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

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

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

with the inverse,

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.

Look at the instantaneous power:

Transformed system Stator Rotor Shaft Use electrical angle:

Energy conversion

Change to conservation of power Electrical system: Mechanical system:

Coupling field summation This assumes a lossless coupling field.

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

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.

Look at this in the transformed variables

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

Equate coefficients

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

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).

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

Change of variables in integration

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:

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!

So: