ELECTRICAL MACHINES Contents 1 Introduction Power converters and electrical machines Rotating power converters Static power converters The role of electromechanical power conversion Principles of operation Magnetic and current circuits Rotating electrical machines Reversible machines Significance and typical applications Variables and relations of rotational movement Notation and system of units Target knowledge and skills Basic characteristics of electrical machines Equivalent circuits Mechanical characteristics Transient processes in electrical machines Mathematical model Adopted approach and analysis steps Prerequisites Notes on converter fed variable speed machines Remarks on high efficiency machines Remarks on iron and copper usage 2 Electromechanical energy conversions Lorentz force Mutual action of parallel conductors Electromotive force in a moving conductor Generator mode Reluctant torque Reluctant force Force on conductors in electrical field Change of permittivity Piezoelectric effect Magnetostriction 3 Magnetic and electrical coupling field Converters based on electrostatic field Charge, capacitance, and energy Source work, mechanical work, and field energy Force expression Conversion cycle Energy density of electrical and magnetic field Coupling field and transfer of energy Converter involving magnetic coupling field Linear converter Rotational converter Rotational converter Back electromotive force 4 Magnetic circuits Analysis of magnetic circuits Flux conversion law Generalized relation between magnetic Field H and induction B The flux vector Magnetizing characteristics of ferromagnetic materials Magnetic resistance of the circuit Energy in a magnetic circuit Energy in a magnetic circuit Reference direction of the magnetic circuit Losses in magnetic circuits Hysteresis losses Losses due to eddy currents Total losses in magnetic circuit The methods of reduction of iron losses Eddy current in laminated ferromagnetic 5 Rotating electrical machines Magnetic circuit of rotating machines Mechanical access The windings Slots in magnetic circuit The position and notation of winding axis Conversion losses Magnetic field in air gap Field energy, size, and torque 6 Modeling electrical machines The need for modeling Problems of modeling Conclusion Neglected phenomena Distributed energy and distributed parameters Neglecting parasitic capacitances Neglecting iron losses Neglecting iron nonlinearity Power of electrical sources Electromotive force Voltage balance equation Leakage flux Energy of the coupling field Power of electromechanical conversion Torque expression Mechanical subsystem Kinetic energy Model of mechanical subsystem Balance of power in electromechanical converters Equations of mathematical model 7 Single-fed and double-fed converters Analysis of single-fed converter Variation of self-inductance The expressions for power and torque Analysis of double-fed converter Variation of mutual inductance Torque expression Average torque Conditions for generating nonzero torque Magnetic poles Direct current and alternating current machines Torque as a vector product Position of the flux vector in rotating machines Rotating field Types of electrical machines Direct current machines Induction machines Synchronous machines 8 Magnetic field in the air gap Stator winding with distributed conductors Sinusoidal current sheet Components of stator magnetic field Axial component of the field Tangential component of the field Radial component of the rotor field Survey of components of the rotor Magnetic field Convention of representing magnetic Field by vector 9 Energy, flux, and torque Interaction of the stator and rotor fields Energy of air gap magnetic field Electromagnetic torque The torque expression Turn flux and winding flux Flux in one stator turn Flux in one rotor turn Winding flux Winding flux vector Winding axis and flux vector Vector product of stator and rotor flux vectors Conditions for torque generation Torque-size relation Rotating magnetic field System of two orthogonal windings System of three windings 10 Electromotive forces Transformer and dynamic electromotive forces Electromotive force in one turn Calculating the first derivative Of the flux in one turn Summing electromotive forces of individual Conductors Voltages balance in one turn Electromotive force waveform Root mean square value Of electromotive forces Electromotive force in a winding Concentrated winding Distributed winding Chord factor Belt factor Harmonics suppression of winding belt Electromotive force in distributed winding Individual harmonics Peak and rms of winding electromotive force 11 Introduction to DC machines Construction and principle of operation Construction of the stator Separately excited machines Current in rotor conductors Mechanical commentator Rotor winding Communication Operation of commentator Making the rotor winding Problems with commutation Rotor magnetic field Current circuits and magnetic circuits Magnetic circuits Direct and quadrature axis Vector representation Resultant fluxes Resultant flux of the machine Electromotive force and electromagnetic torque Electromotive force in armature winding Torque generation Torque and electromotive force expressions Calculation of electromotive force E Calculation of torque 12 Modeling and supplying DC machines Voltage balance equation for excitation winding Voltage balance equation in armature winding Changes in rotor speed Mathematical model DC machine with permanent magnets Block diagram of the model Torque control Steady-state equivalent circuit Mechanical characteristics Stable equilibrium Properties of mechanical characteristics Speed regulation DC generator Topologies of DC machine power supplies Armature power supply requirements Four quadrants in T- and U-I diagrams The four-quadrant power converter Pulse-width modulation Current ripple Topologies of power converters 13 Characteristics of DC machines Rated voltage Mechanical characteristics Natural characteristics Rated current Thermal model and intermittent operation Rated flux Rated speed Field weakening High-speed operation Torque and power in field weakening Flux change Electromotive force change Current change Torque change Power change The need for field-weakening operation Steady-state operating area Power losses and power balance Power of supply Losses in excitation winding Losses armature winding Power of electromechanical conversion Iron losses Mechanical losses Losses due to rotation Mechanical power Rated and declared values Nameplate data Current ripple Frequency control Field weakening Reversal of frequency-controlled Induction machines Steady state and transient operating area Steady state operating limits RI compensation Critical speed Construction of induction machines Mains-supplied machines Variable frequency induction Machines 14 Synchronous machines Principle of operation Stator windings Revolving field Torque generation Construction of synchronous machines Stator magnetic circuit Construction of the rotor Supplying the excitation winding Excitation with rotating transformer Permanent magnet excitation Characteristics of permanent magnets Magnetic circuits with permanent magnets Surface mounts and buried magnets Characteristics of permanent magnet machines 15 Mathematical model of synchronous machine Modeling synchronous machines Magneto motive force Two-phase equivalent Clarke 3 /2 transform Inductance matrix and voltage balance equations Park transform Inductance matrix in dq frame Vectors as complex numbers Voltage balance equations Electrical subsystem of isotropic machines Torque in isotropic machines Anisotropic rotor Reluctant torque Reluctance motor 16 steady-state operation Voltage balance equations at steady state Equivalent circuit Peak and rms values of currents and voltages Phasor diagram of isotropic machine Phasor diagram of anisotropic machine Torque in anisotropic machine Torque change with power angle Mechanical characteristics Synchronous machine supplied from stiff network Operation of synchronous generators Increase of turbine power Increase in line frequency Reactive power and voltage changes Changes in power angle 17 Transients in synchronous machines Electrical and mechanical time constants Hunting of synchronous machines Damped LC circuit Damping of synchronous machines Damper winding Short circuit of synchronous machines DC component Calculation of Isc1 Calculation of Isc2 Calculation of Isc3 Transient and sub transient phenomena Interval 1 Interval 2 Interval 3 18 Variable frequency synchronous machines Inverter-supplied synchronous machines Torque control principles Current control principles Field weakening Transient and steady-state operating area
ELECTRICAL MACHINES
Contents
1 Introduction
Power converters and electrical machines
Rotating power converters
Static power converters
The role of electromechanical power conversion
Principles of operation
Magnetic and current circuits
Rotating electrical machines
Reversible machines
Significance and typical applications
Variables and relations of rotational movement
Notation and system of units
Target knowledge and skills
Basic characteristics of electrical machines
Equivalent circuits
Mechanical characteristics
Transient processes in electrical machines
Mathematical model
Adopted approach and analysis steps
Prerequisites
Notes on converter fed variable speed machines
Remarks on high efficiency machines
Remarks on iron and copper usage
2 Electromechanical energy conversions
Lorentz force
Mutual action of parallel conductors
Electromotive force in a moving conductor
Generator mode
Reluctant torque
Reluctant force
Force on conductors in electrical field
Change of permittivity
Piezoelectric effect
Magnetostriction
3 Magnetic and electrical coupling field
Converters based on electrostatic field
Charge, capacitance, and energy
Source work, mechanical work, and field energy
Force expression
Conversion cycle
Energy density of electrical and magnetic field
Coupling field and transfer of energy
Converter involving magnetic coupling field
Linear converter
Rotational converter
Back electromotive force
4 Magnetic circuits
Analysis of magnetic circuits
Flux conversion law
Generalized relation between magnetic
Field H and induction B
The flux vector
Magnetizing characteristics of ferromagnetic materials
Magnetic resistance of the circuit
Energy in a magnetic circuit
Reference direction of the magnetic circuit
Losses in magnetic circuits
Hysteresis losses
Losses due to eddy currents
Total losses in magnetic circuit
The methods of reduction of iron losses
Eddy current in laminated ferromagnetic
5 Rotating electrical machines
Magnetic circuit of rotating machines
Mechanical access
The windings
Slots in magnetic circuit
The position and notation of winding axis
Conversion losses
Magnetic field in air gap
Field energy, size, and torque
6 Modeling electrical machines
The need for modeling
Problems of modeling
Conclusion
Neglected phenomena
Distributed energy and distributed parameters
Neglecting parasitic capacitances
Neglecting iron losses
Neglecting iron nonlinearity
Power of electrical sources
Electromotive force
Voltage balance equation
Leakage flux
Energy of the coupling field
Power of electromechanical conversion
Torque expression
Mechanical subsystem
Kinetic energy
Model of mechanical subsystem
Balance of power in electromechanical converters
Equations of mathematical model
7 Single-fed and double-fed converters
Analysis of single-fed converter
Variation of self-inductance
The expressions for power and torque
Analysis of double-fed converter
Variation of mutual inductance
Average torque
Conditions for generating nonzero torque
Magnetic poles
Direct current and alternating current machines
Torque as a vector product
Position of the flux vector in rotating machines
Rotating field
Types of electrical machines
Direct current machines
Induction machines
Synchronous machines
8 Magnetic field in the air gap
Stator winding with distributed conductors
Sinusoidal current sheet
Components of stator magnetic field
Axial component of the field
Tangential component of the field
Radial component of the rotor field
Survey of components of the rotor
Magnetic field
Convention of representing magnetic
Field by vector
9 Energy, flux, and torque
Interaction of the stator and rotor fields
Energy of air gap magnetic field
Electromagnetic torque
The torque expression
Turn flux and winding flux
Flux in one stator turn
Flux in one rotor turn
Winding flux
Winding flux vector
Winding axis and flux vector
Vector product of stator and rotor flux vectors
Conditions for torque generation
Torque-size relation
Rotating magnetic field
System of two orthogonal windings
System of three windings
10 Electromotive forces
Transformer and dynamic electromotive forces
Electromotive force in one turn
Calculating the first derivative
Of the flux in one turn
Summing electromotive forces of individual
Conductors
Voltages balance in one turn
Electromotive force waveform
Root mean square value
Of electromotive forces
Electromotive force in a winding
Concentrated winding
Distributed winding
Chord factor
Belt factor
Harmonics suppression of winding belt
Electromotive force in distributed winding
Individual harmonics
Peak and rms of winding electromotive force
11 Introduction to DC machines
Construction and principle of operation
Construction of the stator
Separately excited machines
Current in rotor conductors
Mechanical commentator
Rotor winding
Communication
Operation of commentator
Making the rotor winding
Problems with commutation
Rotor magnetic field
Current circuits and magnetic circuits
Magnetic circuits
Direct and quadrature axis
Vector representation
Resultant fluxes
Resultant flux of the machine
Electromotive force and electromagnetic torque
Electromotive force in armature winding
Torque generation
Torque and electromotive force expressions
Calculation of electromotive force E
Calculation of torque
12 Modeling and supplying DC machines
Voltage balance equation for excitation winding
Voltage balance equation in armature winding
Changes in rotor speed
DC machine with permanent magnets
Block diagram of the model
Torque control
Steady-state equivalent circuit
Stable equilibrium
Properties of mechanical characteristics
Speed regulation
DC generator
Topologies of DC machine power supplies
Armature power supply requirements
Four quadrants in T- and U-I diagrams
The four-quadrant power converter
Pulse-width modulation
Current ripple
Topologies of power converters
13 Characteristics of DC machines
Rated voltage
Natural characteristics
Rated current
Thermal model and intermittent operation
Rated flux
Rated speed
Field weakening
High-speed operation
Torque and power in field weakening
Flux change
Electromotive force change
Current change
Torque change
Power change
The need for field-weakening operation
Steady-state operating area
Power losses and power balance
Power of supply
Losses in excitation winding
Losses armature winding
Iron losses
Mechanical losses
Losses due to rotation
Mechanical power
Rated and declared values
Nameplate data
Frequency control
Reversal of frequency-controlled
Steady state and transient operating area
Steady state operating limits
RI compensation
Critical speed
Construction of induction machines
Mains-supplied machines
Variable frequency induction
Machines
14 Synchronous machines
Principle of operation
Stator windings
Revolving field
Construction of synchronous machines
Stator magnetic circuit
Construction of the rotor
Supplying the excitation winding
Excitation with rotating transformer
Permanent magnet excitation
Characteristics of permanent magnets
Magnetic circuits with permanent magnets
Surface mounts and buried magnets
Characteristics of permanent magnet machines
15 Mathematical model of synchronous machine
Modeling synchronous machines
Magneto motive force
Two-phase equivalent
Clarke 3 /2 transform
Inductance matrix and voltage balance equations
Park transform
Inductance matrix in dq frame
Vectors as complex numbers
Voltage balance equations
Electrical subsystem of isotropic machines
Torque in isotropic machines
Anisotropic rotor
Reluctance motor
16 steady-state operation
Voltage balance equations at steady state
Equivalent circuit
Peak and rms values of currents and voltages
Phasor diagram of isotropic machine
Phasor diagram of anisotropic machine
Torque in anisotropic machine
Torque change with power angle
Synchronous machine supplied from stiff network
Operation of synchronous generators
Increase of turbine power
Increase in line frequency
Reactive power and voltage changes
Changes in power angle
17 Transients in synchronous machines
Electrical and mechanical time constants
Hunting of synchronous machines
Damped LC circuit
Damping of synchronous machines
Damper winding
Short circuit of synchronous machines
DC component
Calculation of Isc1
Calculation of Isc2
Calculation of Isc3
Transient and sub transient phenomena
Interval 1
Interval 2
Interval 3
18 Variable frequency synchronous machines
Inverter-supplied synchronous machines
Torque control principles
Current control principles
Transient and steady-state operating area
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