Analysis of Synchronous Machines

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).

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