# Matrix Analysis of Electrical Machinery

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Matrix Analysis of Electrical Machinery, Second Edition is a 14-chapter edition that covers the systematic analysis of electrical machinery performance. This edition discusses the principles of various mathematical operations and their application to electrical machinery performance calculations. The introductory chapters deal with the matrix representation of algebraic equations and their application to static electrical networks. The following chapters describe the fundamentals of different transformers and rotating machines and present torque analysis in terms of the currents based on the principle of the conservation of energy. A chapter focuses on a number of linear transformations commonly used in machine analysis. This edition also describes the performance of other electrical machineries, such as direct current, single-phase and polyphase commutator, and alternating current machines. The concluding chapters cover the analysis of small oscillations and other machine problems. This edition is intended for readers who have some knowledge of or are concurrently studying the physical nature of electrical machines.

## Table of Contents

Preface to the Second Edition

Preface to the First Edition

1 Introduction

Conventions

Units

Parameters

2 Elements of Matrix Algebra

Matrix Representation of Simultaneous Equations

Multiplication of Matrices

Application of Matrices to the Solution of Simultaneous Linear Equations—Inversion

Singular Matrices

The Transpose and Inverse of a Product

Alternative Methods of Inversion

Compound Matrices

Linear Transformation

Reduction to Diagonal Form

The Advantages of Matrices

Types of Matrix

Differentiation and Integration of a Matrix

3 Application of Matrix Algebra to Static Electrical Networks

Laplace Transform Equations

Notation

Linear Transformation in Electrical Circuit Analysis

Choice of Transformations—Invariance of Power

Transformation of Voltage and Impedance for Invariant Power with a Given Current Transformation

4 Transformers

The Two-Winding Transformer

Parameters

The Three-Winding Transformer

Parameters

More Complicated Magnetic Circuits

5 The matrix Equations of the Basic Rotating Machines

The Matrix Equation of the Basic Commutator Machine

Matrix Equations of Slip-Ring and Squirrel-Cage Machines

The Matrix Equation of the Balanced Two-phase Machine with a Uniform Air-Gap (Induction Machine)

The Matrix Equation of the Balanced Two-phase Revolving-Arniature Salient-Pole Machine (Synchronous Machine)

The Form of the Transformed Impedance Matrix

Limitations of the Method

6 The Torque Expressions

The Energy Stored in the Magnetic Fields

Torque Expressions

Derivation of the Torque Expression from the Equation v = Ri+p(Li)

The Transformation of ∂L/∂θ

Derivation of the Torque Expression from the Equation v = Ri+Lpi+Gθi

The Mean Steady-State Torque in A.C. Machines

Direction of Torque

7 Linear Transformations in Circuits and Machines

Resolution of Rotor M.M.F.

Transformation between Two Sets of Stationary Axes (Brush-Shifting Transformation)

The Equivalence of Three-phase and Two-phase Systems

The Transformation from Three-phase to Two-phase Axes (a, b, c to α, β, o)

The Transformation from Three-phase Axes to Symmetrical Component Axes (a, b, o to p, n, o)

The Transformation from Two-phase Axes to Symmetrical Component Axes (α, β, o to p, n, o)

Steady-State and Instantaneous Symmetrical Components

Transformation from Two-phase Rotating Axes to Stationary Axes (α, β o to d, q, o)

Transformation from Three-phase Rotating Axes to Stationary Axes (a, b, c to d, q, o)

The Transformation from Stationary Axes to Forward and Backward Axes (d, q, o to f, b, o)

The Transformation of Stator Winding Axes

Physical Interpretation of Various Sets of Axes

Rotor

Stator

8 The Application of Matrix Techniques to Routine Performance Calculations

The Establishment of the Transient Impedance Matrix

Phasor Diagrams and Equivalent Circuits

Phasor Diagrams

Equivalent Circuits

Interconnections between Machines or between Machines and other Circuit Elements

Closed Circuits

Specific Types of Problem

(a) Given the Terminal Voltages of all Windings carrying Current, to Find the Currents

(b) Given the Terminal Voltages of all Windings carrying Current, to Find the Terminal Voltages of the Open-Circuited Windings

(c) Given the Terminal Voltages of Some Windings and the Currents in the Others, to Find the Remaining Currents and Voltages

(d) Given the Terminal Voltages to Find the Torque in Terms of the Speed or Phase Angle

(e) Given the Terminal Voltages, to Find the Speed and/or Currents at a Given Torque

The Analysis of Three-phase Machines

The Effects of Zero-Sequence Currents

9 D.C. and Single-phase Commutator Machines

The Series Commutator Machine

The Shunt Commutator Machine

The D.C. Shunt Motor

The D.C. Shunt Generator on Open Circuit

The D.C. Shunt Generator on Resistance Load

Parameters of D.C. Machines

Shunt and Separately Excited D.C. Machines

D.C. Series Machine

The Repulsion Motor

Steady-State Performance in Complex Terms

Steady-State Instantaneous Currents and Torque

10 The Steady-State Performance of Polyphase Machines

The Balanced Polyphase Induction Machine

Currents

Equivalent Circuit

Torque

Balanced Terminal Voltage

Unbalanced Terminal Voltage

Parameters

The Unbalanced Two-phase Induction Machine

Equivalent Circuit

Currents

Torque

Single-phase Operation of Induction Machine

Three-phase Machine

Two-phase Machine

Single-phase Machine

Parameters

The Polyphase Synchronous Machine with a Uniform Air-Gap and no Damper Windings

Phasor Diagram

Field Current Required for Given Armature Terminal Conditions

Torque

Parameters

The Polyphase Synchronous Machine with Salient Poles and no Damper Windings

Open-Circuit Condition

Short-circuit Condition

On Balanced Load as a Generator

Phasor Diagram

Torque

Parameters

11 Transient and Negative-Sequence Conditions in A.C. Machines

Balanced Induction Machine Transients

Sudden Application of Terminal Voltage

The Synchronous Machine with Salient Poles and No Damper Windings

Sudden Three-phase Short Circuit from Open Circuit

Field Current

Armature Currents

Impedance to Negative Sequence

Impedance to Negative Sequence Current

Impedance to Negative Sequence Voltage

The Synchronous Machine with Salient Poles and Damper Windings

Balanced Steady-State Conditions

Transient Conditions

Sudden Three-phase Short Circuit from Open Circuit

Impedance to Negative Sequence

Parameters

12 Small Oscillations

Separately Excited D.C. Machine

Balanced Induction Machine

Synchronous Machine

13 Miscellaneous Machine Problems

The Metadyne Generator with Its Quadrature Brushes Displaced

The Ferraris-Arno Phase Converter

The Polyphase Induction Machine with a Single-phase Secondary Circuit

Torque

Power Selsyns

The Single-phase Performance of the Synchronous Generator with a Uniform Air-Gap and No Damping Circuits

14 Conclusion

Appendices

A. Restriction on Rotor Windings

B. Torque under Saturated Conditions

Torque in Multi-circuit Device

Calculation of Torque

C. Definitions of Systems of Axes

D. Trigonometric Formula

Laplace Transforms

Exercises

Hints and Answers to Exercises

References

Index

## Product details

- No. of pages: 368
- Language: English
- Copyright: © Pergamon 2013
- Published: January 1, 1974
- Imprint: Pergamon
- eBook ISBN: 9781483137292

## About the Author

### N. N. Hancock

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