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Analysis of Synchronous Machines, Second Edition

Analysis of Synchronous Machines, Second Edition by T.A. Lipo
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Winding Distribution in an Ideal Machine


The Winding Function

Calculation of the Winding Function

Multipole Winding Configurations

Inductances of an Ideal Doubly Cylindrical Machine

Calculation of Winding Inductances

Mutual Inductance Calculation—An Example

Winding Functions for Multiple Circuits

Analysis of a Shorted Coil—An Example

General Case for C Circuits

Winding Function Modifications for Salient-Pole Machines

Leakage Inductances of Synchronous Machines

Practical Winding Design

Reference Frame Theory


Rotating Reference Frames

Transformation of Three-Phase Circuit Variables to a Rotating Reference Frame

Stationary Three-Phase r–L Circuits Observed in a d–q–n Reference Frame

Matrix Approach to the d–q–n Transformation

The d–q–n Transformation Applied to a Simple Three-Phase Cylindrical Inductor

Winding Functions in a d–q–n Reference Frame

Direct Computation of d–q–n Inductances of a Cylindrical Three-Phase Inductor

The d–q Equations of a Synchronous Machine


Physical Description

Synchronous Machine Equations in the Phase Variable or as-, bs-, cs- Reference Frame

Transformation of the Stator Voltage Equations to a Rotating Reference Frame

Transformation of Stator Flux Linkages to a Rotating Reference Frame

Winding Functions of the Three-Phase Stator Windings in a d–q–n Reference Frame

Winding Functions of the Rotor Windings

Calculation of Stator Magnetizing Inductances

Mutual Inductances between Stator and Rotor Circuits

d–q Transformation of the Rotor Flux Linkage Equation

Power Input

Torque Equation

Summary of Synchronous Machine Equations Expressed in Physical Units

Turns Ratio Transformation of the Flux Linkage Equations

System Equations in Physical Units Using Hybrid Flux Linkages

Synchronous Machine Equations in Per Unit Form

Steady-State Behavior of Synchronous Machines


d–q Axes Orientation

Steady-State Form of Park’s Equations

Steady-State Torque Equation

Steady-State Power Equation

Steady-State Reactive Power

Graphical Interpretation of the Steady-State Equations

Steady-State Vector Diagram

Vector Interpretation of Power and Torque

Phasor Form of the Steady-State Equations

Equivalent Circuits of a Synchronous Machine

Solutions of the Phasor Equations

Solution of the Steady-State Synchronous Machine Equations Using MathCAD

Open-Circuit and Short-Circuit Characteristics

Saturation Modeling of Synchronous Machines Under Load

Construction of the Phasor Diagram for a Saturated Round-Rotor Machine

Calculation of the Phasor Diagram for a Saturated Salient-Pole Synchronous Machine

Zero Power Factor Characteristic and the Potier Triangle

Other Reactance Measurements

Steady-State Operating Characteristics

Calculation of Pulsating and Average Torque during Starting of Synchronous Motors

Transient Analysis of Synchronous Machines


Theorem of Constant Flux Linkages

Behavior of Stator Flux Linkages on Short-Circuit

Three-Phase Short-Circuit, No Damper Circuits, Resistances Neglected

Three-Phase Short-Circuit from Open Circuit, Resistances and Damper Windings Neglected

Short-Circuit from Loaded Condition, Stator Resistance and Damper Winding Neglected

Three-Phase Short-Circuit from Open Circuit, Effect of

Resistances Included, No Dampers

Extension of the Theory to Machines with Damper Windings

Short-Circuit of a Loaded Generator, Dampers Included

Vector Diagrams for Sudden Voltage Changes

Effect of Exciter Response

Transient Solutions Utilizing Modal Analysis

Comparison of Modal Analysis Solution with Conventional Methods

Unsymmetrical Short-Circuits

Power System Transient Stability



Torque Angle Curves

Mechanical Acceleration Equation in Per Unit

Equal Area Criterion for Transient Stability

Transient Stability Analysis

Transient Stability of a Two Machine System

Multi-Machine Transient Stability Analysis

Types of Faults and Effect on Stability

Step-by-Step Solution Methods Including Saturation

Machine Model Including Saturation

Summary-Step-by-Step Method for Calculating Synchronous Machine Transients

Excitation Systems and Dynamic Stability


Generator Response to System Disturbances

Sources of System Damping

Excitation System Hardware Implementations

IEEE Type 1 Excitation System

Excitation Design Principles

Effect of the Excitation System on Dynamic Stability

Naturally Commutated Synchronous Motor Drives


Load Commutated Inverter (LCI) Synchronous Motor Drives

Principle of Inverter Operation

Fundamental Component Representation

Control Considerations

Starting Considerations

Detailed Steady-State Analysis

Time Step Solution

Sample Calculations

Torque Capability Curves

Constant Speed Performance

Comparison of State Space and Phasor Diagram Solutions

Extension of d–q Theory to Unbalanced Operation


Source Voltage Formulation

System Equations to Be Solved

System Formulation with Non-Sinusoidal Stator Voltages

Solution for Currents

Solution for Electromagnetic Torque

Example Solutions

Linearization of the Synchronous Machine Equations


Park’s Equations in Physical Units

Linearization Process

Transfer Functions of a Synchronous Machine

Solution of the State Space and Measurement Equations

Design of a Terminal Voltage Controller

Design of a Classical Regulator

Computer Simulation of Synchronous Machines


Simulation Equations

MATLAB® Simulation of Park’s Equations

Steady-State Check of Simulation

Simulation of the Equations of Transformation

Simulation Study

Consideration of Saturation Effects

Air Gap Saturation

Field Saturation

Approximate Models of Synchronous Machines


Appendix 1: Identities Useful in AC Machine Analysis

Appendix 2: Time Domain Solution of the State Equation

Appendix 3: Three-Phase Fault

Appendix 4: TrafunSM

Appendix 5: SMHB Synchronous Machine Harmonic Balance

CRC Press; June 2012
600 pages; ISBN 9781439880685
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Title: Analysis of Synchronous Machines, Second Edition
Author: T.A. Lipo
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