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