brushless motor fundamentals

Brushless Motor Fundamentals

4.3~4.7. Series and Parallel Connections

2 월 22 일 곽규열

4.3 Multiple Phases

• 단상모터는 모든 지점에서 토크를 발생시키지 못하므로 잘 쓰이지 않는다 .

• 180ºE 에서의 back EMF 와 토크는 영이 된다 . 이 지점에서 토크는 토크를 만들 수 없다 .

• 모터가 위의 지점에서 정지상태가 된다면 , 외부에서 물리적인 힘을 주지 않는다면 로터는 회전을 하지 않는다 .

• 위의 문제점을 제거하고 일정한 토크를 발생시키기 위해서 단상보다 많은 상을 이용한다 .

• 대부분의 모터는 삼상을 가지고 있다 .

• 삼상보다 많은 상은 흔한 경우가 아니다 . 높은 출력을 요하는 출력장치의 경우에만 삼상보다 많은 상을 사용한다 .

4.3 Multiple Phases

Fig 1. A three phase motor

• the zero crossings of the back EMF and torque → (360ºE)/3 → 120ºE or 60ºM

• Phase A, phase B, and phase C 모두같은파형을가진다 .

• Phase A and B 는 120ºE 만큼의오프셋을가진다 .

마찬가지로 phase A and C는 -120ºE 만큼오프셋을가진다 .

4.3 Multiple Phases

Fig 2. Back EMF and torque waveform for a three phase motor

4.4 Design Variations

Fractional Pitch Coils

→ When the coil pitch differs from 180ºE, the winding is called a fractional pitch winding.

Fig 3. Motor having on fractional pitch coil

/c c p

coil pitch factor


Fig 4. Motor with rotor at 60ºE

The flux linkage increases from the minimum to zero


Fig 5. Motor with rotor at 120ºE

• The flux linkage reaches a maximum

• The flux linkage remains at the maximum until θe = 180º

• It starts decreasing through zero to the minimum again


Fig 6. Flux linkage and back EMF for the fractional pitch coil case

Fig. 5

Fig. 4

Fig. 3


Fractional Pitch Magnets

→ Magnets seldom span a full pole pitch of 180ºE because flux at the transitions between North and South poles not contribute to torque

Fig 7. A motor having fractional pitch magnets

ro m stA R LConsidering the flux in the air gap over the magnet surface only.

No flux crosses the air gap over the gaps between the magnets


Fig 8. Flux linkage and back EMF for the fractional pitch magnet case


Fractional Slot Motor

• Nspp = Ns/Nm/Nph slot per pole per phase

• When Nspp = an integer, the motor is said be an integral slot motor

• When Nspp ≠ an integer, the motor is said be a fractional slot motor

• There is a distinction between a motor having fractional windings and a fractional slot motor. The first characterizes the windings; the second characterizes the slots that contain windings


Fig 9. A fractional slot motor

Ns = 15 slots,

Nm = 4 magnet poles,

And Nph = 3 phases

→ Nspp = 1.25 and

the angular slot pitch

θs = 360ºM/Ns = 24ºM or 48ºE


Fig 10. Phase A winding for a 4 poles, 15 slot motor

There are 3 phases, 15 slots, and each coil fills a net one slot

→ Ncph = Ns/Nph

= 15/3 =5


Fig 11. Flux linkage and back EMF of coil Ca

Offset

θab=4·48ºE=192ºE

θac=7·48ºE=336ºE

θad=8·48ºE=384ºE

θae=11·48ºE=528ºE

Coil Cb and Ce are woung in the opposite direction


Fig 12. Sum of coil back EMFs to get net winding back EMF

0

0

( ) ( ) ( 192 ) ( 336 )

( 384 ) ( 528 )

( ) ( 12 ) ( 24 )

( 24 ) ( 12 )

a a a

a a

a a a

a a

e e e e

e e

e e e

e e

4.5 Coil Resistance

• Multiple coils connected together to form phases are a basic part of all motors.

• Coils have two electrical properties, namely resistance and inductance

• Resistance is a property of all materials. It represents a measure of how much the material the flow of current.

( )

lR

Awhere the resistivity of the material m

l the material length along the direction of current flow

4.5 Coil Resistance

In general, material resistivity is a function of temperature.

0 0

0

( ) ( )[1 ( )]T T T T

where T temperature

T a base temperture

the thermal resistivity coefficient

4.5 Coil Resistance

Coils in motors are most commonly composed of multiple turns of round insulated wire

• The centermost circle is the bare conductor

• The next outer layer is the wire insulation, which is commonly available in three thickness, single, double, and triple

• The final layer is an optional layer of bonding material

Fig 13. Wire cross section

4.5 Coil Resistance

Several standards exist for classifying wire according to diameter. The most common standard is American Wire Gage(AWG)

8.24865(0.890526)

int

( )

Gwb

wb

d

where G the eger wire gage

d the bare wire diameter mm

As the gage increases, the diameter decreases.

log8.24865

log 0.890526

wbd

G

4.5 Coil Resistance

Because AWG is based on a geometric progression, wire gages are related to each other by ratios

Fig 14. Relative wire resistance versus wire gage

4.5 Coil Resistance

• The above figure plots resistance relative to a wire having any gage G to wires having gages G, G+1, etc.

• A wire of gage G+3 has twice the resistance of a wire of gage G. So two wires of gage G+3 taken in parallel have the same resistance as on wire of gage G

• At G+1, resistance is approximately 26% greater than that at G. So increasing the wire gage by one, increases resistance and I2R losses by 26% provided current remains constant.

• At G-1, resistance decreases approximately to about 79% of that at G. So, decreasing the wire gage by one, decrease the I2R losses for fixed current to 79% of what they are at G.

4.6 Coil Inductance

• Inductance is not usually a critically parameter in brushless permanent magnet motors.

• Inductance determines the time constant of the windings.

• When a coil is placed in stator slots, its inductance changes dramatically compared to its inductance when surrounded by air.

•Mutual inductance exists between the coils in a given phase as well as between the coils in different phase.

• Mutual inductance between coils in a given phase is considered here, but mutual inductance between coils in different phases is not. Because mutual inductance between phases is small relative to self inductance

2 21 1

2 2VLi H dV

4.6 Coil Inductance

When coils are placed in slots, the coil inductance has three distinct components due to the three distinct area where significant magnetic field is created by coil current.

→ The air gap, the slots, and the end turns

Air Gap Inductance

→ The air gap inductance component is due to the flux crossing the air gap

→ Consider the magnetic circuit model that the MMF is produced by the stator coils and ignore the flux source in the magnet model

4.6 Coil Inductance

Fig 15. Magnetic circuit model for the computation of air gap inductance

Rg = the air gap reluctance over one pole pitch

Rm = the magnet reluctance

Ni = The MMF source associated with each coil

4.6 Coil Inductance

Letting the outer ring be the reference node, identifying Fr as the center node, and setting the sum of the fluxes leaving the center node to zero

1 2 3 4 0

0r r r r

g m g m g m g m

F Ni Ni F F Ni Ni F

R R R R R R R R

Fr = 0

1 1 1 1g m

Ni

R R

4.6 Coil Inductance

The net flux linked by all four coils

2

1 2 3 4( ) m

g m

N N iN

R R

2m

gg m

N NL

i R R

202 st rog

m

R

L RL N

lg

C

(2 / )

g st p ro

m st ro

A L R

N L R

◄

The air gap inductance is proportional to the rotor surface area

4.6 Coil Inductance

Fig 16. Slot leakage flux

Slot Leakage Inductance

→ Coil current produces a magnetic field that crosses from one side of a slot of a slot to the other.

• This figure depicts a slot in a linear motor.

• This figure includes narrow slot opening between shoes that taper back to the stator teeth

4.6 Coil Inductance

The inductance component that results from the magnetic field that crosses the slot in the y-direction is commonly called the slot leakage inductance.

(2 )( )y

sb s

N i xH x

w d

Where, the slot is assumed to contain two coil sides each having N turns

• The field intensity Hy is zero at the slot bottom because no current is enclosed.

• As x increases, more current is enclosed.

• When all the current is enclosed at x=ds, the field intensity reaches its maximum value equal of Hy=Ni/wsb.

4.6 Coil Inductance

20 (2 )3

s stca

sb

d LL N

w

Where, Lst = the axial length of the slot

Matching this expression to the fundamental relationship Lca=N2P

0

3s st

sb

d LP

w

← Coil areas

← P = the effective permeance of the slot

Including the shoe area → The field intensity is constant over the shoe area

2 0 0 01 (2 )

3 ( ) / 2s st t st sh st

ssb so sb so

d L d L d LL N

w w w w

4.6 Coil Inductance

End Turn Inductance

→ The end turn inductance is created by the magnetic field that surrounds a coil after it leaves one slot and before it enters another slot

Fig 17. Magnetic field about a cylindrical conductor

2

iH

r

20 ln2e

c

l N rL

R

When r>Rc

4.6 Coil Inductance

Fig 17. End turn geometry approximation

20

1 ln4 2cp cp

e

s

NL

A

Where, τcp = the mean coil pitch

r = τcp

As = cross-sectional area

4.6 Coil Inductance

Since there are 2Nm end turn bundles per phase winding and there is no mutual coupling between the end turns of other coils in the same phase, the total end turn inductance per phase is

20 ln2 2

m cp cpe

s

N NL

A

The net phase winding inductance with all coils connected in series is the sum of the three fundamental components,

ph g s eL L L L

4.7 Series and Parallel Connections

• In the preceding sections, back EMF, resistance and inductance were analyzed under the assumption that all coils in a phase are connected in series

• The connection in series is the majority of motor designs

• When all coils are connected in series, the phase back EMF is simply the sum of the individual coil back EMFs.

• When coils are connected in parallel, the coil back EMFs can create circulating currents that contribute to I2R losses but do not provide beneficial torque production.


Fig 18. Two coils connected in parallel

The two inductances and two resistances add, and the two back EMF sources subtract


• The combined back EMF e1-e2 is equal to zero and no current ic circulate around the loop from one coil to the other

• If the individual coil back EMFs are not instantaneously identical, the combined back EMF e1-e2 is nonzero and current ic circulates around the loop independent of any current applied to the parallel coils during motor design

• To avoid circulating currents and their associated loss, only coils having identical back EMFs can be connected in parallel

• For most motor designs this is not possible and therefore parallel-connected coils do not appear often in practice

• In the unusual case when the number of turns cannot be decreased to lower the back EMF amplitude, parallel coil connections must be accepted


Fig 19. Coil resistances in series and in parallel

2

/ 1eqp

eqs

R R n

R nR n

brushless motor fundamentals

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