by two components which are 90° apart. In the case of asymmetrically built machines these components are called “d” (for direct) and “q” (for quadrature). The transformation from the stator system to the rotor system can be shown to be:

)2(

2

1

2

1

2

1)120sin()120sin()sin(

)120cos()120cos()cos(

3

2i

0

d

°+−°−−−°+°−

=

b

y

r

q

i

i

i

xxx

xxx

i

i

where x=x0+wt with x0 being the angle between the axis of phase R and the d-axis at wt=0°. As can be seen from Equation (2), under symmetrical conditions i0 ill be zero because the three currents sum up to zero [7]. The physical picture behind this transformation is that one can imagine one coil wound around the d-axis and another around the q-axis as represented by Figure 2-4 [8]:

Figure 2-4: Space phasor diagram

16

By using the transformation the currents in the coils of the d-and q-axis can be calculated. Both systems r, y, b and d, q, 0 must represent the same space phasor at the end. Because the rotor-and-stator systems represent the same current space phasor the d-and-q axis currents can also be obtained by applying Pythagoras [7]:

( )

∠+==

d

qqddqs i

iiiII arctan22

(3)

This technique provides very good results for detecting 3 three phase balance faults with detection times that are almost immediately since it allows the control scheme to a balanced three phase system as a DC quantity [9]. A 3-phase, 4 pole, 380V machine with single layer windings was kept at a constant rms value of 16 A and modelled using a finite element software. After each case study a table has been drawn up firstly varying wt and keeping x0 constant and then varying x0 and keeping wt constant. Only the rotor quantities are taken into consideration as we are dealing with the dq0 transformations. A legend has been drawn up to illustrate some similarities between the rotor currents.

Legend wt and x0 at 90°, 180°and 270°

magnitudes are negatively equal peak magnitude magnitudes are equal

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III. RESULTS

A. Symmetrical system A symmetrical system was analysed to establish a reference. Figure 3-1 illustrate the phasor diagram with peak values for ir = 16A∠0°, iy=16A∠-120°, ib=16A∠120° at wt = 0° and x0= 0°.

( ) ( )11208*21208*216*23

2 °−∠−°∠−=sI

Is = 22.63∠0° A

)2(

12016*2

12016*2

016*2

2

1

2

1

2

1)120sin()120sin()0sin(

)120cos()120cos()0cos(

3

2i

0

d

°∠°−∠

°∠

°+−°−−°−°°−°

=

i

iq

id = 22.63 A iq = 0 A i0 = 0 A

( ) ( )362.22

0arctan062.22 22

∠+== dqs II

Is = 22.63∠0° A

Figure 3-1: Phasor diagram at wt=0° and x0=0°

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Figures 3-2 and 3-3 illustrate how the current space phasor changes as a function of the angle between the red phase and the d-axis.

Figures 3-4 & 3-5 represents the space phasors taken over a period of 360° and plotted as a function of time.

It can be seen that as x0 changes the dq currents changes in magnitude, but remain constant.

Figure 3-2: Phasor diagram at wt=30° and x0=0° Figure 3-3: Phasor diagram at wt=0° and x0=30°

Figure 3-4: Threé phase current system at x0=0° Figure 3-5: Threé phase current system at x0=30°

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TABLE I.

Symmetrical system

x0=0° id iq i0 Is Is wt=0° id iq i0 Is wt=0 22.63 0 0 22.63 – 0i 22.63∠0° x0=0 22.63 0.00 0 22.63∠0°

wt=10 22.63 0 0 22.28 + 3.93i 22.63∠10° x0=10 22.28 -3.93 0 22.63∠0°

wt=20 22.63 0 0 21.26 + 7.74i 22.63∠20° x0=20 21.26 -7.74 0 22.63∠0°

wt=30 22.63 0 0 19.6 +11.31i 22.63∠30° x0=30 19.60 -11.31 0 22.63∠0°

wt=40 22.63 0 0 17.33 +14.54i 22.63∠40° x0=40 17.33 -14.54 0 22.63∠0°

wt=50 22.63 0 0 14.54 +17.33i 22.63∠50° x0=50 14.54 -17.33 0 22.63∠0°

wt=60 22.63 0 0 11.31 +19.6i 22.63∠60° x0=60 11.31 -19.60 0 22.63∠0°

wt=70 22.63 0 0 7.74 +21.26i 22.63∠70° x0=70 7.74 -21.26 0 22.63∠0°

wt=80 22.63 0 0 3.93 +22.28i 22.63∠80° x0=80 3.93 -22.28 0 22.63∠0°

wt=90 22.63 0 0 0 +22.63i 22.63∠90° x0=90 0.00 -22.63 0 22.63∠0°

wt=100 22.63 0 0 -3.93 +22.28i 22.63∠100° x0=100 -3.93 -22.28 0 22.63∠0°

wt=110 22.63 0 0 -7.74 +21.26i 22.63∠110° x0=110 -7.74 -21.26 0 22.63∠0°

wt=120 22.63 0 0 -11.31 +19.6i 22.63∠120° x0=120 -11.31 -19.60 0 22.63∠0°

wt=130 22.63 0 0 -14.54 +17.33i 22.63∠130° x0=130 -14.54 -17.33 0 22.63∠0°

wt=140 22.63 0 0 -17.33 +14.54i 22.63∠140° x0=140 -17.33 -14.54 0 22.63∠0°

wt=150 22.63 0 0 -19.6 +11.31i 22.63∠150° x0=150 -19.60 -11.31 0 22.63∠0°

wt=160 22.63 0 0 -21.26 + 7.74i 22.63∠160° x0=160 -21.26 -7.74 0 22.63∠0°

wt=170 22.63 0 0 -22.28 + 3.93i 22.63∠170° x0=170 -22.28 -3.93 0 22.63∠0°

wt=180 22.63 0 0 -22.63 + 0i 22.63∠180° x0=180 -22.63 0.00 0 22.63∠0°

wt=190 22.63 0 0 -22.28 – 3.93i 22.63∠190° x0=190 -22.28 3.93 0 22.63∠0°

wt=200 22.63 0 0 -21.26 – 7.74i 22.63∠200° x0=200 -21.26 7.74 0 22.63∠0°

wt=210 22.63 0 0 -19.6 -11.31i 22.63∠210° x0=210 -19.60 11.31 0 22.63∠0°

wt=220 22.63 0 0 -17.33 -14.54i 22.63∠220° x0=220 -17.33 14.54 0 22.63∠0°

wt=230 22.63 0 0 -14.54 -17.33i 22.63∠230° x0=230 -14.54 17.33 0 22.63∠0°

wt=240 22.63 0 0 -11.31 -19.6i 22.63∠240° x0=240 -11.31 19.60 0 22.63∠0°

wt=250 22.63 0 0 -7.74 -21.26i 22.63∠250° x0=250 -7.74 21.26 0 22.63∠0°

wt=260 22.63 0 0 -3.93 -22.28i 22.63∠260° x0=260 -3.93 22.28 0 22.63∠0°

wt=270 22.63 0 0 -0 -22.63i 22.63∠270° x0=270 0.00 22.63 0 22.63∠0°

wt=280 22.63 0 0 3.93 - 22.28i 22.63∠280° x0=280 3.93 22.28 0 22.63∠0°

wt=290 22.63 0 0 7.74 -21.26i 22.63∠290° x0=290 7.74 21.26 0 22.63∠0°

wt=300 22.63 0 0 11.31 -19.6i 22.63∠300° x0=300 11.31 19.60 0 22.63∠0°

wt=310 22.63 0 0 14.54 -17.33i 22.63∠310° x0=310 14.54 17.33 0 22.63∠0°

wt=320 22.63 0 0 17.33 -14.54i 22.63∠320° x0=320 17.33 14.54 0 22.63∠0°

wt=330 22.63 0 0 19.6 -11.31i 22.63∠330° x0=330 19.60 11.31 0 22.63∠0°

wt=340 22.63 0 0 21.26 – 7.74i 22.63∠340° x0=340 21.26 7.74 0 22.63∠0°

wt=350 22.63 0 0 22.28 – 3.93i 22.63∠350° x0=350 22.28 3.93 0 22.63∠0°

wt=360 22.63 0 0 22.63 – 0i 22.63∠360° x0=360 22.63 0.00 0 22.63∠0°

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B. Magnitude asymmetry in one phase In this case study the red phase current is reduced. The phasor diagram is illustrated in Figure 3-6. Figure 3-7 represents the current system. ir = 10A∠0°, iy=16A∠-120°, ib=16A∠120° at wt = 30° and x0= 0°.

By having an asymmetrical system the sum of the phase currents will not equal zero and this result in a zero component i0. It can be seen that Is do not move along the reference axis (wt).

Figure 3-6: Phasor diagram at wt = 30° and x0= 0°

Figure 3-7: Threé phase current system at x0= 0°

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TABLE II.

magnitude asymmetry in 1 phase

x0=0° id iq i0 Is Is wt=0° id iq i0 Is

wt=0 16.97 0 -2.83 16.97 – 0i 16.97∠0° x0=0 16.97 0 -2.83 16.97∠0°

wt=10 17.14 0.97 -2.79 16.71 + 3.93i 17.17∠13.23° x0=10 16.71 -2.95 -2.83 16.97∠0°

wt=20 17.63 1.82 -2.66 15.95 + 7.74i 17.72∠25.89° x0=20 15.95 -5.80 -2.83 16.97∠0°

wt=30 18.38 2.45 -2.45 14.7 +11.31i 18.55∠37.57° x0=30 14.70 -8.49 -2.83 16.97∠0°

wt=40 19.31 2.79 -2.17 13 +14.54i 19.5∠48.2° x0=40 13.00 -10.91 -2.83 16.97∠0°

wt=50 20.29 2.79 -1.82 10.91 +17.33i 20.48∠57.81° x0=50 10.91 -13.00 -2.83 16.97∠0°

wt=60 21.21 2.45 -1.41 8.49 +19.6i 21.36∠66.58° x0=60 8.49 -14.70 -2.83 16.97∠0°

wt=70 21.97 1.82 -0.97 5.8 +21.26i 22.04∠74.74° x0=70 5.80 -15.95 -2.83 16.97∠0°

wt=80 22.46 0.97 -0.49 2.95 +22.28i 22.47∠82.46° x0=80 2.95 -16.71 -2.83 16.97∠0°

wt=90 22.63 0 0 0 +22.63i 22.63∠90° x0=90 0 -16.97 -2.83 16.97∠0°

wt=100 22.46 -0.97 0.49 -2.95 +22.28i 22.47∠97.54° x0=100 -2.95 -16.71 -2.83 16.97∠0°

wt=110 21.97 -1.82 0.97 -5.8 +21.26i 22.04∠105.26° x0=110 -5.80 -15.95 -2.83 16.97∠0°

wt=120 21.21 -2.45 1.41 -8.49 +19.6i 21.36∠113.42° x0=120 -8.49 -14.70 -2.83 16.97∠0°

wt=130 20.29 -2.79 1.82 -10.91 +17.33i 20.48∠122.19° x0=130 -10.91 -13.00 -2.83 16.97∠0°

wt=140 19.31 -2.79 2.17 -13 +14.54i 19.5∠131.8° x0=140 -13.00 -10.91 -2.83 16.97∠0°

wt=150 18.38 -2.45 2.45 -14.7 +11.31i 18.55∠142.43° x0=150 -14.70 -8.49 -2.83 16.97∠0°

wt=160 17.63 -1.82 2.66 -15.95 + 7.74i 17.73∠154.11° x0=160 -15.95 -5.80 -2.83 16.97∠0°

wt=170 17.14 -0.97 2.79 -16.71 + 3.93i 17.17∠166.77° x0=170 -16.71 -2.95 -2.83 16.97∠0°

wt=180 16.97 0 2.83 -16.97 + 0i -16.97∠0° x0=180 -16.97 0 -2.83 16.97∠0°

wt=190 17.14 0.97 2.79 -16.71 – 3.93i 17.17∠-166.77° x0=190 -16.71 2.95 -2.83 16.97∠0°

wt=200 17.63 1.82 2.66 -15.95 – 7.74i 17.73∠-154.11° x0=200 -15.95 5.80 -2.83 16.97∠0°

wt=210 18.38 2.45 2.45 -14.7 -11.31i 18.55∠-142.43° x0=210 -14.70 8.49 -2.83 16.97∠0°

wt=220 19.31 2.79 2.17 -13 -14.54i 19.5∠-131.8° x0=220 -13.00 10.91 -2.83 16.97∠0°

wt=230 20.29 2.79 1.82 -10.91 -17.33i 20.48∠-122.19° x0=230 -10.91 13.00 -2.83 16.97∠0°

wt=240 21.21 2.45 1.41 -8.49 -19.6i 21.36∠-113.42° x0=240 -8.49 14.70 -2.83 16.97∠0°

wt=250 21.97 1.82 0.97 -5.8 -21.26i 22.04∠-105.26° x0=250 -5.80 15.95 -2.83 16.97∠0°

wt=260 22.46 0.97 0.49 -2.95 -22.28i 22.47∠-97.54° x0=260 -2.95 16.71 -2.83 16.97∠0°

wt=270 22.63 0 0 0 -22.63i 22.63∠-90° x0=270 0 16.97 -2.83 16.97∠0°

wt=280 22.46 -0.97 -0.49 2.95 -22.28i 22.47∠-82.46° x0=280 2.95 16.71 -2.83 16.97∠0°

wt=290 21.97 -1.82 -0.97 5.8 -21.26i 22.04∠-74.74° x0=290 5.80 15.95 -2.83 16.97∠0°

wt=300 21.21 -2.45 -1.41 8.49 -19.6i 21.36∠-66.58° x0=300 8.49 14.70 -2.83 16.97∠0°

wt=310 20.29 -2.79 -1.82 10.91 -17.33i 20.48∠-57.81° x0=310 10.91 13.00 -2.83 16.97∠0°

wt=320 19.31 -2.79 -2.17 13 -14.54i 19.5∠-48.2° x0=320 13.00 10.91 -2.83 16.97∠0°

wt=330 18.38 -2.45 -2.45 14.7 -11.31i 18.55∠-37.57° x0=330 14.70 8.49 -2.83 16.97∠0°

wt=340 17.63 -1.82 -2.66 15.95 – 7.74i 17.72∠-25.89° x0=340 15.95 5.80 -2.83 16.97∠0°

wt=350 17.14 -0.97 -2.79 16.71 – 3.93i 17.17∠-13.23° x0=350 16.71 2.95 -2.83 16.97∠0°

wt=360 16.97 0 -2.83 16.97 – 0i 16.97∠0° x0=360 16.97 0 -2.83 16.97∠0°

22

C. Magnitude asymmetry in all phases In this case study the current in the red phase and yellow phases are reduced causing an imbalance in the magnitudes. Figure 3-8 illustrates the phasor diagram and Figure 3-9 represents the current system. ir = 10A∠0°, iy=6A∠-120°, ib=16A∠120° at wt = 30° and x0= 0°.

It can be seen that the dq0 currents peak at different angles than the previous case study.

Figure 3-8: Phasor diagram at wt = 30° and x0= 0°

Figure 3-9: Threé phase current system at x0= 0°

23

TABLE III.

magnitude asymmetry in all phases

x0=0° id iq i0 Is Is wt=0° id iq i0 Is

wt=0 14.61 4.08 -0.47 14.61 + 4.08i 15.17∠15.6° x0=0 14.61 4.08 -0.47 15.17∠15.6° wt=10 16.04 4.00 -1.17 15.1 + 6.72i 16.53∠23.99° x0=10 15.10 1.48 -0.47 15.17∠15.6° wt=20 17.35 3.43 -1.84 15.13 + 9.16i 17.69∠31.19° x0=20 15.13 -1.16 -0.47 15.17∠15.6° wt=30 18.38 2.45 -2.45 14.7 +11.31i 18.55∠37.57° x0=30 14.70 -3.77 -0.47 15.17∠15.6° wt=40 19.02 1.17 -2.99 13.82 +13.13i 19.06∠43.53° x0=40 13.82 -6.27 -0.47 15.17∠15.6° wt=50 19.19 -0.24 -3.43 12.52 +14.54i 19.19∠49.27° x0=50 12.52 -8.57 -0.47 15.17∠15.6° wt=60 18.86 -1.63 -3.77 10.84 +15.51i 18.92∠55.05° x0=60 10.84 -10.61 -0.47 15.17∠15.6° wt=70 18.07 -2.82 -4.00 8.83 +16.01i 18.28∠61.12° x0=70 8.83 -12.34 -0.47 15.17∠15.6° wt=80 16.92 -3.68 -4.10 6.56 +16.03i 17.32∠67.74° x0=80 6.56 -13.68 -0.47 15.17∠15.6° wt=90 15.56 -4.08 -4.08 4.08 +15.56i 16.09∠75.31° x0=90 4.08 -14.61 -0.47 15.17∠15.6°

wt=100 14.13 -4.00 -3.94 1.48 +14.61i 14.68∠84.22° x0=100 1.48 -15.10 -0.47 15.17∠15.6° wt=110 12.82 -3.43 -3.68 -1.16 +13.22i 13.27∠95.01° x0=110 -1.16 -15.13 -0.47 15.17∠15.6° wt=120 11.79 -2.45 -3.30 -3.77 +11.43i 12.04∠108.25° x0=120 -3.77 -14.70 -0.47 15.17∠15.6° wt=130 11.15 -1.17 -2.82 -6.27 + 9.29i 11.21∠124.02° x0=130 -6.27 -13.82 -0.47 15.17∠15.6° wt=140 10.98 0.24 -2.26 -8.57 + 6.87i 10.98∠141.28° x0=140 -8.57 -12.52 -0.47 15.17∠15.6° wt=150 11.31 1.63 -1.63 -10.61 + 4.24i 11.43∠158.22° x0=150 -10.61 -10.84 -0.47 15.17∠15.6° wt=160 12.10 2.82 -0.95 -12.34 + 1.48i 12.43∠173.16° x0=160 -12.34 -8.83 -0.47 15.17∠15.6° wt=170 13.25 3.68 -0.24 -13.68 – 1.32i 13.74∠-174.49° x0=170 -13.68 -6.56 -0.47 15.17∠15.6° wt=180 14.61 4.08 0.47 -14.61 – 4.08i 15.17∠-164.4° x0=180 -14.61 -4.08 -0.47 15.17∠15.6° wt=190 16.04 4.00 1.17 -15.1 – 6.72i 16.53∠-156.01° x0=190 -15.10 -1.48 -0.47 15.17∠15.6° wt=200 17.35 3.43 1.84 -15.13 – 9.16i 17.69∠-148.81° x0=200 -15.13 1.16 -0.47 15.17∠15.6° wt=210 18.38 2.45 2.45 -14.7 -11.31i 18.55∠-142.43° x0=210 -14.70 3.77 -0.47 15.17∠15.6° wt=220 19.02 1.17 2.99 -13.82 -13.13i 19.06∠-136.47° x0=220 -13.82 6.27 -0.47 15.17∠15.6° wt=230 19.19 -0.24 3.43 -12.52 -14.54i 19.19∠-130.73° x0=230 -12.52 8.57 -0.47 15.17∠15.6° wt=240 18.86 -1.63 3.77 -10.84 -15.51i 18.92∠-124.95° x0=240 -10.84 10.61 -0.47 15.17∠15.6° wt=250 18.07 -2.82 4.00 -8.83 -16.01i 18.28∠-118.88° x0=250 -8.83 12.34 -0.47 15.17∠15.6° wt=260 16.92 -3.68 4.10 -6.56 -16.03i 17.32∠-112.26° x0=260 -6.56 13.68 -0.47 15.17∠15.6° wt=270 15.56 -4.08 4.08 -4.08 -15.56i 16.09∠-104.69° x0=270 -4.08 14.61 -0.47 15.17∠15.6° wt=280 14.13 -4.00 3.94 -1.48 -14.61i 14.68∠-95.78° x0=280 -1.48 15.10 -0.47 15.17∠15.6° wt=290 12.82 -3.43 3.68 1.16 -13.22i 13.27∠-84.99° x0=290 1.16 15.13 -0.47 15.17∠15.6° wt=300 11.79 -2.45 3.30 3.77 -11.43i 12.04∠-71.75° x0=300 3.77 14.70 -0.47 15.17∠15.6° wt=310 11.15 -1.17 2.82 6.27 – 9.29i 11.21∠-55.98° x0=310 6.27 13.82 -0.47 15.17∠15.6° wt=320 10.98 0.24 2.26 8.57 – 6.87i 10.98∠-38.72° x0=320 8.57 12.52 -0.47 15.17∠15.6° wt=330 11.31 1.63 1.63 10.61 – 4.24i 11.43∠-21.78° x0=330 10.61 10.84 -0.47 15.17∠15.6° wt=340 12.10 2.82 0.95 12.34 – 1.48i 12.43∠-6.84° x0=340 12.34 8.83 -0.47 15.17∠15.6° wt=350 13.25 3.68 0.24 13.68 + 1.32i 13.74∠5.51° x0=350 13.68 6.56 -0.47 15.17∠15.6°

wt=360 14.61 4.08 -0.47 14.61 + 4.08i 15.17∠15.6° x0=360 14.61 4.08 -0.47 15.17∠15.6°

24

D. Angle asymmetry In this case study the stator current angles are changed with respect to the red phase their values are; ir = 16A∠0°, iy=16A∠300°, ib=16A∠150° at wt = 0° and x0= 0°. Figure 3-11 represents the current system at x0=0°

It can be seen that the dq0 angles are not dependent of these stator angles, but only their magnitudes (Figure 3-10). The dq0 angles are dependant to the angular velocity and the angle between the red phase and the d-axis.

Figure 3-10: Phasor diagram at wt = 0° and x0= 0°

Figure 3-11: Threé phase current system at x0= 0°

25

TABLE IV.

Angle asymmetry

x0=0° id iq i0 Is Is wt=0° id iq i0 Is

wt=0 17.85 17.85 4.78 17.85 +17.85i 25.24∠45° x0=0 17.85 17.85 4.78 25.24∠45°

wt=10 20.43 17.39 5.19 17.1 +20.67i 26.83∠50.4° x0=10 20.67 14.48 4.78 25.24∠45°

wt=20 22.69 16.08 5.44 15.83 +22.87i 27.81∠55.31° x0=20 22.87 10.67 4.78 25.24∠45°

wt=30 24.38 14.07 5.52 14.07 +24.38i 28.15∠60.01° x0=30 24.38 6.53 4.78 25.24∠45°

wt=40 25.27 11.61 5.44 11.9 +25.14i 27.81∠64.67° x0=40 25.14 2.20 4.78 25.24∠45°

wt=50 25.27 8.99 5.19 9.36 +25.14i 26.83∠69.58° x0=50 25.14 -2.20 4.78 25.24∠45°

wt=60 24.38 6.53 4.78 6.53 +24.38i 25.24∠75.01° x0=60 24.38 -6.53 4.78 25.24∠45°

wt=70 22.69 4.53 4.23 3.51 +22.87i 23.14∠81.27° x0=70 22.87 -10.67 4.78 25.24∠45°

wt=80 20.43 3.22 3.55 0.38 +20.67i 20.67∠88.95° x0=80 20.67 -14.48 4.78 25.24∠45°

wt=90 17.85 2.76 2.76 -2.76 +17.85i 18.06∠98.79° x0=90 17.85 -17.85 4.78 25.24∠45°

wt=100 15.27 3.22 1.89 -5.82 +14.48i 15.61∠111.9° x0=100 14.48 -20.67 4.78 25.24∠45°

wt=110 13.00 4.53 0.96 -8.7 +10.67i 13.77∠129.19° x0=110 10.67 -22.87 4.78 25.24∠45°

wt=120 11.31 6.53 0 -11.31 + 6.53i 13.06∠150° x0=120 6.53 -24.38 4.78 25.24∠45°

wt=130 10.42 8.99 -0.96 -13.59 + 2.2i 13.77∠170.8° x0=130 2.20 -25.14 4.78 25.24∠45°

wt=140 10.42 11.61 -1.89 -15.45 – 2.2i 15.61∠-171.9° x0=140 -2.20 -25.14 4.78 25.24∠45°

wt=150 11.31 14.07 -2.76 -16.84 – 6.53i 18.06∠-158.81° x0=150 -6.53 -24.38 4.78 25.24∠45°

wt=160 13.00 16.08 -3.55 -17.71 -10.67i 20.68∠-148.93° x0=160 -10.67 -22.87 4.78 25.24∠45°

wt=170 15.27 17.39 -4.23 -18.05 -14.48i 23.14∠-141.26° x0=170 -14.48 -20.67 4.78 25.24∠45°

wt=180 17.85 17.85 -4.78 -17.85 -17.85i 25.24∠-135° x0=180 -17.85 -17.85 4.78 25.24∠45°

wt=190 20.43 17.39 -5.19 -17.1 -20.67i 26.83∠-129.6° x0=190 -20.67 -14.48 4.78 25.24∠45°

wt=200 22.69 16.08 -5.44 -15.83 - 22.87i 27.81∠-124.69° x0=200 -22.87 -10.67 4.78 25.24∠45°

wt=210 24.38 14.07 -5.52 -14.07 -24.38i 28.15∠-119.99° x0=210 -24.38 -6.53 4.78 25.24∠45°

wt=220 25.27 11.61 -5.44 -11.9 - 25.14i 27.81∠-115.33° x0=220 -25.14 -2.20 4.78 25.24∠45°

wt=230 25.27 8.99 -5.19 -9.36 -25.14i 26.83∠-110.42° x0=230 -25.14 2.20 4.78 25.24∠45°

wt=240 24.38 6.53 -4.78 -6.53 -24.38i 25.24-104.99∠° x0=240 -24.38 6.53 4.78 25.24∠45°

wt=250 22.69 4.53 -4.23 -3.51 -22.87i 23.14∠-98.73° x0=250 -22.87 10.67 4.78 25.24∠45°

wt=260 20.43 3.22 -3.55 -0.38 -20.67i 20.67∠-91.05° x0=260 -20.67 14.48 4.78 25.24∠45°

wt=270 17.85 2.76 -2.76 2.76 -17.85i 18.06∠-81.21° x0=270 -17.85 17.85 4.78 25.24∠45°

wt=280 15.27 3.22 -1.89 5.82 -14.48i 15.61∠-68.1° x0=280 -14.48 20.67 4.78 25.24∠45°

wt=290 13.00 4.53 -0.96 8.7 -10.67i 13.77∠-50.81° x0=290 -10.67 22.87 4.78 25.24∠45°

wt=300 11.31 6.53 0 11.31 – 6.53i 13.06∠-30° x0=300 -6.53 24.38 4.78 25.24∠45°

wt=310 10.42 8.99 0.96 13.59 – 2.2i 13.77∠-9.2° x0=310 -2.20 25.14 4.78 25.24∠45°

wt=320 10.42 11.61 1.89 15.45 + 2.2i 15.61∠8.1° x0=320 2.20 25.14 4.78 25.24∠45°

wt=330 11.31 14.07 2.76 16.84 + 6.53i 18.06∠21.19° x0=330 6.53 24.38 4.78 25.24∠45°

wt=340 13.00 16.08 3.55 17.71 +10.67i 20.68∠31.07° x0=340 10.67 22.87 4.78 25.24∠45°

wt=350 15.27 17.39 4.23 18.05 +14.48i 23.14∠38.74° x0=350 14.48 20.67 4.78 25.24∠45°

wt=360 17.85 17.85 4.78 17.85 +17.85i 25.24∠45° x0=360 17.85 17.85 4.78 25.24∠45°

26

E. Angle and magnitude asymmetry in one phase In this case study the current in the red phase is reduced and the angles of the blue and yellow phases are changed. Figures 3-12 and 3-13 illustrates the phasor diagrams at different angles.ir = 10A∠0°, iy=16A∠300°, ib=16A∠150°

Figure 3-14 represents the three phase current system for this investigation.

One can see that by having angle asymmetry the peak values of Is is much higher that magnitude asymmetry.

Figure 3-14: Threé phase current system at x0= 0°

Figure 3-12: Phasor diagram at wt = 0° and x0= 0° Figure 3-13: Phasor diagram at wt =3 0° and x0= 0°

27

TABLE V.

angle and magnitude asymmetry in 1 phase

x0=0° id iq i0 Is Is wt=0° id iq i0 Is

wt=0 12.19 17.85 1.95 12.19 +17.85i 21.62∠55.67° x0=0 12.19 17.85 1.95 21.62∠55.67°

wt=10 14.94 18.36 2.40 11.52 +20.67i 23.66∠60.87° x0=10 15.10 15.46 1.95 21.62∠55.67°

wt=20 17.70 17.90 2.78 10.51 +22.87i 25.17∠65.32° x0=20 17.56 12.60 1.95 21.62∠55.67°

wt=30 20.14 16.52 3.07 9.18 +24.38i 26.05∠69.37° x0=30 19.48 9.36 1.95 21.62∠55.67°

wt=40 21.95 14.40 3.27 7.56 +25.14i 26.25∠73.26° x0=40 20.81 5.84 1.95 21.62∠55.67°

wt=50 22.94 11.78 3.37 5.72 +25.14i 25.78∠77.18° x0=50 21.51 2.13 1.95 21.62∠55.67°

wt=60 22.96 8.98 3.37 3.7 +24.38i 24.66∠81.37° x0=60 21.55 -1.63 1.95 21.62∠55.67°

wt=70 22.03 6.34 3.26 1.57 +22.87i 22.92∠86.07° x0=70 20.94 -5.35 1.95 21.62∠55.67°

wt=80 20.25 4.18 3.06 -0.6 +20.67i 20.68∠91.66° x0=80 19.69 -8.90 1.95 21.62∠55.67°

wt=90 17.85 2.76 2.76 -2.76 +17.85i 18.06∠98.79° x0=90 17.85 -12.19 1.95 21.62∠55.67°

wt=100 15.10 2.25 2.38 -4.84 +14.48i 15.27∠108.48° x0=100 15.46 -15.10 1.95 21.62∠55.67°

wt=110 12.34 2.71 1.93 -6.76 +10.67i 12.63∠122.36° x0=110 12.60 -17.56 1.95 21.62∠55.67°

wt=120 9.90 4.08 1.41 -8.49 + 6.53i 10.71∠142.43° x0=120 9.36 -19.48 1.95 21.62∠55.67°

wt=130 8.08 6.21 0.86 -9.95 + 2.2i 10.19∠167.53° x0=130 5.84 -20.81 1.95 21.62∠55.67°

wt=140 7.10 8.83 0.28 -11.11 – 2.2i 11.33∠-168.8° x0=140 2.13 -21.51 1.95 21.62∠55.67°

wt=150 7.07 11.63 -0.31 -11.94 – 6.53i 13.61∠-151.33° x0=150 -1.63 -21.55 1.95 21.62∠55.67°

wt=160 8.00 14.26 -0.89 -12.4 -10.67i 16.36∠-139.29° x0=160 -5.35 -20.94 1.95 21.62∠55.67°

wt=170 9.78 16.42 -1.44 -12.48 -14.48i 19.12∠-130.75° x0=170 -8.90 -19.69 1.95 21.62∠55.67°

wt=180 12.19 17.85 -1.95 -12.19 -17.85i 21.62∠-124.33° x0=180 -12.19 -17.85 1.95 21.62∠55.67°

wt=190 14.94 18.36 -2.40 -11.52 -20.67i 23.66∠-119.13° x0=190 -15.10 -15.46 1.95 21.62∠55.67°

wt=200 17.70 17.90 -2.78 -10.51 -22.87i 25.17∠-114.68° x0=200 -17.56 -12.60 1.95 21.62∠55.67°

wt=210 20.14 16.52 -3.07 -9.18 -24.38i 26.05∠-110.63° x0=210 -19.48 -9.36 1.95 21.62∠55.67°

wt=220 21.95 14.40 -3.27 -7.56 -25.14i 26.25∠-106.74° x0=220 -20.81 -5.84 1.95 21.62∠55.67°

wt=230 22.94 11.78 -3.37 -5.72 -25.14i 25.78∠-102.82° x0=230 -21.51 -2.13 1.95 21.62∠55.67°

wt=240 22.96 8.98 -3.37 -3.7 -24.38i 24.66∠-98.63° x0=240 -21.55 1.63 1.95 21.62∠55.67°

wt=250 22.03 6.34 -3.26 -1.57 -22.87i 22.92∠-93.93° x0=250 -20.94 5.35 1.95 21.62∠55.67°

wt=260 20.25 4.18 -3.06 0.6 -20.67i 20.68∠-88.34° x0=260 -19.69 8.90 1.95 21.62∠55.67°

wt=270 17.85 2.76 -2.76 2.76 -17.85i 18.06∠-81.21° x0=270 -17.85 12.19 1.95 21.62∠55.67°

wt=280 15.10 2.25 -2.38 4.84 -14.48i 15.27∠-71.52° x0=280 -15.46 15.10 1.95 21.62∠55.67°

wt=290 12.34 2.71 -1.93 6.76 -10.67i 12.63∠-57.64° x0=290 -12.60 17.56 1.95 21.62∠55.67°

wt=300 9.90 4.08 -1.41 8.49 – 6.53i 10.71∠-37.57° x0=300 -9.36 19.48 1.95 21.62∠55.67°

wt=310 8.08 6.21 -0.86 9.95 – 2.2i 10.19∠-12.47° x0=310 -5.84 20.81 1.95 21.62∠55.67°

wt=320 7.10 8.83 -0.28 11.11 + 2.2i 11.33∠11.2° x0=320 -2.13 21.51 1.95 21.62∠55.67°

wt=330 7.07 11.63 0.31 11.94 + 6.53i 13.61∠28.67° x0=330 1.63 21.55 1.95 21.62∠55.67°

wt=340 8.00 14.26 0.89 12.4 +10.67i 16.36∠40.71° x0=340 5.35 20.94 1.95 21.62∠55.67°

wt=350 9.78 16.42 1.44 12.48 +14.48i 19.12∠49.24° x0=350 8.90 19.69 1.95 21.62∠55.67°

wt=360 12.19 17.85 1.95 12.19 +17.85i 21.62∠55.67° x0=360 12.19 17.85 1.95 21.62∠55.67°

28

F. Angle and magnitude asymmetry in all phases In this case study the stator currents in the red and yellow phases are reduced and the angles of the yellow and blue phases. Figures 3-15 and 3-16 illustrate the phasor diagrams at different angles. ir = 10A∠0°, iy=6A∠300°, ib=16A∠150°

Figure 3-17 represents the three phase current system for this investigation.

As the magnitudes of the stator currents are decreased so will the magnitudes of the dq0 currents decrease.

Figure 3-15: Phasor diagram at wt = 0° and x0= 0°

Figure 3-16: Phasor diagram at wt = 30° and x0= 0°

Figure 3-17: Threé phase current system at x0= 0°

29

TABLE VI.

angle and magnitude asymmetry in all phases x0=0° id iq i0 Is Is wt=0° id iq i0 Is

wt=0 14.55 13.76 -0.40 14.55 +13.76i 20.03∠43.4° x0=0 14.55 13.76 -0.40 20.03∠43.4°

wt=10 17.01 12.66 -0.63 14.55 +15.43i 21.21∠46.68° x0=10 16.71 11.03 -0.40 20.03∠43.4°

wt=20 18.95 10.79 -0.83 14.12 +16.62i 21.81∠49.65° x0=20 18.38 7.96 -0.40 20.03∠43.4°

wt=30 20.14 8.36 -1.01 13.26 +17.31i 21.81∠52.55° x0=30 19.48 4.65 -0.40 20.03∠43.4°

wt=40 20.42 5.67 -1.16 11.99 +17.47i 21.19∠55.54° x0=40 19.99 1.19 -0.40 20.03∠43.4°

wt=50 19.76 3.05 -1.27 10.36 +17.1i 19.99∠58.79° x0=50 19.89 -2.30 -0.40 20.03∠43.4°

wt=60 18.25 0.82 -1.35 8.42 +16.21i 18.27∠62.55° x0=60 19.19 -5.72 -0.40 20.03∠43.4°

wt=70 16.06 -0.77 -1.38 6.22 +14.83i 16.08∠67.25° x0=70 17.91 -8.96 -0.40 20.03∠43.4°

wt=80 13.47 -1.51 -1.37 3.82 +13i 13.55∠73.62° x0=80 16.08 -11.93 -0.40 20.03∠43.4°

wt=90 10.77 -1.32 -1.32 1.32 +10.77i 10.85∠83.01° x0=90 13.76 -14.55 -0.40 20.03∠43.4°

wt=100 8.31 -0.22 -1.23 -1.22 + 8.22i 8.31∠98.44° x0=100 11.03 -16.71 -0.40 20.03∠43.4°

wt=110 6.37 1.65 -1.10 -3.73 + 5.42i 6.58∠124.54° x0=110 7.96 -18.38 -0.40 20.03∠43.4°

wt=120 5.19 4.08 -0.94 -6.13 + 2.45i 6.6∠158.21° x0=120 4.65 -19.48 -0.40 20.03∠43.4°

wt=130 4.90 6.77 -0.75 -8.34 - 0.59i 8.36∠-175.95° x0=130 1.19 -19.99 -0.40 20.03∠43.4°

wt=140 5.56 9.39 -0.54 -10.29 - 3.62i 10.91∠-160.62° x0=140 -2.30 -19.89 -0.40 20.03∠43.4°

wt=150 7.07 11.63 -0.31 -11.94 - 6.53i 13.61∠-151.33° x0=150 -5.72 -19.19 -0.40 20.03∠43.4°

wt=160 9.26 13.21 -0.07 -13.22- 9.25i 16.13∠-145.02° x0=160 -8.96 -17.91 -0.40 20.03∠43.4°

wt=170 11.85 13.95 0.17 -14.1 -11.68i 18.31∠-140.36° x0=170 -11.93 -16.08 -0.40 20.03∠43.4°

wt=180 14.55 13.76 0.40 -14.55 -13.76i 20.03∠-136.6° x0=180 -14.55 -13.76 -0.40 20.03∠43.4°

wt=190 17.01 12.66 0.63 -14.55 -15.43i 21.21∠-133.32° x0=190 -16.71 -11.03 -0.40 20.03∠43.4°

wt=200 18.95 10.79 0.83 -14.12 -16.62i 21.81∠-130.35° x0=200 -18.38 -7.96 -0.40 20.03∠43.4°

wt=210 20.14 8.36 1.01 -13.26 -17.31i 21.81∠-127.45° x0=210 -19.48 -4.65 -0.40 20.03∠43.4°

wt=220 20.42 5.67 1.16 -11.99 -17.47i 21.19∠-124.46° x0=220 -19.99 -1.19 -0.40 20.03∠43.4°

wt=230 19.76 3.05 1.27 -10.36 -17.10i 19.99∠-121.21° x0=230 -19.89 2.30 -0.40 20.03∠43.4°

wt=240 18.25 0.82 1.35 -8.42 -16.21i 18.27∠-117.45° x0=240 -19.19 5.72 -0.40 20.03∠43.4°

wt=250 16.06 -0.77 1.38 -6.22 -14.83i 16.08∠-112.75° x0=250 -17.91 8.96 -0.40 20.03∠43.4°

wt=260 13.47 -1.51 1.37 -3.82 -13i 13.55∠-106.38° x0=260 -16.08 11.93 -0.40 20.03∠43.4°

wt=270 10.77 -1.32 1.32 -1.32 -10.77i 10.85∠-96.99° x0=270 -13.76 14.55 -0.40 20.03∠43.4°

wt=280 8.31 -0.22 1.23 1.22 - 8.22i 8.31∠-81.56° x0=280 -11.03 16.71 -0.40 20.03∠43.4°

wt=290 6.37 1.65 1.10 3.73 - 5.42i 6.58∠-55.46° x0=290 -7.96 18.38 -0.40 20.03∠43.4°

wt=300 5.19 4.08 0.94 6.13 - 2.45i 6.6∠-21.79° x0=300 -4.65 19.48 -0.40 20.03∠43.4°

wt=310 4.90 6.77 0.75 8.34 + 0.59i 8.36∠4.05° x0=310 -1.19 19.99 -0.40 20.03∠43.4°

wt=320 5.56 9.39 0.54 10.29 + 3.62i 10.91∠19.38° x0=320 2.30 19.89 -0.40 20.03∠43.4°

wt=330 7.07 11.63 0.31 11.94 + 6.53i 13.61∠28.67° x0=330 5.72 19.19 -0.40 20.03∠43.4°

wt=340 9.26 13.21 0.07 13.22 + 9.25i 16.13∠34.98° x0=340 8.96 17.91 -0.40 20.03∠43.4°

wt=350 11.85 13.95 -0.17 14.1 +11.68i 18.31∠39.64° x0=350 11.93 16.08 -0.40 20.03∠43.4°

wt=360 14.55 13.76 -0.40 14.55 +13.76i 20.03∠43.4° x0=360 14.55 13.76 -0.40 20.03∠43.4°

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IV. ANALYSIS OF RESULTS

CASE A Under symmetrical conditions and for x0=0° the magnitude of the current space phasor (Is) and “direct “current (id) are equal and constant independent of wt, and iq=i0=0. As the angle (x0) between the red phase and d-axis changes, there will be a current induced into the quadrature axis (iq). The current space phasor has the maximum magnitude. The “direct” curent peaks every 180° starting at 0° and it can be seen that iq peaks 90° after id. Because of the symmetry in the system there will be no zero sequence currents. CASE B For asymmetry in one phase and for x0=0°, it can be seen that id first reaches its peak at 90° and then peaks every 180° afterwards. The current in the quadrature axis (iq) peaks 45° before and after id peaks. Because of the asymmetry a zero sequence current is present and peaks every 180° starting at 0°. An important observation can be seen that as id peaks iq=i0=0. As can be seen that firstly i0 peaks at wt=0°, 45° later id peaks at 45° and another 45° later, id peaks. After the first 90° of rotation this sequence change, starting now id peaking, iq and then i0 peaking at 180°. Now this all starts again and keeps on going for the duration of the asymmetry in the one phase. By changing x0 and keeping wt=0°, the current space phasor (Is) stays at the reference angle but is not at its maximum. id peaks at x0=0° and then every 180° subsequently. iq peaks 90° after id. As can be seen from the table, when id peaks iq is equal to zero and visa versa. i0 stays constant. CASE C When magnitude asymmetry in all phases occurs, Is and id will peak (minimum and maximum) at the same angles but will fluctuate in magnitude elsewhere. id first peaks at 50° and then peaks 180° subsequently. iq starts peaking at 0° and then every 90° subsequently. By varying x0 and keeping wt=0°, id peaks at 20° and then subsequently 180°. iq peaks 90° after id peaks. i0 and Is are constant and it can be seen that Is are not on the reference angle. CASE D For angle asymmetry it can be seen that iq does not peak 90° after id. id peaks at 45° and then every 180° subsequently. iq peaks at 0° and then every 180° subsequently. It can be seen that iq peaks 45° before id. i0 peaks at 30° and then peaks 180° subsequently. By varying x0 and keeping wt=0°, id peaks at 45° and every 180° after that. iq peaks 90° after id. i0 and Is stays constant. CASE E For angle and magnitude asymmetry in 1 phase, id peaks at 60° and then every 180° subsequently. iq peaks at 10° and then every 180° subsequently. i0 peaks at 50° and then every 180° subsequently. By varying x0 and keeping wt=0°, id peaks at 60° and then every 180° after that. iq peaks 90° after id. i0 and Is stays constant.

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CASE F For angle and magnitude asymmetry in all phases, id peaks at 40° and then every 180° afterwards. iq peaks at 170° and then every 180° subsequently. i0 peaks before iq at 70°. By varying x0 and keeping wt=0°, id peaks at 40° and then every 180° subsequently. iq peaks 90° after id. i0 and Is stays constant. Without a q-axis current the net torque of the motor will be zero. Since the d and q currents are not constant, a pulsating torque is produced by the motor [10].

V. CONCLUSION After investigating the various case studies it was shown that the dq0 gives various results. Firstly, for any type of asymmetry, there will be a zero sequence current produced and the current space phasor will not move with respect to the reference angle (wt). To distinguish between magnitude and angle asymmetry; with magnitude asymmetry the angle between the peaks for id and i0 are 180° and iq are 90° (Case B and C). When angle asymmetry and angle with magnitude asymmetry occurs the angle between the peaks for id, iq, and i0 are all 180° (Case D, E and F). One can see that when angle asymmetry occurs the dq0 current peaks are much higher in the case of a symmetrical systm. At this stage it is very difficult to determine any resemblance between angle asymmetry and angle with magnitude asymmetry.

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VI. REFERENCES [1] Anon, DQ0 transformation. Wikipedia. Available: www.wikipedia.com/dq0-transformation,(Feb 2010) [2] T. Demiray (March 2006). Comparison of Phasor Dynamics Approach to different Modeling Techniques in a common Simulation Framework, (Version 10) [online]. Available: www.EEH - Power Systems Laboratory.com/ Comparison of Phasor Dynamics Approach to different Modeling Techniques in a common Simulation Framework [3] Professor Ali Keyhani ( March 2010). “Synchronous Machine Modelling: Derivation of the dq0 Equations of an Idealized Three-Phase Synchronous Machine.” OHIO State University, Machine Department [online]. Availible: www.ohiostate.com/machines/derivationofthedq0equations [4] M.R. Islam (May 2009). “COGGING TORQUE, TORQUE RIPPLE AND RADIAL FORCE ANALYSIS OF PERMANENT MAGNET SYNCHRONOUS MACHINES”, The Graduate Faculty of The University of Akron. [online] www.akronuniversity.com/machines [5] M. Olszewski (January 2006). “MODELING RELUCTANCE-ASSISTED PM MOTORS”, U.S. Department of Energy, FreedomCAR and Vehicle Technologies, EE-2G. [online] www.oakridgenationallaboratory.com/permanentmagnetmotors [6] X. Tu a, L.-A. Dessaint a,∗, M. El Kahel b, A. Barry c, (2006). “Modeling and experimental validation of internal faults in salient pole synchronous machines including space harmonics”, Published by Elsevier B.V. on behalf of IMACS. [online] www.sciencedirect.com/ Modeling and experimental validation of internal faults in salient pole synchronous machines including space harmonics [7] E. Voss, The Voss lectures on electrical machines IV, 12th edition, Cape Peninsula University of Technology, 2010 [8] K.P. Kovacs. “ Symmetrische Komponenten in Wechselstrommaschinen”. Birkhauser Verlag Basel und Stuttgart, 1962 [9] J. Aguinaga, (February 2008). “ Study of static transfer switches”, Helsinki University of technology, faculty of electronics, communication and automation. [online] www.helsinki.com/machines

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[10] Brian A. Welchko* Jackson Wai Thomas M. Jahns Thomas A. Lipo, (October 2004). “Magnet Flux Nulling Control of Interior PM Machine Drives for Improved Response to Short-Circuit Faults”, in Conf. Rec. IEEE IAS, 2004, October Seattle. [online] www.sciencedirect.com/Machinedrives