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a THESIS

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LIDQARY
Michigan State
University

 

 

 

This is to certify that the
thesis entitled

PRELIMINARY DESIGN CONSIDERATIONS FOR

INTEGRATING A COMPOSITE IMPELLER IN A PERMANENT

M.S.

 

MAGNET BRUSH LESS MOTOR

presented by

Mark Hubert Mouland

has been accepted towards fulfillment
of the requirements for the

degree in Mechanical Engineering

 

M044

Major Professor’s Signature
(Mg/Wu 2
I

Date

MSU is an affirmative-action, equal-opportunity employer

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DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/07 p:lClRC/DateDue.indd-p.1

 

PRELIMINARY DESIGN CONSIDERATIONS FOR
INTEGRATING A COMPOSITE IMPELLER IN A PERMANENT
MAGNET BRUSHLESS MOTOR
By

Mark Hubert Mouland

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirement
for the degree of
MASTER OF SCIENCE
MECHANICAL ENGINEERING

2007

ABSTRACT
PRELIMINARY DESIGN CONSIDERATIONS FOR
INTEGRATING A COMPOSITE IMPELLER IN A PERMANENT
MAGNET BRUSHLESS MOTOR
By
MARK HUBERT MOULAND

Utilizing water as a refrigerant in an air conditioning device is a complex problem
that requires various innovative ideas in order to create a suitable compressor that can
successfully obtain the extremely high compression ratios required. This compressor
requires multiple stages of impellers, which in order to withstand the high forces must be
a wound composite created on a winding machine.

The largest challenge is getting the impellers to spin at the extremely high Speed
necessary. It is proposed using an innovative design in which the entire impeller acts as
the shaft of a permanent magnet brushless motor. By placing magnets around the
circumference of the shroud, or integrating them in the shroud itself, the entire impeller
can act as one large shaft in the center of a motor device.

This thesis aims to look at the background research required to create the motor
for such a device. Included are discussions on basic refi'igeration cycles, information
regarding the composite impeller, an overview of magnetics, as well as a numerous
equations, and data regarding overall permanent magnet motor design.

All of these data are subsequently integrated together in order to propose
numerous designs for a motor solution. These designs are accompanied. by graphs and

charts to outline various trends present using multiple design parameters. The data are

then used with a weighted design matrix in order to select a preliminary design solution.

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................. iv
LIST OF FIGURES ............................................................................................................ v
NOMENCLATURE ........................................................................................................ viii
INTRODUCTION .............................................................................................................. 1
BACKGROUND ................................................................................................................ 3
Water as a Refrigerant: ................................................................................................... 3
Composite Impellers: ...................................................................................................... 9
Brushless Permanent Magnet Motors: .......................................................................... 12
Magnetics: ..................................................................................................................... 40
Frequency Drives: ......................................................................................................... 46
HIGH SPEED MOTOR PARAMETERS ........................................................................ 48
Magnetic Selection ........................................................................................................ 48
Impeller Considerations ................................................................................................ 52
General Motor and High Speed Considerations ........................................................... 58
DESIGN CALCULATIONS AND OPTIMIZATION ..................................................... 65
CONCLUSION ................................................................................................................. 82
Appendix A: Winding Diagram ....................................................................................... 86
Appendix B: Motor Plots ................................................................................................. 87
Appendix C: Motor Calculations ..................................................................................... 93
REFERENCES ................................................................................................................. 95

iii

LIST OF TABLES

Table 1. Refrigerant Comparison Chart ............................................................................. 5
Table 2. Magnet Choices ................................................................................................. 49
Table 3: Wire Gauge Sizes .............................................................................................. 60
Table 4: Design Constraints in the Spreadsheet .............................................................. 62
Table 5: Motor Final Data ................................................................................................ 63
Table 6: Stage Characteristics .......................................................................................... 64
Table 7. Optimum Design Characteristics ....................................................................... 71
Table 8. Weighted Design Matrix .................................................................................... 71
Table 9. Final Design Summary and Characteristics ....................................................... 80
Table C1. Final Design Inputs ......................................................................................... 93
Table C2. Final Design Calculations ............................................................................... 94

iv

LIST OF FIGURES

Figure 1. Ideal Refrigeration Cycle ................................................................................... 4
Figure 2. Water Chiller Examples ..................................................................................... 6
Figure 3. Schematic of Current Chiller Design ................................................................. 6
Figure 4. Schematic of Proposed Chiller Design ............................................................... 7
Figure 5. Composite Winding Machine ........................................................................... 10
Figure 6. Example of a Composite Woven Impeller ....................................................... 10
Figure 7. Trapezoidal Back EMF ..................................................................................... 12
Figure 8. Simple 1 Phase 1 Magnet Pair Motor ............................................................... 13
Figure 9. Sinusoidal Torque Characteristic ..................................................................... 14
Figure 10. Example of 2-Magnet, 3-Phase Motor ........................................................... 15
Figure 11. Permeance Coefficient Vs. Bg/Br ................................................................... 18
Figure 12. Permeance Coefficient Diagram ..................................................................... 19
Figure 13. Nomenclature for Various Dimensions. ......................................................... 22
Figure 14. Simplified 1D model. ..................................................................................... 22
Figure 15. Distributed Winding ....................................................................................... 24
Figure 16. Winding Diagram ........................................................................................... 25
Figure 17. Stator Slot and Teeth Nomenclature ............................................................... 26
Figure 18. Slot Nomenclature .......................................................................................... 27
Figure 19. Winding Diagram ........................................................................................... 30
Figure 20. Efficient Wire Packing Method ...................................................................... 32
Figure 21. Slot correction factor diagram ........................................................................ 33

Figure 22.
Figure 23.
Figure 24.
Figure 25.
Figure 26.
Figure 27.
Figure 28.
Figure 29.
Figure 30.
Figure 31.
Figure 32.
Figure 33.
Figure 34.
Figure 35.
Figure 36.
Figure 37.
Figure 38.
Figure 39.
Figure 40.
Figure 41.
Figure 42.
Figure 43.

Figure 44.

Tooth Head Diagram ...................................................................................... 35

Winding Length Diagram ............................................................................... 36
Hysteresis Loop .............................................................................................. 4O
Flux Diagram .................................................................................................. 41
Halbach magnetized 4 magnet rotor, versus a parallel_magnetized rotor ........ 43
Magnetic flux from a Halbach array. ............................................................. 44
Two segments per pole Halbach machine, and three segments _per pole ....... 45
Half Bridge Drive ........................................................................................... 46
Full H-Bridge Drive ....................................................................................... 47
Total rotor losses for slotless stator for different retaining sleeves ................. 52
Micro magnet fabrication on a substrate ......................................................... 54
Hysteresis loop of the micro bonded magnet. ................................................. 55
Mold Alignment and Curing ........................................................................... 56
Flux density variation for Halbach array with an air cored rotor. ................... 59
Flux density variation for Halbach array with an iron cored rotor ................. 59
Dynamic Motor Diagram ................................................................................ 63
Efficiency Plot ................................................................................................. 65
Efficiency Vs. Wire Diameter ......................................................................... 66
Motor Constant vs. Permeance Coefficient, Ferrite Magnet ........................... 67
Efficiency vs. Permeance Coefficient, Ferrite Magnet ................................... 68
Mass vs. Permeance Coefficient, Ferrite Magnet ............................................ 68
Windings Needed for a Ferrite 6 Magnet Motor ............................................. 69
Fill Factor vs Permeance Coefficient .............................................................. 70

vi

 

Figure 45. Efficiency vs Rotor Radius for a Ferrite Magnet ............................................ 73
Figure 46. Efficiency vs Rotor Radius for a Magnetic Resin Array ................................. 73
Figure 47. Mass vs Rotor Radius for a Ferrite Magnet ..................................................... 74
Figure 48. Mass vs Rotor Radius for a Magnetic Resin Array ......................................... 74
Figure 49. Mass vs Rotor Radius for a F ern'te Magnet ..................................................... 75
Figure 50. Efficiency vs Rotor Radius for a Ferrite .......................................................... 76
Figure 51. Mass vs Current for a Magnetic Resin and Ferrite .......................................... 77
Figure 52. Efficiency vs Current for a Magnetic Resin and Ferrite .................................. 78
Figure 53. Final Motor Values .......................................................................................... 79
Figure 54. Procedure Flow Chart ...................................................................................... 83
Figure Al. Winding Chart for 3 phase 6 magnet motor ................................................... 86
Figure B]. Mass vs Rotor Radius for a Ferrite Magnet, 6 Magnets ................................. 87
Figure B2. Motor Constant vs Rotor Radius for a Ferrite Magnet, 6 Magnets ................ 87
Figure B3. Efficiency vs Rotor Radius for a Ferrite Magnet, 6 Magnets ......................... 88
Figure B4. Mass vs Rotor Radius for a Ferrite Magnet, 4 magnets .................................. 88
Figure B5. Motor Constant vs Rotor Radius for a Ferrite Magnet, 4 Magnets ................ 89
Figure B6. Efficiency vs Rotor Radius for a Ferrite Magnet, 4 Magnets ......................... 89
Figure B7. Mass vs Rotor Radius for a Magnetic Resin Array, 6 Magnets ..................... 90
Figure B8. Motor Constant vs Rotor Radius for a Magnetic Resin, 6 Magnets ............... 90
Figure B9. Efficiency vs Rotor Radius for 3 Magnetic Resin, 6 Magnets ........................ 91
Figure B10. Mass vs Rotor Radius for a Magnetic Resin Array, 4 Magnets ................... 91
Figure Bl 1. Motor Constant vs Rotor Radius for a Magnetic Resin, 4 Magnets ............. 92
Figure B12. Efficiency vs Rotor Radius for a Magnetic Resin, 4 Magnets ...................... 92

vii

fl:
fm

gslot

NOMENCLATURE

Area of the Stator

Area of the Windable Slot

Air-gap Flux Density

Residual Induction

Stator Yoke Flux Density

Tooth F lux Density

Flux Concentration Factor
Tangential Component of Fluid Flow
Diameter of the Wire

Windable Slot Depth

Windable Slot Depth inner Diameter
Windable Slot Depth outer Diameter
Shoe Tip Depth

Shoe Taper Depth

back emf

Shaft Energy

Electrical Frequency

Mechanical Frequency

Air-gap length

Air-gap length in the Slot Opening

Magnetic Field Intensity

viii

current

Leakage Factor

Motor Constant

Reluctance Factor

Slot Correction Factor

Lamination Stacking Factor

Bare Wire Slot Fill Factor

Length of the Magnet (radial direction)

Length of the stator (motor axial length)

Mmagne, Total mass of the magnets in the motor

Mstator

.2

.2
a

Total mass of the stator

Total mass of the wires, magnets, and stator
Total mass of the wire windings for all phases
Number of wire turns

Number of Magnet Poles

Number of Stator Teeth

Nominal Coil Span

Power

Permeance Coeflicient

Eddy Current Losses

Power losses in slot wiring (12R losses)
Power required to spin impeller

Radius to outer magnet edge

ix

9:

fl,

,0
Pmagner
Psraror
Pwrre
(P:
45mm

00m

Radius to rotor outer edge

Slot Resistance

Torque

Tangential Velocity of Impeller
Width of the slot at the bottom
Width of the slot at the top
Bottom Slot Width

Slot Opening Width

Width of the Stator Yoke

Stator Tooth Width

Mechanical Position
Electrical Position
Angular Tooth width at air I-gap
Angular Slot Pitch
Recoil Permeability
Resistivity of the wire
Magnet Material Density
Stator Material Density
Wire Material Density
Tooth Flux

Total Stator F lux

Angular Speed

INTRODUCTION

The goal of this thesis is to design a motor drive system for a compressor in a
water as a refiigerant air conditioning unit. Instead of having an impeller in the
compressor driven on a shaft, the instead goal is to make the entire impeller the “shaft” of
an electric motor. This means that magnets are on the circumference of the impeller and
are driven by a stator, which surrounds the impeller.

The design of a brushless permanent magnet motor to drive this system is a
complex process that integrates numerous disciplines in order to obtain a suitable design.
Overall, this thesis aims to be an introductory paper into this design methodology for
creating a motor, as well as calculating and choosing possible solutions. This design can
be divided into a few major challenges: (1) Determining the magnetic materials to be
used in the impeller and their integration into the impeller, (2) Designing a motor for the
selected magnetic material that achieves the desired combination of torque and rotational
speed, and (3) Comparing numerous design parameters and seeing how they affect the
overall efficiency, mass and motor constant.

A simple background in some engineering aspects that are necessary to fully
understand the problem will first be discussed. These include a basic look at the
refrigeration cycle and the challenges behind water as a coolant. A few basics about
magnets are also covered. This includes choices of magnet to be used in the design and
also different arrangement and array types for the magnets. This includes Halbach arrays
as well as their advantages and disadvantages. Next, the composite impeller is

introduced, explaining how it is created. This particular discussion includes how the

outer shroud of the impeller can be potentially magnetized as an alternative to the
traditional method of attaching them onto the rotor.

Thereafter, relevant basic equations regarding the design of electric motors are
presented. This covers simple winding theory, as well as pole design criteria, which
includes determining the back electromotive force (emf) and air-gap flux necessary to
obtain a given rotational speed. These equations are used with numerous simplifying
assumptions to provide a simple design procedure that is followed in an excel
spreadsheet.

Lastly, this thesis combines the above backgrounds, into preliminary design
considerations for an electric motor that can drive the composite impeller. This is
accomplished using data previously regarding the compressor specifics. The values for
impeller sizes, power, and speed are all obtained through previous work. Thus, this thesis
only focuses on the design of the innovative motor drive system. The goal is to use all of
the information acquired in research to obtain a functional design that address all of the
problems and constraints the compressor will face. Upon completion of this
optimization, the final motor design for each of the compressor stages will be presented,
as well as a general methodology to be used. These calculations will be supported with
numerous charts and graphs that Show key trends to be utilized in the future.

Although this paper’s key considerations are for the applications relating to water
as a coolant, it should be noted that a magnetically driven impeller has wide range
applications in the design of ducted fans and even in turbines and propulsion devices

suitable for use in unmanned, aerial vehicles.

BACKGROUND

Water as a Refrigerant:

Refrigerants have been a major environmental issue, most particularly in the past
decade. Although the magnitude of the problem was unknown before, it is now widely
known that many refiigerants actively contribute to the depletion of the ozone layer, as
well as help promote global warming.

Older refrigerants such as R-12 had high ozone depletion potentials (ODP) and
also high Global Warming Potentials (GWP). These have been replaced by safer
alternatives, such as R134a (zero ODP) and R22. However, both of these still promote
global warming.

Thus, it is important that alternative refrigerants are created. Water is one of the
best choices as a refrigerant that could replace both of these.

Aside from being much safer for the environment, water has been shown to even
save up to 30% on energy savings. [9] Thus, with the current energy crisis in the United
States, water would lessen our dependence on foreign oil. In California alone, if 5% of
all air conditioner units were water based, this would yield energy savings of 405 MW a
year, out of 27,000 MW used for Air Conditioning. [2]

A basic ideal refrigeration cycle is fairly simple to understand and consists of four
major parts. The compressor is a unit that adds enthalpy to the fluid flow and is the
driving force behind the cycle by increasing the pressure and temperature of the coolant.
Next comes the condenser, which takes superheated vapor from the outlet of the

compressor and condenses it in a constant pressure process to a saturated liquid. This is

the process that rejects heat into the surroundings. The expansion valves expand this
saturated liquid, causing a very quick drop in pressure at constant entropy. Finally, the
evaporator is what takes heat in from the surroundings and creates the cooling effect.

The following diagram shows these 4 components.

 
  
  
 
   
 

 

Expansion
Valve

Compressor

Pressure

Evaporator

Enthalpy

Figure 1. Ideal Refrigeration Cycle

The low evaporator pressure means that the mass flow rate through the
compressor will be very small, and thus low power (around 250 Watts), is required to run

the machine. The values for numerous refrigerants can be plotted and compared.

Table l. Refrigerant Comparison Chart [9]

 

Ref. Cap. Comp. out.
Ref ODP GWP (kj kg“ ') COP rt temp. ("C)
R718 0“ 0+ 2309 5. 70 10.0 223
R71 7 0" 0* 102.9 5. 90 3-03 99
R12 1: 51500: 106.5 5.70 2.33 55
R22 0034‘ 1900": 145.6 5.60 2.3.5 07
R290 0‘ 20: 250.1 5.42 2.66 54
R134a 0? 1600* 131.9 5.54 3.18 62
R1 52a 0‘ 190: 2255.0 5.88 3.15 52

From this table, it is important to note a few characteristics. First, as stated
earlier, the ODP and GWP of water is 0, which again means it has no harmful effects on
the environment. COP represents the coefficient of performance, which in essence is

how effective the refiigeration device is, the COP is simply calculated using

CoolingEflect
CompressorWork

COP =

 

(1)

From figure 1, the COP is simply found by looking at the enthalpy change of the
evaporator (the x — axis value) and the enthalpy change of the compressor. The chart
shows that the COP of water is equal to or comparable to other common refiigerants.
Also, the refrigeration capacity of water is much higher than its counterparts. Perhaps the
most important data to pull from this table is the symbol it which represents the pressure
ratio needed. Water requires a relatively high pressure ratio by the compressor and this is
the most important challenge to overcome. This is why special care is taken in designing
a drive system that can run a compressor to obtain such high ratios. Currently, there are a
few commercial units being used worldwide that use water as the coolant and one can be

seen below.

 

Figure 2. Water Chiller Examples

The primary drawback to these units, however, is their very large size. The high
pressure ratios are only obtained using large radial compressors with diameters in the
region of 1 meter. [9] The schematic of the above design can be seen below. In the
diagram, the 2 blades represent 2 radial compressors, and the evaporator (what generates

the cooling effect) is seen on the right.

a l ’1'— “Com rector l
a “”1124: {kind tag: ‘1:
fo'rdpréer I,\ 1’

A s I "

     
     

  

      
  

/ ,_
Compressor
. 1st S age
Evaporato
.5°C

 
     

I

    
   

 

Coollng Water Chilled Water
Cycle - Cycle

Figure 3. Schematic of Current Chiller Design

This large size means this device only has commercial and large scale
applications. With most homes using AC during the summer months, this represents a
huge untapped market. It would be advantageous to take these large devices and try to
scale them down. This would allow water as a refrigerant to go from primarily
commercial and large scale uses to a device that can comfortably sit in someone’s home
or backyard.

Two routes are generally proposed to create a small, home device. The first route,
which is the topic of this paper, is to use a series of axial compressors to achieve such
high pressure ratios. The second route uses a condensing wave rotor to aide in the
compression. A schematic of the former case with the axial compressor stages can be

seen below.

 

 

Figure 4. Schematic of Proposed Chiller Design

The primary feature of this design is the series of four axial, compressor stages

arranged in series. These blades will be spinning at extremely high speeds, and this

thesis aims to devise a drive system to rotate these blades at high velocity. This can be
accomplished in a novel approach that integrates the impellers into an electric motor,
using the fact that the compressor blades are all composite and easily wound on a
machine.

The alternative approach to using axial, compressor stages is implementing a
wave rotor in the design. This has the benefit of helping to compress the fluid more,
without the need of having numerous axial fans arranged in a series. The biggest
drawbacks with a wave rotor are the precise complexities needed to setup such a device.

In the next section, we will briefly discuss the background behind the composite

wound impellers used in the compressor.

Composite Impellers:

Water has many challenges regarding its use in a compact design. These include
high, compressor, outlet temperatures. The solution is the design of a high-speed,
multistage, counter-rotating, axial compressor. AS opposed to very large rotors, (the
large size means the tip speed will be very high at low RPMS), this design uses a small
radius, spinning at extremely high speeds. An analogy that helps explain this is Euler’s

Equation.

’5 = uZCuZ _ ulcul (2)

In the equation above, u; is the tangential velocity at the impeller tip (outlet), 3" is
the shaft work, and cu; is the tangential component of the fluid flow velocity. As the
radius of the impeller increases, the tip speed increases, which increases the shaft work.

Because of these high rotational speeds, (on the magnitude of thousands of rpm),
the impeller itself must be of extremely high quality, so that the forces imposed on the
impeller don’t cause it to tear itself apart. A composite impeller solves this by being both
strong and lightweight.

The considered impeller can be generated on a commercial winding machine and
have a radius of around 6cm. It can be made of several continuous fibers that are wet
with polymer matrix material during the winding process and harden together to form a
single piece. The winding machine is a Simple device that takes carbon strands and

winds them around objects, much like thread winding on a spindle. By precisely

calibrating and programming this machine, it can be made to perform a winding pattern
that creates an impeller.
An example of this winding machine can be seen below. These machines are

commercially available through numerous companies.

 

WSH - Super Hornet Winder

  

 

 

 

Figure 5. Composite Winding Machine

The biggest advantage of this winding machine is that it can make prototypes
rapidly and quickly. As opposed to prototyping in a machine shop, modifications to the

design can be made fast. An example of a woven impeller can be seen in fig. 6.

 

Figure 6. Example of a Composite Woven Impeller

10

Another aspect of this design is the resin bath in which the carbon fibers are
dipped. After the impeller is wound, the resin can be left to set and harden. It is
proposed that perhaps various materials, (such as powdered magnets), could be mixed in
with the resin bath. This means that when the composite sets and hardens, magnetic
materials will already be integrated in the design. By merely magnetizing this layer, the
entire impeller could become a weak magnet. Also, instead of integrating magnetic
material in the resin, magnets could be wound in with the fibers during the winding

process.

11

Brushless Permanent Magnet Motors:

Brushless permanent magnet electric motors function on the basic principles that
a current flowing through wound wires in a stator will generate an induced magnetic field
that interacts with permanent magnets attached to a rotor. The stator is the part of the
motor that houses and surrounds the rotor and also uses the windings to create the
electromagnetic field. The rotor is the part of the motor that rotates and generally has
permanent magnets mounted on its surface. The rotor is most commonly a shaft of some
sort, used to drive some device. The most important component of the motor, therefore,
is the torque that it generates at a certain rotational speed. Generally, for motor design,
high torque is the ideal output. For our purposes, however, it is known that the impeller
blades require very low power (250 W).

The motor design considered here is a direct current (DC) brushless permanent
magnet motor. DC in this case means that the motor will have a trapezoidal back
electromotive force (instead of a sinusoidal current in a pure alternating current AC

motor). An example of the back electromotive force used in our application can be seen

 

 

below
Back EMF
9 Current
3
E
13 0.

 

 

 

 

 

 

 

 

Figure 7. Trapezoidal Back EMF

12

At its simplest, a motor can consist of a single 2-pole (north and south) magnet as
a rotor, surrounded by 2 stator slots with windings. Each pair of windings makes up a
motor phase. When current is applied to the windings, a magnetic field is created
through the windings and thus interacts with the magnet poles at the rotor in the center.

By reversing the current, the polarization of the windings reverses.

 

Figure 8. Simple 1 Phase 1 Magnet Pair Motor [4]

In the above diagram, the 2 magnet poles in the middle represent the “rotor”, and
the circular outer shroud represents the stationary “stator”. Using this simple diagram,
we can generate a basic torque characteristic of the motor. Intuitively, since the rotor
magnets are closest to the magnetic field of the windings, they will receive the lowest
force because the force is effectively locking the magnet in place. Conversely, as the

magnets move further away from the stator teeth, the torque acting on the rotor will

increase. This torque characteristic follows a sinusoidal relationship and can be seen

below.

-'t.:"‘2

 

nfz

 

 

.EZSZI [SIZE]

 

 

 

 

 

 

 

W '5
§_ 51

S
I!

Figure 9. Sinusoidal Torque Characteristic [4]

Although the preceding example shows only 2 magnet poles and one phase, any
combination of phases and magnet poles can be created so long as the number of rotor
magnet poles is an even number and the number of stator phases is greater or equal to 1.

An example of a 2-magnet pole, 3-phase motor can be seen below.

14

 

Figure 10. Example of 2-Magnet, 3-Phase Motor [4]

As the number of phases and poles increases, the torque characteristic on the
rotor changes as well. An important distinction to make is the mechanical position of the
rotor, and the electrical position. The mechanical position 0,, of a rotor is defined by the

number of poles facing the airgap N,,., and the electrical position 1%

N
o =—'" *6 3
E 2 m ()

So, if a motor with a 4-magnet pole (2-magnet pairs) rotor does one entire
electrical rotation, it has only clone 180 degrees of mechanical rotation. The electrical

frequency f, mechanical frequency fin are related to each other through the number of

poles.

f. = prn. (4)

Thus, for a certain mechanical rotational speed, the electrical frequency required
increases with the number of magnet poles. This is where the motor design is often
constrained by the limitations of the input device (frequency converter) that is used to
drive the motor. Therefore, fewer magnet poles are used for the design of high-speed
motors to keep the electrical frequency down. The drawback for using fewer magnet
poles, however, is that the torque production is much lower. The choice of number of
magnet poles also ties into the efficiency and overall losses of the design.

For an efficient design of such a brushless motor, it is desirable to minimize the
losses. Numerous losses and numerous variables are present. A motor constant has been

derived to aid in the optimization of a motor. [4]

 

Km = 138 Rm JwaL,,Nm A, / p (5)

With this equation, the objective is to maximize the value of the motor constant
Km, where Bg represents the airgap flux density, Rm the rotor outside radius, wa the bare
wire slot fill factor, L,, the length of the stator, Nm the number of magnet poles, A, the
area of the stator, and p the resistivity of the wires. The slot, fill factor is a measure of
how well the wires fill the slots they are wound in. Because typical wires are round, it is
impossible to have a slot with 100% of the volume filled by wires.

One of the most important parts of this equation is the air-gap flux density. This
represents the magnetic field that is present in the region between the stator and the rotor.

A high, flux density means a stronger magnetic field is present. This air-gap, flux density

16

is what is essentially driving the rotor and this is critical in the design of a motor. The

flux can be found by

g f
1+K,E5-
P

where K, is the leakage factor, C¢ is the flux concentration factor, PC is the perrneance
coefficient and ,u, is the relative permeability. [4] The remanence B,, is based on the
magnetic material chosen. The remanence can be found from tables regarding data on
various magnets, which are readily available in online resources. The flux concentration

factor is calculated by
C, = Am / A g (7)

where Am is the magnet surface area and A g is the surface area of the gap. In equation (4),
the leakage factor K; is generally less than 1 and is difficult to obtain analytically;
however, it can be determined quite easily and accurately using the finite element
method. It is usually around .9 to 1.0. [3] The reluctance factor K, represents the
resistance that the non magnetic materials have, analogous to electrical resistance in an
electric circuit. This value is generally in the range of 1.0 to 1.2. Much like the leakage
factor, the reluctance is difficult to determine. The value )1, represents the relative recoil
permeability of the material, which is the permeability of the material, relative to free
space (47r'10'7 H/m). This value is typically between 1.0 and 1.2. These values represent

quantities that are common for most electrical motors. Intuitively, the flux in the air-gap

17

Bg will be less than the residual induction of the magnet 3,. Also, the air-gap flux can’t
exceed this value, as this represents the upper bound on Bg.

The quantity PC, is the permeance coefficient, and is found by

 

P= ’" (8)

where g is the length of the gap and 1m is the magnet length. The permeance coefficient is
determined based on the geometry of the surfaces. The value for the permeance
coefficient is usually from 4 to 6 for magnet designs. This is due to the fact that this
range aims to optimize the air-gap flux. Keeping all other values constant, the following

graph shows how the permeance coefficient can affect the air- gap flux.

 

 

 

 

 

 

F— _. 5.- - E i . ___-_. - _ __. _.. .._-_.,-~-S
| 1
l
l
l 0.75 —
l
E
3, 0.5 --
l in
l
l 0.25 «
I
l
l
l
l 0 . ' ’
l o 2 4 6 a
Permeance Coefficient

 

 

Figure 11. Permeance Coefficient Vs. Bg/Br

l8

At low values of permeance coefficient the magnet is very thin and thus the air-
gap flux is small in comparison to the residual flux. As the magnet gets thicker and the
permeance coefficient increases, the effective air-gap flux increase slowly becomes
negligible. Thus, you are adding too much magnet to your design and obtaining not
much net change on the air- gap flux. This value B5./B, can more simply be called the flux
density. The larger the permeance coefficient the higher the flux density will be. For any
motor a high flux density is desirable.

For simplicity, the air-gap can be modeled as a rectangular prism. [3] The

diagram below shows the nomenclature used for calculating the permeance coefficient.

| |/__

 

ll
Am

 

 

 

 

Figure 12. Permeance Coefficient Diagram

The flux concentration factor C4) is calculated by dividing the area of the magnet
facing the gap and the area of the air-gap. For cases where the magnet is directly
touching the air-gap, the flux concentration factor is simply 1. However, it takes on a
different value if a retaining sleeve is between the magnet face and the air-gap.

These preceding equations and values can be used to get a simple value of the
magnetic flux in the airgap. With the airgap flux calculated, the back emf for the motor

eb can be determined using equation 9 and equation 10.

19

e, =2NmNB L R a) (9)

gstrom

The angular speed is com, N is the number of wire turns, and Rm is the radius to the
outside of the rotor. For a sleeveless rotor, Rm is the same as Rm. Lastly, an equation that

relates the torque generated by the motor to the back emf can be shown as

*.
ITI 2% (10)
a)

m

where i is the current flowing through the wound wires.

The torque generated by the motor can be related to power by

 

T = x (11)

This gives the maximum torque possible (no losses) with a given input power.
The air-gap flux can also be used to determine the width of the stator teeth. The primary
concern is ensuring that the flux flowing into the tooth is below the saturation level of the
ferromagnetic material. [4] Exceeding the saturation level of the ferromagnetic material
means the material is permanently altered and will have a degradation in magnetization.
The flux on one stator tooth €15, can be calculated from total flux that is acting over all the

stator teeth (Dmm/

20

B 27rR L
¢INotal = g N r0 Sf (12)

S S

 

115.:

where N, represents the number of stator teeth. This can be related to finding the teeth

width

27rRmBg
w, = —— (13)
NS KS! Bf

where K,, is the lamination stacking factor. [4]

Lamination stacking is very important. The stator of a motor is not a solid piece
of metal but rather thin sheets of metal which are pressed together. Between these sheets
thin layers of insulating material are used. This means there is a lot mechanical and
electrical resistance in the axial direction of the stator. This helps force most of the flux
in the stator towards the center of the motor and not axially through the stack.

K,, is the ratio of the area of the steel, to the total area of the tooth in
perpendicular direction. This value is generally in the range of 0.8 0.99. The width of

the stator yoke can similarly be found using

anB
w... =___,__ (14)
- NmK B

st sy

The following diagrams shall aid in further understanding the dimensional

nomenclature.

21

 

Figure 13. Nomenclature for Various Dimensions. [4]

It should be noted that the airgap flux neglects gaps due to slots in the stator. A
diagram that models the magnetic flux through the various components of the motor can

be seen in figure 14.

 

 

 

 

 

 

Flt
stator run-rent sheet V
r—‘
‘ x
airgap IV
sleeve 1]]
magnet 5::I1::3:223:31:Sigifljfii:Iii:1:312:22:::::::I:::::::::Ii: II
shalt -:-:-:-:3:3:.:3:-:-:-:-:-:-:-:1:3:~:.:- I

 

 

 

Figure 14. Simplified 1D model. [6]

22

This diagram shows the various components of a flux calculation and that all
aspects of the motor can have an affect on the air-gap flux. This is the crucial coupling
for the energy transmission from the stator to the rotor. For the calculations, we will be
neglecting the shaft and will have a Sleeveless motor. Also, a lot of internal resistances
are accounted for in the various constants.

The next major aspect of motor design is determining the winding of the coils in
the slots. The number of windings and how the motor is wound helps determine the
shape of the stator teeth. Earlier it was mentioned that the number of Slots in a motor is
designated with the value N,. This value is dependent on the number of phases in the
motor. As a general rule of thumb, the number of slots in a motor must be a multiple of

the number of phases in the motor.

—*k (15)

The preceding equation relates the number phases Nm/2, to the number of slots N3,
using an integer called k. There are two types of winding methods that can be used,
concentrated winding or distributed winding.

Concentrated winding is similar to the winding in figure 10. The windings of
each phase are isolated from one another. More common, however, is the method of
distributed windings. [4] In this method, instead of a winding going around only one

tooth, windings span across multiple teeth. This grouping of multiple teeth under one

23

winding produces a pole for that coil. The following diagram illustrates this much more

clearly.

 

Figure 15. Distributed Winding [4]

In this diagram, it is clear to see that part A represents a single phase of the motor.
This can be applied to any motor, with a different number of phases and magnet poles.

This introduces a new term called the coil span. The coil span outlines over how
many stator teeth the coil will cover. The coil span can be found by relating the number

of stator teeth N, and the number of magnet poles N,,..

 

NM = 5 (16)

In the preceding equation, Nm, is the nominal coil span. For a 3-phase motor,
each of the back emf must be 120° out of phase for each other. To simplify winding

further, various literature and software is available that has common winding diagrams

24

available. This eliminates the need of figure out the winding diagram for your particular
case. In the appendix, an example of a winding diagram is given.
The following table is an excerpt from the appendix diagram, and simply shows

how the coils are to be wound.

 

 

 

C oil Coil Phase A Phase 8 Phase C
No. Angle, °E In Out In Out In Out
I 0 1 4 5 8 3 6
2 0 l 16 5 2 3 18
3 0 7 4 ll 8 9 6
4 0 7 10 11. H 9 12
5 0 13 10 l7 I4 15 12
6 0 t 13 16 17 2 15 18

 

 

 

 

 

Figure 16. Winding Diagram [4]

This table is for a 6-magnet pole, 18-slot motor. All of the slots are numbered
from 1 to 18 and correspond to the values in the table.

Another consideration regarding winding involves how the winding goes around
each tooth in the stator. The coils wound can either be closer to the stator yoke (the
outrnost part of the motor, termed the “slot bottom”) or closer to the rotor, the “slot top”.
Each slot must house two sets of windings, with one being in the slot bottom and the slot
top. This difference is small enough to be negligible.

With a basic understanding of windings we can move on to calculate more
dimensional characteristics of the motor. These are related to the stator teeth dimensions

and the size of the stator slots.

25

 

 

 

 

_,H

Figure 17. Stator Slot and Teeth Nomenclature

The first thing we can calculate is the width of the slot opening ww. This opening
exists so that the wires used in the windings can be easily wound around the teeth. Thus,
the value is merely dependent on the size of the wires used. The slot opening should be 2
to 3 times the size of the covered wire diameter.[4] For a nice size, we can simply say

that

w” = 2mm (17)

where Dm-re is the diameter of the wire gauge chosen. The value for d5), is similarly
arbitrarily chosen. This is the shoe tip depth. For simplicity the shoe tip depth will be the
same as the slot opening. Next we need the shoe taper depth d,. Similarly, this can be
chosen to be 1/2 the size of the shoe tip depth d,;,.

With these dimensions chosen, we can start working towards finding the windable
slot depth d,. As the name implies, this length indicates the amount of room there will be

for the windings to fit in. Earlier in figure 15, we looked into distributed winding. Using

26

this figure it is apparent that each slot will have to fit 2 sets of windings. These can be
wound either in the slot top or the slot bottom. Either way, this implies that each slot will
have to accommodate 2*N windings, where N is the number of windings required. This
fact will be used when calculating how the windings will fit in each slot.

First, we need to determine the radius of the stator at the beginning of d,, which is
the location where the windings will begin. Using the diagram regarding the tooth

dimensions and figure 13, we can find this value by using,

d,1 = Rm +g+als,l +11, (18)

where d3), signifies the start of the windable slot depth. Next, to simplify our winding
method, we will assume that the slot area will be a perfect square instead of an arc.
Given the fact that there will be a relatively small difference in the arc lengths in our

design, this is reasonable. The following diagram shows the new nomenclature.

 

 

ds1

 

 

"81

Figure 18. Slot Nomenclature

27

With d3] solved for, we can calculate the value for the slot width w,,. This is

found by using

W _ 27:02,, +g+dsh +d,)—Ns *w,
51 _ N

S

 

(19)

This equation solves for the circumference of the stator at the point d, 1, then
subtracts the total thickness of all the stator teeth and finally divides it by the number of
slots. Now we can begin to determine how the windings will fit in the slot.

In order to wind this, we will say that the windings for the left tooth in figure 18
will cover both the slot top and bottom, but will remain in the left half of the slot opening.
Thus, the slot width we will be working with, will be 1/2 the value for wsl. After this
initial slot winding is finished there should be ample room to rewind each slot such that
the slot bottoms and tops are wound.

A simple understanding of standard wire gauges is required and will be explained
a little later. For now, the diameter of the wires is going to be merely described as Dwire.

We can start out the slot winding analysis with the value N, which in equation 9
represented the total number of windings that are needed for each stator tooth. We also
know these windings must fit in a slot of width w, 1/2, with a height of d,. Also, we can
only have integer values of N because it is impossible to do fractional windings. Thus,
the value for N will be rounded up. We also will associate a wire gauge diameter called
Dwgm, with this winding number.

First, lets determine how many windings we can fit going across in a straight line

of length w, 1/2. This is Simply done using

28

]\I __ yvhl
OCTOSS _ 2*D

More

(20)

which gives us the number of wires that can fit across in the length w31/2. It is important
to note, that if the value for Nam” is a decimal, it means that the last wire will only be a
fractional part of a wire. Therefore, we must round down this value of Nam”. Now we
have a value that shows us how many wires can fit in a row of our slot. With this
determined, we can work up, and figure out how many vertical rows of wires we need.

This is done using a similar equation.

 

N = (21)

Similarly to before, this value can come out to be a decimal, which means we need a
fractional number of vertical rows. In order to ensure we have the correct number of
windings N in our device, we round this value up. The final result is a number that gives
us the number of winding rows and columns we need in order to fit the windings in half

of a slot. The diagram below shows the previous equations explained in more detail.

29

 

 

Figure 19. Winding Diagram

In this picture the circles that are over the black line represent wires that would be
the fractional amount. Thus, in a row, the number of wires is rounded down from 3.75
wires to 3. Similarly, in the vertical value needed to be rounded as well, to accommodate
the 1 wire that is leftover. Once this is completed however, this winding method can be
shifted to wire each phase in either the top half of the slot or the bottom half. (This
example wound it in the left of the slot, and the right of the slot) This is consistent with
the method mentioned earlier.

With the winding completed, the end goal of determining d, can finally be
completed using the following equation,

d,=N *D

up wire

(22)

which simply multiplies the number of rows needed, by the diameter of the wire.

For this case, we simply modeled the area of the slot as a square that is found by

using:

30

A = w,l *d, (23)

slot

However, the actual Slot is roughly trapezoidal in shape, bounded on the top and

bottom by 2 arcs. More accurately, the slot area can be determined by [4]

Astor = F”— [(Rs0 _ WS)’)2 _ (Rm + g + d5” + d! )2 ]— (24)

wh *(Rso — wsy —Rro —g-dsh ‘d:)

t

This value of the Slot area can be used to determine the slot fill factor. The slot
fill factor shows how densely packed the wires are in the slot and how much of the slot

consists of wire instead of empty space. The Slot fill factor is found by

_2N*.25*7rD 2

Kw}, A wire (25)

 

slot

Due to the fact that wires are round and not square, a fill factor of 1.0 is
impossible to obtain. Also, layering the wires like we did in figure 19 is the least
efficient way to pack in the wires. The most efficient way would look something like in

the figure below.

31

:ggg vs fl

Figure 20. Efficient Wire Packing Method

 

By doing this type of packing, the slot fill factor increases by a few percent. For
the sake of simplicity in calculation, the row/column model in figure 19 will be used.
This is a more conservative approach. Also, we are not taking into account whether the
wire is covered with insulation or bare. However, in [4], it was mentioned that with
reasonable costs it is possible to obtain a value of wa in the range of 50% fairly easily.

Another key aspect of motor design is the slot correction factor. Earlier in
equation 6 we determined the flux across the air-gap. This calculation assumed a
constant air-gap in the motor. However, we know that this is not true. Periodically, a
rotor point will pass by a slot opening (with length Wm), which means the effective air-
gap at this point increases, thus changing the air-gap flux calculation. When passing this
slot opening the airgap now becomes the length from the rotor to the inside of the stator.

This is shown in the following diagram.

32

 

 

 

 

Figure 21. Slot correction factor diagram

This new opening width is denoted as g,, and represents this new air-gap length
which occurs as the rotor passes the slot opening. Or, we can say that 6, represents the
angle for the tooth head and slot opening and 19, represents the angle of the tooth head.
The difference of these two angle measures is the angle of the opening of the slot. We
can then say that the air-gap g, is a function of the angle, g(19). This can be integrated into

equation 6 to give us an equation for the slot correction factor[4].

 

1+ P”
K
K 6 = ’ R 26
g Kr/JR

The value ofg(19) ranges from the normal air-gap value of g, (in which case that fi'action
becomes 1), to a max value of g,,,,. The value of gm, is simply calculated by one of the

following 2 equations.

33

gslot 2 (R30 _ wsy )— Rm = g + dsh + dt + d3 (27)

This function can be plotted on a graph of K31 versus 6, where 6 is the current
location over the tooth and slot opening you are at.

With this gm, value, there will also be a permeance coefficient associated with it.
Much like in figure 12, we can use equation 8 to find the permeance coefficient for this

air-gap. This is given by a slightly modified equation.

P — I": (28)

c _
g slot Cd)

 

Because of the increased air-gap this permeance coefficient will be much smaller than its
counterpart.

With equations for all of the dimensions determined, we can easily calculate an
estimate for the total mass of the magnets, stator and wire windings. This is done
considering the fact that the stator and rotor are essentially hollowed out cylinders. The
densities of the magnets, stator material and wire are all known, thus a simple volume
calculation will lead to the masses.

To find the volume of the stator, the key values are the stator inner radius R,,-,
stator outer radius Rm, and the teeth dimensions d,, d,, d,;,, w,0. With the slot area
determined in equation 24, this further increases the simplicity of the calculations.

First we can calculate the volume of the stator, starting from the windable slot

depth all, going out to the outer stator. This is done using

34

V = [7412,02 —ar,,2 )— N, * A5,”, [*1, (29)

stator 1

where l,,, signifies the length of the stator. Next, we need to add in the volume of the
teeth that lay below the windable slot area. To simplify this, we can model these as

squares and triangles, like below.

dshIl II“1t
171

I W“: I
Figure 22. Tooth Head Diagram

Finding the volume of the tooth using this diagram is done using:

V = (thdxll +1/2*(Wlh — lb)*df + Wfb *dl)*NS *ISI (30)

stator 2

Using both of these equations, we can get the total volume of the stator by adding them

together. Finally, using the density of the stator material p,, we can obtain the mass.

M stator = (Vstarorl + VstatorZ ) * pstaror (3 l)

Next, the mass of the magnets can be obtained. Because the magnets are a thin
ring around the rotor this is a very easy calculation and is the same calculation for a

hollowed out cylinder. The density of the material is signified by p,,,.

Mag... = In *(R..’ -R,’)*I.. rpm... (32)

Lastly, the mass of the wire can be calculated. This is done by estimating the wire
length, and assuming a constant wire diameter. To estimate the length of the wire we can
use a few facts. First, the wire can be modeled as being wound in a square around the
stator teeth. Because we would be using distributed winding, this means the windings

will span a few teeth. Thus, we can model it like below.

 

Figure 23. Winding Length Diagram

Used in conjunction with figure 17, we can obtain a simple expression to calculate

the total length of wire in 1 winding as follows

L = 2(1, + 2w, + 3w,,,) (33)

winding

36

Using this equation, finding the total length of all the wires in the stator is found
by using,

= 3L *N*N,, (34)

wires winding
where the 3 represents the 3 phases of the motor. To find the length for 1 phase, we

would Simply divide the preceding equation by 3. Finally, this equation can be used to

find the total mass of the wire.

M = L *1/4*”Dwire2 *pwr're (35)

where MW, signifies the total mass of all the wires and windings in the motor. This

merely represents a rough estimate. Finally, we can integrate all these mass equations

together to get an initial estimate as to the total weight of the motor.

M =M ,+M +M

wires stator magnets

(36)

total

This value can be used to help optimize our design by trying to minimize the overall mass
of the device.

Finally, we can integrate all the preceding equations to obtain a simple efficiency
calculation. Any motor designed will not operate at 100% efficiency. Losses due to heat,

resistances in the wire, eddy currents and cogging torque will aim to reduce the

37

effectiveness of the motor. At extremely high speeds these effects become more and
more prominent and need to be addressed.

One source of loss in a motor is common electrical losses in the wiring. Any
conducting material is not perfect and has an internal resistance in it. Given the fact that
our wires are not superconductors, there will be losses present. These power losses (in

the form of heat), can be determined using the simple equation below

 

Riot = 12 * Rslot (37)
where Rm, is the resistance of the slot. This resistance is calculated by[4]
N2
R... pl" (38)
KM) As

Combining these two equations together gives us a simple expression for the losses from
wire resistances in the Slot. It should be noted that these losses are fairly small in
magnitude compared to other loss calculations.

Next, we can calculate the eddy current losses in the motor. Simply put, eddy
currents are circulating flows of electrons generated inside of the conductor. This
circulating flow creates its own magnetic field that opposes the magnetic field being
generated in the stator.

In [10], the eddy current losses were shown to be calculated by

38

P = ntzo),2(D

6‘

wire / 2)4 (L
8p

wires / 3) (39)

 

All of the loss equations can be added together to obtain the total power losses in the

motor. The efficiency can be determined by

 

E fficienc _ Pstage _ I)losses *100
y - (40)

stage
This, in conjunction with the total weight, gives us two important parameters towards

obtaining a lightweight and efficient motor design. Other losses can be more accurately

determined by using the finite element method to model the motor.

39

Magnetics:

The design of a motor hinges on the proper selection of a magnetic material to be
used in the rotor. For the particular design introduced here, the permanent magnets will
be modeled as located on the surface of the impeller.

A permanent magnet is one that is capable of maintaining a magnetic field
without excitation and magnetomotive force (mmf). In other words, this is an object that
generates a magnetic field without the continual need of inputs. A permanent magnet is
created by placing a metal alloy in a large mmf field. Upon removing the mmf field, a
residual magnetic field value B, will remain. If a subsequent mmf is applied to the
magnet, the operating point of the magnet will move. [11] This can be plotted on a chart,

and will form a hysteresis loop.

 

 

 

B
B
x a I y

81 b c
32 d I e
T..- I

I I

I I

I I

I l

I I

I I

I l a}.

Figure 24. Hysteresis Loop [11]

40

A hysteresis loop shows how the magnetic field value B, relates to an input
magnetic field intensity H. As long as the magnetic field intensity H does not exceed the
initial magnetic field value 3,, the magnet is considered to be reasonably permanent. The
hysteresis loop is important because it shows the operating points of the magnet.[3] The
slope of the 8-H curve at any single point is known as the relative differential
permeability and is simply the ratio of magnetic flux density to magnetic field intensity.

The diagram below is helpful in showing how the flux flows through a typical motor.

 

Figure 25. Flux Diagram [4]

In motor design, there are a few primary classes of magnets used. These include
Alnicos, Ferrites/Ceramics and Rare earth metals.

Alnicos have the advantage of maintaining high magnetic flux over large air-gaps,
but have tendencies to be demagnetized. Ferrites are generally the most popular due to
their low cost and high electrical resistance. The last type of magnets are the rare-earth

magnets. Although these are best choice for most applications, they have relatively

41

higher costs associated with them. Rare-earth magnets generally contain iron, nickel and
cobalt combined with other rare earth elements.

Alnico’s have the primary advantage of having low temperature coefficients. An
alnico magnet has a temperature coefficient of around Br, -.02%/°C and a maximum
temperature of 520°C.[3] Alnico’s generally have higher remanences than ferrite
magnets, but are usually less than various rare earth magnets. These magnets derive their
name from the fact they contain aluminum, nickel, and cobalt. Alnico’s are also very
brittle and should not be used as structural components in any device. Alnico’s can be
either cast, sintered or bonded.

Being cast is Similar to any other casting process. This is where you pour molten
metal into a mold. Sintered magnets are created by pressing small particles of the magnet
in a press and then sintering them to create a solid magnet. Bonding is a die pressing
process where magnetic particles are mixed with an epoxy binder which are then heat
cured. Most magnets can be created using these 3 methods. Sintered magnets have a
higher residual induction than bonded magnets due to the fact a bonded magnet contains
the epoxy binder.

Ferrites have relatively high temperature coefficients. These are usually around
3,, -.2%/°C, with a maximum service temperature of around 400°C. Ferrites are
extremely popular due to their low cost, however, they have comparatively lower residual
inductions than their counterparts.

Rare earth magnets, are the most expensive magnets but have the highest residual
inductions. They are generally made from Cobalt and Samarium (SmCo) or Neodymium,

Iron and Boron (NdFeB). For a SmCo magnet, the temperature coefficient is around 3,,

42

-.O3%/C up to -.45%/°C, with a maximum temperature of 300 to 350°C. An NdFeB
magnet has a temperature coefficient of 3,, -.O9%/C up to -.15%/°C, with a maximum
service temperature of 250°C.[3] SmCo magnets thus have a higher temperature
resistance but also a much higher cost associated with them.

Beside the proper arrangement of the stator windings, the design of a brushless
permanent magnet motor needs to consider how the magnets on the rotor are arranged.
The rotor magnets can be arranged in various ways.

First, the magnets can be arranged in array where all the magnets are charged

radially as shown to the right in figure 26

   

PM rot a»
Figure 26. Halbach magnetized 4 magnet rotor (left), versus a parallel
magnetized rotor (right).

This magnetic configuration is usually the simplest to implement and costs the
least.

However, a Halbach array on the left in figure 26, arranges the magnets
differently to produce a more ideal magnetic field.

In a Halbach array, the magnetic flux generated by the magnets is concentrated on

one side of the array and is effectively cancelled out on the other.

43

 

Figure 27. Magnetic flux from a Halbach array. [17]

Such an array has numerous advantages. First, they have a sinusoidal air-gap
field distribution. This is different than a square wave field distribution obtained with
discrete radial magnets. Also, an array of this nature has negligible cogging torque.
Cogging torque is when motor permanent magnets try to align themselves with a
maximum number of ferromagnetic materials.[l6] This results the torque pulsations felt
when the motor shaft is rotated by hand. [4]

As shown in figure 26, a brushless permanent magnet motor can utilize a Halbach
array arranged in a circle to surround the rotor. In a simple Halbach array this sequence
requires only two magnets for each pole. However, by utilizing more magnets in each

sequence, the magnetic field generated by the Halbach array becomes more ideal.[l6]

 

Figure 28. Two segments per pole Halbach machine (left), and three segments
per pole (right)[l6]

The airgap flux from an ideal Halbach array can be calculated as outlined in [12].

This equation is given below.

-ZBr—l—f;[1—[:r] ][[ ]_[%’L] +[R7mjp :lcos(p6)

_ - (41)

2;) 2,03 2p
R. _ m _ _ 51
Kin) (1 #.>+(1+#.)[ ] (1+fl.)+(1 #.)[R]

m S

— d _ —

 

an".

 

airgap '—

>6

 

 

 

 

 

’° I

In this equation, p is the number of magnet pole pairs, (NM/2), r represents the
current radius you are at (Rm<r<R,) and 6 is represents what angle around the rotor you
are currently at. Also, K is a value that is 1 for an iron cored rotor and (l -u,)(l+p,) for an

air cored rotor.

45

Frequency Drives:

To drive the electric motor an alternating current must be supplied to all three
phases of the motor. The higher the rotational Speed of the motor, the more expensive the
fi'equency drive will have to switch the currents.

Earlier, the concept of electrical frequency was introduced. This is dependent
upon the number of magnet pole pairs in the motor. The electrical frequency is the
frequency at which the current must switch in order to have the motor rotate at its desired
mechanical frequency.

In order to obtain the trapezoidal back emf in the motor, a square wave pulse of
current is required. There are few types of circuits that can be used to supply current to a
phase. These include a half bridge, a full h-bridge topology and a Y-connection.

A half bridge is much cheaper in cost, and only supplies current to one phase at a
time. Thus, if phase 1 has current supplied to it, the 2"d and 3rd phases are not being used.
The trade off of this however, is a much lower torque production[4]. This is present in
low power applications where cost needs to be minimized and torque production

efficiency isn’t important.

 

RP]! Rphé Rph
Lph Ii Lph it Lpr. is

 

 

 

 

O

I i t

Figure 29. Half Bridge Drive

 

46

The firll h-bridge topology is much more complex. It allows independent control
of each phase current. [4] This is much more complex than the preceding example. This
requires many more transistor switches and is more expensive. In the following diagram,

this represents a full H-bridge that would control one phase.

 

l
31\ Lph 32\

an “ an

Figure 30. Full H-Bridge Drive

 

 

 

The most common type of topology used is a Y-connection configuration.
However, the current of 2 of the phases is dependent on the current of the first phase.
Thus, each phase cannot be controlled independently, like in a full h-bridge. But this
topology allows for more torque production, since all phases are running and allows it a

reasonable cost.

47

HIGH SPEED MOTOR PARAMETERS

Magnetic Selection

Above, basics regarding magnets and magnet arrangements were discussed. The
compressor in consideration shall operate at high efficiency, with a high rotational speed.
Each magnet choice and arrangement Should be heavily weighed and critiqued.

Ideally, with a composite impeller created on a winding machine, the best choice
for a magnet would be one that could be integrated into the impeller itself. The best
choice may be the addition of magnetic powder in the matrix material (resin) at the outer
diameter of the impeller.

The first step is deciding which magnets should be used in the rotor. Various
references on brushless magnet design generally favor the use of the rare earth magnet
'NdFeB[8, 9, 10]. Although rare earth magnets were generally associated with much
higher costs, advances in the manufacturing of NdFeB magnets now have driven the cost
down further. [3] The drawback from this magnet is that it is strongly temperature
dependent. It is easily demagnetized at high temperatures and cooling the magnets is
extremely important for the utilization of NdFeB.

For the application of being used to rotate an axial impeller in a small or medium
size R718 chiller, the torque requirements are very low (typically <0.l l Nm).
Therefore for this purpose, NdFeB magnets generate an unnecessary strong magnetic
field. F urtherrnore, NdFeB magnets are relatively more expensive than other magnet

choices. Due to the fact that this impeller motor will eventually have commercial

48

applications, the utilization of these magnets may increase the price of the refrigeration or
air conditioning unit.

Ferrite magnets seem to be the ideal choice for this application. First, ferrites are
cheap to manufacture which helps keep the price of the motor unit down.

Another consideration is the temperature range of the magnets. Ferrites have a
maximum service temperature up to 400°C. Also, ferrites are most economical in low
horsepower motors due to their cheap price and low Br. They also have low eddy current
losses. [3]

For the overall design, various magnet types will be investigated. These can be
seen in the table below

Table 2. Magnet Choices

Magnets Br Density kglm‘3 BrlDensity
Sin. Fe 0.35 4900 7.14E-05
Sin. NdFeB 1.2 7500 16E-05
Bonded NdFeB 0.5 7500 6.66E-05
Sin. Alnico 0.75 5700 13E—05
Bonded Alnico 0.35 5700 6.14E-05
Sin SmCo 0.9 7500 12E-05
Bonded SmCo 0.55 7500 7.333E-05
Magnet Paste 0.1 4900 2.04E-05

Next, the array type needs to be investigated. Two primary types of arrangements
exist, a simple radial charging where each magnet pole is charged in only the radial
direction, or a Halbach array where the two or more magnet segments make up a magnet.
The Halbach array has the advantage of reducing the cogging torque. Also, a Halbach
array is self shielding and requires less iron backing. [12] In [6], it was shown that for
high speed permanent motor applications, a Halbach magnetized rotor has less rotor

losses than a radially magnetized rotor.

49

Earlier, the calculations used assumed radial magnetization, which gave a perfect
square wave magnetic field in the air-gap. Even though a Halbach array would be the
best choice for this high speed motor, this simplifying calculation will still be used.

The next major criteria relating to the magnet arrangement is the number of
magnets in a sequence per pole. As stated before, the more magnets used in a Halbach
sequence, the more ideal the magnet is.

In [16], a method has been described, where instead of numerous magnets (each
individually magnetized) being used to create the Halbach array, a high performance
nearly ideal Halbach array can be created. This is accomplished by orienting anisotropic
NdFeB powder during the injection molding process and magnetizing them with the
required Halbach field.

However for a ferrite magnet, this may not be possible, thus the array may need to
be created of numerous discrete magnets. A compromise needs to be found in terms of
the number of magnets in the sequence per pole versus an ideal Halbach array.

Lastly, the total of magnet pole pairs should be chosen. For such a high Speed
application, the number of magnet pole pairs should be minimized. A lower magnet pole
number, however, also necessitates thicker back iron on the stator and rotor in order to
limit magnetic saturation. [14] A lower pole number also drives the cost down of the
frequency generator that will be required to drive the motor. As the number of magnet
pole pairs increases, the electrical frequency of the motor increases. (remember, the
mechanical frequency will remain constant). For our design, we will only investigate low

magnet pole pair numbers. So in our design spreadsheet, only 2 magnet pole pairs, and 3

50

magnet pole pairs will be considered. This should be sufficient to keep the electrical

frequency low enough, such that the cost of a frequency drive will be lower.

51

Impeller Considerations

Some of the concerns regarding the integration of the motor into the impeller are
whether or not the impeller itself can be magnetized or whether a ring of magnets needs
to be affixed on it. Given that the lifespan of the motor is intended to be years, the
magnets used for the motor should be resilient magnets that won’t degrade noticeably in
quality over such a long period of time.

Due to high centrifugal forces for rotors of relative large diameters, the magnets
must use a retaining sleeve around them to attach them to the rotor. In many cases, the
magnet is simply glued to the rotor. This is usually favored since magnets are brittle and
not malleable. However, high rotational speeds mean that the glue may not be
sufficiently strong. This retaining sleeve should be nonmagnetic and ideally would not

contribute many losses. A sleeve with some conduction leads to eddy currents that cause

rotor losses. [1]

 

 

 

 

 

m T r l’
80 > Dm = 18 mm Copper
KS 3 IO Will]

70 » «
g 60 Stainless
l; Steel
(1)
a 6m
.0
5‘3 40 . .
9 .
.5 Titanium
6 30* 1
I'-

2" ' only PM

10 .

o , .. l a a a

0 20 4D 60 80 11:0 120

Speed (krpm)

Figure 31. Total rotor losses for slotless stator for different retaining sleeve
materials.[l]

52

Figure 31 shows losses as they depend on the material used in the retaining
sleeve. This material selection is also related to costs and to how efficient the rest of the
design is.

So far, we’ve only discussed using permanent magnets arrayed around the
perimeter of the rotor but there is another innovative method that is being considered.

As stated earlier, the composite impeller is created using a winding machine. The
carbon fibers, prior to being wound, are dipped in a solution of a liquid resin. After
setting, the resin hardens which gives the impeller its strength.

Generally this resin solution is used as a binding and strengthening agent to keep
the rotor together, however, various materials can be mixed in with the resin as well. It is
theoretically possible to integrate magnets into the resin, so that when the impeller is
wound it already has magnets throughout its design.

A magnetic powder can be used with the resin and subsequently magnetized.
Currently, magnetic hardening resin has been applied to micro fabrication techniques.

In [5], a method was described that uses resin bonded magnets in Micro-electrical
mechanical-systems applications (MEMS). The paper explains how difficult it is to
produce magnets on a substrate. First, a powered Ba-ferrite or Sr-ferrite substance was
used. This is due to the fact that the powder size of these magnets is sufficiently small
(on the scale of 1 pm), as opposed to 10 um for various rare earth magnets. The method

of placing the magnet on a substrate can be seen below.

53

_ » Photon-stat
I 5’ I I __I
(a) amateur: (1))

Paste

 

  

h‘licromaehined resin-bonded magnet

 

 

(c) ' («9

Figure 32. Micro magnet fabrication on a substrate.[5]

First, (a) shows the substrate (in this case a silicon wafer), with nothing on it.
Next, a mold was placed on top of the substrate, with gaps that were to be filled with the
magnet. In (c), the magnetic resin/paste was placed on the substrate and finally, in (d),
the mold was taken away leaving only the magnet.

The key thing to note in this method is that the resin magnet mix was a room
curable epoxy. Generally, when created powdered bonded magnets, the magnet needs to
be heat cured and pressed. But this experiment shows promise that eventually we can
make a weak magnet that needs no pressing and minimal curing.

Thus, this is extremely advantageous in that no heat treating or pressure are
needed in order to obtain the final magnet. The hysteresis loop of this magnet can be

seen below.

54

 

10000 15000
H (00)

 

 

.5001

Figure 33. Hysteresis loop of the micro bonded magnet.[S]

This loop shows the magnetic characteristics of the micro bonded magnet. When
being used in the application the composite impeller, the film on each of the carbon
ribbons could essentially be considered a micro-bonded magnet.

In our design, we can use a simple way to model a magnet obtained this way.
First of all, we know that the magnetic layer will be a mix of magnetic powder, epoxy,
and carbon fibers. So, we can model it as essentially a weak magnet with perhaps a small
fraction of the residual induction a pure magnet would have. The value would be at best
an educated guess and estimate.

The next primary concern is how to magnetize the magnets such that they can be
used for our application. This is an extremely difficult process, given the way the

impeller is wound.

55

In [13], a method was described in which nanoparticle polymers were induced by
an electric field. Although this was not used to magnetize the composite, it could be used
as a stepping stone as a method for magnetizing the resin powder.

In the experiment, the carbon nanotubes were in an epoxy resin and were induced
in an electric field. This was used to increase the strength of the composite. The curing

method used a permanent magnet and a mold seen below.

 

 

 

 

’/ ‘ '1/1/1‘1/[1/3

 

 

Figure 34. Mold Alignment and Curing.[13]

This diagram shows the mold used. The epoxy resin and nanoparticle mix was
placed into the mold at the top. This mold was then placed between the permanent
magnets seen below, and allowed to cure for 24 hours. The end result was a small wafer

with its particles aligned.

56

The biggest problem with this experiment is that particles were not magnetized,
only aligned. Also, the curing time for this experiment (hours and not seconds), is a
drawback to our process. But, this shows that an epoxy nanoparticle mix can indeed be
aligned using a permanent magnet.

For our purposes, this could mean that a ribbon, with epoxy/resin/magnet coating
on it, could be pulled through a magnetic field device like the one seen above and be
magnetized in this manner. The end result would be a wound impeller with multiple
magnets on it.

These two experiments together both show that the process of magnetizing the
carbon fiber ribbons is possible. First we showed that a magnetic powder resin mix can
be formed on a substrate and second we showed that an epoxy nanoparticle mix can be
aligned using a permanent magnet. Perhaps by using a powerful electromagnet, we can
more quickly align the magnetic particles in the resin and obtain a relatively strong
magnet. An important factor to note, is that the flux to density ratio of this will be the
worst. Because this magnet does not exist, we will arbitrarily assign it a flux Br of .lT,

and a density of 4900 kg/mA3.

57

General Motor and High Speed Considerations

When looking at the thickness of the magnets, figure 35 from [12] is extremely
helpful. The chart shows flux densities for an air cored Halbach arrayed rotor and how it
relates to pole pair numbers. On the chart, the value Rr is analogous to the nomenclature
Rm used in this paper. For a high-speed low-torque motor, (3 or 2 pole pairs), the most
desirable designs on the chart have a small rotor to magnet radius ratio. With the design
here (in which we have a large rotor, and a small magnet layer), this is not feasible and
the values are likely to be in the lower flux regions on this chart. Figure 36 shows the
flux density with an iron core. At low pole pair numbers the chart shows that the flux
density is much higher for all radius ratios. This shows that maybe even a thin layer of
back iron inside the magnet ring could increase the flux density if necessary.

Basically, with an air cored rotor (such as ours), the flux density is very low for
low pole pair numbers. We need to make sure we have a relatively thicker magnet layer

to ensure that our design has ample flux.

58

 

+Rr/Rm=0.0
*Rr/RnFOJ
*erRrrFOA
+erRm=06
+W.7
-2 ' -o- erRm=0.8

+RriRm=09
0 "“'j""tfvf"rtt""vw

0 5 10 15 20 25 30
Pole Pair Number

 

Flux Density

 

 

 

Figure 35. Flux density variation for varying magnet thickness in a Halbach array
with an air cored rotor.[12]

 

1.2 W
l 4

E .8-1

C

    

,6 a ‘ *W.“

| ' *RriRmOl
4 ‘ *erRnFOA
‘ +RriRm=06

Flint Deusrty

+RriRm=07
-2 " + erRm=0.8
+ We

0 V ' V V U V V U V I V V T V I V 1 T U I V ' V V U V V V '
0 5 10 15 20 25 30
Pole Pair Number

 

 

 

Figure 36. Flux density variation for varying magnet thickness in a Halbach array
with an iron cored rotor. [12]

Another important issue in the design of a high speed motor is the wiring in the
stator. The wiring used will be a copper wire. The sizing of copper wires is standardized

and called the American Wire Gauge (AWG). The AWG assigns a value to different

59

wire thicknesses. For example, No. 10 AWG corresponds to a wire 2.989mm in
diameter. As the number increases, the wire thickness decreases. The following table

shows some wire gauges that are used in the spreadsheet for this project.

Table 3: Wire Gauge Sizes
AWG diameter (mm)

These are present in a drop down menu on the spreadsheet. All of these wires

will have the same resistivity p, which will be 1.7*10"-8 Ohm*m.

Copper 0 9.53
Copper 01 8.487
Copper 02 7.558
Copper 04 5.994
Copper 06 4.753
Copper 08 3.77
Copper 10 2.989
Copper 12 2.371
Copper 14 1.88
Copper 16 1.491
Copper 18 1.182
Copper 20 0.938

60

MOTOR DESIGN SPREADSHEET USER’S GUIDE

Supplementing this thesis is a spreadsheet that was used to obtain all of the
results. The Spreadsheet followed the equations listed earlier and calculates all the
needed values in a systematic manner.

The most difficult thing to implement into the spreadsheet are loss calculations.
Many losses can only be calculated using the finite element analysis and other intensive
calculations.

The other most important thing regarding the spreadsheet is the weight
calculation. This is a rough estimate and uses the methodology explained earlier.

So, the Spreadsheet aims to quantify the final efficiency of the given design, the
dimensions of the final design and the weight. These output values are intended to be
used with other considerations (Halbach arrays, retaining sleeves), in order to obtain a
design matrix that systematically determines the best design for this application.

First, the spreadsheet has a list of parameters that can be varied in the design, they
are:

1. Number of Magnets

2. Current in Wiring

3. Magnetic Material Properties (Residual Induction, density)

4. Stator Material

5. Wire Gauge

6. Permeance Coefficient.

61

For simplification, all other parameters will be held constant. Many of these
values are defined by compressor calculations carried out earlier, or are constants for
motors that are widely known. To simplify changing these values, the wire gauge and

magnet type are present as a drop down menu. The next diagram shows some of the

motor constants.

Table 4: Desiin Constraints in the Spreadsheet

    
   

Number of Magnets Nm - __ 6

Magnet Pole Pairs rim/2 ~ ' 3

Phases ' - - 3

Total Slots _ Ne - . 18
Current 1‘ amps 0.5
Stator Length Let m 0.04
Air Gap Length g, _ I‘m 13.001

Matenal Chorces

Magnet Density pmagnel kg’m’IS
Residual Induction . . _ Br WT 0.35
Stator Material - - lron
Stator Density 7 potato: Rymna , 7874
Tooth Saturation Density Bi ‘ T 1.8
Copper Wire Gauge _ - - __ Copper 18 AWG
Wire Resistivity p Ohm'm OWN 7_
Wire Density pairs kg’m‘3 8%0
Wire Thickness M19 or 0 1101152 ..
Lamination Stacking Factor , Kat - 0.8
Leakage Factor Kl - 0.9
Relative Recoil Factor ur - 1.05 _,
Permeability Free Space uo H/m 1.255643%
Reluctance Factor Kr l

The shaded boxes indicate areas where the value can be changed or modified.
Another key feature of the spreadsheet is a plot of the motor that is dynamically obtained.
As values are modified and changed, the diagram updates, which gives us a quick way to

see what the motor looks like. An example of this output is seen below.

62

Figure 37. Dynamic Motor Diagram

The diagram only shows 2 magnet sections, and 6 stator teeth. This is constant
regardless of how many magnets are actually present in the motor. The primary benefit
of this diagram is showing if the motor looks “normal” and realistic. Some combinations
of values give you a stator tooth that is extremely thin and centimeters long. Obviously,
this cannot be constructed and common sense will dictate we must change the design.

One table in the spreadsheet summarizes some of the important data.

   
    

 

 
   

         

 

  

able 5: Motor Final Data
E. . .. .37 .- w- - .... rmi i T ‘2 6ii iii: . . 1:,"- iii-“ray is; " "it J. 33573:?
., .. . Grammar Stages
Name Variable Units I ll Ill IV
Windings N - 9 1 1 1 1 12
Back EMF eb - 488.6 533.4 580.8 633

lR"2 losses W 0.00139 0.0017 0.0017 0.00186
Eddy Current Losses W 21.0104 11 .3575 17.2572 22.8105
Total Loss W 21.0118 11.3592 17.2589 22.8124
Efficiency % 91 .3992 95.7408 94.0569 92.7923
Motor Constant Km - 9.05778 16.4455 14.0993 13.0654
Total Mass kg 2.391 57 3.72808 3.37402 2.72207
Slot Fill Factor 0.41947 0.72272 0.41669 0.50175

63

The chart shows the values for each stage and is key to our optimization. For this
project the initial values such as required rotational speed, rotor torque and rotor diameter
are all fixed based on previous work done. These are summarized below

Table 6: Stage Characteristics
Radius Speed Power

Stage (m) (rpm) (W)
1 0.05804 67222 244.3
2 0.089478 40107 266.7
3 0.074457 51240 290.4
4 0.066423 60979 316.5

These can be expanded upon by looking at how changes in the rotor radius and

speed (with constant power) affect efficiency and mass.

64

DESIGN CALCULATIONS AND OPTIMIZATION

The most important fact of any high speed motor is that losses increase as the
rotational speed increases. In other words, as the rpm of the motor increases, you are

fighting off losses more and more. The following chart shows how rotational speed can

affect the overall efficiency for this design.

100 r_\

92 -1 —~ _-___.___ 4———-—

 

 

 

 

Efficiency (7.:

 

 

 

 

0 20000 40000 60000
Speed (RPM)

 

 

Figure 38. Efficiency Plot

This was determined for one of the stages of this compressor design. Using the
data given in the spreadsheet section of this report, we can start the optimization process.

One of the design parameters we selected was the use of a Halbach array. This
was to give us a more ideal air-gap flux, as well as allow us to neglect the cogging torque.
By reducing the cogging torque effects, we can in turn increase the overall efficiency of
the design. The spreadsheet however, does not accurately account for a Halbach array.

There will be numerous assumptions in our calculations, to summarize them, they

are:

65

1. No back iron behind the magnetic ring

2. No Halbach Array Calculation

3. No Cogging Torque

4. No retaining sleeve present over the magnets
5. Efficiency calculations omit Core Losses

6. Modified Winding Method

First, we can look at the sintered ferrite magnet. We can optimize this by finding
the highest motor constant, the highest efficiency, or the lowest weight. These can be
found by crawling through the wire thicknesses, adjusting the air-gap and changing the
magnet thickness via the permeance coefficient. The current will be held at .5 amps.

First, we can choose different wire types. The wires will be wound and stacked
using the method explained earlier and shown in figure 19. The following plot shows

how wire gauge affects the efficiency.

 

 

 

 

 

 

 

0.0008 0.001 1 0.0014 0.0017 0.002
Wire Demeter (m)

 

 

 

Figure 39. Efficiency Vs. Wire Diameter

66

This chart shows that for a simple calculation, a smaller wire gauge will lead to
fewer losses in our eddy current and copper wire losses. Because our wires are relatively
short, the losses due to the internal resistance in the wires are negligible. Having a
thinner wire makes more sense due to the way the device will wound. However, this
device has a high electrical frequency which can lead to losses in smaller wires.

As a simple solution, picking the wire with a diameter of .001182m (Wire Gauge
18), gives us both a high efficiency. Picking too small a wire gives us a winding scheme
in which all the windings take place in 1 row, which isn’t practical.

Using the spreadsheet we can input all our initial values and vary the permeance
coefficient to see how it affects the efficiency, mass and motor constant. We will be
looking at a design with 6 magnets and 3 magnet pole pairs, as well as comparing it to a

4-magnet 2-pole pair solution. The following 3 charts show this data for a ferrite magnet.

 

"mmifiég'fiegj-

10l— fl ,___ _, . -‘___ _fl___ -__ 4
9.8 1H - w - Z3, __fi_ M!“ __ JL". '6 Magnetslw
/ .

 

 

Motor Constant

 

 

 

 

 

 

l
‘ .
l Permeance Coefficient g
l

Figure 40. Motor Constant vs. Permeance Coefficient, Ferrite Magnet

67

r_ L

 

95.5 f

92.5

Efficiency

91.5

90.5

94.5 ..

93.5 .

 

 

 

 

 

 

 

Penmanoe Coefficient

J
— 7 A i f — “f flinging—T
3::- - 6 Magnetsl .
_ _ _ __ L...__ n _. ___ *_ m
N
\ \ / ~— 5“ K a
4 4.5 5 5.5 6

 

 

Figure 41. Efficiency vs. Permeance Coefficient, Ferrite Magnet

 

 

 

 

4 Magnets'

 

 

 

 

 

Permeance Coefficient

 

Figure 42. Mass vs. Permeance Coefficient, Ferrite Magnet

In general, these charts show that for two designs with the same magnet thickness

(permeance coefficient), the 4 magnet design has much more mass.

Thus, a thinner

magnet ring is needed in order for a 4 magnet design to be on par with the 6 magnet

design. This thinner magnet ring comes with a trade off of more windings needed. The

68

second chart shows that the optimum permeance coefficient for this ferrite magnet motor
with these input condition is at 4.8. Permeance coefficient should be from 4 to 6 and is
visible in figure 11.

The sudden jumps present in the graphs has to do with the way the winding
scheme works out (as visible in figure 19) and corresponds with 1 less winding being
needed due to the amount of magnets present. The following graph can outline this

better.

 

 

 

 

 

 

12 - , #fi Am ,___ A, v n _,_ ,__-
f Rounded Windings
11 l —- -Fractiona| Winding Line
£110- \ 4 —-———— —— ~—
1:
= \
E , - __ _____ 1-, _-

:7
‘l
ll
l
L/
/
‘—

 

 

 

Permeance Coefficient

 

 

 

Figure 43. Windings Needed for a Ferrite 6 Magnet Motor

This stair step pattern occurs due to the fact we cannot have fractional windings in
the design. Similarly, we can use this data to find the slot fill factor. This is using the

winding scheme mentioned before, where windings are arranged in simple rows.

69

 

 

Fill Factor

 

 

8 10 12 14
Permeance Coefficient

 

 

 

Figure 44. Fill Factor vs. Permeance Coefficient

The max points on this graph occur when a winding row is completely filled. By
adding one more winding, the entire stator has to get bigger (larger diameter), in order to
accommodate this extra winding. Of course, this represents a winding method that would
not be seen in normal practice and also omits the fact a human hand will be winding the
machine. However, it outlines how important the fill factor is. This slot fill factor chart
can be created for multiple wire diameters. By keeping in mind that the permeance
coefficient must be from 4-6, a chart that integrates varying wire diameter, fill factor and
permeance coefficient can help ensure the slot fill factor is maximized.

Using this constrained view we can run through quick calculations on all the
different magnet types. This was accomplished by varying the permeance coefficient
from 4 to 6, and seeing how efficiency, motor constant and mass change. The permeance
coefficient value presented represents a reasonable slot fill factor as well as the highest

efficiency. This table constitutes the first run through of the data and will aim to narrow

70

down the magnet choices. This table represents the first stage of the compressor, and all

of these have 18 wire gauge, .5 amp current and 6 magnets.

Table 7. Optimum Design Characteristics

Permeance Motor

Coefficient Efficiency (%) Constant (k) Mass (kg)
Sin. Fe 4.8 91.40 9.058 2.392
Sin. NdFeB 4 65.34 9.866 4.594
Bonded NdFeB 5.6 87.31 8.966 2.832
Sin. Alnico 5.6 80.30 8.966 3.486
Bonded Alnico 4.8 91.40 9.058 2.450
Sin SmCo 4 74.94 9.866 3.639
Bonded SmCo 4 86.57 9.044 2.607
Magnet Paste 4 97.70 9.044 2.478

These values can be put into a weighted design matrix, in order to determine the

best possible design solution.

Table 8. Weighted Design Matrix

Cost Weight Efficiency Motor Constant Temp. Stability Total

Weighting 10 5 5 5 5 30

01 Sin. Fe 10 4 4 2 1 21

O2 Sin. NdFeB 3 1 1 5 4 14
03 Bonded NdFeB 1 4 3 3 4 15
04 Sin. Alnico 7 2 2 1 3 15
05 Bonded Alnico 5 4 4 2 3 18
06 Sin SmCo 3 2 2 5 5 17
07 Bonded SmCo 1 3 4 1 5 14
08 Magnet Resin 7 4 5 1 1 18

This design uses a few categories to help find a solution. First, cost basically
looks at how much the magnet is to create, or in the case of the magnetic paste, how
expensive the technology might be to implement. Weight, efficiency, and motor constant
were all obtained from the spreadsheet optimization. Temperature stability looks at how

well the magnets can cope with high temperatures, but also how easily they degrade

71

when present in higher temperatures. The discussion earlier in the magnetics section was
used to come to these values.

From this simple analysis, the ferrite magnet comes out with the highest score.
Although earlier it was mentioned that high speed motors generally favor rare earth
magnets, the biggest reason why a ferrite would be better is that we are not making a
design that requires large torque. Also, this calculation was carried out with a relatively
low current. Another choice that is noteworthy to look over is the magnetic resin
solution. Again, this solution modeled the magnetized shroud/resin mix as an extremely
weak magnet.

Also, these calculations were created with 3 magnet pole pairs instead of 2.
Generally high speed devices use only 1 or 2 magnet pole pairs, in order to keep the cost
of the frequency drive down. It is possible that by having such a large radius, the design
might not work as well due to the magnets being too big and spanning too large of an arc
length.

These next set of plots will use a constant tip speed. We will assume a rotor of
radius 5cm, spinning at 60,000 rpm. For these numbers, it works out to a tip speed of
314.15 m/s. As we increase the radius, the rotational speed will slow down to maintain
the constant tip velocity. This can give us an idea as to how radius will affect the motor
design. First, let’s look at the mass of the motor with a 6 magnet ferrite magnet, and

compare it to a plot of the magnetic resin.

72

98 1

 

 

 

 

 

 

 

 

 

P.C.=
94 n- __ W__. _ in. r
I a ' a ‘
l C , un- "'
i .2 ‘— 4' ’- 1
0 ‘- n l
s E .— r
90 — -- 4 ». -' -— s — a — wwfl— — l
i r l
.. I i 4Magnets ;
,. i- - -6 Magnets l
as .
0.04 0.045 0.05 0.055 0.06
Rotor Radiue(m) i

 

Figure 45. Efficiency vs. Rotor Radius for a Ferrite Magnet

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r*“ “—* —— —— — F“ *“T—WT
100 1
RC. =5 -

99 “—9 — __ ‘4
1 I
; e l
l 5 / - _ .— ,. I! II
'5 93 _ ' ‘- __ .. «,4. -___.__ ___m l

'. E a. I- ‘— — i
i 97 w ’ f ’ I
,- .fi—_ “7 ‘7' “1 4 Magnets ,
l i- - -6 Magnets 1
96 ‘ .

0.04 0.045 0.05 0.055 0.06 l

5 RotorRadius(m) ‘
L l

 

Figure 46. Efficiency vs. Rotor Radius for a Magnetic Resin Array

These plots show a permeance coefficient of 5. It shows that the efficiency of a 4
magnet motor is higher than a 6 magnet motor when the magnet thickness is held

constant. Also, as the rotor radius increases, the efficiency increases as well. Also, the

73

magnetic resin has a higher overall efficiency for a given rotor radius. The next set of
plots will show the mass of the motor.

l

 

 

   

 

 

 

 

 

 

 

l 3.4
l l ” Triggers: P.C.=5
._' - -6Magnets
A28- . .._.
f ‘ ¢
2
2.2 -- a ‘k
v ’ v
i i .
h ‘ .r
l ~ ._ . —'
i 1.6 g

i 0.04 0.045 0.05 0.055 0.06
Rotor Radius (m)

 

Figure 47. Mass vs. Rotor Radius for a Ferrite Magnet

 

 

 

 

4 Magnets;

. 3.2 _ __.__ _
l L' - .6 Magnetsl / 4
i 7’ v

 

 

 

 

 

 

0.04 0. 045 0. 05 0.055 0.06
Rotor Radius (m)

 

Figure 48. Mass vs. Rotor Radius for a Magnetic Resin Array

74

The first ferrite plot shows that the mass increases as the rotor radius increases.
The second plot shows more of a saw pattern. This occurs due to the fact the stator for
the magnetic resin requires many more windings. Thus, by increasing the rotor radius,
you are increasing the circumference of the motor at the slot opening. This in turn means
that as the rotor radius increases, more windings can fit across in a row. This stair step
pattern takes a dip when a row is eliminated by the increasing radius. Since the ferrite
magnet has less windings, however, this occurs much less frequently. Also, at lower
rotor radius the ferrite magnet is lighter. As the rotor radius increases the magnet resin
approach has less mass.

For the preceding rotor radius calculations, we kept the tip speed constant. Now,

we can see how increasing the tip speed will affect the design.

 

 

 

 

 

  

 

 

 
 

 

 

 

 

 

 

f'_"" ' " "—7.
i 2.8 , .
. 4
2.5
l
5 2
I 5 2.2
E s /
' 19 l- - -TrpSpeed315m/s

‘ ‘gi— -TrpSpeed 275mls

1', np Speed 355m/s
1.6 T l l
0.04 0.045 0.05 0.055 0.06
Rotor Radiue(m)

 

 

Figure 49. Mass vs. Rotor Radius for a Ferrite Magnet

75

 

 

 

 

 

 

 

 

 

 

 

93.5 —— e Emmi m.v_n__,_m___ ...—--_:"'_
/ ’ .- ’
i E ,— "Z « -*
E .5 91 TE—‘m ‘7 ;— _ *T 77 f; ‘ ’ "T
E / v
‘ / ‘ I I T 5 315m/ ‘
88.5— ’4! «—_ -_...__#__1f" " " Ip peed sfi
P —- -Trp Speed 275mls
L Tip Speed 355m/sj
86 g . -
N 0.04 0.045 0.05 0.055 0.06
l Rotor Radiuc(m)

 

 

Figure 50. Efficiency vs. Rotor Radius for a Ferrite

By increasing the tip speed you are increasing the angular fi'equency. This
increase in angular fi'equency causes our efficiency calculation to decrease. Another
thing that a higher tip speed does is decrease the torque from equation 11. Since angular
frequency is equal to the tip speed divided by the radius, with a constant power the torque
will go down. A lower torque means that less windings are needed, thus the mass will be
less. The mass plot shows this in a way. The sudden dips are places where a winding
row is eliminated. Although the 355m/s plot isn’t the lowest for the range shown, over
the long run, it will have much lower mass. This decrease in mass is not without a cost.
Increasing the tip speed at constant radius increases the angular frequency, which
increases the cost of the frequency drive needed to run the motor. Using equation 11, and

rearranging, we obtain the following equation:

T=P*— (42)

76

 

This simple relation can tell us how velocity and power affect the torque and thus
the overall motor. Increasing the power by a factor of 2 is the same as decreasing the
velocity by a factor of 2. It is very easy to relate these parameters together.

Next, we can look at how changing the current can affect mass and efficiency.

All of the following plots were calculated at a permeance coefficient of 5.

 

[__-.___ _ _,* ___.___
. 6
* I

5

 

 

Ferrite

______, \ ”we; ‘ '7’ L_:-- — Magnetic ReSinE
2 7 5‘;

 

 

.h.

 

 

 

 

Mace (kg)
OJ

 

 

E 1 F T l +
0 0.4 0.8 1.2 1.6 2
Current (amps)

 

 

 

Figure 51. Mass vs. Current for a Magnetic Resin and Ferrite

This plot shows that for lower current values the magnetic resin has a much
higher mass. However, as the current is ramped up this becomes much lighter than the
ferrite. At higher currents, the masses will tail off as the number of windings slowly

reduces to 1 winding.

77

 

 

 

 

 

 

 

 

 

 

E
90 ,, — * --—- A *— E
E
E .2
0 -n, ”if-..
E 70 .. Ferrite E-
E —- - Magnetic Resin
1 60 dm ____ ______ __ ,v, fig__; _ _.__f:_r:__lnfl:.A:—:-——— . '
E E
50 . . .
o 0.4 0.8 1.2 1.6 2 E

Current (amps)
E

Figure 52. Efficiency vs. Current for 8 Magnetic Resin and Ferrite

This figure shows how efficiency greatly increases as the current increases. For
any design the current should be kept high to minimize losses. The magnetic resin has
higher efficiency for a given current.

The preceding plots and also the plots in the appendix can give us a good idea as
how to factors vary and can affect each other. Using some of this information, we can
begin to find the best design. The current will be purposely kept from .5 to 1 amps. This
best design will look at a ferrite magnet.

In figure 45, it shows that for a 4 magnet motor the efficiency is always much
higher. A 4 magnet motor also has the advantage of driving down the cost of the
frequency drive. Since we are dealing with such high speeds, driving down the electrical
frequency is beneficial.

This means however, that the mass of the design will be much higher compared to
a 6 magnet motor (with a fixed permeance coefficient). Since we want to keep the air-

gap constant, this means to have a competitive mass, we need to use the lower end of the

78

 

permeance coefficient to obtain a thinner magnet ring. This trade off means we’ll have
less air-gap flux (based on figure 11). The biggest problem with only 4 magnets is the
size of the slot. Since or rotor is relatively large, too few magnets will mean the slot size
will be too big. A thinner magnet also means that we’ll require more windings. More
windings can be counterbalanced by more current if necessary. However, because of the
low torque requirements for this motor the number of windings should be still easy to
manage.

We should also keep figure 44 in mind and recognize we need to maximize the
slot fill factor. Using this information and the spreadsheet we can come up with a
preferred design. Due to the fact each stage has its own power, radius and rotational
speed requirements, the optimization will be carried out 4 times. ‘

The optimization will utilize the assumptions explained earlier and will use the

data already presented. The following figure shows what the final motor will look like.

Figure 53. Final Motor Values

This table will show what the various motor dimensions are, as well as what the

key characteristics are.

79

“Table_ 9. Final Desi

 

N— L —‘l-—l I—-a|

 

-- CompressorStages ., 7,
Name Variable Units I ll III IV

Rotor Radius Rr m 0.05804 0.08948 0.07446 0.06642
Magnet Radius Rm m 0.06284 0.09408 0.07946 0.07042

Air-Gap g m 0.001 0.001 0.001 0.001

Magnet Length Lm m 0.0048 0.0046 0.005 0.004
Stator Inner Radius Rs m 0.06384 0.09508 0.08046 0.07142
Width Stator Yoke wsy m 0.00664 0.00987 0.00846 0.00719
Stator Yoke Inner - m 0.0721 1 0.10099 0.08873 0.0797
Stator Outer Radius Rso m 0.07876 0.11086 0.09719 0.08688
Width Slot Opening wso m 0.00236 0.00236 0.00236 0.00236
Bottom Slot Width wsb m 0.01992 0.03082 0.02572 0.02257
Shoe Tip Depth dsh m 0.00236 0.00236 0.00236 0.00236
Shoe Taper Depth dt m 0.00118 0.00118 0.00118 0.00118
Windable Slot Depth ds m 0.00473 0.00236 0.00473 0.00473
Width Tooth Body wtb m 0.00443 0.00658 0. 00564 0.00479
' Tooth Head Width - _ m1“ 001992 3.91...- 03082 0. 02572 _ 02257

 

 

. ‘araseermwrm-«em “T511. 9 ’-, ' . ..
. -- .. a. t. ' fi’ - -- .4. » I
-.- .~ 1 1‘ «45'». a \I“‘ . . _ -‘r ,

Summa

and Characteristics

 

 
 

Compressor Stages H “ ‘

Name Variable Units i ii iii iv
Windings N - 9 1 1 1 1 12
Back EMF eb - 488.6 533.4 580.8 633

IR"2 losses W 0.00139 0.0017 0.0017 0.00186
Eddy Current Losses W 21.0104 1 1.3575 17.2572 22.8105
Total Loss W 21.0118 11.3592 17.2589 22.8124
Efficiency % 91 .3992 95.7408 94.0569 92.7923
Motor Constant Km - 9.05778 16.4455 14.0993 13.0654
Total Mass kg 2.39157 3.72808 3.37402 2.72207
Slot Fill Factor 0.41947 0.72272 0.41669 0.50175

 

These tables give dimensions and values that can be used to create the motor
needed to run the device. The rest of the data from the spreadsheet is in Appendix C.

The motor solution for this would be more efficient and have fewer losses. This is
due to the fact that the electrical frequency of the motor would be lower and thus less
eddy current losses would be induced. For a high speed motor, minimizing any losses is
ideal. Another solution that would be noteworthy to look at is the solution that used a

magnetic resin bath.

80

Although this is the preferred solution in the long run, currently there are too
many unknowns as to how this would be implemented. The weak magnet is possible due
to the low torque required in this design. This design also had the least amount of losses,
and gave the lightest overall design.

Above all, the spreadsheet gives the user a good idea as to what the general
dimensions, magnet requirements as well as input data are needed for the motor design.
This can provide an excellent first step, when numerous design ideas can quickly be

swapped.

81

CONCLUSION

This thesis was able to show how to create an electric motor to serve as the drive
system for an impeller. This topic was discussed in the context of a high speed impeller
in a chiller that uses water as a refrigerant. All of the essential background details for
understanding the basic motor design were presented and explained in their entirety.

Following the procedure presented in the preceding sections gives an approximate
idea of the quality and the dimensions of the motor. However, this methodology does not
take into consideration numerous losses and is not adapted for Halbach arrays.

It is also very difficult to determine how the motor will behave with an impeller
as the backing to the magnet. This perhaps would be similar to a device that is between
an air cored rotor and an iron cored rotor, but above all, is something best left relegated to
a complex finite element analysis.

In general, a simple methodology can be used to describe how the various data
was crawled through using the spreadsheet. This is loosely followed, and outlines how
many of the charts were explained and how data leading into the design matrix was

obtained. The following flow chart shows this simple procedure

82

 

 

Frequency Generebr
(current)

i

Change!"
I: Memeti’ype I”

i

Yes Ehange Perm Codi]

[Selectl «Magnets.

 

 

 

 

 

Thatch Results. use metrbrto
seled best design

Figure 54. Procedure Flow Chart

The data obtained from this data crawling outlined a lot of important trends and
key relationships. Some of these key trends presented in this thesis can be summarized in

the following bulleted list:

1. Air-gap Flux tails off as Permeance Coefficient Increases, ideal range is 4 to 6
2. Higher rotational speeds have lower efficiencies and larger losses
3. Increasing the rotor radius while keeping the tip speed constant increases

efficiency

83

4. Increasing the tip speed while keeping required power constant decreases mass
and decreases efficiency

5. High slot fill factor is ideal

6. The more magnet pairs you have, the lower the mass and efficiency

7. Increasing current lowers the mass. This levels off as the current is further
increased

8. Increasing the current also increases the efficiency

9. Lower pole pair numbers means a lower electrical frequency, which vastly
reduces the cost of the frequency drive.

10. Better quality magnets means more efficient torque production

11. Higher residual induction generally means more mass

This list is very general and should be used with many precise calculations and a
very holistic approach. As with any design process, there is no systematic way to solve
every design problem.

The final design chosen was a motor with 3 magnet pairs utilizing a ferrite magnet
arranged in a Halbach array. This consideration was made based on calculations carried
out on a spread sheet, as well as a design matrix which was used to weigh design criteria.
A separate calculation was carried out for each of the 4 stages of the device. The aim of
each of these stages was to obtain a solution with low mass, a reasonable efficiency, and
a high slot fill factor.

Continuing research might focus on doing a finite element analysis on the motor
design presented here. This would more accurately be able to calculate losses, as well as
learn about how flux travels through the design. This model could include an impeller at

the center.

84

Further studies can be conducted into how the magnets are best integrated into the
rotor and how various sleeves can affect the design. Also the particularities of
magnetizing the composite impeller directly are open to investigations. An optimization
for low cost and small size may be desired too.

A prototype could also be constructed. This would more adequately show how
the technologies discussed might work in practice. The spreadsheet could also be
upgraded to account more accurately for a Halbach array.

Another big thing to further address is the winding method. This could be looked
at by creating a better winding algorithm and looking at the slot openings more closely.
Also, an investigation could be carried out into whether distributed winding would be the
best solution for this application.

Lastly, a comparative study can be completed that compares the merits of this
innovative drive system, versus merely rotating the impeller using a motor, gear box and
a shaft.

Above all this thesis was able to provide the foundation towards the design of a
permanent magnet motor drive system for a compressor in a water as a refrigerant device.
The thesis researched basic key concepts in the numerous disciplines required to tackle
such a problem. This thesis should serve as a guide towards working towards a final
design and as the first preliminary study in a long multi-step process of engineering

design.

85

 

Appendix A: Winding Diagram [4]

 

 

 

 

 

N. l NLlflmEERm/Ral I<._.L 05.1
18 | 6 l 1 II 0.56 0.96 1 l 3
Coil Coil Phase A Phase B Phase C
No. Angle, °E In Out In Out In Out
1 0 1 4 5 8 3 6
2 0 1 16 5 2 3 18
3 o 7 4 11 8 9 6
4 o 7 10 11 14 9 12
5 0 13 10 17 14 15 12
6 0 13 16 17 2 15 18
7
8
9
10
11
12

 

 

 

Figure Al. Winding Chart for 3 phase 6 magnet motor

86

 

 

Appendix B: Motor Plots

 

 

 

 

.' ' ‘ E
0.04 0.045 0.05 0.055 0.06 E
Rotor Radius (m)

 

Figure Bl. Mass vs Rotor Radius for a Ferrite Magnet, 6 Magnets

 

"recreate... "

 

 

 

 

0.04 0.045 0.05 0.055 0.06
Rotor Radius (m)

 

 

Figure B2. Motor Constant vs Rotor Radius for a Ferrite Magnet, 6 Magnets

87

 

 

 

 

 

 

 

 

t C n - i ,
96
93.5 - —-
5'
5
3 91 4 » , .
E
in
88.5
86 _ . .
0.04 0.045 0.05 0.055 0.06
Rotor Radius (m)
g

 

Figure B3. Efficiency vs Rotor Radius for a Ferrite Magnet, 6 Magnets

 

 

 

 

 

 

 

0.04 0.045 0.05 0.055 0.06
ibtor lhdiue (m) 1

Figure B4. Mass vs Rotor Radius for a Ferrite Magnet, 4 magnets

88

 

 

A 8 Motor Constant

 

 

 

 

0.04 0.045 0.05 0.055 0.06
\ Rotor Radius (m)

 

Figure B5. Motor Constant vs Rotor Radius for a Ferrite Magnet, 4 Magnets

 

 

98

 

 

 

 

 

0.04 0.045 0.05 0.055 0.06
Rotor Radius (m)

 

Figure B6. Efficiency vs Rotor Radius for a Ferrite Magnet, 4 Magnets

89

 

 

 

 

 

 

 

 

 

 

0.04 0.045 0.05 0.055 0.06
Rotor Radius (m)

 

Figure B7. Mass vs Rotor Radius for a Magnetic Resin Array, 6 Magnets

 

-. .___
I

\

A.

on:

MotorWEonstant

 

 

 

 

 

 

 

 

 

0.04 0.045 0.05 0.055 0.06
Rotor Radius (m)

 

Figure B8. Motor Constant vs Rotor Radius for a Magnetic Resin, 6 Magnets

90

 

 

 

 

 

 

 

 

 

 

 

i 99
' 5‘
C
.2
O
E
97 ,9 - - - ,
"’ :- '- "P.C.=41
{-— -P.C.=5é
L P.C.=6;
‘ % I T I 7M“
\ 0.04 0.045 0.05 0.055 0.06

Rotor Radius (m)

 

Figure B9. Efficiency vs Rotor Radius for a Magnetic Resin, 6 Magnets

 

3.5

 

3.25 «

 

 

 

Mm (k9)

 

 

 

2.75

 

 

 

 

0.04 0.045 0.05 0.055 0.06
Rotor Radius (m)

 

i
t

Figure B10. Mass vs Rotor Radius for a Magnetic Resin Array, 4 Magnets

91

 

 

 

 

Motor Constant A

 

 

 

 

 

0.04 0.045 0.05 0.055 0.06
Rotor Radius (m)

 

Figure Bl 1. Motor Constant vs Rotor Radius for a Magnetic Resin, 4 Magnets

 

 

 

 

 

 

0.04 0.045 0.05 0.055 0.06
Rotor Radius (m)

 

Figure BIZ. Efficiency vs Rotor Radius for a Magnetic Resin, 4 Magnets

92

 

Appendix C: Motor Calculations

Table C1. Final Desiin Iniuts

Number of Magnets Nm - 6
Magnet Pole Pairs Nm/2 - 3
Phases - - 3
Total Slots Ns - 1 8
Current i amps 0.5
Stator Length Lst m 0.04
Air-gap Length 7, ., g‘ , 7“ .. __ m ‘7 0.001
_ ‘ . . MaterialCholcos ;.. , r . , ,
Material Choices - - 01 Sin. Fe
Magnet Density pmagnet kglm"3 4900
Residual Induction Br T 0.35
Stator Material - - Iron
Stator Density pstator kglm"3 7874
Tooth Saturation Density Bt T 1.6
Copper Wire Gauge - - Copper 18 AWG
Wire Resistivity p Ohm*m 0.000000017
Wire Density pwim kg/m"3 8960
Wire Thickness Dwire m 0.001182
Lamination Stacking Factor Kst - 0.8
Leakage Factor KI - 0.9
Relative Recoil Factor ur - 1.05
Permeability Free Space uo Hlm 1.25664E-06
Reluctance Factor Kr - 1

93

 
   

Table C27. Final De Calualtion __ a

. , _ J . , .. Calculations .. ‘ . ,, _ . _. -... f . - . . - 1... . »
Name Variable Units I II III IV
Stage rpm rpm 67222 40107 51240 60979
Angular Speed wm rad/s 7039.47 4199.99 5365.84 6385.7
Electrical Ang Speed we rad/s 21 1 18.4 12600 16097.5 19157.1
Rotor Radius Rr m 0.05804 0.08948 0.07446 0.06642
Stage Power P W 244.3 266.7 290.4 316.5
Stage Torque T nm 0.0347 0.0635 0.05412 0.04956
Magnet Radius Rm m 0.06284 0.09408 0.07946 0.07042
Magnet Length Im m 0.0048 0.0046 0.005 0.004
Stator Length Ist m 0.04 0.04 0.04 0.04
Air-gap Length 9 m 0.001 0.001 0.001 0.001
Flux Concentration Factor Co - 1 1 1 1
Calculated Perm Coef Pc - 4.8 4.6 5 4
Back emf eb V 488.6 533.4 580.8 633
Air-gap Flux 89 T 0.25846 0.25646 0.26033 0.2495
Flux Density - 0.73846 0.73274 0.7438 0.71287
Windings N 8.90308 10.9662 10.9016 11.7533
Windings Rounded N - 9 1 1 1 1 12
Total Flux Ototal T 0.00408 0.00606 0.0052 0.00442
Slot Opening wso m 0.00236 0.00236 0.00236 0.00236
Tooth Width wt m 0.01992 0.03082 0.02572 0.02257
Shoe Thickness dsh m 0.00236 0.00236 0.00236 0.00236
Shoe Taper Length dt m 0.00118 0.00118 0.00118 0.00118
Radius to tooth inner - - 0.06739 0.09862 0.084 0.07497
Width of the Tooth Body wtb 0.00443 0.00658 0.00564 0.00479
Width Stator Yoke wsy m 0.00664 0.00987 0.00846 0.00719
Minimum Slot Width 0.01909 0.02785 0.02368 0.02138
Width for 1 set of windings 0.00955 0.01392 0.01184 0.01069
Max winding rows — - 8 11 10 9
Winding Radial Rows 2 1 2 2
Windable Slot Depth ds m 0.00473 0.00236 0.00473 0.00473
Slot Area (estimate) - m"2 9.4E-05 6.7E-05 0.00012 0.0001
Slot Fill Factor wa - 0.41947 0.72272 0.41669 0.50175
Air-gap in Slot gs m 0.00927 0.00691 0.00927 0.00927
Slot Correction Factor Ksl 0.4024 0.47657 0.41051 0.3676
Slot Permeance Ps m 0.51 758 0.6657 0.53914 0.43131
Slot Resistance Rslot 0.00558 0.00682 0.00682 0.00744
lR"2 losses Pslot 0.00139 0.0017 0.0017 0.00186
Wire Length - m 9.87914 15.2371 13.7658 13.9867
wire resistance Rwire ohm 0.00018 0.00028 0.00025 0.00026
Eddy Current Losses. Pe W. 21.0104 11.3575 17.2572 22.8105
Total Losses 21.01 18 11.3592 17.2589 22.8124
Efficiency 91 .3992 95.7408 94.0569 92.7923
Motor Constant Km - 9.05778 16.4455 14.0993 13.0654
Total Mass - kg 2.39157 3.72808 3.37402 2.72207

94

 

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[1 7] http://www.matchrockets.com/ether/Halbach.html

96

 

   

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