Page 61 - Handbook of Electrical Engineering

40 HANDBOOK OF ELECTRICAL ENGINEERING

which may be represented by a simple linear function,

ω = ω o − kT (2.47)

where k is a positive number in the order of 1.0 pu equal to the open-loop slope, and ω o is the shaft
speed at no-load.
Reference 7 discusses the slope k in Chapter 2, Section 2.3.1.
Assume that the turbine is designed to deliver unit torque at unit speed, therefore,

1.0 = ω o − k(1.0) = ω o − k (2.48)

From which ω o = 1 + k and so (2.47) becomes,

1 + k − ω
ω = 1 + k − kT or T = (2.49)
k
The speed can now be related to the shaft power rather than the torque,

1 + k − ω
P = ω (2.50)
k
Or in the form of a quadratic equation,

2
0 = ω − (1 + k)ω + kP (2.51)
The two roots of which are,

2
1 + k (1 + k) − 4kP 1/2
ω 1,2 = ± (2.52)
2 2
The positive root applies to the stable operating region, whilst the negative root applies to the
unstable region after stalling occurs.
For example assume k = 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 speed ω (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)