Application of Frequency and Modal Analyses for Predicting the Air Gap Flux and
Resulting Pressure Distributions within a Permanent Magnet Motor
by
David M. Fellegara
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
Master of Engineering
Major Subject: Mechanical Engineering
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
August, 2013
i
CONTENTS
LIST OF FIGURES .......................................................................................................... iv
LIST OF ACRONYMS .................................................................................................... vi
LIST OF SYMBOLS ....................................................................................................... vii
ACKNOWLEDGMENT ................................................................................................ viii
ABSTRACT ..................................................................................................................... ix
1. Introduction.................................................................................................................. 1
2. Background .................................................................................................................. 2
2.1
Modal Analysis .................................................................................................. 2
2.1.1
Example of the Application of Modal Analysis ..................................... 2
2.1.2
Normal and Complex Mode Shapes ...................................................... 4
2.2
Modal Analysis and Its Applicability to Understanding Air Gap Flux
Distributions ....................................................................................................... 5
2.3
Overview of Permanent Magnet Motor Machinery Design ............................... 6
2.3.1
Stator Core ............................................................................................. 6
2.3.2
Motor Rotor............................................................................................ 8
3. Theory / Methodology ............................................................................................... 10
3.1
Overview of Permanent Magnet Motor Principles of Operation ..................... 10
3.2
Permanent Magnet Motor Source Mechanisms ............................................... 10
3.3
3.2.1
Rotor Generated Machine Native Gap Flux Distribution Components 11
3.2.2
Stator Generated Machine Native Flux Distribution Components ...... 17
3.2.3
Effect of DC Offset on the Air Gap Flux Distribution......................... 24
3.2.4
Interaction between the Rotor and Armature Air Gap Flux Components
.............................................................................................................. 26
Calculation of Air Gap Electromagnetic Pressure Distribution ....................... 31
4. Results and Discussion .............................................................................................. 34
4.1
Validation of the Predicted Spatial and Temporal Gap Flux and Pressure
Distributions ..................................................................................................... 34
ii
4.1.1
4.2
Flux2D by Magsoft .............................................................................. 34
Comparison on EM FEA Results to Predicted Air Gap Flux & Pressure
Distributions ..................................................................................................... 36
5. Conclusion ................................................................................................................. 41
List of References ............................................................................................................ 42
Appendix A - Calculation of Air Gap Flux and Pressure Distributions and Plotting Script
................................................................................................................................... 44
iii
LIST OF FIGURES
Figure 1. Superposition of Individual Single Degree of Freedom Mode Shapes Results
in the Overall Structural Response of the Bell [1] ............................................................. 3
Figure 2. Simplified Depiction of Normal and Complex Mode Shapes [3] ..................... 5
Figure 3. Stator Lamination [4] ........................................................................................ 7
Figure 4. Stacked, Unwound, Stator Core [5] .................................................................. 7
Figure 5. Stacked and Wound Stator Core [6] .................................................................. 8
Figure 6.
Example of Surface Mounted (bottom) and Embedded (top) Permanent
Magnet Rotors ................................................................................................................... 9
Figure 7. N=6 Spatial Distribution of the Fundamental Rotor Flux Field for a 12 Pole
Machine ........................................................................................................................... 13
Figure 8. Frequency-Wavenumber Diagram Depicting the Rotor Generated Air Gap
Flux Components ............................................................................................................. 14
Figure 9. N=18 Spatial Distribution of the First Rotor Harmonic (3E 3P) Flux Field for
a 12 Pole Machine ........................................................................................................... 16
Figure 10. Stator Winding and Field Representation ..................................................... 18
Figure 11. Frequency-Wavenumber Diagram Depicting the Stator Winding Generated
Air Gap Flux Components ............................................................................................... 21
Figure 12. Phasor Diagram Depicting Vector Sum of Perfectly Sinusoidal Input Phase
Currents............................................................................................................................ 22
Figure 13. Spatial Distribution of the Armature Flux Field for a 12 Pole Machine ....... 23
Figure 14. Frequency-Wavenumber Diagram Depicting the DC Offset Generated Air
Gap Flux Components ..................................................................................................... 26
Figure 15. Illustration of Uniform Flux Distribution Being Concentrated at the Stator
Teeth ................................................................................................................................ 27
Figure 16. Permeance Variation Across the Stator Teeth Over Two Poles.................... 28
Figure 17. Frequency-Wavenumber Diagram Depicting the Air Gap Flux Distribution
as a Result of Rotor and Stator Interaction ...................................................................... 30
Figure 18. Frequency-Wavenumber Diagram Depicting the Complete Air Gap Flux
Distribution ...................................................................................................................... 31
iv
Figure 19. Frequency-Wavenumber Diagram Depicting the Resultant Air Gap Pressure
Distribution ...................................................................................................................... 33
Figure 20. 12 Pole, 6 Slots per Pole, Flux2D EM FEA Model ...................................... 36
Figure 21. Frequency-Wavenumber Diagram for the Rotor Generated Air Gap Flux
Components, EM FEA Result ......................................................................................... 37
Figure 22. Frequency-Wavenumber Diagram for the Stator Winding Generated Air Gap
Flux Components, EM FEA Results ............................................................................... 38
Figure 23. Frequency-Wavenumber Diagram Depicting the Complete Air Gap Flux
Distribution ...................................................................................................................... 39
Figure 24. Frequency-Wavenumber Diagram of the Air Gap Pressure Distribution, EM
FEA Model Results.......................................................................................................... 40
v
LIST OF ACRONYMS
AC
Alternating Current
EM FEA
Electromagnetic Finite Element Analysis
MMF
Magnetomotive Force
vi
LIST OF SYMBOLS
ω
Angular Frequency
θ
Angular Position (degrees)
i
Current (amps)
I
Current (amps)
I
Current (amps)
Φ
Magnetic Flux (Wb)
dB
Decibels
B
Flux (Tesla)
Hz
Frequency (Hz = 1 / seconds)
m
Integer
n
Integer
MMF
Magnetomotive Force (Ampere - turns)
P
Number of Pole Pairs
RCircuit
Reluctance (Ampere-Turns / Wb)
S
Number of Stator Slots
µ0
Permeability of Free Space (4πx10-7)
P
Permeance
psi
Pressure (lbs/in2)
R
Rotational Harmonic or Rotational Rate (Hz)
T
Tesla
t
Time
N
Wavenumber
vii
ACKNOWLEDGMENT
I would like to thank Professor Ernesto Gutierrez-Miravete for his support throughout
the completion of my Master’s Project. I would also like to thank my colleagues for
sharing their knowledge of Flux2D software. Lastly, I would like to thank my family
and friends for their continued support throughout my academic career.
viii
ABSTRACT
Some synchronous wound permanent magnet motors require substantial amplitudes of
time harmonics in their input current waveforms in order to make full use of the flux
waveform generated by the permanent magnets. The role of these harmonics can be
understood from the perspective of classical sine wave motor methods by generalizing
sine wave motor magnetomotive force and flux expressions to include all of the relevant
Fourier components, as opposed to just the fundamental which is typically studied. The
purpose of this paper is to investigate, and ultimately predict, the role of temporal and
spatial harmonics in the production of torque in permanent magnet motors from the
standpoint of sine wave motor theory as well as predicting the magnetic and electric
fields generated by the rotating array of permanent magnets and synchronous windings
of a permanent magnet machine and determining their distribution in space and their
behavior with frequency.
ix
1. Introduction
In synchronous wound permanent magnet electric motors, the alternating current within
the stator windings produces a traveling magnetic wave in the air gap of the machine.
This is interchangeably referred to as the armature field, or the stator field. Each
magnetic pole on the rotor is attracted to its opposite pole in the armature field. This
force of attraction between the armature field and magnetic field of the rotor mounted
magnets causes the rotor of the motor to spin synchronously with the armature field.
The resulting flux field in the air gap of the motor represents a linear superposition of
both the armature and rotor fields.
This resultant flux field can be decomposed into Fourier components. The product of
this Fourier decomposition is a frequency-wavenumber spectrum that is rich in
harmonics due to things like rotor pole and stator slot discretization and imperfect
current waveforms within the stator windings.
These discrete Fourier components
interact to create radial and torsional pressures, or forces, which ultimately produce
torque and often generate additional structural vibrations on the machine.
Likewise, Fourier decomposition of the individual armature and rotor fields can be
obtained to identify the contribution of the two fields to the resulting linear superposition
of the two fields (i.e. the resultant flux and pressure fields).
In order to fully understand the electromagnetic characteristics of a of a permanent
magnet motor and the electromagnetic design impacts on structural vibrations and torque
production of a machine, the Fourier components that interact to generate the air gap
flux and pressure distributions that act on the rotor and stator need to be evaluated and
well understood. As this paper will discuss, modal and frequency analyses can be used
to aid in understanding, and predicting, these complex machine electromagnetic gap flux
characteristics.
1
2. Background
In order to be able to apply modal analysis methodologies and theories to understanding
and predicting air gap flux distributions within a rotating permanent magnet motor, one
must have a thorough understanding of modal analysis and how it can be applied to flux
distributions.
2.1 Modal Analysis
A simple definition of modal analysis can be made by comparing it to frequency domain
analysis.
In frequency domain analysis, a complex time domain signal can be
decomposed into a set of individual sine or cosine waves, each with a specific frequency
and amplitude, with a certain phase relationship relative to other individual sine or
cosine waves. In modal analysis, a complex deflection pattern can be decomposed or
resolved into a set of individual mode shapes, each with a specific frequency and
damping. In this report, both frequency and modal analysis will be utilized to evaluate,
and ultimately predict, the modal, also referred to as spatial, and frequency, also referred
to as temporal, components of an air gap flux distribution within a rotating permanent
magnet motor.
2.1.1
Example of the Application of Modal Analysis
For completeness, it is worth reviewing how a structural response can be represented in
different domains, how modal descriptions relate to descriptions in the spatial, temporal,
and frequency domains, and how these structural response representations in the
different domains apply to gap flux distributions as opposed to structural responses. For
example, the response of a bell, which is a lightly damped structure, can be analyzed or
evaluated using modal analysis methodologies.
When struck, a bell produces an
acoustic response containing a limited number of pure tones. The associated vibration
response has exactly the same pattern, and the bell dissipates the energy from an impact
by vibrating at particular discrete frequencies. In Figure 1, each column shows the
response of the bell represented in the different domains. In the physical domain, the
2
complex geometric deflection pattern of the bell can be represented by a set of simpler,
independent deflection patterns, or mode shapes. In the time domain, the vibration
response of the bell can be represented by a set of decaying sinusoids over time. Lastly,
in the frequency domain, analysis of the time domain response gives a frequency
spectrum containing a series of resonances, which are the superposition of the individual
response spectra. In all three domains, the overall response of the bell is the result of
individual modal responses superimposed with each other.
Each individual modal
response can be defined by the modal frequency, modal damping, and model
displacement or mode shape, which together forms a complete description of the
dynamic characteristics of the bell.
Figure 1. Superposition of Individual Single Degree of Freedom Mode Shapes Results
in the Overall Structural Response of the Bell [1]
As discussed in the example above, a mode shape is a deflection pattern associated with
a particular modal or resonant frequency. It is a parameter which defines a deflection
3
pattern as if that mode existed in isolation from all others within a system. In this
example, the actual physical displacement at any point will always be a combination of
all the mode shapes of the system. However, a mode shape is an inherent dynamic
property of a system. It represents the relative displacements of all parts of the system
for that particular resonant or modal frequency [1].
2.1.2
Normal and Complex Mode Shapes
Modal displacements, or more simply, mode shapes can be divided into two categories:
normal modes and complex modes. Normal modes of a system are often defined as
patterns of motion in which all parts of the system move sinusoidally with the same
frequency and with fixed phase relation [2]. However, for the purposes of this paper, a
mode shape holds that same definition but in respect to a flux profile rather than a profile
of displacements. Normal modes are characterized by the fact that all parts of the
◦
system are moving either in phase, or 180 out of phase with each other. The modal
displacements are therefore real and are positive or negative. Normal mode shapes can
be thought of as standing waves with fixed node lines. An air gap flux distribution can
be evaluated as a superposition of its normal modes, or normal flux profiles. The modes
are normal meaning they can move independently. Therefore, the excitation of one
mode will not cause, or cross couple, resulting in flux in a different mode shape. As
discussed previously, a mode can be characterized by a modal frequency and a modal
shape. The mode shape, N, is numbered according to the number of waves, or a single
sinusoidal wave, in the flux distribution. An example of an N=6 is presented in Figure 7
and discussed later on in Section 3.2.1. Understanding normal modes is important in
understanding and predicting air gap flux distributions. Complex modes can have any
phase relationship between the different parts of the system. The modal displacements
are complex and can have complex phasing associated with them. Complex mode
shapes can be considered as propagating waves with no stationary node lines. An
example of normal and complex modes can be seen in Figure 2 where the multiple lines
reflect displacements of the beam at different instances in time.
4
Normal Mode
Complex Mode
Standing Wave
Propagating Wave
Node Point
Figure 2. Simplified Depiction of Normal and Complex Mode Shapes [3]
2.2 Modal Analysis and Its Applicability to Understanding Air Gap
Flux Distributions
It is important to note that, in the examples provided previously, structural displacements
of a bell and a cantilevered beam were used to describe modal analysis and the
relationship between the spatial, temporal, and frequency domains.
Structural
displacements are easier to visualize when discussing or describing modal analyses
because they are real and tangible. However, these analysis methodologies can be
further extended to analyze not only modal displacements, but electric motor air gap flux
distributions as well. In this paper, an extension of these analysis methodologies are
applied to predicting the air gap flux and pressure distributions in permanent magnet
motors, and the resulting force distribution on both the rotor and stator physical
structures resulting in structural vibrations.
5
2.3 Overview of Permanent Magnet Motor Machinery Design
In a permanent magnet motor, the rotor is the moving part on which the magnets are
mounted which turns the shaft to deliver mechanical power or torque. The stator of a
permanent magnet motor is the stationary portion of the motor which contains copper
windings in which currents flow. The air gap of the machine refers to the area between
the rotating rotor and the stationary stator.
2.3.1
Stator Core
Generally speaking, the stator core is the same for all alternating current permanent
magnet motors. The stator core consists of a stack of thin sheets of ferromagnetic steel
commonly referred to as stator core laminations, Figure 1. The stator core laminations
are usually on the order of 20 mils thick and are electrically insulated from each other.
The laminations are often held together under an axially applied compression load on the
stator core. An example of a stacked, unwound, stator core can be seen in Figure 4.
Each stator core lamination consists of three subcomponents; the stator tooth, the stator
slot, and the stator backiron.
These three subcomponents can be seen graphically
depicted in Figure 3 and Figure 4.
6
Stator Backiron
Stator Slot
Stator Tooth
Figure 3. Stator Lamination [4]
Stator Backiron
Stator Slot
Stacked
Laminations
Stator Tooth
Figure 4. Stacked, Unwound, Stator Core [5]
7
Copper coils are wound within the slots of the stator core which carry current to induce a
rotating magnetic field on the stator. Coils consist of a certain number of turns of copper
conductors. Each coil belongs to a certain phase of the machine. The individual coils
are then connected together to form closed circuits within the machine. The collection
of stator coils and connections is often referred to as the stator windings. An example of
a stacked and wound stator core can be seen in Figure 5.
Stator
Windings
Figure 5. Stacked and Wound Stator Core [6]
2.3.2
Motor Rotor
For permanent magnet machines, the rotor can be one of two designs. Either a surface
mounted magnet or an embedded permanent magnet rotor. For surface mounted magnet
rotors, the permanent magnets are simply attached to the surface of a steel rotor rim. For
an embedded permanent magnet motor, the motor rotor consists of two key elements, the
rotor poles and the permanent magnets themselves. The rotor poles are constructed
8
much like the stator core in the sense that the poles are comprised of laminations that are
stacked together to create a pole the length of the machine. The benefit if an embedded
permanent magnet rotor is increased efficiency due to its ability to concentrate the rotor
flux field through the pole heads. For both rotor topologies, the magnets are arranged to
have opposing polarities as viewed circumferentially around the rim of the rotor. Both
an embedded permanent magnet (top) and surface mounted (bottom) rotors can be seen
in Figure 6.
Figure 6. Example of Surface Mounted (bottom) and Embedded (top) Permanent
Magnet Rotors
9
3. Theory / Methodology
3.1 Overview of Permanent Magnet Motor Principles of Operation
In synchronous wound permanent magnet electric motors, the alternating current within
the stator windings produces a traveling magnetic wave, or flux field, in the air gap of
the machine. This is interchangeably referred to as the armature field, or the stator field.
Each magnetic pole on the rotor is attracted to its opposite pole in the armature field.
This force of attraction between the armature field and magnetic field of the rotor
mounted magnets causes the rotor of the motor to spin synchronously with the armature
field. The resulting flux field in the air gap of the motor represents a linear superposition
of the armature and rotor fields. In addition to those that produce torque, the armature
and rotor fields have additional flux components that are dependent on the fundamental
design characteristics of a machine such as the discretization of the rotor poles and the
stator windings.
3.2 Permanent Magnet Motor Source Mechanisms
Alternating Current (AC) permanent magnet motors create gap flux distributions which
result in air gap pressure forces through a large variety of mechanism which must be
well understood and, to a certain extent, quantified by analyses, in order to develop
electric machine designs which meet their life and vibration requirements and machinery
health and fatigue standards. These air gap generated forces are often categorized into
two distinct categories; machine native forces and forces resulting from current
energized windings. Machine native forces are defined as those produced by the ideally
manufactured machine either when the machine is at open circuit (i.e. no currents in the
stator windings / un-energized stator windings) and at loads when the stator windings are
energized with a single frequency, perfectly sinusoidal current waveform. Ideal motor
native forces originate from the discrete geometric design features that shape the gap
flux distribution (e.g. rotor poles, slot geometry, and stator winding topology).
Analytical modeling approaches to predict these sources are discussed in the following
sections and validation of these numerical modeling analyses is discussed in Section 4.
10
The second set of sources, or current distortion induced forces, are those produced by the
machine at loaded conditions in response to distortions in the currents supplied to the
motor (i.e. non-ideal perfectly sinusoidal input currents).
In the simplest view, a
permanent magnet motor can be idealized as an actuator that converts amps on the input
terminals into shear stress (i.e. torque forces) and radial forces. Therefore, almost any
current energizing the windings will produce some sort of air gap force within the
machine. This paper will focus on predicting the former of these sources rather than the
later as those associated with distortion on the input currents are much more difficult to
analytically compute due to their strong dependence on machine characteristics such as
winding topology of the distributed phase currents within the wound stator core and
number of phases within the machine. However, a single current related source, that
being a DC offset in the input phase currents, will be touched upon as an example of
how currents effect the overall gap flux distribution within a machine due to its lack of
dependence on some of the machine characteristics identified above. One additional
source, that being the potential for manufacturing asymmetries which could potentially
affect the overall gap flux distribution, will not be discussed in this paper. The following
sections discuss how both the armature and rotor flux fields can be predicted for
machine native sources as well as the sources associated with a DC offset on the input
phase currents.
3.2.1
Rotor Generated Machine Native Gap Flux Distribution Components
As previously stated, torque is produced by the interaction of stator armature and rotor
magnetic fields of like spatial pole number. It is known that the air gap flux density due
to the rotor magnets contains space harmonics, which may, in some machines, be
substantial. The spatial distribution of the rotor generated air gap flux density can be
seen in Equation (1) where θ is the angular position in the air gap, in electrical degrees,
as rotated circumferentially around the machine [8]. It should be noted that B refers to
either the radial or torsional components of air gap flux.
= 1 cos
− 3 cos3
+ 5 cos5
− 7 cos7
+ 9 cos9
− ⋯ (1)
11
It is important to note that these spatial harmonics, as presented in Equation (1), rotate
synchronously with the rotor, as identified by the lack of time dependence on the
distribution. The effect of time variation can be accounted for by rewriting Equation (1)
in a traveling wave form, as seen in Equation (2), where ω is the angular frequency of
the fundamental component of flux variation at any fixed point in the stator, and t is
time.
, = 1 cos − − 3 cos3
− 3 + 5 cos5
− 5
− 7 cos7
− 7 + 9 cos9
− 9 − ⋯
(2)
Any of these traveling wave components may contribute to the production of torque if
there exists a stator field, or stator magnetomotive force (MMF), traveling with the same
number of spatial poles (i.e. stator generated air gap flux component with the same
spatial order). An MMF is defined as any physical driving (i.e. motive) force that
produces magnetic flux [9]. If the angular frequency of such a stator generated MMF
wave is the same as that of rotor field flux wave, the result is steady motor torque.
Otherwise, the relative motion of the rotor field and stator MMF waves results in
unsteady, or pulsating, torque.
As presented in Equation (2), the fundamental rotor flux term is the result of the first
component in the Equation (2) Fourier series representing the rotor generated gap flux
distribution. The fundamental rotor flux is often the largest magnitude rotor generated
flux component and exists in the spatial domain at one times the number of pole pairs
(i.e. 1*P) and rotates in the frequency domain at one times the electrical line frequency
(i.e. 1*E). For example, Figure 7 shows the spatial distribution of the rotor fundamental
flux for a machine that has 12 poles (i.e. 6 pole pairs). The N=6, or one times the
number of pole pairs, P, spatial pattern can be seen in Figure 7 by the six sine waves that
describe the flux distribution as rotated circumferentially around the machine from θ=0
to θ=360.
12
+
-
Θ=0
+
+Θ
+
-
-
+
+
-
+
Figure 7. N=6 Spatial Distribution of the Fundamental Rotor Flux Field for a 12 Pole
Machine
Harmonics of the rotor fundamental flux exist at higher spatial and temporal orders and
are the result of the additional Fourier components in the Equation (2) Fourier series.
These harmonics are usually much smaller in amplitude than the fundamental as they
result in unsteady, or pulsating/parasitic, torque if there is no stator field, or stator MMF,
traveling with a comparable number of spatial poles. The amplitude of these rotor pole
harmonics can be reduced with simple machine design changes like rotor pole shape in
an effort to improve machine efficiency.
The objective of standard sine wave motor design is to eliminate, or minimize to the
extent possible, all components of B(θ,t) and the stator generated MMF wave except for
the fundamental of each. The interaction of the two fundamentals produces the desired
steady torque. However, in some machine designs, many of the harmonic traveling
13
waves are left in the B(θ,t) and, in some cases, attempts are made to produce
corresponding traveling MMF waves with the stator windings to exploit the presence of
the various harmonic components of B(θ,t) for steady torque production.
The 1*E, 1*P rotor fundamental flux and the higher order harmonics of the rotor
fundamental can be visualized to exist on a frequency-wavenumber diagram as depicted
in Figure 8. A description of the frequency-wavenumber diagram can be found in in
Section 3.2.1.1.
10E
4
2
1
10E
3.8
5E
3.6
5E
3.4
Amplitude (dB re: Tesla)
0E
3.2
10P
5P
5P
10P
3
Fundamental
Rotor Flux
(1E , 1P)
5E
Frequency
(E = Electrical Line Frequency)
Rotor
Harmonics
2.8
2.6
5E
10E
2.4
2.2
10E
3
4
2
10P
5P
0P
5P
10P
Circumferential Order
(P = Number of Pole Pairs)
Figure 8. Frequency-Wavenumber Diagram Depicting the Rotor Generated Air Gap
Flux Components
3.2.1.1 Understanding the Frequency-Wavenumber Diagram
The frequency-wavenumber diagram of a machines air gap flux distribution is a useful
tool for discussing motor source mechanisms.
14
A thorough understanding of this
diagram, also sometimes referred to as an air gap flux colormap, is key to identifying
source mechanisms for a given motor design. An example of such a diagram was
presented in Figure 8 above.
The y-axis corresponds to frequency, in most cases normalized by the fundamental
electrical line frequency. The x-axis corresponds to circumferential spatial order, in
most cases normalized by the number of pole pairs, P, within a machine. The color scale
is a measure of the flux amplitude, often in units of decibels (dB) relative to a Tesla.
Intense red colors signify a high amplitude flux component.
For example, the
fundamental rotor generated flux component, discussed in Section 3.2.1 above, that
exists at 1*E, 1*P, is the result of the rotating embedded magnets which are responsible
for torque production within the machine.
It can also be seen that the rotor pole
harmonics, whose amplitudes are dictated by the coefficients B1,3,5… of Equation (2), are
rapidly decaying with harmonic along the diagonal.
It should be noted that the
amplitudes depicted in Figure 8 are merely for demonstration and do not represent
predicted or measured air gap flux amplitudes.
To aid in understanding the frequency-wavenumber diagram consider Figure 7 above.
Figure 7 is simply a slice along the horizontal axis of Figure 8 at one times the electrical
line frequency (i.e. 1*E). Figure 8has this slice highlighted with a darker gray stripe. It
can be seen that the grey stripe passes through the rotor fundamental at one times the
number of pole pairs, 1*P, which, in the case of a 12 pole machine is 6. Therefore,
Figure 7 depicts an N=6 mode shape as rotated circumferentially around the machine.
Figure 9 depicts what the gap flux spatial distribution would be for the first rotor
harmonic (i.e. 3*E 3*P).
In this case, Figure 9 depicts an N=18, or 3*P, spatial
distribution. The modes identified here are synonymous with those discussed in Section
2.1 when applied to modal analysis of physical displacements of a structure. However,
in this case, the mode number reflects a flux distribution pattern as opposed to the
physical displacement of a structure.
15
Θ=0
+Θ
Figure 9. N=18 Spatial Distribution of the First Rotor Harmonic (3E 3P) Flux Field for
a 12 Pole Machine
It is important to understand that the spatial distribution for the rotor fundamental and
harmonics only contain spatial content, or mode shapes, from a single, normal mode.
This can be seen by the lack of additional terms in Figure 8 along the highlighted 1*P
and additional rotor harmonic lines. Later, it will be seen that the gap flux spatial
distribution is really the superposition of all modes along a single frequency slice in the
frequency-wavenumber diagram.
In this example, the result is relatively straight
forward because of the lack of spatial content at other frequencies.
However, as
additional source terms are predicted, and rotor and armature field interactions
accounted for, additional spatial content will be added to the frequency-wavenumber
diagram.
A detailed understanding the frequency-wavenumber diagram shows clearly how the
rotor flux distributions of Equation (2) discussed in Section 3.2.1 and the stator flux
distributions discussed later in Section 3.2.2 are traveling waves that can be
16
characterized by an infinite Fourier series of sinusoidal functions with specific
combinations of time-space harmonics and wave amplitudes. Although the N=6 and
N=18 modes are depicted as stationary in Figure 7 and Figure 9, in reality they are
rotating at n*E, where E is the fundamental electrical line frequency.
As a note, the time-space, or frequency-wavenumber, flux distribution is real, meaning
quadrants 1 and 3 are a complex conjugate pair, as are quadrants 2 and 4. For this
reason, a complete Fourier description is provided with just two of the four quadrants of
the diagram.
3.2.2
Stator Generated Machine Native Flux Distribution Components
It is well known that current flowing through a winding produces a directed magnetic
field. In a permanent magnet motor, the magnetic field is confined within the stator iron
(i.e. stator backiron and stator teeth), except where it fringes across the air gap of the
machine. The magnetic potential, or MMF, is expressed as MMF=N*i, where N is the
number of turns in the winding and i is the electrical current. The resultant MMF around
the circuit depends on the magnetic flux, Φ, and the net reluctance, or electrical
impedance of the stator windings, RCircuit. The magnetic equivalent to Ohm’s law is that
MMF=Φ*RCircuit [10]. Therefore, the reluctance, or impedance, is similar to resistance,
which sets the electrical relationship between voltage and current. The stator iron has a
much lower reluctance when compared to the reluctance of air in the air gap resulting in
the magnetic flux to be concentrated in the stator iron and the net circuit reluctance to be
almost completely dominated by the reluctance of the air gap (i.e. RCircuit ≈ RAir Gap).
The stator field is produced through superposition from multiple phase windings. Figure
10 shows a simple example of the superposition of three current phases in a stator
winding topology. It should be noted that the square wave character of the generated
field will emphasize odd harmonics in the Fourier series used to characterize the
generated flux waveform discussed in the following paragraphs.
17
Various winding
architectures, including increased number of electrical phases, are commonly used to
improve the quality of the stator field depicted in Figure 10.
Backiron
Linear
Superposition
Square Wave Character
Figure 10. Stator Winding and Field Representation
For a three phase stator winding, where the phase belts are displaced in space by 120o,
and for a motor with P pole pairs, then the spatial components of the stator MMF due to
current in each of the three phases are as follows in Equations (3), (4), & (5) where ia, ib,
and ic, are the instantaneous values of the phase currents, and Kn includes information on
the number of turns, harmonic pitch and distribution factors, etc [11]. It can be seen that
the third space harmonic, K3cos(3(Pθ)), is a special case in that the components due to
all three phases are spatially in phase.
18
= 1 cos
− 3 cos3
+ 5 cos5
− 7 cos7
+ ⋯ (3)
= 1 cos
− 120 − 3 cos3
− 120 + 5 cos5
− 120
− 7 cos7
− 120 + ⋯ = 1 cos
− 120 − 3 cos3
+ 5 cos5
+ 120
− 7 cos7
− 120 + ⋯ (4)
= 1 cos
+ 120 − 3 cos3
+ 120 + 5 cos5
+ 120
− 7 cos7
+ 120 + ⋯ = 1 cos
+ 120 − 3 cos3
+ 5 cos5
− 120
− 7 cos7
+ 120 + ⋯ (5)
The stator field, or armature field, generated by the stator coils is a linear superposition
of flux induced by the three electrical winding phases. The stator discretization leads
conceptually to a square wave reconstruction of the desired armature field.
Each
contribution of the three individual fields can be expressed as a Fourier series of space
harmonics. As previously stated, due to the square nature of the generated waveform
caused by the stator discretization, odd harmonics will be emphasized in the Fourier
series.
As stated above, in Equations (3), (4), & (5), Kn relates the amplitudes of the Fourier
components back to motor design parameters such as number of turns, number of
windings, distribution factors, gap size, etc. These coefficients can be estimated from
complex analytics and then potentially be ‘calibrated’ using finite element software. For
the purpose of this paper, estimating the amplitude of these coefficients will not be
evaluated. But rather, the focus being on predicting the spatial and temporal harmonic
content for a given machine design, in this case a machine with 12 poles and 6 stator
slots per pole.
Knowing the phase currents contribution to the air gap flux distribution are given by
Equations (3) - (5), and assuming they are perfect sinusoidal inputs with an amplitude I,
Equations (6) & (7), recalling the trigonometric substitution of Equation (8), the
19
resultant armature flux field distribution in the air gap as a result of three phase currents
simplifies to Equation (9). It is worth noting that imbalances in the phase currents (e.g.
phase to phase imbalance or a DC offset in the phase currents) will add additional
content to the Fourier series representation of the gap flux distribution due to energized
stator windings. An example of how a DC offset in the phase currents effects the gap
flux distribution will evaluated in Section 3.2.3.
! = cos# + Φ ! = cos# − 120 + Φ ! = cos# + 120 + Φ = = Φ = Φ = Φ = 0
(6)
(7)
1
cos% cos& = 'cos% − & + cos% + &(
2
(8)
)* = + + 3!
= +1 cos
− # + 5 cos5
+ # − 7 cos7
− # ,
2
(9)
The spatial distribution of the stator winding generated MMF for a machine that has 12
poles is identical to the rotor fundamental traveling wave component of Equation (2),
which results in the production of torque.
Harmonics of the stator fundamental flux exist at higher spatial orders and are the result
of the additional components in the Equation (9) Fourier series.
Like the rotor
harmonics discussed above in Section 3.2.1, these harmonics are usually much smaller in
amplitude than the fundamental. The 1*E 1*P fundamental stator flux and the higher
order harmonics of the stator fundamental can be visualized to exist on the frequencywavenumber diagram as depicted in Figure 11. Unlike the rotor flux harmonics, the
stator flux harmonics all exist at the same frequency, one times the fundamental
electrical line frequency of the machine, as expected based on Equation (9).
20
1
10E
3.8
5E
1 st & 2 nd Phase
Belt Harmonics
3.6
5E
3.4
Amplitude (dB re: Tesla)
0E
3.2
10P
5P
5P
10P
3
2.8
Fundamental
Stator Flux5E
5E
Frequency
(E = Electrical Line Frequency)
10E
4
2
2.6
10E
2.4
2.2
10E
3
4
2
10P
5P
0P
5P
10P
Circumferential Order
(P = Number of Pole Pairs)
Figure 11. Frequency-Wavenumber Diagram Depicting the Stator Winding Generated
Air Gap Flux Components
Because the armature and rotor fields are of like spatial order and rotating at the same
frequency, the fundamental MMF traveling wave of Equation (9) interacts with the
fundamental of the rotor field flux of Equation (2) to produce most of the steady
machine torque.
In the case of perfect sinusoidal input currents, this interaction
produces all of the steady torque.
As expected, the discretization of the three phase stator winding generated stator field
leads to odd space harmonics in the air gap flux distribution. Under the assumption of
pure sinusoidal input current, or perfect power, additional current related harmonics are
self-canceling resulting in no additional terms at other electrical harmonic frequencies as
seen in Figure 12. Knowing that the current amplitudes in all three phases are equal,
Equation (7), and knowing the relative phasing between the three current phases, phase
21
shifted 120o from each other, a simple phasor diagram can be produced to visualize the
vector sum of the individual contributions. This can be seen in Figure 12.
ic
ia
ib
Figure 12. Phasor Diagram Depicting Vector Sum of Perfectly Sinusoidal Input Phase
Currents
As identified in Figure 11, the fundamental can be seen present at one times the input
electrical line frequency and is the result of the first component in the Fourier series of
Equation (9). Higher harmonics, also identified in Figure 11, are the result the additional
odd harmonic present in the Fourier series due to the discretization of the stator
representation.
These remaining Fourier components are traveling flux waves,
alternately co-rotating or counter-rotating with the fundamental stator flux. Co-rotating
waves are designated as positive (+) traveling waves and counter-rotating waves are
designated as negative (-) traveling waves.
Like what was presented for the rotor fundamental flux, Figure 13 replicates that of
Figure 7 by being a single slice of the frequency wavenumber diagram along the
horizontal 1*E line highlighted in Figure 11 with a darker grey line.
The spatial
distribution of the air gap flux no longer consists of a single mode shape but rather, the
superposition of multiple mode shapes, in this case, 1*P, 5*P, 7*P, and 11*P. Like the
rotor harmonics, the first and second phase belt harmonics of the stator fundamental flux
decay rapidly in amplitude.
22
Θ=0
+Θ
Figure 13. Spatial Distribution of the Armature Flux Field for a 12 Pole Machine
The assumption of perfect sinusoidal power leads to a relatively simple analytical
expression that is presumably dominated by the stator fundamental term of Equation (9)
(i.e. K1 > K3, K5, K7…) by design because, as discussed above, it is the term that
interacts with the rotor fundamental to produce torque.
As discussed below in Section 3.3, when calculating the air gap pressure distribution, the
fifth and seventh space harmonic traveling waves of the stator MMF of Equation (9),
commonly referred to as the first and second phase belt harmonics, interact with the
fifth and seventh traveling waves of the rotor flux of Equation (2), to produce torque
pulsation at an angular frequency of 6ω. In permanent magnet motors, the shape and/or
span of the magnets or poles is designed to reduce the fifth and seventh space harmonics
23
of the rotor flux, and the stator coil pitch is usually selected to reduce K5 and K7 of the
stator MMF of Equation (9), thereby minimizing the torque pulsation, or the parasitic
torque reducing the overall efficiency of a machine. Additional reduction is provided by
the distributed nature of the stator windings. The price paid for these design preferences
is a reduction in amplitude of the fundamental rotor and stator MMFs, accounted for in
alternating current permanent magnet motor machine design terminology as the
composite winding factor, Kw, which accounts for the inopportunity to exploit flux
harmonics for fundamental torque production [12]. Both of these effects result in a
reduction in machine torque density, often defined as ft/lbs. per lb weigh of a machine,
and efficiency.
3.2.3
Effect of DC Offset on the Air Gap Flux Distribution
Up to this point, the input currents inducing the stator flux distribution components have
been assumed to be perfect sinusoidal waveforms. In reality, there are many sources of
current imperfections that will cause distortions in the input currents resulting in
additional Fourier components when analytically predicting the air gap flux distribution
within a permanent magnet motor. One example of this is the presence of a DC offset
on the input phase current waveforms. A DC offset on the phase current waveforms
creates some additional Fourier components in the air gap flux distribution.
As
expected, the resultant stator flux distribution includes some additional space harmonics
all at zero frequency (i.e. due to the fact that, by definition, DC refers to zero frequency
[13]). As before, Equation (3) - (5) and Equation (6), however, with a presumably small
component of additional DC current on the input current waveforms,
, due to
inaccuracies in the controllers providing the input current, the flux field can be written
for each of the individual three phase windings as Equations (10), (11), & (12).
= +! ̅ + cos# ,1 cos
− 3 cos3
+ 5 cos5
− 7 cos7
+ ⋯ 24
(10)
= +! ̅ + cos# − 120,1 cos
− 120 − 3 cos3
+ 5 cos5
+ 120 − 7 cos7
− 120 + ⋯ = +! ̅ + cos# + 120,1 cos
+ 120 − 3 cos3
+ 5 cos5
− 120 − 7 cos7
+ 120 + ⋯ (11)
(12)
As expected, the resultant stator flux distribution from the flux induced by each of the
three phase currents includes some new/additional spatial harmonics at zero frequency
and can be expressed by Equation (13) [14].
)* = + + (13)
3!
= +1 cos
− # + 5 cos5
+ # − 7 cos7
− # + ⋯,
2
3! ̅
+ +−3 cos3
+ 6 cos6
− 9 cos9
+ 12 cos12
− ⋯,
2
In Equation (13), the components highlighted in red represents the contributions from
the perfectly sinusoidal input current waveforms, or perfect power, while those
highlighted in blue represent new contributions from the DC offset on the input
waveforms. In the frequency-wavenumber domain, these resulting DC flux components
lie along the 0E line (i.e. DC) as seen in Figure 14.
25
10E
4
2
1
10E
3.8
3.6
5E
5E
3.4
Amplitude (dB re: Tesla)
0E
3.2
10P
5P
5P
10P
3
2.8
5E
Frequency
(E = Electrical Line Frequency)
DC Triplens
2.6
5E
10E
2.4
2.2
10E
3
4
2
10P
5P
0P
5P
10P
Circumferential Order
(P = Number of Pole Pairs)
Figure 14. Frequency-Wavenumber Diagram Depicting the DC Offset Generated Air
Gap Flux Components
Figure 14 shows that the addition flux content due to a DC offset in the input current
waveforms results in flux harmonics at DC current and at 3*nP spatial orders, as
expected based on Equation (13). For a three phase machine, these components are
often referred to as the DC triplens.
3.2.4
Interaction between the Rotor and Armature Air Gap Flux Components
The segmentation of the stator teeth introduces an additional periodicity into the air gap
reluctance which modulates the fundamental rotor flux adding additional spatial and
temporal content to the air gap flux distribution. This additional content is caused by the
movement of the rotor fundamental magnetic field past the stator teeth. Alignment of
the magnetic field with the stator tooth produces a peak of magnetic attraction between
26
the rotor and the stator, while alignment of the rotor field between two stator teeth
results in a minimum of attraction. Figure 15 provides a visual example illustrating a
uniform flux distribution being concentrated at the stator teeth.
Backiron
Figure 15. Illustration of Uniform Flux Distribution Being Concentrated at the Stator
Teeth
This variation in magnetic attraction between the rotor and the stator affects the torque
that the machine produces. When the rotor magnetic field is aligned with a stator tooth,
the rotor is in a preferred angular position relative to the stator. That is, torque is
required to turn the rotor away from this position. Likewise, the amplitude of the air gap
magnetic field is at a peak when the rotor field is aligned with a tooth, and is at a
minimum when the rotor field is between two teeth. The pulsation in amplitude of the
air gap flux field as the rotor rotates produces a radial force that peaks simultaneously at
all points around the circumference of the air gap at the slot passing frequency. The slot
passing frequency occurs at a frequency S*Rn, where S is the number of stator slots in
the machine, R is the rotational rate in Hz, and n is an integer. The rotation rate is
related to the fundamental electrical line frequency, E, and the number of pairs of poles
P in the machine, E=PR. Therefore, the stator slot passing frequency may also be
27
expressed in electrical terms as SEn/P. The quantity S/P is referred to as the number of
slots per pair of poles in the machine.
This interaction, or modulation, of the fundamental rotor flux and harmonics due to the
segmentation of the stator teeth can conceptually be modeled, for the case of perfect
sinusoidal input current, by introducing a spatial weighting function from a square wave
variation in the gap reluctance near the teeth. Figure 16 illustrates this air gap reluctance
variation over two poles as a permeance variation, P, across the stator teeth for a
machine with six teeth or slots per pole.
P
P
0
θ
Figure 16. Permeance Variation Across the Stator Teeth Over Two Poles
Mathematically, this permeance variation can be expressed or approximated by
Equations (14) and (15) [15].
= ∗ 0 = ∗ 1))1 , ≈ +
3 + 41 ∗ cos6
+ 43 ∗ cos18
+ 45 ∗ cos30
+ ⋯ ,
∗ +1 cos
− # − 3 cos3
− 3# + 5 cos5
− 5# − 7 cos7
− 7# + 9 cos9 − 9# − ⋯ ,
28
(14)
(15)
Equation (15) can be rewritten to segregate the spatial and temporal components that
result from perfect power and those resulting from the spatial modulation of the rotor
fundamental (i.e. sidebands of the rotor harmonics). This can be seen in Equation (16).
≈ 3 ∗ +1 cos
− # − 3 cos3
− 3# + 5 cos5
− 5# − 7 cos7
− 7# + 9 cos9 − 9# − ⋯ ,
+
(16)
41 cos6
1 cos1 − 6
− # + 1 cos1 + 6
− # 2
+ 3 cos3 − 6
− 3# + 3 cos3 + 6
− 3# + 5 cos5 − 6
− 5# + 5 cos5 + 6
− 5# + ⋯ In Equation (16), the components highlighted in red represents the contributions from
perfect power while those highlighted in blue represent the interaction between the stator
teeth and the rotor MMF which results in an air gap flux distribution that has many
additional Fourier components that arise as sidebands off the rotor field harmonics as
seen in Figure 17.
These additional harmonics are commonly referred to as slot
modulation of the rotor pole harmonics.
29
10E
4
2
1
10E
3.8
5E
5E
3.6
3.4
Amplitude (dB re: Tesla)
0E
3.2
10P
5P
5P
10P
3
2.8
5E
Frequency
(E = Electrical Line Frequency)
Slot Modulation of the
Rotor Pole Harmonics
2.6
5E
10E
2.4
2.2
10E
3
4
2
10P
5P
0P
5P
10P
Circumferential Order
(P = Number of Pole Pairs)
Figure 17. Frequency-Wavenumber Diagram Depicting the Air Gap Flux Distribution
as a Result of Rotor and Stator Interaction
Finally, using the Fourier series expressions of Equations (2), (9), & (15), the complete
air gap flux distribution for a 12 pole machine with 6 slots per pole can be predicted,
assuming perfect input power, as the linear superposition of the individual components
discussed above. This would result in an air gap flux distribution that looks like that
presented in Figure 18.
30
10E
4
2
1
10E
3.8
5E
5E
3.4
Amplitude (dB re: Tesla)
0E
3.2
10P
5P
5P
10P
3
2.8
5E
Frequency
(E = Electrical Line Frequency)
3.6
2.6
5E
10E
2.4
2.2
10E
3
4
2
10P
5P
0P
5P
10P
Circumferential Order
(P = Number of Pole Pairs)
Figure 18. Frequency-Wavenumber Diagram Depicting the Complete Air Gap Flux
Distribution
It can be seen that Figure 18 contains all of the components from the multiple sources
discussed previously in this section, with the exception of DC offset terms.
3.3 Calculation of Air Gap Electromagnetic Pressure Distribution
The radial and torsional forces (i.e. pressures) that act on both the rotor and stator are
determined by the air gap flux distribution. As shown above in Section 3.2, the air gap
flux distribution can be written as a simplified Fourier series, Equation (17), depending
on whether predicting the rotor or armature fields and their interactions.
31
7 =∞
, = 6
8=∞
6 7 ,8 ∗ cos8 − 7# (17)
7 =−∞ 8=−∞
However, the radial and torsional electromagnetic air gap pressures are proportional to
the square of the air gap flux distribution and can be written as follows in Equation (18).
=
1
6 6 6 6 7 1 ,8 1 ∗ 7 2 ,8 2 cos81 − 82 − 71 − 72 # 4;
71 72 81 82
(18)
+ cos81 + 82 − 71 + 72 # Lastly, because electromagnetic forces and torques are produced by the squaring of the
gap flux distributions, Equation (18) can be simplified to well known Maxwell’s stress
equations [17]. Equation (19) are the Maxwell stress tensor equations relevant to this
calculation where Bn is the flux density normal to the stator bore, Bt is the flux density
tangential to the stator bore, and µ0 is the permeability of free space. The radial gap
pressure, Pn, is also referred to as the normal pressure and the torsional gap pressure, Pt,
is also referred to as the shear stress of the machine and is a measure of the torque
capable for a given machine design to produce.
8 =
82 − 2 2;0
=
8 ∗ ;0
(19)
The resulting radial and torsional gap pressure distributions can also be visualized using
the same frequency-wavenumber diagram discussed in Section 3.2.1.1 above. However,
the squaring of the normal and torsional gap flux distributions results in a frequencywavenumber spectrum that is very rich in harmonics due to the interactions that take
place during the squaring of the radial and torsional flux components as seen in Figure
19.
32
10E
4
2
1
10E
3.8
5E
5E
3.4
Amplitude (dB re: Tesla)
0E
3.2
10P
5P
5P
10P
3
2.8
5E
Frequency
(E = Electrical Line Frequency)
3.6
2.6
5E
10E
2.4
2.2
10E
3
4
2
10P
5P
0P
5P
10P
Circumferential Order
(P = Number of Pole Pairs)
Figure 19. Frequency-Wavenumber Diagram Depicting the Resultant Air Gap Pressure
Distribution
33
4. Results and Discussion
4.1 Validation of the Predicted Spatial and Temporal Gap Flux and
Pressure Distributions
To validate the predictive capabilities of the Section 3.2 and 3.3 methodologies, an
Electromagnetic Finite Element Analysis (EM FEA) model of a 12 pole machine with
six slots per pole was developed and executed to compare the results against those of
Sections 3.2 and 3.3 used to predict the spatial and temporal components of the air gap
flux and corresponding pressure distributions for a machine with a comparable number
of poles and slots. For this assessment, air gap flux distribution estimates were based on
numerical models of a representative motor. Machine forcing functions, or sources,
associated with ideal native sources were computed using a single pole EM FEA model.
The use of EM FEA enables for a detailed characterization of the air gap flux
distribution for both the individual flux components as well as the flux and pressure
components resulting from the additional interaction between the rotor and stator fields.
4.1.1
Flux2D by Magsoft
‘Flux2D is a finite element software application used for electromagnetic and thermal
physics simulations, both in 2D and 3D. Flux can handle the design and analysis of any
electromagnetic device. Featuring a large number of functionalities, including extended
multi-parametric analysis, advanced electrical circuit coupling and kinematic coupling, it
is suitable for static, harmonic, and transient analysis. Flux is suitable for designing,
analyzing and optimizing a variety of devices and applications’ [18].
Flux2D is finite element software written by Magsoft that is used to model
electromagnetic circuits and electromagnetic machines. A physical application of the
Flux2D software is its ability to model transient magnetics which is ideal for modeling
rotating electromagnetic machinery allowing for the calculation of things like flux
density, magnetic fields, saturation states, machine inductance, machine torque, and
power factors. It does this by physically modeling a rotating air gap within a machine
which requires solutions to the kinematic equations. The basic process for developing
34
an EM FEA model in Flux2D is similar to other finite element software and can be
summarized in four basic steps. First, the geometry of the machine was generated which
is dictated by the overall topology of the motor being modeled. That geometry was then
meshed and physical material properties were then associated with the meshed geometry
components. Second, an external electric circuit was created and then coupled to the
meshed geometry. Specifically, the external electric circuit was coupled to the geometry
elements that represent the coils of the machine which account for the phase currents
flowing through the stator windings which, as discussed above, produce the stator field.
Third, the Flux2D software parameterized and solved for the flux distributions
accounting or the kinematic effects of the rotating air gap. The last step of the process
was post-processing at which point the magnetic potential at each point along the
extraction radius at the rotating air gap was extracted and post-processed into a two
dimensional matrix that represents the flux components at each node along the extraction
radius as a function of time (i.e. 2D matrix of [space/location x time]). Applying a two
dimensional Fast Fourier Transform (2D FFT) on the [space/location x time] matrix
results in the frequency-wavenumber gap flux distribution identical to that discussed in
detail in Section 3.2.1.1. Then, using the Maxwell stress tensor equations of Equation
(18), the resulting two dimensional gap pressure profiles can be computed and compared
to the expected gap pressure profile of Section 3.3. A depiction of the model developed
and executed in Flux2D is presented in Figure 20.
35
Figure 20. 12 Pole, 6 Slots per Pole, Flux2D EM FEA Model
4.2 Comparison on EM FEA Results to Predicted Air Gap Flux &
Pressure Distributions
As discussed above in Section 4.1, an EM FEA model was built and executed to
compare to the predicted results of Sections 3.2 and 3.3. First, a model was constructed
that included magnets and the magnetic rotor poles only and a smooth stator bore with
no slot, tooth discretization was assumed to verify the results of Section 3.2.1. Figure 21
presents the results of these analyses in the same frequency-wavenumber diagram as
discussed previously.
36
4
14
13
12
3.8
11
10
Rotor Pole
Harmonics
9
8
3.6
6
3.4
5
Amplitude (dB re: Tesla)
Frequency
(E = Electrical
Line Frequency)
Electrical Line Frequency
(nE)
7
4
3
3.2
2
1
0
3
-1
-2
Funda mental
Rotor Flux
(1E , 1P)
-3
-4
-5
2.8
-6
2.6
-7
-8
-9
2.4
-10
-11
-12
2.2
-13
-14
-15
2
-15 -14 -13 -12 -11 -10 -9
-8
-7
-6
-5
-4
-3 -2 -1 0
1
2
3
Spatial Harmonic Order (nP)
4
5
6
7
8
9
10 11 12 13 14
Circumferential Order
(P = Number of Pole Pairs)
Figure 21. Frequency-Wavenumber Diagram for the Rotor Generated Air Gap Flux
Components, EM FEA Result
Figure 21 compares favorably with the predicted gap flux distribution of Figure 8. The
results of the EM FEA model show the same fundamental rotor flux at one times the
electrical line frequency and at one times the number of pole pairs along with the
additional higher order harmonics of Equation (2).
Next, the EM FEA model was modified to predict only the gap flux terms associated
with the energized stator winding. To do this, the rotor was modeled as a simplified
circumferential non-permeable piece of steel and the stator slot, tooth, and winding
topologies were explicitly modeled and coupled to an external electric circuit. The
results of this EM FEA model can be seen in Figure 22.
37
4
14
13
12
3.8
11
10
9
3.6
8
1 st & 2 nd Phase
Belt Ha rmonics
6
3.4
5
4
3
Amplitude (dB re: Tesla)
Frequency
(E = Electrical
Line Frequency)
Electrical Line Frequency
(nE)
7
3.2
2
1
0
3
-1
-2
-3
2.8
-4
-5
-6
Funda mental
Stator Flux
-7
2.6
-8
-9
2.4
-10
-11
-12
2.2
-13
-14
-15
-15 -14 -13 -12 -11 -10 -9
-8
-7
-6
-5
-4
-3 -2 -1 0
1
2
3
Spatial Harmonic Order (nP)
4
5
6
7
8
9
10 11 12 13 14
2
Circumferential Order
(P = Number of Pole Pairs)
Figure 22. Frequency-Wavenumber Diagram for the Stator Winding Generated Air Gap
Flux Components, EM FEA Results
As with above, the results presented in Figure 22 compare favorably with the predicted
distribution presented in Figure 11. Again, the fundamental stator flux is present at 1*E,
1*P and the first and second phase belt harmonics can be seen to exist along the
fundamental 1*E frequency at higher spatial orders.
Lastly, the two models discussed previously were combined to produce a model that has
both the rotor and armature field producing components present. In other words, the
explicitly modeled rotor of the first model was coupled with the second model and used
to replace the non-permeable rotor of the second model resulting in an EM FEA model
that had the rotating magnetic components as well as the stator tooth, slot, and winding
38
components. The resulting gap flux distribution from this full coupled model can be
seen in Figure 23.
4
14
13
12
3.8
11
Slot Modula tion of the
Rotor Pole Harmonics
10
9
3.6
8
6
3.4
5
4
3
Amplitude (dB re: Tesla)
Frequency
(E = Electrical
Line Frequency)
Electrical Line Frequency
(nE)
7
3.2
2
1
0
3
-1
-2
-3
2.8
-4
-5
-6
2.6
-7
-8
-9
1st & 2nd Phase
Belt Harmonics
2.4
-10
-11
-12
2.2
-13
-14
-15
-15 -14 -13 -12 -11 -10 -9
-8
-7
-6
-5
-4
-3 -2 -1 0
1
2 3
Spatial Harmonic Order (nP)
4
5
6
7
8
9
10 11 12 13 14
2
Circumferential Order
(P = Number of Pole Pairs)
Figure 23. Frequency-Wavenumber Diagram Depicting the Complete Air Gap Flux
Distribution
As anticipated based on Section 3.2 and Equations (2), (9), & (15), the EM FEA result of
Figure 23 compare favorably with those of Figure 18 above. It should be noted that the
air gap flux distributions presented previously in this section reflect radial flux
components only. Additional plots depicting the torsional flux components could have
also been presented, however, due to their strong similarity to the radial distributions,
only a single set of figures has been presented.
39
To verify the Maxwell stress tensor methodology of calculating air gap pressure
distributions from the flux distribution results previously presented, Equation (19) was
utilized along with the calculated radial and torsional gap flux from the EM FEA model
to calculate the air gap pressure distribution for comparison back to Figure 19 above.
The results of this calculation can be seen in Figure 24.
4
14
13
12
3.8
11
10
9
3.6
8
6
3.4
5
4
3
Amplitude (dB re: psi)
Frequency
(E = Electrical
Line Frequency)
Electrical Line Frequency
(nE)
7
3.2
2
1
0
3
-1
-2
-3
2.8
-4
-5
-6
2.6
-7
-8
-9
2.4
-10
-11
-12
2.2
-13
-14
-15
-15 -14 -13 -12 -11 -10 -9
-8
-7
-6
-5
-4
-3 -2 -1 0
1
2
3
Spatial Harmonic Order (nP)
4
5
6
7
8
9
10 11 12
13
14
2
Circumferential Order
(P = Number of Pole Pairs)
Figure 24. Frequency-Wavenumber Diagram of the Air Gap Pressure Distribution, EM
FEA Model Results
Again, the results of the predicted air gap pressure distribution of Figure 19 compare
favorably with the results of the EM FEA model presented in Figure 24.
40
5. Conclusion
The analysis approaches and methodologies discussed in Sections 3.2 and 3.3 have
shown, based on the results of Section 4.2, to be capable of accurately predicting the role
of temporal and spatial harmonics in the production of torque in permanent magnet
motors as well as predicting the magnetic and electric fields generated by the rotating
array of permanent magnets and synchronous windings of a permanent magnet machine,
specifically, determining their distribution in space and their behavior with frequency.
41
List of References
[1]
“The Fundamentals of Modal Testing: Application Note 243-3” Agilent
Technologies: Innovating the HP Way, Retrieved June, 21 2013, from
http://www.modalshop.com/techlibrary/Fundamentals of Modal Testing.pdf
[2]
“Normal
Mode”,
Wikipedia,
Retrieved
http://en.wikipedia.org/wiki/Normal_mode
[3]
Blevins, Robert D., “Formulas for Natural Frequency and Mode Shape”, Krieger,
Florida, 1984
[4]
Stator
Lamination,
Retrieved
June
23,
2013,
http://www.tradeindia.com/fp1409208/Low-Voltage-High-Power-StatorLamination.html
[5]
Stacked Stator Image, Retrieved June 23, 2013, from http://image.made-inchina.com/2f0j00WMtQrlLosCuc/Motor-Stator.jpg
[6]
Wound,
Stacked,
Stator,
Retrieved
June,
23,
2013,
from
http://www.ronsmotor.com/wp-content/uploads/2011/01/Stator-Rewinds1640x480.jpg
[7]
Fitzgerald, A. E., Kingsley, Jr., Charles, Umans, Stephen D., “Electric
Machinery”, Sixth Edition, New York, NY: McGraw-Hill; 2003
[8]
Z.Q. Zhu, D. Howe, E. Bolte, and B. Ackermann, “Instantaneous Magnetic Field
Distribution in Brushless Permanent Magnet DC Motors, Part I: Open Circuit
Field”, IEEE Transactions on Magnetics, Vol. 29, No. 1, January, 1993
[9]
“Magnetomotive Force”, Wikipedia, Retrieved
http://en.wikipedia.org/wiki/Magnetomotive_force
[10]
“Magnetic Circuit”, Wikipedia, Retrieved
http://en.wikipedia.org/wiki/Magnetic_circuit
[11]
Z.Q. Zhu, D. Howe, “Instantaneous Magnetic Field Distribution in Brushless
Permanent Magnet DC Motors, Part II: Armature-Reaction Field”, IEEE
Transactions On Magnetics, Vol. 29, No. 1, January, 1993
[12]
V.B. Honsinger, “The Fields and Parameters of Interior Type AC Permanent
Magnet Machines: Asynchronous Operation”, IEEE Trans. Power Apparatus &
Systems, Vol. PAS-101, No. 4, April, 1982
[13]
“Direct
Current”,
Wikipedia,
Retrieved
http://en.wikipedia.org/wiki/Direct_current
42
June,
23
June,
August,
July,
2013,
from
from
30
2013,
from
10
2013,
from
7
2013,
from
[14]
V.B. Honsinger, “Performance of Polyphase Permanent Magnet Machines”,
IEEE Trans. Power Apparatus & Systems, Vol. PAS-99, No. 4, July/August,
1980
[15]
Z.Q. Zhu, D. Howe, “Instantaneous Magnetic Field Distribution in Brushless
Permanent Magnet DC Motors, Part III: Effect of Stator Slotting”, IEEE
Transactions On Magnetics, Vol. 29, No. 1, January, 1993
[16]
Z.Q. Zhu, D. Howe, “Instantaneous Magnetic Field Distribution in Brushless
Permanent Magnet DC Motors, Part IV: Magnetic Field on Load”, IEEE
Transactions On Magnetics, Vol. 29, No. 1, January, 1993
[17]
“Maxwell Stress Tensor”, Wikipedia, Retrieved July, 20 2013, from
http://en.wikipedia.org/wiki/Maxwell_stress_tensor
[18]
“Flux 11”, Magsoft Corporation, Retrieved July 15, 2013, from http://magsoftflux.com/products/flux
43
Appendix A - Calculation of Air Gap Flux and Pressure Distributions
and Plotting Script
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Purpose:
This script processes and plots the air gap flux distribution
%
results of the EM FEA model. This script also calculates the
%
associated pressure distribution and plots that as well.
%
% Revision:
00, July 21, 2013 --> Initial Release
%
% Usage:
Loads EM FEA results and initiates plotting and processing.
%
outputData = calcBandP(bnorm_time , btan_time)
%
% Inputs:
Radial and Torsional air gap flux distributions from output of
%
of the EM FEA model
%
- EM FEA outputs (i.e. inputs to this script) are two
%
dimensional matrices formated as follows:
%
- [Air Gap Circumferential Location x Time]
%
% Outputs:
Figures of processed radial air gap flux and and pressure
%
distributions for a 12 pole machine with 6 slots per pole.
%
% Functions
% Called:
None
%
% =============================================================
%% Process & Plot EM FEA Results
%
clear all; close all; home; set(0,'defaultfigurewindowstyle','docked');
%
% __________________
% Load Time Series Flux
% --> Only Populated Rotor
bnorm_time = load('C:\Users\dfellega\Desktop\RPI\Final Project\Flux\rotor_norm_flux.mat');
bnorm_time = load('C:\Users\dfellega\Desktop\RPI\Final Project\Flux\rotor_tan_flux.mat');
%
% --> Only Wound Stator
% bnorm_time = load('C:\Users\dfellega\Desktop\RPI\Final Project\Flux\stator_norm_flux.mat'
% bnorm_time = load('C:\Users\dfellega\Desktop\RPI\Final Project\Flux\stator_tan_flux.mat');
%
% --> Complete Machine
% bnorm_time = load('C:\Users\dfellega\Desktop\RPI\Final Project\Flux\norm_flux.mat'
% bnorm_time = load('C:\Users\dfellega\Desktop\RPI\Final Project\Flux\tan_flux.mat');
%
outputData = calcBandP(bnorm_time , btan_time);
disp(outputData);
%
% return
%
% ==================================================================
44
%% This Function Process' and Plots the EM FEA Results
%
function outputData = calcBandP(bnorm_time , btan_time);
%
% ___________________
% Calc. [Wn. x Freq.] Flux
bnorm_fft = (1/size(bnorm_time,1)) * (2/size(bnorm_time,2)) * fftshift(fft2(bnorm_time(:,:)));
btan_fft = (1/size(btan_time,1)) * (2/size(btan_time,2)) * fftshift(fft2(btan_time(:,:)));
%
% __________________
% Maxwell Stress Tensor
pnorm_time = (1/(2*((4*pi)*10^-7))) * (bnorm_time.^2 - btan_time.^2);
ptan_time = (1/((4*pi)*10^-7)) * (bnorm_time .* btan_time);
%
% ___________
% Convert to PSI
pnorm_time = pnorm_time .* (1.45037738E-4);
ptan_time = ptan_time .* (1.45037738E-4);
%
% ______________________
% Calc. [Wn. x Freq.] Pressure
pnorm_fft = (1/size(pnorm_time,1)) * (2/size(pnorm_time,2)) * fftshift(fft2(pnorm_time(:,:)));
ptan_fft = (1/size(ptan_time,1)) * (2/size(ptan_time,2)) * fftshift(fft2(ptan_time(:,:)));
%
% ___
% Plot
figure; clf
imagesc(20*log10(abs(bnorm_fft(83:112,82:111)))); caxis([-80 20]);
set(gca,'ytick',1:30,'yticklabel',14:-1:-15,'ygrid','on','ydir','normal')
set(gca,'xtick',1:30,'xticklabel',-15:14,'xgrid','on')
xlabel('Spatial Harmonic Order (nP)'); ylabel('Electrical Line Frequency (nE)');
%
figure; clf
imagesc(20*log10(abs(pnorm_fft(83:112,82:111)))); caxis([-80 20]);
set(gca,'ytick',1:30,'yticklabel',14:-1:-15,'ygrid','on','ydir','normal')
set(gca,'xtick',1:30,'xticklabel',-15:14,'xgrid','on')
xlabel('Spatial Harmonic Order (nP)'); ylabel('Electrical Line Frequency (nE)');
%
outputData = 'Processing Complete...';
%
% ==================================================================
45