Homework # 8
4-3. A three-phase four-pole winding is installed in 12 slots on a stator. There are 40 turns of wire in each slot of the
windings. All coils in each phase are connected in series, and the three phases are connected in Δ. The flux per pole in
the machine is 0.060 Wb, and the speed of rotation of the magnetic field is 1800 r/min.
(a) What is the frequency of the voltage produced in this winding?
(b) What are the resulting phase and terminal voltages of this stator?
SOLUTION
(a) The frequency of the voltage produced in this winding is
n P (1800r / min)(4 poles )
fe = m =
= 60 Hz
120
120
(b) There are 12 slots on this stator, with 40 turns of wire per slot. Since this is a four-pole machine, there are two sets
of coils (4 slots) associated with each phase. The voltage in the coils in one pair of slots is
E A = 2π ⋅ N C ⋅ φ ⋅ f = 2π (40)(0.060)(60) = 640V
There are two sets of coils per phase, since this is a four-pole machine, and they are connected in series, so the
total phase voltage is
Vφ = 2(640) = 1280V
Since the machine is Δ-connected, VL = Vφ = 1280 V
5-27. A 25-MVA three-phase 13.8-kV two-pole 60-Hz synchronous generator was tested by the open-circuit test, and its
air-gap voltage was extrapolated with the following results:
Open-circuit test
Field current, A
Line voltage, kV
Extrapolated air-gap voltage, kV
320
13.0
15.4
365
13.8
17.5
380
14.1
18.3
475
15.2
22.8
570
16.0
27.4
475
1550
570
1885
The short-circuit test was then performed with the following results:
Short-circuit test
Field current, A
Armature current, A
320
1040
365
1190
380
1240
The armature resistance is 0.24 Ω per phase.
(a) Find the unsaturated synchronous reactance of this generator in ohms per phase and in per-unit.
(b) Find the approximate saturated synchronous reactance XS at a field current of 380 A. Express the answer both in
ohms per phase and in per-unit.
(c) Find the approximate saturated synchronous reactance at a field current of 475 A. Express the answer both in ohms
per phase and in per-unit.
(d) Find the short-circuit ratio for this generator.
SOLUTION
(a) The unsaturated synchronous reactance of this generator is the same at any field current, so we will look at it at a
field current of 380 A. The extrapolated air-gap voltage at this point is 18.3 kV, and the short-circuit current is
1240 A. Since this generator is Y-connected, the phase voltage is Vφ = 18.3 / 3 = 10,566 V and the armature
current is IA = 1240 A. Therefore, the unsaturated synchronous reactance is
10,566
X Su =
= 8.52 Ω
1240
The base impedance of this generator is
3Vφ ,base 2 3(7967V) 2
=
= 7.62 Ω
Z base =
Sbase
25MVA
Therefore, the per-unit unsaturated synchronous reactance is
8.52
X Su , pu =
= 1.12 Ω
7.62
(b) The saturated synchronous reactance at a field current of 380 A can be found from the OCC and the SCC. The
OCC voltage at IF = 380 A is 14.1 kV, and the short-circuit current is 1240 A. Since this generator is Y-connected,
the corresponding phase voltage is Vφ = 14.1/ 3 = 8141 V and the armature current is IA = 1240 A. Therefore,
the saturated synchronous reactance is
8141
= 6.57 Ω
1240
and the per-unit unsaturated synchronous reactance is
6.57
X Su , pu =
= 0.862 Ω
7.62
X Su =
(c) The saturated synchronous reactance at a field current of 475 A can be found from the OCC and the SCC. The
OCC voltage at IF = 475 A is 15.2 kV, and the short-circuit current is 1550 A. Since this generator is Y-connected,
the corresponding phase voltage is Vφ = 15.2 / 3 = 8776 V and the armature current is IA = 1550 A. Therefore, the
saturated synchronous reactance is
8776
= 5.66 Ω
1550
and the per-unit unsaturated synchronous reactance is
5.66
X Su , pu =
= 0.743 Ω
7.62
X Su =
(d) The rated voltage of this generator is 13.8 kV, which requires a field current of 365 A. The rated line and armature
current of this generator is
25 MVA
IL =
= 1046 A
3(13.8 kV)
The field current required to produce a short-circuit current of 10465 A is about 320 A. Therefore, the
short-circuit ratio of this generator is
365
SCR =
= 1.14
320