40 HANDBOOK OF ELECTRICAL ENGINEERING
which may be represented by a simple linear function,
ω=ωo−kT ( 2. 47 )wherekis a positive number in the order of 1.0 pu equal to the open-loop slope, andωois the shaft
speed at no-load.
Reference 7 discusses the slopekin Chapter 2, Section 2.3.1.
Assume that the turbine is designed to deliver unit torque at unit speed, therefore,1. 0 =ωo−k( 1. 0 )=ωo−k( 2. 48 )From whichωo= 1 +kand so (2.47) becomes,
ω= 1 +k−kTorT=1 +k−ω
k( 2. 49 )
The speed can now be related to the shaft power rather than the torque,
P=
(
1 +k−ω
k)
ω( 2. 50 )Or in the form of a quadratic equation,
0 =ω^2 −( 1 +k)ω+kP ( 2. 51 )The two roots of which are,
ω 1 , 2 =1 +k
2±
(
( 1 +k)^2 − 4 kP
2) 1 / 2
( 2. 52 )
The positive root applies to the stable operating region, whilst the negative root applies to the
unstable region after stalling occurs.
For example assumek= 1 .5. Table 2.4 shows the values of the two roots for an increase in
shaft power.
Table 2.4. Open-loop steady state speed-power char-
acteristic of a gas turbine (k= 1 .5)
Shaft power Shaft speedω(per unit)
P(per unit)
Positive root Negative root
0.0 2.5 0.0
0.5 2.151 0.349
0.75 1.911 0.589
1.00 1.500 1.000
1.04 1.250 1.250
1. 04 +(unstable)