Lecture 2 Synchronous Machines Synchronous Machine Dynamic Equations
- Slides: 12
Lecture 2 Synchronous Machines Synchronous Machine Dynamic Equations Professor: Ali Keyhani
Synchronous Machine Dynamic Equations The machine dynamic equations referred to the stator (Motor operation) 2
Synchronous Machine Dynamic Equations 3
Synchronous Machine Dynamic Equations Figure A Equivalent Circuits of a Two-phase synchronous machine with reference frame fixed in rotor. 4
Synchronous Machine Dynamic Equations Using Park’s transformation (AIEE Trans. Vol. 48, July 1929) Note that r – means that reference frame is fixed in the rotor s – stator variable 5
Synchronous Machine Dynamic Equations We can write the following equations from the d-q axis by inspection: From the d-q axis equivalent circuits, the flux linkage equations can also be written by inspection: 6
Synchronous Machine Dynamic Equations The rotor angle 7
Synchronous Machine Dynamic Equations where, : the angular displacement of a stator phase voltage, generally : the electrical angular velocity of the rotor : the electrical angular velocity of the terminal voltages The when the time zero position is selected so that the fundamental component is maximum at (i. e. a cosine with ) is positive for motor operation. is negative for generator operation. 8
Synchronous Machine Dynamic Equations For generator operation: Motor operation: 9
Synchronous Machine Dynamic Equations For motor operation, the torque angle is defined to be negative. The d-q axis dynamic equations are the same as before. By definition For steady-state operation: Motor operation For motor operation, the torque angle is defined to be negative. The negative convention is chosen for motor operation because as motor is loaded, the rotor slows down and angle between the rotor field with respect to resultant field increases with respect to horizontal reference in the clockwise direction. 10
Synchronous Machine Dynamic Equations Transient condition at the steady-state Reverse convention is used for generator operation. That is the toque angle is defined to be positive. Note that the torque angle is the angle between the axes of the rotor field and the resultant field in the machine. The resultant field is the vector sum of stator (armature) and rotor field. For generator operation 11
Synchronous Machine Dynamic Equations The torque angle is defined to be positive for generator operation. To obtain the d-q axis dynamic equations, replace with and 12