Wind power technology - Wind Power Integration: Connection and System Operational Aspects - page 68

Then from (3.11), the torque is

V s
ð R s þ R r = s Þ

T ¼ 3 p f
2

R r
s w s

ð 3 : 13 Þ

2 2

þð X s þ X r Þ

The relationship between torque and slip is readily estimated if the small stator
resistance R s is neglected. The torque may then be written as

sV s
ð R 0 r þ s 2 X 2

R r
w s

sR r
R 0 r þ s 2 X 2

T ¼ 3 p f
2 Þ ¼ k

ð 3 : 14 Þ

where

k ¼ 3 p f

V s
w s 2

X ¼ X s þ X r

It may be seen from (3.14) that, when the slip s is small, the torque is directly
proportional to it. Also, when the slip is large, the torque will be given approxi-
mately by

R r
sX 2

T ¼ k

The torque will therefore be inversely proportional to slip for larger values of slip.
It follows that there must be an intermediate value of slip for which the torque
is a maximum. This value of slip may be obtained by differentiating the torque with
respect to slip and equating to zero:

ds ksR r ð R 0 r þ s 2 X 2

Þ 1

dT
ds ¼

d

Þ 1 Þ 2 ¼ kR r ð R 0 r þ s 2 X 2 ksR r ð R 0 r þ s 2 X 2 2 sX 2

¼ k R r ð R 0 r þ s 2 X 2

Þ 2 s 2 R r X 2
ð R 0 r þ s 2 X 2

2

Þ

R 0 r s 2 R r X 2
ð R 0 r þ s 2 X 2

¼ k

2

Þ

¼ 0

The torque will therefore be a maximum when

R 0 r s 2 R r X 2

¼ 0

ð 3 : 15 Þ

R r
X

∴ s ¼