synchronous generator pitch and distribution factors
DESCRIPTION
Lecture notes on AC machineryTRANSCRIPT
The Stator Coil Pitch and Distribution Factors
The Coil Pitch
In some generators, the coil sides does not exactly correspond to adjacent North (N) and South (S) poles of the rotor. If they do, they are called full-pitch coils.
If the sides of the coils span an arc whose angle is less than the angle spanned by adjacent N & S poles, these coils are called fractional-pitch coil.
The physical or mechanical angle spanned by adjacent N & S poles known as the pole pitch (ρp) is dependent on the number of poles (P) of the rotor but are always 180 electrical degrees apart.
P
o
p
360 (mechanical degrees)
op 180 (electrical degrees)
Hence, for P = 2, the mechanical pitch and electrical pitch of adjacent poles are numerically the same.
The Coil Pitch
NN
S
S
ρp
A full-pitch coil a-a’
a
a’
NN
S
S
ρp
A fractional-pitch coil a-a’
a
a’coil
pitch
coilpitch
coil a-a’ containing several turns
The Coil Pitch
In reality, the air-gap flux density distribution is not sinusoidal. It’s contaminated by harmonics.
Air-gap flux density:B(α)= BM cos (ωt – α)
stator
rotor
air gap
α
BM
rotor is moving in this direction
a’ ab’c c’ b a’
Perfect sinusoidal air-gap flux density(coils are full-pitched)
The Coil Pitch
In reality, the air-gap flux density distribution is not sinusoidal. It’s contaminated by harmonics.
Air-gap flux density:B(α)= BM cos (ωt – α)
stator
rotor
air gap
α
BM
rotor is moving in this direction
a’ ab’c c’ b a’
Non-sinusoidal air-gap flux density(coils are full-pitched)
The Coil Pitch
In reality, the air-gap flux density distribution is not sinusoidal. It’s contaminated by harmonics.
If the air-gap flux contains harmonics, so as the induced voltage and current in the stator windings which is an undesirable condition.
Introducing fractional-pitch coils in stator windings can suppress some of these harmonics and improved the shape of the induced voltage and current in the stator winding.
EA
t
The phase voltage induced on a generator due to non-sinusoidal flux
distribution
The Coil Pitch
Conversion of mechanical degrees to electrical degrees and vice versa:
NN
S
S
ρp
The pole pitch of a 4-pole machine is 90o mechanical and always 180o electrical
If we let De the pitch in electrical degrees and Dm, the corresponding pitch in mechanical degrees, then
me DP
D2
Note that if P = 2, De = Dm.
The Coil Pitch
The coil pitch ρc is defined as the span of the coil sides in electrical or mechanical degrees.
Sometimes the span is given as a fraction of the pole pitch.
If the pitch of the coil in mechanical degrees is given by θm, then its pitch in electrical degrees is,
2
Pmc
Fractional-pitch coils are also known as chorded windings.
(electrical degrees)
The Coil Pitch (ρc):
Checkpoint 1
A 3-phase, 8-pole alternator has full-pitch coils. What is the coil pitch a) in electrical degrees b) in mechanical degrees? If the coil pitch is 7/8 of the pole pitch, what are the pitches in electrical and mechanical degrees?
Solution:
a)
oomc
oo
m
P180
2
845
2
458
360
(electrical degrees)
oc 45 (mechanical degrees)b)
Checkpoint 1
Solution:
c)
oomc
oo
m
P5.157
2
8375.39
2
375.398
7
8
360
oc 375.39
(electrical degrees)
(mechanical degrees)
a-b
c-d BM
ω
Bv
B
v
eba in segment ba:
eba = (v x B) • l = (vB sin 90o) l cos 0o
eba = v BM l cos (ωt – 90o-½ρc)
dc
ab
l
+
-
edc
eba
eind
+ -
Effect of Coil Pitch on Induced Voltage
ρc
90-½ρc
90-½ρc
A 2-pole generator with chorded coil a-b-c-d (the coil pitch ρc in electrical degrees is also equal to its mechanical degrees)
- +
Note: this induced voltage must be directed into this page.
a-b
c-d BM
ω
Bv
B
v
edc in segment dc :
edc = (v x B) • l = (vB sin 90o) l cos 0o
edc = v BM l cos (ωt – 90o+½ρc)
dc
ab
l
+
-
edc
eba
eind
+ -
Effect of Coil Pitch on Induced Voltage
ρc
90-½ρc
90-½ρc
A 2-pole generator with chorded coil a-b-c-d (the coil pitch ρc in electrical degrees is also equal to its mechanical degrees)
- +
Note: this induced voltage must be directed out of this page
a-b
c-d BM
ω
Bv
B
v
segments bc and ad will have no induced voltage since they don’t cut any flux.
dc
ab
l
+
-
edc
eba
eind
+ -
Effect of Coil Pitch on Induced Voltage
ρc
90-½ρc
90-½ρc
A 2-pole generator with chorded coil a-b-c-d (the coil pitch ρc in electrical degrees is also equal to its mechanical degrees)
- +
Total induced voltage on coil:
Effect of Coil Pitch on Induced Voltage
tlvB
tlvBtlvB
eee
cM
co
Mco
M
dcbaind
cos2
sin2
)90cos()90cos( 21
21
te cind cos
2sin
Since 2vBMl = φω, :
Note: ρc is the coil pitch in electrical degrees which is also the pitch in mechanical degrees for a 2-pole machine.
Effect of Coil Pitch on Induced Voltage
tke pind cos where,
Notes: 1) ρc is the coil pitch in electrical degrees which is also
the pitch in mechanical degrees for a 2-pole machine.2) If ρc = 180o electrical, kp = 1 and eind = φω cosωt, which is
the original formula for the induced voltage on a coil for full-pitched coil.
3) ω is the mechanical angular velocity of the prime mover.
2sin c
p
pk
is called the pitch factor.
If coil has NC number of turns,
tkNe pCind cos
Assignment
Research and explain how harmonics in the generator generated voltage can be minimized by manipulating the pitch ρc of the coil in the stator?
The following information is known about the simple four-pole generator shown below. The peak flux density of the rotor magnetic field is 0.2 T, and the mechanical rate of rotation of the shaft is 3600 rpm. 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.a) What is the RMS line voltage generated by the
generator if the coil is full-pitched?b) If the coil is 7/8 of the pole pitch, what is the RMS
line voltage?c) What is the mechanical pitch of the coil in b)?d) What is the frequency of the generated voltage?
Illustrative Problem 9
§ Solution:
Illustrative Problem 9
a) The RMS value is given by:
22
sin cC
A
NE
For full-pitch coil, ρc = 180o e. Also NC =15, φ = 2rlBM,
VE
E
A
A
1202
2180
sin)60/23600)(2.03.05.0(15
§ Solution:
Illustrative Problem 9
b)
VE
E
A
A
69.1172
2)8/7(180
sin)60/23600)(2.03.05.0(15
mechP
oc 75.78
4
2)8/7(180
2
c)
The Coil Distribution Factor
The windings associated with each phase of the a generator are not concentrated in a single pair of slots on the stator surface.
Each coil (which consists of several turns) are distributed among several adjacent pair of slots.
The spacing in degrees between adjacent slots on a stator is called the pitch factor γ , expressed in either mechanical or electrical degrees.
The Coil Distribution Factor
NS
a1
a2
c4
c3
c2
c1a’3
a’4a’2
a’1b3
b4
b1 b2
a3
a4 b’3
b’4
b’2b’1
c’3
c’4
c’2
c’1
Phase belt orPhase group
A simple double-layer full-pitch distributed winding for a two-pole AC generator
All coil sides of a given phase are placed in adjacent slots. These coils sides are know as phase belt or phase group.
Note that phase “a” has 4 coils and they occupy 2 slots per phase belt.
Pitch of coil b1
The Coil Distribution Factor
A double-layer fractional-pitch AC winding for a 2-pole generator
a1
a4
b’1
c3
c4
c1c2
a’4a’2
a’3b3
a’1
b1 b4
a3
c’1a2
b’4
b’2b’3
b2
c’4
c’2
c’3
Phase belt orPhase group
NS
Pitch of coil b1
The Coil Distribution Factor
Coil Distribution Factor (kd):
When a phase winding is distributed over several adjacent slots, the actual induced voltage Vφd of that winding is less than the induced voltage Vφnd if the windings were concentrated (non-distributed) on a pair of slots.
The ratio of Vφd and Vφnd is called the distribution factor
nd
dd V
Vk
For a winding occupying n slots per phase belt spaced γ degrees apart, the distribution factor is given by,
)2/sin(
)2/sin(
n
nkd
The Coil Distribution Factor
The Induced Voltage Including Distribution Effects:
The induced voltage including distribution effects is given by,
tkfkNe
tkkNe
dpCind
dpCind
cos2
cos
Hence, the RMS voltage per phase is given by,
fkNfkkNkfkN
E wCdpCdpC
A
222
2
where kw = kp kd is called the winding factor.
A simple two-pole, 3-phase, Y-connected synchronous machine stator is used to make a generator. It has a double-layer coil construction, with four stator coils per phase distributed as shown below.
Illustrative Problem 10
N
S
a1
a4
b’1
c3
c4
c1c2
a’4a’2
a’3b3
a’1
b1 b4
a3
c’1a2
b’4
b’2b’3
b2
c’4
c’2
c’3
Phase belt orPhase group
Coil pitchfactor
Each coil has 10 turns. The winding have an electrical pitch of 150 degrees. The rotor (and magnetic field) is rotating at 3000 rpm and the flux per pole in this machine is 0.019 Wb.a) What is the slot pitch of this
stator in mechanical degrees?
b) How many slots do the coils of this stator span?
c) What is the magnitude of the phase voltage of one phase and the terminal voltage?
Solution:
Illustrative Problem 10
a) oo
3012
360 (both electrical and mechanical)
b) 5180
150
2
12
o
o
spancoil
c) 2n
9659.0)2/30sin(2
)2/302sin(
o
o
dk
9659.02
150sin
o
pk
VE
E
A
A
157
)50)(019.0)(9659.0)(9659.0)(10)(4(2
END OF SESSION