Induction Motor Analysis

Master Thesis

Time-stepping FE-analysis of a 22kW Variable Impedance Induction Motor (VZIM) and thermal analysis

Mehdi Hajinoroozi

Advisors: Prof. Dr.-Ing. habil. h.c. Andreas Binder Dipl.-Ing. Hooshang Gholizad

Novemebr 2011

Summary Variable Impedance Induction Motor consists of a sectionalized rotor, with 3 different sections and each section has its own bars and end-rings which have different geometry and materials, in order to achieve high starting torque and high efficiency at the rated operating  point together with high breakdown torque. In order to investigate the electromagnetic and thermal characteristics of this motor (22 kW VZIM), three different motors VZIM1, VZIM2 and VZIM3 with the same stator but different rotors have been considered and analyzed. Electromagnetic and thermal analyses of a 22kW Variable Impedance (Z) Induction Motor  (VZIM) are the aim of this master thesis. Finite element method is a precise and useful approach to analyze electrical machines, therefore the electromagnetic analysis of the VZIM motor is done using  Flux2D software and the thermal analysis is done by  ANSYS  software. Furthermore all the calculation results obtained by finite element method and relevant tools are compared with analytical calculations, carried out by  KLASYS  program. In order to do steady state AC and time stepping electromagnetic analysis of the 22kW VZIM motor, first the geometries of three assumed independent motors VZIM1, VZIM2 and VZIM3 are generated in  Flux2D and with assigning the materials and meshing the geometries, the models are solved and relevant torque-slip, fundamental stator phase current-slip, input power-slip, efficiency-slip and power factor-slip characteristics are calculated by  Flux2D and compared with the results of the  KLASYS software. In addition, for thermal calculations due to the existence of the measured values of the 5.5kW VZIM motor, first the 5.5kW motor is analyzed with the  ANSYS  tool and the results are compared with measured values, afterwards the 22kW VZIM motor is analyzed with  ANSYS  tool and the results are compared with a simplified thermal equivalent circuit model results. To carry out the simulation with  ANSYS , first the geometries of three assumed independent motors VZIM1, VZIM2 and VZIM3 are generated and after assigning the materials and meshing the geometries, the models are solved with assigning the loss densities of different  parts of the motors as heat sources. At the end the values of analytical calculations and finite element method are compared to make sure that the temperature rise in the stator winding does not exceed the thermal limit of the insulations.

ii

Contents Table of Contents List of Symbols List of Figures List of Tables

1

Introduction

1.1 1.2 1.3 1.4 1.5

Preface Electromagnetic and thermal analysis of the 22kW VZIM Introduction to finite element method Electrical and mechanical parameters and dimensions of the 22kW VZIM Procedure of the project

2 2.1 2.2 2.3 2.4 2.5 2.6 2.6.1 2.6.2 2.6.3 2.6.4 2.6.5 2.6.6

Geometry and mesh generation with Flux2D Introduction Geometry creation Meshing the geometry Material assignment Electrical circuit for the motor

Stator resistance Inductance of stator winding overhang Rotor end-ring leakage inductance Resistance of the end-Ring Segment VZIM1 Resistance of the end-Ring Segment VZIM2 Resistance of the end-Ring Segment VZIM3

11 11 12 14 15 16 16 16 17 17 17 18

3 3.1 3.2 3.2.1 3.2.2 3.2.3

Steady state electromagnetic analysis Introduction Steady state AC analysis of the 22kW VZIM Steady state AC analysis of the VZIM1 Steady state AC analysis of the VZIM2 Steady state AC analysis of the VZIM3

20 21 21 26 29

4 4.1 4.2 4.3 4.4 4.5 4.6 4.6.1 4.6.2 4.6.3

Time stepping analysis of VZIM Introduction Stator and rotor field spatial harmonics Asynchronous harmonic torques Synchronous harmonic torques Derived slips used for time stepping analysis Time-stepping analysis of induction motors Time-stepping analysis of VZIM1 Time-stepping analysis of VZIM2 Time-stepping analysis of VZIM3

32 32 32 33 34 35 35 40 45

Calculation of values of circuit’s elements

iii

1 3 3 6 10

4.7 4.8 4.9 4.9.1

Instantaneous Torque wave forms at different rotor speeds Synchronous harmonic torque assessment in VZIM Power losses in different parts of the motor Losses of VZIM at different speeds

50 54 61 63

5 5.1 5.2 5.2-1 5.2-2 5.3 5.3.1 5.3.2 5.3.3 5.3.4

Thermal analysis Preface Temperature rise calculation by thermal equivalent circuits Calculation of  ο ஼௨ and ο ௘ for VZIM2 (22kW) Calculation of  ο ஼௨ and ο ௘ for VZIM3 (22kW)  Numerical calculation of temperature temperature rise Thermal analysis of 5.5kW VZIM2 Thermal analysis of 5.5kW VZIM3 Thermal analysis of 22kW VZIM2 Thermal analysis of 22kW VZIM3

65 65 69 70 71 72 75 79 82

6

Conclusion

87

7

Bibliography

88

8 8.1 8.2 8.3 8.4

Appendix Appendix I Appendix II Appendix III Appendix IV

89 91 109 111

iv

List of Symbols ௜

 B  H   A ௦௜

 f  h ௗ ௙ ௣ ௪ ௣

l   L  L ௕ ௕

m ௠ ௟  ௧௛ ௛

M  n ௦ ௖

 p  P  q Q  R  s 

t  u,U    ሷ  ,   ሷ ௎ V 

ˍ

T A/m Vs/m m Hz m m m m m m m m m m m  N.m. 1/s W Ohm m s V A m m S/m Τଷ Ohm.m Vs/(Am ) -

number of parallel baranches of winding  parallel wires per turn magnetic flux density magnetic field strength Magnetic vector potential inner stator diameter  electric frequency height of stator slot opening distribution factor  slot fill factor, frequency coefficient  pitching factor  winding factor  Pole pitch axial length self-inductance overall length  permeance /unit length length of the winding overhang number of phases mean diameter of the end-ring length of the end-ring segment cross section area of end-ring end-ring thickness end-ring height torque rotational speed number of turns per phase number of turns per coil number of pole pairs  power  number of slots per pole and phase number of slots electric resistance and thermal resistance slip slot opening time electric voltage current/voltage transformation ratio magnetic voltage (m.m.f.) circumference co-ordinate air gap width electric conductivity density electric resistivity ordinal number of rotor space harmonics magnetic permeability ordinal number of stator space harmonics v

௧௛

ο ஼௨ ௘

 j i

ሺ ሻ ௛ ௘ ௠

d  ௧௛

1/s K/W Ԩ K W W W/(ଶ ) A/ଶ A W Ws/(ଶ ଷ ) 1/(Ohm.m)) T m/s m W/(m.)

electric angular frequency Heat resistance temperature temperature rise Copper losses iron losses heat transfer coefficient current density current volume density of instantaneous power loss hysteresis coefficient coefficient of losses in excess peak value of the magnetic flux density wind speed over stator outer surface and winding overhang thickness of slot insulation and stator and rotor lamination Thermal conductivity

vi

List of Figures 1.1-1 1.1-2 1.4-1 1.4-2 1.4-3 2.2-1 2.2-2 2.2-3 2.3-1 2.3-2 2.4-1 2.5-1 3.1-1 3.1-2 3.2.1-1

3.2.1-2 3.2.1-3 3.2.1-4 3.2.1-5 3.2.2-1 3.2.2-2 3.2.2-3 3.2.2-4 3.2.3-1 3.2.3-2 3.2.3-3 3.2.3-4

4.3-1 4.4-1

Cross-section of variable impedance induction motor, rotor position at stand still [6] Cross-section of variable impedance induction motor rotor position at nominal speed [6] Cross-section of Variable Impedance Induction motor (VZIM1), Prepared by  ANSYS  Cross-section of Variable Impedance Induction motor (VZIM2), Prepared by  ANSYS  Cross-section of Variable Impedance Induction motor (VZIM3), Prepared by  ANSYS  Created geometry of VZIM1, by Flux2D Created geometry of VZIM2, by Flux2D Created geometry of VZIM3, by Flux2D Generated mesh of VZIM1 in the air-gap using Flux2D Generated mesh of VZIM1 using Flux2D  B-H characteristic of M270-50A iron sheets, which are used in stator and rotor  lamination [7] Star connected stator electrical circuit of the motor, by Flux2D  Nonlinearity relation of  B  B and H [5] Static and equivalent  B- H   H curves for different cases [5] Torque-slip characteristics of VZIM1, comparison of the steady state calculation results of  Flux2D  Flux2D and the analytical calculations by KLASYS  Torque-slip characteristics of VZIM1, comparison of skewed and unskewed rotor, obtained by KLASYS  Phase current-slip characteristic of VZIM1 , comparison of the steady state calculation results of  Flux2D  Flux2D and the analytical results of  KLASYS   KLASYS 

2

Normal component of air-gap flux density for one pole pair of VZIM1 at a slip of 0.0253, calculated by  Flux2D Numerically calculated flux lines in VZIM1 at a slip of 0.0253, obtained  by Flux2D

24

Torque-slip characteristics of VZIM2, comparison of the steady state calculation results of  Flux2D  Flux2D and the analytical calculations by KLASYS  Phase current-slip characteristic of VZIM2 , comparison of the steady state calculation results of  Flux2D  Flux2D and the analytical results of  KLASYS   KLASYS 

26

 Normal component of air-gap air-gap flux density for one pole pair pair of VZIM2 at a slip of 0.0253, calculated by  Flux2D  Numerically calculated flux lines in VZIM2 VZIM2 at a slip of 0.0253, obtained obtained by  Flux2D

27

Torque-slip characteristics of VZIM3, comparison of the steady state calculation results of  Flux2D  Flux2D and the analytical calculations by KLASYS  Phase current-slip characteristic of VZIM3 , comparison of the steady state calculation results of  Flux2D  Flux2D and the analytical results of  KLASYS   KLASYS 

29

 Normal component of air-gap air-gap flux density for one pole pair pair of VZIM3 at a slip of 0.0253, calculated by  Flux2D  Numerically calculated flux lines in VZIM3 VZIM3 at a slip of 0.0253, obtained obtained by  Flux2D

30

Asynchronous harmonic torques of the 5th and 7th stator field harmonics, which are superimposed on fundamental asynchronous torque [2] Typical effects of synchronous and asynchronous harmonic torques in inductio n machines [2]

33

vii

2 8 9 9 11 11 12 12 13 14 15 20 21 22 23 23

25

27

28

30

31

34

4.6.1-1

Torque-slip characteristic of unskewed VZIM1, calculated with Flux2D timestepping, the effects of the synchronous harmonic torques at slips ൌ

ͲǤͺͷ͹, 4.6.1-2

ൌ ͳ and

ൌ ͳǤͲ͹ͳ of the the motor are observable

Torque-slip characteristic of unskewed VZIM1, calculated with KLASYS , the

effects of the synchronous harmonic torques at slips ൌ ͳǤͲ͹ͳ of the the motor are observable 4.6.1-3

4.6.1-4 4.6.1-5 4.6.1-6 4.6.1-7 4.6.1-8 4.6.2-1

36

ൌ ͲǤͺͷ͹ ,

36

ൌ ͳ and

Torque-slip characteristic of unskewed VZIM1, time-stepping and AC analysis results obtained by Flux2D, the harmonic torque effects at lower speeds of t he motor are obvious Stator phase current of unskewed VZIM1, time stepping anal ysis results calculated by Flux2D and KLASYS  Input power-slip of VZIM1, comparison of the analysis results calculated by  Flux2D and KLASYS  Output power-slip of VZIM1, comparison of the analysis results calculated by  Flux2D and KLASYS  Efficiency-slip of VZIM1, comparison of the analysis results calculated by  Flux2D and KLASYS  Power factor-slip of VZIM1, comparison of the analysis results calculated by  Flux2D and KLASYS  Torque-slip characteristic of unskewed VZIM2, obtained by KLASYS and Flux2D time-stepping calculations, the harmonic torque effects at lower speeds of the

38

38 38 39 39 40 41

motor are observable 4.6.2-2

Torque-slip characteristic of unskewed VZIM2, calculated with KLASYS , the

effects of the synchronous harmonic torques at slips ൌ ͳǤͲ͹ͳ of the the motor are observable 4.6.2-3

4.6.2-4 4.6.2-5 4.6.2-6 4.6.2-7 4.6.2-8 4.6.3-1

ൌ ͳ and

4.6.3-4 4.6.3-5 4.6.3-6

42

43 43 44 44 45 46

ൌ ͳǤͲ͹ͳ of the the motor are observable

Torque-slip characteristic of unskewed VZIM3, calculated with KLASYS , the

effects of the synchronous harmonic torques at slips ൌ ͳǤͲ͹ͳ of the the motor are observable 4.6.3-3

ൌ ͳ and

Torque-slip characteristic of unskewed VZIM2, time-stepping and AC analysis results obtained by Flux2D, the harmonic torque effects at lower speeds of the motor are obvious Stator phase current of unskewed VZIM2, comparison of the time stepping analysis results calculated by Flux2D and KLASYS  Input power-slip of VZIM2, comparison of the analysis results calculated by  Flux2D and KLASYS  Output power-slip of VZIM2, comparison of the analysis results calculated by  Flux2D and KLASYS  Efficiency-slip of VZIM2, comparison of the analysis results calculated by  Flux2D and KLASYS  Power factor-slip of VZIM2, comparison of the analysis results calculated by  Flux2D and KLASYS  Torque-slip characteristic of unskewed VZIM3, calculated with Flux2D timestepping, the effects of the synchronous harmonic torques at slips ൌ

ͲǤͺͷ͹, 4.6.3-2

ൌ ͲǤͺͷ͹ ,

41

ൌ ͲǤͺͷ͹ ,

ൌ ͳ and

Torque-slip characteristic of unskewed VZIM3, time-stepping and AC analysis results obtained by Flux2D, the harmonic torque effects at lower speeds of t he motor are obvious Stator phase current of unskewed VZIM2, comparison of the analysis results calculated by Flux2D and KLASYS  Input power-slip of VZIM3, comparison of the analysis results calculated by  Flux2D and KLASYS  Output power-slip of VZIM3, comparison of the analysis results calculated by  Flux2D and KLASYS  viii

46

47

47 48 48

4.6.3-7 4.6.3-8 4.7-1 4.7-2 4.7-3 4.7-4 4.7-5 4.7-6 4.7-7 4.8-1 4.8-2 4.8-3 4.8-4 4.8-5 4.8-6 4.8-7

4.8-8 4.8-9 4.8-10 4.9.1-1 4.9.1-2 4.9.1-3 5.2-1 5.2-2

5.2-3 5.3.1-1 5.3.1-2 5.3.1-3 5.3.1-4 5.3.1-5 5.3.2-1 5.3.2-2 5.3.2-3 5.3.2-4 5.3.2-5

Efficiency-slip of VZIM3, comparison of the analysis results calculated by  Flux2D and KLASYS  Power factor of VZIM3, comparison of the analysis results calculated by Flux2D and KLASYS  Torque-time characteristic of VZIM2 at 1462rpm, calculated by Flux2D, torque oscillates around a constant mean value after reaching the steady state Torque-time characteristic of VZIM2 at 1462rpm, calculated by Flux2D, zoomed view after steady state is reached Torque-time characteristic of VZIM2 at 1200rpm, calculated by Flux2D, torque oscillates around a constant mean value after reaching the steady state Torque-time characteristic of VZIM2 at 1200rpm, calculated by Flux2D, zoomed view after a steady state is reached Torque-time characteristic of VZIM2 at 525rpm, calculated by Flux2D, torque oscillates around a constant mean value Torque-time characteristic of VZIM2 at 525rpm, calculated by  Flux2D, zoomed view after a steady state is reached Torque-time of VZIM2 at 205.5rpm, calculated by Flux2D, where break down slip of the 7th asynchronous torque harmonic happens at slip 0.863 or 205.5rpm

49

First and second positions of the rotor bars with fixed stator position to calculate the synchronous harmonic torques, prepared by  Flux2D Torque-time of unskewed VZIM2 at ൌ ͲǤͺͷ͹ , calculated by Flux2D, rotor 

54

 position step 1 Torque-time of unskewed VZIM2 at ൌ ͲǤͺͷ͹ , calculated by Flux2D, rotor   position step 5 Torque-time of unskewed VZIM2 at ൌ ͲǤͺͷ͹ , calculated by Flux2D, rotor   position step 7 Variation of synchronous torque at ൌ ͲǤͺͷ͹ as a function of rotor position Variation of synchronous torque at ൌ ͳ a function of rotor position Variation of synchronous torque at ൌ ͳǤͲ͹ͳ as a function of rotor position Torque-time of unskewed VZIM2 at speed 224.28 rpm, calculated b y Flux2D Torque-time of unskewed VZIM2 at speed 10 rpm, calculated b y Flux2D Torque-time of unskewed VZIM2 at speed -97.14 rpm, calculated b y Flux2D

Comparison between losses of VZIM1 at speed equal to 1390.9 calculated with Flux2D and KLASYS  Comparison between losses of VZIM2 at speed equal to 1455.8 calculated with Flux2D and KLASYS  Comparison between losses of VZIM3 at speed equal to 1474.65 calculated with Flux2D and KLASYS  The simplified thermal equivalent network for an i nduction motor  Output power-slip characteristics of 22kW VZIM2 and VZIM3 and the average of  the powers Calculated output power of the VZIM motor wit h the supper-position and equivalent circuit methods 2D model of the 5.5kW V ZIM2 and meshing, by ANSYS  3D model of the 5.5kW VZIM2, by ANSYS  3D model meshing of the 5.5kW V ZIM2, by ANSYS  The thermal solution of the 5.5kW V ZIM2 under nominal operation, by ANSYS  The calculated temperature in the winding overhang of VZIM2, by ANSYS  2D model of the 5.5kW VZIM3 and meshing, by ANSYS  3D model of the 5.5kW VZIM3, by ANSYS  3D model meshing of the 5.5kW V ZIM3, by ANSYS  The thermal solution of the 5.5kW V ZIM3 under nominal operation, by ANSYS  The calculated temperature in winding overhang of 5.5kW VZIM3, by ANSYS  ix

49 50 51 51 52 52 53 53

54 55 55 56 56 56 58 59 59 63 64 64 66 67 68 72 72 73 74 75 75 76 76 78 78

5.3.3-1 5.3.3-2 5.3.3-3 5.3.3-4 5.3.3-5

5.3.4-1 5.3.4-2 5.3.4-3 5.3.4-4 5.3.4-5

2D model of the 22kW VZIM2 and meshing, by ANSYS  3D model of the 22kW VZIM2 , by ANSYS  3D model meshing of the 22kW VZIM2 , by ANSYS  The thermal solution of the 22kW VZIM2 under nominal operation, by ANSYS  The calculated temperature in the winding overhang of 22kW VZIM2 , by ANSYS  2D model of the 22kW VZIM3 and meshing, by ANSYS  3D model of the 22kW VZIM3 , by ANSYS  3D model meshing of the 22kW VZIM3 , by ANSYS  The thermal solution of the 22kW VZIM3 under nominal operation, by ANSYS  The calculated temperature in the winding overhang of 22kW VZIM3 , by ANSYS 

x

79 79 80 81 82 82 83 83 85 85

List of Tables 1.4-1 1.4-2 1.4-3 1.4-4 1.4-5 1.4-6 2.4-1 2.6-1 4.8-1

4.8-2 4.8-3 4.8-4 4.8-5 4.8-6 4.8-7 4.8-8 4.8-9 4.8-10

4.8-11

4.8-12

4.8-13

4.8-14

4.9-1

Stator dimensions Stator winding details The rotor dimensions dimensions and parameters parameters of VZIM1 The rotor dimensions dimensions and parameters parameters of VZIM2 The rotor dimensions dimensions and parameters parameters of VZIM3 VZIM material details  B-H characteristic data of the iron sheet type M270-50A [7] Electrical values of VZIM stator circuit Maximum, minimum and peak to peak values of synchronou s torque for  VZIM2 in two different methods of calculation (rotation and oscillation methods) for the slip ൌ ͲǤͺͷ͹ Maximum, minimum and peak to peak values of synchronou s torque for  VZIM2 in two different methods of calculation (rotation and oscillation methods) for the slip ൌ ͳ Maximum, minimum and peak to peak values of synchronou s torque for  VZIM2 in two different methods of calculation (rotation and oscillation methods) for the slip ൌ ͳǤͲ͹ͳ Maximum, minimum and peak to peak values of synchronou s torque for  VZIM1 (oscillation method) Maximum, minimum and peak to peak values of synchronous torque for  VZIM3 (oscillation method) Maximum, minimum and peak to peak values of synchronous torque for  VZIM1, at the slip ൌ ͲǤͺͷ͹, the comparison of the calculated values by  KLASYS and Flux2D Maximum, minimum and peak to peak values of synchronous torque for  VZIM1, at the slip ൌ ͳ, the comparison of the calculated values by KLASYS  and Flux2D Maximum, minimum and peak to peak values of synchronous torque for  VZIM1, at the slip ൌ ͳǤͲ͹ͳ, the comparison of the calculated values by  KLASYS and Flux2D Maximum, minimum and peak to peak values of synchronous torque for  VZIM2, at the slip ൌ ͲǤͺͷ͹, the comparison of the calculated values by  KLASYS and Flux2D Maximum, minimum and peak to peak values of synchronous torque for  VZIM2, at the slip ൌ ͳ, the comparison of the calculated values by KLASYS  and Flux2D Maximum, minimum and peak to peak values of synchronous torque for  VZIM2, at the slip ൌ ͳǤͲ͹ͳ, the comparison of the calculated values by  KLASYS and Flux2D

6 6 7 7 7 8 14 19 57 57 58 58 58 60 60 60 60 60 60

Maximum, minimum and peak to peak values of synchronous torque for  61 VZIM3, at the slip ൌ ͲǤͺͷ͹, the comparison of the calculated values by  KLASYS and Flux2D Maximum, minimum and peak to peak values of synchronous torque for  61 VZIM3, at the slip ൌ ͳ, the comparison of the calculated values by KLASYS  and Flux2D Maximum, minimum and peak to peak values of synchronous torque for  61 VZIM3, at the slip ൌ ͳǤͲ͹ͳ, the comparison of the calculated values by  KLASYS and Flux2D

Values of specific total losses according to the data sheet of the M27050A [7], and calculated power losses, in frequencies 50Hz, 100Hz and 200Hz xi

62

5.2-1 5.2-2

5.2.1-1 5.2.2-1 5.2.2-2

5.3.1-1 5.3.1-2

5.3.2-1 5.3.2-2

5.3.2-3 5.3.3-1 5.3.3-2 5.3.4-1 5.3.4-2 5.3.4-3 5.3.4-4 Appendix II Appendix II Appendix II Appendix II Appendix II Appendix II Appendix II Appendix III Appendix III Appendix III

Heat transfer coefficient for one side of the winding overhang close to the centrifugal mechanism Losses in different parts of the VZIM2 and VZIM3 in the speed equal to 1467.36 rpm Thermal conductivities of the materials used in VZIM2 thermal model Thermal conductivities of the materials used in VZIM3 thermal model Temperature rise in the stator winding of VZIM2 and VZIM3 calculated based on simplified thermal equivalent circuit in the nominal operating speed equal to 1467.36 rpm Loss densities in different parts of the 5.5kW VZIM2 Heat transfer coefficient at different boundary conditions of the 5.5kW VZIM2 Loss densities in different parts of the 5.5kW VZIM3 Heat transfer coefficient at different boundary conditions of the 5.5kW VZIM3 Temperature rise in stator slot of 5.5kW VZIM2 and VZIM3 calculated by  ANSYS  Loss densities in different parts of the 22kW VZIM2 Heat transfer coefficient at different boundary conditions of the 22kW VZIM2 Loss densities in different parts of the 22kW VZIM3 Heat transfer coefficient at different boundary conditions of the 22kW VZIM3 Temperature rise in stator slot of 22kW VZIM2 and VZIM3 calculated by  ANSYS  Temperature rise in stator slot of 22kW VZIM2 and VZIM3 calculated by  ANSYS  Synchronous harmonic torque slip

Stator ordinal numbers and related asynchronous harmonic torque slips Stator ordinal numbers and related asynchronous harmonic torque slips Stator ordinal numbers and related asynchronous harmonic torque slips Slips which are used to perform time-stepping analysis of VZIM1 Slips which are used to perform time-stepping analysis of VZIM2 Slips which are used to perform time-stepping analysis of VZIM3 Losses in VZIM1 calculated by Flux2D and KLASYS  Losses in VZIM2 calculated by Flux2D and KLASYS  Losses in VZIM3 calculated by Flux2D and KLASYS 

xii

67 68 69 70 71 73 74 76 77 78 80 81 83 84 85 86 91 92 96 101 106 107 108 109 109 110

Chapter 1: Introduction

1.1 Preface Due to the robustness and reliability with low production and maintenance costs of the line operated induction motors, this type of machines are used in more than 80% of motor  applications worldwide. Making use of current displacement effect by designing induction motors with deep bar or  double cage rotors results in desirable starting torque and efficiency at nominal speed operation, but reduction of breakdown torque is not favorable which happens in this case  because of increased rotor leakage reactance. One possible solution is Variable Impedance Induction Motor (VZIM). The Variable Impedance Induction Motor has high starting torque, in addition to high efficiency at the nominal operating point. This type of induction motor includes a sectionalized cage rotor with different bar shapes for each sub-cage section is used. At standstill, sub-cages with high resistances are placed inside the stator bore to increase the starting torque and reduce the starting current. By increasing the rotational speed, a centrifugal mechanism moves the rotor in the axial direction and brings the other sections of  the rotor cage with low resistances inside the stator bore to reduce the rotor losses at rated speed. This leads to increased efficiency at nominal operation [6]. Figures 1.1-1 and 1.1-2 show the cross section view of a 3-section VZIM motor at standstill and nominal speed. The rotor length is 1.5 times the stator length, and it is composed of three different sub-cages (“A (“ A”: starting sub-cage, sub-cage, “B”: middle subsub -cage and “C”: “C” : final sub-cage). At standstill, the subcages “A” and “B” are inside the rotor bore; means they are active at start-up of the motor). By increasing the speed, a centrifugal axial rotor shifter mechanism pulls the rotor in the axial direction and brings the third section of the rotor cage inside the stator bore. At the nominal speed the sub-cage sub- cage “A” is put completely out of the stator bore and only the sub- cages “B” and “C” are active.

-1-

Fig 1.1-1: Cross-section of variable impedance induction motor, rotor position at stand still [6]

Fig 1.1-2: Cross-section of variable impedance induction motor rotor position at nominal speed [6]

-2-

1.2 Electromagnetic and thermal analysis of the 22kW VZIM

A 4 pole, 22kW VZIM operating with a nominal voltage of 400V and a nominal frequency of  50Hz has been designed. The rotor has 3 sections with different rotor bar shapes. The first rotor section has round bars, the middle cage has deep bars and the final cage has wedge bars. The electromagnetic characteristics and thermal effects of the electrical losses in this motor  has been analyzed and investigated. With the help of  Flux2D  Flux2D and ANSYS  which are suitable tools for electromagnetic and thermal analysis and are based on finite element method, the motor’s characteristics and features are calculated and the results of the electromagnetic analysis prepared by  Flux2D are compared with analytical calculations which mostly are obtained by  KLASYS tool. In this analysis it has been assumed that there are three motors with the same stator but different rotor bars, therefore three motors VZIM1, VZIM2 and VZ IM3 are analyzed. 1.3 Introduction to finite element method [4]

In engineering and science there are many physical phenomena which can be described with Partial Differential Equations (PDE), solving these equations with analytical methods for  arbitrary shapes is almost impossible. The Finite Element Method (FEM) is a numerical approach by which these PDE can be approximately solved. Solving a PDE by FEM method is done by dividing the calculation domain in finite elements in which a known variation of the physical values is assumed. This variation is usually a  polynomial variation with an arbitrary power but in practical applications the maximal degree Three is used. In conclusion the field distribution is assumed to be a polynomial variation of  first order in most of the cases, of second order in more rare situation and extremely rare of  the third order. To have a good FEM calculation some general rules must be followed. After dividing the calculation domain in finite elements with meshing the area of calculation, first of all the finite elements have to cover the entire calculation domain, the elements must not overlap and the nodes of one element cannot be found on the lines of the adjacent elements but must have the same position with the adjacent elements nodes.

ൌൌ Ͳ                      ሺሺͳǤͳͳǤͳǤǤ͵͵െͳሻെͳെʹሻെʹሻሻ ൌ ή                                                        ሺ ͳ Ǥ ͵ െ ͵ሻ ሺͳǤൌ͵െͳሻ                                                             ሺͳǤͳǤ͵െͶሻെͶ ሻ ሺ ሻሻ ൌ Ͳ ሺͳǤ͵െʹሻ ሺͳǤ͵െ͵ሻ ሺͳǤ͵െͶሻ

In case of magneto static problems, the field is described by the following Maxwell equations,

and the constitutive law for material, From

we can write B as,

 because it is always true

From , and static problem can be obtained,

, where

is called the magnetic vector potential.

the differential equation that describes a magneto

-3-

      ሺఓଵ ሻ ൌ                  ൌ                                 ൌ  ሺͳǤͳǤ͵െͷሻെͷ ሻ ൌ ͲǤ                                                          ሺͳǤͳǤ͵െͶ ሻ ሺ ǡ ǡ ሻ ሺͳǤ͵െͶሻ ሺͳǤ͵െͶ ሻ ௖ Each field is defined by its sources , and by related eddies may be chosen free. The coulomb gauge is often used as

, so

Further, if a constant vector  is added to , still and are valid, so the magnetic vector potential is defined by choosing the value for one node from domain. This is necessary in order to obtain a correct solution. The unknown of the problem is the value of  in the grid nodes and the field source is given by the current density in the elements volumes (areas for 2D).

ሺͳǤ͵െ͵ሻ

After solving the linear system of equations the values of the flux density  B are obtained. Afterwards the field strength  H  due to is calculated and by integration over the domain the energy and generated force can also b e calculated. In FEM the stored magnetic energy is calculated by integrating the energy density as a volume integral for 3D-FEM or as an area integral for 2D-FEM. The volume energy density is defined as: .

ൌሺൌሺ ή ሻΤʹ ௡೐ Ǥ Ǥ ൌ න ൌ  න ʹ ൌ  ෍௘ୀଵන ʹ Ǥ                           ሺͳǤͳǤ͵െͷ ሻ ൌ ൌ ή ௡೐ Ǥ ௡೐ሺͳǤ௘͵ήെͷሻ௘ ή ௘ ௡೐ ௘ଶ ൌ௘ୀଵ෍ න ʹ ൌ  ෍௘ୀଵ ʹ ൌ  ෍௘ୀଵ ʹ ή ௘ ή Ǥ                  ሺͳǤͳǤ͵െ͸ሻെ͸ ሻ ሺͳǤ͵െ͸ሻൌ ͳ ௡೐ Ǥ ௡೐ ௘ଶ ൌ௘ୀଵ෍ න ʹ ൌ ෍௘ୀଵ ʹ ή ௘ ή ௘ Ǥ                                   ሺͳǤͳǤ͵െ͹ሻ ሺͳǤ ͵ െ͹ሻ   Ȁ  ሺͳǤ ͵ െ͸ሻ  The energy density can be integrated on the calculation domain for the 3D-FEM as it follows

As

is calculated as derivative of  , is constant within each element. Via also  H  is constant within each element. So considering that inside one element the values for  B can be transformed into:  B and H are constant,

For 2D-FEM the equation model in the z direction equal with

is valid, if we consider that we have a length of the :

The energy in is measured in , whereas the energy in is measured in . In order to get the value of the stored magnetic energy in a volume from a 2D-FEM calculation, one has to multiply the calculated value with the length of the model in the zdirection. For the magneto static calculation the magnetic field is considered to be constant and no eddy currents are induced in the conductive materials. This stationary situation is sufficient to describe only the problems with slow varying fields and problems where the effects of the variation of the fields are considered negligible. For the rest of the problems consideration of  -4-

the variation of the fields is necessary and calculation in time domain or in frequency domain is applied. The time domain calculation is actually a succession of static calculations for  different time moments. The time variable cannot be considered continuously and has to be considered using a certain time step. Each static calculation that is performed during a transient calculation is called a non-linear iteration and for each iteration the results from the  previous iteration are considered as starting point. The variation of magnetic field between two iteration steps determines the variation speed of the magnetic field. In order to calculate this influence the Maxwell  equations must be written in a form that considers the magnetic field variation in time

    ൌൌ Ͳ                                                      ሺͳሺǤͳ͵Ǥ͵െെͻሻͺሻ ൌെ                                                   ሺͳǤ͵ െ ͳͲሻ ൌൌ ˍήή Ǥ                             ሺሺͳǤͳͳǤǤ͵͵െͳͳሻ െͳͳെ ͳʹሻሻ    ൌሺͳǤ͵െͳ͵ሻ     ሺͳǤ  ͵ െͳͳሻ                                               ሺͳǤ͵ െ ͳ͵ሻ ൌെ ൌെ                                     ሺͳǤ͵െͳͶሻ and the constitutive law for material

Like for magneto static problems we can write the flux density as a curl of vector potential Using

in

we obtain

which is the electric field strength determined by the time-varying field. It represents the electromotive force induced in the conductor due to the magnetic field variation. From we can calculate the current density induced by the magnetic field variation using

ሺͳǤ͵െͳͶሻ ሺൌͳǤ͵െͳʹሻ െˍή                                                   ሺͳǤ͵ െ ͳͷሻ ൌ  ൅ ௘

The current density has 2 components: -The source component that is given as entry data for the problem at the beginning of the calculation. It is represented by the current density in the conductors that are exciting the  primary magnetic field model. model. -The component that is induced due to the magnetic field variation .

௘

൬ͳ ൰ ൌ  െ ˍ ή Ǥ                                       ሺͳǤ͵ െ ͳ͸ሻ

Considering these two components results in

This is the differential equation to be solved by the FE program, in a time-step solution. In  practice with usage of a finite element tool there are some steps to solve the problem. The geometry is prepared, then geometry must be meshed and afterwards material are assigned to the regions, finally by linking an external electrical circuit the model is ready to be solved. To create the geometry, symmetry and periodicity must be taken into account. If the geometry -5-

has periodicity, type of the periodicity (symmetric or anti-symmetric conditions) must be defined. 1.4 Electrical and mechanical parameters and dimensions of the 22kW VZIM

Due to three rotor sections the names VZIM1 for the first cage, VZIM2 for the middle cage and VZIM3 for the last cage, are assigned to each part of the rotor. Stator dimensions are given in Table 1.4-1 . The stator winding details, which has a star  connection are listed in Table 1.4-2. Tables 1.4-3, 1.4-4 and 1.4-5 list the rotor dimensions and parameters of the VZIM1, VZIM2 and VZIM3. The rotor sub-cages are not skewed. Table 1.4-6 consists of the material details used i n VZIM. Table 1.4-1: Stator dimensions Description Stack length Inner stator diameter  Outer stator diameter  Air gap width  Number of stator slots Width of stator slot opening Height of stator slot opening Stator slot heit Stator slot width Radius of stator slot opening

parameter

௘௦௜௦௜ ௦௢ ௦ ௦ସ௦ ௦௦

Dimension mm mm mm mm mm mm mm mm mm

Value 240 170 270 0.45 36 3.1 0.737 22.6 10.277 3.606

parameter m  p q

Dimension mm mm -

Value 3 2 3 12 1 Single layer  Full pitch 72 226.6 9 1 Star 

Table 1.4-2: Stator winding details Description  Number of phases  Number of pole pairs  Number of slots per pole and phase  Number of turns per coil  Number of parallel branches Type of winding Winding pitch  Number of turns per phase Length of overhang  Number of parallel wires per turn Diameter of conductor  Winding connection

௖

a -

௦௕ ௜஼௨ -

-6-

Table 1.4-3: The rotor dimensions dimensions and parameters of VZIM1 Description Outer rotor diameter  Inner rotor diameter   Number of rotor bars Bar shape Bar material Width of rotor slot opening Height of rotor slot opening Rotor bar diameter  End-ring thickness End-ring height

parameter

௥௢௥௜ ௥ ௥ସ ௥௘௥ ௘௥ -

Dimension mm mm mm mm mm mm mm

Value 169.1 80 28 Round Bronze 4.5 1 10.2 4.3 44

Dimension mm mm mm mm mm mm mm mm

Value 169.1 80 28 Deep Copper  4 1 8.2 9.2 4.6 44

Dimension mm mm mm mm mm mm mm mm

Value 169.1 80 28 Inverse wedge Copper  2 1 17.7 10.2 6 44

Table 1.4-4: The rotor dimensions dimensions and parameters of VZIM2 Description Outer rotor diameter  Inner rotor diameter   Number of rotor bars Bar shape Bar material Width of rotor slot opening Height of rotor slot opening Height of rotor bar  Width of rotor bar  End-ring thickness End-ring height

parameter

௥௢௥௜ ௥ ௥ସ ௥௥ ௘௥௘௥ -

Table 1.4-5: The rotor dimensions dimensions and parameters of VZIM3 Description Outer rotor diameter  Inner rotor diameter   Number of rotor bars Bar shape Bar material Width of rotor slot opening Height of rotor slot opening Height of rotor bar  Width of rotor bar  End-ring thickness End-ring height

parameter

௥௢௥௜ ௥ ௥ସ ௥௥ ௘௥௘௥ -

-7-

Table 1.4-6: VZIM material details

ԨԨ

Description Conductivity of bronze at 110 Conductivity of copper at 110 Lamination type Conductivity of lamination Loss at 50Hz, 1T Loss at 50Hz, 1.5T

ˍˍ஻௥௢௡௭௘ ௖௢௣௣௘௥ ˍ௟௔ଵ଴௠௜௡

parameter

-

ଵହ

ିଵିଵ ିଵ

Dimension S S S W/kg W/kg

Value 1.69E+7 4.22E+7 M270-50A 1.818E+6 1.07 2.52

Cross-sections of the motors have been depicted in figures 1.4-1, 1.4-2 and 1.4-3. The first motor VZIM1 has round rotor bars made of bronze and second and third motor m otor VZIM2 and VZIM3 have rotor bars made of copper.

Fig 1.4-1: Cross-section of Variable Impedance Induction motor (VZIM1), prepared by ANSYS 

-8-

Fig 1.4-2: Cross-section of Variable Impedance Induction motor (VZIM2), prepared by ANSYS 

Fig 1.4-3: Cross-section of Variable Impedance Induction motor (VZIM3), prepared by ANSYS  -9-

1.5 Procedure of the project

In order to simplify the simulation, three separate induction motors have been considered, in this simulation VZIM1, VZIM2 and VZIM3 each one has a full length rotor and stator. With the help of  Flux2D  Flux2D, electromagnetic analysis and with  ANSYS , thermal analysis has been done. Two types of electromagnetic analysis have been carried out: -Steady state magnetic AC analysis -Time stepping analysis. All results from Flux2D have been compared with the analytical results of  KLASYS   KLASYS tool.

- 10 -

Chapter 2: Geometry and mesh generation with Flux2D

2.1 Introduction

Electromagnetic analysis based on finite element method is done with  Flux2D. In this chapter  chapter  the creation of geometry, mesh generation, electrical circuit determination and material assignment is explained. 2.2 Geometry creation

The variable impedance induction motor which is analyzed has 4-poles, 36-stator slots and 28 rotor bars, three different motors with the same stator but different rotors are being considered. Based on this consideration 3 models has been created. Because of simplicity and according to Anti-cyclic boundary condition, ¼ of the motor is modeled. Three models which have been prepared as VZIM1, VZIM2, and VZIM3 are depicted in Fig 2.2-1, Fig 2.2-2 and Fig 2.2-3.

Fig 2.2-1: Created geometry of VZIM1, by Flux2D

- 11 -

Fig 2.2-2: Created geometry of VZIM2, by  Flux2D

Fig 2.2-3: Created geometry of VZIM3, by Flux2D

2.3 Meshing the geometry

An automatic mesh generator is in charge of meshing the faces. With the help of the mesh  points, it is possible to have fine and dense meshing in the areas which the higher accuracy is needed. Fine mesh in the air-gap, at the top of the rotor bars, and the stator teeth, is necessary  because of accuracy of the results, air-gap must be fine meshed, to have accurate torque calculation. The top of the rotor bars near the air gap are densely meshed, to taking into the account the current displacement effect. In addition the stator teeth must be meshed fine enough to consider the saturation effect because of high flux density in this area. In Fig.2.3-1 and Fig.2.3-2 generated mesh for VZIM1 is shown. The fine mesh in the air-gap air -gap is observable in Fig.2.3-1.

Fig 2.3-1: Generated mesh of VZIM1 in t he air-gap using Flux2D

- 12 -

Fig 2.3-2: Generated mesh of VZIM1 using Flux2D

- 13 -

2.4 Material assignment

Stator and rotor lamination in 22kW VZIM are made of M270-50A [7].  B (H) curve and values are shown in Fig.2.4-1 Fi g.2.4-1 and table 2.4-1. Table 2.4-1: B-H characteristic data of the iron sheet type M270-50A [7]  B/Tesla

A/  )  H/( A/ 0 21.73374 43.46748 73.65323 111.3854 158.5506 217.5072 291.2029 383.3224 498.472 642.4088 822.3299 1047.231 1328.358

 B/Tesla 1.516379 1.534915 1.556988 1.582977 1.613156 1.647613 1.686154 1.728191 1.772635 1.817807 1.861388 1.900418 1.931347 1.955117

 H/( A /   ) 2119.027 2668.102 3354.447 4212.377 5284.791 6625.307 8300.953 10395.51 13013.71 16286.45 20377.38 25491.05 31883.13 43860.55

0 0.454861 0.7016061 0.902243 1.048599 1.157766 1.240593 1.304297 1.353865 1.392849 1.42385 1.448816 1.469241 1.486302

B-H Characteristic of M270-50A iron iron sheets 2.5

2

1.5

  a    l   s   e    T    /    B

1

0.5

0 0

50 00

1 0000

15 0 00

20 000

2 500 0

30 0 00

35 000

40 00 0

45 000

50 000

H/A ^(−૚)

Fig.2.4-1: B-H characteristic of M270-50A iron sheets, which are used in stator and rotor lamination [7] - 14 -

VZIM1 has rotor bars which are made of bronze, VZIM2 and V ZIM3 have rotor bars made of  copper. This special alloy of bronze has the electrical resistivity equal to 0.588E-07 Ohm.m. Copper has the electrical resistivity resistivit y equal to 0.237E-07 Ohm.m. 2.5 Electrical circuit of the motor

Three phase winding in 22kW VZIM is star connected. The electrical circuit which is created in Flux2D, is depicted in Fig.2.5-1.

Fig.2.5-1: Star connected stator electrical circuit of the motor, by Flux2D

The resistances R1, R2 and R3 represent the winding overhang resistances; also the inductances L1, L2 and L3 represent the overhang inductances. In addition stator phase windings under stator slots are shown as BPA, BPB and BMC. VA, VB and VC are three phase voltage sources with 230 V and 120 degree phase shift with each other; moreover Q1 represents the squirrel cage rotor which is connected to common  point of voltage sources. sources.

- 15 -

2.6 Calculation of values of circuit’s elements

The values of the circuit components are calculated and assigned to the circuit. In this section the values of the stator resistance, inductance of stator winding overhang, rotor end-ring leakage inductance and resistance of the end-ring segments are calculated. 2.6.1 Stator resistance

௦

 ஻஼ ஻஻ ଵ ଶ ଷ ͳ ൉ʹ൉ሺ ൅ ሻ ௦ ௘ ௕ ൌ ௦ ˍ௖௨ ௖௨ ൉             ିଵିଵ                          ሺʹǤ͸Ǥͳ െ ͳሻ ሺ ሻ ൌ ሺʹͲԨሻ ʹͲԨሻ ή ሺͳ ൅ ణ ή ο ሻǡ ణ ൌ ͳΤ ʹͷͷ ǡ ο ൌ െʹͲԨǡ ൌ Τͳ ˍǤ ሺʹǤ͸Ǥͳെʹሻ

Stator winding resistance per phase is sum of resistance of the coil under stator slots (also ) and winding overhang (also and ). Resistance per phase is calculated according to [1]:

஻஺

Ԩ ିଷ ͳ ൉ ʹ ൉ ͳ ൉ ʹ ൉ ͳ ͹ʹ൉ ʹ ൉ ʹ ͶͲή ͳ Ͳ ௦ ௘ ௦ ௘       ஻஺ǡଶ଴Ԩ ൌˍ௖௨ ௖௨ ൉ ൌˍ௖௨ Ͷ ή ௖௨ଶ ή ൌൌͷ͹ήͳͲି଺ Ͷ ή ͳଶ ή ͳͲି଺ ήͻήͳ ൌ ͲǤͲͺͷ͹͹ Ԩ      ஻஺ǡଵଵ଴Ԩ ൌʹ͵ͷ൅ͳͳͲ ʹ͵ʹ͵ͷͷ ൅ʹͲ൅ ʹͲ ஻஺ǡଶ଴Ԩ ൌ ͲǤͳͳ͸ȳǤ Ԩ ିଷ ͳ ൉ ʹ ൉ ͳ ൉ ʹ ൉ ͳ ͹ʹ൉ ʹ ൉ ʹ ʹ͸Ǥ ͸ ή ͳ Ͳ ௦ ௕ ௦ ௕      ଵǡଶ଴Ԩ ൌˍ௖௨ ௖௨ ൉ ൌˍ௖௨ Ͷ ή ௖௨ଶ ή ൌൌͷ͹ήͳͲି଺ Ͷ ή ͳଶ ή ͳͲି଺ ήͻήͳ ൌ ͲǤͲͺͲͻͺ Ԩ      ଵǡଵଵ଴Ԩ ൌʹ͵ͷ൅ͳͳͲ ʹ͵ʹ͵ͷͷ ൅ʹͲ൅ ʹͲ ଵǡଶ଴Ԩ ൌ ͲǤͳͲͻͷȳǤ ఙ௕ ൌ ଴ ௦ଶ ʹ ௕ ௕                                              ሺʹǤ͸Ǥʹ െ ͳሻ ௕ ௕ ௪ ൌ௕ ͲǤͲǤ͵͹ ൉ ൉ ൤ ͳ൅ͲǤ ͻ ൉ ௕௕ െͳͲή ൅ͳͲή ௪൨                                 ሺʹǤ͸Ǥʹെʹሻ Resistance of one phase under stator slot at 20

equals to:

and the resistance of one phase under slot of stator in 110

Resistance of the winding overhang at 20

is given by:

is calculated as below:

and the resistance of winding overhang at 110

is equal to:

2.6.2 Inductance of stator winding overhang

Inductance of the winding overhang is calculated according to [1], using following equation:

where is the length of the winding overhang and calculated as below:

௪

is the permeance /unit length which is

where is the diagonal of the cross-section of the coils in the winding overhang and is calculated as following:

- 16 -

ଶ ଶ ͻ ή ͳ ௜ ௖௨ ௖ ඨ  ௙ ൌඨ ͲǤͶʹ͸ή ͳʹ ൌ ͳͷǤͻ                   ሺʹǤ͸Ǥʹ െ ͵ሻ ௪ ൌඨ  ௕ ʹʹʹ͸Ǥ൉ ʹ͸Ǥ͸͸െͳͲή ͳ ͷǤ ͻ ௕ ൌ ͲǤͲǤ͵͹ ൉ ͲǤ͸͹ ൉ ൤ͳ ൅ ͲǤͻ ʹʹ͸Ǥ ൅ͳͲήͳͷǤͻ൨ ൌ ͲǤʹͺ͹Ǥ ʹ ି଻ ଶ ൌ Ͷ ή ή ͳͲ ή ͹ʹ ʹͶǤ ఙ௕ ʹ ʹʹ͸Ǥ͸ήͳͲିଷ ή ͲǤʹͺ͹ ൌ ͲǤͶʹͶ

According to [1] for a single phase winding with q=3, c is equal to 0.67 and as below:

is calculated

Therefore the inductance of the winding overhang equals to:

2.6.3 Rotor end-ring leakage inductance

                                ሺ ʹ Ǥ ͸ Ǥ ͵ െ ͳሻ ఙ௕௥ ൌ ଴ ௥ଶ ʹ ௕ ௕Ǥ       ௗ గ   ଵ଻଴గ ௕ ൌ ͲǤͳʹ ௕ ൌ ௣ ଶൌ ଶ௣ೞ೔ ൌଵ଻଴గସ ൌ ͳ͵͵Ǥͷ ఙ௕௥ ൌ  Ͷ ή ή ͳͲି଻ ή ൬ͳʹ൰ ή ʹʹ ήͲǤͳʹήͳ͵͵ǤͷήͳͲ ିଷ ൌ ͷǤͷǤͲ͵ ή ͳͲିଽିଽǤ

Rotor end-ring leakage inductance is calculated according to [1] as following:

where

and

, hence:

2.6.4 Resistance of the end-ring segment of VZIM1

௠ ௜ ௥ ௢௥ ௠ ൌ ௢௥ ൅ʹ ௜௥ ൌͳ͸ͻǤͳʹ൅ͺͳ ൌ ͳʹͷǤ ௥ ௠ ൌ௟ ௠௥ ൌ ήʹͺͳʹͷൌൌͶǤ͵ͳͶǤͲʹǡ ௧௛ ௛ ൌ ͶͶǡ  ൌൌ ௧௛ ή ௛ ൌ ͳͺͻǤʹଶ ିଷ ͳͶǤ Ͳ ʹή ͳ Ͳ ௟ ο  ൌ፺஻௥ǡଵଵ଴Ԩ  ൌͳ͹ǤͺήͳͲ଺ ήͳͺͻǤʹήͳͲି଺ ൌ ͶǤͳ͸ ή ͳͲି଺ି଺ȳ Ǥ               

First of all the mean diameter of the end-ring the inner diameter  and the outer diameter 

With the number of the rotor bars and the end-ring segment is calculated as below:

is calculated which is the mean value of  of the end-ring:

the mean diameter of end-ring, the length l ength of 

the end-ring thickens is and the end-ring height is ring thickness and height, the cross section area of end-ring is calculated as: . Then resistance of the end-ring is calculated as:

using endusing

2.6.5 Resistance of the end-Ring Segment VZIM2

The mean diameter of the end-ring diameter  and the outer diameter 

௜௥

௠௢௥

is calculated using the mean value of the inner  of the end-ring:

- 17 -

௠ ൌ ௢௥ ൅ʹ ௜௥ ൌͳ͸ͻǤͳʹ൅ͺͳ ൌ ͳʹͷ ௥ ௠ ௟ ൌ ௠௥ ൌ ήʹͺͳʹͷ ൌ ͳͶǤͲʹ ௧௛ ൌ ͶǤ͸ ௛ ൌ ͶͶ  ൌൌ ௧௛ ή ௛ ൌԨʹͲʹǤͶଶ ʹ ʹͷͷ ͷͷ ˍ௖௨ǡଵଵ଴Ԩ ൌʹ͵ʹ͵ͷʹ͵ͷ൅ʹͲ ˍ ൌ ͷ ൅ͳͳ൅ ͳͳͲͲ ௖௨ǡଶ଴Ԩ ͵Ͷͷ ͷ͹ήିଷͳͲ଺ ൌ ͶʹǤʹǤͶ ͳ͵ήͳ͵ ή ͳͲ଺ ൗ ο  ൌ፺௖௨ǡଵଵ଴Ԩ௟  ൌͶʹǤͳ͵ήͳͶǤͳͲͲ଺ ʹήήʹͳͲʹǤͲ ͶήͳͲି଺ ൌ ͳǤͳǤ͸ͶͶ͸ͶͶ ή ͳͲି଺ି଺ȳ     

With the number of the rotor slots and the end-ring segment is calculated as:

the mean diameter of end-ring, the length of 

end-ring thickens is, and the end-ring height is ring thickness and height, the cross section area of end-ring is calculated as: conductivity of copper in 110

.using end-

is calculated as:

Then resistance of the end-ring is calculated as:

2.6.6 Resistance of the end-ring segment VZIM3

௠ ௜௥ ௢௥ ൅ ௜௥ ͳ͸ͻǤͳ൅ͺͳ௢௥ ௠ ൌ ʹ ൌ ʹ ൌ ͳʹͷ ௥ ௠ ൌ௟ ௠௥ ൌ ήൌʹͺͳʹͷ͸ൌͳͶǤͲʹ ௧௛ ௛ ൌ ͶͶ  ൌൌ ௧௛ ή ௛ ൌԨʹ͸Ͷଶ ʹ ʹͷͷ ͷͷ ˍ௖௨ǡଵଵ଴Ԩ ൌʹ͵ʹ͵ͷʹ͵ͷ൅ʹͲ ˍ ൌ ͷ ൅ͳͳ൅ ͳͳͲͲ ௖௨ǡଶ଴Ԩ ͵Ͷͷ ͷ͹ήିଷͳͲ଺ ൌ ͶʹǤʹǤͶ ͳ͵ήͳ͵ ή ͳͲ଺ ൗ ο  ൌ፺௖௨ǡଵଵ଴Ԩ௟  ൌൌͶʹǤͳ͵ήͳͶǤͳͲͲ଺ʹήήʹͳͲ͸ͶήͳͲି଺ ൌ ͳǤͳǤʹ͸ͳʹ͸ͳ ή ͳͲି଺ି଺ȳ                

The mean diameter of the end-ring diameter  and the outer diameter 

is calculated using the mean value of the inner  of the end-ring,

with the number of the rotor slots and the end-ring segment is calculated as:

the mean diameter of end-ring, the length of 

end-ring thickens is and the end-ring height is thickness and height, the cross section area of end-ring is calculated as: conductivity of copper in 110

.using end-ring

is calculated as:

Then resistance of the End-Ring is calculated as:

All the values derived from formulas in sections 2.6.4 to 2.6.6 are shown in Table 2.6-1.

- 18 -

Table 2.6-1: Electrical values of VZIM stator circuit

஻஺ଵ ଶ ஻஻ଷ ǡ ஻஼ ଵఙ௕௥ ଶ ଷ οο ଵଶ ο ଷ Parameter , , , , ,

value 0.116 Ω 0.1095 Ω 0.424 mH

ͷǤͶǤͲͳ͵ή͸ήͳͳͲͲିଽି଺ି଺ȳ ͳǤͳǤ͸ʹͶͶή͸ͳήͳͳͲͲି଺ȳȳ

description Resistance per phase of the coil un der stator slot Resistance per phase of winding overhang Inductance pre phase of winding overhang End-ring leakage inductance Resistance of the end-ring segment, VZIM1 Resistance of the end-ring segment, VZIM2 Resistance of the end-ring segment, VZIM3

- 19 -

Chapter 3: Steady state electromagnetic analysis

3.1 Introduction

In steady state electromagnetic analysis, it is assumed that variables are changing purely sinusoidal with time and there are no harmonics. Due to the saturation effect of magnetic material, electromagnetic flux density  B and electromagnetic field strength  H , could not be sinusoidal at the same time. If the electric circuit had voltage source the B would be considered as sinusoidal variable and therefore because of   B (H) curve,  H  would not be a sinusoidal variable with time. Fig.3.1-1 shows that how  B and H curves change if one of them is sinusoidal.

Fig. 3.1-1: Nonlinearity relation of  B  B and H [5]

The above contradictions are dealt by using an equivalent  B(H) characteristic, which is different for the original  B(H) characteristic and based on the energy equivalence method, as shown in Fig.3.1-2. For the two extreme cases of  B sinusoidal and  H  sinusoidal the  B(H) curve is modified and shown in Fig.3.1-2 based on the above method. The results obtained with the equivalent curves calculated in the two extreme cases most often include the exact result. The equivalent curve can equally be calculated by means of a linear combination  between these two extreme cases as shown in Fig.3.1-2. Only the numerical values of the  post-processed quantities, that depend on energy, are correct. The instantaneous values computed by this analysis are approximations as they are sinusoidal.

- 20 -

Fig. 3.1-2: Static and equivalent  B- H   H curves for different cases [5]

3.2 Steady state AC analysis of the 22kW VZIM

As previously was mentioned the VZIM motor is considered as three different motors with the same stator and different rotor bars. Therefore three different motors VZIM1, VZIM2 and VZIM3 are analyzed and the results are compared with analytical calculations obtained from  KLASYS tool. 3.2.1 Steady state AC analysis of the VZIM1 Fig.3.2.1-1 shows the torque-slip characteristic of VZIM1, which compares the steady state AC analysis results calculated by  Flux2D and  KLASYS . It is apparent that the values calculated by  KLASYS  are higher than values obtained using  Flux2D. In lower slips the difference between the values of  KLASYS  and  Flux2D are less than 10% and in slips higher  than 0.3 the difference reaches to around 12%. In addition the breakdown torque in both calculation methods happens around slip equal to 0.65, for the unskewed rotor cage. The calculation results for the skewed rotor with one stator slot pitch obtained by  KLASYS  shows that the breakdown slip is close to one as depicted in Fig.3.2.1.2. As the rotor is equivalently skewed by shifting the sub-cages in the circumferential direction with respect to each other   by half a stator slot pitch, , which is equivalent to one sator slot  pitch skewing, therefore in the design phase, the optimization has been done for the skewed rotor case to achieve the maximum starting torque at stand still.

Ƚ௦௞௘௪ ൌ ͵͸ͲȀ͵͸Ȁʹ ൌ ͷι - 21 -

The stator phase current-slip characteristics of VZIM1 which are prepared by  Flux2D and  KLASYS  are depicted in Fig.3.2.1-3. Maximum deviation between results of the  KLASYS  and  Flux2D is less than 4%, which shows a good precision.

ι

In Fig.3.2.1-4 the normal component of the air-gap flux density of VZIM1 at a slip of 0.0253 for one pole pair or 180 mechanical degrees has been depicted.

ିଵ

Fig.3.2.1-5 shows the numerically calculated flux lines in VZIM1 by 2 pole pairs at a slip of  0.0253 or 1462.05 , obtained by Flux2D.

Fig.3.2.1-1: Torque-slip characteristics of unskewed VZIM1, comparison of the steady st ate calculation results of  Flux2D  Flux2D and the analytical calculations by KLASYS 

- 22 -

Fig.3.2.1-2: Torque-slip characteristics of VZIM1, comparison of skewed and unskewed rotor, obtained by KLASYS 

Fig.3.2.1-3: Phase current-slip current-slip characteristic of unskewed VZIM1 , comparison of the steady state calculation results of  Flux2D  Flux2D and the analytical results of  KLASYS   KLASYS 

- 23 -

Fig.3.2.1-4: Normal component of air gap flux density in the center of the air gap, for one  pole pair of VZIM1 at a slip of 0.0253, 0.0253, calculated by  Flux2D

- 24 -

Fig.3.2.1-5: Numerically calculated calculated flux lines in VZIM1 at a slip of 0.0253, 0.0253, obtained by  Flux2D

- 25 -

3.2.2 Steady state AC analysis of the VZIM2

Fig.3.2.2-1 shows the torque-slip characteristic of VZIM2, which compares the steady state AC analysis results calculated by  Flux2D and  KLASYS . It is apparent that the values calculated by KLASYS are higher than values obtained using  Flux2D. The difference between values of  KLASYS  and  Flux2D are around 15%. In addition the breakdown torque in both calculation methods happens at around a sli p of 0.3. In Fig.3.2.2-2 stator phase current-slip characteristics of VZIM2 which are calculated by  Flux2D and  KLASYS  are shown. In lower slips the difference of the values in  KLASYS  and  Flux2D is around 12%, but in higher values of slips the difference is lower than 4%.

ι

In Fig.3.2.2-3 the normal component of air-gap flux density of VZIM2 at a slip of 0.0294 for  one pole pair or 180 mechanical degrees has been depicted.

ିଵ

Fig.3.2.2-4 shows the numerically calculated flux lines in VZIM2 at a slip of 0.0294 or  1455.8 , obtained by Flux2D.

Fig.3.2.2-1: Torque-slip characteristics of unskewed VZIM2, comparison of the steady state calculation results of  Flux2D  Flux2D and the analytical calculations by KLASYS 

- 26 -

Fig.3.2.2-2: Phase current-slip characteristic of unskewed VZIM2 , comparison of the steady state calculation results of  Flux2D  Flux2D and the analytical results of  KLASYS   KLASYS 

Fig.3.2.2-3: Normal component of air gap flux density in the center of the air gap, for one pole

 pair of VZIM2 at a slip of 0.0253, 0.0253, calculated by  Flux2D

- 27 -

Fig.3.2.2-4: Numerically calculated flux lines in VZIM2 at a slip of 0.0253, obtained by

 Flux2D

- 28 -

3.2.3 Steady state AC analysis of the VZIM3

Fig.3.2.3-1 shows the torque-slip characteristic of VZIM3, which compares the steady state AC analysis results calculated by  Flux2D and  KLASYS . It is apparent that the values calculated by KLASYS are higher than values obtained using  Flux2D. The difference between values of  KLASYS  and  Flux2D are around 12%. In addition the breakdown torque in both calculation methods happens at around a sli p of 0.2. In Fig.3.2.3-2 stator phase current-slip characteristics of VZIM3 which are prepared by  Flux2D and  KLASYS  are shown. In lower slips the difference of the values in  KLASYS  and  Flux2D is around 11%, but in higher values of slips the difference is lower than 3%.

ι

In Fig.3.2.3-3 the normal component of the air gap flux density of VZIM3 at a slip of 0.0169 for one pole pair or 180 mechanical degrees has been depicted.

ିଵ

Fig.3.2.3-4 shows the numerically calculated flux lines in VZIM3 at a slip of 0.0169 or  1474.65 , obtained by Flux2D.

Fig.3.2.3-1: Torque-slip characteristics of unskewed VZIM3, comparison of the steady state calculation results of  Flux2D  Flux2D and the analytical calculations by KLASYS 

- 29 -

Fig.3.2.3-2: Phase current-slip characteristic of unskewed VZIM3 , comparison of the steady state calculation results of  Flux2D  Flux2D and the analytical results of  KLASYS   KLASYS 

Fig.3.2.3-3: Normal component of air gap gap flux density in the center of the air gap, for one pole

 pair of VZIM3 at a slip of 0.0253, 0.0253, calculated by  Flux2D

- 30 -

Fig.3.2.3-4: Numerically calculated flux lines in VZIM3 at a slip of 0.0253, obtained by

 Flux2D

- 31 -

Chapter 4: Time stepping analysis

4.1 Introduction

Despite the fact that stator voltage and current are sinusoidal, spatial distribution of flux density in the air gap is not sinusoidal but step-like, as winding is located in slots. In steady state analysis only the fundamental sine wave of air gap flux density distribution was considered which gives a rough estimation of motor characteristics, but more precise and accurate results are obtained by consideration of higher space harmonics of the stator and rotor field distribution. 4.2 Stator and rotor field spatial harmonics

The step-like air gap flux density distribution generated by stator,

ఋǡ௦ሺ ௦ǡ ሻ

along stator 

௦ ஶ ఋǡ௦ሺ ௦ǡ ሻ ൌ௩ୀଵǡହି෍ǡ଻ǡǥ ఋǡ௩ ή  ቆ ௣ ௦ െ ௦ ቇ                          ሺͶǤʹെͳሻ ൌ ͳ ൅ ʹ ௦ ൌ ͳ ൅ ͸ ൌ ͳǡͳǡ െͷǡ͹ǡ െͳͳǡെͳͳǡͳ͵ǡെͳ͹ǡെͳ͹ǡͳͻǡͳͻǡെʹ͵ǡെʹ͵ǡʹͷǡͷǡ െʹͻǡെʹͻǡ͵ͳǡ    ሺͶǤͶǤʹ െ ʹሻ

circumference co-ordinate

ǣ

is represented by Fourier series [2]

ǣ ൌ ͲǡͲǡ േͳേͳǡǡ േʹǡേʹǡ േ͵ǡേ͵ǡ ǥ

Furthermore the step-like air gap flux density distribution generated by rotor 

ఋǡ௥ ሺ ௥ǡ ሻ

along

௥ ஶ ఋǡ௦ሺ ௦ǡ ሻ ൌఓୀଵ෍ ఋǡఓ ή  ቆ ௣ ௥ െ ௥ ቇ                               ሺͶǤʹെ͵ሻ ൌ ͳ ൅ሺ ௥Τ ሻ ή                                                 ሺͶǤʹ െ Ͷሻ

rotor circumference co-ordinate

ǣ

is represented by Fourier series [2]

ǣ ൌ ͲǡͲǡ േͳേͳǡǡ േʹǡേʹǡ േ͵േ͵ǡǡ ǥ .

4.3 Asynchronous harmonic torques

௥௩

Rotor harmonic currents produce not only additional cage losses, but also due to Lorentz forces with stator field harmonic additional torque, which is called asynchronous harmonic torque. For the special case this is the asynchronous torque of   KLOSS 

ఋ௦௩

ൌ ͳ

- 32 -

function. The stator harmonic field induces the rotor harmonic current and the rotor harmonic current produces torque with the stator harmonic field.

௩

௩௩ ൌ൏ ͲͲ ௩൐Ͳ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ௥൅ ௥௛௩ሻ                                          ሺͶǤ͵ െ ͳሻ ௥ఙ௩ ൅ ௥௛௩ ௥

Asynchronous harmonic torque follows a  KLOSS function, depending on harmonic slip , at this torque is zero. At this torque is positive and adds to fundamental torque. At the torque is negative and is breaking the machine. At harmonic break down slip:

torque reaches positive and negative maximum values. As down slip is small.

harmonic break 

Fig.4.3-1: Asynchronous Asynchronous harmonic torques of the 5 th and 7th stator field harmonics, which are superimposed on fundamental asynchronous torque [2]

4.4 Synchronous harmonic torques Rotor field harmonic of step like air gap flux density distribution, excited by rotor  fundamental current , Will also produce parasitic torque with the stator field harmonics. Like asynchronous harmonic torque, the condition for constant torque generation is: -the same wave length -the same velocity (means also same direction of movement) of the stator and rotor field wave. Therefore conditions for slip where synchronous harmonic torque occurs are as following:

௥

ൌ ൌ ͳ

- 33 -

ൌ െ ൌ െ൅ ͳͳǤ

Fig.4.4-1: Typical effects of synchronous and asynchronous harmonic torques in induction machines [2]

Time stepping simulation of the motors is done in the slip range from 0 to 2. In between of  these 2 operating points some important slips corresponding to synchronous and asynchronous torques are taken into account. Asynchronous and synchronous harmonic torques happen in slips that have been calculated in Appendix II and they are considered in the simulation of the motors (VZIM1, VZIM2, and VZIM3). Fig.4.4-1 shows typical effects of synchronous and asynchronous harmonic torques in induction machines. 4.5 Derived slips used for time stepping analysis

The sets of slips to perform the time stepping analysis of VZIM1, VZIM2 and VZIM3 motors have been derived and are given in Table V, Table VI and Table VII in Appendix II. In these sets of slips the asynchronous and synchronous harmonic torque slips and breakdown slips of  asynchronous harmonic torques are included.

- 34 -

4.6 Time-stepping analysis of induction motors

Time-stepping analysis of the VZIM motor has been done to observe the transients and harmonic effects on the characteristics of the motor. When the time step is in the order of  1/32th of the electrical cycle, fairly accurate and good results are provided. In our analysis the motor is line fed and the frequency is 50Hz, and the corresponding electrical cycle is 1/50=0.02s; therefore with consideration of 40 time steps per one cycle, would be an acceptable time step. Time stepping analysis has been done for some speeds of the motor, in the range of -1500rpm to 1500rpm corresponding to the slips given in Table II, Table III and Table IV in Appendix II. In this analysis the friction and windage losses and load inertia are neglected.

ο ൌ ͲǤͲʹΤ ͶͲൌ

ͷ ିସ

4.6.1 Time-stepping analysis of VZIM1

Time stepping analysis of the VZIM1 motor has been done in different speeds which are given in Table V in Appendix II. The results consist of the output torque, input phase current, input power, efficiency, and power factor which have been compared with the results of the ,  KLASYS  tool. The synchronous harmonic torques which happen at slips, and are calculated with  Flux2D as explained in the section 4.8 of this chapter. are

ൌ ͲǤͺͷ͹ ൌ ͳ ൌ ͲǤͺͷ͹

ൌ ͳǤͲ͹ͳ ൌ ͳ ൌ ͳǤͲ͹ͳ ൌ ͲǤͺͷ͹ ൌ ͳ ൌ ͳǤͲ͹ͳ ൌ ͳ ൌ ͳǤͲ͹ͳ

In Fig.4.6.1-1 the torque slip characteristic of VZIM1, derived from time stepping analysis with  Flux2D, is shown. The effects of the synchronous harmonic torques at slips , and are clearly observed. In Fig.4.6.1-2 the torque slip characteristic of  VZIM1, derived from  KLASYS , is shown. The effects of the synchronous harmonic torques at slips , and and are clearly observed. are

ൌ ͲǤͺͷ͹

The values of synchronous torque calculated by  Flux2D and KLASYS  at slips , and , are compared in Table 4.8-6, Table 4.8-7 and Table 4.8-8. It is noticeable that the synchronous torque values calculated by  KLASYS  are in most of the cases much higher than the values calculated by  Flux2D. The values of  Flux2D  Flux2D are more reasonable and acceptable.

- 35 -

ൌ ͲǤͺͷ͹ ൌ ͳ ൌ ͳǤͲ͹ͳ

Fig.4.6.1-1: Torque-slip characteristic of unskewed VZIM1, calculated with Flux2D time-stepping, the

effects of the synchronous harmonic torques at slips the motor are observable

,

and and

of the of

ൌ ͲǤͺͷ͹ ൌ ͳ ൌ ͳǤͲ͹ͳ

Fig.4.6.1-2: Torque-slip characteristic of unskewed VZIM1, calculated with KLASYS , the effects of 

the synchronous harmonic torques at slips are observable

,

and and

of the the motor  of

In Fig.4.6.1-3 the torque slip characteristics calculated by time stepping and steady state AC analysis are compared. The effects of the field harmonics lead to considerable deviations in the calculated values. In Fig.4.6.1-4 the fundamental phase current prepared by  Flux2D and  KLASYS are compared. The results match very well and maximum difference is less than 5.5%. - 36 -

Fig.4.6.1-5 shows the input power-slip characteristic of VZIM1 obtained from  Flux2D and  KLASYS . At lower slips the calculated results of  KLASYS   KLASYS are lower than  Flux2D and at higher  slips results of  KLASYS   KLASYS are higher. The values match well and the maximum difference is less than 5%. In addition Fig.4.6.1-6 shows the output power-slip characteristic of VZIM1 obtained from  Flux2D and KLASYS. Fig.4.6.1-7 shows the efficiency calculated by  Flux2D and  KLASYS . Maximum efficiency achieved from  Flux2D results is 91.41% at a slip of 0.3 and maximum efficiency calculated  by KLASYS is 90.5% at the slip equal to 0.3. The results match very well. Fig.4.6.1-8 compares the power factor values obtained from  Flux2D and  KLASYS . The maximum power factor calculated by  KLASYS  equals 0.924 which at slip equal to 0.12 happens, and in  Flux2D maximum power factor is equal to 0.962 which at slip equal to 0.1 happens, which is higher than  KLASYS maximum power factor.

Fig.4.6.1-3: Torque-slip characteristic of unskewed VZIM1, time-stepping and AC analysis results obtained by Flux2D, the harmonic torque effects at lower speeds of the motor  are obvious

- 37 -

Fig.4.6.1-4: Stator phase current of unskewed VZIM1, time steppi ng analysis results calculated by  Flux2D and KLASYS 

Fig.4.6.1-5: Input power-slip of unskewed VZIM1, comparison of t he analysis results calculated by  Flux2D and KLASYS 

- 38 -

Fig.4.6.1-6: Output power-slip of unskewed V ZIM1, comparison of the analysis results calculated by  Flux2D and KLASYS 

Fig.4.6.1-7: Efficiency-slip of unskewed VZIM1, comparison of the analysis results calculated by  Flux2D and KLASYS 

- 39 -

Fig.4.6.1-8: Power factor-slip of unskewed VZIM1, comparison of the analysis results calculated by  Flux2D and KLASYS 

4.6.2 Time-stepping analysis of VZIM2

Similar to VZIM1, time-stepping analysis of VZIM2 has been done with  Flux2D and results have been compared with results of the  KLASYS  tool. In this analysis the considered slips similarly vary from 0 to 2 and in between additional 50 slips have been considered for  calculations, which have been given in table VI in Appendix II. The synchronous harmonic torques which happen at slips, , and and are calculated with  Flux2D as are explained in the section 4.8 of this chapter.

ൌ ͲǤͺͷ͹ ൌ ͳ ൌ ͳǤͲ͹ͳ

ൌ ͲǤͺͷ͹

In Fig.4.6.2-1 the torque slip characteristic of VZIM2, derived from time stepping analysis with  Flux2D, is shown. The effects of the synchronous harmonic torques at slips , and are clearly observed. In Fig.4.6.2-2 the torque slip characteristic of  VZIM2, derived from  KLASYS , has been shown. The effects of the synchronous harmonic torques at slips , and and are clearly observed. are

ൌ ͳ ൌ ͳǤͲ͹ͳ ൌ ͲǤͺͷ͹ ൌ ͳ ൌ ͳǤͲ͹ͳ ൌ ͳ ൌ ͳǤͲ͹ͳ

ൌ ͲǤͺͷ͹

The values of synchronous torque calculated by  Flux2D and KLASYS  at slips , and , are compared in Table 4.8-9, Table 4.8-10 and Table 4.8-11. It is noticeable that the synchronous torque values calculated by  KLASYS  are in most of the cases much higher than the values calculated by  Flux2D. The values of  Flux2D  Flux2D are more reasonable and acceptable.

- 40 -

Fig.4.6.2-1: Torque-slip characteristic of unskewed VZIM2, obtained by KLASYS and Flux2D timestepping calculations, the harmonic torque effects at lower speeds of the motor are observable

ൌ ͲǤͺͷ͹ ൌ ͳ ൌ ͳǤͲ͹ͳ

Fig.4.6.2-2: Torque-slip characteristic of unskewed VZIM2, calculated with KLASYS , the effects of 

the synchronous harmonic torques at slips are observable

,

and and

of the the motor  of

To observe the effect of the harmonics, the torque-slip characteristic of VZIM2 prepared from time-stepping and steady state AC analysis have been depicted in Fig.4.6.2-3. A considerable

- 41 -

difference happens at the slip 0.857 and 1.071 due to the synchronous harmonics at these  points. Fig.4.6.2-4 depicts the comparison between fundamental stator phase current calculated by  KLASY  and  Flux2D. The difference between results is below 10% which mostly happens at high values of slip ( ).

൐ͳ

Fig.4.6.2-5 shows the input power-slip characteristic of VZIM2, calculated by  KLASYS  and  Flux2D. At higher slips the values of  Flux2D are around 11% lower than values of  KLASYS . In addition Fig.4.6.2-6 shows the output power-slip characteristic of VZIM2 obtained from  Flux2D and KLASYS  Fig.4.6.2-7 depicts the efficiency-slip characteristic of VZIM2, prepared by  KLASYS  and  Flux2D. The efficiency calculated by  Flux2D at rated speed of VZIM2 (1455.8 rpm) is equal to 0.9104 is higher than that of  KLASYS   KLASYS which is equal to 0.905. Fig.4.6.2-8 depicts the power factor-slip characteristic of VZIM2, prepared by  KLASYS  and  Flux2D.

Fig.4.6.2-3: Torque-slip characteristic of unskewed VZIM2, time-stepping and AC analysis results obtained by Flux2D, the harmonic torque effects at lower speeds of the motor  are obvious

- 42 -

Fig.4.6.2-4: Stator phase current of unskewed VZIM2, comparison of the time stepping analysis results calculated by Flux2D and KLASYS 

Fig.4.6.2-5: Input power-slip of unskewed VZIM2, comparison of t he analysis results calculated by  Flux2D and KLASYS 

- 43 -

Fig.4.6.2-6: Output power-slip of unskewed V ZIM2, comparison of the analysis results calculated by  Flux2D and KLASYS 

Fig.4.6.2-7: Efficiency-slip of unskewed VZIM2, comparison of the analysis results calculated by  Flux2D and KLASYS 

- 44 -

Fig.4.6.2-8: Power factor-slip of unskewed VZIM2, comparison of the analysis results calculated by  Flux2D and KLASYS 

4.6.3 Time-stepping analysis of VZIM3

Similar to VZIM1 and VZIM2, time-stepping analysis of VZIM3 has been done with  Flux2D and results have been compared with the results of the  KLASYS  tool. In this analysis the considered slips similarly vary from 0 to 2 and in between additional 50 slips have been considered for calculations, which are given in table VII in Appendix II. The synchronous harmonic torques which happen at slips, , and and are calculated with are  Flux2D as explained in the section 4.8 of this chapter.

ൌ ͲǤͺͷ͹ ൌ ͳ ൌ ͳǤͲ͹ͳ

ൌ ͲǤͺͷ͹

In Fig.4.6.3-1 the torque slip characteristic of VZIM3, derived from time stepping analysis with  Flux2D, is shown. The effects of the synchronous harmonic torques at slips , and are clearly observed. In Fig.4.6.3-2 the torque slip characteristic of  VZIM3, derived from  KLASYS , is shown. The effects of the synchronous harmonic torques at slips , and and are clearly observed. are

ൌ ͳ ൌ ͳǤͲ͹ͳ ൌ ͲǤͺͷ͹ ൌ ͳ ൌ ͳǤͲ͹ͳ ൌ ͳ ൌ ͳǤͲ͹ͳ

ൌ ͲǤͺͷ͹

The values of synchronous torque calculated by  Flux2D and KLASYS  at slips , and , are compared in Table 4.8-12, Table 4.8-13 and Table 4.8-14. It is noticeable that the synchronous torque values calculated by  KLASYS  are in most of the cases much higher than the values calculated by  Flux2D. The values of  Flux2D  Flux2D are more reasonable and acceptable. To observe the effect of the t he harmonics, the torque-slip characteristic of VZIM3 prepared from time-stepping and steady state AC analysis have been depicted in Fig.4.6.2-2. A considerable difference happens at the slip equal to 0.857 and 1.071 due to the synchronous harmonics at these points.

Fig.4.6.3.3 shows the comparison between fundamental phase current prepared by  KLASYS  and  Flux2D. The difference between the results is below 5% which mostly happens at high slip values ( ).

൐ͳ

- 45 -

ൌ ͲǤͺͷ͹ ൌ ͳ ൌ ͳǤͲ͹ͳ

Fig.4.6.3-1: Torque-slip characteristic of unskewed VZIM3, calculated with Flux2D time-stepping, the

effects of the synchronous harmonic torques at slips the motor are observable

,

and and

of the of

ൌ ͲǤͺͷ͹ ൌ ͳ ൌ ͳǤͲ͹ͳ

Fig.4.6.3-2: Torque-slip characteristic of unskewed VZIM3, calculated with KLASYS , the effects of 

the synchronous harmonic torques at slips are observable

,

- 46 -

and and

of the the motor  of

Fig.4.6.3-3: Torque-slip characteristic of unskewed VZIM3, time-stepping and AC analysis results obtained by Flux2D, the harmonic torque effects at lower speeds of the motor  are obvious

Fig.4.6.3-4: Stator phase current of unskewed VZIM2, comparison of the analysis results calculated by  Flux2D and KLASYS 

Fig.4.6.3-4 depicts the input power-slip characteristic of VZIM3, prepared by  KLASYS  and  Flux2D. At higher slips the values of  Flux2D are around 4% lower than values of  KLASYS . In addition Fig.4.6.3-5 shows the output power-slip characteristic of VZIM3 obtained from  Flux2D and KLASYS.

- 47 -

Fig.4.6.3-5: Input power-slip of unskewed VZIM3, comparison of t he analysis results calculated by  Flux2D and KLASYS 

Fig.4.6.3-6: Output power-slip of unskewed V ZIM3, comparison of the analysis results calculated by  Flux2D and KLASYS 

Fig.4.10.3-6 depicts the efficiency-slip characteristic of VZIM3, prepared by  KLASYS  and  Flux2D. The efficiency calculated by  Flux2D at rated speed of VZIM3 (1474.65 rpm) is equal to 0.912 is lower than that of  KLASYS   KLASYS which is equal to 0.925.

- 48 -

Fig.4.6.3-7: Efficiency-slip of unskewed VZIM3, comparison of the analysis results calculated by  Flux2D and KLASYS 

Fig.4.6.3-7 depicts the input power factor-slip characteristic of VZIM3, prepared by  KLASYS  and  Flux2D.The maximum power factor is equal to 0.956 which happens at a slip equal to 0.025 in  Flux2D calculated results and in case of  KLASYS the maximum power factor is equal to 0.937 which happens at a slip equal to 0.03.

Fig.4.6.3-8: Power factor of unskewed VZIM3, comparison of the analysis results calcul ated by  Flux2D and KLASYS 

- 49 -

4.7 Instantaneous torque wave forms at different rotor speeds

In order to draw the torque-slip characteristic, the average value of the torque over one period of the stator electrical cycle has been considered. In order to observe the oscillations and ripple of the torque over time in some sample operating speeds of the motors, the torque-time characteristics are shown in Fig.4.7-1 to Fig.4.7-7. From these figures the influence of the harmonics on the torque characteristics is clearly observable, the lower the speed, the higher the influence of the harmonics and therefore the ripple on the torque-time characteristics increases. At the speed equal to 1462 in Fig.4.7-1 the average torque is lower than the average torque at the speed equal to 1200 in Fig.4.7-4 and the ripple has also lower amplitude. At the speed equal to 205.5 rpm in Fig.4.7-7 the influence of  the synchronous harmonic torques is quite clear. This operating point where the peak of the th 7 asynchronous harmonic torque occurs, is quite near the slip where the synchronous harmonic torque happens. Therefore the result is a quite oscillating torque-time characteristic.

ൌ ͲǤͺͷ͹

Fig.4.7-1: Torque-time characteristic of unskewed VZIM2 at 1462rpm, calculated by Flux2D, torque oscillates around a constant mean value after reaching the steady state

- 50 -

Fig.4.7-2: Torque-time characteristic of unskewed VZIM2 at 1462rpm, calculated by Flux2D, zoomed view after steady state is reached

Fig.4.7-3: Torque-time characteristic characteristic of unskewed VZIM2 at 1200rpm, 1200rpm, calculated by Flux2D, torque oscillates around a constant mean value after reaching the steady state

- 51 -

Fig.4.7-4: Torque-time characteristic of unskewed VZIM2 at 1200rpm, calculated by Flux2D, zoomed view after a steady state is reached

Fig.4.7-5: Torque-time characteristic of unskewed VZIM2 at 525rpm, calculated by Flux2D, torque oscillates around a constant mean value

- 52 -

Fig.4.7-6: Torque-time characteristic of unskewed VZIM2 at 525rpm, calculated by Flux2D, zoomed view after a steady state is reached

Fig.4.7-7: Torque-time of unskewed VZIM2 at 205.5rpm, calculated by Flux2D, where break down slip of the 7th asynchronous torque harmonic happens at slip 0.863 or 205.5rpm

- 53 -

4.8 Synchronous harmonic torque assessment in VZIM

ͳǤͲ͹ͳ

ൌ ͲǤͺͷ͹ ൌ ͳ ൌ

Synchronous harmonic torques of VZIM, happen at the slips, , and . In this section the torque time characteristics of VZIM1, VZIM2 and VZIM3 at these slips are studied. For VZIM2 the synchronous harmonic torques are studied and derived by varying the rotor position with respect to a fixed stator position, for a rotor slot pitch (we call this method the rotation method). The step for each position is taken as one tenth of a rotor  slot pitch i.e. = 360/28/10 = 1.2857 mechanical degrees. The average torque for a stator   period is calculated for each of the eleven steps. Fig.4.8.1 shows first and second positions of  the rotor bars with fixed stator position.

Fig.4.8-1: First and second positions of the rotor bars with fixed stator position to calculate the

synchronous harmonic torques, prepared by  Flux2D

ൌ ͲǤͺͷ͹ ൌ ͳ ൌ ͳǤͲ͹ͳ

Fig.4.8.2 to Fig.4.8.4 show the variation of the torque for the rotor position steps 1, 5 and 7 of  VZIM2 at the slip respectively. It is clear that the average value of the synchronous torque at each step is a constant value. Similar procedure has been done for  VZIM2 at slips and and , and the torque time values have been calculated for  eleven positions. The calculated synchronous torques, for three different slips of VIM2 are shown in Fig.4.8-5 to Fig.4.8-7.

Fig.4.8-2: Torque-time of unskewed VZIM2 at 1

ൌ ͲǤͺͷ͹

, calculated by Flux2D, rotor position step

- 54 -

Fig.4.8-3: Torque-time of unskewed VZIM2 at 5

Fig.4.8-4: Torque-time of unskewed VZIM2 at 7

ൌ ͲǤͺͷ͹

, calculated by Flux2D, rotor position step

ൌ ͲǤͺͷ͹

, calculated by Flux2D, rotor position step

- 55 -

500 400 300   m  .    N    /   e200   u   q   r   o    T 100 0 0

20

40

-100

60

80

10 0

ൌ ͲǤͺͷ͹

Rotor position at percentage of rotor slot pitch

Fig.4.8-5: Variation of synchronous torque at

as a function of rotor position as

580 560 540   m  .    N    / 520   e   u   q500   r   o    T480 460 440 0

20

40

60

80

100

Rotor position at percentage of rotor slot pitch Fig.4.8-6: Variation of synchronous torque at

ൌ ͳ

a function of rotor position

350 330   m  . 310    N    /   e   u   q   r 290   o    T

270 250 0

20

40

60

80

100

Rotor position at percentage of rotor slot pitch Fig.4.8-7: Variation of synchronous torque at

ൌ ͳǤͲ͹ͳ

as a function of rotor position as

- 56 -

Furthermore the synchronous torque of the VZIM2 has been calculated with another method (we call this method the oscillation method). The torque time characteristics of VZIM2 have  been calculated in three different speeds and each speed is 10 rpm higher than the speed where synchronous harmonic torque happens. For VZIM2 the synchronous harmonic torques happen at 214.28 rpm ( ), 0 rpm ( ), ) and -107.1428 ( ). Therefore the ). torque time characteristic of the VZIM2 at speeds 224.28 rpm, 10 rpm and -97.1428 are calculated. As the oscillation of the torque time characteristic occurs because of the synchronous harmonic torque; therefore the maximum and minimum of the variation of the average torque in the torque time characteristic, equals to the maximum and minimum values of the synchronous harmonic torques.

ൌ ͲǤͺͷ͹ ൌ ͳ

ൌ ͳǤͲ͹ͳ

Fig.4.8.8 to Fig.4.8.10 show the torque time characteristic of VZIM2 at three different speeds 224.28 rpm, 10 rpm and -97.1428. At speed 224.28 rpm the maximum value of the average torque is 438.4 Nm and the minimum value is -20.88 Nm and the peak to peak value is 459.28Nm. On the other hand the maximum value of s ynchronous harmonic torque calculated  by changing the position of the rotor related to fixed stator method is 448.24 Nm and the minimum value is -20.8 Nm with the peak to peak value equal to 469.04 Nm. The difference  between two methods of synchronous torque calculation of peak to peak value is around 2.1% at speed 224.28 rpm. The values are summarized in Table 4.8-1. The torque values for speeds 10 rpm and -97.1428 rpm are calculated in similar way and are shown in Table 4.8-2 and Table 4.8-3. Clearly the calculated values with two different methods (rotation and oscillation methods) have a maximum difference of 7.66%. Due to the reasonable differences of the results of two calculation methods for VZIM2, the calculation of the maximum and minimum and peak to peak values of synchronous torques at the slips where synchronous torques of VZIM1 and VZIM2 happen are done by the oscillation method due to the low calculation time of this method. Table 4.8-4 and Table 4.8-5 show the values of synchronous torques for VZIM1 and VZIM3, calculated by oscillation method in three different slips, where synchronous harmonic torques happen ( , and ). ).

ൌ ͲǤͺͷ͹ ൌ

ͳ ൌ ͳǤͲ͹ͳ

ൌ ͲǤͺͷ͹

Table 4.8-1: Maximum, minimum and peak to p eak values of synchronous torque for VZIM2 in two different methods of calculation (rotation and oscillation methods) for the slip Maximum torque (N.m) Minimum torque(N.m torque(N.m)) Peak to peak value (N.m)

Oscillation method

Rotation method

Difference %

438.4 -20.88 459.28

448.24 -20.8 469.04

2.19% 0.3% 2.08%

ൌ ͳ

Table 4.8-2: Maximum, minimum and peak to p eak values of synchronous torque for VZIM2 in two different methods of calculation (rotation and oscillation methods) for the slip Maximum torque (N.m) Minimum torque(N.m torque(N.m)) Peak to peak value (N.m)

Oscillation method

Rotation method

Difference %

313.32 244.56 68.76

326.84 262.16 64.68

4.1% 6.7% 5.9%

- 57 -

ൌ ͳǤͲ͹ͳ

Table 4.8-3: Maximum, minimum and peak to peak values of synchronous torque for VZIM2 in two different methods of calculation (rotation and oscillation methods) for the slip Maximum torque (N.m) Minimum torque(N.m torque(N.m)) Peak to peak value (N.m)

Oscillation method

Rotation method

Difference %

574.2 470.88 103.32

556.8 461.4 95.4

3.03% 2.01% 7.66%

Table 4.8-4: Maximum, minimum and peak to p eak values of synchronous torque for VZIM1 (oscillation method) Maximum torque (N.m) Minimum torque(N.m torque(N.m)) Peak to peak value (N.m)

ൌ ૙Ǥ૙Ǥ ૡ૞ૠૡ૞ૠ 605.56 130.8 474.76

ൌ ૚

456.24 406.28 49.96

ൌ ૚Ǥ૚Ǥ ૙ૠ૚૙ૠ૚

629.72 499.8 129.92

Table 4.8-5: Maximum, minimum and peak to p eak values of synchronous torque for VZIM3 (oscillation method) Maximum torque (N.m) Minimum torque(N.m torque(N.m)) Peak to peak value (N.m)

ൌ ૙Ǥ૙Ǥ ૡ૞ૠૡ૞ૠ 370 -42.44 412.44

ൌ ૚

300.88 185.68 114.32

Fig.4.8-8: Torque-time of unskewed VZIM2 at speed 224.28 rpm, calculated by Flux2D

- 58 -

ൌ ૚Ǥ૚Ǥ ૙ૠ૚૙ૠ૚

458.32 351.72 107.6

Fig.4.8-9: Torque-time of unskewed VZIM2 at speed 10 rpm, calculated by Flux2D

Fig.4.8-10: Torque-time of unskewed VZIM2 at speed -97.14 rpm, calcul ated by Flux2D

In addition, the values of synchronous harmonic torques for VZIM1, VZIM2 and VZIM3 in three different slips, where synchronous harmonic torques happen are calculated by  KLASYS . In Table 4.8-6 to Table 4.8-14 the calculated values with  KLASYS and  Flux2D are compared. It is noticeable that the synchronous torque values calculated by  KLASYS  are in the most of  the cases much higher than the values calculated by  Flux2D.The values of  Flux2D are more reasonable and acceptable.

- 59 -

ൌ ͲǤͺͷ͹

Table 4.8-6: Maximum, minimum and peak to p eak values of synchronous torque for VZIM1, at t he slip , the comparison of the calculated values by KLASYS and Flux2D

Flux2D  KLASYS  Difference

ൌ ͳ

Maximum torque (N.m)

Minimum torque (N.m)

Peak to peak value (N.m)

606.56 1165.215 558.655

130.8 -208.695 338.659

474.76 1373.91 899.15

Table 4.8-7: Maximum, minimum and peak to p eak values of synchronous torque for VZIM1, at t he slip , the comparison of the calculated values by KLASYS and Flux2D

Flux2D  KLASYS  Difference

ൌ ͳǤͲ͹ͳ

Maximum torque (N.m)

Minimum torque (N.m)

Peak to peak value (N.m)

456.24 526.085 69.845

406.28 473.915 67.635

49.96 52.17 2.21

Table 4.8-8: Maximum, minimum and peak to p eak values of synchronous torque for VZIM1, at t he slip , the comparison of the calculated values by KLASYS and Flux2D

Flux2D  KLASYS  Difference

ൌ ͲǤͺͷ͹

Maximum torque (N.m)

Minimum torque (N.m)

Peak to peak value (N.m)

629.72 899.99 270.27

499.8 230.427 269.373

129.92 669.564 539.644

Table 4.8-9: Maximum, minimum and peak to p eak values of synchronous torque for VZIM2, at t he slip , the comparison of the calculated values by KLASYS and Flux2D

Flux2D  KLASYS  Difference

ൌ ͳ

Maximum torque (N.m)

Minimum torque (N.m)

Peak to peak value (N.m)

438.4 1083.33 644.93

-20.88 -316.67 296.67

459.28 1400 940.72

Table 4.8-10: Maximum, minimum and peak to peak values of synchronous torque for VZIM2, at the slip , the comparison of the calculated values by KLASYS and Flux2D

Flux2D  KLASYS  Difference

ൌ ͳǤͲ͹ͳ

Maximum torque (N.m)

Minimum torque (N.m)

Peak to peak value (N.m)

313.32 474.995 161.675

244.56 341.665 70.105

68.76 133.33 64.57

Table 4.8-11: Maximum, minimum and peak to peak values of synchronous torque for VZIM2, at the slip , the comparison of the calculated values by KLASYS and Flux2D

Flux2D  KLASYS  Difference

Maximum torque (N.m)

Minimum torque (N.m)

Peak to peak value (N.m)

574.2 624.995 50.795

470.88 291.665 179.215

103.32 333.33 230.01

- 60 -

ൌ ͲǤͺͷ͹

Table 4.8-12: Maximum, minimum and peak to peak values of synchronous torque for VZIM3, at the slip , the comparison of the calculated values by KLASYS and Flux2D

Flux2D  KLASYS  Difference

ൌ ͳ

Maximum torque (N.m)

Minimum torque (N.m)

Peak to peak value (N.m)

370 907.13 537.13

-42.44 -464.29 421.85

412.44 1371.42 958.98

Table 4.8-13: Maximum, minimum and peak to peak values of synchronous torque for VZIM3, at the slip , the comparison of the calculated values by KLASYS and Flux2D

Flux2D  KLASYS  Difference

ൌ ͳǤͲ͹ͳ

Maximum torque (N.m)

Minimum torque (N.m)

Peak to peak value (N.m)

300.88 358.925 58.045

185.68 198.215 12.535

114.32 160.71 46.39

Table 4.8-14: Maximum, minimum and peak to peak values of synchronous torque for VZIM3, at the slip , the comparison of the calculated values by KLASYS and Flux2D

Flux2D  KLASYS  Difference

Maximum torque (N.m)

Minimum torque (N.m)

Peak to peak value (N.m)

458.32 353.57 104.75

351.72 246.43 105.29

106.6 107.14 0.54

4.9 Power losses in different parts of the motor

In order to determine the efficiency of the motor, it is necessary to calculate the losses in different parts of the motor, these loss components are: -stator and rotor ohmic losses -friction and windage losses -iron losses (mainly in stator iron) -additional no-load losses such as tooth pulsation and surface losses -additional load losses such as stator and rotor eddy current losses in conductors. In simulation by Flux2D the mechanical losses are neglected. Tables I, II and III in Appendix III show the iron losses which include eddy-current losses and hysteresis losses in addition to the stator and rotor ohmic losses, which have been calculated  by Flux2D and compared with the results of  KLASYS   KLASYS tool.

ሺ ሻ ൌ ሺ ሻΤ ଶ ଶ ଷଶ  ሺ ሻ ൌ ௛ ௠ଶ ൅ ͳʹ ൭ ሺ ሻ൱ ൅ ௘ ൭ ሺ ሻ൱     ሺଷሻ              ሺͶǤͻെͳሻ

In  Flux2D regarding the iron losses, the volume density of the instantaneous power loss , is written as [5]:

which is composed of losses by hysteresis, classical losses and losses in excess, where - 61 -

௛௘ ௠

- is the coefficient by hysteresis - is the coefficient of losses in excess - is the conductivity of the material -d is the thickness of the lamination - is the peak value of the magnetic flux density Within the frame of the flux computation,

-

௙௘

ଶ൭

ଶ ሺሻ

ሺ ሻ ൌ ൦௛ ௠ଶ ൅ ͳʹ ൱ ൅

ଷଶ  ௘ ൭ ሺ ሻ൱ ൪ ή ௙௘ ሺଷሻ          ሺͶǤͻെʹሻ

is the stack fill factor (close to 1), this coefficient considers the electrical insulation of the

lamination of the magnetic core.

ሺ ሻ ൌ ሺ ሻΤ ଶ ଶ ଶ ଶ ଷଶ  ሺ ሻ ൌ ௛ ௠ ൅ ͸ ሺ ௠ ሻ ൅ ௘ሺ ௠ ሻ ήͺǤ͸͹ሺଷሻ               ሺͶǤͻെ͵ሻ

In addition, in steady state AC magnetic application the volume density of the average power  is:

where: - is the coefficient by hysteresis - is the coefficient of losses in excess - is the conductivity of the material -d is the thickness of the lamination - f   f is frequency - is the peak value of the magnetic flux density

௛௘

௠

௛

௘

Coefficients , and are necessary to calculate the iron losses with  Flux2D, according to the data sheet of the M270-50A [7] which is the material of the stator and rotor lamination and with consideration of the equal to 1.5T and with solving , in three different frequencies 50Hz, 100Hz and 200Hz the values of  , and are calculated.

௠

௛

ሺ ௦ሻ ൌ ͹ͺͲͲ୫୏୥୏ሺ୥యሻ

ሺͶǤ ͻ െ͵ሻ ௘

The values of the specific total losses in frequencies 50Hz, 100Hz and 200Hz, according to the data sheet of the M270-50A [7] are given in Table 4.9-1. In addition the density of the iron sheet M270-50A is

and the power loss is

ሺ ሻ ൌ ௦ ή

, the calculated

values of the power losses for the frequencies 50Hz, 100Hz and 200Hz and are shown in Table 4.9-1. Table 4.9-1: Values of specific total losses according to the data sheet of the M270-50A [7], and calculated power losses, in frequencies 50Hz, 100Hz and 200Hz

ሺሺ ሻሺ ൗൗ ሻ૜ሻ

ʹǤͳͻ͹͵Ͷͷ͵ 

͸Ǥͷʹʹ͸Ͳ͹

50Hz

100Hz

- 62 -

200Hz

19.45

ͳͷͳ͹ͳͲ

ʹͷ͸ ଶଷ ௛௘ ൌൌ ͲǤͻ͹Ǥ͹ʹ͸͸

after the calculations the following results are obtained: (Ws/( (Ws/( ))

ൌ ʹͲ͵Ͷ͵͹ͷǤͲ͵͵ ଷ ή ሺΤሻଵǤଵǤହ

((1/(Ohm.m)), 1/(Ohm.m)), (W/( (W/( )) When these values are available with the help of the  Flux2D iron losses are calculated. 4.9.1 Losses of VZIM at different speeds

In this section the losses of VZIM1, VZIM2 and VZIM3 in three different speeds which have  been calculated by  Flux2D and  KLASYS  are compared. Here the components of stator and rotor iron and copper losses have been considered. Tables I, II and III in Appendix III show the losses of VZIM1, VZIM2 and VZIM3. Apparently there are differences between losses calculated by  KLASYS  and  Flux2D, at 22kW nominal output power operation. In VZIM2 the stator copper losses are around 14%, the rotor  copper losses about 30% and stator iron losses about 11%, calculated by  Flux2D are higher  than values calculated by  KLASYS. In VZIM3 the stator copper losses are around 18%, the rotor copper losses about 30% and stator iron losses about 21%, calculated by  Flux2D are higher than values calculated by  KLASYS. Fig.4.9.1-1 shows the comparison between losses of VZIM1 at rated speed equal to 1390.9 rpm calculated with  Flux2D and  KLASYS , Fig.4.9.1-2 shows the comparison between losses of VZIM2 at the rated speed equal to 1455.8 rpm calculated with  Flux2D and  KLASYS  and Fig.4.9.1-3 shows the comparison between losses of VZIM3 at the rated speed equal to 1474.65 rpm calculated with  Flux2D and KLASYS.

Fig.4.9.1-1: Comparison between losses of VZIM1 at speed equal to 1390.9 rpm calculated with Flux2D and KLASYS 

- 63 -

Fig.4.9.1-2: Comparison between losses of VZIM2 at speed equal to 1455.8 rpm calculated with Flux2D and KLASYS 

Fig.4.9.1-3: Comparison between losses of VZIM3 at speed equal to 1474.65 rpm calculated with Flux2D and KLASYS 

- 64 -

Chapter 5: Thermal analysis 5.1 Preface

Calculating the temperature rise in electrical machines which occurs due to electrical and mechanical losses, is one of the important steps in design of electrical machines. According to Arrhenius’ law, velocity of chemical decomposition of materials incre ases exponentially with temperature and for solid insulation materials Montsinger’s rule is valid, that the insulation life span  L decreases by 50% with increase of temperature by . Due to the high sensitivity of insulation materials to over-temperature, thermal classes for different types of  insulation materials are defined, which gives the maximum permissible temperature limit in hot spot of insulation. For example in class F maximum temperature rise is 105 for  the machines in the power range of  [1].

ͷ ୒ ൑ ͷ

ͳͲ ο ሺሻ 

Cooling system of electrical machines influences their thermal utilization and with a high efficient cooling system, power per mass of electrical machines can be raised. There are different possibilities to propel the coolant flow in electrical machines. In case of motors without fan, cooling is done due to the natural convection and heat radiation. With shaft mounted fan motors the speed of air flow depends on the velocity of motor. The external fan is another way for cooling the electrical machine. According to the second fundamental law of thermodynamics, the natural heat flow is only  possible from a hot to a cold region. region. Basic principles for heat transfer are -Conduction, by heat conducting materials -Convection, by moving coolants like air or water  -Radiation, which does not need any medium for heat transfer. Calculating the temperature in electrical machines may be done either by numerical methods or equivalent circuits. Calculation by numerical method in this project has been done by  ANSYS tool.

5.2 Temperature rise calculation by thermal equivalent circuits

To have an estimation of temperature rise in the stator winding simplified thermal networks are used.

- 65 -

In a simplified thermal network only copper and iron losses in the stator and thermal resistances between copper and iron due to the slot insulation, heat convection from winding overhang to air and heat convection from the stator iron to air are considered. Stator copper  losses in the winding ( ) and stator iron losses in the stator iron stack ( ) are loss sources. Fig.5.2-1 shows the simplified thermal equivalent network for an induction motor.

஼௨

௘

Fig.5.2-1: The simplified thermal equivalent network for an induction motor 

௧௛ଵ ௧௛ଵ ൌ ͳΤሺ ሻ  ௧௛ଶ ൌ Τ൫ ௜௦ ൯

Heat resistances are, between stator iron and ambient cooling air, given by convection, according to [1] where is the surface of the stator iron housing and is the heat transfer coefficient which describes the cooling effect of flowing coolant.  between slot conductor copper and iron stack, mainly determined by heat resistance of the slot insulation, according to [1]

, where



is the slot surface,

௧௛ଶ ௜௦ ௧௛ଷ

is the

thermal conductivity of the slot insulation and is the thickness of the slot insulation. and are thermal resistances between winding overhang and surrounding air given by convection. Due to special structure of the VZIM, as this motor is totally enclosed, at one side of the winding overhang, the heat transfer coefficient is considered as for natural convection and heat radiation and at the other side, the rotating centrifugal mechanism leads to a better cooling due to the convection. Therefore in order to calculate the heat transfer  coefficient, with try and error and comparison with measured values, is estimated for the 5.5kW VZIM ( ). For the 22kW VZIM due to similar structure with the ). 5.5kW VZIM, a proportional ratio of  was considered, the calculations are presented in Appendix IV and the values are shown in Table 5.2-1.

௧௛ସ

ൌ ͳͷ

ൌ ͷͲȀሺȀሺʹሻ

- 66 -

Table 5.2-1: Heat transfer coefficient for one side of the winding overhang close to the centrifugal mechanism Winding overhang

VZIM2 22kW

VZIM3 22kW

Heat transfer coefficient

63.32

62.86

Ȁሺ ૛ ሻ

ο ஼௨ ο ௘

ο ൌ െ ௔௠௕ Ԩ ο ௧௛ଷ஼௨ ൅ ο ௧௛ସ஼௨ ൅ ο ஼௨ െ௧௛ଶο ௘ ൌൌ ஼௨ǡ஼௨ǡ௦                                ሺͷǤʹ െ ͳሻ ο ௧௛ଵ௘ െ ο ஼௨ െ௧௛ଶο ௘ ൌ ௘ǡ௦                                       ሺͷǤʹ െ ʹሻ

and are unknown temperature rises, which is the temperature difference between motor local temperature and ambient temperature of surrounding air. In this calculation ambient temperature temperature is considered to be 20 . With consideration of the steady state temperature rise only two algebraic linear equations have to be solved,

As VZIM2 and VZIM3 are active in rated operating speed of the motor, thermal calculations of VZIM2 and VZIM3 are done in an operating speed which the average of the output power   produced by VZIM2 and VZIM3 V ZIM3 in this speed is equal to 22kW.Therefore as it is depicted in Fig.5.2-2, the average of output power of VZIM2 and VZIM3 versus slip is calculated and at a speed of 1467.36 rpm the average power is equal to 22kW. The losses of VZIM2 and VZIM3 in this speed (1467.36 rpm) have been calculated and are shown in Table 5.2-2. Based on those losses the temperature rise and for VZIM2 and VZIM3 are calculated in the following sections.

ο ஼௨ ο ௘

Power-Slip characteris ch aracteristic tic 80000 60000

   W40000     /    r    e    w 20000    o    p    t 0    u    p    t -20000    u    O

VZIM2 power VZIM3 power 0

0. 5

1

1. 5

2

2.5

Average power

-40000 -60000

Slip

Fig.5.2-2: Calculated output power-slip characteristics of 22kW V ZIM2 and VZIM3 and the average of the powers

- 67 -

Table 5.2-2: Losses in different parts of the VZIM2 and VZIM3 in the speed equal to 1467.36 rpm

VZIM2 VZIM3 Average

ǡ

/W 492.073 1395.6 943.83

ǡ

ǡ

//W 466.2 760.52 613.36

/W 192.6 159.64 162.62

ǡ

/W 31.36 33.8 32.58

/W 16894.2 27374.75 22134.47

In calculation of the average output power in Fig.5.2-2, it is assumed that the equivalent output power of the VZIM motor can be estimated using super-position method. More accurate calculations based on an equivalent circuit model [6] show that the super-position method overestimates the equivalent torque and output power of the VZIM motor, especially at lower speeds. But it is quite accurate at the rated operating region of the motor, as shown in Fig.5.2-3. As here, only the calculation of the output power in the operating region of the VZIM motor is required, hence more simple method of super-positioning is used.

Fig.5.2-3: Calculated output power of the VZIM motor with the supper-position and equivalent circuit methods

- 68 -

5.2-1 Calculation of 

ο ο and

for VZIM2 (22kW)

The materials used in construction of VZIM2 and the related thermal conductivities are shown in Table 5.2.1-1. Table 5.2.1-1: Thermal conductivities of the materials used in VZIM2 thermal model Material Air Iron stack Insulation 0.031 40 0.2 Thermal conductivity, W/(m.K)

௧௛ଵ ൌ Τͳሺ ሻ

Copper 380

After determination of materials, calculation of thermal resistances is done as following. is the heat resistance between the stator iron stack and the ambient cooling air, is the surface of the stator iron housing and is the heat transfer coefficient, is

 Τ ଶ ଷ ൌ ͳͷ

ήͳͲି଺ ଶ

Ȁሺଶሻ Τ

equal to 203472 , and according to [1] is calculated by ( in , in ) for moving air over bare metallic hot surface. The speed of wind flow over the stator body, based on an empirical rule [9], is equal to 70 percent of the linear speed of the top point of the fan blade, therefore

ൌͳͲͲ͹Ͳ ʹ͸Ͳ ൌͳͲͲ͹Ͳ ήͳ͸͸ǤʹͷήͳͲିଷ ή ʹ ήͳ͸ͲͶ͸͹Ǥ͵͸ ൌ ͳ͹Ǥͺ͹͹ Ǥ ൌ௧௛ଵ ൌͳͷଶͳΤΤሺଷ ൌሻͳͷͷͳൌή ͳ͹ͳ͹ǤͳΤሺǤͳͲʹǤͳͺͲʹǤ͹ଶΤଷͷൌʹήʹͳͲ͵Ͷ͹ʹ ͲʹǤͷʹȀሺȀሺͳͲʹି଺ሻሻ ൌ ͲǤͲͶ͹ͻ͵ Ǥ  ௧௛ଶ ൌ Τ൫ ௜௦ ൯  ௜௦ ି଺ ሻ ͲିଷିଷΤሺͲǤͲǤʹͳͲήͳ͵ͻͲͲǤ Ͷ ͵͸ ή  ͳ Ͳ ൌ ͲǤ Ͳ ͹ͳͻͶ Ȁ ௧௛ଶ௧௛ଷ ൌൌ ͳΤΤሺ൫ ௜௦ሻ൯ൌൌͳΤሺͲͳͷͳǤʹͷήήͻͳʹͻͶͶ ି଺ሻି଺ൌ ͲǤ͹ͳȀ  Ȁ ͳΤሺ Ͳ͹ሻൌ ͳΤሺ͸͵Ǥ͵ʹήͻʹͻͶͶ ͳͲ ሻ ൌ ͲǤͳ͹ȀȀ ௧௛ସ஼௨ǡ௦ ൌൌ ͶͻʹǤ ௘ǡ௦ ൌ ͳͻʹǤ͸ οο ௘஼௨ ൌൌ ͳͻǤ ͵ͷǤ͸͵͸ͳ Ǥ 

 R is the radius of the fan blade and n is the rotational speed of the VZIM2 with 22kW output  power. Consequently heat transfer transfer coefficient can be calculated as: ,

where

is slot surface,

is thermal conductivity of slot insulation and

is the thickness of the slot insulation.

According to equations (5.2-1) and (5.2-2) we get:

The maximum temperature rise for class F of insulations is 105 . Here the value is much lower than the maximum allowable temperature rise; therefore insulation is in the safe side. The reason for this low temperature rise is the fact that the middle cage (VZIM2) is working under rated load (16.89kW instead of 22kW) at the rated speed of 1467.36 rpm. The final temperature rise will be the t he average value of both VZIM2 and VZ IM3.

- 69 -

5.2-2 Calculation of 

ο ο and

for VZIM3 (22kW)

The materials used in construction of VZIM3 and the related thermal conductivities are shown in Table 5.2.2-1. Table 5.2.2-1: Thermal conductivities of the materials used in VZIM3 thermal model Material Air Iron stack Insulation Thermal 0.031 40 0.2 conductivity, W/(m.K)

Copper

380

௧௛ଵ ൌ ͳΤሺ ሻ

Calculations are similar to calculations for VZIM2 in the previous section. is the heat resistances between the stator iron stack and the ambient cooling air, is the surface of the stator iron housing and is the heat transfer coefficient, is equal to 203472

 Τ ൌ ͳͷଶ ଷ Ȁሺଶሻ

ͳͲି଺ ଶ ൌͳͲͲ͹Ͳ ʹ͸Ͳ ൌͳͲͲ͹Ͳ ήͳ͸͸ǤʹͷήͳͲିଷ ή ʹ ήͳ͸ͲͶ͸͹Ǥ͵͸ ൌ ͳ͹Ǥͺ͹͹ Ǥ ൌ௧௛ଵ ൌͳͷଶͳΤΤሺଷ ൌሻͳͷͷͳൌή ͳ͹ͳ͹ǤͳΤሺǤͳͲʹǤͳͺͲʹǤ͹ଶΤଷͷൌʹήʹͳͲ͵Ͷ͹ʹ ͲʹǤͷʹȀሺȀሺͳͲʹି଺ሻሻ ൌ ͲǤͲͶ͹ͻ͵ȀȀǤ ௧௛ଶ ൌ Τ൫ ௜௦ ൯  ௜௦ ି଺ ሻ ͲିଷିଷΤሺͲǤͲǤʹͳͲήͳ͵ͻͲͲǤ Ͷ ͵͸ ή  ͳ Ͳ ൌ ͲǤ Ͳ ͹ͳͻͶ Ȁ ௧௛ଶ௧௛ଷ ൌൌ ͳΤΤሺ൫ ௜௦ሻ൯ൌൌͳΤሺͲͳͷͳǤʹͷήήͻͳʹͻͶͶ ି଺ሻି଺ൌ ͲǤ͹ͳȀ  Ȁ ͳΤሺ ͸ሻൌ ͳΤሺ͸ʹǤͺ͸ήͻʹͻͶͶ ͳͲ ሻ ൌ ͲǤͳ͹ȀȀ ௧௛ସ஼௨ǡ௦ ൌൌ ͳ͵ͻͷǤ ௘ǡ௦ ൌ ͳͷͻǤ͸Ͷ οο ௘஼௨ ൌൌ ͶͲǤ ͻͲǤͺͷͳ͹ Ǥ ο ஼௨ 

ήΤ

, and according to [1] can be calculated by ( in , in ) for moving air over bare metallic hot surface. The speed of wind flow over the stator body,  based on an empirical rule [9], is equal to 70 percent of the linear speed of the top point of the fan blade, therefore

 R is the radius of the fan blade and n is the rotational speed of the VZIM2 with 22kW output  power. Consequently heat transfer transfer coefficient can be calculated: ,

where

is slot surface,

is thermal conductivity of slot insulation and

is the thickness of slot insulation.

According to equations (5.2-1) and (5.2-2) we get:

The calculated temperature rise in the winding is lower than the temperature rise limit 105 for class F insulations used in this motor. Therefore the insulation operates in the safe side, although the motor operates at overload condition (27.37kW instead of 22kW). Table 5.2.2-2 shows the temperature rise in the stator winding and iron of VZIM2 and VZIM3 calculated based on simplified thermal equivalent circuit and the average values.

- 70 -

Table 5.2.2-2: Temperature rise in the stator winding of VZIM2 and VZIM3 calculated based on simplified thermal equivalent circuit in the nominal operating speed equal to 1467.36 rpm

οο

͵ͷǤͳͻǤ͸͵͸ͳ

ͻͲǤͶͲǤͷͺ͹ͳ

VZIM2

VZIM3

͸ʹǤ͵ͲǤʹͻ͵ͷͶ ͸ʹǤͻͶ

Average value

The average value of the temperature rise of VZIM2 and VZIM3, which is equal to is considered as the temperature rise of the 22kW VZIM motor, which is well below the temperature rise limit of the used insulation materials (Thermal class F).

,

5.3 Numerical calculation of temperature rise

In  ANSYS  as a finite element tool, first of all the geometry of the model has to be prepared, assigning the materials and meshing the areas are next steps. The loss density is used as heat source, therefore based on the different losses of the motor, the loss densities for different  parts of the motor are calculated using the formula , where  P is the calculated losses in each part, and V is the volume in which the losses occur.

ൌ Τ

Electromagnetic analysis of 22kW VZIM is done with  Flux2D in previous chapters and just 2D model of the VZIM motors was prepared but in this section for thermal analysis 3D models of VZIM2 and VZIM3 are built to consider the thermal effect of the windingoverhangs and end-rings. As usual, the number of the stator teeth and rotor teeth are not equal, hence a symmetry can be considered if for the simplification, the number of rotor teeth is assumed to be 36 instead of the 28 which is the real number of rotor teeth. With this simplification simulation of a half of the stator and rotor slots is enough to do the thermal analysis of the VZIM motor. In order to take into account the extra losses which appear due to the simplification in this new model, the rotor copper losses are multiplied by the factor  compensate the extra losses added to the simulating model.

ଶ଼ଷ଺

to

In this section, first of all the numerical calculation of the temperature rise based on the finite element method with  ANSYS  tool for the 5.5kW VZIM is done and compared with measured values, because the 5.5kW VZIM was already manufactured and tested. Afterwards the numerical calculation of temperature rise in 22kW VZIM has been carried out using  ANAYS  tool. In three dimensional models considered for thermal analysis of 5.5kW and 22kW VZIM, the convection heat transfer at the stator outer surface, the outer surface of the winding overhang and the end rings are considered as the boundary conditions of the thermal model. For  simplification the heat transfer due to heat conduction, through the motor shaft is neglected. Clearly due to the thermal conduction between the motor shaft and rotor, in case of modeling the motor shaft, the calculated rotor temperature would be lower than values are presented with the simplified model in following sections.

- 71 -

5.3.1 Thermal analysis of 5.5 kW VZIM2

First a 2D geometry of 5.5kW VZIM2 is prepared and after assigning the material and meshing the geometry, by extruding the 2D model, the 3D model is constructed. In Fig.5.3.1-1 the 2D model and meshing of VZIM2 are depicted, the 3D model and meshing are depicted in Fig.5.3.1-2 and Fig.5.3.1-3.

Fig.5.3.1-1: 2D model of the 5.5kW VZIM2 and meshing, by ANSYS 

Fig.5.3.1-2: 3D model of the 5.5kW VZIM2, by ANSYS 

- 72 -

Fig.5.3.1-3: 3D model meshing of the 5.5kW VZIM2, by ANSYS 

Loss densities in all volumes of VZIM2 have been calculated and assigned to the simulation model before solving the thermal model. Table 5.3.1-1 shows the thermal loss densities in VZIM2 5.5kW. Table 5.3.1-1: Loss densities in different parts of the 5.5kW VZIM2 Stator Winding Rotor bars winding overhang 0.000453318 0.00023955 0.00082892 Loss density W/

૜

Stator iron

Rotor iron

0.000045299

0.000000566

The air flow in three areas including, the stator outer surface, winding overhang and end-rings surfaces are considered and the heat transfer coefficients due to the convection are calculated for those areas. - The heat transfer coefficient for the stator outer surface is calculated as below:

ൌൌ ͹ͲͳͷଶΤଷʹൌ ͳͷͷͳൌ͹Ͳή ͳ͵ͳ͵ǤǤήͲͳͶͶʹʹǤଶΤଷ ͵ൌͷήͳͺͲ͵ǤିଷͳͳήȀሺʹȀሺήͳʹͶͷͷǤሻ ʹ ൌ ͳ͵ǤͲͶͶͶͶ Ǥ ͳͲͲ ͸Ͳ ͳͲͲ ͸Ͳ  ͳͷȀሺȀሺʹሻ ൌ ͷͲȀሺȀሺʹሻ ൌ ͷͲȀሺȀሺʹሻ

ൌ

-The heat transfer coefficients for winding overhangs are considered as below: For one side of the motor the heat transfer coefficient over winding overhang is for natural convection and heat radiation, but for the other side of the motor,  because of the rotating centrifugal mechanism, the heat transfer coefficient for winding overhang was drawn with try and error, considering the value of the maximum temperature rise of the winding overhang, already measured. By choosing the heat transfer coefficient , the calculated and measured values match well together, hence consideration of  is acceptable. is

- 73 -

ൌ ͳͷȀሺȀሺʹሻ

-The heat transfer coefficient for the end-ring surface is considered as below: As in area near the end ring air is nearly not moving, therefore considered.

is

Table 5.3.1-2 shows the heat transfer coefficient at different boundary conditions of the 5.5kW VZIM2. Table 5.3.1-2: Heat transfer coefficient at different boundary conditions o f the 5.5kW VZIM2

Stator outer surface

End ring

Winding overhang far from centrifugal mechanism

83.11

15

15

ሺ Ȁሺ ૛ ሻ Ԩ

Winding overhang close to centrifugal mechanism

50

Ԩ  

Fig.5.3.1-4 shows the thermal solution of the 5.5kW VZIM2, temperature changes from 84.5 on the stator body to 147.89 inside the rotor. For thermal class F of insulations the maximum allowable temperature rise is 105 . For VZIM2 the maximum temperature rise in the stator winding is 72.28 as it has been shown in Fig.5.3.1-5, and assures that insulation lies in a safe side. The measured value of the maximum overhang temperature rise is 67.7 , which is 6.3% lower than the finite element calculated value. The ambient temperature is considered to be 20 .

Ԩ

Fig.5.3.1-4: The thermal solution of the 5.5kW VZIM2 under nominal operation, by ANSYS 

- 74 -



Fig.5.3.1-5: The calculated temperature in the winding overhang of VZIM2, by ANSYS 

5.3.2 Thermal analysis of 5.5kW VZIM3

Similar to 5.5kW VZIM2, first a 2D geometry of 5.5kW VZIM3 is prepared and after  assigning the material and meshing the geometry, by extruding the 2D model, the 3D model is constructed. In Fig.5.3.2-1 2D model and meshing of 5.55kW VZIM3 are depicted, also in Fig.5.3.2-2 and Fig.5.3.2-3 the 3D model and meshing are depicted.

Fig.5.3.2-1: 2D model of the 5.5 kW VZIM3 and meshing, by ANSYS 

- 75 -

Fig.5.3.2-2: 3D model of the 5.5kW VZIM3, by ANSYS 

Fig.5.3.2-3: 3D model meshing of the 5.5kW VZIM3, by ANSYS 

Loss densities in all volumes of VZIM3 have been calculated and assigned to the simulation model before solving the thermal model. Table 5.3.2-1 shows the thermal loss densities in 5.5kW VZIM3. Table 5.3.2-1: Loss densities in different parts of the 5.5kW VZIM3 Stator Winding Rotor bars winding overhang 0.00047015 0.00024844 0.0003707 Loss density W/

૜

Stator iron

Rotor iron

0.00004544

0.000000461

The air flow in three areas including the stator outer surface, winding overhang and end-rings surfaces are considered and the heat transfer coefficient due to the convection are calculated for those areas. - 76 -

ൌൌ ͹ͲͳͷଶΤଷʹൌ ͳͷͷͳൌ͹Ͳή ͳ͵ͳ͵ǤǤήͳͳ͹ଶΤʹʹǤଷ ൌ͵ͷήͺͳ͵ǤͲ͸ିଷͷȀሺήȀሺʹʹήͳͶ͹Ͳሻ ൌ ͳ͵Ǥͳ͹͹Ǥ ͳͲͲ ͸Ͳ ͳͲͲ ͸Ͳ  ͳͷȀሺȀሺʹሻ ൌ ͷͲȀሺȀሺʹൌሻ ͷͲȀሺȀሺʹሻ ൌ ͳͷȀሺȀሺʹሻ - The heat transfer coefficient for the stator outer surface is calculated as below:

ൌ

- The heat transfer coefficients for wi nding overhangs are considered as below: For one side of the motor the heat transfer coefficient over winding overhang is , for the other side of the motor, because of the rotating centrifugal mechanism, the heat transfer coefficient for winding overhang was drawn with try and error, considering the value of the maximum temperature rise of the winding overhang was already measured,  by consideration of heat transfer coefficient the calculated values and measured values match well together, hence consideration of  is acceptable. -The heat transfer coefficient for the end-ring surface is considered as below: As in this area air is nearly not moving, therefore is considered.

Table 5.3.2-2 shows the heat transfer coefficient at different boundary conditions of the 5.5kW VZIM3. Table 5.3.2-2: Heat transfer coefficient at different boundary condi tions of the 5.5kW VZIM3

Stator outer surface

End ring

Winding overhang far from centrifugal mechanism

83.65

15

15

ሺ Ȁሺ ૛ ሻ

Winding overhang close to centrifugal mechanism

50

Ԩ

Ԩ 

Fig.5.3.2-4 shows the thermal solution of the VZIM3, temperature changes from 80.17 on the stator body to 121.969 inside the rotor. For thermal class F of insulations the maximum allowable temperature is 105 . For VZIM3 the maximum temperature rise in stator winding overhang is 63.35 as it has been shown in Fig.5.3.2-5, and assures that insulation lies in a safe side. The measured value of the maximum overhang temperature rise is 67.7 , which is 6.42% higher than the calculated value by finite element method. The ambient temperature is considered to be 20 .

 Ԩ



- 77 -

Fig.5.3.2-4: The thermal solution of the 5.5kW VZIM3 under nominal operation, by ANSYS 

Fig.5.3.2-5: The calculated temperature in winding overhang of 5.5kW VZIM3, by ANSYS 

Table 5.3.2-3 shows the temperature rise in stator slot of 5.5kW VZIM2 and VZIM3 stator  slot calculated by ANSYS. Table 5.3.2-3: Temperature rise in stator slot of 5.5kW VZIM2 and VZIM3 calculated by ANSYS 

ο

͹ʹǤʹͺ

͸͵Ǥ͵ͷ

VZIM2

VZIM3

͸͹Ǥͺͳ ͸͹Ǥͺͳ

Average value

The average value of the temperature rise of VZIM2 and VZIM3 which is equal to , is considered as the temperature rise of the 5.5kW VZIM motor. The measured value of the maximum overhang temperature is which obviously shows a very good fitting  between the measured value and calculated values of temperature rise by  ANSYS tool.

͸͹Ǥ͹

- 78 -

5.3.3 Thermal analysis of 22kW VZIM2

First a 2D geometry of 22kW VZIM2 is prepared and after assigning the material and meshing the geometry, by extruding the 2D model, the 3D model is constructed. In Fig.5.3.31 2D model and meshing of it are depicted, also in Fig.5.3.3-2 and Fig.5.3.3-3 the 3D model and its meshing are depicted.

Fig.5.3.3-1: 2D model of the 22kW VZIM2 and meshing, by ANSYS 

Fig.5.3.3-2: 3D model of the 22kW VZIM2 , by ANSYS 

- 79 -

Fig.5.3.3-3: 3D model meshing of the 22kW VZIM2 , by ANSYS 

Loss densities in all volumes of VZIM2 have been calculated and assigned to the simulation model before solving the thermal model. Table 5.3.3-1 shows the thermal loss densities in 22kW VZIM2. Table 5.3.3-1: Loss densities in different parts of the 22kW VZIM2 Stator Winding Rotor bars winding overhang Loss density 0.000169 0.0001074 0.0007152 W/

૜

Stator iron

Rotor iron

0.00002834

0.000007678

The air flow in three areas including the stator outer surface, winding over hang and end-rings surfaces are considered and the heat transfer coefficient due to the convection are calculated for those areas. -The heat transfer coefficient for the stator outer surface is calculated as below:

ൌൌ ͹ͲͳͷଶΤଷʹൌ ͳͷͷͳൌ͹Ͳή ͳ͹ͳ͹ǤǤήͺͳ͹ଶΤ͸͸Ǥଷ ൌʹͷήͳͳͲʹǤͲିଷͷʹήȀሺʹȀሺήͳʹͶ͸͹Ǥሻ ͵͸ ൌ ͳ͹Ǥͺ͹͹ ͳͲͲ ͸Ͳ ͳͲͲ ͸Ͳ  ʹ ൌ ͳͷȀሺȀሺ ሻ ൌ ͸͵Ǥ͵ʹȀሺȀሺʹሻ ൌ ͳͷȀሺȀሺʹሻ

-The heat transfer coefficients for winding overhangs are considered as below: At one side, the heat transfer coefficient for winding overhang is , for the other side of the motor due to the rotating centrifugal mechanism, as already was discussed . -The heat transfer coefficient for the end-ring surface is considered as below: As in this area air is nearly not moving, therefore is considered.

Table 5.3.3-2 shows the heat transfer coefficient at different boundary conditions of the 22kW VZIM2.

- 80 -

Table 5.3.3-2: Heat transfer coefficient at different boundary conditions o f the 22kW VZIM2

ሺ Ȁሺ ૛ ሻ Ԩ 

Stator outer surface

End ring

Winding overhang far from centrifugal mechanism

102.52

15

15

Winding overhang close to centrifugal mechanism

63.32

Ԩ  

Fig.5.3.3-4 shows the thermal solution of the 22kW VZIM2, temperature changes from 59.7929 on the stator body to 150.95 inside the rotor. For thermal class F of insulations the maximum allowable temperature rise is 105 . For VZIM2 the maximum temperature rise in the stator winding overhang is 42.73 as it has been shown in Fig.5.3.3-5, and assures that insulation lies in a safe side. The calculated value of the stator winding temperature rise is 35.31 which is 17.3% lower than finite element calculated value.

Fig.5.3.3-4: The thermal solution of the 22kW VZIM2 under nominal operation, by ANSYS 

- 81 -

Fig.5.3.3-5: The calculated temperature in the winding overhang of 22kW VZIM2 , by ANSYS 

5.3.4 Thermal analysis of 22kW VZIM3

Similar to 22kW VZIM2, first a 2D geometry of 22kW VZIM3 is prepared and after  assigning the material and meshing the geometry, by extruding the 2D model, the 3D model is constructed. In Fig.5.3.4-1 2D model and meshing of 22kW VZIM3 are depicted, also in Fig.5.3.4-2 and Fig.5.3.4-3 the 3D model and meshing are depicted.

Fig.5.3.4-1: 2D model of the 22kW VZIM3 and meshing, by ANSYS 

- 82 -

Fig.5.3.4-2: 3D model of the 22kW VZIM3 , by ANSYS 

Fig.5.3.4-3: 3D model meshing of the 22kW VZIM3 , by ANSYS 

Loss densities in all volumes of VZIM3 have been calculated and assigned to the simulation model before solving the thermal model. Table 5.3.4-1 shows the thermal loss densities in 22kW VZIM3. Table 5.3.4-1: Loss densities in different parts of the 22kW VZIM3 Stator Winding Rotor bars winding overhang Loss density 0.00048 0.000304 0.000606 W/

૜

- 83 -

Stator iron

Rotor iron

0.00002349

0.00000934

The air flow in three areas including the stator outer surface, winding over hang and end-rings surfaces are considered and the heat transfer coefficient due to the convection are calculated for those areas. -The heat transfer coefficient for the stator outer surface is calculated as below:

ൌൌ ͹ͲͳͷଶΤଷʹൌ ͳͷͷͳൌ͹Ͳή ͳ͹ͳ͹ǤǤήͺͳ͹ଶΤ͸͸Ǥଷ ൌʹͷήͳͳͲʹǤͲିଷͷʹήȀሺʹȀሺήͳʹͶ͸͹Ǥሻ ͵͸ ൌ ͳ͹Ǥͺ͹͹ ͳͲͲ ͸Ͳ ͳͲͲ ͸Ͳ  ൌ ͳͷȀሺȀሺʹሻ ൌ ͸ʹǤͺ͸ȀሺȀሺʹሻ ൌ ͳͷȀሺȀሺʹሻ

-The heat transfer coefficients for winding overhangs are considered as below: At one side, the heat transfer coefficient for winding overhang is , for the other side of the motor due to the rotating centrifugal mechanism, as already was discussed . - The heat transfer coefficient for the end-ring surface is considered as below: As in this area air is nearly not moving, therefore is considered.

Table 5.3.4-2 shows the heat transfer coefficient at different boundary conditions of the 22kW VZIM3. Table 5.3.4-2: Heat transfer coefficient at different boundary conditi ons of the 22kW VZIM3

ሺ Ȁሺ ૛ ሻ Ԩ

Stator outer surface

End ring

Winding overhang far from centrifugal mechanism

102.52

15

15

Ԩ 

Winding overhang close to centrifugal mechanism

62.86

Fig.5.3.4-4 shows the thermal solution of the 22kW VZIM3, temperature changes from 98.7877 on the stator body to 224.19 inside the rotor. For thermal class F of insulations the maximum allowable temperature rise is 105 . For VZIM3 the temperature in stator  winding overhang is 85.72 as it has been shown in Fig.5.3.4-5, and assures that insulation lies in a safe side. The calculated value of the stator winding temperature rise is 90.57 which is 5.5% higher than calculated value by finite element method.



- 84 -



Fig.5.3.4-4: The thermal solution of the 22kW VZIM3 under nominal operation, by ANSYS 

Fig.5.3.4-5: The calculated temperature in the winding overhang of 22kW VZIM3 , by ANSYS 

Table 5.3.4-3 shows the temperature rise in stator slot of 22kW VZIM2 and VZIM3 stator slot calculated by ANSYS. Table 5.3.4-3: Temperature rise in stator slot of 22kW V ZIM2 and VZIM3 calculated by ANSYS 

ο

ͶʹǤ͹͵

ͺͷǤ͹ʹ

VZIM2

VZIM3

- 85 -

͸ͶǤʹʹ

Average value

͸ͶǤʹʹ

The average value of the temperature rise of VZIM2 and VZIM3 ( ) is considered as the temperature rise of the 22kW VZIM motor which i s well below the maximum temperature rise for the insulation class F ( ). The mean value of the calculated values of the maximum overhang temperature with simplified equivalent thermal circuit is , which shows a 1.99% difference between finite element method calculations and the calculations  based on the simplified thermal circuits. circuits.

ͳͲͷ

͸ʹǤͻͶ

In order to summarize the thermal calculations of 5.5kW and 22kW VZIM, the values of the stator maximum winding temperature rise and the rotor maximum temperature rise are shown in Table 5.3.4-4. The thermal analyses of 5.5kW VZIM2 and VZIM3 motors are done in the operation point of the motor where the output power of each motor is 5.5kW. But the thermal analyses of 22kW VZIM2 and VZIM3 motors are done in an operation point of the motor, where the average output power of 22kW motors (VZIM2 and VZIM3) is 22kW. Therefore according to the previous description and calculation in section 5.2, as it is shown in Table 5.2-2, the 22kW VZIM2 is analyzed in an operation point, that its output power is 16.89kW and the 22kW VZIM3 is analyzed in an operation point, where its output power is 27.37kW. Therefore the 22kW VZIM3 motor with the output power 27.37kW is clearly overloaded, therefore the temperature rise of the rotor and stator of 22kW VZIM3 is relatively high. As  previously mentioned, the average temperature rise of VZIM2 and VZIM3 in the rated operating point of the VZIM is important and shows the temperature rise of the real motor. Table 5.3.4-4: Temperature rise in stator slot of 22kW V ZIM2 and VZIM3 calculated by ANSYS 

5.5kW VZIM2 Stator max. winding temperature rise Rotor max. temperature rise

5.5kW VZIM3

Average value of  5.5kW VZIM

22kW VZIM2

22kW VZIM3

Average value of  22kW VZIM

͹ʹǤʹͺ ͸͵Ǥ͵ͷ ͸͹Ǥͺͳ ͶʹǤ͹͵ ͺͷǤ͹ʹ ͸ͶǤʹʹ ͳͶ͹Ǥͺͻͷ ͳʹͳǤͻ͸ͻ ͳ͵ͶǤͻ͵ʹ ͳ͵ͲǤͻͷ ʹͲͶǤͳͻ ͳ͸͹Ǥͷ͹

- 86 -

Chapter 6: Conclusion Electromagnetic and thermal analyses of Variable Impedance Induction Motor (VZIM) have  been done for a 4 pole 22kW motor. The electromagnetic analyses show that the expected goals of the high starting torque and high efficiency at the rated operating point of the VZIM motor are fulfilled. Electromagnetic analysis of the 22kW VZIM motor is carried out with consideration of three independent motors VZIM1, VZIM2 and VZIM3, using the finite element software  Flux2D and the results are compared with the analytical calculations performed by  KLASYS  tool. In the steady state AC analysis the torque-slip, stator phase current-slip, normal air-gap flux density and flux lines in the electromagnetically active parts of the motors are drawn. The results of the analytical and finite element method, are compared which shows a good match  between them. Moreover in time stepping analysis torque-slip, fundamental stator phase current-slip, input power-slip, efficiency-slip and power factor-slip characteristics are drawn and compared. Comparison of torque-slip characteristics in steady state AC and time stepping analysis show that the  KLASYS  tool always predicts higher values than  Flux2D. The efficiency-slip, phase current-slip, input power-slip and power factor-slip characteristics of the VZIM motor in time stepping and steady state AC analysis, calculated by  Flux2D and KLASYS  match very well in all slips. Thermal calculations have been done for VZIM with the help of the  ANSYS  tool which is a finite element method tool. Comparing the results obtained by  ANSYS  and simplified thermal circuits of the 22kW VZIM motor shows a low difference about 1.99%, and the results assure that the insulation of the stator winding always stays in a safe side, and the temperature never  exceeds the critical values. In addition, the average value of the temperature rise of VZIM2 and VZIM3 ( calculated by  ANSYS , is considered as the temperature rise of the 22kW VZIM motor which is well below the maximum temperature rise for the Thermal Class F( ).

ͳͲͷ

͸ͶǤʹʹሻ

- 87 -

Bibliography [1] Binder, A. ,‘‘ , ‘‘CAD CAD and system dynamics of electrical machines ,’’ Lecture script, TU Darmstadt, 2010. [2] Binder, A. ,‘‘ , ‘‘Motor Motor development for electrical drive systems , ’’ Lecture script, TU Darmstadt, 2010. [3] Binder, A. ,‘‘ , ‘‘Electrical Electrical Machines and Drives, ’’ Lecture script, TU Darmstadt, 2009. [4] Binder, A. ; Funieru, B. ‚‘‘ ‚‘‘Design Design of Electrical Machines and Actuators with Numerical Field Calculations, ’’ Seminary notes, TU Darmstadt, 2010. [5] Flux2D Application‚‘‘ Application ‚‘‘Induction Induction machine tutorial calculations in Flux2D, ’’ Cedrat corporation, 2011. [6] Gholizad, H. ; Binder, A. , “Analytical Modeling of Variable Impedance Induction Motors’’ Motors ’’,, in proc. IEEE IEMDC , 15-18 May 2011, Niagara Falls, Canada, pp. 1504-1509. [7] Data sheet, Power Core M270-50A, Elektroband NO / NGO electrical steel. [8] Xyptras, J. ; Hatziathanassiou, V. , “Thermal analysis of an electrical machine taking into the account the iron losses and the deep bar effect ,’’  IEEE Transactions on Energy Conversion vol. 14, no. 4, December 1999, pp. 996-1003. [9] Huai, Y. ; Melnik, R. ; Thogersen, P. , “Computational analysis of temperature rise  phenomena in induction motor  s’’ ’’,, Applied Thermal engineering vol. 23, no. 7, May 2003, pp. 779-795.

- 88 -

8. Appendix

8.1 Appendix I: Calculation of magnetizing main and stray inductances

Magnetizing main inductance

௛ǡஶ ௛ǡஶ ିଷ ିଷ ʹ ή ͵ ʹͶͲ ή ͳͲ ή ͳ͵ ͳ͵͵Ǥ ͵Ǥ Ͷ ͷ ͷή ή ͳͲ ି଻ ଶ ଶ ή ͳͲ ή ͹ʹ ή ͲǤͻ͸ ή ଶ ʹ ή ͲǤͲǤͶͷ ή ͳͲିଷ

for infinite iron permeability is only determined by air 

gap. According to [1] the magnetizing main i nductance

is calculated as:

ʹ ଶ ଶ ൌ ௛ǡஶ ଴ ௪ ଶ ௣ ൌ Ͷ ൌ ͲǤ ͳ ʹͻͻͷ  ൌ௪ ͵ቀቀ͸ͳͺቁቁ ൌ ͲǤͻ͸ ͳ͵͵ǤͶͷ ൉ ͳͲିଷ ௣ ൌ ʹ௦௜ ൌ ͳ͵͵Ǥ ͳʹͻͻǤͻǤͷ ή ͳͲͳͲିସ ൌ ͳ͸Ǥ͵͹͵͹ ή ͳͲିସ ൌ ͳ͸Ǥ͵͹͵͹ ఙǡ௢௦ ൌ ௦ǡ௢ ௦௛ ൌ ͲǤͲͳʹ͸ ή ͳʹͻ ଶ ଶ ൉ ͷ ൉ ͻ ൅ ͳ െ ͳ ൌ ͲǤͲͳʹ͸Ǥ ͷ ൅ ͳ ௦ ଶ ൌ ሺ ሻ ൉ െ ͳ ൌ ቀ ቁ ௦ǡ௢ ͵ ௪௦ǡଵ ͸ ௦ଶ ͵ ൉ ͲǤͻ͸ͻ͸ ͸ ൉ ͻ ᇱఙǡ௢௥ ൌ ௥ǡ௢ ௛ ൌ ͲǤͲͳ͸ ή ͳʹͻͻǤͷ ή ͳͲିସ ൌ ʹͲǤ͹ͻʹ ή ͳͲିସ ௥ǡ௢ ൌ ሺሺ ൉ͳ Τ ௥ሻሻଶ െ ͳ ൌ ሺʹ ൉ͳ Τʹͺʹͺሻሻ ଶ െ ͳ ൌ ͲǤͲͳ͸ ൬ ʹ ൉ Τʹͺ ൰ ൉ Τ௥ ଶ ʹ ͹ʹ ଶ ି଻ ʹʹ͸ǤǤ͸ ή ͳͲͳͲିଷ ൉ ͲǤʹͺ͹ ௦ǡఙ௕ ൌ ͲǤͶʹͶ ௦ǡఙ௕ ൌ ଴ ௦ ௕ ௕ ൌ Ͷ ή ͳͲ ή ʹ ή ʹ ൉ ʹʹ͸ ʹʹ͸Ǥ Ǥ ͸ െ ͳͲ ή ͳͷ ͳͷǤ Ǥ ͻ ൌ௕ ͲǤͲǤ͵͹ ൉ ൉ ൤ͳ൤ͳ൅൅ ͲǤͻ ൉ ௕௕ െ൅ ͳͲͳͲ ήή ௪௪൨ ൌ ͲǤ͵͹൉͵͹ ൉ ͲǤ͸͹ ൉ ൤ͳ ൅ ͲǤͻ ൉ʹʹ͸ ൨ ൌ ͲǤʹͺ͹ ʹʹ͸Ǥ ʹʹ͸ Ǥ ͸ ൅ ͳͲ ή ͳͷ ͳͷǤ Ǥ ͻ ଶ ௜ ௖௨ ௪ ൌ ඨ  ௙ ௖ ൌ ඨ ͻͲǤ൉ ͳͶʹ͸൉ ͳʹ ൌ ͳͷǤͻ ௕ ൌ ͲǤͳʹ ௕ ൌ ௣ ଶʹ ʹ ͳ ଶ ି଻ ͳʹ ൉ ͳ͵͵ͳ͵͵ǤǤͷ ൉ ͳͲିଷ ൌ ͷǤͲ͵൉Ͳ͵ ൉ ͳͲିଽ ௥ǡఙ௕ ൌ ଴ ௦ ௕ ௕ ൌ Ͷ ή ͳͲ ൉ ൬ʹ൰ ή ʹ ൉ ͲǤͳʹ൉ ʹͶͲ ή ͳͲିଷ ൌ ͺǤʹ͸ͳ ή ͳͲିସ ௦ఙ ൌ ଴ ௦ଶ ௣Ǥଶ௤ೞ ௦ ௘ ൌ Ͷ ή ͳͲି଻ ൉ ሺ͹ʹ͹ʹሻሻଶ ൉ ଶήଶଷ ή ͳǤͷͺ͸ ή ʹͶͲ

Stator harmonic stray inductance is calculated as:

where stator harmonic stray coefficient is calculated:

where rotor stray harmonic inductance is calculated as: Rotor harmonic stray coefficient is calculated:

Inductance of stator winding overhang is calculated as below:

End-ring stray inductance of the rotor cage is calculated: ,

Slot stray inductance of stator winding is calculated:

- 89 -

ିଷ ିଷ ͳ͹Ǥ Ͷ Ͳͺ൉ Ͳͺ ൉ ͳ ͳͲ Ͳ ͲǤ ͹ ͵͹ ൉ ൉ͳͲ ͳͲ ଵ ସ ௦ ൌ ͵ ௔௩ ൅ ͲǤͲǤ͸ͺͷͺͷ൅൅  ൌ ͵ ൉ͺǤ ିଷ ൅ ͲǤͲǤ͸ͺͷͺͷ ൅ ͵Ǥͳ ൉ ͳͲͳͲିଷ ൌ ͳǤͷͺ͹ ൉ ͺǤ ͹ ͹Ͷ൉ Ͷ ൉ ͳ ͳͲ Ͳ ௔௩ ൌ ௜ ൅ʹ ௔ ൌ ͹Ǥʹͳʹ ൅ʹͳͲǤʹ͹͹ ή ͳͲିଷ ൌ ͺǤ͹ͶͶͷ ή ͳͲିଷǤ Rotor slot stray inductance of VZIM1 (Round bar) calculation:

Rotor slot stray inductance of VZIM1 (round bar) is calculated according to [1]

ఙǡ௕௔௥ ൌ ଴Ǥ ௥Ǥ ௘ ൌ Ͷ ή ͳͲି଻ ൉ ͲǤͺͶ͵͵ ൉ ʹͶͲ ή ͳͲିଷ ൌ ͲǤʹͷʹ ή ͳͲି଺ ǡ௥௢௨௡ௗ ൌ෥ ͳ ିଷ ିଷ ͳͲǤ ʹ ή ͳ ͳͲ Ͳ ͳ൉ͳͲ ସ ௥ ൌ ቆͳͷ  ൅ ͲǤͶ͹ቇ ൉ ǡ௥௢௨௡ௗ ൅  ൌ ቆͳͷ ൉ ͶǤͷ ή ͳͲିଷ ൅ ͲǤͶ͹ቇ ൉ ǡ௥௢௨௡ௗ ൅ ͶǤ ͷ ൉ͳͲିଷ

ൌ ͲǤͺͶ͵͵Ǥ

Rotor slot stray inductance of VZIM2 (Deep bar) calculation:

Rotor slot stray inductance of VZIM2 (Deep bar) is calculated according to [1]

Ͷʹͺͺ ൉ʹͶͲ ൉ ʹͶͲ ή ͳͲିଷ ൌ ͲǤͳ͸͵͸ ή ͳͲି଺ ఙǡ௕௔௥ ൌ ଴Ǥ ௥Ǥ ௘ ൌ Ͷ ή ͳͲି଻ ൉ ͲǤͷͶʹͺͺ ൌ௥ ൌ ଵήඥ ඥ ή௦ ൌή ௥ͳήͳήή ͷͲͷ଴Ͳή ˍൌ௖௨ͷͲ ൌ ͺǤ͵ʹ ή ͳͲͳͲିଷሺሺξ ξ ήʹ ሻή െͷͲήሺͶሺήʹ ήሻͳͲି଻ ή ͷǤͷ͵Ǥ͹ ή ͳͲͳͲ଻ൌሺሺͲǤͳǤͺ͹͸ͻ͵͵ͺ͹ሻ ሻ ሺ ሻ ͵ͺ͹ െ   ሺ ͳǤ ͹ ͵ͺ͹ ͵ͺ͹ሻ ൌ  ൌ ͲǤͻͺͷͺ  ൌ ʹ ሺʹ ሻ െሺ ሺ ሻ ሺ ሻ ሺ ሻ  ʹ ͳǤ ͹ ͵ͺ͹ ሺ  ͳǤ ͹ ͵ͺ͹ ͵ͺ͹ሻ െ   ሺ ͳǤ ͹ ͵ͺ͹ ͵ͺ͹ሻ ͺǤ ʹ ͳ ଵ ସ ൌ ൉ ൅ ൌ ൉ ͲǤ ͻ ͻͺͷͺ ͺ ͷͺ ൅ ௥ ͵    ͵ ή ͻǤͻǤʹ Ͷ ή ͳͲିଷ ൌ ͲǤͷͶʹͺͺǤ

where:

(At standstill) (At

Rotor slot stray inductance of VZIM3 (Wedge bar) calculation:

Rotor slot stray inductance of VZIM2 (Wedge bar) is calculated according to [1]

ିଷ ൌ ͲǤʹͺʹ ή ͳͲି଺ ൉ ͲǤ ͻ ͵ͷͺ ൉ ʹͶͲ ή ͳͲ ఙǡ௕௔௥ ൌ ଴Ǥ ௥Ǥ ௘ ൌ Ͷ ή ͳͲି଻ ͳ͹Ǥ ͹ ͳ ଵ ସ ൌ ൉ ൅ ൌ ൉ ͲǤ ͹ ͵ ൅ ௥ ͵ ௘ ǡ௪௘ௗ௚௘  ͵ ή ͻǤͺͺʹͺ ʹ ൌ ͲǤͻ͵ͷͺ

where:

ൌ௥ ൌ ଵήඥ ඥ ή௦ ൌή ௥ͳήͳήή ͷͲͷ଴Ͳή ˍൌ௖௨ͷͲ ൌ ͳ͹Ǥ͹ ή ͳͲͳͲିଷξ ξ ήή ͷͲͷͲ ή ͶήͶ ή ή ͳͲି଻ ή ͷǤ͹ ή ͳͲͳͲ଻ ൌ ͳǤͺ͹͸ͷǤ (At standstill) (At

- 90 -

8.2 Appendix II: a) Calculation of slips, where a synchronous harmonic torque of  VZIM1, VZIM2, VZIM3 happens

In order to derive the slips that synchronous harmonics of the VZIM occur, a ccording to [2], Stator ordinal numbers are derived as,

ൌ ͳ൅ ͳ൅ ʹ

ൌ ͳ ൅ ͸ ൌ ͳǡͳǡെͷǡ͹ െͷǡ͹ǡǡ െͳͳǡͳ͵ǡെͳ͹ ͳ͵ǡ െͳ͹ǡͳͻ ǡͳͻǡǡ െʹ͵ െʹ͵ǡʹͷǡ ǡʹͷǡെʹͻ െʹͻǡ͵ͳ ǡ͵ͳ Rotor ordinal numbers are derived as, ൌ ͳ ൅ ௥ ൌ ͳ ൅ ʹͺʹ ൌ ͳǡͳǡെͳ͵ǡ െͳ͵ǡͳͷǡെʹ͹ ͳͷǡ െʹ͹ǡʹͻǡ ǡʹͻǡെͶͳ െͶͳ Condition for slip where synchronous harmonic occurs is derived as, ൌ ൌͳ ൌ ൌͳ ൌͳ ൌെ ൌ െ൅ ͳͳ ൌ െ ൌ ͳ͵ ൌ െ൅ ͳͳ ൌ ͳͳ͵͵ െ൅ ͳͳ ൌ ͲǤͺͷ͹ െ ͳ ൌ ͳǤͲ͹ͳ ൌ െ ൌ െʹͻ ൌ െ൅ ͳͳ ൌ െʹͻ െʹͻ ൅ ͳ ௦

After deriving the slips, the related speeds of synchronous harmonic torque is calculated as

ൌ ௦௬௖௦௬௡െ ൌ ሺͳ െ ሻ ௦௬௡ ൌ ሺͳ െ ͲǤͺ ͲǤͺͷ͹ ͷ͹ሻͳͷͲͲ ൌ ʹͳͶǤͶ ൌ ሺͳ െ ሻ ௦௬௡ ൌ ሺͳ െ ͳǤͲ ͳǤͲ͹ͳ ͹ͳሻሻͳͷͲͲ ൌ െͳͲ͸Ǥͷ

In Table I synchronous harmonic torque slips which are the same for VZIM1, VZIM2, and VZIM3 are shown. Table I: Synchronous harmonic torque slip Harmonic 13 -29

slip 0.857 1.071

Rotor speed(rpm) 214.5 -106.5

b) Calculation of the slips which asynchronous harmonic torque happens in VZIM1 Slips that asynchronous harmonic torques of the VZIM1 occur are calculated according to [2]. Stator ordinal numbers are derived as

ൌ ͳ൅ ͳ ൅ ʹ ௦ ൌ ͳ ൅ ͸ ൌ ͳǡെͷǡ͹ ͳǡ െͷǡ͹ǡǡ െͳͳǡͳ͵ǡെͳ͹ ͳ͵ǡ െͳ͹ǡͳͻ ǡͳͻǡǡ െʹ͵ െʹ͵ǡʹͷǡ ǡʹͷǡെʹͻ െʹͻǡ͵ͳ ǡ͵ͳǡǡ െ͵ͷ െ͵ͷǡ͵͹ ǡ͵͹ǤǤ The asynchronous harmonic slips are derived as ൌ ͳ െ ͳǤ

- 91 -

Table II: Stator ordinal numbers and related asynchronous harmonic torque slips 1 -5 7 -11 13 -17 19 -23 25 -29 31 -35 37 0 1.2 0.857 1.09 0.92 1.0588 0.947 1.043 0.96 0.96 1.034 0.96 0.96 1.028 0.07 0.072 2 Harmonic break down slips, are calculated for each asynchronous harmonic torque, according to [2] as

௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻǡ where ௥ఙ௩ is calculated as ௥ఙ௩ ൌ ሺͳΤ ௩ଶ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ .

For all ordinal numbers the breakdown slips are calculated as following:

ൌ െͷ: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ʹ ௣ ௘ ௥ ଶ ଶ ൌ ή ௥௛௩ ଴ ௥ ௪௥ ଶ ή ଶ ή ή ଶ ʹ ή ʹͺ ͳ͵͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ Ǥͷή ή ͳ ͳͲ Ͳ ି଻ ି଺ ൌ ʹǤͷ ʹǤͷ͵ ͵ ൉ ͳͲ ௥௛ǡିହ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή ʹͷ ή ʹ ή ିଷ ͲǤͶͷͷ ήͳͲ ͲǤͶ ή ͳͲ ሺሺ ή ή Τ ௥ ሻ ൌ   ሺሺെͷ ή ʹ ή Τʹͺ ʹͺሻሻ ൌ ͲǤͺͲ͵ ௩ିୀହ ൌ ήଶ ή Τ ௥ െͷήή ʹ ή Τʹͺ ଶ െͷ ʹǤͷ͵ήή ͳͲ ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ௥ఙǡିହ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤͺͲ͵ሻ െ ͳሻ ή ʹǤͷ͵ ൌ ʹǤ͸͸ ʹǤ͸͸ ή ͳͲ ͳͲି଺ ିସ ʹǤͳ͸ ʹǤͳ ͸ ൉ ͳ ͳͲ Ͳ ௥ ௩ିୀହǡ௕ ൌ േ ௦ ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺʹǤͷ ʹǤͷ͵͵ ή ͳͲ ͳͲି଺ ൅ ʹǤ͸͸ ʹǤ͸͸ήή ͳͲ ͳͲି଺ሻ ൌ േͲǤͳ͵ ൌͳെͳെ ௩ ௩ ൌ ͳ െ ሺͳ െ ሻ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െെͷͲǤͳ͵ ൌ ͳǤͳ͹Ͷ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅െͷͲǤͳ͵ ൌ ͳǤʹʹ͸ For  ൌ ͹: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ʹ ͶͲ ή ήͳͲ ͳͲ ି଻ ି଺ ൌ ʹǤʹ ʹǤʹͻͷ ͻͷ൉ ൉ ͳ ͳͲ Ͳ ௥௛ǡ଻ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή Ͷͻ ή ʹ ή ିଷ ͲǤͶͷ ή ͳͲ ሺ Τ ሻ ሺ Τ ሻ ሺ ή ή ௥ ൌ  ͹ ή ʹ ή ʹͺ ൌ ͲǤ͸͵͸  ௩ୀ଻ ൌ ή ଶή Τ ௥ ͹ ή ʹ ή Τʹͺ ଶ ͲǤ͸͵͸ሻሻ െ ͳሻ ή ʹǤʹͻͷ ʹͻͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ௥ఙǡିହ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤ͸͵͸ ൌ ͶǤ͸Ͷ ͶǤ͸Ͷ ή ͳͲ ͳͲି଺ For 

- 92 -

ିସ ʹǤ͹ ʹǤ͹ͷ͸ ͷ͸ ൉ ൉ͳͲ ͳͲ ௩ୀ଻ǡ௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺͶǤ͸Ͷ ή ͳͲି଺ ൅ ʹǤʹͻͷ ή ͳͲି଺ሻ ൌ േͲǤͲͻͻ ͳ െ ௩ ሺ ሻ ൌ ͳ െ ͳ െ ൌ ͳ െ ௩ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െ ͹ͲǤͲͻͻ ൌ ͲǤͺ͹ͳ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅ ͲǤͲͻͻ ͹ ൌ ͲǤͺͶ͵ For  ൌ െͳͳ: ௥ ௩ିୀଵଵǡ௕ ൌ േ ௦ ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ି଻ ି଺ ൌ ͲǤͷ ͲǤͷʹͶ ʹͶ ൉ ͳͲ ௥௛ǡ଻ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή ͳʹͳ ή ʹ ή ିଷ ͲǤͶͷͷ ή ͳͲ ͲǤͶ ͳͲ ሺሺ ή ή Τ ௥ሻ ൌ   ሺሺͳͳ ή ʹ ή Τʹͺ ʹͺሻሻ ൌ ͲǤʹͷʹ ௩ୀ଻ ൌ ή ଶή Τ ௥ ͳͳ ή ʹ ή Τʹͺ ଶ ͲǤʹͷʹሻሻ െ ͳሻ ή ͲǤͷʹͶ ͷʹͶ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ௥ఙǡିହ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤʹͷʹ ൌ ͺǤͻͻ ͺǤͻͻ ή ͳͲ ͳͲି଺ ିସ ʹǤ ͳ ͳ͸ ͸ ൉ ൉ͳͲ ͳͲ ௥ ௩ିୀଵଵǡ௕ ൌ േ ௦ ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺͲǤͷ ͲǤͷʹͶ ʹͶ ή ͳͲି଺ ൅ ͺǤͻͻ ή ͳͲି଺ሻ ൌ േͲǤͲ͹ʹ ൌͳെͳെ ௩ ௩ ൌ ͳ െ ሺͳ െ ሻ ͲǤͲ͹ʹ ൌ ͳǤͲͺͶ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െെͳͳ ͲǤͲ͹ʹ ൌ ͳǤͲͻ͹ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅െͳͳ ௥

ൌ ͳ͵: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ି଻ ି଺ ൌ ͲǤ͵ ͲǤ͵͹ͷ ͹ͷ ൉ ͳͲ ௥௛ǡଵଷ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή ͳ͸ͻ ή ʹ ή ିଷ ͲǤͶͷͷ ή ͳͲ ͲǤͶ ͳͲ ሺ Τ ሻ ሺ Τ ሻ ሺ  ή ή ሺ  ͳ͵ ή ʹ ʹή ή ʹͺሻ ʹͺ ௥ ൌ ௩ୀଵଷ ൌ ήଶ ή Τ ௥ ͳ͵ ή ʹ ή Τʹͺ ൌଶ ͲǤͲ͹͸ ͲǤͲ͹͸ሻሻ െ ͳሻ ή ͲǤ͵͹ͷ ή ͳͲି଺ ൅ ͲǤͲ ͲǤͲͳʹ͹ ͳʹ͹ ή ͳͲିସ ௥ఙǡଵଷ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤͲ͹͸ ൌ ͸Ǥͷͺͳ ή ͳͲିହ ିସ ʹǤͳ͸ ʹǤͳ ͸ ൉ ൉ͳͲ ͳͲ ௥ ௩ୀଵଷǡ௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺ͸Ǥͷͺͳ ή ͳͲି଺ ൅ ͲǤ͵͹ ͲǤ͵͹ͷͷ ή ͳͲି଺ሻ ൌ േͲǤͲͳ ൌͳെͳെ ௩ ௩ ൌ ͳ െ ሺͳ െ ሻ

For 

- 93 -

ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െͳ͵ͲǤͲͳ ൌ ͲǤͻʹ͵ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅ͳ͵ͲǤͲͳ ൌ ͲǤͻʹ͵ ൌ െͳ͹: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ି଻ ି଺ ൌ ͲǤʹͳͻ ൉ ͳͲ ௥௛ǡିଵ଻ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή ʹͺͻ ή ʹ ή ିଷ ͲǤͶͷ ή ͳͲ ሺሺ ή ή Τ ௥ ሻ ൌ   ሺሺͳ͹ ή ʹ ή Τʹͺ ʹͺሻሻ ൌ െͲǤͳ͸͵ ௩ିୀଵ଻ ൌ ήଶ ή Τ ௥ ͳ͹ ή ʹ ή Τʹͺ ଶ ͲǤͳ͸͵ሻሻ െ ͳሻ ή ͲǤʹͳͻ ʹͳͻ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ௥ఙǡିଵ଻ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤͳ͸͵ ൌ ͻǤʹͻ ͻǤʹͻ ή ͳͲ ͳͲି଺ ିସ ʹǤ ͳ ͳ͸ ͸ ൉ ൉ͳͲ ͳͲ ௥ ௩ିୀଵ଻ǡ௕ ൌ േ ௦ ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺͻǤʹͻή ʹͻ ή ͳͲ ͳͲି଺ ൅ ͲǤʹͳͻ ή ͳͲି଺ሻ ൌ േͲǤͲ͹ʹ ൌͳെͳെ ௩ ௩ ൌ ͳ െ ሺͳ െ ሻ ͲǤͲ͹ʹ ൌ ͳǤͲͷͶ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െെͳ͹ ͲǤͲ͹ʹ ൌ ͳǤͲ͸͵ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅െͳ͹ For  ൌ ͳͻ: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ି଻ ି଺ ൌ ͲǤͳ ͲǤͳ͹ͷ ͹ͷ ൉ ͳͲ ௥௛ǡଵଽ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή ͵͸ͳ ή ʹ ή ିଷ ͲǤͶͷͷ ή ͳͲ ͲǤͶ ͳͲ ሺ Τ ሻ ሺ Τ ሻ ሺ  ή ή ሺ  ͳͻ ή ʹ ή ʹͺ ௥ ൌ ௩ୀଵଽ ൌ ήଶ ή Τ ௥ ͳͻ ή ʹήʹ ή Τʹͺ ൌଶ െͲǤʹͳͳ ௥ఙǡଵଽ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤʹͳͳሻ െ ͳሻ ή ͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͷǤͲʹͷ ή ͳͲି଺ ିସ ʹǤͳ͸ ʹǤͳ ͸ ൉ ൉ͳͲ ͳͲ ௥ ௩ୀଵଽǡ௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͷǤͲʹͷ ή ͳͲି଺ሻ ൌ േͲǤͳ͵ʹ ൌͳെͳെ ௩ ௩ ൌ ͳ െ ሺͳ െ ሻ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െͳͻͲǤͳ͵ʹ ൌ ͲǤͻͷͶ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅ͳͻͲǤͳ͵ʹ ൌ ͲǤͻͶ For 

- 94 -

ൌ െʹ͵: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ି଻ ି଺ ൌ ͲǤͳ ͲǤͳʹ ʹ ൉ ൉ͳͲ ͳͲ ௥௛ǡିଶଷ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή ͷʹͻ ή ʹ ή ିଷ ͲǤͶͷ ή ͳͲ ሺ Τ ሻ ሺ Τ ሻ  ή ή  ʹ͵ ή ʹ ή ʹͺ ௥ ൌ ௩ିୀଶଷ ൌ ήଶ ή Τ ௥ ʹ͵ ή ʹ ή Τʹͺ ൌଶ െͲǤͳ͹Ͷ ͲǤͳ͹Ͷሻሻ െ ͳሻ ή ͲǤͳ͹ͷ ͳ͹ͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ௥ఙǡିଶଷ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤͳ͹Ͷ ൌ ͷǤͳͳ ͷǤͳͳ ή ͳͲ ͳͲି଺ ିସ ʹǤͳ͸൉ͳͲ ௥ ௩ିୀଶଷǡ௕ ൌ േ ௦ ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺͲǤͳʹή ͳʹ ή ͳͲ ͳͲି଺ ൅ ͷǤͳͳ ͷǤͳͳήή ͳͲ ͳͲି଺ሻ ൌ േͲǤͳ͵ͳͷ ൌͳെͳെ ௩ ௩ ൌ ͳ െ ሺͳ െ ሻ ͲǤͳ͵ͳͷ ൌ ͳǤͲ͵͹ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െെʹ͵ ͲǤͳ͵ͳͷ ൌ ͳǤͲͶͻ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅െʹ͵ For  ൌ ʹͷ: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ି଻ ൌ ͲǤͳͲͳͷ ൉ ͳͲି଺ ௥௛ǡଶହ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή ͸ʹͷ ή ʹ ή ͲǤͶͷͷ ή ͳͲ ͲǤͶ ͳͲିଷ ሺ ή ή Τ ௥ ሻ ൌ  ሺʹͷ ή ʹήʹ ή Τʹͺ ሻ ൌ െͲǤͳͳͳ ሺ  ሺ ʹͺሻ ൌ ௩ୀଶହ ήଶ ή Τ ௥ ʹͷ ή ʹ ή Τʹͺ ଶ ͲǤͲͳʹ͹ ͳʹ͹ ή ͳͲିସ ௥ఙǡଶହ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤͳͳͳሻ െ ͳሻ ή ͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͲǤͲ ൌ ͻǤͶ ή ͳͲି଺ ିସ ʹǤͳ͸ ʹǤͳ ͸ ൉ ͳ ͳͲ Ͳ ௥ ௩ୀଶହǡ௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺͲǤͳͲͳͷ ή ͳͲି଺ ൅ ͻǤͶ ή ͳͲ ͳͲି଺ሻ ൌ േͲǤͲ͹ʹ ൌͳെͳെ ௩ ௩ ൌ ͳ െ ሺͳ െ ሻ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െʹͷͲǤͲ͹ʹ ൌ ͲǤͻ͸ʹ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅ʹͷͲǤͲ͹ʹ ൌ ͲǤͻͷͺ For 

ൌ െʹͻ: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅

For 

௥௛௩ ሻ - 95 -

ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ʹ ͶͲ ή ήͳͲ ͳͲ ି଻ ି଻ ൌ Ͷ ൉ ͳͲ ൬ ή ͳ൰ ή ή ൌ ͲǤ͹ͷͶ ൉ ͳͲ ௥௛ǡିଶଽ ଶ ିଷ ʹ ή ͺͶͳ ή ʹ ͲǤͶͷ ή ͳͲ ሺሺ ή ή Τ ௥ሻ ൌ   ሺሺʹͻ ή ʹ ή Τʹͺ ʹͺሻሻ ൌ ͲǤͲ͵Ͷʹ ௩ିୀଶଽ ൌ ήଶ ή Τ ௥ ʹͻ ή ʹ ή Τʹͺ ଶ ͲǤͲͳʹ͹ ʹ͹ ή ͳͲିସ ௥ఙǡିଶଽ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤͲ͵Ͷʹሻ െ ͳሻ ή ͲǤ͹ͷͶ ή ͳͲି଻ ൅ ͲǤͲͳ ൌ ͸Ǥͷ͸ ͸Ǥͷ͸ ή ͳͲ ͳͲିହ ିସ ʹǤ ͳ ͳ͸ ͸ ൉ ൉ͳͲ ͳͲ ௥ ௩ିୀଶଽǡ௕ ൌ േ ௦ ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺ͸Ǥͷ ͸Ǥͷ͹͵ ͹͵ ή ͳͲିହሻ ൌ േͲǤͲͳ ൌͳെͳെ ௩ ௩ ൌ ͳ െ ሺͳ െ ሻ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െ ͲǤͲͳ ൌ ͳǤͲ͵Ͷ

െʹͻ ͲǤͲͳ ൌ ͳǤͲ͵Ͷͺ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅െʹͻ

All slips, in which asynchronous harmonic torques happen, and the related slips of the  breakdown harmonic torques, in addition to selected slips for time ti me stepping calculation of the VZIM1, are shown in Table V. c) Calculation of the slips, where an Asynchronous harmonic torque happens in VZIM2

Similar to VZIM1 all the calculations have been done for VZIM2. Stator ordinal numbers are derived as

ൌ ͳ൅ ͳ ൅ ʹ ௦ ൌ ͳ ൅ ͸ ൌ ͳǡെͷǡ͹ ͳǡ െͷǡ͹ǡǡ െͳͳǡͳ͵ǡെͳ͹ ͳ͵ǡ െͳ͹ǡͳͻ ǡͳͻǡǡ െʹ͵ െʹ͵ǡʹͷǡ ǡʹͷǡെʹͻ െʹͻǡ͵ͳ ǡ͵ͳǡǡ െ͵ͷ െ͵ͷǡ͵͹ ǡ͵͹ǤǤ The asynchronous harmonic slips are derived as ൌ ͳ െ ͳǤ Table III: Stator ordinal numbers and related asynchronous harmonic torque slips 1 -5 7 -11 13 -17 19 -23 25 -29 31 -35 37 0 1.2 0.857 1.09 0.92 1.0588 0.947 1.043 0.96 0.96 1.034 0.96 0.96 1.028 0.97 0.972 2 Harmonic break down slips are calculated for each asynchronous harmonic torque, according to [2] as

௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻǡ where ௥ఙ௩ is calculated as ௥ఙ௩ ൌ ሺͳΤ ௩ଶ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ,

For all ordinal numbers the breakdown slips are calculated as following:

ൌ െͷ: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅

௥௛௩ ሻ - 96 -

ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ Ǥͷή ή ͳ ͳͲ Ͳ ି଻ ି଺ ൌ Ͷ ൉ ͳͲ ൬ ή ͳ൰ ή ή ൌ ʹǤͷ ʹǤͷ͵ ͵ ൉ ͳͲ ௥௛ǡିହ ଶ ିଷ ʹ ή ʹͷ ή ʹ ͲǤͶͷͷ ήͳͲ ͲǤͶ ή ͳͲ ሺሺ ή ή Τ ௥ ሻ ൌ   ሺሺെͷ ή ʹ ή Τʹͺ ʹͺሻሻ ൌ ͲǤͺͲ͵ ௩ିୀହ ൌ ήଶ ή Τ ௥ െͷήή ʹ ή Τʹͺ ଶ െͷ ʹǤͷ͵ήή ͳͲ ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ௥ఙǡିହ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤͺͲ͵ሻ െ ͳሻ ή ʹǤͷ͵ ൌ ʹǤ͸͸ ʹǤ͸͸ ή ͳͲ ͳͲି଺ ିହ ͻǤʹͳ ͻǤʹ ͳ ൉ ͳ ͳͲ Ͳ ௥ ௩ିୀହǡ௕ ൌ േ ௦ ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺʹǤͷ ʹǤͷ͵͵ ή ͳͲ ͳͲି଺ ൅ ʹǤ͸͸ ʹǤ͸͸ήή ͳͲ ͳͲି଺ሻ ൌ േͲǤͲͷ͸ͷ ൌͳെͳെ ௩ ௩ ൌ ͳ െ ሺͳ െ ሻ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െ ͲǤͲͷ͸ͷ ൌ ͳǤͳͺͺ͹

െͷ ͲǤͲͷ͸ͷ ൌ ͳǤʹͳͳ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅ െͷ ൌ ͹: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ʹ ͶͲ ή ήͳͲ ͳͲ ି଻ ି଺ ൌ ʹǤʹ ʹǤʹͻͷ ͻͷ൉ ൉ ͳ ͳͲ Ͳ ௥௛ǡ଻ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή Ͷͻ ή ʹ ή ିଷ ͲǤͶͷ ή ͳͲ ሺ Τ ሻ ሺ Τ ሻ  ή ή ௥ ൌ  ͹ ή ʹ ή ʹͺ ൌ ͲǤ͸͵͸ ௩ୀ଻ ൌ ή ଶή Τ ௥ ͹ ή ʹ ή Τʹͺ ଶ ͲǤ͸͵͸ሻሻ െ ͳሻ ή ʹǤʹͻͷ ʹͻͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ௥ఙǡିହ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤ͸͵͸ ൌ ͶǤ͸Ͷ ͶǤ͸Ͷ ή ͳͲ ͳͲି଺ ିହ ͻǤʹͳ ͻǤʹ ͳ ൉ ͳ ͳͲ Ͳ ௥ ௩ୀ଻ǡ௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺͶǤ͸Ͷ ή ͳͲି଺ ൅ ʹǤʹͻͷ ή ͳͲି଺ሻ ൌ േͲǤͲͶ͵͸ ൌͳെͳെ ௩ ௩ ൌ ͳ െ ሺͳ െ ሻ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െ ͲǤͲͶ͵͸ ͹ ൌ ͲǤͺ͸͵͵ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅ ͲǤͲͶ͵͸ ͹ ൌ ͲǤͺͷ ൌ െͳͳ: ௥ ௩ିୀଵଵǡ௕ ൌ േ ௦ ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ି଻ ି଺ ൌ ͲǤͷ ͲǤͷʹͶ ʹͶ ൉ ͳͲ ௥௛ǡ଻ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή ͳʹͳ ή ʹ ή ିଷ ͲǤͶͷͷ ή ͳͲ ͲǤͶ ͳͲ - 97 -

ሺ ή ή Τ ௥ሻ ൌ ሺͳͳ ή ʹήʹ ή Τʹͺሻ ൌ ͲǤʹͷʹ ௩ୀ଻ ൌ ή ଶή Τ ௥ ͳͳ ή ʹ ή Τʹͺ ଶ ͲǤʹͷʹሻሻ െ ͳሻ ή ͲǤͷʹͶ ͷʹͶ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ௥ఙǡିହ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤʹͷʹ ൌ ͺǤͻͻ ͺǤͻͻ ή ͳͲ ͳͲି଺ ିସ ʹǤ ͳ ͳ͸ ͸ ൉ ൉ͳͲ ͳͲ ௥ ௩ିୀଵଵǡ௕ ൌ േ ௦ ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺͲǤͷ ͲǤͷʹͶ ʹͶ ή ͳͲି଺ ൅ ͺǤͻͻ ή ͳͲି଺ሻ ൌ േͲǤͲ͵ʹ ͳ െ ௩ ሺ ሻ ൌ ͳ െ ͳ െ ൌ ͳ െ ௩ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െ ͲǤͲ͵ʹ ൌ ͳǤͲͺͺ െͳͳ ͳ െ ͳ ൅ ͲǤͲ͵ʹ ൌ ͳǤͲͻ͵ ൌ ͳ െ ௩ ൌ ͳ െ െͳͳ

ൌ ͳ͵: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ି଻ ି଺ ൌ ͲǤ͵ ͲǤ͵͹ͷ ͹ͷ ൉ ͳͲ ௥௛ǡଵଷ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή ͳ͸ͻ ή ʹ ή ିଷ ͲǤͶͷͷ ή ͳͲ ͲǤͶ ͳͲ ሺሺ ή ή Τ ௥ ሻ ൌ   ሺሺͳ͵ ή ʹήʹ ή Τʹͺ ʹͺሻሻ ൌ ͲǤͲ͹͸ ௩ୀଵଷ ൌ ήଶ ή Τ ௥ ͳ͵ ή ʹ ή Τʹͺ ଶ ͲǤͲ͹͸ሻሻ െ ͳሻ ή ͲǤ͵͹ͷ ή ͳͲି଺ ൅ ͲǤͲ ͲǤͲͳʹ͹ ͳʹ͹ ή ͳͲିସ ௥ఙǡଵଷ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤͲ͹͸ ൌ ͸Ǥͷͺͳ ή ͳͲିହ ିହ ͻǤʹͳ ͻǤʹ ͳ ൉ ൉ͳͲ ͳͲ ௥ ௩ୀଵଷǡ௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺ͸Ǥͷͺͳ ή ͳͲି଺ ൅ ͲǤ͵͹ ͲǤ͵͹ͷͷ ή ͳͲି଺ሻ ൌ േͲǤͲͲͶͶ͵ʹ ͳ െ ௩ ሺ ሻ ൌ ͳ െ ͳ െ ൌ ͳ െ ௩ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െ ͲǤͲͲͶͶ͵ʹ ͳ͵ ൌ ͲǤͻʹ͵Ͷ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅ ͲǤͲͲͶͶ͵ʹ ͳ͵ ൌ ͲǤͻʹʹ͹ ൌ െͳ͹: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ି଻ ି଺ ൌ Ͷ ൉ ͳͲ ൬ ή ͳ൰ ή ή ൌ ͲǤʹͳͻ ൉ ͳͲ ௥௛ǡିଵ଻ ଶ ିଷ ʹ ή ʹͺͻ ή ʹ ͲǤͶͷ ή ͳͲ ሺሺ ή ή Τ ௥ ሻ ൌ   ሺሺͳ͹ ή ʹ ή Τʹͺ ʹͺሻሻ ൌ െͲǤͳ͸͵ ௩ିୀଵ଻ ൌ ήଶ ή Τ ௥ ͳ͹ ή ʹ ή Τʹͺ ଶ ͲǤͳ͸͵ሻሻ െ ͳሻ ή ͲǤʹͳͻ ʹͳͻ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ௥ఙǡିଵ଻ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤͳ͸͵ ൌ ͻǤʹͻ ͻǤʹͻ ή ͳͲ ͳͲି଺ - 98 -

ିହ ͻǤ ʹ ʹͳ ͳ ൉ ൉ͳͲ ͳͲ ௩ିୀଵ଻ǡ௕ ൌ േ ௦ ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺͻǤʹͻή ʹͻ ή ͳͲ ͳͲି଺ ൅ ͲǤʹͳͻ ή ͳͲି଺ሻ ൌ േͲǤͲ͵Ͳͺ ൌͳെͳെ ௩ ௩ ൌ ͳ െ ሺͳ െ ሻ ͲǤͲ͵Ͳͺ ൌ ͳǤͲͷ͹ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െെͳ͹ ͲǤͲ͵Ͳͺ ൌ ͳǤͲ͸Ͳ͸ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅െͳ͹ ௥

ൌ ͳͻ: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ʹ ௣ ௘ ௥ ଶ ଶ ൌ ή ௥௛௩ ଴ ௥ ௪௥ ଶ ή ଶ ή ή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ି଻ ି଺ ൌ ͲǤͳ ͲǤͳ͹ͷ ͹ͷ ൉ ͳͲ ௥௛ǡଵଽ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή ͵͸ͳ ή ʹ ή ିଷ ͲǤͶͷͷ ή ͳͲ ͲǤͶ ͳͲ ሺሺ ή ή Τ ௥ ሻ ൌ   ሺሺͳͻ ή ʹ ή Τʹͺሻ ൌ െͲǤʹͳͳ ௩ୀଵଽ ൌ ήଶ ή Τ ௥ ͳͻ ή ʹήʹ ή Τʹͺ ଶ ͲǤʹͳͳሻሻ െ ͳሻ ή ͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ௥ఙǡଵଽ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤʹͳͳ ൌ ͷǤͲʹͷ ή ͳͲି଺ ିହ ͻǤʹͳ ͻǤʹ ͳ ൉ ൉ͳͲ ͳͲ ௥ ௩ୀଵଽǡ௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͷǤͲʹͷ ή ͳͲି଺ሻ ൌ േͲǤͲͷ͸Ͷ ͳ െ ௩ ሺ ሻ ൌ ͳ െ ͳ െ ൌ ͳ െ ௩ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െ ͳͻͲǤͲͷ͸Ͷ ൌ ͲǤͻͷ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅ ͲǤͲͷ͸Ͷ ͳͻ ൌ ͲǤͻͶͶ ൌ െʹ͵: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ି଻ ି଺ ൌ ͲǤͳ ͲǤͳʹ ʹ ൉ ൉ͳͲ ͳͲ ௥௛ǡିଶଷ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή ͷʹͻ ή ʹ ή ିଷ ͲǤͶͷ ή ͳͲ ሺሺ ή ή Τ ௥ ሻ ൌ   ሺሺʹ͵ ή ʹ ή Τʹͺ ʹͺሻሻ ൌ െͲǤͳ͹Ͷ ௩ିୀଶଷ ൌ ήଶ ή Τ ௥ ʹ͵ ή ʹ ή Τʹͺ ଶ ͳ͹ͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ௥ఙǡିଶଷ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤͳ͹Ͷሻ െ ͳሻ ή ͲǤͳ͹ͷ ൌ ͷǤͳͳ ͷǤͳͳ ή ͳͲ ͳͲି଺ ିହ ͻǤʹͳ൉ͳͲ ௥ ௩ିୀଶଷǡ௕ ൌ േ ௦ ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺͲǤͳʹή ͳʹ ή ͳͲ ͳͲି଺ ൅ ͷǤͳͳ ͷǤͳͳήή ͳͲ ͳͲି଺ሻ ൌ േͲǤͲͷ͸ ͳ െ ௩ ሺ ሻ ൌ ͳ െ ͳ െ ൌ ͳ െ ௩ - 99 -

ͲǤͲǤͲͷ͸ ൌ ͳǤͲͶͳ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െെʹ͵ ͲǤͲͷ͸ ൌ ͳǤͲͶͷ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅െʹ͵ ൌ ʹͷ: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ʹ ௣ ௘ ௥ ଶ ଶ ൌ ή ௥௛௩ ଴ ௥ ௪௥ ଶ ή ଶ ή ή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ି଻ ൌ ͲǤͳͲͳͷ ൉ ͳͲି଺ ௥௛ǡଶହ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή ͸ʹͷ ή ʹ ή ͲǤͶͷͷ ή ͳͲ ͲǤͶ ͳͲିଷ ሺሺ ή ή Τ ௥ ሻ ൌ   ሺሺʹͷ ή ʹήʹ ή Τʹͺ ʹͺሻሻ ൌ െͲǤͳͳͳ ௩ୀଶହ ൌ ήଶ ή Τ ௥ ʹͷ ή ʹ ή Τʹͺ ଶ ͲǤͳͳͳሻሻ െ ͳሻ ή ͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͲǤͲ ͲǤͲͳʹ͹ ͳʹ͹ ή ͳͲିସ ௥ఙǡଶହ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤͳͳͳ ൌ ͻǤͶ ή ͳͲି଺ ିହ ͻǤʹͳ ͻǤʹ ͳ ൉ ͳ ͳͲ Ͳ ௥ ௩ୀଶହǡ௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺͲǤͳͲͳͷ ή ͳͲି଺ ൅ ͻǤͶ ή ͳͲ ͳͲି଺ሻ ൌ േͲǤͲ͵Ͳͺ ͳ െ ௩ ሺ ሻ ൌ ͳ െ ͳ െ ൌ ͳ െ ௩ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െ ʹͷͲǤͲ͵Ͳͺ ൌ ͲǤͻ͸ͳ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅ ͲǤͲ͵Ͳͺ ʹͷ ൌ ͲǤͻͷͺ͹ ൌ െʹͻ: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ʹ ͶͲ ή ήͳͲ ͳͲ ି଻ ି଻ ൌ ͲǤ͹ͷͶ ൉ ͳͲ ௥௛ǡିଶଽ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή ͺͶͳ ή ʹ ή ିଷ ͲǤͶͷ ή ͳͲ ሺሺ ή ή Τ ௥ሻ ൌ   ሺሺʹͻ ή ʹ ή Τʹͺ ʹͺሻሻ ൌ ͲǤͲ͵Ͷʹ ௩ିୀଶଽ ൌ ήଶ ή Τ ௥ ʹͻ ή ʹ ή Τʹͺ ଶ ͲǤͲ͵Ͷʹሻሻ െ ͳሻ ή ͲǤ͹ͷͶ ή ͳͲି଻ ൅ ͲǤͲͳ ͲǤͲͳʹ͹ ʹ͹ ή ͳͲିସ ௥ఙǡିଶଽ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤͲ͵Ͷʹ ൌ ͸Ǥͷ͸ ͸Ǥͷ͸ ή ͳͲ ͳͲିହ ିହ ͻǤ ʹ ʹͳ ͳ ൉ ൉ͳͲ ͳͲ ௥ ௩ିୀଶଽǡ௕ ൌ േ ௦ ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺ͸Ǥͷ ͸Ǥͷ͹͵ ͹͵ ή ͳͲିହሻ ൌ േͲǤͲͲͶͶ͸ ͳ െ ௩ ሺ ሻ ൌ ͳ െ ͳ െ ൌ ͳ െ ௩ ͲǤͲͲͶͶ͸ ൌ ͳǤͲ͵Ͷ͵ʹ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െെʹͻ ͲǤͲͲͶͶ͸ ൌ ͳǤͲ͵Ͷ͸͵ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅ െʹͻ - 100 -

All slips, in which asynchronous harmonic torques happen, and the related slips of the  breakdown harmonic torques, in addition to selected slips for time ti me stepping calculation of the VZIM2 are shown in Table VI. d) Calculation of the slips where an Asynchronous harmonic torque happens in VZIM3

Similar to VZIM1and VZIM2 all the calculations have been done for VZIM3. Stator ordinal numbers are derived as

ൌ ͳ൅ ͳ ൅ ʹ ௦ ൌ ͳ ൅ ͸ ൌ ͳǡെͷǡ͹ ͳǡ െͷǡ͹ǡǡ െͳͳǡͳ͵ǡെͳ͹ ͳ͵ǡ െͳ͹ǡͳͻ ǡͳͻǡǡ െʹ͵ െʹ͵ǡʹͷǡ ǡʹͷǡെʹͻ െʹͻǡ͵ͳ ǡ͵ͳǡǡ െ͵ͷ െ͵ͷǡ͵͹ ǡ͵͹ǤǤ The asynchronous harmonic slips are derived as, ൌ ͳ െ ͳǤ Table IV: Stator ordinal numbers and related asynchronous harmonic torque slips 1 -5 7 -11 13 -17 19 -23 25 -29 31 -35 37 0 1.2 0.857 1.09 0.92 1.0588 0.947 1.043 0.96 0.96 1.034 0.96 0.96 1.028 0.97 0.972 2 Harmonic break down slips are calculated for each asynchronous harmonic torque, according to [2] as

௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻǡ where ௥ఙ௩ is calculated as ௥ఙ௩ ൌ ሺͳΤ ௩ଶ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ .

For all ordinal numbers the breakdown slips are calculated as following:

ൌ െͷ: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ Ǥͷή ή ͳ ͳͲ Ͳ ି଻ ൌ ʹǤͷ͵ ʹǤͷ͵ ൉ ͳͲି଺ ௥௛ǡିହ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή ʹͷ ή ʹ ή ͲǤͶͷͷ ήͳͲ ͲǤͶ ή ͳͲିଷ ሺሺ ή ή Τ ௥ ሻ ൌ   ሺሺെͷ ή ʹ ή Τʹͺ ʹͺሻሻ ൌ ͲǤͺͲ͵ ௩ିୀହ ൌ ήଶ ή Τ ௥ െͷήή ʹ ή Τʹͺ ଶ െͷ ͲǤͺͲ͵ሻሻ െ ͳሻ ή ʹǤͷ͵ ʹǤͷ͵ήή ͳͲ ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ௥ఙǡିହ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤͺͲ͵ ൌ ʹǤ͸͸ ʹǤ͸͸ ή ͳͲ ͳͲି଺ ିହ ͵ǤͺͶ ͵Ǥͺ Ͷ ൉ ͳ ͳͲ Ͳ ௥ ௩ିୀହǡ௕ ൌ േ ௦ ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺʹǤͷ ʹǤͷ͵͵ ή ͳͲ ͳͲି଺ ൅ ʹǤ͸͸ ʹǤ͸͸ήή ͳͲ ͳͲି଺ሻ ൌ േͲǤͲʹ͵ͷ ൌͳെͳെ ௩ ௩ ൌ ͳ െ ሺͳ െ ሻ ͲǤͲʹ͵ͷ ൌ ͳǤͳͻͷ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െെͷ ͲǤͲʹ͵ͷ ൌ ͳǤʹͲͻ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅ െͷ - 101 -

ൌ ͹: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ʹ ͶͲ ή ήͳͲ ͳͲ ି଻ ି଺ ൌ ʹǤʹ ʹǤʹͻͷ ͻͷ൉ ൉ ͳ ͳͲ Ͳ ௥௛ǡ଻ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή Ͷͻ ή ʹ ή ିଷ ͲǤͶͷ ή ͳͲ ሺ Τ ሻ ሺ Τ ሻ  ή ή  ͹ ή ʹ ή ʹͺ ௥ ൌ ௩ୀ଻ ൌ ή ଶή Τ ௥ ͹ ή ʹ ή Τʹͺ ൌ ͲǤ͸͵͸ ͲǤ͸͵͸ሻሻଶ െ ͳሻ ή ʹǤʹͻͷ ʹͻͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ௥ఙǡିହ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤ͸͵͸ ͶǤ͸Ͷ ή ͳͲି଺ ିହ ͵ǤͺͶ ͵Ǥͺ Ͷ ൉ ͳ ͳͲ Ͳ ௥ ௩ୀ଻ǡ௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺͶǤ͸Ͷ ή ͳͲି଺ ൅ ʹǤʹͻͷ ή ͳͲି଺ሻ ൌ േͲǤͲͳ͹͹ ൌͳെͳെ ௩ ௩ ൌ ͳ െ ሺͳ െ ሻ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െ ͲǤͲͳ͹͹ ͹ ൌ ͲǤͺͷͻ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅ ͲǤͲͳ͹͹ ͹ ൌ ͲǤͺͷͷ ൌ െͳͳ: ௥ ௩ିୀଵଵǡ௕ ൌ േ ௦ ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ି଻ ൌ ͲǤͷʹͶ ͲǤͷʹͶ ൉ ͳͲି଺ ௥௛ǡ଻ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή ͳʹͳ ή ʹ ή ͲǤͶͷͷ ή ͳͲ ͲǤͶ ͳͲିଷ ሺ ή ή Τ ௥ሻ ൌ ሺͳͳ ή ʹ ή Τʹͺሻ ൌ ͲǤʹͷʹ ௩ୀ଻ ൌ ή ଶή Τ ௥ ͳͳ ή ʹ ή Τʹͺ ଶ ͲǤʹͷʹሻሻ െ ͳሻ ή ͲǤͷʹͶ ͷʹͶ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ௥ఙǡିହ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤʹͷʹ ൌ ͺǤͻͻ ͺǤͻͻ ή ͳͲ ͳͲି଺ ିହ ͵Ǥ ͺ ͺͶ Ͷ ൉ ൉ͳͲ ͳͲ ௥ ௩ିୀଵଵǡ௕ ൌ േ ௦ ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺͲǤͷ ͲǤͷʹͶ ʹͶ ή ͳͲି଺ ൅ ͺǤͻͻ ή ͳͲି଺ሻ ൌ േͲǤͲͳʹͻ ൌͳെͳെ ௩ ௩ ൌ ͳ െ ሺͳ െ ሻ ͲǤͲͳʹͻ ൌ ͳǤͲͺͻ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െെͳͳ ͲǤͲͳʹͻ ൌ ͳǤͲͻͳ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅െͳͳ ൌ ͳ͵: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅

௥௛௩ ሻ - 102 -

ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ି଻ ି଺ ൌ Ͷ ൉ ͳͲ ൬ ή ͳ൰ ή ή ൌ ͲǤ͵ ͲǤ͵͹ͷ ͹ͷ ൉ ͳͲ ௥௛ǡଵଷ ଶ ିଷ ʹ ή ͳ͸ͻ ή ʹ ͲǤͶͷͷ ή ͳͲ ͲǤͶ ͳͲ ሺሺ ή ή Τ ௥ ሻ ൌ   ሺሺͳ͵ ή ʹήʹ ή Τʹͺ ʹͺሻሻ ൌ ͲǤͲ͹͸ ௩ୀଵଷ ൌ ήଶ ή Τ ௥ ͳ͵ ή ʹ ή Τʹͺ ଶ ͲǤͲͳʹ͹ ͳʹ͹ ή ͳͲିସ ௥ఙǡଵଷ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤͲ͹͸ሻ െ ͳሻ ή ͲǤ͵͹ͷ ή ͳͲି଺ ൅ ͲǤͲ ൌ ͸Ǥͷͺͳ ή ͳͲିହ ିହ ͵ǤͺͶ ͵Ǥͺ Ͷ ൉ ൉ͳͲ ͳͲ ௥ ௩ୀଵଷǡ௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺ͸Ǥͷͺͳ ή ͳͲି଺ ൅ ͲǤ͵͹ ͲǤ͵͹ͷͷ ή ͳͲି଺ሻ ൌ േͲǤͲͲͳͺ͸ ൌͳെͳെ ௩ ௩ ൌ ͳ െ ሺͳ െ ሻ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െ ͲǤͲͲͳͺ͸ ൌ ͲǤͻʹ͵ʹ

ͳ͵ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅ ͲǤͲͲͳͺ͸ ͳ͵ ൌ ͲǤͻͳ͸ͺ ൌ െͳ͹: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ି଻ ି଺ ൌ ͲǤʹͳͻ ൉ ͳͲ ௥௛ǡିଵ଻ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή ʹͺͻ ή ʹ ή ିଷ ͲǤͶͷ ή ͳͲ ሺ Τ ሻ ሺ Τ ሻ ሺ  ή ή ሺ  ͳ͹ ή ʹ ή ʹͺሻ ʹͺ ௥ ൌ ௩ିୀଵ଻ ൌ ήଶ ή Τ ௥ ͳ͹ ή ʹ ή Τʹͺ ൌଶ െͲǤͳ͸͵ ʹͳͻ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ௥ఙǡିଵ଻ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤͳ͸͵ሻ െ ͳሻ ή ͲǤʹͳͻ ൌ ͻǤʹͻ ͻǤʹͻ ή ͳͲ ͳͲି଺ ିହ ͵Ǥ ͺ ͺͶ Ͷ ൉ ൉ͳͲ ͳͲ ௥ ௩ିୀଵ଻ǡ௕ ൌ േ ௦ ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺͻǤʹͻή ʹͻ ή ͳͲ ͳͲି଺ ൅ ͲǤʹͳͻ ή ͳͲି଺ሻ ൌ േͲǤͲͳʹͻ ൌͳെͳെ ௩ ௩ ൌ ͳ െ ሺͳ െ ሻ ͲǤͲͳʹͻ ൌ ͳǤͲͷͺ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െെͳ͹ ͲǤͲͳʹͻ ൌ ͳǤͲͷͻ͸ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅െͳ͹ ൌ ͳͻ: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ଶ ή ʹ ௥ ή ௣ ௘ ௥௛௩ ൌ ଴ ௥ଶ ௪௥ ଶή ଶή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ି଻ ൌ ͲǤͳ͹ͷ ͲǤͳ͹ͷ ൉ ͳͲି଺ ௥௛ǡଵଽ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή ͵͸ͳ ή ʹ ή ͲǤͶͷͷ ή ͳͲ ͲǤͶ ͳͲିଷ - 103 -

ሺ ή ή Τ ௥ ሻ ൌ ሺͳͻ ή ʹήʹ ή Τʹͺሻ ൌ െͲǤʹͳͳ ௩ୀଵଽ ൌ ήଶ ή Τ ௥ ͳͻ ή ʹήʹ ή Τʹͺ ଶ ͲǤʹͳͳሻሻ െ ͳሻ ή ͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ௥ఙǡଵଽ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤʹͳͳ ൌ ͷǤͲʹͷ ή ͳͲି଺ ିହ ͵ǤͺͶ ͵Ǥͺ Ͷ ൉ ൉ͳͲ ͳͲ ௥ ௩ୀଵଽǡ௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͷǤͲʹͷ ή ͳͲି଺ሻ ൌ േͲǤͲͲʹ͵͹ ൌͳെͳെ ௩ ௩ ൌ ͳ െ ሺͳ െ ሻ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െ ͲǤͲͲʹ͵͹ ൌ ͲǤͻͶ͹Ͷ ͳͻ ͳ െ ͳ ൅ ͲǤͲͲʹ͵͹ ൌ ͳ െ ௩ ൌ ͳ െ ͳͻ ൌ ͲǤͻͶ͸͸

ൌ െʹ͵: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ʹ ௣ ௘ ௥ ଶ ଶ ൌ ή ௥௛௩ ଴ ௥ ௪௥ ଶ ή ଶ ή ή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ି଻ ି଺ ൌ ͲǤͳ ͲǤͳʹ ʹ ൉ ൉ͳͲ ͳͲ ௥௛ǡିଶଷ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή ͷʹͻ ή ʹ ή ିଷ ͲǤͶͷ ή ͳͲ ሺ ή ή Τ ௥ ሻ ൌ ሺʹ͵ ή ʹ ή Τʹͺሻ ൌ െͲǤͳ͹Ͷ ௩ିୀଶଷ ൌ ήଶ ή Τ ௥ ʹ͵ ή ʹ ή Τʹͺ ଶ ͲǤͳ͹Ͷሻሻ െ ͳሻ ή ͲǤͳ͹ͷ ͳ͹ͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ௥ఙǡିଶଷ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤͳ͹Ͷ ൌ ͷǤͳͳ ͷǤͳͳ ή ͳͲ ͳͲି଺ ିହ ͵ǤͺͶ൉ͳͲ ௥ ௩ିୀଶଷǡ௕ ൌ േ ௦ ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺͲǤͳʹή ͳʹ ή ͳͲ ͳͲି଺ ൅ ͷǤͳͳ ͷǤͳͳήή ͳͲ ͳͲି଺ሻ ൌ േͲǤͲʹ͵ͷ ͳ െ ௩ ሺ ሻ ൌ ͳ െ ͳ െ ൌ ͳ െ ௩ ͲǤͲʹ͵ͷ ൌ ͳǤͲͶʹ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െെʹ͵ ͲǤͲʹ͵ͷ ൌ ͳǤͲͶͶ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅െʹ͵ ൌ ʹͷ: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ʹ ௣ ௘ ௥ ଶ ଶ ൌ ή ௥௛௩ ଴ ௥ ௪௥ ଶ ή ଶ ή ή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ି଻ ି଺ ൌ ͲǤͳͲͳͷ ൉ ͳͲ ௥௛ǡଶହ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή ͸ʹͷ ή ʹ ή ିଷ ͲǤͶͷͷ ή ͳͲ ͲǤͶ ͳͲ ሺሺ ή ή Τ ௥ ሻ ൌ   ሺሺʹͷ ή ʹήʹ ή Τʹͺ ʹͺሻሻ ൌ െͲǤͳͳͳ ௩ୀଶହ ൌ ήଶ ή Τ ௥ ʹͷ ή ʹ ή Τʹͺ ଶ ͲǤͳͳͳሻሻ െ ͳሻ ή ͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͲǤͲ ͲǤͲͳʹ͹ ͳʹ͹ ή ͳͲିସ ௥ఙǡଶହ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤͳͳͳ ൌ ͻǤͶ ή ͳͲି଺ - 104 -

ିହ ͵Ǥͺ ͵ǤͺͶ Ͷ ൉ ͳ ͳͲ Ͳ ௩ୀଶହǡ௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺͲǤͳͲͳͷ ή ͳͲି଺ ൅ ͻǤͶ ή ͳͲ ͳͲି଺ሻ ൌ േͲǤͲͳʹͻ ൌͳെͳെ ௩ ௩ ൌ ͳ െ ሺͳ െ ሻ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െ ʹͷͲǤͲͳʹͻ ൌ ͲǤͻ͸Ͳͷ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅ ͲǤͲͳʹͻ ʹͷ ൌ ͲǤͻͷͻͷ ൌ െʹͻ: ௥ ௩௕ ൌ േ ௦ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ʹ ௣ ௘ ௥ ଶ ଶ ൌ ή ௥௛௩ ଴ ௥ ௪௥ ଶ ή ଶ ή ή ଶ ʹ ή ʹͺ ͳ͵ ିଷ ή ʹͶͲ ିଷ ͳ ͳ͵͵Ǥͷ ͵Ǥͷή ή ͳͲ ʹ ͶͲ ή ήͳͲ ͳͲ ି଻ ି଻ ൌ ͲǤ͹ͷͶ ൉ ͳͲ ௥௛ǡିଶଽ ൌ Ͷ ൉ ͳͲ ൬ʹ ή ͳ൰ ή ଶ ή ͺͶͳ ή ʹ ή ିଷ ͲǤͶͷ ή ͳͲ ሺሺ ή ή Τ ௥ሻ ൌ   ሺሺʹͻ ή ʹ ή Τʹͺ ʹͺሻሻ ൌ ͲǤͲ͵Ͷʹ ௩ିୀଶଽ ൌ ήଶ ή Τ ௥ ʹͻ ή ʹ ή Τʹͺ ଶ ͲǤͲ͵Ͷʹሻሻ െ ͳሻ ή ͲǤ͹ͷͶ ή ͳͲି଻ ൅ ͲǤͲͳ ͲǤͲͳʹ͹ ʹ͹ ή ͳͲିସ ௥ఙǡିଶଽ ൌ ሺͳΤ ௩ െ ͳሻ ή ௥௛௩ ൅ ௥ఙ ൌ ሺͳΤሺͲǤͲ͵Ͷʹ ൌ ͸Ǥͷ͸ ͸Ǥͷ͸ ή ͳͲ ͳͲିହ ିହ ͵Ǥ ͺ ͺͶ Ͷ ൉ ൉ͳͲ ͳͲ ௥ ௩ିୀଶଽǡ௕ ൌ േ ௦ ሺ ௥ఙ௩ ൅ ௥௛௩ ሻ ൌൌ േ ʹ ή ή ͷͲ ή ሺ͸Ǥͷ ͸Ǥͷ͹͵ ͹͵ ή ͳͲିହሻ ൌ േͲǤͲͲͳͺ͹ͷ ͳ െ ௩ ሺ ሻ ൌ ͳ െ ͳ െ ൌ ͳ െ ௩ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ െ ͲǤͲͲͳͺ͹ͷ െʹͻ ൌ ͳǤͲ͵ͶͶͳͺ ൌ ͳ െ ͳ െ ௩ ൌ ͳ െ ͳ ൅ ͲǤͲͲͳͺ͹ͷ െʹͻ ൌ ͳǤͲ͵Ͷ͵ͺͳ ௥

All slips, in which asynchronous harmonic torques happen, and the related slips of the  breakdown harmonic torques, in addition to selected slips for time ti me stepping calculation of the VZIM2, are shown in Table VII.

- 105 -

Table V: Slips, which are used to perform time-stepping analysis of VZIM1 Slip Speed(rpm) 0.00001 1500 0.01 1485 0.02 1470 0.0253 1462 0.03 1455 0.04 1440 0.05 1425 0.06 1410 0.08 1380 0.1 1350 0.12 1320 0.15 1275 0.2 1200 0.3 1050 0.45 825 0.65 525 0.843 235.5 0.857 214.5 0.871 193.5 0.917 124.5 0.92 120 0.923 115.5 0.94 90 0.947 79.5 0.954 69 0.958 63 0.96 60 0.962 57 0.987 19.5 1 0 1.013 -19.5 1.034 -51 1.0344 -51.6 1.0348 -52.2 1.037 -55.5 1.043 -64.5 1.054 -81 1.0588 -88.2 1.063 -94.5 1.071 -106.5 1.077 -115.5 1.084 -126 1.09 -135 1.096 -144 1.174 -261 1.2 -300 1.226 -339 1.33 -495 1.66 -990 2 -1500

- 106 -

Table VI: Slips, which are used to perform time-stepping analysis of VZIM2 Slip Speed(rpm) 0.00001 1500 0.01 1485 0.02 1470 0.0253 1462 0.03 1455 0.04 1440 0.05 1425 0.06 1410 0.08 1380 0.1 1350 0.12 1320 0.15 1275 0.2 1200 0.3 1050 0.45 825 0.65 525 0.85 225 0.857 214.5 0.8633 205.05 0.9166 125.1 0.92 120 0.9234 114.9 0.944 84 0.947 79.5 0.95 75 0.9587 61.95 0.96 60 0.961 58.5 0.995 7.5 1.0044 -6.6 1.013 -19.5 1.0344 -51 1.03432 -51.48 1.03463 -51.94 1.041 -61.5 1.043 -64.5 1.045 -73.5 1.057 -85.5 1.0588 -88.2 1.0606 -90.9 1.0643 -96.45 1.071 -106.5 1.077 -115.5 1.088 -132 1.09 -135 1.093 -139.5 1.1887 -283.05 1.211 -316.5 1.33 -495 1.66 -990 2 -1500 - 107 -

Table VII: Slips, which are used to perform time-stepping analysis of VZIM3 Slip Speed(rpm) 0.00001 1500 0.01 1485 0.02 1470 0.0253 1462 0.03 1455 0.04 1440 0.05 1425 0.06 1410 0.08 1380 0.1 1350 0.12 1320 0.15 1275 0.2 1200 0.3 1050 0.45 825 0.65 525 0.855 217.5 0.859 211.5 0.8633 205.05 0.9168 124.5 0.92 120 0.9232 115.2 0.9466 80.1 0.947 79.5 0.9474 78.9 0.9595 60.75 0.96 60 1.34 -51 1.03438 -51.57 1.0344 -51.6 1.034418 -51.62 1.042 -63 1.043 -64.5 1.044 -66 1.058 -87 1.0588 -88.2 1.0596 -89.4 1.071 -106.5 1.089 -133.5 1.09 -135 1.091 -136.5 1.191 -286.5 1.2 -300 1.209 -313.5 1.33 -495 1.66 -990 2 -1500

- 108 -

   3    M    I    Z    V    d   n   a    2    M    I    Z    V  ,    1    M    I    Z    V    f   o   s   e   s   s   o    l    d   e    t   a    l   u   c    l   a    C   :    I    I    I   x    i    d   n   e   p   p    A    3  .    8

   W     /    2    1    4    2    r    2    4    2    9    9  ,    5    3    3    5    6    4    7    e  ,  ,  ,  ,  ,     f    5    0    6    4  ,    8    5    6  ,    2    S    2    2    2    2    1    0    3    1    S    1    O    L

   W     /    5    r    2    4    6    5    2    6    8    0    2    5    1    7    7  ,    0    e    3  ,  ,  ,  ,     f    6    9    1  ,    8  ,    3  ,    3    S    7    7    1    S    3    0    3    0    3    7    3    1    O    L

   W    8     /    2    7    8    2    1    s    2  ,    1    6    5  ,    8  ,  ,  ,    2    0    e  ,  ,    9     f    1    1    6    9    8    6    8    1    5    S    6    1    2    1    1    3    7    3    S    1    1    1    O    L

   W    2    6    6     /    9    5    2    1    1    7    9    s    8    0  ,  ,  ,    8    0    0    7    8    e  ,  ,  ,    9  ,  ,     f    0    7    1    7    7    8    1    3    S    0    7    S    2    1    0    1    2    8    9    7    2    1    2    O    L

   W   4    8    8    4  ,     /    4  ,    0    6    6  ,    3  ,    3    2    4  ,    6    8    7  ,    9    5  ,    1    u    3    3    5    6    c    7    4    7    3    1    8    S    9    6    9    2    0    5    5    4    4    9    2    S    1    1    1    3    5    1    4    O    L

   2    W    2    1    9    2     /    5  ,    7    7    1  ,    8    3    2    4    2  ,    7    2  ,  ,  ,    1    8    u    2    2    3    5  ,    0    c    3    7    5    3    3    2    8    3    S    3    5    2    4    9    6    3    9    S    8    0    5    8    6    4    O    6    L

   W    7    8    1    6     /  ,    2  ,    9    6    2  ,  ,    1    7    0    4    1    7    3  ,  ,  ,    0    0    u    1    4    4    4    0    c    6    4    2    4    6    S    4    9    6    8    3    3    1    9    3    S    9    7    7    1    2    6    2    5    O    L

   W    5    3    8    9    7     /  ,    8  ,    3    8    0    9  ,  ,    1    5    0    8    7  ,  ,  ,    9    3    1    u    5    4  ,    6    0    7    c    2    9    1    2    8    4    S    7    0    6    5    5    1    2    S    8    7    5    4    6    6    9    6    5    7    O    L

   y    c    3    4    5    8    2    3    n    6    5    9    5    0    0    7    0    6    e    8    7    9    1    2    4    5    i  ,    8  ,  ,    c    0  ,    2  ,  ,    9    i  ,  ,    0    0    0    0    0     f   -      0    0     f    E

   y    c    4    2    9    4    7    8    n    0    5    5    9    0    0    1    3    0    e    1    9    2    i    3    2    3    9  ,    9  ,  ,    c  ,    1  ,    9  ,    i  ,    0  ,    0    0    0    0    0     f   -      0     f    0    E

   m    N    8    2    2    4    6    6    7    4    5     /  ,    7  ,    5  ,  ,    6    5  ,    e    4  ,    3  ,    0  ,    3    5    u    8    4    5    1    4    4    5    3    0    q    5    4    1    5    5    r    4    4    3    o    1    T

   m    N   2    3    4    1    6    1    6    3     /  ,    7  ,    3  ,  ,    0  ,  ,  ,  ,    e    5    5    8    0    1    6    8    6    u    4    0    4    0    2    4    2    1    3    q    r    1    1    1    1    3    5    2    3    o    T

   3

   9

   2

   8    4

   2

   3

   7

   6

   2

   2

   0

   3    2  ,    0  ,

   1

   7

   8

   2

   2

   1

   8

   6

   2    5    2    5    2    2  ,  ,  ,  ,  ,    W   5    0    0    5  ,    6  ,    8  ,    3    3    8     /    t    1    8    1    9    0    3    1    6    u    2    4    0    0    9    1    4    1    o    0    8    1    6    1    0    9    7    4    P    2    1    0    1    7    3    3     -

   4    6    1  ,  ,    6  ,    3  ,    W   2  ,    7    2    2    2  ,    3    7    2     /  ,    5    t    0    9    5    9    9    7    2    9    3    u    6    4    7    8    1    8    3    o    1    0    1    7    2    9    1    8    6    P    2    2    9    4    2    5     -

   7

   2    6    2    4    A    4  ,    0    5    1    3    6    8    8  ,  ,     /  ,  ,  ,    7     '    0    1    3      S    2  ,    5    2    2    0    0    8    3    8    I      Y    2    7    3    1    1

   1    2    5    8    7    1    A    1    4  ,  ,  ,    5    4    6    3  ,  ,  ,  ,    7    8  ,    7      S     /     '    3    6    9    6    4    0    0    8    3    2      Y    I    3    3    3    2    2    2    3    3

   1

   8    0    1  ,    0  ,  ,    0  ,    d    9    8    W   8   n     /    8    5    0    4    0    8    0    7    0   a    2    1    0    0    3    7    4    3    n    1    4    4    1    7    7    2    9    3      D    i    7    0      2    P    2    2    2    1    1    4    1    3

   5    5    5    2    9    F    7    7    1    0    9    6    1    7    6   y     (    0    6    s    7    8    5    9    9    8  ,    b    o  ,  ,  ,  ,    8  ,  ,  ,    0    0    d    C    0    0    0    0    0    0   e    t   a    l    9    7   u    6    6    9    6    4    9  ,    9  ,   c    A    9  ,    4    3  ,  ,    3    l  ,  ,  ,    8     /   a    1    7    7    5    6    0    0    0    6   c    I    3    3    1    1    0    2    9    0    3    2    2    1    M    I    Z    m    9    V    p    0  ,    r    2    5     /    0    0    6   n    2    i     d    5    9    4    5    1    3   s    e    1      e   e    1   s    p   s    S   o    L   :     d    D    s    D    s    D    s    D    s    I    o    2    y    2    y    2    y    2    y   e     h    s    x    s    x    s    l    x    s    a    x    a    u    l    a    u    l    a    b    t    u    l    u    l     l     l     l   a    e    l    T    M   F    K    F    K    F    K    F    K

   4    4    6   y    i    F    7    6    7    7    3    0    1    4    1    b     (    9    9    6    5    s  ,    9  ,    9  ,    7  ,    5  ,  ,  ,  ,    d    o    0    0    0    0    0    0    0    0   e    C    t   a    l   u    2    4    6   c    3    6    2    8    l    0    0  ,    3  ,    A    1  ,    4  ,    9    3   a     /  ,  ,    2  ,    0    9   c    1    5  ,    4    6    8    0    6    2    2    I    3    3    1    2    8    3    3    3    3    2    3    M    I    Z    m    V    p    8    0  ,    r    2   n     /    5    0    5    i    6    2     d    5    5    4   s    e    5    1    4    1   e      e    1   s   s    p   o    S    L   :    I    s    D    s    D    s    D    s    I     d    o    D    2    y    2    y    2    y    2    y   e     h    l    x    s    x    s    x    s    x    s    a    a    a    a    b    t    u    u    u    u    l     l     l     l     l     l     l   a    e    l    T    M   F    K    F    K    F    K    F    K

     S      A      L      K

   3

   7    5    7

 ,  ,    3  ,    0    9    0    4    9    d    W   3  ,    2    4    3    1    0   n     /    5    5    9    8    2    2    4    3    1   a    n    6    7    i    0    6    1    5    2    6    3    3    8    1    1    1    2      D    P    5    2    2    9

     2    x    u      l    i      F     )

   1

   1

     S      A      L      K

   x    u      l      F     )

   3

   1

   3

   1

   1

     9    0    1   -

   W     /    8    2    7    2    4    6    9    r    6    5    8    4    9    9  ,    5    0    9    e  ,  ,  ,    7     f  ,    6    9    7    2  ,    3  ,    8  ,    6    S    3    0    1    1    3    0    3    7    S    2    1    O    L    W   4     /    6    8    4    2    9    2    2    s    2  ,    2  ,    1    4  ,  ,    2  ,    2    e    1  ,    8    3  ,  ,     f    7    0    2    7    2    6    7    S    0    1    2    1    8    7    S    4    1    6    1    1    2    1    O    L    8    2    W   6    1    2    4     /  ,    5  ,    5    7    9    0    2    1    2  ,  ,    4  ,    6  ,    2    8  ,    u  ,    8    6    5    7    7    3    c    7    3    8    2    4    7    5    7    S    8    3    9    7    4    8    6    6    S    4    9    0    3    3    7    O    3    6    L    5    W   5    3    5    7    7    7     /    9    9  ,  ,  ,  ,    6    8    1    0    1    3  ,    7  ,  ,    6    0    7    u    1    3    3  ,    8    4    2    1    c    7    7    4    7    7    5    S    0    3    5    8    4    7    S    9    7    8    1    2    6    9    8    1    6    O    7    L

   y    c    5    9    9    2    2    2    7    n    2    1    2    0    0    9    6    9    0    e    i    2    3    9    9    9    9    0    1  ,  ,    c  ,  ,  ,  ,  ,  ,    i    0    0    0    0    0    0    0    0     f   -       f    E    m    N   2    4    8    4    4    9    3    3    4     /  ,    8  ,    0  ,  ,    4  ,  ,  ,    e    1  ,    5    0    7    7    8    1    u    3    2    2    8    7    4    0    0    3    q    1    3    2    1    r    4    2    2    2    o    1    T    4

   8

   8

   9

   3    3    2  ,  ,  ,    1    2    3  ,  ,    W   7  ,    9    8    0    7     /    1  ,    4    3    3    9    t    4    6    4    6    7    7    7    8    u    0    3    7    7    7    7    2    8    o    2    9    1    2    2    0    6    3    P    2    1    3    1    2    3        1

   5

   1    8    1    9    5    A    4    3  ,    7  ,  ,      S     /    9  ,  ,    3  ,    0  ,  ,    2    6    8      Y     '    3    7    5    0    1    3    5    0    1      S    2    3      A    I    3    3    5    4    3    3    3    3      L      K    7    7    1    d    1    0    6  ,    0  ,    0  ,    0  ,   n  ,    6    0    7    W   6    1    0    1    0   a     /    8    0    0    3    9    8    0    4    3    6    9    4    3    5    8    4    6      D    n    i    1    2      2    P    4    0    3    0    1    2    2    1    2    3    2    x

   u      l      F     )

   2

   1

   1

   5   y    i    7    2    3    4    2    7    F    2    8    b     (    1    5    3    9    8    2    0    5    9    9    9    5    5    5    5    s    d  ,  ,  ,    9  ,  ,  ,  ,  ,   e    o    0    0    0    0    0    0    0    t    c    0   a    l   u    9   c    1    l    2    3    4    2    6  ,    6    0   a    A    2  ,    9  ,   c     /    6  ,    0  ,    6  ,  ,    8    7    2    6    3    2    4    3    1    3    1    4    3    I    6    0    3    3    3    5    3    3    M    I    Z    V    m    5   n    p    0    r    6    i    2  ,    5    0    6   s     /    4    2     d    5    4   e    7    5    e    1   s    1    4     s    e    1   o    p    L    S   :    I    I    I     d    D    s    D    s    D    s    D    s    o    2    y    2    y    2    y    2    y   e     h    s    x    s    x    s    l    x    s    b    t    a    x    a    u    l    a    u    l    a    u    l    u     l   a    e    l     l     l     l    T    M   F    K    F    K    F    K    F    K

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8.4 APPENDIX IV: Estimation of the heat transfer coefficient at the winding-overhang close to the centrifugal mechanism

Although VZIM motor is totally enclosed, at one side of the winding overhang, heat transfer  coefficient is considered as ൌ ͳͷȀሺʹ ሻ for the natural convection and heat radiation, but at the other side of the motor, the rotating centrifugal mechanism leads to a better cooling due to the forced convection. Therefore in order to calculate the heat transfer coefficient for this winding overhang close to the centrifugal mechanism, with try and error and comparison with measured values, is calculated for 5.5kW VZIM ( ൌ ͷͲȀሺʹ ሻ ) and for 22kW VZIM due to the similar structure with 5.5kW VZIM, a proportional value of  is considered. In the following, it is described how the value of heat transfer coefficient is derived. Measured values at the rated load thermal test for 5.5kW VZIM shows that the maximum temperature rise for the winding overhang is 67.7 . According to this value a finite element model of 5.5kW VZIM is prepared using  ANSYS , and the value of  is adjusted, and finally with a heat transfer coefficient about ൌ ͷͲȀሺʹ ሻ the calculated values, based on the finite element method with  ANSYS  and measured values are matched together. For 5.5kW VZIM2 (nominal speed for 5.5kW VZIM2 is equal to 1455.2rpm) with consideration of the centrifugal mechanism as a standard fan, the heat transfer coefficient is equal to ൌ

ͺ ଷΤସ ൌ ͺሺ ͺሺͳ͵ǤͲͶͶ ͳ͵ǤͲͶͶሻሻଷΤସ ൌ ͷͶǤͻȀሺʹ ሻ, besides for 22kW VZIM2 motor (nominal speed for 22kW VZIM2 is equal to 1467.36 rpm) the heat transfer coefficient for winding overhang with considering the mechanical part of the motor, as a standard fan, is equal to

ൌ ͺ

ଷΤସ

ൌ

ͺሺͳ͹Ǥͺ͹ ͺሺ ͳ͹Ǥͺ͹ሻሻଷΤସ ൌ ͸ͻǤͷ͵Ȁሺʹ ሻ. Therefore due to the similarity of the 2 types of the motor  (5.5kW and 22kW) the applied value of  for VZIM2 22kW is ൌͷͲ ΤͷͶǤͻ ή ͸ͻǤͷ͵ ൌ ʹ ͸͵Ǥ͵ʹ Ȁሺ ሻ . For 5.5kW VZIM3 (nominal speed for 5.5kW VZIM3 is equal to 1470rpm) with consideration of the centrifugal mechanism as a standard fan, the heat transfer coefficient is equal to ൌ ͺ ଷΤସ ൌ ͺሺ ͺሺͳ͵Ǥͳ͹ ͳ͵Ǥͳ͹ሻሻଷΤସ ൌ ͷͷǤ͵ Ȁሺ ʹ ሻ . Besides for 22kW VZIM3 motor  (nominal speed for 22kW VZIM3 is equal to 1467.36rpm) the heat transfer coefficient for the winding overhang, with considering the mechanical part of the motor as a standard fan, is equal to ൌ ͺ ଷΤସ ൌ ͺሺͳ͹Ǥͺ͹ሻଷΤସ ൌ ͸ͻǤͷ͵Ȁሺʹሻ. Therefore due to the similarity of  the 2 types of the motor (5.5kW and 22kW) the applied value of  for VZIM2 22kW is ʹ ൌͷͲ ΤͷͷǤ͵ ή ͸ͻǤͷ͵ ൌ ͸ʹǤͺ͸ Ȁሺ ሻ .

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