ABSTRACT
This thesis presents the stability analysis of three horse-power squirrel cage induction motor for enhanced dynamic performance. It includes the machine Simulink modelling under dynamic condition, its steady state analysis and stability study. Simulink software is used for step by step dynamic modelling of the machine. Matlab program is also developed for the study of the steady state behaviour of the machine, this includes comparative analysis of the torque-speed characteristics of the machine under transient and steady-state, the torque-speed characteristics of the machine at various values of rotor resistance and also the torque-slip characteristics of the machine at various slip ranges. For the stability studies, the ordinary differential equations that described the motor electrical and mechanical models is linearized for the computation of the eigenvalues, transfer functions, poles and zeros. From the transfer function developed the step response of change in electromagnetic torque divided by change in load torque at different operating slip is analysed. The transfer function is also used to analyse the change in electromagnetic torque divided by change in stator voltage at different operating slip. The zero and pole plots are also presented on the plane. The results from the dynamic modelling of the machine reveal the oscillations or ripples experienced by the machine at start-up. The torque-speed graph at various values of rotor resistance shows that the maximum torque is independent of rotor resistance, but the starting torque, the speed at which the maximum torque is developed and the duration at which it is sustained depend on the rotor resistance. The study of the torque-slip characteristics of the machine at various slip ranges shows that at the slip range of zero to one, the machine is in the motoring mode. Considering the slip range of -2 to 2, the machine experiences both the motoring mode (slip range of 0 to 2) and the generating mode (slip range of -2 to 0). The step response graphs, the eigenvalues and the zero-pole plots are used for the stability analysis of the machine for enhanced dynamic performance. The results show that the machine is stable at the two different cases presented when the steady state slip values are 0, 0.03, 0.1, 0.4 and 0.6, but is unstable at steady state slip values of 0.8 and 1.0. This study gives an idea of better operating regions of the induction machine for enhanced dynamic performance. The stability analysis is carried out using eigenvalue method which takes care of absolute stability of the machine. An improvement can be achieved by the use of other stability control techniques such as Nyquist plot and Root locus which can account for the motor’s relative stability.
TABLE OF CONTENTS
Title Page. . . . . . . . . . i
Declaration Page . . . . . . . . ii
Certification . . . . . . . . iii
Dedication . . . . . . . . . iv
Acknowledgments . . . . . . . . v
Table of Contents . . . . . . . . vi
List of Tables. . . . . . . . . . vii
List of Plates. . . . . . . . . . ix
List of Figures. . . . . . . . . xi
List of Symbols and Abbreviations . . . . . . xiv
Abstract. . . . . . . . . . xvi
CHAPTER 1: INTRODUCTION
1.1 Background of Study . . . . . . . 1
1.2 Problem Statement . . . . . . . 2
1.3 Aim and Objectives . . . . . . . 2
1.4 Significance of the Study . . . . . . 2
1.5 Scope of Study . . . . . . . 3
1.6 Justification of Study . . . . . . . 3
1.7 Overview of Thesis . . . . . . . 3
CHAPTER 2: LITERATURE REVIEW
2.0 Brief History of Induction Motors . . . . . 5
2.1 Types of Electric Motors . . . . . . 6
2.2 Insulation of Electric Machines . . . . . 7
2.5.1 Maximum Allowable Temperatures of Various Types of Insulation . 7
2.3 Parts of Induction Motor . . . . . . 8
2.4 Operating Principle of Induction Motor . . . . 10
2.5 Stability Analysis of Induction Motors . . . . 11
2.6 Review of Related Works . . . . . . 11
CHAPTER 3: MATERIALS AND METHODS
3.1 Materials . . . . . . . . 17
3.2 Methods . . . . . . . . 17
3.3 Simulation of Induction Machine . . . . . 18
3.3.2 The Induction Motor Model . . . . . . 18
3.4 Dynamic Model of Three-Phase Squirrel Cage Motor . . 19
3.4.2 Matlab/Simulink Representation of the Dynamic Model . . 23
3.5 Steady State Analysis . . . . . . . 27
3.6 Linearization of Three Phase Induction Machine . . 29
3.7 Transfer Function of Three-Phase Induction Machine . . 32
CHAPTER 4: RESULTS AND DISCUSSION
4.1 Simulink Model Results . . . . . . 33
4.2 Steady-State Analysis Results. . . . . . 38
4.3 Steady State Torque-Speed Graph and Dynamic Model Torque-Speed Graph Compared . . . . . . . . 42
4.4 Torque-Speed Analysis . . . . . . 43
4.4.1 Steady State Electromagnetic Torque-Speed Graph Produced at Various Values of Rotor Resistance . . . . . . . 43
4.4.2 Steady State Analysis of Electromagnetic Torque against Slip at Various Ranges of Slip Values . . . . . . . . 44
4.5 Stability Analysis of the Displacement Behaviour of the Machine about an Operating Point . . . . . . . 46
4.5.1 The Stability Analysis Using Step Response of Change in Electromagnetic Torque Divided By Change in Load Torque . . . . . 46
4.5.2 The Stability Analysis Using Step Response of the Change in Electromagnetic Torque Divided By Change in Stator Voltage . . . 49
4.5.3 Stability Analysis When Step Response of Change in Electromagnetic Torque Divided By Change in Load Torque and Step Response of Change in Electromagnetic Torque Divided By Change in Stator Voltage at Different Operating Slip are Compared . . . . . . 52
4.5.4 Stability Analysis Using Eigenvalues, Zeros and Poles . . 55
4.5.5 Stability Analysis by Plotting Zeros and Poles on a Cartesian Plane. 61
CHAPTER 5: CONCLUSION AND RECOMMENDATIONS
5.1 Conclusion . . . . . . . . 70
5.2 Recommendations . . . . . . . 71
References . . . . . . . . 72
List of Tables
2.1 Classes of Insulation and Their Maximum Permissible Temperature. . 7
3.1 Three Horse-Power Induction Motor Parameters . . . 23
4.1 Eigenvalues, Zeros, Poles and Gain of Transfer Function of Change in Electromagnetic Torque Divided by Change in Load Torque at Various Values of Slip . . . . . . . . . 55-58
4.2 Eigenvalues, Zeros, Poles and Gain of Transfer Function of Change in Electromagnetic Torque Divided by Change in Stator Voltage at Various Values of Slip . . . . . . . . . 58-60
List of Plates
2.1 Stator of Induction Motor . . . . . . 8
2.2 Squirrel Cage Rotor: Squirrel-cage rotor . . . . 9
2.3 Wound rotor with rheostat . . . . . . 10
List of Figures
2.1 Types of Electric Motors . . . . . . 6
3.1 Idealized Circuit Model of a Three-Phase Induction Machine . 19
3.2 Model of Induction Machine in Q-Axis . . . . 22
3.3 Model of Induction Machine in D-Axis . . . . 23
3.4 Simulink Representation of the Dynamic Model . . . 24
3.5 Voltage Transformation from Three-Phase to Two-Phase . . 25
3.6 Simulink Representation of the Flux Linkage . . . 25
3.7 Simulink Model of Stator Current, Rotor Currents and Flux Linkages in DQ-Axis . . . . . . . . . . 26
3.8 Simulink Model of Electromagnetic Torque and the Rotor Angular Speed 26
3.9 Equivalent Circuit of an Induction Motor . . . . 27
3.10 Thevenin Equivalent Circuit . . . . . . 27
4.1 Graph of Stator Phase A Current against Time . . . 34
4.2 Graph of Stator Phase B Current against Time . . . 34
4.3 Graph of Stator Phase C Current against Time . . . 35
4.4 Graph of Rotor Phase A Current against Time . . . 35
4.5 Graph of Rotor Phase B Current against Time . . . 36
4.6 Graph of Rotor Phase C Current against Time . . . 36
4.7 Graph of Electromagnetic Torque against Time . . . 37
4.8 Graph of Speed against Time . . . . . . 37
4.9 Graph of Electromagnetic Torque against Speed . . . 38
4.10 Graph of Input Power against Speed . . . . . 39
4.11 Graph of Power Factor against Speed. . . . . 40
4.12 Graph of Stator Current against Speed . . . . 40
4.13 Graph of Output Power against Speed . . . . 41
4.14 Graph of Efficiency against Speed . . . . . 41
4.15 Graph of Developed Electromagnetic Torque against Speed . 42
4.16 Graphs of Electromagnetic Torque against Speed from Dynamic Model and from Computation of Steady State Parameters Compared 43
4.17 Graph of Steady State Electromagnetic Torque against Speed at Various Values of Rotor Resistance . . . . . . 44
4.18 Graph of Steady State Electromagnetic Torque-Speed at Slip Range of Zero to One . . . . . . . 45
4.19 Graph of Steady State Electromagnetic Torque-Speed at Slip Range of -2 to 2 . . . . . . . 45
4.20 Step Response of Change in Electromagnetic Torque Divided by Change in Load Torque for Slip Value of zero . . . . 47
4.21 Step Response of Change in Electromagnetic Torque Divided by Change in Load Torque for Slip Value of 0.03 . . . . 47
4.22 Step Response of Change in Electromagnetic Torque Divided by Change in Load Torque for Slip Value of 0.1 . . . . . 48
4.23 Step Response of Change in Electromagnetic Torque Divided by Change in Load Torque Slip Value of 0.4 . . . . 48
4.24 Step Response of Change in Electromagnetic Torque Divided by Change in Load Torque for Slip Value of 0.6. . . . 49
4.25 Step Response of Change in Electromagnetic Torque Divided by Change in Stator Voltage for Slip Value of zero . . . 50
4.26 Step Response of Change in Electromagnetic Torque Divided by Change in Stator Voltage for Slip Value of 0.03 . . . 50
4.27 Step Response of Change in Electromagnetic Torque Divided by Change in Stator Voltage for Slip Value of 0.1 . . . 51
4.28 Step Response of Change in Electromagnetic Torque Divided by Change in Stator Voltage for Slip Value of 0.4 . . . 51
4.29 Step Response of Change in Electromagnetic Torque Divided by Change in Stator Voltage for Slip Value of 0.6 . . . 52
4.30 Step Response of change in Electromagnetic Torque divided by change in Load Torque at Various Values of Slip . . . 53
4.31 Graph of change in Electromagnetic Torque divided by change in Stator Voltage against Time at Various Values of Slip . . 54
4.32 Step Response of change in Electromagnetic Torque divided by change in Stator Voltage at Various Values of Slip . . . 54
4.33 Zero and Pole Plots of Slip =0 for Change in Electromagnetic Torque Divided by Change in Load Torque . . . . 61
4.34 Zero and Pole Plots of Slip =0.03 for Change in Electromagnetic Torque Divided by Change in Load Torque . . . . 62
4.35 Zero and Pole Plots of Slip =0.1 for Change in Electromagnetic Torque Divided by Change in Load Torque . . . . 62
4.36 Zero and Pole Plots of Slip =0.4 for Change in Electromagnetic Torque Divided by Change in Load Torque. . . . . 63
4.37 Zero and Polar Plots of Slip=0.6 for Change in Electromagnetic Torque Divided by Change in Load Torque . . . . 63
4.38 : Zero and Pole Plots of Slip =0.8 for Change in Electromagnetic Torque Divided by Change in Load Torque . . . . 64
4.39 : Zero and Pole Plots of Slip =1.0 for Change in Electromagnetic Torque Divided by Change in Load Torque . . . . . 64
4.40 : Zero and Pole Plots of Slip=0 for Change in Electromagnetic Torque Divided by Change in Stator Voltage . . . . . 65
4.41 : Zero and Pole Plots of Slip=0.03 for Change in Electromagnetic Torque Divided by Change in Stator Voltage . . . . 65
4.42 : Zero and Pole Plots of Slip=0.1 for Change in Electromagnetic Torque Divided by Change in Stator Voltage . . . . . 66
4.43 : Zero and Pole Plots of Slip =0.4 for Change in Electromagnetic Torque Divided by Change in Stator Voltage . . . . 66
4.44 : Zero and Pole Plots of Slip =0.6 for Change in Electromagnetic Torque Divided by Change in Stator Voltage . . . . . 67
4.45 : Zero and Pole Plots of Slip=0.8 for Change in Electromagnetic Torque Divided by Change in Stator Voltage . . . . . 67
4.46 : Zero and Pole Plots of Slip =1.0 for Change in Electromagnetic Torque Divided by Change in Stator Voltage . . . . 68
LIST OF SYMBOLS AND ABBREVIATIONS
P = number of poles
⍵b = motor angular electrical base frequency
⍵r = rotor angular electrical speed d-axis = direct axis component
q-axis = quadrature axis component
Rs = stator resistance
Xls = stator leakage reactance
Pf = power factor
Rr = rotor referred resistance
Xlr = rotor leakage reactance
Xm = magnetizing reactance
Te = electromagnetic torque
TL = externally applied mechanical load torque
J = Moment of inertia
Vqs = q-axis stator voltage
Vds = d-axis stator voltage
Vdr = d-axis referred rotor voltage
Vqr = q-axis referred rotor voltage
Iqs = q-axis stator current
Ids= d-axis stator current
Iqr =q-axis referred rotor current
Idr =d-axis referred rotor current
P=d/dt
Ѱ= flux linkage
Ns = Synchronous speed
N = Mechanical speed
PI = Proportional Integral controller
TS-FEM = Time Stepping Finite Element Method
e.m.f = Electromotive force
FOC = Field Oriented Control
DTC = Direct Torque Control
Ias = stator Current
Zin= Stator impedance
S=Slip
𝑉𝑡ℎ= Thevenin voltage
Zth= Thevnin impedance
𝑟𝑡ℎ= Real part of the Thevenin impedance
𝑋𝑡ℎ=Imaginary part of the Thevenin impedance
= small change in q- and d-axis referred rotor current
= small change in q- and d-axis stator voltage
= small change in d-axis referred rotor voltage𝛥⍵𝑟 = change in rotor angular electrical speed
𝑆𝑜= steady-state slip
CHAPTER 1
INTRODUCTION
1.1 BACKGROUND OF STUDY
Induction machine is one of the most widely used electrical machines for energy conversion both in household and industries. It is very rugged, has low maintenance requirement and is cost effective. It has sufficiently high efficiency, since brushes which can cause frictional losses are not needed. It is self-starting and has high percentage of energy consumption of the total electrical energy generated. The application of this multiphase induction machine can be found in pump, steel mill, and hoist drives. It is also used as motor controlled drive in vehicles, air conditioning systems, and in wind turbines. The single phase of this machine can be used in household appliances like refrigerators and even bench tools (Krause et al, 2013; Theraja, 2006).
When induction machine is used in wind power generation, it operates as induction generator, the direction of power flow is from wind power driving the shaft of the machine, thereby, supplying power to the local power grid (Keyhani, 2011).
Three phase induction motor consists of stator and the rotor. In the construction of three phase motor, the stator is slotted and its winding is arranged in the slot such that when connected to three phase supply, it produces magnetic flux with a constant magnitude and rotates. As it rotates, it cuts through the rotor conductor and induces electromotive force. The induced electromotive force sets up current in the rotor conductors and the direction of the current is determined by Lenz’s law (Theraja, 2006; Therib, 2017).
Induction machines always experience transient disturbances particularly during starting and addition of load which affect normal operation of the machine. This transient behaviour of the induction machine as a result of transient disturbance has an effect on the overall performance of the system connected to it. Hence, the need for its stability analysis for enhanced dynamic operation.
1.2 PROBLEM STATEMENT
Induction machines experience transient disturbances which affect their performance as well as the whole system it is a component part. Hence, the need to study the behaviour of this machine during steady state and transient; and also the stability performance of induction machine. The study will give an idea of the normal dynamic and steady state operation of the machine and also helps in determining a better operating regions where the induction machine can re-establish steady state operation after an occurrence of disturbance for enhanced dynamic performance.
1.3 AIM AND OBJECTIVES
The aim of this thesis is to explore the stability analysis of induction machine for enhanced dynamic performance.
The objectives of this work are as follow:
1) To develop the d-q model equations for the induction motor.
2) To develop the Simulink model for the machine under dynamic condition.
3) To develop the matlab programs for the computation of the induction motor steady state behaviour.
4) To develop Matlab program for the stability studies and validate the results with pole-zero plots.
1.4 SIGNIFICANCE OF THE STUDY
The result achieved from the study will give an insight to the steady state behaviour and transient behaviour of the induction machine. Also the stability analysis of the induction machine will give an idea of better operating regions where the induction machine can re-establish steady state operation after occurrence of disturbance for enhanced dynamic performance.
1.5 SCOPE OF STUDY
This work is limited to the development of Simulink model of three horse-power (3hp) squirrel cage Induction Machine under dynamic condition; and development of Matlab program for computation of the motor steady state parameters and its stability studies.
1.6 JUSTIFICATION OF STUDY
The normal operation of induction machine most times is affected when there is disturbance in the network in which it is connected to. This disturbance can occur when there is addition of load, change of load or even fluctuation in voltage. Hence, there is need to study the dynamic behaviour, steady state behaviour and stability behaviour of induction machine. This will give an idea of a better operating region where the performance of the induction machine can be more reliable and efficient. In this work, step response graph, eigenvalue method and representation of zero and pole plots in a Cartesian plane will be used in predicting the stability behaviour of the three horse power squirrel cage induction machine.
1.7 OVERVIEW OF THESIS
Chapter one is the introductory chapter that introduces the general knowledge of the project. It explains the need for carrying out this research. It captures the aim and objectives of the research work, significance of the study, scope and finally the overview of the work.
Chapter two is on literature review which includes types of electric motors, insulation of electric motors, the brief history of induction motors, parts of induction motor, operating principle of induction motor, stability analysis of induction motors and review of related works.
Chapter three is on materials and methods. It discusses the materials used for the work as well as the methods used. This chapter covers simulation of induction machine under dynamic condition, steady state analysis, linearization of three phase induction machine and transfer function of three-phase induction machine.
Chapter four presents the results obtained from the research work and detailed discussion of the results. These include the Matlab/Simulink simulation results of three horse-power squirrel cage induction motor, results from matlab program developed for computation of steady-state parameters of the machine and results obtained from Matlab program developed for stability studies.
Chapter five focuses on conclusion and recommendation.
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