1
Electric Motors and Drives - Third Edition
Solutions to Review Questions - Chapter 10
1)
The synchronous speed of all a.c. motors (induction, reluctance, synchronous) is governed by
120 f
the equation N S 
. Hence for synchronous speed of 300 rev/min when the supply
p
frequency is 60 Hz, a winding with 24 poles will be required.
2)
In all mains-fed a.c. machines the voltage should be adjusted in proportion to the frequency in
order to keep the magnitude of the air-gap flux density wave constant. Hence the voltage at 50
V
V
50
 350 Volts.
Hz is given by 50  60 , i.e. V50  420 
50
60
60
3)
With no shaft projecting at either end the set-up is clearly not intended to provide any
mechanical output power, so we can deduce that one of the machines is intended to run as a
motor and drive the other as a generator. The arrangement is in fact a bi-directional frequency
changer, for allowing 50 Hz apparatus to be supplied from a 60 Hz system, or vice-versa.
The 10-pole machine has a synchronous speed of 600 rev/min at 50 Hz, while the 12-pole
machine also has synchronous speed of 600 rev/min when it is supplied at 60 Hz. Both
machines can work as motor of generator, so power can be transferred in either direction
between the 50 Hz and 60 Hz systems.
The machines could be separated and used as synchronous motors. With a 50 Hz supply, the
speeds available are 500 and 600 rev/min, while these increase to 600 and 720 rev/min with a
60 Hz supply.
4)
The steady-state speed of the rotor is constant because it is locked to the synchronously-rotating
field. But when the load torque is increased there is an immediate deceleration of the rotor that
causes it to drop back in relation to the rotating field. In so doing the torque produced on the
rotor by the field increases, so the deceleration reduces until a new equilibrium position is
reached where the speed of the rotor is again exactly synchronous but the torque has increased.
A stroboscope illuminating the shaft allows an observer to see the increase in load-angle, which
normally occurs rapidly so that the slight and transient departure from synchronous speed is
barely perceptible.
5)
In a synchronous machine operating in the steady state the magnetic field produced by the stator
windings rotates at the same speed as the rotor, so that from the point of view of an observer on
the rotor there is no ‘flux cutting’ and therefore no motional e.m.f. The only time that there will
be motional e.m.f. in the rotor is during run-up, or when there is a transient load change. This is
important in only a limited number of cases, for example under fault conditions in very large
synchronous machines connected to the power system.
6)
The first thing that we should note is that because there is no mechanical load on the shaft of the
motor, the only mechanical power being produced is that associated with any frictional or
windage losses, which must be very small. We can deduce that the electrical input power must
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be small, so the fact that at one value of rotor d.c. excitation the a.c. current in the stator
becomes very small should not come as a surprise.
To go further we must do as suggested and make use of the equivalent circuit (Figure 10.3) and
phasor diagram. We are told that the machine is a large one, so we can make the assumption
that the stator resistance is negligible without having much effect on the result.
The phasor diagram is shown in Figure 10A. This is based on Figure 10.4, but with the load
torque reduced to zero and the losses neglected, in which case the input power becomes zero.
The locus of the stator current as the rotor excitation is varied is shown by the dotted line.
Fig 10A here
Because the power is zero (or at least very small), the real or in-phase component of the stator
current is always almost zero: it has been taken as zero in Figure 10A for the sake of simplicity.
However, the stator current (I) must be such as to satisfy Kirchoff’s law (that the applied
voltage V must equal the sum of the induced e.m.f. E and the volt-drop across the synchronous
reactance. Hence we see that when E is small (Fig 10A(a)), the stator current is large and
lagging the voltage by 90°, so that the machine looks like an inductance when viewed from the
supply.
In the extreme case when there is no rotor excitation (E = 0), the stator current has its maximum
V
value of
, and in this condition we see that all of the MMF required to produce the air-gap
XS
flux is provided by the stator. In this condition the machine has its minimum apparent
inductance. (If we were talking about an induction motor with an identical stator to the one
here, we would call the current the magnetising current, and we would refer to the stator
inductive reactance as the magnetising reactance, rather than the synchronous reactance.)
When the rotor excitation (E) is increased the stator current reduces (i.e. the apparent
inductance increases) until, when E = V, the stator current is zero. Further increase in E causes
the current to increase again, but his time it leads the voltage by 90°, and the machine therefore
looks like a capacitor when viewed from the system.
Synchronous machines operating in this mode (i.e. without any mechanical power output) were
widely used at one time as ‘synchronous compensators’ in power systems. They allow the
power-factor of a region to be optimised by being set to have the desired value of inductive or
capacitive reactance.
7)
During run-up the acceleration is directly proportional to the torque and inversely proportional
to the inertia. Hence if the inertia is doubled, the acceleration will be halved and the time taken
to reach steady-state speed will increase by a factor of two.
In the steady-state there is no acceleration and therefore no torque associated with inertia.
Provided that we increase the load torque gradually, so that the accelerating component of
torque is negligible, the inertia will have no effect on the load angle at which the maximum
torque is developed (i.e. the pull-out condition).
3
EV
sin  , and so for a given
XS
supply voltage V and excitation e.m.f. E, the pull-out torque occurs when the load-angle
( ) reaches 90°. As expected, the torque expression does not contain inertia.)
(The expression for torque in asynchronous machine is T 
8)
We can answer this question most directly by making use of the torque expression quoted at
EV
sin  . This expression assumes that the stator resistance has
the end of question 7, i.e. T 
XS
negligible influence on the torque, which will be true because we are told that the motor is
large.
We are asked about what happens when we change the rotor excitation (i.e. E), but,
paradoxically we will be better off if we begin by noting what does not change. The main point
to note is that the mechanical load torque is unaltered, the steady-state speed remains equal to
the synchronous speed, and therefore the torque produced by the motor must remain unchanged.
The supply voltage V is naturally unchanged.
Hence of we denote the original condition by suffix 1 and the new condition by suffix 2 we can
EV
E V
E
1
sin 40 ,   2  25.4.
write T  1 sin 1  2 sin  2 ,  sin  2  1 sin 1 
XS
XS
E
1.5
9)
The answer to question 5 makes clear that in the steady-state there are no motional e.m.f.’ s
induced in the rotor body or windings, and therefore it is possible for the rotor body to be made
of solid steel, rather than laminations. Most large synchronous machines (such as those used as
alternators in power plants) must have solid forged rotors in order to achieve the required
mechanical strength, but many small ones are made of laminations because it is easier to punch
the slots for the winding than to machine them in a solid rotor.
10)
Increasing the number of phases makes it easier to achieve the desired sinusoidal spatial
distribution of the rotating field, and makes for better utilisation of the conductors, but the down
side is that there are more leads and terminals. There is however no fundamental reason, and
the current state-of-the-art seems to favour three.
11)
There is very little difference form the user’s perspective. The principal internal difference is
that the term brushless d.c. tends to be used to describe a motor in which the windings produce
a rectangular m.m.f. wave, and the windings are fed with rectangular current waves, whereas a
self-synchronous motor will have sinusoidally-distributed windings fed with sinusoidal
currents.
12)
A unipolar drive circuit is one in which the current can only flow in one direction, for example
that shown in chapter 2, Figure 2.4, where the current can only flow downwards through the
load. In a reluctance motor, the torque does not depend on the direction of current in the
winding, so a simple unipolar supply is satisfactory.
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A bipolar drive circuit can supply either positive or negative current: for example in Figure
2.13, if transistors 1 and 4 are turned on, the current will flow one way through the load, while
if transistors 2 and 3 are turned on instead, the current will flow the opposite way through the
load. Clearly the bipolar supply is more complex than the unipolar supply: it requires more
switching devices and is therefore more costly.
A further advantage of the unipolar circuit is that there is no danger of an inadvertent shortcircuit across the supply, because the winding is always in series with the switching device. In
contrast, in a bipolar supply there is a danger that if, by mistake, two switching devices in the
same ‘leg’ are turned on simultaneously, a short-circuit will occur and the devices will be
destroyed by the excessive current that will flow.
13)
The ‘dump’ resistor is provided in drives which cannot continuously convert mechanical energy
into electrical energy. The majority of drives seldom require prolonged regeneration, and do not
justify the additional expense of a fully-regenerative capability. But most drives need to be able
to provide rapid braking (e.g. when the target speed is suddenly lowered), and therefore they are
designed so that they can produce negative (braking) torque, and absorb the excess kinetic
energy, rather than return it to the mains. The energy is dissipated in a resistor, which therefore
gets hot. The resistor will have been designed for operation on an intermittent basis, and
continuous operation is unlikely to be permissible, except perhaps at a very low torque.
14)
All electromechanical energy converters are inherently capable of operating in motor or
generator mode. In practice, the nature of the supply system determines whether bi-directional
energy flow is possible. All mains-fed a.c. motors can generate; the cheapest power-electronic
converters have no capacity for regeneration; more expensive ones can absorb short-term
reverse energy flows, but not return the energy to the supply; and the top of the range models
will allow continuous generation.
- End of Solutions Chapter 10 -