Dynamics of Saturated Electric Machines

This book is a result of the author's work which was initiated about a decade ago and which, in the meantime, has resulted in his Ph.D. Thesis and several technical papers. The book deals with accurate modeling of electric machines during transient and steady states, a topic which has been usually avoided in the literature. The modeling techniques herein take into account all machine peculiarities, such as the type and connection of its windings, slotting, and saturation in the iron core. A special emphasis in the book is given to the exact physical interpretation of all phenomena which influence the machine's transient behavior. Besides the Introduction, the book has five chapters. The second chapter describes basic concepts of the magnetic equivalent circuit theory and has examples of magnetic equivalent circuits of several types of machines with their node potential equations. In the third chapter the transform matrices w' and w" of A.C. wind­ ings are derived. These matrices playa very important role in the magnetic equivalent circuit theory because they connect the quantities from the ma­ chine's magnetic equivalent circuit, branch fluxes, and mmfs with the ma­ chine's phase currents and fluxes.

1 Introduction.- 2 Magnetic Circuits.- 2.1 Analogies Between Electric and Magnetic Circuits.- 2.2 Permeances in Magnetic Circuits.- 2.2.1 Constant Permeances.- 2.2.2 Parametric Nonlinear Permeances.- 2.2.2.1 Unskewed Machine with Different Number of Stator and Rotor Slots.- 2.2.2.2 Influence of Skewing.- 2.2.2.3 Rotor Eccentricity.- 2.2.2.4 Complete Air Gap Permeance.- 2.2.2.5 Influence of Fringing Flux.- 2.2.3 Inherently Nonlinear Permeances.- 2.2.3.1 B-H Curve Approximation.- 2.2.3.2 Influence of Variable Geometry.- 2.2.4 Convention About Symbols.- 2.3 Sources in Magnetic Circuits.- 2.3.1 Magnetomotive Force of a Coil in a High-Permeability Medium.- 2.3.2 Magnetomotive Force of a Distributed Winding.- 2.3.3 Permanent Magnets in Magnetic Circuits.- 2.3.4 Convention About Symbols.- 2.4 Force and Torque.- 2.5 Magnetic Equivalent Circuits of Electromagnetic Devices.- 2.5.1 Singly and Multiply Excited Iron Core.- 2.5.2 Windings in Slots.- 2.5.3 Elementary Pole Pair.- 2.5.3.1 The Number of Stator Parallel Circuits Restriction.- 2.5.3.2 Stator Fractional Slot Winding Restriction.- 2.5.3.3 The Number of Rotor Slots per Pole Restriction.- 2.5.3.4 Summary of Conditions for the Number of Elementary Pole Pairs.- 2.5.3.5 The Number of Stator and Rotor Slots per Elementary Pole Pair.- 2.5.4 Induction Machine.- 2.5.5 Salient Pole, Wound Rotor Synchronous Machine.- 2.5.6 Permanent Magnet Synchronous Machine.- 2.5.7 Stepper and Switched Reluctance Motor.- 2.5.8 Shaded Pole Induction Machine.- 2.5.9 Conductive Plate in External Magnetic Field.- 2.6 Magnetic Equivalent Circuit Solution Procedure.- 2.6.1 Node Potential Equations and Permeance Matrices.- 2.6.2 k-Winding, n-Limb Transformer.- 2.6.3 Induction Machine.- 2.6.3.1 Simple Model.- 2.6.3.2 Complex Model.- 2.6.4 Wound Rotor Synchronous Machine.- 2.6.5 Permanent Magnet Synchronous Machine.- 2.6.6 Stepper and Switched Reluctance Motor.- 2.6.7 Shaded Pole Induction Machine.- 3 Winding Transform Matrices.- 3.1 Winding Topology Matrix.- 3.1.1 Single-Layer Windings.- 3.1.1.1 Concentric Winding.- 3.1.1.2 Distributed Winding.- 3.1.2 Double-Layer Winding.- 3.2 Number of Turns per Coil Matrix.- 3.3 Coil Currents Matrix.- 3.4 Slot Ampere-Turns Matrix.- 3.4.1 Single-Layer Winding.- 3.4.1.1 Concentric Winding.- 3.4.1.2 Distributed Winding.- 3.4.2 Double-Layer Winding.- 3.4.2.1 One-Layer Model.- 3.4.2.2 Two-Layer Model.- 3.5 Tooth Magnetomotive Force Matrix.- 3.6 Magnetomotive Force Transform Matrix w?.- 3.7 Coil Flux Matrix.- 3.7.1 Single-Layer Winding.- 3.7.1.1 Concentric Winding.- 3.7.1.2 Distributed Winding.- 3.7.2 Double-Layer Winding.- 3.7.2.1 One-Layer Model.- 3.7.2.2 Two Layer Model.- 3.8 Number of Turns per Elementary Phase Matrix.- 3.9 Elementary Phase Flux Matrix.- 3.9.1 Single-Layer Windings.- 3.9.1.1 Concentric Winding.- 3.9.1.2 Distributed Winding.- 3.9.2 Double-Layer Windings.- 3.10 Flux Transform Matrix w?.- 3.11 Summary of Winding Transform Matrices.- 3.12 Squirrel Cage Connection Matrices.- 4 Extended System of Machine Algebraic Equations.- 4.1 Squirrel Cage Induction Machine — A Simple Model.- 4.2 Squirrel Cage Induction Machine — A Complex Model.- 4.3 Wound Rotor Synchronous Machine.- 4.4 Permanent Magnet Synchronous Machine.- 4.5 Stepper and Switched Reluctance Motor.- 4.6 Shaded Pole Induction Motor.- 5 Machine System of Differential Equations and Complete System of Algebraic Equations.- 5.1 Stator Differential Equations of an A.C. Machine.- 5.1.1 Voltage-Fed m-Phase Machine.- 5.1.1.1 Polygon Connection.- 5.1.1.2 Star without Common Neutral Connection.- 5.1.1.3 Star with Common Neutral Connection.- 5.1.2 Current-Fed m-Phase Machine.- 5.1.2.1 Polygon Connection.- 5.1.2.2 Star without Common Neutral Connection.- 5.1.2.3 Star with Common Neutral Connection.- 5.1.3 Summary of Stator Winding Connection Types.- 5.1.4 Voltage Equations of Other Types of A.C. Machines.- 5.1.5 Failures in a Voltage-Fed A.C. Machine Stator Winding.- 5.1.5.1 Voltage-Fed m-Phase Machine; Polygon Connection.- 5.1.5.2 Voltage-Fed m-Phase Machine; Star without Common Neutral Connection.- 5.1.5.3 Voltage-Fed m-Phase Machine; Star with Common Neutral Connection.- 5.1.6 Failures in a Current-Fed A.C. Machine Stator Winding.- 5.1.6.1 Current-Fed m-Phase Machine; Polygon Connection.- 5.1.6.2 Current-Fed m-Phase Machine; Star without Common Neutral Connection.- 5.1.6.3 Current-Fed m-Phase Machine; Star with Common Neutral Connection.- 5.2 Rotor Differential Equations of an A.C. Machine.- 5.2.1 Squirrel Cage Winding Voltage Differential Equations.- 5.2.2 Wound Rotor Voltage Differential Equations.- 6 Unique System of Machine Equations.- 6.1 Influence of Series-Added Impedance.- 6.1.1 Inductors in Series with a Machine.- 6.1.2 Capacitors in Series with a Machine.- 6.2 k-Winding, n-Limb Transformer.- 6.3 Induction Machine — a Simple Model.- 6.3.1 Voltage-Fed Machine.- 6.3.2 Current-Fed Machine.- 6.4 Squirrel Cage Induction Machine — a Complex Model.- 6.4.1 Voltage-Fed Machine.- 6.4.2 Current-Fed Machine.- 6.5 Wound Rotor Synchronous Machine.- 6.6 Permanent Magnet Synchronous Machine.- 6.7 Step and Switched Reluctance Motor.- 6.8 Shaded Pole Induction Machine.- 6.9 Initial Conditions Definition.- Appendix I - Matrix Operators.- Appendix II - Vector and Matrix Differentiation.- Appendix III - Applications.- III.1. Squirrel Cage Induction Machine.- III.1.1 Starting Characteristics of the Basic Machine.- III.1.2 Basic Machine with Additional Inertia on the Shaft.- III.1.3 Basic Machine with a Decreased Number of Rotor Slots and an Additional Inertia on the Shaft.- III.1.4 Basic Machine with Linear B-H Curve and Additional Inertia.- III.1.5 Wye-Connected Basic Machine with Additional Inertia.- III.1.6 Reversing of the Basic Machine with Additional Inertia.- III.1.7 Basic Machine with Blocked Rotor.- III. 1.8 Basic Machine with Full Pitch Stator Winding and Additional Inertia.- III.1.9 Wye-Connected Basic Machine with 4/6 Winding Pitch.- III.1.10 Basic Machine with Unskewed Rotor and Additional Inertia.- III.1.11 Unsymmetric Connection of the Basic Machine.- III.1.12 Basic Machine in Steinmetz Connection.- III.1.13 Basic Machine with Additional Inertia Fed from a Six-Step Voltage Source Inverter.- III.1.14 Wye-Connected Basic Machine with Additional Inertia Fed from a Six-Step Voltage Source Inverter.- III.1.15 Basic Machine with Additional Inertia Fed from a Current Source.- III.1.16 Capacitor Braking of the Basic Machine.- III.1.17 D.C. Current Braking of the Basic Machine.- III.2. Permanent Magnet Synchronous Machine.- III.2.1 Starting of the Basic Machine without Permanent Magnets.- III.2.2 Starting of the Basic Machine.- III.2.3 Loading of the Basic Machine.- III.2.4 Heavy Loading — Loss of Synchronism.