Brushless Motor Fundamentals

1
Brushless Motor Fundamentals
4.34.7. Series and Parallel Connections 2?22? ???
2
4.3 Multiple Phases

• ????? ?? ???? ??? ????? ???? ? ??? ???.
• 180ºE ??? back EMF ? ??? ?? ??. ? ???? ??? ???
  ?? ? ??.
• ??? ?? ???? ????? ???, ???? ???? ?? ?? ???? ???
  ??? ?? ???.
• ?? ???? ???? ??? ??? ????? ??? ???? ?? ?? ????.
• ???? ??? ??? ??? ??.
• ???? ?? ?? ?? ??? ???. ?? ??? ??? ????? ????
  ???? ?? ?? ????.

3
4.3 Multiple Phases

• the zero crossings of the back EMF and torque ?
  (360ºE)/3 ? 120ºE or 60ºM
• Phase A, phase B, and phase C ?? ?? ??? ???.
• Phase A and B ? 120ºE ??? ???? ???. ????? phase
  A and C? -120ºE ?? ???? ???.

Fig 1. A three phase motor
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4.3 Multiple Phases
Fig 2. Back EMF and torque waveform for a three
phase motor
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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
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4.4 Design Variations
The flux linkage increases from the minimum to
zero
Fig 4. Motor with rotor at 60ºE
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4.4 Design Variations

• The flux linkage reaches a maximum
• The flux linkage remains at the maximum until ?e
  180º
• It starts decreasing through zero to the
  minimum again

Fig 5. Motor with rotor at 120ºE
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4.4 Design Variations
Fig. 5
Fig. 4
Fig. 3
Fig 6. Flux linkage and back EMF for the
fractional pitch coil case
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4.4 Design Variations
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
Considering the flux in the air gap over the
magnet surface only. No flux crosses the air gap
over the gaps between the magnets
Fig 7. A motor having fractional pitch magnets
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4.4 Design Variations
Fig 8. Flux linkage and back EMF for the
fractional pitch magnet case
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4.4 Design Variations

• 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

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4.4 Design Variations
Ns 15 slots, Nm 4 magnet poles, And Nph 3
phases ? Nspp 1.25 and the angular slot pitch
?s 360ºM/Ns 24ºM or 48ºE
Fig 9. A fractional slot motor
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4.4 Design Variations
There are 3 phases, 15 slots, and each coil fills
a net one slot ? Ncph Ns/Nph
15/3 5
Fig 10. Phase A winding for a 4 poles, 15 slot
motor
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4.4 Design Variations
Offset ?ab448ºE192ºE ?ac748ºE336ºE ?ad848º
E384ºE ?ae1148ºE528ºE Coil Cb and Ce are
woung in the opposite direction
Fig 11. Flux linkage and back EMF of coil Ca
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4.4 Design Variations
Fig 12. Sum of coil back EMFs to get net winding
back EMF
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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.

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4.5 Coil Resistance
In general, material resistivity is a function of
temperature.
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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
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4.5 Coil Resistance
Several standards exist for classifying wire
according to diameter. The most common standard
is American Wire Gage(AWG)
As the gage increases, the diameter decreases.
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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
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4.5 Coil Resistance

• The above figure plots resistance relative to a
  wire having any gage G to wires having gages G,
  G1, etc.
• A wire of gage G3 has twice the resistance of a
  wire of gage G. So two wires of gage G3 taken in
  parallel have the same resistance as on wire of
  gage G
• At G1, 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.

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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

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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
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4.6 Coil Inductance
Rg the air gap reluctance over one pole
pitch Rm the magnet reluctance Ni The MMF
source associated with each coil
Fig 15. Magnetic circuit model for the
computation of air gap inductance
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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
Fr 0
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4.6 Coil Inductance
The net flux linked by all four coils
?
The air gap inductance is proportional to the
rotor surface area
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4.6 Coil Inductance
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

Fig 16. Slot leakage flux
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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.
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 xds, the
  field intensity reaches its maximum value equal
  of HyNi/wsb.

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4.6 Coil Inductance
? Coil areas
Where, Lst the axial length of the slot
Matching this expression to the fundamental
relationship LcaN2P
? P the effective permeance of the slot
Including the shoe area ? The field intensity is
constant over the shoe area
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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
When rgtRc
Fig 17. Magnetic field about a cylindrical
conductor
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4.6 Coil Inductance
Fig 17. End turn geometry approximation
Where, tcp the mean coil pitch r
tcp As cross-sectional area
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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
The net phase winding inductance with all coils
connected in series is the sum of the three
fundamental components,
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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.

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4.7 Series and Parallel Connections
Fig 18. Two coils connected in parallel
The two inductances and two resistances add, and
the two back EMF sources subtract
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4.7 Series and Parallel Connections

• 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

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4.7 Series and Parallel Connections
Fig 19. Coil resistances in series and in parallel