machinery fundamentals

8/7/2019 machinery fundamentals

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[email protected] Ch4 - Electrical Machines by Chapman Page 1 of 16

Chapter 4

AC Machinery Fundamentals

Contents

Simple Loop in a uniform Magnetic Field

The Voltage Induced in a Simple Rotating Loop

Alternative way to express induced voltage

Torque Induced in a Current-Carrying Loop

The Rotating Magnetic Field

Induced Voltage in AC Machines

AC Machine Power Flows and Losses

Power Flow Diagram

Voltage Regulation and Speed Regulation


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Simple Loop in a uniform Magnetic Field

Front view View of coil.

(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.

(a) (b)

(c)


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The Voltage Induced in a Simple Rotating Loop

eind = (v x B) l1. Segment ab: 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 4-2 b. The quantity v x B points into the page, which is the same direction as seg-

ment ab. Therefore, induced voltage on this segment of the wire is eba = (v x B) l = vBl sin ab into the page

2. Segment bc: In the first half of this segment, the quantity v x B points into the page, and in the second half ofthis segment, the quantity v x B points out of the page. Since the length l is in the plane of the page, v x B is per-

pendicular to l for both portions of the segment. Therefore, the voltage in segment be will be zero: ecb = 0

3. Segment cd: 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 quantityv 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

edc = (v x B) I = vBIsin cd out of the page

4. Segment da. Just as in segment bc, v x B is perpendicular to l. Therefore, the voltage in this segment will be

zero too: ead = 0

Total induced voltage on the loop eind is the sum of the voltages on each of its sides: vBl sin ab + vBl sin ad

Note that ab = 180 - cd and recalling the trigonometric identity sin = sin (180 -).

Therefore, the induced voltage becomes eind = 2vBL sin

The resulting voltage eind is shown as a function of time in Figure 4-3.


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Alternative way to express induced voltage

= wtAlso, tangential velocity v of the edges of the loop can be expressed as

v = rw (4-7)

where ris the radius from axis of rotation out to the edge of the loop, and

w

is the angular velocity of the loop.Substituting these expressions into Eq. (4-6) gives

eind = 2rwBI sin wt(4-8)

Notice also that the area A of the loop is just equal to 2rl,

Therefore, eind = AB w sin wt (4-9)

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.

max = AB (4-10)Therefore, the final form of the voltage equation is

eind = maxw sin wt (4-11)


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Torque Induced in a Current-Carrying Loop

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.

Figure 4-4:

A current-carrying loop in a uniform

magnetic field.

(a) Front view;

(b) view of coil

(a) (b)

ab = ilB.rSinab

cd= ilB.rSincd


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cont

Force on each segment of the loop will be given by Equation (1-43),

F = i(l x B) (1-43)where i = magnitude of current in the segment

l = length of segment, with direction of l in direction of current flowB = magnetic flux density vector.

The torque on that segment will then be given by

T = (force applied) (perpendicular distance)

= (F) (r sin ) = rF sin (1-6)where is the angle between the vectorr and the vectorF. The direction of the

torque is clockwise if it would tend to cause a clockwise rotation and counter-

clockwise if it would tend to cause a counterclockwise rotation.


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cont

Total induced torque on the loop ind is sum of the torques on each of its sidesind = rilB sin ab + rilB sin cdNote that ab = cd, so the induced torque becomes

ind = 2rilB sin (4-17)

The resulting torque ind is shown in Figure 4-6. Note that the torque is maxi-

mum 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.


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The Rotating Magnetic Field

If two magnetic fields are present, 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 a 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.

The fundamental principle of ac machine operation is that if a 3-phase set of currents, each of

equal magnitude and differing in phase by 120, flows in a 3-phase winding, then it will pro-

duce a rotating magnetic field of constant magnitude.

FIGURE 4-8

(a) A simple 3-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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Concept of the rotating magnetic field

Since B =H

Coil currents

Magnetic field

Flux densities

Intensities


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cont


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cont

Although the direction of the magnetic field has changed, the magnitude is constant.

Magnetic field is maintaining a constant magnitude while rotating in a counterclockwise

direction.

If the current in any of two of the windings is reversed, the direction of rotation of the

magnetic field will be reversed

where fe = frequency of the field (current) in Hz (typically 60 Hz)

nm = mechanical speed of rotation in revs per minute

P = number of poles pairs

Relationship between Electrical frequency and speed of rotation of magnetic field


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Magnetomotive Force and Flux Distribution On AC Machines

(a) AC machine with a cylindrical rotor.

(b) AC machine with a salient-pole rotor FIGURE 4-14

(b) The magnetomotive force distribu-

tion resulting from the winding, com-

pared to an ideal distribution.


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Induced Voltage in AC Machines


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AC Machine Power Flows and Losses

The efficiency of an ac machine is defined by

The losses that occur in ac machines can be divided into four categories:

1. Electrical or copper losses (I2R losses)

2. Core losses

3. Mechanical losses

4. Stray load losses

The mechanical power that is converted is given by:


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Power Flow Diagram

(a) The power-flow diagram of a 3-phase ac generator.

(b) The power-flow diagram of a 3-phase ac motor.


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Voltage Regulation and Speed Regulation

Voltage regulation(VR) is a measure of the ability of a generator to keep a

constant voltage at its terminals as load varies.

where Vnl is the no-load terminal voltage of the generator and

Vfl is the full-load terminal voltage of the generator

Speed regulation (SR) is a measure of the ability of a motor to keep a constant

shaft speed as load varies.

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