I introduction 2

Introduction to AC Machines Dr. Suad Ibrahim Shahl
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ELECTRICAL MACHINES II
Lecturer: Dr. SSuuaadd IIbbrraahhiimm SShhaahhll
Syllabus
I. Introduction to AC Machine
II. Synchronous Generators
III. Synchronous Motors
IV. Three-Phase Induction Machines
V. Three-Phase Induction Motors
VI. Induction Generators
VII. Induction Regulators
Recommended Textbook :
1) M.G.Say
Alternating Current Machines
Pitman Pub.
2) A.S. Langsdorf
Theory of AC Machinery
McGRAW-HILL Pub.

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I. Introduction to AC Machines
Classification of AC Rotating Machines
•Synchronous Machines:
•Synchronous Generators: A primary source of electrical energy.
•Synchronous Motors: Used as motors as well as power factor compensators (synchronous condensers).
•Asynchronous (Induction) Machines:
•Induction Motors: Most widely used electrical motors in both domestic and industrial applications.
•Induction Generators: Due to lack of a separate field excitation, these machines are rarely used as generators.
• Generators convert mechanical energy to electric energy.
Energy Conversion
• Motors convert electric energy to mechanical energy.
• The construction of motors and generators are similar.
• Every generator can operate as a motor and vice versa.
• The energy or power balance is :
– Generator: Mechanical power = electric power + losses
– Motor: Electric Power = Mechanical Power + losses.

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AC winding design
The windings used in rotating electrical machines can be classified as
 Concentrated Windings
• All the winding turns are wound together in series to form one multi-turn coil
• All the turns have the same magnetic axis
• Examples of concentrated winding are
– field windings for salient-pole synchronous machines
– D.C. machines
– Primary and secondary windings of a transformer
 Distributed Windings
• All the winding turns are arranged in several full-pitch or fractional-pitch coils
• These coils are then housed in the slots spread around the air-gap periphery to form phase or commutator winding
• Examples of distributed winding are
– Stator and rotor of induction machines
– The armatures of both synchronous and D.C. machines
Armature windings, in general, are classified under two main heads, namely,
 Closed Windings
• There is a closed path in the sense that if one starts from any point on the winding and traverses it, one again reaches the starting point from where one had started
• Used only for D.C. machines and A.C. commutator machines
 Open Windings
• Open windings terminate at suitable number of slip-rings or terminals
• Used only for A.C. machines, like synchronous machines, induction machines, etc
Some of the terms common to armature windings are described below:
1. Conductor. A length of wire which takes active part in the energy- conversion process is a called a conductor.
2. Turn. One turn consists of two conductors.
3. Coil. One coil may consist of any number of turns.
4. Coil –side. One coil with any number of turns has two coil-sides.

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The number of conductors (C) in any coil-side is equal to the number of turns (N) in that coil.
One-turn coil two-turn coil multi-turn coil
5. Single- layer and double layer windings.
 Single- layer winding
• One coil-side occupies the total slot area
• Used only in small ac machines one coil-side per slot
 Double- layer winding
• Slot contains even number (may be 2,4,6 etc.) of coil-sides in two layers
• Double-layer winding is more common above about 5kW machines
Two coil –sides per slot
4-coil-sides per slot
Coil- sides
Coil- sides
Coil - sides
Overhang
Top layer
Bottom layer

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The advantages of double-layer winding over single layer winding are as follows:
a. Easier to manufacture and lower cost of the coils
b. Fractional-slot winding can be used
c. Chorded-winding is possible
d. Lower-leakage reactance and therefore , better performance of the machine
e. Better emf waveform in case of generators
6. Pole – pitch. A pole pitch is defined as the peripheral distance between identical points on two adjacent poles. Pole pitch is always equal to 180o
7. Coil–span or coil-pitch. The distance between the two coil-sides of a coil is called coil-span or coil-pitch. It is usually measured in terms of teeth, slots or electrical degrees. electrical.
8. Chorded-coil.
 If the coil-span (or coil-pitch) is equal
 in case the coil-pitch is to the pole-pitch, then the coil is termed a full-pitch coil.
less
 if there are S slots and P poles, then pole pitch 푸푸=푺푺 푷푷 slots per pole than pole-pitch, then it is called chorded, short-pitch or fractional-pitch coil
 if coil-pitch 풚풚=푺푺 푷푷 , it results in full-pitch winding
 in case coil-pitch 풚풚<푺푺 푷푷 , it results in chorded, short-pitched or fractional-pitch
Full-pitch coil Short-pitched or chorded coil
N
S
Coil span
Pole pitch
N
S
Coil span
Pole pitch

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In AC armature windings, the separate coils may be connected in several different manners, but the two most common methods are lap and wave
In polyphase windings it is essential that
 The generated emfs of all the phases are of equal magnitude
 The waveforms of the phase emfs are identical
 The frequency of the phase emfs are equal
 The phase emfs have mutual time-phase displacement of 휷휷=ퟐퟐퟐퟐ 풎풎 electrical radians. Here m is the number of phases of the a.c. machine.
Phase spread
Where field winding on the rotor to produce 2 poles and the stator carries 12 conductors housed in 12 slots.
3-phase winding - phase spread is 120
o
A
B
C
E1
E2
E3
E4
E5
E6
E7
E8
E9
E10
E11
E12
1
2
3
4
5
6
7
8
9
10
11
12
N
S

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Time phase angle is 120o between EA, EB and E
C
 Maximum emf Em
 Zero emf induced in conductor 4 (conductor 4 is cutting zero lines of flux) induced in conductor 1ቀ퐸퐸1=퐸퐸푚푚 √2ቁR
 the emf generated in conductor 7 is maximum (conductor 7 is cutting maximum lines of flux from S pole)
 the polarity of emf in conductor 7 will be opposite to that in conductor 1, 푬푬ퟕퟕ=푬푬풎풎 √ퟐퟐ , opposite to E1
 similarly the emfs generated in conductors 2, 3, 5, 6 and in conductor 8 to 12 can be represented by phasors E
2, E3 , E5 , E6 and E8 to E12
 the slot angle pitch is given by 훾훾=180표표 푆푆푆푆푆푆푆푆푆 푝푝푝푝푝푝 푝푝푝푝푝=180표표 6=30표표
 if
푏푏푏푏푏푏푏푏 푒푒푒푒푒 표표표표 푐푐푐푐 ퟏퟏ 푖푖푖 푐 푡 푏푏푏푏푏푏푏푏 푒푒푒푒푒 표표표표 푐푐푐푐 ퟐퟐ 푓 푒푒푒푒푒 표표표표 푐푐푐푐 ퟐퟐ 푖푖푖 푐 푡 푓 푒푒푒푒푒 표표표표 푐푐푐푐 ퟑퟑ 푏푏푏푏푏푏푏푏 푒푒푒푒푒 표표표표 푐푐푐푐 ퟑퟑ 푖푖푖 푐 푡 푏푏푏푏푏푏푏푏 푒푒푒푒푒 표표표표 푐푐푐푐 ퟑퟑ ቑ 퐸퐸퐴퐴=퐸퐸1+퐸퐸2+퐸퐸3+퐸퐸4
Similarly, 퐸퐸퐵퐵=퐸퐸5+퐸퐸6+퐸퐸7+퐸퐸8 퐸퐸퐶퐶=퐸퐸9+퐸퐸10+퐸퐸11+퐸퐸12
 the phase belt or phase band may be defined as the group of adjacent slots belonging to one phase under one pole-pair
Conductors 1, 2, 3 and 4 constitute first phase group
Conductors 5, 6, 7 and 8 constitute second phase group
Conductors 9, 10, 11 and 12 constitute third phase group
 the angle subtended by one phase group is called phase spread, symbol σ
휎휎=푞푞푞 =4×30표표 where
푞푞=푛 표표표표 푠 푝푝푝푝푝푝 푝푝푝푝푝 푝푝푝푝푝푝 푝푝ℎ푠=푆푆 푃푃푃
EA
EB
EC
E1
E2
E3
E4
E12
E11
E10
E9
E5
E6
E7
E8

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Sequence of phase-belts (groups)
Let
12-conductors can be used to obtain three-phase single – layer winding having a phase spread of 60o
 coil pitch or coil span y = pole pitch τ = 푆푆 푃푃=122=6 (휎휎=60표표)
 for 12 slots and 2 poles, slot angular pitch γ =30o
 for 휎휎=60표표 , two adjacent slots must belong to the same phase
A
B
C
E1
E2
E3
E4
E5
E6
E7
E8
E9
E10
E11
E12
1
2
3
4
5
6
7
8
9
10
11
12
N
S
A′
B′
C′
3-phase winding, phase spread is 60o

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(a)
(b)
Phase spread of 60o
(b) Time-phase diagram for the emfs generated in (a) , 12 slots,2 pole winding arrangement
A
B
C
E1
E7
E2
-E8
E5
E6
E9
E10
-E11
-E12
-E4
-E3
120o
1
2
3
4
5
6
7
8
9
10
11
12
b
b
a
a
c
c
d
d
120o
120o
γ=30o
A
A′
A
C′
C′
B
B
C
C
A′
B′
B′
B1
A1
C1
B2
A2
C2

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Double Layer Winding
 synchronous machine armatures and induction –motor stators above a few kW, are wound with double layer windings
 if the number of slots per pole per phase 풒풒=푺푺 풎풎풎 is an integer, then the winding is called an integral-slot winding
 in case the number of slots per pole per phase, q is not an integer, the winding is called fractional-slot winding. For example
 a 3-phase winding with 36 slots and 4 poles is an integral slot winding, because 푞푞=363×4=3 푖푖푖 푎 푖푖푖푖푖푖
 a 3-phase winding with 30 slots and 4 poles is a fractional slot winding, because 푞푞=303×4=52 푖푖푖 푛 푎 푖푖푖푖푖푖
 the number of coils C is always equal to the number of slots S, C=S
1- Integral Slot Winding
Example: make a winding table for the armature of a 3-phase machine with the following specifications:
Total number of slots = 24 Double – layer winding
Number of poles = 4 Phase spread=60
Coil-span = full-pitch
o
(a)Draw the detailed winding diagram for one phase only
(b) Show the star of coil-emfs. Draw phasor diagram for narrow-spread(σ=60o) connections of the 3-phase winding showing coil-emfs for phases A and B only.
Solution: slot angular pitch, 훾훾=4×180표표 24=30표표
Phase spread, 휎휎=60표표
Number of slots per pole per phase, 푞푞=243×4=2
Coil span = full pitch = 244=6

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(a)
Detailed double layer winding diagram for phase A for 3-phase armature having 24 slots, 4 poles, phase spread 60
o

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(c) The star of coil emfs can be drawn similar to the star of slot emfs or star of conductor emfs
Phasor diagram showing the phasor sum of coil-emfs to obtain phase voltages A and B

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2. integral slot chorded winding
 Coil span (coil pitch) pole pitch (y τ)
 The advantages of using chorded coils are:
 To reduce the amount of copper required for the end-connections (or over hang)
 To reduce the magnitude of certain harmonics in the waveform of phase emfs and mmfs
 The coil span generally varies from 2/3 pole pitch to full pole pitch
Example. Let us consider a double-layer three-phase winding with q = 3, p = 4, (S = pqm = 36 slots), chorded coils y/τ = 7/9
The star of slot emf phasors for a double-layer winding p = 4 poles,
q = 3 slots/pole/phase, m = 3, S = 36

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Double-layer winding: p = 4 poles, q = 3, y/τ = 7/9, S = 36 slots.

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3. Fractional Slot Windings
If the number of slots qof a winding is a fraction, the winding is called a fractional slot winding. Advantages of fractional slot windings when compared with integral slot windings are:
1. a great freedom of choice with respect to the number of slot a possibility to reach a suitable magnetic flux density
2. this winding allows more freedom in the choice of coil span
3. if the number of slots is predetermined, the fractional slot winding can be applied to a wider range of numbers of poles than the integral slot winding the segment structures of large machines are better controlled by using fractional slot windings
4. this winding reduces the high-frequency harmonics in the emf and mmf waveforms
Let us consider a small induction motor with p = 8 and q = 3/2, m = 3. The total number of slots S = pqm = 8*3*3/2= 36 slots. The coil span y is y = (S/p) = (36/8) = 4slot pitches
Fractionary q (q = 3/2, p = 8, m = 3,S = 36) winding- emf star,

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 The actual value of q for each phase under neighboring poles is 2 and 1, respectively, to give an average of 3/2
Fractionary q (q = 3/2, p = 8, m = 3, S = 36) winding
slot/phase allocation coils of phase A
Single – Layer Winding
 One coil side occupies one slot completely, in view of this, number of coils C is equal to half the number of slots S, 푪푪=ퟏퟏ ퟐퟐ푺푺
 The 3-phase single –layer windings are of two types
1. Concentric windings
2. Mush windings
Concentric Windings
 The coils under one pole pair are wound in such a manner as if these have one center
 the concentric winding can further be sub-divided into
1. half coil winding or unbifurcated winding
2. Whole coil winding or bifurcated winding

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Half coil winding
For phase A only
 The half coil winding arrangement with 2-slots per pole per phase and for σ=60o
 A coil group may be defined as the group of coils having the same center
 The number of coils in each coil group = the number of coil sides in each phase belt (phase group)
 The carry current in the same direction in all the coil groups
whole coil winding
For phase A only
 The whole coil winding arrangement with 2-slots per pole per phase
 The number of coil sides in each phase belt (here 4) are double the number of coils (here 2) in each coil group
 There are P coil groups and the adjacent coil groups carry currents in opposite directions
Example. Design and draw (a) half coil and (b) whole coil single layer concentric windings for a 3-phase machine with 24-slots, 4-poles and 60o phase spread.

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Solution: (a) half coil concentric winding
푆푆푆푆푆푆푆푆푆 푎 푝푝푝ℎ,훾훾=4×180표표 24=30표표
퐹퐹퐹 푝푝푝ℎ 표표표 푝푝푝푝푝 푝푝푝ℎ=244=6 푠 푝푝푝ℎ푒
Half-coil winding diagram for 24 slots, 4 poles, 60o phase spread single layer concentric winding (two – plane overhang)

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(b) Whole-coil concentric winding
For slot pitch γ = 30o phase spread σ = 60o
 The number of coils per phase belt = 2,
 The number of coils in each coil group = 1
 The pole pitch=6
 The coil pitch of 6 slot pitches does not result in proper arrangement of the winding
 In view of this, a coil pitch of 5 is chosen
Whole-coil winding arrangement of 24 slots, 4 poles, 60o phase spread, single layer concentric winding (three-plane overhang)
Mush Winding
 The coil pitch is the same for all the coils
 Each coil is first wound on a trapezoidal shaped former. Then the short coil sides are first fitted in alternate slots and the long coil sides are inserted in the remaining slots
 The number of slots per pole per phase must be a whole number
 The coil pitch is always odd

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For example, for 24 slots, 4 poles, single-layer mush winding, the pole pitch is 6 slots pitches. Since the coil pitch must be odd, it can be taken as 5 or 7. Choosing here a coil pitch of 5 slot pitches.
Single – layer mush winding diagram for 24 slots, 4 poles and 60o phase spread
H.W: Design and draw
1. 3-phase, 24-slots, 2-poles single-layer winding (half coil winding)
2. a.c. winding: 3-phase, 4 -pole, 24- slots, double layer winding with full pitch coils (phase B phase C)
3. a.c. winding: 3-phase, 4 -pole, 24- slots, double layer winding with chorded coils y/τ = 5/6
4. 10 -pole, 48- slots, fractional 3-phase double layer winding

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 When balanced 3-phase currents flow in balanced 3-phase windings, a rotating magnetic field is produced.
Rotating Magnetic Field
 All 3-phase ac machines are associated with rotating magnetic fields in their air-gaps.
For example, a 2-pole 3-phase stator winding
 The three windings are displaced from each other by 120o along the air-gap periphery.
 Each phase is distributed or spread over 60o (called phase-spread σ=60o)
 The 3-phase winding a, b, c is represented by three full pitched coils, aa′ , bb′ , cc′
 For instance, the concentrated full-pitched coil aa′ represents phase a winding in all respects
 A current in phase a winding establishes magnetic flux directed along the magnetic axis of coil aa′
 Positive currents are assumed to be flowing as indicated by crosses in coil-sides a′ , b′ , c′

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Magnetic flux plot
 At the instant 1, the current in phase a is positive and maximum Im
 At the instant 2, 풊풊풂풂=푰푰풎풎 ퟐퟐ , 풊풊풃풃=푰푰풎풎 ퟐퟐ and 풊풊풄풄=−푰푰풎풎 ቀ풊풊풃풃=풊풊풄풄=−푰푰풎풎 ퟐퟐቁ
 At the instant 3, 풊풊풂풂=−푰푰풎풎 ퟐퟐ , 풊풊풃풃=푰푰풎풎 and 풊풊풄풄=−푰푰풎풎 ퟐퟐ
 The 2 poles produced by the resultant flux are seen to have turned through further 60o
 The space angle traversed by rotating flux is equal to the time angle traversed by currents
 The rotating field speed, for a P-pole machine, is
 ퟏퟏ 푷푷 ퟐퟐൗ revolution in one cycle
 풇풇 푷푷 ퟐퟐൗ revolutions in f cycles
Production of rotating magnetic field illustrated by magnetic flux plot

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 풇풇 푷푷 ퟐퟐൗ revolutions in one second [because f cycles are completed in one second]
Here f is the frequency of the phase currents. If ns
푛푠=푓푃푃2⁄=2푓푃푃 denotes the rotating field speed in revolutions per sec, then
Or
푁푁푠=120푓푝푝 푟.푝푝.푚푚 [The speed at which rotating magnetic field revolves is
called the Synchronous speed]
Space phasor representation
 When currents ia , ib , ic
Production of rotating magnetic field illustrated by space phasor m.m.fs. flow in their respective phase windings, then the three stationary pulsation m.m.fs 퐹퐹푎ഥ ,퐹퐹푏푏തതത , 퐹퐹푐ഥ combine to give the resultant m.m.f. 퐹퐹푅푅തതത which is rotating at synchronous speed.
 At the instant 1,
푖푖푎=퐼퐼푚푚  푠 푝푝ℎ푎 퐹퐹ത 푎=푚푚푚푚푚푚푚푚푚푚푚 푚푚.푚푚.푓.퐹퐹푚푚
푖푖푏푏=푖푖푐=−퐼퐼푚푚 2  푡ℎ푒 푚푚.푚푚.푓.푝푝ℎ푎 퐹퐹ത 푏푏 =퐹퐹ത 푐=퐹퐹푚푚 2
The resultant of m.m.fs. 푭푭ഥ 풂풂 ,푭푭ഥ 풃풃 , 푭푭ഥ 풄풄 is 푭푭ഥ 푹푹 and its magnitude is given by
The vertical component of 푭푭ഥ 풃풃 푭푭ഥ 풄풄 cancel each other.
퐹퐹푅푅=퐹퐹푚푚+ 2퐹퐹푚푚 2cos60표표 = 32 퐹퐹푚푚

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 At the instant 2,
푖푖푎=푖푖푏푏=퐼퐼푚푚 2 푖푖푐=−퐼퐼푚푚 푡ℎ푒 푚푚.푚푚.푓.푝푝ℎ푎 퐹퐹ത =퐹퐹ത 푏푏=퐹퐹푚푚 2 푠 푝푝ℎ푎 퐹퐹ത =푚푚푚푚푚푚푚푚푚푚푚 푚푚.푚푚.푓.퐹퐹푚푚 The resultant m.m.f. 퐹퐹푅푅 =32퐹퐹푚푚 [it rotate by a space angle of 60o
 At the instant 3, clockwise]
푖푖푎=푖푖푐=−퐼퐼푚푚 2 푖푖푏푏=퐼퐼푚푚 The resultant m.m.f. 퐹퐹푅푅 =32퐹퐹푚푚 [The resultant m.m.f. has turned through a further space angle of 60o
Sinusoidal rotating mmf wave creates in phase sinusoidal rotating flux density wave in the air gap; the peak value of B- wave is given by from its position occupied at instant 2]
Where g is air-gap length Example: Prove that a rotating magnetic field of constant amplitude is produced when 3-phase balanced winding is excited by three-phase balanced currents. Solution: three – phase balanced currents given by
A constant-amplitude rotating m.m.f. or rotating field is produced in the air-gap of a three-phase machines at synchronous speed
------ (1)

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The three mmfs Fa , Fb and Fc can be expressed mathematically as
Angle α is measured from the axis of phase a The mmf of phase a can be expressed as
Similarly, for phases b c,
The resultant mmf 퐹퐹푅푅(훼훼,푡) can be obtained by adding the three mmfs given by Eqs. (1), (2) and (3).
------ (2)
------ (3)
------ (4)
------ (5)

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Eq.(5), therefore, reduces to
It can be shown that Eq.(6) represents a travelling mmf wave of constant amplitude ퟑퟑ ퟐퟐ푭푭풎풎
H.W: A three-phase, Y-connected winding is fed from 3-phase balanced supply, with their neutrals connected together. If one of the three supply leads gets disconnected, find what happens to the m.m.f. wave .
But
mmf
------ (6)
At
At
At

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• A wire loop is rotated in a magnetic field.
Electromotive Force (EMF) Equation
– N is the number of turns in the loop
– L is the length of the loop
– D is the width of the loop
– B is the magnetic flux density
– n is the number of revolutions per seconds
• A wire loop is rotated in
a magnetic field.
• The magnetic flux through
the loop changes by the position
• Position 1 all flux links with
the loop
• Position 2 the flux linkage
reduced
• The change of flux linkage
induces a voltage in the loop
• The induced voltage is an ac voltage
• The voltage is sinusoidal
• The rms value of the induced voltage loop is:
The r.m.s value of the generated emf in a full pitched coil is
퐸퐸=퐸퐸푚푚푚푚푚 √2 , where 퐸퐸푚푚푚푚푚=휔휔푟푁푁∅=2휋휋휋∅ [∅=퐵퐵퐵퐵퐵퐵]
∴ 퐸퐸=퐸퐸푚푚푚푚푚 √2=√2 휋휋 푓∅=4.44푓∅
()()tLDBtωcos=Φnπω2= ()()()[]()tLDBNdttdLDBNdttdNtVωωω sincos== Φ= 2 ωLDBNVrms=
E
E

28
Winding Factor (Coil Pitch and Distributed Windings)
Pitch Factor or Coil Pitch
The ratio of phasor (vector) sum of induced emfs per coil to the arithmetic sum of induced emfs per coil is known as pitch factor (Kp) or coil span factor (Kc) which is always less than unity.
Let the coil have a pitch short by angle θ electrical space degrees from full pitch and induced emf in each coil side be E,
• If the coil would have been full pitched, then total induced emf in the coil would have been 2E.
• when the coil is short pitched by θ electrical space degrees the resultant induced emf, ER
퐸퐸푅푅=2퐸퐸cos휃휃 2 in the coil is phasor sum of two voltages, θ apart
Pitch factor, 푲푲풑풑=푷푷푷푷푷푷푷푷푷푷푷 풔풔풔풔 풐 풄풄풄풄풄 풔풔풔풔풔 풆푨푨푨푨푨푨푨푨푨 풔풔풔풔 풐 풄풄풄풄풄 풔풔풔풔풔 풆= ퟐퟐퟐ퐜퐜퐜퐜퐜퐜휽휽 ퟐퟐ ퟐퟐퟐ=퐜퐜퐜퐜퐜퐜휽휽 ퟐퟐ
Example. The coil span for the stator winding of an alternator is 120o. Find the chording factor of the winding.
Solution: Chording angle, 휃휃=180표표−푐 푠=180표표−120표표=60표표
Chording factor, 퐾퐾푝푝=cos휃휃 2=cos60표표 2=0.866
E
E
E
휽휽 ퟐퟐ
휽휽 ퟐퟐ
휽휽

29
The ratio of the phasor sum of the emfs induced in all the coils distributed in a number of slots under one pole to the arithmetic sum of the emfs induced(or to the resultant of emfs induced in all coils concentrated in one slot under one pole) is known as breadth factor (K
Distribution Factor
b) or distribution factor (Kd
퐾퐾푑= 퐸퐸퐸퐸퐸 푖푖푖 푖푖푖 푑 푤푤푤퐸퐸퐸퐸퐸 푖푖푖 푖푖푖 푡ℎ푒 푤푤푤 푤푤푤 ℎ푎푎푎푎 푏푏푏 푐)
=푃푃ℎ푎 푠 표표표표 푐 푒퐴퐴퐴ℎ푚푚푚 푠 표표표표 푐 푒
 The distribution factor is always less than unity.
 Let no. of slots per pole = Q and no. of slots per pole per phase = q
Induced emf in each coil side = E
Angular displacement between the slots, 훾훾=180표표 푄푄
c
 The emf induced in different coils of one phase under one pole are represented by side AC, CD, DE, EF… Which are equal in magnitude (say each equal Ec ) and differ in phase (say by γo) from each other.
γ
γ
γ/2
γ/2
γ/2
qγ
A
B
C
D
E
F
E
E
E
E
E
O

30
If bisectors are drawn on AC, CD, DE, EF… they would meet at common point (O). The point O would be the circum center of the circle having AC, CD, DE, EF…as the chords and representing the emfs induced in the coils in different slots.
EMF induced in each coil side, 퐸퐸푐=퐴퐴퐴=2푂푂푂sin훾훾 2
퐴퐴퐴ℎ푚푚푚 푠=푞푞×2×푂푂푂sin훾훾 2
∴ The resultant emf, 퐸퐸푅푅=퐴퐴퐴=2×푂푂푂sin퐴퐴퐴2=2×푂푂푂sin푞푞푞2
distribution factor, 푘푑=푃푃ℎ푎 푠 표표표표 푐 푒퐴퐴퐴ℎ푚푚푚 푠 표표표표 푐 푒
= 2×푂푂푂sin푞푞푞2 푞푞×2×푂푂푂sin훾훾 2= 퐬퐬퐬퐬퐬풒풒휸휸 ퟐퟐ 풒풒퐬퐬퐬퐬퐬휸휸ퟐퟐ
Example. Calculate the distribution factor for a 36-slots, 4-pole, single layer 3- phase winding.
Solution: No. of slots per pole, 푄푄=364=9
No. of slots per pole per phase, 푞푞=푄푄 푁푁푁 표표표표 푝푝ℎ푎=93=3
Angular displacement between the slots, 훾훾=180표표 푄푄=180표표 9=20표표
Distribution factor, 퐾퐾푑= sin푞푞푞2 푞푞sin훾훾 2= sin3×20표표 23sin20표표 2=13sin30표표 sin10표표=0.96

31
Example1. A 3-phase, 8-pole, 750 r.p.m. star-connected alternator has 72 slots on the armature. Each slot has 12 conductors and winding is short chorded by 2 slots. Find the induced emf between lines, given the flux per pole is 0.06 Wb.
Solution:
Flux per pole, ∅=0.06 푊푊푊
푓=푝푝푝60=4×75060=50 퐻퐻퐻퐻
Number of conductors connected in series per phase,
푍푍푠=푁푁푁 표표표표 푐푐푐푐 푝푝푝푝푝푝 푠 ×푛 표표표표 푠 푁푁푁푏푏푏 표표표표 푝푝ℎ푎
=12×723=288
Number of turns per phase, 푇푇=푍푍푠2=2882=144
Number of slots per pole, 푄푄=728=9
Number of slots per pole per phase, 푞푞=푄푄 3=93=3
Distribution factor, 퐾퐾푑= sin푞푞푞2 푞푞sin훾훾 2= sin3×20표표 23sin20표표 2=13sin30표표 sin10표표=0.96
Chording angle, 휃휃=180표표×29=40표표
Pitch factor, 퐾퐾푝푝=cos휃휃 2=cos40표표 2=cos20표표=0.94
Induced emf between lines, 퐸퐸퐿=√3×4.44×퐾퐾푑×퐾퐾푝푝×∅×푓×푇푇 =ඥ3×4.44×0.96×0.94×0.06×50×144=2998 푉푉

32
 the variation of magnetic potential difference along the air –gap periphery is of rectangular waveform and of magnitude 12푁푁푁
Magnetomotive Force (mmf) of AC Windings M.m.f. of a coil
 The amplitude of mmf wave varies with time, but not with space
 The air –gap mmf wave is time-variant but space invariant
 The air –gap mmf wave at any instant is rectangular
Mmf distribution along air-gap periphery
The fundamental component of rectangular wave is found to be 퐹퐹푎1= 4 휋휋 ∙ 푁푁푁2cos훼훼=퐹퐹1푝푝cos훼훼
Where
α = electrical space angle measured from the magnetic axis of the stator coil

33
Here F1p , the peak value of the sine mmf wave for a 2-pole machine is given by 퐹퐹1푝푝=4 휋휋∙푁푁푁2 퐴퐴퐴 푝푝푝푝푝푝 푝푝푝푝푝 When i=0  F1p =0 i=Imax
 The mmf distribution along the air gap periphery depends on the nature of slots, winding and the exciting current =√ퟐퟐ푰푰 For 2-pole machine 퐹퐹1푝푝푝=4 휋휋∙푁푁√2퐼퐼 2 퐴퐴퐴 푝푝푝푝푝푝 푝푝푝푝푝 For p-pole machine 퐹퐹1푝푝푝=4 휋휋∙푁푁√2퐼퐼 푃푃 퐴퐴퐴 푝푝푝푝푝푝 푝푝푝푝푝 M.m.f of distributed windings
 The effect of winding distribution has changed the shape of the mmf wave, from rectangular to stepped
Developed diagram and mmf wave of the machine
(each coil has Nc turns and each turn carries i amperes)

34
Example: a 3-phase, 2-pole stator has double-layer full pitched winding with 5 slots per pole per phase. If each coil has Nc turns and i is the conductor current, then sketch the mmf wave form produced by phase A alone.
A 3-phase, 2-pole stator with double-layer winding having 5 slots per pole per phase
 For any closed path around slot 1, the total current enclosed is 2Nci ampere
 Magnetic potential difference across each gap is ퟏퟏ ퟐퟐ [ퟐퟐ푵푵풄풄풊풊]=푵푵풄풄풊풊
 The mmf variation from −푵푵풄풄풊풊 to +푵푵풄풄풊풊 at the middle of slot 1
 The mmf variation for slot 1′ is from +푵푵풄풄풊풊 to −푵푵풄풄풊풊
 The mmf variation for coil 11′ is of rectangular waveform with amplitude ±푁푁푐푖푖 . similarly, the rectangular mmf waveforms of amplitude ±푁푁푐푖푖 are sketched for the coils 22′ , …, 55′
 The combined mmf produced by 5 coils is obtained by adding the ordinates of the individual coil mmfs.
 The resultant mmf waveform consists of a series of steps each of height
ퟐퟐ푵푵풄풄풊풊 = (conductors per slot) (conductor current)
 The amplitude of the resultant mmf wave is ퟓퟓ푵푵풄풄풊풊 .

35
Mmf waveforms

36
Harmonic Effect
 The flux distribution along the air gaps of alternators usually is non- sinusoidal so that the emf in the individual armature conductor likewise is non-sinusoidal
 The sources of harmonics in the output voltage waveform are the non- sinusoidal waveform of the field flux.
 Fourier showed that any periodic wave may be expressed as the sum of a d-c component (zero frequency) and sine (or cosine) waves having fundamental and multiple or higher frequencies, the higher frequencies being called harmonics.
The emf of a phase due to the fundamental component of the flux per pole is:
퐸퐸푝푝ℎ1=4.44푓퐾퐾푤푤1푇푇푝푝ℎ∅1
Where 퐾퐾푤푤1=푘푑1.퐾퐾푝푝1 is the winding factor. For the nth harmonic
퐸퐸푝푝ℎ푛=4.44푛푤푤푤푇푇푝푝ℎ∅푛
The nth harmonic and fundamental emf components are related by
퐸퐸푝푝ℎ푛퐸퐸푝푝ℎ1=퐵퐵푛퐾퐾푤푤푤퐵퐵1퐾퐾푤푤1
The r.m.s. phase emf is:
퐸퐸푝푝ℎ=ට൫퐸퐸푝푝ℎ12+퐸퐸푝푝ℎ32+∙∙∙+퐸퐸푝푝ℎ푛2൯
 All the odd harmonics (third, fifth, seventh, ninth, etc.) are present in the phase voltage to some extent and need to be dealt with in the design of ac machines.
 Because the resulting voltage waveform is symmetric about the center of the rotor flux, no even harmonics are present in the phase voltage.
 In Y- connected, the third-harmonic voltage between any two terminals will be zero. This result applies not only to third-harmonic components but also to any multiple of a third-harmonic component (such as the ninth harmonic). Such special harmonic frequencies are called triplen harmonics.

37
 The pitch factor of the coil at the harmonic frequency can be expressed as
퐾퐾푝푝푝=cos푛2 where n is the number of the harmonic
Elimination or Suppressed of Harmonics
Field flux waveform can be made as much sinusoidal as possible by the following methods:
1. Small air gap at the pole centre and large air gap towards the pole ends
2. Skewing: skew the pole faces if possible
3. Distribution: distribution of the armature winding along the air-gap periphery
4. Chording: with coil-span less than pole pitch
5. Fractional slot winding
6. Alternator connections: star or delta connections of alternators suppress triplen harmonics from appearing across the lines
For example, for a coil-span of two-thirds ቀ23푟ቁ of a pole pitch 퐶퐶퐶−푠,∝= 23×180표표=120표표 (푖푖푖 푒 푑) 퐶퐶ℎ표표표 푎,휃휃=180표표−훼훼=180표표−120표표=60표표
퐾퐾푝푝1=cos 푛2=cos60표표 2=cos30표표=0.866
For the 3rd harmonic: 퐾퐾푝푝3=cos3×60표표 2=cos90표표=0;
Thus all 3rd (and triplen) harmonics are eliminated from the coil and phase emf . The triplen harmonics in a 3-phase machine are normally eliminated by the phase connection.
Example: An 8-pole, 3-phase, 60o spread, double layer winding has 72 coils in 72 slots. The coils are short-pitched by two slots. Calculate the winding factor for the fundamental and third harmonic.
Solution: No. of slots per pole, 푄푄=728=9
No. of slots per pole per phase, 푞푞=푄푄 푚푚=93=3

38
퐶퐶퐶 푠,∝= 180표표×푐 푠 푖푖푖 푡 표표표표 푠푁푁푁.표표표표 푠 푝푝푝푝푝푝 푝푝푝푝푝 = 180표표(9−2) 9=140표표
퐶퐶ℎ표표표 푎,휃휃=180표표−푐 푠=180표표−140표표=40표표
For the fundamental component 퐷 푓,퐾퐾푑= sin푞푞푞2⁄ 푞푞sin훾훾2⁄ = sin3×20표표 23sin20표표 2=0.96 푃푃푃ℎ 푓,퐾퐾푝푝=cos 휃휃 2=cos40표표 2=0.94 푊푊푊 푓,퐾퐾푤푤=퐾퐾푑×퐾퐾푝푝=0.96×0.94=0.9
For the third harmonic component (n=3) 퐷 푓,퐾퐾푑3= sin푛2⁄ 푞푞sin푛2⁄ = sin3×3×20표표 23sin3×20표표 2=0.666 푃푃푃ℎ 푓,퐾퐾푝푝3=cos3휃휃 2=cos3×40표표 2=0.5 푊푊푊 푓,퐾퐾푤푤3=퐾퐾푑3×퐾퐾푝푝3=0.666×0.5=0.333
Example3: Calculate the r.m.s. value of the induced e.m.f. per phase of a 10-pole, 3-phase, 50Hz alternator with 2 slots per pole per phase and 4 conductors per slot in two layers. The coil span is 150o .the flux per pole has a fundamental component of 0.12Wb and a third harmonic component.
Solution: No. of slots/pole/phase, 푞푞=2
No. of slots/pole, 푄푄=푞푞푞=2×3=6
No. of slots/phase =2푝푝푝=10×2=20
No. of conductors connected in series, 푍푍푠=20×4=80
No. of series turns/phase, 푇푇=푍푍푠2=802=40

39
Angular displacement between adjacent slots, 훾훾=180표표 푄푄=180표표 6=30표표 퐷 푓,퐾퐾푑= sin푞푞푞2⁄ 푞푞sin훾훾2⁄ = sin2×30표표 22sin30표표 2=0.966 퐶퐶퐶 푠 푓,퐾퐾푝푝=cos 휃휃 2=cos(180표표−150표표) 2=cos15표표=0.966
Induced emf per phase (fundamental component),
퐸퐸푝푝ℎ1=4.44 퐾퐾푑퐾퐾푝푝∅푓
=4.44×0.966×0.966×0.12×50×40=994.4 푉푉
For third harmonic component of flux 퐷 푓,퐾퐾푑3= sin푞푞푞2⁄ 푞푞sin푛2⁄ = sin2×3×30표표 23sin3×30표표 2=0.707 퐶퐶퐶 푠 푓,퐾퐾푝푝3=cos3(180표표−150표표) 2=cos45표표=0.707
퐹퐹퐹퐹퐹,푓3=3×푓=3×50=150
Flux per pole, ∅3=13×0.12×20100=0.008 푊푊푊
Induced emf per phase (third harmonic component)
퐸퐸푝푝ℎ3=4.44 퐾퐾푑3퐾퐾푝푝3∅3푓3푇푇
=4.44×0.707×0.707×0.008×150×40=106.56 푉푉
Induced emf per phase,
퐸퐸푝푝ℎ=ට퐸퐸푝푝ℎ12+퐸퐸푝푝ℎ32=ඥ(994.4)2+(106.56)2=1000 푉푉

40
H.W
1. Three-phase voltages are applied to the three windings of an electrical machine. If any two supply terminals are interchanged, show that the direction of rotating mmf wave is reversed,through its amplitude remains unaltered.
2. A 3-phase 4-pole alternator has a winding with 8 conductors per slot. The armature has
a total of 36 slots. Calculate the distribution factor. What is the induced voltage per phase
when the alternator is driven at 1800 RPM, with flux of 0.041 Wb in each pole?
(Answer. 0.96, 503.197 Volts/phase)
3. A 10 MVA ,11 KV,50 Hz ,3-phase star-connected alternator s driven at
300 RPM. The winding is housed in 360 slots and has 6 conductors per
slot, the coils spanning (5/6) of a pole pitch. Calculate the sinusoidally distributed
flux per pole required to give a line voltage of 11 kV on open circuit, and the full- load current per conductor. (Answer. 0.086 weber , 524.864 Amps)
4. A three phase four pole winding is excited by balanced three phase 50 Hz currents. Although the winding distribution has been designed to minimize the harmonics, there remains some third and fifth spatial harmonics. Thus the phase A mmf can be written as
Similar expressions can be written for phase B (replacing θ by θ -120o and ωt by ωt- 120o) and phase C (replacing θ by θ +120o and ωt by ωt+120o
Derive the expression for the total three phase mmf, and show that the fundamental and the 5).
th harmonic components are rotating

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