1. ECE 576 POWER SYSTEM DYNAMICS AND STABILITY
Lecture 4
Synchronous Machine Dynamics
Professor Pete Sauer
Department of Electrical and Computer Engineering
© 2000 University of Illinois Board of Trustees, All Rights Reserved

2. Synchronous machines 3 bal. windings (a,b,c) – stator
Field winding (fd) on rotor
Damper in “d” axis
(1d) on rotor
2 dampers in “q” axis
(1q,2q) on rotor

3. Fundamental laws
Kirchhoff’s Voltage Law, Ohm’s Law, Faraday’s Law, Newton’s Second Law
Stator
Rotor
Shaft

5. with the inverse,

6. Not symmetric if T is not power invariant.
Note: This transformation is not power invariant. This means that some unusual things will happen when we use it.
Example: If the magnetic circuit is assumed to be linear
(symmetric)
Not symmetric if T is not power invariant.

7. Look at the instantaneous power:

8. Transformed system
Stator
Rotor
Shaft
Use electrical angle:

9. Energy conversion

10. Change to conservation of power
Electrical system:
Mechanical system:

12. Coupling field summation
This assumes a lossless coupling field.

13. For independent states , a, b, c, fd, 1d, 1q, 2q

14. Equate coefficients etc.
There are eight such “reciprocity conditions for this model.
These are key conditions – i.e. the first one gives an expression for the torque in terms of the coupling field energy.

15. Look at this in the transformed variables

18. For independent states shaft, d, q, o, fd,1d, 1q, 2q

19. Equate coefficients

20. Look at second derivatives:
This “non-symmetry” is due to the transformation

21. Will show that:
So that:
Which gives us an expression for torque in terms of flux linkage and current (for linear or nonlinear magnetic model).

22. Notice the important difference between “abc” variables and “qdo” variables:

24. Change of variables in integration

25. This is a path integral.
Because the coupling field is lossless, it can be shown that the path of integration is arbitrary.
Start with de-energized system so that:

26. Integrate theta first (while the electrical system is off).
Wf stays at zero because the torque and all electrical variables are still zero.
Assume: iq, id, io, ifd, i1d, i1q, i2q are independent of shaft (current/flux linkage relationship is independent of shaft).
Then Wf will be independent of theta also!

27. So: