Generators - Wind Energy Systems for Electric Power Generation - page 42

3.3.2 Performance Equations and Equivalent Circuits

3.3.2.1 Model Assumptions

Synchronous generator inductors are built either with salient poles or as turbo-type
rotors. The inductor field generated by either excitation winding or permanent mag-
nets assigns a specified direction to the rotor called the pole axis. Apart from round
rotor machines the saliency gives rise to a variable, periodic air gap reluctance long
the circumference.
Usually the two-axis theory, also called Park' theory, serves to describe the syn-
chronous machine behaviour. It features an orthogonal coordinate system with the
axes d (pole axis, direct axis) and q (intermediate axis, quadrature axis).
The steady-state performance can be discussed using equivalent circuits of
lumped parameters. For the case of a turbo-type machine with constant air-gap,
In this case equal reluctance in q - and d -axis provides reactance symmetry, X q =
X d . Figure 3.15 shows two equivalent circuits suitable to describe stady state at
synchronous speed, one with ideal emf and series impedance, the other with ideal
current source and parallel impedance. In the figure U p is the emf induced by the
rotor flux (inductor voltage), while I f is the equivalent field current referred to ar-
mature side. Note that X d = X md + X σ1 .

Fig. 3.15 Equivalent circuits of the turbo-type synchronous machine ( a ) with voltage source U p ;
( b ) with current source I f

3.3.2.2 Operation at Given Stator Voltage

In synchronous machine analysis a quantity of special importance is the load
When the machine is running at synchronous speed,

ω 1 / z p with
impressed terminal voltage U 1 , the voltage and torque equations of a three-phase
turbo-type machine are given by:
U 1 = R 1 I 1 + jX d I 1 + U p

Ω = Ω syn =

U p = jX hd I f = U p · e j ϑ

;

U 1 | · U p
Ω syn ·

(3.18)

3 |

T = − sin ϑ

X d

where X d = X md + X σ 1 is the direct-axis synchronous reaktance, I f denotes the ex-
citation current referred to stator side, for a magnetically homogenous machine,
X q = X d . Load angle angle

ϑ is defined as the electrical angle between terminal