Problems
θ=
Cn
2π
CN
353
ð6:80Þ
where
Cn = instantaneous count
CN = total counts between two positive going zero crossings of efa
The count CN also represents the speed of the rotor and can be used as the speed signal.
The speed signal can also be obtained from the derivative of the position signal θ, as shown in
Fig. 6.47b.
This simple method of position estimation is suitable for mainly constant speed applications
where the machine speed changes slowly, such as in pumps and fans. This sensorless drive
system will not work at zero or near zero speed, because the back emf, efa , is zero or almost
zero. This drive can be started by two methods:
Method 1: In this method, the drive is started by turning on switches of the inverter so that
current flows in one phase and returns through another phase. This will produce a torque
and turn the motor.
Method 2: In some cases, this drive can also be started by applying low-frequency freerunning three-phase sine-modulated PWM voltages with gradually increasing frequency.
In either method, after the motor starts rotating, the control can be switched to the sensorless
controller of Fig. 6.47b.
The above is a very simple scheme for a sensorless drive. It is presented to give the concept of
sensorless drives. A great deal of research has been carried out to overcome the problems
associated with sensorless drives. More information on sensorless drives can be obtained
from the IEEE press book, Sensorless Control of AC Motor Drives, by Rajashekara, Kawamura,
and Matsuse.
PROBLEMS
6.1
In a factory, the following are the loads:
Induction motors:
1000 hp
0.7 average power factor
0.85 average efficiency
Lighting and heating load: 100 kW
A 3φ synchronous motor is installed to provide 300 hp to a new process. The synchronous motor
operates at 92% efficiency. Determine the kVA rating of the synchronous motor if the overall
factory power factor is to be raised to 0.95. Determine the power factor of the synchronous motor.
6.2
A 3φ, 60 Hz supply and two 3φ synchronous machines are available. Determine the speed, and a
suitable number of poles, for each synchronous machine to provide:
(a) A 3φ, 180 Hz supply.
(b)
A 3φ, 500 Hz supply.
354
chapter 6 Synchronous Machines
6.3
The following test data are obtained for a 3φ, 195 MVA, 15 kV, 60 Hz star-connected synchronous machine.
Open-circuit test:
If ðAÞ
VLL ðkVÞ
150
3.75
300
7.5
450
11.2
600
13.6
750
15
900
15.8
1200
16.5
Short-circuit test:
If = 750 A; Ia = 7000 A
The armature resistance is very small.
(a) Draw the open-circuit characteristic, the short-circuit characteristic, the air gap line, and the
modified air gap line.
(b)
Determine the unsaturated and saturated values of the synchronous reactance in ohms and
also in pu.
(c)
Find the field current required if the synchronous machine is to deliver 100 MVA at rated
voltage, at 0.8 leading power factor.
(d)
Find the voltage regulation of the synchronous generator for the load of part (c). Voltage
regulation (VR) is defined as follows:
VR =
6.4
Vt jload removed −Vt jwith load
× 100%
Vt jwith load
The following test results are obtained for a 3φ, 25 kV, 750 MVA, 60 Hz, 3600 rpm, star-connected
synchronous machine at rated speed.
I f ðAÞ
V LL ðkVÞ
open-circuit test
I a ðAÞ
short-circuit test
V LL ðkVÞ
air gap line
1500
25
10,000
30
(a) Determine the number of poles of the synchronous machine.
6.5
(b)
Determine the unsaturated and saturated values of the synchronous reactance in ohms and
per unit.
(c)
The short-circuit test is performed at constant field current (1500 A) but at different speeds—
1000 rpm, 2000 rpm, 3000 rpm, and 3600 rpm. Determine the short-circuit current at these
speeds.
(d)
Determine the field current if the synchronous machine delivers rated MVA to an infinite bus
at 0.9 lagging power factor.
The synchronous machine of Example 6.2 is connected to a 3φ, 14 kV, 60 Hz infinite bus and
draws 5 MW at 0.85 leading power factor.
(a) Determine the values of the stator current ðIa Þ, the excitation voltage ðEf Þ, and the field
current ðIf Þ. Draw the phasor diagram.
(b)
If the synchronous motor is disconnected from the infinite bus without changing the field
current, determine the terminal voltage before the speed decreases.
Problems
355
6.6
Repeat Problem 6.5 if the power factor is lagging.
6.7
A 3φ synchronous condenser has Xs = 1:2 pu. The maximum field current is limited to 2.5 times
the rated field current. The rated field current produces rated terminal voltage at open circuit.
Determine the maximum reactive power that the synchronous condenser can provide.
6.8
A three-phase, 250 hp, 2300 V, 60 Hz, Y-connected nonsalient rotor synchronous motor has a
synchronous reactance of 11 ohms per phase. When it draws 165:8 kW, the power angle is 15
electrical degrees. Neglect ohmic losses.
(a) Determine the excitation voltage per phase, Ef .
(b)
Determine the supply line current, Ia .
(c)
Determine the supply power factor.
(d)
If the mechanical load is thrown off and all losses become negligible,
(i) Determine the new line current and supply power factor.
(ii) Draw the phasor diagram for the condition in ðiÞ.
(iii) By what percent should the field current If be changed to minimize the line current?
6.9
A 3 ϕ, 11 kV, 60 Hz, 25 MVA, Y-connected cylindrical-rotor synchronous machine generator has
Ra = 0:45 Ω per phase and Xs = 4:5 Ω per phase. The generator delivers the rated load at 11 kV and
0.85 lagging power factor. Determine the excitation voltage, Ef , for this operating condition.
6.10
Repeat Problem 6.9 if the power factor is leading.
6.11
The synchronous machine in Problem 6.9 is operated as a synchronous motor and it is drawing
rated MVA at 0.85 lagging power factor. Determine the excitation voltage, Ef , for this operating
condition.
6.12
A 3 ϕ, 10 kVA, 220 V, Y-connected synchronous generator has Ra = 0:25 Ω per phase and
Xs = 5:0 Ω per phase. Determine the excitation voltage, Ef , when the generator is delivering full
load at power factor of
(a) 0.85 lagging.
(b)
1.0 unity.
(c)
0.8 leading.
6.13
The synchronous generator of Problem 6.12 has a field resistance of Rf = 5:0 Ω, core loss of
250 W and rotational loss of 200 W.
Determine the efficiency of the generator in all cases of operating conditions of Problem 6.4.
The excitation current is 8 A at unity power factor and assume that excitation voltage Ef varies
linearly with the excitation current.
6.14
A 3 ϕ, 25 MVA, 15 kV, 60 Hz synchronous generator has Xs = 1:25 pu and Ra = 0:02 pu.
Calculate
6.15
(a)
Actual values of Xs and Ra in ohm.
(b)
Total full load copper losses.
A 3φ, 2000 kVA, 11 kV, 1800 rpm synchronous generator has a resistance of synchronous
reactance of 15 ohms per phase.
356
chapter 6 Synchronous Machines
(a) The field current is adjusted to obtain the rated terminal voltage at open circuit.
(i) Determine the excitation voltage Ef .
(ii) If a short circuit is applied across the machine terminals, find the stator current.
(b)
The synchronous machine is next connected to an infinite bus. The generator is made to
deliver the rated current at 0.8 power factor lagging.
(i) Determine the excitation voltage Ef .
(ii) Determine the percentage increase in the field current relative to the field current of
part (a).
(iii) Determine the maximum power the synchronous machine can deliver for the excitation current of part (b). Neglect Ra .
6.16
A 3φ, 120 MVA, 12 kV, 60 Hz, two-pole, 0.85 lagging PF, Y-connected, steam turbine–driven
alternator has a stator resistance of Ra = 0:015 pu and a synchronous reactance of Xs = 0:85 pu.
(a)
Determine the synchronous speed.
(b)
Determine Ra and Xs in ohms.
(c)
Determine the excitation voltage ðEf Þ if the alternator delivers power to an infinite bus at the
rated condition. Draw the phasor diagram.
(d)
At full-load (rated) condition, the efficiency is 92%. At this condition, determine
(i) The power lost in the armature resistance.
(ii) The rotational loss.
(iii) The torque in newton-meter applied to the shaft by the steam turbine prime mover.
6.17
6.18
A 3φ, 14 kV, 10 MVA, 60 Hz, two-pole, 0:85 PF lagging, star-connected, synchronous generator
has Xs = 20 Ω per phase and Rs = 2 Ω per phase. The generator is connected to an infinite bus.
(a)
Determine the excitation voltage at the rated condition. Draw the phasor diagram for this
condition.
(b)
Determine the torque angle at the rated condition.
(c)
If the field current is kept constant, determine the maximum power the generator can supply.
(d)
For the condition in part (c), determine the generator current and the power factor. Draw the
phasor diagram for this condition. Neglect Rs for parts (c) and (d). Find actual values in parts
(a) to (d) rather than pu values.
A 3φ, 20 kVA, 208 V, four-pole star-connected synchronous machine has a synchronous reactance of Xs = 1:5 Ω per phase. The resistance of the stator winding is negligible. The machine is
connected to a 3φ, 208 V infinite bus. Neglect rotational losses.
(a) The field current and the mechanical input power are adjusted so that the synchronous
machine delivers 10 kW at 0.8 lagging power factor. Determine the excitation voltage ðEf Þ
and the power angle ðδÞ.
(b)
The mechanical input power is kept constant, but the field current is adjusted to make the
power factor unity. Determine the percent change in the field current with respect to its
value in part (a).
Problems
6.19
357
A 3φ, 25 kV, 500 MVA, 60 Hz alternator has a synchronous reactance Xs = 1:5 pu. The alternator
is connected to an infinite 25 kV bus, through a feeder of reactance 0:25 pu, as shown in Fig.
P6.19. The terminal voltage of the alternator is maintained at 25 kV for any loading, by means of a
voltage regulator that adjusts the field current.
(a) Draw the phasor diagram.
(b)
Determine the current and the power factor of the alternator.
(c)
Determine the excitation voltage of the alternator.
FIGURE P6.19
6.20
For the alternator power system of Problem 6.19, determine the maximum power in MW that can
be transmitted over the feeder before synchronism is lost, if
(a) The voltage regulator is used to maintain the alternator terminal voltage at 25 kV.
(b)
6.21
The voltage regulator is not used, and the excitation current is kept constant at a value that
makes the excitation voltage 25 kV.
The nameplate of a Y-connected synchronous motor has the following information.
Hp 20,000, RPM 1800, PF 1.0
Volts 6600, Ampere 1350, Phase 3, Frequency 60
Excitation voltage 120, Amp 5.5
The per-unit synchronous reactance is Xs = 0:95, and the per-unit resistance is Ra = 0:012.
(a)
Determine the number of poles of the synchronous motor.
(b)
Determine Xs and Ra in ohms.
(c)
For rated (full-load) condition
(i) Determine the output torque in newton-meter.
(ii) Determine the efficiency.
(iii) Determine the rotational loss.
(iv) Determine the power loss in the field circuit.
(v) Determine Ef .
6.22
A 3φ, 1 MVA, 2300 V, 60 Hz synchronous machine has negligible stator resistance and a saturated synchronous reactance Xs = 1:25 Ω at rated terminal voltage. The efficiency of the machine
is 0.95 at rated speed. The machine is connected to an infinite bus.
(a)
Determine Xs in pu.
(b)
Determine the excitation voltage and the power angle when the machine operates as a
synchronous motor at 0.85 lagging power factor and delivers 500 hp.
(c)
The field current is now reduced by 40%, keeping the power output the same as in (b). Find
the stator current and the power factor. Will the motor lose synchronism?
358
chapter 6 Synchronous Machines
6.23
A 3φ, 10 MVA, 2300 V, 60 Hz synchronous machine has Xs = 0:9 pu and negligible stator
resistance. The machine is connected to an infinite bus. If Vt = 2300=0 V and Ef = 3450=120 V,
(a) Is the machine operating as a generator or motor?
(b)
6.24
Determine the power transfer (MW) and the power factor of the machine. Draw the phasor
diagram.
A 3φ, 2300 V, 60 Hz, 12-pole, Y-connected synchronous motor has 4.5 ohms per phase synchronous reactance and negligible stator winding resistance. The motor is connected to an infinite
bus and draws 250 amperes at 0.8 power factor lagging. Neglect rotational losses.
(a) Determine the output power.
(b)
6.25
6.26
Determine the power to which the motor can be loaded slowly without losing synchronism.
Determine the torque, stator current, and supply power factor for this condition.
A 1 MVA, 3φ, 2300 V, 60 Hz, 10 pole, star-connected cylindrical-rotor synchronous motor is
connected to an infinite bus. The synchronous reactance is 0:8 pu. All losses may be neglected.
The synchronous motor delivers 1000 hp, and the motor operates at 0.85 power factor leading.
(a)
Determine the excitation voltage Ef .
(b)
Determine the maximum power and torque the motor can deliver for the excitation current
of part (a).
(c)
The power output is kept constant at 1000 hp, and the field current is decreased. By what
factor can the field current of part (a) be reduced before synchronism is lost?
An M–G set consisting of a synchronous generator and a synchronous motor is shown in Fig.
P6.26. The ratings of the machines are:
Synchronous generator: 3φ, 1 MVA, 2300 V, 60 Hz, 0:85 PF lagging, Xs = 0:9 pu
Synchronous motor: 3φ, 500 kVA, 2300 V, 60 Hz, 0:85 PF leading, Xs = 0:8 pu
FIGURE P6.26
The generator is equipped with a voltage regulator, which maintains the terminal voltage at the
rated value. The motor delivers 500 hp, and its field current is adjusted to make it operate at unity
power factor.
(a) Determine the synchronous reactance in ohms.
6.27
(b)
Determine the excitation voltage of each machine.
(c)
Draw the phasor diagram.
A 3φ cylindrical-rotor synchronous machine and a shunt dc machine are mechanically coupled to
transfer power from a dc source to an ac source and vice versa. The ratings of the machines are
Synchronous machine: 12 kVA, 208 V
Xs = 3:0 Ω
DC machine:
12 kW, 220 V
Neglect all losses.
Problems
359
The dc machine is connected to a 220 V dc bus, and the synchronous machine is connected to a
3φ, 208 V, 60 Hz bus. The excitation of the synchronous machine is made 1:25 pu.
(a) For zero power transfer, determine the armature current in the dc machine and the current
and power factor of the synchronous machine.
6.28
(b)
Eight kilowatts is transferred from the dc bus to the ac bus through the two machines. What
adjustment is necessary? Determine the armature current in the dc machine and the stator
current and power factor of the synchronous machine.
(c)
Repeat part (b) if 8 kW is transferred from the ac bus to the dc bus.
A 3φ, 4:6 kV, 60 Hz, four-pole, Y-connected synchronous machine has the following current
ratings:
Armature current rating = 62:75 A
Field current rating = 15:0 A
Rated voltage synchronous reactance Xs = 1:25 pu
The excitation voltage ðEf Þ at the rated speed is 4:6 kV (line-to-line) when the field current is
7:5 amps.
(a) Determine the kVA rating of the machine.
6.29
6.30
(b)
Construct the capability curve for the machine for generator operation. Use per-unit values.
(c)
Determine the power factor and the power angle for optimum operating conditions; that is,
field heating and armature heating are both equal to their allowable maxima.
A 3φ, 100 MVA, 12 kV, 60 Hz salient pole, synchronous machine has Xd = 1:0 pu, Xq = 0:7 pu,
and negligible stator resistance. The machine is connected to an infinite bus and delivers 72 MW
at 0.9 power factor lagging.
(a)
Determine the excitation voltage and the power angle. Draw the phasor diagram with Vt as
reference.
(b)
Determine the maximum power the synchronous generator can supply if the field current is
made zero. Determine the machine current and power factor for this condition. Draw the
phasor diagram.
A salient pole synchronous machine delivers rated power to a load of unity power factor. The daxis and q-axis reactances are
Xd = 0:95 pu,
Xq = 0:45 pu
The power angle is δ = 25 .
The stator winding resistance is negligible.
6.31
(a)
Determine the excitation voltage ðEf Þ and the terminal voltage ðVt Þ.
(b)
Determine Ia , Id , and Iq , and draw the phasor diagram.
A 3φ, 40 MVA, 11 kV, 60 Hz, salient pole, synchronous machine has Xd = 1:5 pu; Xq = 1:0 pu,
and negligible stator resistance. The machine is connected to an infinite bus, and the field current
is adjusted to make the excitation voltage equal to the bus voltage. Determine the maximum value
of the steady-state power that the machine can supply. Find the stator current ðIa Þ and the power
factor at this maximum power condition. Draw the phasor diagram corresponding to this case.
360
chapter 6 Synchronous Machines
6.32
A 3φ, 200 MVA, 11 kV, 60 Hz, 200 rpm hydro generator has Xd = 1:45 pu, Xq = 0:85 pu, and
negligible stator resistance.
6.33
(a)
The generator is connected to a 3φ, 11 kV, 60 Hz infinite bus and delivers 100 MVA at 0.8
lagging power factor. Determine the power angle δ and the excitation voltage Ef . Draw the
phasor diagram.
(b)
The field excitation of the generator is now slowly reduced until the generator reaches its
static stability limit. For this condition, determine the power angle δ, excitation voltage Ef ,
machine current, and power factor. Draw the phasor diagram.
A 3φ salient pole synchronous machine has reactances Xd = 1:2 pu and Xq = 0:6 pu. Neglect
armature resistance losses.
(a) The machine operates as a synchronous motor and draws 0:8 pu of power at a power factor
of 0.8 leading.
(i) Determine the power angle δ and the excitation voltage Ef in pu and draw the phasor
diagram.
(ii) Determine the power due to excitation and that due to saliency of the machine.
(b)
The machine operates as a synchronous generator and delivers 0:8 pu of power at the PF of
0.8 leading. Determine the excitation voltage Ef in pu and the power angle δ.
6.34
Repeat Problem 6.29 if the synchronous machine operates as a motor.
6.35
A 3φ synchronous machine has the following parameters:
Xd = 0:9 pu,
Xq = 0:65 pu
The field current of the synchronous machine is adjusted to produce an open-circuit voltage
of 1 pu, and the machine is synchronized to an infinite bus. Determine the maximum perunit torque that can be applied slowly without losing synchronism. Find the stator current ðIa Þ
and the power factor at this maximum torque condition. Draw the phasor diagram corresponding
to this case.
6.36
6.37
(a)
A 3φ, cylindrical-rotor synchronous machine has Xs = Xd = 0:9 pu and negligible stator
resistance. The machine delivers rated power to an infinite bus. Determine the minimum
value of the excitation voltage in pu that will keep the machine in synchronism.
(b)
Now consider a 3φ, salient pole synchronous machine that has Xd = 0:9 pu, Xq = 0:6 pu, and
negligible stator resistance. Determine the minimum value of the excitation voltage in pu
that will keep the machine in synchronism while delivering rated power to the infinite bus.
A 3φ, 480 V, 125 hp, 0:85 PF leading, 60 Hz, four-pole, star-connected, synchronous motor has
Ls = 3:85 mH and Ra ! 0. The speed of this motor is controlled over the range 300 to 1800 rpm
using a cycloconverter, as shown in Fig. 6.31b.
(a) Determine the range of supply frequency variation.
(b)
Determine Ef at the rated condition.
(c)
Determine the maximum power the motor can deliver at
(i) rated speed.
(ii) lowest speed.
Assume Vt =f to remain constant over the speed range.
Problems
6.38
361
A 3φ, 5 hp, 208 V, four-pole, 60 Hz, star-connected synchronous motor has negligible stator
resistance, and Ls = 20 mH and Ef = 1:9 f , where f is the frequency. The base speed is the
synchronous speed corresponding to f = 60 Hz, and the terminal voltage at base speed ðnb Þ is 208
(L–L).
(a) Determine the base speed in rpm, and the ratio Vt =f at the base speed.
6.39
6.40
(b)
If for n ≤ nb , Vt =f is kept constant at the value obtained in part (a), determine the maximum
torque the motor will develop at different speeds.
(c)
At the base speed, determine Ia , power factor, and power for the maximum-torque condition.
Draw the phasor diagram for this condition.
An LSM-propelled magnetically levitated train takes 5 MW power at a cruising speed
of 500 km=hr. There are ten (10) superconducting magnets (SCM) for propulsion, and the pole
pitch is 50 cm. Determine
(a)
The frequency of the 3φ supply for the cruising speed.
(b)
The thrust produced in kN at the cruising speed.
In a three-phase switched reluctance motor having six stator poles and four rotor poles, the pole
widths for the stator and rotor poles are the same, with θs = θr = 40 . The currents in the phase
windings are applied when the inductances are increasing. Draw qualitatively the current and
torque waveforms of each phase and the total torque developed by the motor for a square wave
current having
(a) 40 width.
6.41
(b)
30 width.
(c)
20 width.
A 3φ, 10 MVA, 14 kV, 60 Hz synchronous machine has negligible stator winding resistance and a
synchronous reactance of 16:5 ohms per phase. The machine is connected to an infinite bus
ð3φ, 14 kV, 60 HzÞ and delivers power to a mechanical load. The rotational losses can be
neglected. The magnetization characteristics of the machine are shown in Fig. E6.2.
Write a computer program to study the variation of motor terminal current ðIa Þ and power
factor (PF) with field current ðIf Þ. The program should yield
(a) A computer printout in tabular form showing the variation of Ia , PF with If for input power
of 5 MW, 10 MW, and 15 MW.
(b)
A plot of Ia , PF versus If .