Matrix Analysis of Electrical Machinery

Matrix Analysis of Electrical Machinery

2nd Edition - January 1, 1974

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  • Author: N. N. Hancock
  • eBook ISBN: 9781483137292

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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




    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


    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


    The Three-Winding Transformer


    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



    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


    Equivalent Circuit


    Balanced Terminal Voltage

    Unbalanced Terminal Voltage


    The Unbalanced Two-phase Induction Machine

    Equivalent Circuit



    Single-phase Operation of Induction Machine

    Three-phase Machine

    Two-phase Machine

    Single-phase Machine


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

    Phasor Diagram

    Field Current Required for Given Armature Terminal Conditions



    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



    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


    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


    Power Selsyns

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

    14 Conclusion


    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


    Hints and Answers to Exercises



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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