1. Two-pole,3-phase,wye-connected,salient-pole synchronous machine
SYNCHRONOUS MACHINES
Two-pole,3-phase,wye-connected,salient-pole synchronous machine

2. SYNCHRONOUS MACHINES
In abc reference frame, voltage equations can be written as

3. flux linkage equations
SYNCHRONOUS MACHINES
flux linkage equations

4. flux linkage equations
SYNCHRONOUS MACHINES
flux linkage equations
Referring all rotor variables to the stator windings

5. SYNCHRONOUS MACHINES
Referring all rotor variables to the stator windings

6. SYNCHRONOUS MACHINES
Referring all rotor variables to the stator windings

7. SYNCHRONOUS MACHINES
TORQUE EQUATION IN MACHINE VARIABLES

8. SYNCHRONOUS MACHINES
SWING EQUATION

9. SYNCHRONOUS MACHINES
Stator Voltage Equations in Arbitrary Reference-frame Variables
The rotor voltage equations are expressed only in the rotor reference frame:

10. SYNCHRONOUS MACHINES
The flux linkage equations may be expressed as:
The sinusoidal terms are constant, independent of  and r only if  = r

11. Voltage Equations In Rotor Reference-frame variables: park's Equations
SYNCHRONOUS MACHINES
Therefore, the time-varying inductances are eliminated from the voltage equations only if the reference frame is fixed in the rotor.
Voltage Equations In Rotor Reference-frame variables: park's Equations

12. SYNCHRONOUS MACHINES

13. SYNCHRONOUS MACHINES
Park's voltage equations are often written in expanded form:
Flux linkages in expanded form:

14. SYNCHRONOUS MACHINES
The Equivalent q-axis Circuits:

15. SYNCHRONOUS MACHINES
The Equivalent d-axis Circuits:

16. SYNCHRONOUS MACHINES
The Equivalent 0-axis Circuits:

17. SYNCHRONOUS MACHINES
It is often convenient to express the voltage and flux linkage equations in terms of reactances rather than inductances:

18. SYNCHRONOUS MACHINES
Also, it is convenient to define:

19. SYNCHRONOUS MACHINES
If we select the currents as independent variables:

20. SYNCHRONOUS MACHINES
If we select the flux linkages per second as independent variables:

21. SYNCHRONOUS MACHINES
Torque Equations in Substitute Variables:

22. SYNCHRONOUS MACHINES Rotor Angle :
it is convenient to relate the position of the rotor of a synchronous machine to a voltage or to the rotor of another machine.
The electrical angular displacement of the rotor relative to its terminal voltage is defined as the rotor angle,
The rotor angle is the displacement of the rotor generally referenced to the maximum positive value of the fundamental component of the terminal voltage of phase a:

23. SYNCHRONOUS MACHINES Rotor Angle :
It is important to note that the rotor angle is often used as the argument in the transformation between the rotor and synchronously rotating reference frames
The rotor angle is often used in relating torque and rotor speed (if e is constant):

24. SYNCHRONOUS MACHINES
PER UNIT SYSTEM
Base voltage:
the rms value ofthe rated phase voltage for the abc variables
the peak value for the qd0 variables.
Base power:
When considering the machine separately, the power base is selected as its volt-ampere rating.
When considering power systems, a system power base (system base) is selected
Once the base quantities are established, the corresponding base current and base impedance may be calculated.
Base torque is the base power divided by the synchronous speed of the rotor:

25. SYNCHRONOUS MACHINES
The torque expressed in per unit:

26. For balanced conditions: r is constant and equal to e
SYNCHRONOUS MACHINES
ANALYSIS OF STEADY-STATE OPERATION:
For balanced conditions:
the 0s quantities are zero.
r is constant and equal to e
the rotor windings do not experience a change of flux linkages
the current is not flowing in the short-circuited damper windings
the time rate of change of all flux linkages neglected

27. For balanced conditions:
SYNCHRONOUS MACHINES
ANALYSIS OF STEADY-STATE OPERATION:
For balanced conditions:

28. SYNCHRONOUS MACHINES Hence: and if it is noted that: Then:
ANALYSIS OF STEADY-STATE OPERATION:
Hence:
and if it is noted that:
Then:

29. ANALYSIS OF STEADY-STATE OPERATION
It is convenient to define the last term on the right-hand side as (excitation voltage):
if rs is neglected, the expression for the balanced steady-state electromagnetic torque in per unit can be written as:

30. DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE IN INPUT TORQUE

31. DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE IN INPUT TORQUE
Dynamic performance of a hydro turbine generator during a step increase in input torque from zero to rated:

32. DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE IN INPUT TORQUE
Torque versus rotor angle characteristics

33. DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE IN INPUT TORQUE

34. DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE IN INPUT TORQUE
Dynamic performance of a steam turbine generator during a step increase in input torque from zero to 50% rated.

35. DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE IN INPUT TORQUE
Torque versus rotor angle characteristics

36. DYNAMIC PERFORMANCE DURING A 3 PHASE FAULT AT THE MACHINE TERMINALS
a hydro turbine generator

37. DYNAMIC PERFORMANCE DURING A 3 PHASE FAULT AT THE MACHINE TERMINALS
Torque versus rotor angle characteristics:

38. DYNAMIC PERFORMANCE DURING A 3 PHASE FAULT AT THE MACHINE TERMINALS
a steam turbine generator

39. DYNAMIC PERFORMANCE DURING A 3 PHASE FAULT AT THE MACHINE TERMINALS
Torque versus rotor angle characteristics:

40. COMPUTER SIMULATION
Simulation in Rotor Reference Frame
Where:

41. COMPUTER SIMULATION
Simulation in Rotor Reference Frame

42. COMPUTER SIMULATION
Simulation of Saturation

43. COMPUTER SIMULATION
Simulation of Saturation