CAD
Optimization of
PM Machines
Faculty of
Engineering and
Architecture
Final Year Project
Spring Term 05-06
Supervisor:
Dr. Farid Chaaban
Committee:
Dr. Farid Chaaban
Dr. Sami Karaki
Dr. Riad Chedid
Group Memebers:
Al-Mokadem, Reef
Bou Ghannam, Adham
Fares, Dima
Ghoussainy, Rabih
200300782
200302695
200302797
200302246
Abstract
A conventional synchronous machine is to be optimized by utilizing
permanent magnets. A design technique is revealed which transfers this existing
conventional synchronous machine into a surface mounted permanent magnet
machine, using Ferrites and Sintered NdFeB. The conventional machine is first tested
experimentally and then modeled on MagNet, a finite element analysis software. To
ensure validity of the experimental results, a comparison between the experimental
and software results is performed. Afterwards, the conventional synchronous rotor is
replaced by a surface mounted permanent magnet rotor configuration using at first
Ferrites and then NdFeB. The configuration parameters for the optimal design are
determined by split ratio analysis. The split ratio analysis is a mathematical technique
that identifies the optimal ratio of the rotor diameter Dr to stator diameter Do which
corresponds to the maximum output torque. Other machine parameters are changed in
correspondence to this technique; however the size of the optimized machine is
maintained as that of the conventional one. The resulting design is an optimized
permanent magnet synchronous machine that in comparison with the conventional
machine has better electrical and physical parameters.
i
Table of Content
Abstract..........................................................................................................................i
Table of Content...........................................................................................................ii
Table of Illustrations.................................................................................................. iii
Problem Statement......................................................................................................iv
Introduction..................................................................................................................1
Conventional Machine Analysis .................................................................................2
Machine Examination ................................................................................................2
Observations and Dimensions................................................................................5
Experimental Procedure ............................................................................................6
PART ONE ............................................................................................................7
Analysis and Calculations......................................................................................8
Theoretical Approach ................................................................................................9
Electric Loading.....................................................................................................9
Magnetic Loading ................................................................................................10
MagNet Simulation ..................................................................................................12
Permanent Magnets & their Applications...............................................................14
Permanent Magnet Materials ..................................................................................15
Brief history of permanent magnets.....................................................................15
General Properties of Permanent Magnet Materials............................................17
New Permanent Magnet Materials.......................................................................19
Application of Permanent Magnets in Motors.........................................................22
Application of Permanent Magnets in DC-Motors..............................................23
Application of Permanent Magnets in Stepper-Motors .......................................24
Application of Permanent Magnets in Synchronous-Motors...............................25
Permanent Magnet Design ........................................................................................28
Magnet Choice.........................................................................................................28
Rotor Configuration.................................................................................................28
Split Ratio Analysis..................................................................................................31
Torque Equation Derivation ................................................................................31
Split Ratio Equation.............................................................................................32
Magnetic Loading ................................................................................................33
Electric Loading...................................................................................................33
Non-Saturation Criteria........................................................................................33
Geometry Parameters...........................................................................................34
AS Formulation ....................................................................................................35
Torque Formulation .............................................................................................36
Torque Optimization............................................................................................36
Optimization Results ................................................................................................37
Magnet simulation ...................................................................................................39
No load Voltage Calculation................................................................................40
Comparative Analysis................................................................................................43
Cost Analysis ..............................................................................................................45
Conclusion ..................................................................................................................47
References...................................................................................................................48
ACKNOWLEDGEMENT.........................................................................................49
Appendix A - Matlab Code .......................................................................................50
ii
Table of Illustrations
Table of Figures:
Figure 1: Synchronous Motor ........................................................................................2
Figure 2: Synchronous Machine Name Plate.................................................................3
Figure 3:Stator Upper View...........................................................................................4
Figure 4: Stator Coils and Laminations .........................................................................4
Figure 5: Rotor Side View .............................................................................................4
Figure 6: Rotor Coils and Laminations..........................................................................4
Figure 7:Motor Brushes .................................................................................................4
Figure 8: Experiment Connections ................................................................................6
Figure 9:Pole Model ....................................................................................................11
Figure 10: Magnetic Model .........................................................................................11
Figure 11: Equivelant Magnetic Circuit ......................................................................11
Figure 12: Synchronous Motor MagNet Simulation ...................................................13
Figure 13: Flux Lines Path...........................................................................................14
Figure 14: Flux Lines Path...........................................................................................14
Figure 15:Gilbert's Loadstone......................................................................................15
Figure 16: Henry Electromagnet..................................................................................16
Figure 17:Alnico ..........................................................................................................16
Figure 18: Ferrite Permanent Magnet ..........................................................................16
Figure 19: Neodymium-iron-boron..............................................................................17
Figure 20: Samarium Cobalt........................................................................................17
Figure 21: Hysterisis Loop...........................................................................................18
Figure 22: Surface Type Mounted Configuration........................................................29
Figure 23:Rotor Configuration ....................................................................................30
Figure 24: Machine Representation .............................................................................31
Figure 25: Optimized Design Flux Lines using Ceramic Ferrites ...............................39
Figure 26:Optimized Design Flux Lines using Sintered NdFeB .................................39
Table of Graphs:
Graph 1: Experiment 1 Graph........................................................................................8
Graph 2: Flux Variation for Ceramic Ferrites..............................................................40
Graph 3: Flux Variation for Sintered NdFeB ..............................................................41
Graph 4: Ea for Ceramic Ferrites.................................................................................42
Graph 5: Ea for Sintered NdFeB..................................................................................42
Table of Tables:
Table 1:Data of Experiment...........................................................................................7
Table 2: Magnet Properties..........................................................................................28
Table 3: Optimized Machine Parameters.....................................................................38
Table 4: Dimensions Comparison................................................................................44
Table 5: Parameters Comparison .................................................................................44
Table 6: Material Properties.........................................................................................45
Table 7: Volume ..........................................................................................................45
Table 8: Cost ................................................................................................................46
Table 9: Analysis .........................................................................................................46
iii
Problem Statement
A conventional synchronous machine is tested experimentally as an open
circuit generator. The output no load voltage is obtained at rated speed. Also the
torque and electric loading values are obtained relative to the experimental results.
Later, the conventional machine is modeled on MagNet software and the flux value in
the air gap is obtained along with the flux line scheme. This conventional machine is
optimized by converting its wound rotor into a permanent magnet rotor using Ferrites
and NdFeB magnets, for the sake of comparison. The optimization procedure employs
a numerical technique, Split Ratio Analysis which improves machine parameters. The
improvement is revealed in the value of the torque, flux, induced output voltage and
electric loading, keeping the machine size constant.
iv
Introduction
Several efforts have been made so as to come up with efficient, practical and
realistic techniques to optimize old conventional machines into more effective ones.
Higher torque and efficiency are two main desirable features sought after in the new
machines. Permanent magnet materials seemed to help in achieving these objectives.
However introducing a magnet into an electric machine is a difficult mission. This
process would be subjected to several constraints like machine size, life span,
sustainability and cost. Electrical machine researchers have proposed several
mathematical and computer based techniques for this purpose. Procedures followed in
each technique depended on the main objective behind the optimization of the
machine. In this report, an optimization process of an existing conventional machine
is discussed. First, a study of the original machine theoretically and experimentally,
exploiting computer programs and numerical techniques is prepared. Two magnetic
materials, Ferrites and Neodymium Iron Boron, are then introduced separately for the
optimization purpose.
The configuration parameters for the optimal design are
determined by split ratio analysis and computer aided programs. The results of the
optimized prototype are presented comparative to the original conventional machine
characteristics.
1
Conventional Machine Analysis
The optimization of a conventional machine is the main objective behind this
project, thus the choice and examination of an existing machine is a necessity. From
the machines available in the lab, the HPS synchronous machines was most
applicable, since it is smaller in size, easier to disassemble, and could be run as a
motor and generator.
Machine Examination
The objective of examining the machine was to note its major electrical parameters
and dimensions.
Figure 1: Synchronous Motor
The name plate indicated the electrical and mechanical parameters of the machine
selected. According to this plate, the synchronous machine is a three-phase, 380 V,
4pole machine. The stator of the machine has a current of 1.5 A per turn while the
rotor has a current of 0.95 A per turn.
2
Figure 2: Synchronous Machine Name Plate
The motor was later disassembled so as to observe
•
Stator laminations
•
Stator slots
•
Rotor laminations
•
Rotor slots
•
Winding connection and number of turns
•
Brushes and slip rings
3
Figure 3:Stator Upper View
Figure 5: Rotor Side View
Figure 4: Stator Coils and Laminations
Figure 6: Rotor Coils and Laminations
Figure 7:Motor Brushes
4
Observations and Dimensions
At this stage, precise measurements of the machine components dimensions were
taken. Later on, these measurements will allow further analysis of the machine.
1. Stator
Observations
Upon observation, the stator consists of 36 slots. It was also estimated that the
number of turns per slot is 60 slots.
Dimensions
The following dimensions were taken;
•
Length of the stator : 13.6 cm
•
Inner diameter of the stator : 7.0 cm
•
Outer diameter of the stator : 11.1 cm
•
Width of Stator slots: 0.3 cm each
2. Rotor
Observations
Upon observation, the rotor consists of 18 slots. It was also estimated that the
number of turns per slot is 75 turns.
Dimensions
The following dimensions were taken;
•
Length of the rotor : 13.6 cm
•
Inner diameter of the rotor (diameter of the back iron) : 2.35 cm
•
Outer diameter of the rotor : 6.95 cm
•
Width of rotor slots: 0.6 cm each
5
3. Air Gap
The width of the air gap was measured to be 0.5 mm
Experimental Procedure
The main objective was to find the machine's flux φ . This flux will be used to
validate the theoretical and simulated results in future parts of the report. To reach this
goal, an experiment was performed were the machine was running as an open
circuited generator and values of the induced voltage was recorded for different
speeds at constant field current.
The circuit was connected as shown in the figure below and the experiment was
performed according to "Machine's Lab Manual".
Figure 8: Experiment Connections
6
PART ONE
The synchronous generator was mechanically coupled to a prime mover. To change
the speed of the synchronous generator the prime mover speed was altered. This speed
was changed by changing the dc field excitation of the prime mover. The field current
of the synchronous generator is adjusted at its rated value which is equal to 0.95 A.
The no load voltage was measured using a voltmeter and the speed of the rotor was
measured using a stroboscope. The results are summarized in the table below:
Table 1:Data of Experiment 1
Rated Field Current
Synchronous Generator If =0.95A
Ea(V)
ω(rpm)
387
390
394
397
401
404
407
411
414
418
422
430
437
443
445
448
451
455
462
466
471
475
481
488
496
501
504
1240
1250
1260
1275
1285
1295
1310
1320
1330
1345
1360
1385
1410
1430
1440
1450
1460
1470
1490
1510
1525
1540
1560
1575
1605
1625
1635
7
The results were plotted on an excel sheet.
No Load Volatge-Speed Graph
No Load Voltage (V)
600
500
400
y = 0.2957x + 20.357
300
200
100
16
35
16
05
15
60
15
25
14
90
14
60
14
40
14
10
13
60
13
30
13
10
12
85
12
60
12
40
0
Speed (rpm)
Graph 1: Experiment 1 Graph
By linear regression the above graph was approximated into a straight line of equation
Vt = 0.2957ω + 20.357 .
The estimated value of the no load voltage at the rated speed (1500 rpm) is
Ea = 267.8V
Analysis and Calculations
To calculate the flux of the machine, several steps were considered. Noting that
E a = Kφω
[5]
where K is a constant representing the construction of the machine
K=
NC
[5]
2
where N C is the number of turns per coil of a stator .
Knowing that it is a three phase machine therefore the stator has three coils. The stator
also has 36 slots which imply that we have 12 slots per phase. Since each slot contains
8
60 turns as previously mentioned then the total number of turns per coil will be
N C = 720 turns
Therefore: K =
720
= 509 .12 .
2
Finally to calculate the flux of the machine
φ=
Ea
267.8
=
= 3.3 x10−3Wb
K ω 509.12 x1500 x 2π
60
Theoretical Approach
Synchronous machines are machines whose magnetic field current is supplied
by a separate dc power source. In synchronous motors, torque is produced due to the
presence of two magnetic fields. One magnetic field will be produced in the stator and
the other in the rotor .A torque will be induced in the rotor which will cause it to turn
and align itself with the stator magnetic field. The stator magnetic field rotates. The
induced torque in the rotor will cause it to constantly chase the stator magnetic field.[5]
There are two major concepts of any machine design that should be considered,
magnetic loading and electric loading. Together they will produce the torque of the
machine.
1. Electric loading: is the total current per unit periphery of the stator bore.
2. Magnetic loading is the total number of magnetic lines, cut by each conductor,
in one complete revolution. It is the flux that is coming out of the rotor.
Electric Loading
Electric loading depends on the following factors:
9
1. Allowable temperature rise
2. Heat dissipation capability of the motor enclosure
3. Duty cycle
Starting from the definition of electric loading, it will be formulated as: [4]
Q=
Where
JK p AK w
πDa
Da is the rotor diameter
Kp is the packing factor or the ratio of copper area to the total winding area
Kw=fraction of conductors being used to total conductors=1
A is the total winding area
J is the current density
J=
I total in slot
ACu per slot
=
I per turn × N per slot
ACu per slot
=
I per turn × N per slot
Aslot × K P
=
I per turn × N per slot
Astator winding
× KP
36
Therefore,
Q=
I per turn × N per slot × 36 × K w
πDa
=
1.5 × 60 × 36 × 1
= 14.85 KA / m
π 0.0695
Magnetic Loading
In order to find the magnetic loading, an equivalent circuit of the system is obtained,
shown in the following schematics: [5]
10
Figure 10: Magnetic Model
Figure 9:Pole Model
Rs
Ra1
ℑ = Ni
Rr
Ra 2
Figure 11: Equivelant Magnetic Circuit
The obtained equations are:
11
ls =
πd
=
π 0.111
4
4
lr = d = 0.0695m
= 0.0872m
d out − d in
0.111 − 0.07
0.076 = 1.558x10 −3 m 2
l=
2
2
πd
π 0.0695
0.076 = 4.148x10 −3 m 2
Ar = Aa =
l=
4
4
ls
0.0872
Rs =
=
= 222.694KA.turns / Wb
µ r µo As 2000 x 4πx10 −7 x1.558x10 −3
As =
Rr =
Ra =
lr
µ r µ o Ar
=
0.0695
= 6.67 KA.turns / Wb
2000 x 4πx10 −7 x 4.148x10 −3
la
0.000005
=
= 9.59 KA.turns / Wb
µ o Aa 4πx10 −7 x 4.148x10 −3
Rtotal = Rs + Rr + Ra1 + Ra 2 = 248.544 KA.turns / Wb
The number of slots in the rotor is 18 with 75 turns each.
Therefore the number of slots per pole is 4.5 and the number of turns per pole is
75 x 4.5
= 168.75turns ≈ 169turns per pole
2
In conclusion the magnetic loading of the rotor
φ=
ℑ
Ni
169 x0.95
=
=
= 6.45 x10 − 4 Wb
3
Rtotal Rtotal 248.544 x10
MagNet Simulation
12
The synchronous machine is modeled on MagNet Software which is a finite
element analysis software. The finite element analysis will be performed on static 2D.
The choice of 2D will not take into effect the magnetic fields on the boundary of the
machine since it will assume infinite length, but according to literature review this
method will provide accurate results with a tolerance 2%. Thus our choice of 2D
solving is justified.
The machine is represented as a generator by supplying the rotor with
0.95A/turn and supplying no current to the stator. Due to symmetry, one pole of the
generator will be modeled.
The orientation of flux lines are shown in figure1 below.
Figure 12: Synchronous Motor MagNet Simulation
The flux through the center of the above pole is minimal and almost
approaching zero. This is due to the presence of opposing flux fields of equal
magnitudes at the symmetry axis. The opposing field is present due to the right hand
rule, while the equal magnitude is present due to geometrical symmetry and constant
current density through out the copper. Figure 13 will illustrate this phenomenon.
13
Figure 13: Flux Lines Path
On the other hand as we move away from the center, the flux will increase and
it will reach its maximum at the point were the current changes its sense. This is due
to the fact that at this point, all the flux lines have the same sense which is determined
by the right hand rule.
Figure 14: Flux Lines Path
The flux obtained upon simulation is:
φ = 3.918 × 10-3Wb .
A similarity was revealed between the experimental ( φ = 3.3 × 10−3Wb ) and the
simulated ( φ = 3.918 × 10 -3Wb ) values of the flux, which validates the simulation
results.
Permanent Magnets & their
Applications
14
Permanent Magnet Materials
A permanent magnet, just like any other magnet, will produce a magnetic field
of its own once subjected to a strong external magnetic field. However, the very
special characteristic of the permanent magnet is that it will continue to exhibit a
magnetic field even with the external magnetic field being removed. This produced
magnetic field is said to be continuous if the material doesn't experience a change in
the environment. A change in the environment, for example temperature or
demagnetizing field, will redefine the capabilities of the permanent magnet or
sometimes cancel them. Therefore the more the permanent magnet withstands these
changes the better are its capabilities, and the more successful are its applications. [8]
Brief history of permanent magnets
The historical review of permanent magnet allows a
good vision of the development of such materials with
respect to enhancement in their properties, feasibility of
their applications.
Figure 15:Gilbert's Loadstone
The early forms of permanent magnets were described in 1600 by W. Gilbert. They
are called “loadstone with soft iron pole tips” .These are a form of magnetite Fe3O4
that had iron tips that increase attractive forces upon contact .They are used to
magnetize pieces of iron and steel.
In 1825, J. Henry and W. Sturgeon invented the electromagnet.
15
Figure 16: Henry Electromagnet
By the year 1867, German scientists started making ferromagnetic elements from
nonferromagnetic material and nonferromagnetic alloys from ferromagnetic materials
like iron.
In 1901 Heusler alloys were discovered. Heusler alloys contain 10 to 30 percent
manganese and 15 to 19 percent aluminum and copper.
In 1917 cobalt steel alloys were discovered.
In 1931 alnico (Al, Ni ,Co ,Fe ) were discovered.
In 1938 powdered oxides were developed.
Figure 17:Alnico
In the 1950 ferrites (barium ferrite BaOx6Fe2O3 and strontium ferrite SrOx6Fe2O3)
were invented. Hard ferrite (ceramic) magnets were developed in the 1960's as a low
cost alternative to metallic magnets is SrO-6(Fe2O3), strontium hexaferrite.
Figure 18: Ferrite Permanent Magnet
16
In the 1970’s rare earth permanent magnets (samarium-cobalt SmCo and neodymiumiron-boron NdFeB) were developed.
Figure 19: Neodymium-iron-boron
Figure 20: Samarium Cobalt
Since then Rare earth permanent magnets are being increasingly used in machine
industry.
In 2002, NdFeB became the most abundant of all permanent magnets. [8]
General Properties of Permanent Magnet Materials
In designing for a permanent magnet application, several characteristics of that
material are considered. The most important characteristic is the demagnetization
curve which will allow the designer to judge whether the permanent magnet used is
suitable for the application being designed. Also the material properties, the shape of
the magnet and the operating conditions are essential factors that might constrain or
simplify the achievement of a successful design. [8]
Hysteresis Loop
A permanent magnet does not need any excitation winding to produce
magnetic field in an air gap nor does it lead to dissipation of electric power.
Permanent magnets can be described by the B-H hysteresis loop. They are usually
considered to have wide hysteresis loops. During magnetization, an increasing
17
magnetic field is applied to the material until a saturation point is reached. Upon
removing this applied field, a permanent magnet material will not follow the same
path down to flux density = 0, instead, it will retain some of its magnetism. The path
that the permanent magnet follows is called a hysteresis loop and is a key tool in the
quantitative analysis of permanent magnet performance. The amount of magnetization
it retains at zero driving fields is called its remanence. It must be driven back to zero
by a field in the opposite direction; the amount of reverse driving field required to
demagnetize it is called its coercivity. [8]
High remanence means that the magnet can support higher magnetic flux
density in the air gap of the magnetic circuit. While high coercivity means that a
thinner magnet can be used to withstand the demagnetization field.
Figure 21: Hysterisis Loop
Permanent magnets are usually evaluated in the upper left-hand quadrant of
the hysteresis loop that is called demagnetization curve. This curve reveals the state of
the magnet under reversed magnetic field intensity. The application of such a reversed
field reduces the remanence of the magnet, and by reapplying such magnetic field
intensity the flux density will also be reduced causing another minor hysteresis loop.
18
New Permanent Magnet Materials
The permanent magnets that are currently used for electric motors are:
1. Alnicos (Al, Ni, Co ,Fe )
2. Ferrites (barium ferrite BaOx6Fe2O3 and strontium ferrite SrOx6Fe2O3)
3. Hard ferrite SrO-6(Fe2O3), strontium hexaferrite
4. Rare earth permanent magnets (samarium-cobalt SmCo and neodymium-ironboron NdFeB)
Alnico
It has a high magnetic remanent flux density. This advantage will allow a high air gap
magnetic flux density. However, the coercivity is very low and the demagnetization
curve is considered to be non-linear so although it is easy to magnetize alnico, it is
also easy to demagnetize it .They are used in dc commutator motors and in motors
with few Watts up to 150KW, however ferrites became more popular. [8]
The key attributes of Alnico are:
•
Mechanically strong
•
Cast to a variety of shapes
•
Very temperature stable
•
Can change magnetic orientation
•
High Bmax characteristics compared to ceramic materials.
19
Ferrites
A ferrite has a higher coercive force than alnico. Lower remanent magnetic
flux density, low cost and very high electric resistances are its main advantages since
no eddy current losses in the PM volume will occur. Moreover they have an economic
advantage over Alnico. They are commonly used in small DC commutator motors.
The key attributes of Ferrites are
•
Economical
•
Good for simple shapes only
•
Very fragile
•
Require expensive tooling
•
Temperature sensitive (0.2%°C).
Hard Ferrites
Hard Ferrite has normal operating capabilities between -40°C and +250°C. As
temperature increases, remanence decreases whereas coercivity increases. At very low
temperatures there is a risk of permanent demagnetization in magnet systems.
When it appeared that no further significant improvements would be made to ferrite
magnets the search began in 1960’s for other materials with high saturation
magnetization. [1]
Rare Earth Permanent Magnets
It is the newest type of permanent magnets that has been widely used in the
last two decades. It is mostly used in electrical machines. The elements of this type of
magnets are natural minerals that are widely available and used as mixed compounds.
High performance of rare earth magnets has successfully replaced Alnico and ferrite
20
magnets in all applications where the higher efficiency is required. Samarium cobalt
SmCo5 has the advantage of high remanent flux density, high coercive force, linear
demagnetization curve and low temperature coefficients. It is well suited to build
motors with low volume and high power density. It faces a major draw back which is
its high cost due to the lack of Sm and Co.
In 1983, researchers discovered the inexpensive neodymium Nd which
lowered the raw material cost. NdFeB magnets have better magnetic properties than
SmCo5 but only at room temperature. Their disadvantage lie in the fact that their
demagnetization curves mainly the coercive force is strongly temperature dependant.
However this magnet faces corrosion. In year 2002, these magnets showed higher
remanent magnetic flux density and better thermal stability.
The permanent magnet motor was conceived by Howard Robert Johnson
sometime after the 1940s. Allegedly it is a design for a perpetual motion machines.
Reportedly, the device is designed on the principle that a constant imbalance of the
magnetic forces between the rotor and the stator is created. [8]
The key attributes of Samarium cobalt are
•
Quite expensive
•
High Bmax
•
Very good temperature stability
•
Powerful for size
The key attributes of Neodymium Iron Boron are
•
High energy for size
•
More economical than Samarium Cobalt
•
Good in ambient temperature situations
21
•
Relatively high price
•
Corrosion that can result in loss of energy
•
Temperature coefficient of 13% degree centigrade.
Application of Permanent Magnets in Motors
The most important application of permanent magnets is in electric motors.
With the development of these materials and the introduction of the rare earth
magnets, more focus was directed towards the electronic devices, since these magnets
provided a shift towards the electronic evolution of electric motors.
Permanent magnets are widely used in DC Motors. Recent DC motors widely
employ Rare Earth Magnets. Another category of motors where permanent magnets
are used is Stepper Motors .Less widely used, but still effectively employed, are the
Synchronous Motors with permanent magnets.
We note here that since our model is a conventional synchronous motor, our
concern will later be to realize the effectiveness of using a permanent magnet in the
existing motor. [8]
Benefits of using permanent magnets in electrical machines [8]
•
The field excitation circuit in the case of electromagnetic excitations will
exhibit excitation losses due to the energy absorbed by that circuit. However in
the permanent magnet machines no such excitation exists therefore we have an
increase in efficiency
•
Consequently a higher torque or output power is established in the system
•
With the presence of a permanent magnet a higher magnetic flux density will
be established in the air gap
•
Lower complexity in construction
22
•
Permanent magnets implies that the machine is brushless therefore
maintenance is simplified
•
Lower prices for certain types of machines.
Application of Permanent Magnets in DC-Motors
Permanent magnets are widely used in DC motors. In conventional DC
motors, the armature windings provide the technique for controlling the speed of the
motor; also the field winding provides the excitation of the motor. However with the
introduction of permanent magnets into the DC machine, we started having more
efficiency and less complexity in the provision of magnetic field.
Lately, the range of application of these motors broadened especially with the
use of the high energy rare earth magnets, since they are able to produce larger
magnetic fields with a much smaller and much lighter magnet. With these permanent
magnets being used, brushes and commutator segments became unnecessary. The PM
DC motor is also referred to as Brushless DC motor. One of the basic advantages of
having a permanent magnet, especially rare earth magnets, in the DC machine is that
it will eliminate the mechanical switching of the armature current. This switching will
be performed electronically in the presence of the magnet. Also, by using rare earth
magnets, the rotor of the machine will be built with lower inertia; also these magnets
allow a higher air gap flux which means a higher output torque. With the high
coercivity the rare magnet have, an improvement of resistance to demagnetization
from motor's own armature winding is obtained.
DC brushless motors are widely used in automobiles, blowers, starters,
radiator cooling fans, and computer hard disk drives. [8]
23
Application of Permanent Magnets in Stepper-Motors
A stepper motor is known to rotate in a sequence of discrete steps. With this
manner of operation, they are usually digitally controlled. They are commonly used in
application of incremental motion for example printers, plotters, and computer
peripherals.
The mode of operation of a stepper motor requires a variable reluctance to be
established between the stator and the rotor. Also we need an air gap flux to be
produced. In conventional stepper motors, this flux is solely produced by the armature
winding. The use of a permanent magnet will help in the production of this flux and
will therefore increase the air gap flux. With this increase in flux, the torque will
increase, and efficiency is improved.
In hybrid stepper motors, the permanent magnet is built into the rotor. Soft
iron rotor cups will sandwich this magnet. The permanent magnet will help in the
creation a strong holding torque, which is one of the major characteristics of the
stepper motor. The armature winding usually switches the magnetic field to different
angular positions in the air gap; the use of permanent magnet is to maximize the
difference in magnetic fields in the air gap so as to maximize the holding torque.
Alnicos are widely used in hybrid stepper motor. Some designs use rare earth magnets
since they provide lower inertia. [8]
Another type of stepper motors that uses the permanent magnet is called the
can-stack motor. The difference between the hybrid and the can-stack motor is that
the latter has no iron rotor cups, and the permanent magnet is cylindrical with a shaft
passing through it. Ceramic ferrites are commonly used in this motor. Also rare earth
magnets are being used recently. These motors are used in applications large volumes.
24
Application of Permanent Magnets in Synchronous-Motors
With the introduction of permanent magnets to synchronous motors,
commutation was cancelled. The permanent magnet in a synchronous motor will
rotate in synchronous with the armature field. It will produce a maximum torque when
the magnetic field from the permanent magnet and that of the armature are at 90º
difference.
Synchronous motors with permanent magnets are good in applications of
constant supply voltage and constant frequency; they are also good for applications of
variable frequencies. These motors have a better efficiency, higher power factor, and
higher power density. However these machines have a weak starting torque.
Permanent magnet synchronous motors performs like the conventional motors
once the flux from the magnet takes over and allows the synchronization of rotor and
stator. We must note that the initial field excitation is performed by the permanent
magnet itself.
Usually, high coercivity permanent magnets are used in synchronous
machines, because the synchronization speed will force the magnet material to
experience a strong demagnetization field. Ceramic ferrite and rare earth magnets are
mostly employed in synchronous motors. [8]
These motors are commonly used with ratings of 15 KW .They are also
available with ratings up to 746 KW. Recent developments can reach 1MW using rare
earth permanent magnets. [8]
PM synchronous motors have five classical rotor construction configurations: [
1. Merrill's Rotor:
It was the first successful construction of PM synchronous
motor. It is characterized by small power ratings and high frequency.
25
Alnico permanent magnet is used for this configuration. Alnico is
placed on the shaft with the help of aluminum sleeve. It is important to
note that the PM will not be demagnetized since the applied reverse
flux at starting or reversal will only pass through the laminations and
slots and not through the PM.
2. Interior Type PM motor:
This type of motor used for high frequency and high speed. It
has a high protection against demagnetization because the flux line can
pass through the rotor without passing through the PM.
3. Surface PM motors:
This type of motor will have its magnet magnetized radially.
Sometimes an external non ferromagnetic cylinder is used for the
magnetization process. It has a simple construction; however it has a
lower air gap magnetic flux than other types of PM synchronous
motors. Also one of it main disadvantages is that the permanent
magnet of the surface PM motors is not protected against
demagnetization.
4. Inset Type PM motors:
This type has it magnet embedded in shallow slots of the rotor.
Also, its magnet is magnetized radially. Just like the surface PM
motors, this motor requires a non ferromagnetic cylinder. The emf
induced by these motors is lower than that of the surface motor.
26
5. Buried PM motors
These have circumferentially magnetized permanent magnets
that are embedded in deep slots. It needs a non ferromagnetic shaft
since using ferromagnetic shafts will cause a large portion of useless
magnetic flux to go through the shaft. These motors have the largest air
gap magnetic flux density among all the other types. The permanent
magnet in the rotor is protected against armature fields i.e.
demagnetization. However it is relatively complicated in construction.
27
Permanent Magnet Design
Magnet Choice
In permanent magnet synchronous machines the choice of magnet type is of
vital importance. Usually, high coercivity permanent magnets are used, since the
synchronization speed will force the magnetic material to experience a strong
demagnetization field. Ceramic ferrite and neodymium iron boron are mostly
employed in such machines.
Table 2: Magnet Properties [1]
Br (T)
HC (KA/m)
Ceramic Ferrites
(Magnetic Remanence)
0.4
Coercivity
265
1.15
Sintered NdFeB
1.1
700
1.05
µr
As shown in table 2, the magnetic remanence and the coercivity of ceramic
ferrites is lower than that of sintered NdFeB. High remanence means that the magnet
can support higher magnetic flux density in the air gap of the magnetic circuit. While
high coercivity means that a thinner magnet can be used to withstand the
demagnetization field. In this project, the optimization procedure will be implemented
on both types and a later comparison between the two will show the effect of the
magnet type on the overall machine.
Rotor Configuration
As shown in the previous sections, PM synchronous motors have five classical
rotor construction configurations, the Merrill's Rotor, the Interior Type rotor, the
Surface Type rotor, the Inset Type rotor and the Buried type rotor. For the purpose of
this project, the surface type configuration will be selected. This configuration doesn’t
concentrate the flux in a process known as flux focusing as other configurations do.
28
This is vital for this project since flux focusing is normally us with magnets of low
remanence, in order to increase the flux in the air gap. But this will cause undesirable
saturation if it is used with magnets of high remanence such as NdFeB. In addition,
this configuration will have its magnet magnetized radially. Moreover, the equations
that govern the surface type configuration are simpler than other configurations. Thus
the design theory will be simplified. The configuration of the rotor is cylindrical with
no opening in the middle as in the other configurations; this will simplify the
manufacturing and assembly procedure. For this reason, cost will be reduced.
Figure 22: Surface Type Mounted Configuration
The wound rotor of the synchronous machine is replaced by a permanent
magnet rotor using Ceramic Ferrites and NdFeB magnets. The introduction of a
permanent magnet into the machine will increase the magnetic remanence and hence
will improve the magnetic loading which will increase the output torque. Furthermore,
split ratio analysis is applied so as to obtain the optimal machine parameters that will
produce an improved electric loading and thus further augment the output torque.
29
Figure 23 illustrates the rotor configuration along with the machine parameters:
Figure 23:Rotor Configuration
Machine Parameters
La : the length of the machine
As : slot area
Sd : slot depth
tP : tooth pitch
Wt: tooth width
dbi : back iron depth
Ns : total number of slots
Lm : magnet thickness
Lg : air gap
Dr : rotor diameter
Do : stator diameter
Kp : packing factor
Q : electric loading
Bg : air gap magnetic flux density
ab : slot outer width
cd : slot inter width
D1 : distance from center to the beginning of the slot
D2 : distance from center to the ending of the slot
Kp : slot packing factor
Pc : copper losses
30
Split Ratio Analysis
The split ratio analysis [1][2][3] is a technique that optimizes the output torque of
the machine by finding the corresponding rotor to stator diameter. This is performed
by formulating the output torque as a function of the machine parameters, then
differentiating it with respect to the split ratio and setting its value to zero.
Torque Equation Derivation
The force (F) acting on a current-carrying conductor in a magnetic field (B) is
F = B.I .La where I is the current passing through one conductor and La is the active
axial length.
Figure 24: Machine Representation
The torque per conductor is T = F .
Dr
D
= B.I .La . r
2
2
Therefore, the torque for N conductors is:
T = F.
Dr
D
= Bav .I .La . r .N
2
2
31
Due to the fact that the value of the current I varies from one design to another, the
electric loading will be considered as a more suitable parameter to account for Dr and
N.
The electric loading:
Q=
N .I rms
(amps / m)
π .Dr
⇒ I rms =
Q.π .Dr
N
Substituting Irms in the torque expression gives:
T = Bav ×
∴T =
π
2
Q × π × D2
D
× La × 2 × N
2
N
× Q × D22 × Bav × La
Split Ratio Equation
The Torque of any machine is defined by the output power divided by the
angular speed. T =
∴P =
π
2
P
ω
⇒ P = Tω .
Dr2 La Bg Qω where Bg is the magnetic flux density in the air gap which is
approximated to be equal to Bav
2
⎛ Dr ⎞ La
P
Bg Qω
2
=
⎜
⎟
The power to volume ratio:
V
⎝ D0 ⎠ Lo
2
⎛D ⎞ L
T
Thus, the Torque to volume ratio: = 2 ⎜ r ⎟ a Bg Q
V
⎝ D0 ⎠ Lo
Let ξ be the split ratio
ξ=
Dr
Do
32
Thus,
L
T
= 2ξ 2 a Bg Q
V
Lo
In order to optimize the torque, both the electric loading Q and magnetic loading Bg
should be formulated for this specific machine.
Magnetic Loading
It is defined as the magnetic flux density in the air gap Bg =
Br
1 + µr
lg
lm
Electric Loading
The electric loading is defined by Q =
Where J =
Thus Q =
JAs K p N s
π Dr
Pc
As ρ La
K p Ns
π Dr
Pc As
ρ La
Using the geometry of the surface type chosen configuration, which is shown in
figure23.
2
Dr2 ⎤
1 ⎡ ⎛ D0 − 2dbi ⎞
−
As =
π
⎢π ⎜
⎥ − Wt ( Do − dbi − Dr )
⎟
4
4 ⎥⎦
N s ⎣⎢ ⎝
⎠
Non-Saturation Criteria
In addition, the machine should avoid saturation, thus the magnetic field
density in the iron should not exceed Bmax=1.1T. Applying the Continuity Theorem of
the flux, the flux in one slot of the air gap is equal to the flux corresponding to one
slot in the iron of the stator.
33
φairgap = φtooth
Bg × Aairgap / tooth = Bmax × Atooth
Bg × t p × La = Bmax × Wt × La
Wt =
Bg × t p
(First Saturation Criteria)
Bmax
The flux is divided along the magnet in two
directions to two parts each of a value
φmax
2
.
φ slots = φback iron
φback iron =
N s × φ per slot
p
Bmax × d bi =
dbi =
N s × Wt
2p
N s × Bmax ×
Wt
2
p
(Second Saturation Criteria)
Geometry Parameters
Other machine parameters are obtained directly from the geometry of the machine.
tp =
π × Dr
Ns
D1 = Do − 2dbi
D2 = Dr + 2lg
ab =
π × D1
cd =
π × D2
sd =
Ns
− Wt
Ns
− Wt
D1 − D2
2
34
AS Formulation
Using the saturation criteria and substituting them in As, we get:
2
⎤
1 ⎡ ⎡ ( OD − 2dbi ) Dr 2 ⎤ ( OD − dbi − Dr )
⎢π ⎢
.N s .Wt ⎥
−
As =
⎥−
4
4 ⎥⎦
2
N s ⎢ ⎢⎣
⎥⎦
⎣
As =
2
π OD 2 ⎡ ( OD − 2dbi )
⎢
4 N s ⎢⎣
OD 2
−
⎤
Dr 2 ⎤ ⎡⎛ OD π B p Dr Dr ⎞
−
−
⎥ − ⎢⎜
⎟ .N s .Wt ⎥
2
OD ⎥⎦ ⎣⎝ 2
2 Bm p
2 ⎠
⎦
π OD 2 ⎡
2
4 π B p Dr
π 2 B p Dr 2
Dr 2 ⎤
+4
−
As =
⎢1 −
⎥
4 N s ⎣⎢ OD 2 Bm p
4 Bm 2 p 2OD 2 OD 2 ⎥⎦
⎤
D B
π B p Dr B p
1 ⎡ OD B p
π Dr −
π Dr − r p π Dr ⎥
−
⎢
2 Bm p Bm
2 Bm
N s ⎣ 2 Bm
⎦
As =
As =
2
π OD 2 ⎡ Dr 2 ⎛ π 2 B p
⎢
⎜
4 N s ⎣⎢ OD 2 ⎝⎜ p 2 Bm 2
π OD 2
4 Ns
2
⎞ Dr ⎛ 2π B p 2 B p ⎞ ⎤
Bp
2π B p
2
1
+
+
−
+
⎟−
⎜
⎟ + 1⎥
p Bm 2
Bm ⎠⎟ OD ⎝ p Bm Bm ⎠ ⎦⎥
⎡⎣αξ 2 − βξ + λ ⎤⎦
Where
⎡⎛ Bg ⎞ 2 π ⎛ π
⎤
Bg
⎞
α = 2 ⎢⎜
− 1⎥
⎟ + ⎜ + 2⎟ + 2
p⎝ p
Bmax ⎥
⎢⎣⎝ Bmax ⎠
⎠
⎦
β =3
Bg ⎛ π
⎞
⎜ + 1⎟
Bmax ⎝ p ⎠
λ =1
35
Torque Formulation
Substituting the above equations in the Torque equation will lead:
T . y = ξ αξ 2 + βξ + λ
2(1 +
y=
Br
he
ρ
lg
lm
µr )
which is a constant
2.5
o a
D l
kp
Torque Optimization
Optimizing the torque by differentiating the right side of the equation with respect to
the split ratio:
dT
3
= 2αξ 2 + βξ + λ
dξ
2
Setting the above equation to zero, the split ratio corresponding to the optimal Torque
is determined by:
ξ=
−1.5β ±
(1.5β )
2
− 8αλ
4α
36
Code Formulation
To apply the above optimization procedure, a certain algorithm was followed so as to
obtain the optimal machine parameters and output values. The logic behind the
algorithm is shown in the flowchart of figure. The code for this flowchart is simulated
on MatLab (refer to appendix).
Optimization Results
The optimization results of the MatLab code are applied to permanent magnet
designs. The first design used a Ceramic Ferrite magnet while the second design used
a sintered NdFeB magnet. The results are listed in Table 3.
37
Table 3: Optimized Machine Parameters
Machine Parameters
Ceramic Ferrites
Sintered NdFeB
TP
5.7
4.1
Wt
1.3
2.5
dbi
5.8
11.4
D1
99.4
88.3
D2
65.7
46.5
ab
7.4
5.2
cd
4.4
1.5
Sd
16.9
20.9
La
136
136
Lm
5
5
Lg
0.5
0.5
Dr
64.7
45
Do
111
111
Making use of the preceding values, the calculated torque for both magnet designs is
TFerrites = 5.5 N .m , TNdFeB = 9.1 N .m
And the obtained electric loading is:
QFerrites = 17.25kA / m , QNdFeB = 20.5 kA / m
38
Magnet simulation
To test the obtained results, the optimized machine was modeled and
simulated using MagNet software. The flux and induced voltage were then computed.
The initial flux field diagrams are shown below:
Figure 25: Optimized Design Flux Lines using Ceramic Ferrites
Figure 26:Optimized Design Flux Lines using Sintered NdFeB
39
The maximum flux obtained in the air gap for both designs is found to be:
−3
φFerrites = 7.15 × 10 −3Wb and φNdFeB = 16.33 × 10 Wb
No load Voltage Calculation
To calculate the induced no load voltage Ea, the rotor was rotated in steps of 3 º
mechanical. This was performed by the following steps:
1. Sectionalizing the magnet of one pole of the machine into 30 segments
2. Assigning different polarities to each fragment of the pole and rotating the
pole in steps of three degrees.
The flux is recorded for each rotation at a particular point inside the air gap and its
variation for one cycle is graphed.
0.008
0.006
Air Gap Flux (Wb)
0.004
0.002
0
0
60
120
180
240
300
360
-0.002
-0.004
-0.006
-0.008
Electrical Degree
Graph 2: Flux Variation for Ceramic Ferrites
40
0.02
0.015
Air Gap Flux (Wb)
0.01
0.005
0
0
60
120
180
240
300
360
-0.005
-0.01
-0.015
-0.02
Electrical Degree
Graph 3: Flux Variation for Sintered NdFeB
Then, the difference between two consecutive flux values was calculated so as to
obtain the value of dφ . The value of dt was computed using the following relation:
⎛
⎞
3°
dt = ⎜
⎟ × 60sec = 0.33 m sec
⎝ 360°×1500rpm ⎠
Knowing that Ea = N
dφ
dt
and N per pole per phase is 180, the induced voltage is
obtained. The following plot illustrates the induced no load voltage for one electric
cycle for the two designs.
41
800
600
400
Ea (V)
200
0
0
60
120
180
240
300
360
-200
-400
-600
-800
-1000
Electrical Degree
Graph 4: Ea for Ceramic Ferrites
1500
1000
Ea (V)
500
0
0
60
120
180
240
300
360
-500
-1000
-1500
Electrical Degree
Graph 5: Ea for Sintered NdFeB
42
Comparative Analysis
The torque of the machine has increased up to 3 times using Ceramic Ferrite
magnets and 5 times using the Sintered NdFeB magnet while keeping the speed of the
machine constant and maintaining the machine size.
The output torque depends on the magnetic and electric loading of the machine.
This increase in the torque value for both designs is justified by the following reasons:
1. The permanent magnet that replaced the wound rotor coils has a magnetic
remanence of 0.4T for Ceramic Ferrites and 1.1T for Sintered NdFeB rather
than 0.3T of the conventional machine.
2. The stator slots became wider and thinner than that of the conventional
machine for both designs thus allowing more copper concentration per slot
which implies a greater current value and hence an improved value of the
electric loading.
3. Finally, the split ratio technique optimized the dimensions and parameters of
the machine in such a way so that the ratio of the rotor to the stator diameter is
maintained at optimal torque.
43
Table 4: Dimensions Comparison
Conventional
(mm)
Ceramic
Ferrites(mm)
Sintered
NdFeB (mm)
Tooth pitch
Tooth Width
7.8
2.9
5.7
1.3
4.1
2.5
Back iron depth
8.4
5.8
11.4
Distance from center to the
beginning of the slot
36
99.4
88.3
Distance from center to the
ending of the slot
47.1
65.7
46.5
Slot outer width (Stator)
4.08
7.4
5.2
Slot inner width (Stator)
3.01
4.4
1.5
Slot depth
12.1
16.9
20.9
Length of the machine
136
136
136
-
5
5
Air gap length
0.5
0.5
0.5
Rotor diameter
69.5
64.7
45
Stator diameter
111
111
111
Machine Parameters
Magnet thickness
Table 5: Parameters Comparison
Conventional
Ceramic
Ferrites
Output Torque(N.m)
1.9
5.5
9.1
No Load Voltage(V)
267
353.6
909.7
Flux(mWb)
3.9
7.15
16.33
14.85
17.25
20.5
Electric Loading(kA/m)
Sintered
NdFeB
44
Cost Analysis
With the different parameters obtained, the material cost of the different
designs is analyzed and the results are summarized in table:
The cost of the materials (copper, iron, magnets) is the typical values that were
recently recorded.
Table 6: Material Properties
Density
Cost [9]
(103 x kg/m3)
($/kg)
Cost/unit
volume
(103 x $/m3)
Copper
8.9
4.4
39.16
Steel
7.87
1.32
10.3884
Ceramic ferrites
4.9
5.5
26.95
Sintered NdFeB
7.4
77
569.8
Sintered NdFeB
Machine
Ferrite
Machine
Conventional
Machine
Copper
136.99 cm3
195.54 cm3
287.4 cm3
Steel
877.12 cm3
685.75 cm3
597.55 cm3
Ceramic ferrites
-
127.53 cm3
-
Sintered NdFeB
86.62 cm3
-
-
Table 7: Volume
45
Table 8: Cost
Sintered NdFeB
Machine
Ferrite
Machine
Conventional
Machine
Copper
$5
$8
$11
Steel
$9
$7
$6
Ceramic ferrites
$0
$3
$0
Sintered NdFeB
$49
$0
$0
Sintered NdFeB
Machine
Ferrite
Machine
Conventional
Machine
Total Cost
$64
$18
$17
Output Torque
9.10 Nm
5.50 Nm
1.90 Nm
Output Voltage
909.7 V
353.6 V
267 V
Table 9: Analysis
The material cost of each of the three designs is estimated above. The conventional
machine has a lower cost than the other two designs; however this design is restricted
in application since it has a lower torque and lower output voltage than any other
design. Knowing that the size of the machine is maintained constant, the higher output
torque and voltage of the optimized designs will have wider applications and efficient
space utilization. Moreover, the higher cost of the permanent magnet machine that
uses the sintered NdFeB is justified by the increase in the torque by five times.
46
Conclusion
With the availability of computer aided programs, optimization of electric
machines is simplified. Upon using MagNet, the analysis of a conventional
synchronous machine came up with satisfactory conclusions. Moreover experiments
performed in the lab, provided an insight in the study and investigation of the machine
magnetic flux and electrical parameters. The optimization that followed the
conventional machine analysis was performed using split ratio technique. With the
flux in the air gap being improved, the torque of the machine was augmented
significantly, making the machine more efficient in power generation. Moreover, the
optimized machine became a potential candidate for applications requiring high
output power in limited space allocation.
47
References
1. Birch T.S., Chaaban F. B., Howe D., Mellor P.H.,(1991) Topologies for a
Permanent Magnet Generator/Speed Sensor for the ABS on Railway Freight
Vehicles, EMD Conference, London. pp.31-35
2. Campbell, P. (1996) .Permanent Magnet Materials and their Application
3. Chaaban, F.B. (1989) Computer Aided Analysis, Modeling and Experimental
Assessment of Permanent Magnet Machines with Rare Earth Magnets, Ph.D.
Thesis, Liverpool University
4. Chaaban, F.B. (1993) Determination of the optimum Rotor/Stator diameter
ratio of Permanent Magnet Machines, IEEE Trans., pp.521-530.
5. Chapman, J. (2005). Electric Machinery Fundamentals
6. Coren, R. (1989)., Basic Engineering Electromagnetics
7. Elsevier Science Publishing. (1989). Computer-aided analysis and design of
electromagnetic devices
8. Gieras, F., Wing, M. (2002). Permanent Magnet Motor Technology
9. Ronghai Q., Thomas A. (2003) Dual-Rotor, Radial-Flux, Toroidally Wound
Permanent Magnet Machines, IEEE Transactions, Vol.39, No. 6
48
ACKNOWLEDGEMENT
The support of Professor F. Chaaban, American University of Beirut, who
assisted the authors in this report, is gratefully acknowledged.
49
Appendix A - Matlab Code
clc;
clear all;
%Magnet Parameters
ur=1.15;
Br=0.4;
%Motor Parameters
p=4;
Do=0.111;
Ns=36;
kp=0.4;
rho=1.72e-8;
Pc=20;%The total losses of a machine are taken to be 10%
%of the total output power of the machine which is 300W
%Therefore, the total power losses=30 W
%So, the copper losses as 20W since the core losses amd other
% losses constitute a low percentatge of the total losses
%at
rated values.
La= 0.136;
Bmax=1.6;
%Material Costs
CCU=53.4e3;%Copper
CST=7.87e3;%Iron
Cmag=1110e3;%Magnet
% Inital Conditions
lm=0.005;% Magnet Length
lg=0.0005;% Air Gap Length
%Calculating Optimum
Bg=Br/(1+ur*lg/lm);
a=(Bg/Bmax)^2*((pi/p)^2+2*pi/p)+2*Bg/Bmax-1;
b=-Bg/Bmax*(2*pi/p+2);
c=1;
a1=2*a;
b1=3/2*b;
ratio=(-b1-(b1^2-4*a1*c)^0.5)/(2*a1);
Dr=ratio*Do;
tp=pi*(Dr+2*lg)/Ns;
Wt=Bg*tp/Bmax;
db=Wt*Ns/(2*p);
d1=Do-2*db;
d2=Dr+2*lg;
ab=pi*d1/Ns-Wt;
cd=pi*d2/Ns-Wt;
Sd=(d1-d2)/2;
As=Sd*(ab+cd)/2;
Acu=As*kp*Ns;
T=((La*Bg^2*Dr^2*Pc*Acu)/(4*rho))^0.5;
50
J=(Pc/(Acu*La*rho))^0.5;
Q=J*Acu/(pi*Dr);
SSA=pi/4*(Do+d2)*(Do-d2)-As*Ns;
RSA=pi/4*(Dr-2*lm)^2;
SA=SSA+RSA;
Vm=pi*lm*La*(Dr-lm);
VCC=La*Acu;
VS=SA*La;
51