**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

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