Performance and Operation Management - Wind Energy Systems for Electric Power Generation - page 122

where

K D = 2 T k

T N ω re f τ kr
is the initial slope of the per-unit torque characteristic vs. slip,

T N = z p P N
ω N

is the rated torque .

Note that the breakdown-slip is s k = 1 / ( ω re f τ kr ). The relevant characteristic

equation is:

p 2

0

Δω r + p 2 δΔω r + ν Δω r = 0 (6.29)

where

δ = 1 / (2 τ kr )

and

K D
τ kr τ J

0 =

ν

2 2

0 >

For ν δ the solution is conjugate complex:

j

2

p 1 , 2 = − δ ± ν

0 − δ

2 =

− δ ± j ω e

This case is of interest since in most practical the machine, reacting on a dis-
turbance, oscillates at an electromechanical eigenfrequency. This is observed e.g.
during run-up of grid-supplied induction machines.
Example: With parameters T k / T N = 2 , 5 and s k = 0 , 1 the damped eigenfrequency
is f e = ω e / (2 π )=6 , 3Hz at τ J = 1 s and f e = 1 , 3Hz at τ J = 5s.

6.2.3 Synchronous Machine Models

6.2.3.1 Machine with Field Winding on the Rotor

The synchronous machine representation in Fig. 6.5a contains a stator with the
three-phase windings a, b, c, and a rotor, generally of the salient pole type, carrying
a field winding f. The stator quantities may be transformed, in stationary frame, into
α

components, see Fig. 6.5b. Park's transformation leads to the d, q components,
the reference d-direction fixed to the rotor direct (magnetizing) axis. The Figure
does not take care of zero-sequence components. The model
The voltage equations in the three-windings model

, β

α , β , f for the synchronous

machine without damper circuits in the rotor is:
⎡

⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

u u u f R s 00

0 R s 0
00 R f

i i i f ψ α

ψ β
ψ f

d
dt

⎣ ⎦ = ⎣ ⎦ ⎣ ⎦ + ⎣ ⎦

(6.30)