Mesin ac ch4

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MESIN ARUS SEARAH DAN MESIN
SINKRON (ECI 640)
PRIANDA
prianda@hotmail.com
Referensi :
Electric machinery fundamentals, Stephen J. Chapman, Mc Graw Hill, 4th edition.

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CHAPTER - 4
AC MACHINERY FUNDAMENTALS

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AC machines are generators that convert mechanical energy
to ac electrical energy and motors that convert ac electrical
energy to mechanical energy. The fundamental principles of
ac machines are very simple, but unfortunately, they are
somewhat obscured by the complicated construction of real
machines. This chapter will first explain the principles of ac
machine operation using simple examples, and then consider
some of the complications that occur in real ac machines.
There are two major classes of ac machines-synchronous
machines and induction machines. Synchronous machines
are motors and generators whose magnetic field current is
supplied by a separate dc power source, while induction
machines are motors and generators whose field current is
supplied by magnetic induction (transformer action) into their
field windings. The field circuits of most synchronous and
induction machines are located on their rotors.

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This chapter covers some of the fundamentals common to
both types of three-phase ac machines. Synchronous
machines will be covered in detail in Chapters 5 and 6, and
induction machines will be covered in Chapter 7.

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1. A SIMPLE LOOP IN A UNIFORM
MAGNETIC FIELD
We will start our study of ac machines with a simple loop of wire rotating
within a uniform magnetic field. A loop of wire in a uniform magnetic field is
the simplest possible machine that produces a sinusoidal ac voltage. This
case is not representative of real ac machines, since the flux in real ac
machines is not constant in either magnitude or direction. However, the
factors that control the voltage and torque on the loop will be the same as
the factors that control the voltage and torque in real ac machines.
Figure 4-1 shows a simple machine consisting of a large stationary
magnet producing an essentially constant and uniform magnetic field and
a rotating loop of wire within that field. The rotating part of the machine is
called the rotor, and the stationary part of the machine is called the stator.
We will now determine the voltages present in the rotor as it rotates within
the magnetic field.
The Voltage Induced in a Simple Rotating Loop
If the rotor of this machine is rotated, a voltage will be induced in the wire
loop. To determine the magnitude and shape of the voltage, examine
Figure 4-2. The loop of wire shown is rectangular, with sides ab and cd
perpendicular to the plane of the page and with sides bc and da parallel to
the plane of the page. The magnetic field is constant and uniform, pointing
from left to right across the page.

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A SIMPLE LOOP IN A UNIFORM MAGNETIC FIELD
FIGURE 4-1
A simple rotating loop in a uniform magnetic field. (a) Front view; (b) view of coil.
The magnetic field is constant and uniform, pointing from left to right
across the page.
To determine the total voltage etot on the loop, we will examine each
segment of the loop separately and sum all the resulting voltages. The
voltage on each segment is given by Equation (1-45):

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FIGURE 4-2
(a) Velocities and orientations of the sides of the loop with respect to the
magnetic field. (b) The direction of motion with respect to the magnetic field for
side ab. (c) The direction of motion with respect to the magnetic field for side cd.
1. Segment ab. In this segment, the velocity of the wire is tangential to
the path of rotation, while the magnetic field B points to the right, as
shown in Figure 4-2b. The quantity v x B points into the page, which is
the same direction as segment ab. Therefore, the induced voltage on
this segment of the wire is

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2. Segment bc. In the first half of this segment, the quantity v x B points
into the page, and in the second half of this segment , the quantity v x
B points out of the page. Since the length I is in the plane of the page,
v x B is perpendicular to I for both portions of the segment. Therefore
the voltage in segment be will be zero:
3. Segment cd. In this segment, the velocity of the wire is tangential to
the path of rotation, while the magnetic field B points to the right, as
shown in Figure 4-2c. The quantity v x B points into the page, which is
the same direction as segment cd. Therefore, the induced voltage on
this segment of the wire is
4. Segment da. Just as in segment be, v x B is perpendicular to I.
Therefore the voltage in this segment will be zero too:

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The total induced voltage on the loop eind is the sum of the voltages on
each of its sides:
Note that θab = 180o
- θcd, and recall the trigonometric identity sin θ = sin
(180o
- θ). Therefore, the induced voltage becomes
The resulting voltage eind is shown as a function of time in Figure 4-3.
There is an alternative way to express Equation (4-6), which clearly relates
the behavior of the single loop to the behavior of larger, real ac machines.
To derive this alternative expression, examine Figure 4-2 again. If the loop
is rotating at a constant angular velocity ω, then angle θ of the loop will
increase linearly with time. In other words,
Also, the tangential velocity v of the edges of the loop can be expressed
as
where r is the radius from axis of rotation out to the edge of the loop and ω
is the angular velocity of the loop. Substituting these expressions into
Equation (4-6) gives

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Finally, note that the maximum flux through the loop occurs when the loop
is perpendicular to the magnetic flux density lines. This flux is just the
product of the loop's surface area and the flux density through the loop.
Therefore, the final form of the voltage equation is
Thus, the voltage generated in the loop is a sinusoid whose magnitude is
equal to the product of the flux inside the machine and the speed of
rotation of the machine. This is also true of real ac machines. In general,
the voltage in any real machine will depend on three factors:
1.The flux in the machine
2.The speed of rotation
3.A constant representing the construction of the machine (the number of
loops, etc.)

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FIGURE 4-3
Plot of eind versus θ.

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The Torque Induced in a Current-Carrying Loop
Now assume that the rotor loop is at some arbitrary angle θ with respect to
the magnetic field, and that a current i is flowing in the loop, as shown in
Figure 4-4. If a current flows in the loop, then a torque will be induced on
the wire loop. To determine the magnitude and direction of the torque,
examine Figure 4-5. The force on each segment of the loop will be given
by Equation (1-43),
The torque on that segment will then be given by

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FIGURE 4-4
A current-carrying loop in a uniform magnetic field. (a) Front view; (b)
view of coil.

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FIGURE 4-5
(a) Derivation of force and torque on segment ab. (b) Derivation of force and
torque on segment bc. (c) Derivation of force and torque on segment cd. (d)
Derivation of force and torque on segment da.

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where θ is the angle between the vector r and the vector F. The direction
of the torque is clockwise if it would tend to cause a clockwise rotation and
counterclockwise if it would tend to cause a counterclockwise rotation.
1.Segment ab. In this segment, the direction of the current is into the page,
while the magnetic field B points to the right, as shown in Figure 4-5a. The
quantity I x B points down. Therefore, the induced force on this segment of
the wire is
The resulting torque is
2. Segment bc. In this segment, the direction of the current is in the
plane of the page, while the magnetic field B points to the right, as
shown in Figure 4-5b. The quantity I x B points into the page.
Therefore, the induced force on this segment of the wire is
For this segment, the resulting torque is 0, since vectors r and I are
parallel (both point into the page), and the angle θbc is 0.

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3. Segment cd. In this segment, the direction of the current is out of the
page, while the magnetic field B points to the right, as shown in Figure
4-5c. The quantity I x B points up. Therefore, the induced force on this
segment of the wire is
The resulting torque is
4. Segment da. In this segment, the direction of the current is in the
plane of the page, while the magnetic field B points to the right, as
shown in Figure 4-5d. The quantity I x B points out of the page.
Therefore, the induced force on this segment of the wire is
For this segment, the resulting torque is 0, since vectors r and I are
parallel (both point out of the page), and the angle θda is 0.

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The total induced torque on the loop τind is the sum of the torques on each
of its sides:
The resulting torque τind is shown as a function of angle in Figure 4-6. Note
that the torque is maximum when the plane of the loop is parallel to the
magnetic field, and the torque is zero when the plane of the loop is
perpendicular to the magnetic field.
There is an alternative way to express Equation (4-17), which clearly
relates the behavior of the single loop to the behavior of larger, real ac
machines. To derive this alternative expression, examine Figure 4-7. If the
current in the loop is as shown in the figure, that current will generate a
magnetic flux density Bloop with the direction shown. The magnitude of Bloop
will be
Note that θab = θcd, so the induced torque becomes

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where G is a factor that depends on the geometry of the loop.* Also, note
that the area of the loop A is just equal to 2rl. Substituting these two
equations into Equation (4-17) yields the result
FIGURE 4-6
Plot of τind versus θ.
FIGURE 4-7
Derivation of the induced torque
equation. (a) The current in the
loop produces a magnetic flux
density Bloop perpendicular to the
plane of the loop; (b) geometric
relationship between Bloop and Bs.

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where k = AG/µ is a factor depending on the construction of the machine,
Bs is used for the stator magnetic field to distinguish it from the magnetic
field generated by the rotor, and θ is the angle between Bloop and Bs. The
angle between Bloop and Bs can be seen by trigonometric identities to be
the same as the angle θ in Equation (4-1 7).
Both the magnitude and the direction of the induced torque can be
determined by expressing Equation (4-19) as a cross product:
Applying this equation to the loop in Figure 4-7 produces a torque vector
into the page, indicating that the torque is clockwise, with the magnitude
given by Equation (4-19).
Thus, the torque induced in the loop is proportional to the strength of the
loop's magnetic field, the strength of the external magnetic field, and the
sine of the angle between them. This is also true of real ac machines. In
general, the torque in any real machine will depend on four factors:
1.The strength of the rotor magnetic field
2.The strength of the external magnetic field
3.The sine of the angle between them
4.A constant representing the construction of the machine (geometry. etc.)

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2. THE ROTATING MAGNETIC FIELD
In Section 4.1, we showed that if two magnetic fields are present in a
machine, then a torque will be created which will tend to line up the two
magnetic fields. If one magnetic field is produced by the stator of an ac
machine and the other one is produced by the rotor of the machine, then a
torque will be induced in the rotor which will cause the rotor to turn and
align itself with the stator magnetic field.
If there were some way to make the stator magnetic field rotate, then the
induced torque in the rotor would cause it to constantly "chase" the stator
magnetic field around in a circle. This, in a nutshell, is the basic principle of
all ac motor operation.
How can the stator magnetic field be made to rotate? The fundamental
principle of ac machine operation is that if a three-phase set of currents,
each of equal magnitude and differing in phase by 120°,flows in a three-
phase winding, then it will produce a rotating magnetic field of constant
magnitude. The three-phase winding consists of three separate windings
spaced 120 electrical degrees apart around the surface of the machine.
The rotating magnetic field concept is illustrated in the simplest case by an
empty stator containing just three coils, each 120° apart (see Figure 4-8a).
Since such a winding produces only one north and one south magnetic
pole, it is a two pole winding.
To understand the concept of the rotating magnetic field, we will apply a
set of currents to the stator of Figure 4-8 and see what happens at specific
instants of time. Assume that the currents in the three coils are given by
the equations

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THE ROTATING MAGNETIC FIELD
The current in coil aa' flows into the a end of the coil and out the a' end of
the coil. It produces the magnetic field intensity
where 0° is the spatial angle of the magnetic field intensity vector, as
shown in Figure 4-8b. The direction of the magnetic field intensity vector
Haa’(t) is given by the right-hand rule: If the fingers of the right hand curl in
the direction of the current flow in the coil, then the resulting magnetic field
is in the direction that the thumb points. Notice that the magnitude of the
magnetic field intensity vector Haa’(t) varies sinusoidally in time, but the
direction of Haa’(t) is always constant. Similarly, the magnetic field intensity
vectors Hbb’(t) and Hcc’(t) are
The flux densities resulting from these magnetic field intensities are given
by Equation (1-21):
They are

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FIGURE 4-8
(a) A simple three-phase stator. Currents in this stator are assumed positive if
they flow into the unprimed end and out the primed end of the coils. The
magnetizing intensities produced by each coil are also shown. (b) The
magnetizing intensity vector Haa’(t) produced by a current flowing in coil aa'.

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where BM = µHM. The currents and their corresponding flux densities can
be examined at specific times to determine the resulting net magnetic field
in the stator.
For example, at time ωt = 0°, the magnetic field from coil aa’ will be
The magnetic field from coil bb' will be
and the magnetic field from coil cc' will be
The total magnetic field from all three coils added together will be
The resulting net magnetic field is shown in Figure 4-9a.
As another example, look at the magnetic field at time ωt = 90°. At that
time, the currents are

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and the magnetic fields are
The resulting net magnetic field is
The resulting magnetic field is shown in Figure 4-9b. Notice that although
the direction of the magnetic field has changed, the magnitude is constant.
The magnetic field is maintaining a constant magnitude while rotating in a
counterclockwise direction.

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FIGURE 4-9
(a) The vector magnetic field in a stator at time ωt = 0°. (b) The vector
magnetic field in a stator at time ωt = 90°.

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Proof of the Rotating Magnetic Field Concept
At any time t, the magnetic field will have the same magnitude 1.5BM, and
it will continue to rotate at angular velocity ω. A proof of this statement for
all time t is now given.
Refer again to the stator shown in Figure 4-8. In the coordinate system
shown in the figure, the x direction is to the right and the y direction is
upward. The vector x is the unit vector in the horizontal direction, and the
vector ŷ is the unit vector in the vertical direction. To find the total magnetic
flux density in the stator, simply add vectorially the three component
magnetic fields and determine their sum.
The net magnetic flux density in the stator is given by
Each of the three component magnetic fields can now be broken down into
its x and y components.

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Proof of the Rotating Magnetic Field Concept
Combining x and y components yields
By the angle-addition trigonometric identities.
Equation (4-25) is the final expression for the net magnetic flux density.
Notice that the magnitude of the field is a constant 1.5BM and that the
angle changes continually in a counterclockwise direction at angular
velocity ω. Notice also that at ωt = 0°, Bnet = 1.5BM ∠ -90° and that at ωt =
90°, Bnet = 1.5BM ∠ 0°. These results agree with the specific examples
examined previously.

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The Relationship between Electrical Frequency and the Speed of Magnetic Field Rotation
Figure 4-10 shows that the rotating magnetic field in this stator can be
represented as a north pole (where the flux leaves the stator) and a south
pole (where the flux enters the stator). These magnetic poles complete
one mechanical rotation around the stator surface for each electrical cycle
of the applied current. Therefore, the mechanical speed of rotation of the
magnetic field in revolutions per second is equal to the electric frequency
in hertz:
Here fm and ωm are the mechanical speed in revolutions per second and
radians per second, while fe and ωe are the electrical speed in hertz and
radians per second.
Notice that the windings on the two-pole stator in Figure 4-10 occur in the
order (taken counterclockwise)
What would happen in a stator if this pattern were repeated twice within it?
Figure 4-11a shows such a stator. There, the pattern of windings (taken
counterclockwise) is

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FIGURE 4-10
The rotating magnetic field
in a stator represented as
moving north and south
stator poles.

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FIGURE 4-11
(a) A simple four-pole stator
winding. (b) The resulting
stator magnetic poles.
Notice that there are moving
poles of alternating polarity
every 90° around the stator
surface. (c) A winding
diagram of the stator as
seen from its inner surface,
showing how the stator
currents produce north and
south magnetic poles.

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which is just the pattern of the previous stator repeated twice. When a
three-phase set of currents is applied to this stator, two north poles and
two south poles are produced in the stator winding, as shown in Figure 4-
11b. In this winding, a pole moves only halfway around the stator surface
in one electrical cycle. Since one electrical cycle is 360 electrical degrees,
and since the mechanical motion is 180 mechanical degrees, the
relationship between the electrical angle θe and the mechanical angle θm,
in this stator is
Thus for the four-pole winding, the electrical frequency of the current is
twice the mechanical frequency of rotation:
In general, if the number of magnetic poles on an ac machine stator is P,
then there are P/2 repetitions of the winding sequence a-c'-b-a'-e-b'
around its inner surface, and the electrical and mechanical quantities on
the stator are related by

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Also, noting that fm = nm/60, it is possible to relate the electrical frequency
in hertz to the resulting mechanical speed of the magnetic fields in
revolutions per minute. This relationship is

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Reversing the Direction of Magnetic Field Rotation
Another interesting fact can be observed about the resulting magnetic
field. If the current in any two of the three coils is swapped, the direction of
the magnetic field's rotation will be reversed. This means that it is possible
to reverse the direction of rotation of an ac motor just by switching the
connections on any two of the three coils. This result is verified below.
To prove that the direction of rotation is reversed, phases bb' and cc' in
Figure 4-8 are switched and the resulting flux density Bnet is calculated.
The net magnetic flux density in the stator is given by
Each of the three component magnetic fields can now be broken down into
its x and y components:

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Combining x and y components yields
By the angle-addition trigonometric identities,
This time the magnetic field has the same magnitude but rotates in a
clockwise direction. Therefore, switching the currents in two stator phases
reverses the direction of magnetic field rotation in an ac machine.

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Example 4-1. Create a MATLAB program that models the behavior of a
rotating magnetic field in the three-phase stator shown in Figure 4-9.

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When this program is executed, it draws lines corresponding to the three
component magnetic fields as well as a line corresponding to the net
magnetic field. Execute this program and observe the behavior of Bnet.

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3. MAGNETOMOTIVE FORCE AND FLUX
DISTRIBUTION ON AC MACHINES
In Section 4.2, the flux produced inside an ac machine was treated as if it
were in free space. The direction of the flux density produced by a coil of
wire was assumed to be perpendicular to the plane of the coil, with the
direction of the flux given by the right-hand rule.
The flux in a real machine does not behave in the simple manner assumed
above, since there is a ferromagnetic rotor in the center of the machine,
with a small air gap between the rotor and the stator. The rotor can be
cylindrical, like the one shown in Figure 4-12a, or it can have pole faces
projecting out from its surface, as shown in Figure 4-12b. If the rotor is
cylindrical, the machine is said to have nonsalient poles; if the rotor has
pole faces projecting out from it, the machine is said to have salient
poles. Cylindrical rotor or nonsalient-pole machines are easier to
understand and analyze than salient-pole machines, and this discussion
will be restricted to machines with cylindrical rotors. Machines with salient
poles are discussed briefly in Appendix C and more extensively in
References 1 and 2.
Refer to the cylindrical-rotor machine in Figure 4-12a. The reluctance of
the air gap in this machine is much higher than the reluctances of either
the rotor or the stator, so the flux density vector B takes the shortest
possible path across the air gap and jumps perpendicularly between the
rotor and the stator.

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MAGNETOMOTIVE FORCE AND FLUX DISTRIBUTION ON AC MACHINES
To produce a sinusoidal voltage in a machine like this, the magnitude of
the flux density vector B must vary in a sinusoidal manner along the
surface of the air gap. The flux density will vary sinusoidally only if the
magnetizing intensity H (and magnetomotive force ℱ) varies in a
sinusoidal manner along the surface of the air gap (see Figure 4-13).
The most straightforward way to achieve a sinusoidal variation of
magnetomotive force along the surface of the air gap is to distribute the
turns of the winding that produces the magnetomotive force in closely
spaced slots around the surface of the machine and to vary the number of
conductors in each slot in a sinusoidal manner. Figure 4-14a shows such a
winding, and Figure 4-14b shows the magnetomotive force resulting from
the winding. The number of conductors in each slot is given by the
equation
where Nc is the number of conductors at an angle of 0°. As Figure 4-14b
shows, this distribution of conductors produces a close approximation to a
sinusoidal distribution of magnetomotive force. Furthermore, the more
slots there are around the surface of the machine and the more closely
spaced the slots are, the better this approximation becomes.

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FIGURE 4-12
(a) An ac machine with a cylindrical or nonsalient-pole rotor. (b) An ac machine
with a salient-pole rotor.

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FIGURE 4-13
(a) A cylindrical rotor with sinusoidally varying air-gap flux density. (b) The
magnetomotive force or magnetizing intensity as a function of angle a in the air
gap. (c) The flux density as a function of angle α in the air gap.

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FIGURE 4-14
(a) An ac machine with a distributed stator winding designed to produce a
sinusoidally varying airgap flux density. The number of conductors in each slot
is indicated on the diagram. (b) The magnetomotive force distribution resulting
from the winding. compared to an ideal distribution.

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In practice, it is not possible to distribute windings exactly in accordance
with Equation (4-36), since there are only a finite number of slots in a real
machine and since only integral numbers of conductors can be included in
each slot. The resulting magnetomotive force distribution is only
approximately sinusoidal, and higher-order harmonic components will be
present. Fractional-pitch windings are used to suppress these unwanted
harmonic components, as explained in Appendix B.1.
Furthermore, it is often convenient for the machine designer to include
equal numbers of conductors in each slot instead of varying the number in
accordance with Equation (4-36). Windings of this type are described in
Appendix B.2; they have stronger high-order harmonic components than
windings designed in accordance with Equation (4-36). The harmonic-
suppression techniques of Appendix B.1 are especially important for such
windings.

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4. INDUCED VOLTAGE IN AC MACHINES
Just as a three-phase set of currents in a stator can produce a rotating
magnetic field, a rotating magnetic field can produce a three-phase set of
voltages in the coils of a stator. The equations governing the induced
voltage in a three-phase stator will be developed in this section. To make
the development easier, we will begin by looking at just one single-turn coil
and then expand the results to a more general three-phase stator.
The Induced Voltage in a Coil on a Two-Pole Stator
Figure 4-15 shows a rotating rotor with a sinusoidally distributed magnetic
field in the center of a stationary coil. Notice that this is the reverse of the
situation studied in Section 4.1, which involved a stationary magnetic field
and a rotating loop.
We will assume that the magnitude of the flux density vector B in the air
gap between the rotor and the stator varies sinusoidally with mechanical
angle, while the direction of B is always radially outward. This sort of flux
distribution is the ideal to which machine designers aspire. (What happens
when they don't achieve it is described in Appendix B.2.) If α is the angle
measured from the direction of the peak rotor flux density, then the
magnitude of the flux density vector B at a point around the rotor is given
by

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INDUCED VOLTAGE IN AC MACHINES

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FIGURE 4-15
(a) A rotating rotor magnetic field inside a
stationary stator coil. Detail of coil. (b)
The vector magnetic flux densities and
velocities on the sides of the coil. The
velocities shown are from a frame of
reference in which the magnetic field is
stationary. (c) The flux density distribution
in the air gap.

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Note that at some locations around the air gap, the flux density vector will
really point in toward the rotor; in those locations, the sign of Equation (4-
37a) is negative. Since the rotor is itself rotating within the stator at an
angular velocity ωm, the magnitude of the flux density vector B at any
angle α around the stator is given by
The equation for the induced voltage in a wire is
However, this equation was derived for the case of a moving wire in a
stationary magnetic field. In this case, the wire is stationary and the
magnetic field is moving, so the equation does not directly apply. To use it,
we must be in a frame of reference where the magnetic field appears to be
stationary. If we "sit on the magnetic field" so that the field appears to be
stationary, the sides of the coil will appear to go by at an apparent velocity
vrel, and the equation can be applied.

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Figure 4-15b shows the vector magnetic field and velocities from the point
of view of a stationary magnetic field and a moving wire.
The total voltage induced in the coil will be the sum of the voltages induced
in each of its four sides. These voltages are determined below:
1.Segment ab. For segment ab, α = 180°. Assuming that B is directed
radially outward from the rotor, the angle between v and B in segment ab
is 90°, while the quantity v x B is in the direction of I, so
where the minus sign comes from the fact that the voltage is built up
with a polarity opposite to the assumed polarity.
2. Segment bc. The voltage on segment bc is zero, since the vector
quantity v x B is perpendicular to I, so

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3. Segment cd. For segment cd, the angle α = 0°. Assuming that B is
directed radially outward from the rotor, the angle between v and B in
segment cd is 90°, while the quantity v x B is in the direction of l, so
4. Segment da. The voltage on segment da is zero, since the vector
quantity v x B is perpendicular to I, so
Therefore, the total voltage on the coil will be

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Since the velocity of the end conductors is given by v = rωm, Equation (4-
43) can be rewritten as
Finally, the flux passing through the coil can be expressed as φ = 2rlBm
(see Problem 4-7), while ωm = ωe = ω for a two-pole stator, so the induced
voltage can be expressed as
Equation (4-44) describes the voltage induced in a single-turn coil. If the
coil in the stator has Nc turns of wire, then the total induced voltage of the
coil will be
Notice that the voltage produced in stator of this simple ac machine
winding is sinusoidal with an amplitude which depends on the flux φ in the
machine, the angular velocity ω of the rotor, and a constant depending on
the construction of the machine (Nc in this simple case). This is the same
as the result that we obtained for the simple rotating loop in Section 4.1.

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Note that Equation (4-45) contains the term cos ωt instead of the sin ωt
found in some of the other equations in this chapter. The cosine term has
no special significance compared to the sine - it resulted from our choice
of reference direction for α in this derivation. If the reference direction for α
had been rotated by 90° we would have had a sin ωt term.
If three coils, each of Nc turns, are placed around the rotor magnetic field
as shown in Figure 4-16, then the voltages induced in each of them will be
the same in magnitude but wiII differ in phase by 120°. The resulting
voltages in each of the three coils are
Therefore, a three-phase set of currents can generate a uniform rotating
magnetic field in a machine stator, and a uniform rotating magnetic field
can generate a three-phase set of voltages in such a stator.
The Induced Voltage in a Three-Phase Set of Coils

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The Induced Voltage in a Three-Phase Set of Coils
FIGURE 4-16
The production of three-phase voltages
from three coils spaced 120° apart.

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The RMS Voltage in a Three-Phase Stator
The peak voltage in any phase of a three-phase stator of this sort is
Since ω = 2πf, this equation can also be written as
Therefore, the rms voltage of any phase of this three-phase stator is
The rms voltage at the terminals of the machine will depend on whether
the stator is Y- or ∆-connected. If the machine is Y-connected, then the
terminal voltage will be √3 times EA; if the machine is ∆-connected, then
the terminal voltage will just be equal to EA.
Example 4-2. The following information is known about the simple two-
pole generator in Figure 4-16. The peak flux density of the rotor magnetic
field is 0.2 T, and the mechanical rate of rotation of the shaft is 3600 r/min.
The stator diameter of the machine is 0.5 m, its coil length is 0.3 m, and
there are 15 turns per coil. The machine is Y-connected.

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a) What are the three phase voltages of the generator as a function of
time?
b) What is the rms phase voltage of this generator?
c) What is the rms terminal voltage of this generator?

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5. INDUCED TORQUE IN AN AC
MACHINE
In ac machines under normaI operating conditions, there are two magnetic
fields present - a magnetic field from the rotor circuit and another magnetic
field from the stator circuit. The interaction of these two magnetic fields
produces the torque in the machine, just as two permanent magnets near
each other will experience a torque which causes them to line up.
Figure 4-17 shows a simplified ac machine with a sinusoidal stator flux
distribution that peaks in the upward direction and a single coil of wire
mounted on the rotor. The stator flux distribution in this machine is
where Bs is the magnitude of the peak flux density; Bs(α) is positive when
the flux density vector points radially outward from the rotor surface to the
stator surface. How much torque is produced in the rotor of this simplified
ac machine? To find out, we will analyze the force and torque on each of
the two conductors separately.
The induced force on conductor I is
The torque on the conductor is

59
INDUCED TORQUE IN AN AC MACHINE
The induced force on conductor 2 is
The torque on the conductor is
Therefore, the torque on the rotor loop is
Equation (4-52) can be expressed in a more convenient form by examining
Figure 4-18 and noting two facts:
1.The current i flowing in the rotor coil produces a magnetic field of its
own. The direction of the peak of this magnetic field is given by the right-
hand rule, and the magnitude of its magnetizing intensity HR is directly
proportional to the current flowing in the rotor:

60
FIGURE 4-17
A simplified ac machine with
a sinusoidal stator flux
distribution and a single coil
of wire mounted in the rotor.

61
FIGURE 4-18
The components
magnetic flux density
inside the machine of
Figure 4-17.

62
2. The angle between the peak of the stator flux density Bs and the peak
of the rotor magnetizing intensity HR is γ. Furthermore,
By combining these two observations, the torque on the loop can be
expressed as
where K is a constant dependent on the construction of the machine. Note
that both the magnitude and the direction of the torque can be expressed
by the equation
Finally, since BR = µHR, this equation can be reexpressed as
where k = K/µ. Note that in general k will not be constant, since the
magnetic permeability µ varies with the amount of magnetic saturation in
the machine.
Equation (4-58) is just the same as Equation (4-20), which we derived for
the case of a single loop in a uniform magnetic field. It can apply to any ac
machine, not just to the simple one-loop rotor just described.

63
Only the constant k will differ from machine to machine. This equation will
be used only for a qualitative study of torque in ac machines, so the actual
value of k is unimportant for our purposes.
The net magnetic field in this machine is the vector sum of the rotor and
stator fields (assuming no saturation):
This fact can be used to produce an equivalent (and sometimes more
useful) expression for the induced torque in the machine. From Equation
(4-58)
Since the cross product of any vector with itself is zero, this reduces to
so the induced torque can also be expressed as a cross product of BR and
Bnet, with the same constant k as before. The magnitude of this expression
is

64
where δ is the angle between BR and Bnet.
Equations (4-58) to (4-61) will be used to help develop a qualitative
understanding of the torque in ac machines. For example, look at the
simple synchronous machine in Figure 4-19. Its magnetic fields are
rotating in a counterclockwise direction. What is the direction of the torque
on the shaft of the machine's rotor? By applying the right-hand rule to
Equation (4-58) or (4-60), the induced torque is found to be clockwise, or
opposite the direction of rotation of the rotor. Therefore, this machine must
be acting as a generator.

65
FIGURE 4-19
A simplified synchronous
machine showing its rotor
and stator magnetic fields.

66
6. WINDING INSULATION IN AN
AC MACHINE
One of the most critical parts of an ac machine design is the insulation of
its windings. If the insulation of a motor or generator breaks down, the
machine shorts out. The repair of a machine with shorted insulation is
quite expensive, if it is even possible. To prevent the winding insulation
from breaking down as a result of overheating, it is necessary to limit the
temperature of the windings. This can be partially done by providing a
cooling air circulation over them, but ultimately the maximum winding
temperature limits the maximum power that can be supplied continuously
by the machine.
Insulation rarely fails from immediate breakdown at some critical
temperature. Instead, the increase in temperature produces a gradual
degradation of the insulation, making it subject to failure from another
cause such as shock, vibration, or electrical stress. There was an old rule
of thumb that said that the life expectancy of a motor with a given type of
insulation is halved for each 10 percent rise in temperature above the
rated temperature of the winding. This rule still applies to some extent
today.
To standardize the temperature limits of machine insulation, the National
Electrical Manufacturers Association (NEMA) in the United States has
defined a series of insulation system classes. Each insulation system class
specifies the maximum temperature rise permissible for that class of
insulation.

67
WINDING INSULATION IN AN AC MACHINE
There are three common NEMA insulation classes for integral-horsepower
ac motors: B, F, and H. Each class represents a higher permissible
winding temperature than the one before it. For example, the armature
winding temperature rise above ambient temperature in one type of
continuously operating ac induction motor must be limited to 80°C for class
B, 105°C for class F, and 125°C for class H insulation.
The effect of operating temperature on insulation life for a typical machine
can be quite dramatic. A typical curve is shown in Figure 4-20. This curve
shows the mean life of a machine in thousands of hours versus the
temperature of the windings, for several different insulation classes.
The specific temperature specifications for each type of ac motor and
generator are set out in great detail in NEMA Standard MG1-1993, Motors
and Generators. Similar standards have been defined by the International
Electrotechnical Commission (IEC) and by various national standards
organizations in other countries.

68
WINDING INSULATION IN AN AC MACHINE
FIGURE 4-20
Plot of mean insulation life versus winding temperature for various
insulation classes. (Courtesy of Marathon Electric Company.)

69
7. AC MACHINE POWER FLOWS
AND LOSSES
AC generators take in mechanical power and produce electric power, while
ac motors take in electric power and produce mechanical power. In either
case, not all the power input to the machine appears in useful form at the
other end – there is always some loss associated with the process.
The efficiency of an ac machine is defined by the equation
The difference between the input power and the output power of a
machine is the losses that occur inside it. Therefore,
The Losses in AC Machines
The losses that occur in ac machines can be divided into four basic
categories:
1.Electrical or copper losses (I2
R losses)
2.Core losses
3.Mechanical losses
4.Stray load losses

70
AC MACHINE POWER FLOWS AND LOSSES
ELECTRICAL OR COPPER LOSSES. Copper losses are the resistive
heating losses that occur in the stator (armature) and rotor (field) windings
of the machine. The stator copper losses (SCL) in a three-phase ac
machine are given by the equation
where IA is the current flowing in each armature phase and RA is the
resistance of each armature phase.
The rotor copper losses (RCL) of a synchronous ac machine (induction
machines will be considered separately in Chapter 7) are given by
where IF is the current flowing in the field winding on the rotor and RF is the
resistance of the field winding. The resistance used in these calculations is
usually the winding resistance at normal operating temperature.
CORE LOSSES. The core losses are the hysteresis losses and eddy
current losses occurring in the metal of the motor. These losses were
described in Chapter 1. These losses vary as the square of the flux density
(B2
) and, for the stator, as the 1.5th power of the speed of rotation of the
magnetic fields (n1.5
).

71
MECHANICAL LOSSES. The mechanical losses in an ac machine are the
losses associated with mechanical effects. There are two basic types of
mechanical losses: friction and windage. Friction losses are losses caused
by the friction of the bearings in the machine, while windage losses are
caused by the friction between the moving parts of the machine and the air
inside the motor's casing. These losses vary as the cube of the speed of
rotation of the machine.
The mechanical and core losses of a machine are often lumped together
and called the no-load rotational loss of the machine. At no load, all the
input power must be used to overcome these losses. Therefore,
measuring the input power to the stat or of an ac machine acting as a
motor at no load will give an approximate value for these losses.
STRAY LOSSES (OR MISCELLANEOUS LOSSES). Stray losses are
losses that cannot be placed in one of the previous categories. No matter
how carefully losses are accounted for, some always escape inclusion in
one of the above categories. All such losses are lumped into stray losses.
For most machines, stray losses are taken by convention to be 1 percent
of full load.

72
One of the most convenient techniques for accounting for power losses in
a machine is the power-flow diagram. A power-flow diagram for an ac
generator is shown in Figure 4-21a. In this figure, mechanical power is
input into the machine, and then the stray losses, mechanical losses, and
core loses are subtracted. After they have been subtracted, the remaining
power is ideally converted from mechanical to electrical form at the point
labeled Pconv. The mechanical power that is converted is given by
The Power-Flow Diagram
and the same amount of electrical power is produced. However, this is not
the power that appears at the machine's terminals. Before the terminals
are reached, the electrical I2
R losses must be subtracted.
In the case of ac motors, this power-flow diagram is simply reversed. The
power-flow diagram for a motor is shown in Figure 4-21b.
Example problems involving the calculation of ac motor and generator
efficiencies will be given in the next three chapters.

73
The Power-Flow Diagram
FIGURE 4-21
(a) The power-flow diagram
of a three-phase ac
generator. (b) The power-flow
diagram of a three phase ac
motor.

74
8. VOLTAGE REGULATION AND SPEED
REGULATION
Generators are often compared to each other using a figure of merit called
voltage regulation. Voltage regulation (VR) is a measure of the ability of a
generator to keep a constant voltage at its terminals as load varies. It is
defined by the equation
where Vnl is the no-load terminal voltage of the generator and Vfl is the full-
load terminal voltage of the generator. It is a rough measure of the shape
of the generator's voltage-current characteristic - a positive voltage
regulation means a drooping characteristic, and a negative voltage
regulation means a rising characteristic. A small VR is "better" in the sense
that the voltage at the terminals of the generator is more constant with
variations in load.
Similarly, motors are often compared to each other by using a figure of
merit called speed regulation. Speed regulation (SR) is a measure of the
ability of a motor to keep a constant shaft speed as load varies. It is
defined by the equation

75
VOLTAGE REGULATION AND SPEED REGULATION
It is a rough measure of the shape of a motor's torque-speed characteristic
- positive speed regulation means that a motor's speed drops with
increasing load, and a negative speed regulation means a motor's speed
increases with increasing load. The magnitude of the speed regulation tells
approximately how steep the slope of the torque-speed curve is.

76
9. SUMMARY
There are two major types of ac machines: synchronous machines and
induction machines. The principal difference between the two types is that
synchronous machines require a dc field current to be supplied to their
rotors, while induction machines have the field current induced in their
rotors by transformer action. They will be explored in detail in the next
three chapters.
A three-phase system of currents supplied to a system of three coils
spaced 120 electrical degrees apart on a stator will produce a uniform
rotating magnetic field within the stator. The direction of rotation of the
magnetic field can be reversed by simply swapping the connections to any
two of the three phases. Conversely, a rotating magnetic field will produce
a three-phase set of voltages within such a set of coils.
In stators of more than two poles, one complete mechanical rotation of the
magnetic fields produces more than one complete electrical cycle. For
such a stator, one mechanical rotation produces P/2 electrical cycles.
Therefore, the electrical angle of the voltages and currents in such a
machine is related to the mechanical angle of the magnetic fields by

77
SUMMARY
The relationship between the electrical frequency of the stator and the
mechanical rate of rotation of the magnetic fields is
The types of losses that occur in ac machines are electrical or copper
losses (I2
R losses), core losses, mechanical losses, and stray losses.
Each of these losses was described in this chapter, along with the
definition of overall machine efficiency. Finally, voltage regulation was
defined for generators as
and speed regulation was defined for motors as

Mesin ac ch4

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