# Analysis of Synchronous Machines

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## Book Description

**Analysis of Synchronous Machines, Second Edition** is a thoroughly modern treatment of an old subject. Courses generally teach about synchronous machines by introducing the steady-state per phase equivalent circuit without a clear, thorough presentation of the source of this circuit representation, which is a crucial aspect. Taking a different approach, this book provides a deeper understanding of complex electromechanical drives.

Focusing on the terminal rather than on the internal characteristics of machines, the book begins with the general concept of winding functions, describing the placement of any practical winding in the slots of the machine. This representation enables readers to clearly understand the calculation of all relevant self- and mutual inductances of the machine. It also helps them to more easily conceptualize the machine in a rotating system of coordinates, at which point they can clearly understand the origin of this important representation of the machine.

- Provides numerical examples
- Addresses Park’s equations starting from winding functions
- Describes operation of a synchronous machine as an LCI motor drive
- Presents synchronous machine transient simulation, as well as voltage regulation

Applying his experience from more than 30 years of teaching the subject at the University of Wisconsin, author T.A. Lipo presents the solution of the circuit both in classical form using phasor representation and also by introducing an approach that applies MathCAD^{®}, which greatly simplifies and expands the average student’s problem-solving capability. The remainder of the text describes how to deal with various types of transients—such as constant speed transients—as well as unbalanced operation and faults and small signal modeling for transient stability and dynamic stability.

Finally, the author addresses large signal modeling using MATLAB^{®}/Simulink^{®}, for complete solution of the non-linear equations of the salient pole synchronous machine. A valuable tool for learning, this updated edition offers thoroughly revised content, adding new detail and better-quality figures.

## Table of Contents

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**Winding Distribution in an Ideal Machine**

Introduction

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

Introduction

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

Introduction

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

Introduction

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

Introduction

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

Introduction

Assumptions

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

Introduction

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

Introduction

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

Introduction

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

Introduction

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

Introduction

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

## Author(s)

### Biography

**Thomas A. Lipo** received his BEE and MS degrees at Marquette University and his Ph.D from the University of Wisconsin in 1968. After 10 years at the Corporate R&D Center of the General Electric Company in Schenectady. New York, he joined Purdue University as professor in 1978 and subsequently took the same position at the University of Wisconsin in 1980. He was granted the 2004 Hilldale Award, the university’s most prestigious award for scientific achievement. He has published more than 550 technical papers, secured 35 U.S. patents, and written five books in his discipline. He is a Fellow of IEEE and IET (London), and he is also a member of the National Academy of Engineering (USA) and the Royal Academy of Engineering (UK).