Mixed mesh/nodal magnetic equivalent circuit modeling of a

Purdue UniversityPurdue e-Pubs

Open Access Theses Theses and Dissertations

4-2016

Mixed mesh/nodal magnetic equivalent circuitmodeling of a six-phase claw-pole automotivealternatorDaniel C. HorvathPurdue University

Follow this and additional works at: https://docs.lib.purdue.edu/open_access_theses

Part of the Applied Mechanics Commons, and the Power and Energy Commons

This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] foradditional information.

Recommended CitationHorvath, Daniel C., "Mixed mesh/nodal magnetic equivalent circuit modeling of a six-phase claw-pole automotive alternator" (2016).Open Access Theses. 775.https://docs.lib.purdue.edu/open_access_theses/775

https://docs.lib.purdue.edu?utm_source=docs.lib.purdue.edu%2Fopen_access_theses%2F775&utm_medium=PDF&utm_campaign=PDFCoverPages

https://docs.lib.purdue.edu/open_access_theses?utm_source=docs.lib.purdue.edu%2Fopen_access_theses%2F775&utm_medium=PDF&utm_campaign=PDFCoverPages

https://docs.lib.purdue.edu/etd?utm_source=docs.lib.purdue.edu%2Fopen_access_theses%2F775&utm_medium=PDF&utm_campaign=PDFCoverPages

https://docs.lib.purdue.edu/open_access_theses?utm_source=docs.lib.purdue.edu%2Fopen_access_theses%2F775&utm_medium=PDF&utm_campaign=PDFCoverPages

http://network.bepress.com/hgg/discipline/295?utm_source=docs.lib.purdue.edu%2Fopen_access_theses%2F775&utm_medium=PDF&utm_campaign=PDFCoverPages

http://network.bepress.com/hgg/discipline/274?utm_source=docs.lib.purdue.edu%2Fopen_access_theses%2F775&utm_medium=PDF&utm_campaign=PDFCoverPages

https://docs.lib.purdue.edu/open_access_theses/775?utm_source=docs.lib.purdue.edu%2Fopen_access_theses%2F775&utm_medium=PDF&utm_campaign=PDFCoverPages

Graduate School Form30 Updated

PURDUE UNIVERSITYGRADUATE SCHOOL

Thesis/Dissertation Acceptance

This is to certify that the thesis/dissertation prepared

By

Entitled

For the degree of

Is approved by the final examining committee:

To the best of my knowledge and as understood by the student in the Thesis/Dissertation Agreement, Publication Delay, and Certification Disclaimer (Graduate School Form 32), this thesis/dissertation adheres to the provisions of Purdue University’s “Policy of Integrity in Research” and the use of copyright material.

Approved by Major Professor(s):

Approved by:Head of the Departmental Graduate Program Date

Daniel C. Horvath

MIXED MESH/NODAL MAGNETIC EQUIVALENT CIRCUIT MODELING OF A SIX-PHASE CLAW-POLEAUTOMOTIVE ALTERNATOR

Master of Science in Electrical and Computer Engineering

Steven D. PekarekChair

Gregory M. Shaver

Scott D. Sudhoff

Steven D. Pekarek

Venkataramanan Balakrishnan 4/18/2016

MIXED MESH/NODAL MAGNETIC EQUIVALENT CIRCUIT MODELING OF A

SIX-PHASE CLAW-POLE AUTOMOTIVE ALTERNATOR

A Thesis

Submitted to the Faculty

of

Purdue University

by

Daniel C. Horvath

In Partial Fulfillment of the

Requirements for the Degree

of

Master of Science in Electrical and Computer Engineering

May 2016

Purdue University

West Lafayette, Indiana

ii

ACKNOWLEDGEMENTS

First I would like to thank my Advisor, Dr. Steven Pekarek. I am frequently

amazed with his patience both in teaching and in advising. I have been fortunate to work

for him these past years in my undergraduate and graduate career and because of this I

have had fantastic research opportunities and projects that complement my coursework. I

may not be in graduate school had he not asked me to do undergraduate research under

him years ago.

Thank you to my family. My parents David and Kimberly Horvath have always

fostered my curiosity as I grew up – they have always allowed me to freely pursue

projects that would provide a learning opportunity, and I couldn’t ask for better parents.

Sally, my best friend, thank you for your love and support during the past nine years.

Of course I must thank all the students in Power and Energy research area. I am

glad to have such a great group of friends/co-workers. I think our comradery is unique

and special; it is something that every research community needs. Whenever I wish to

discuss a concept or ask a question, I know that there isn’t a single person in our group

who I cannot go to.

Finally, thank you to Remy International, Inc./Borgwarner for the continued

support of this work, especially Dave Fulton, Katie Riley and Greg Holbrook.

iii

TABLE OF CONTENTS

Page

LIST OF TABLES .............................................................................................................. v

LIST OF FIGURES ........................................................................................................... vi

ABSTRACT ..................................................................................................................... viii

INTRODUCTION ....................................................................................................... 1

GEOMETRY OF LUNDELL ALTERNATOR AND

RELUCTANCE/PERMEANCE DERIVATION ........................................................ 3

2.1 Claw Pole Geometry Introduction and Construction .................................................3

2.2 Flux Tube Fundamentals ............................................................................................5

2.3 Stator Flux Tubes .......................................................................................................8

2.3.1 Yoke Flux Tube ............................................................................................. 10

2.3.2 Tooth Flux Tube ............................................................................................ 11

2.3.3 Slot Leakage Flux Tube ................................................................................. 12

2.4 Rotor Flux Tubes ......................................................................................................13

2.4.1 Shaft and Core Flux Tubes ............................................................................ 15

2.4.2 End Disk Flux Tube ....................................................................................... 17

2.4.3 End Piece Flux Tube ...................................................................................... 18

2.4.4 Claw Section Flux Tube ................................................................................ 19

2.4.5 Rotor Leakage Flux Tubes ............................................................................. 22

2.4.6 Permanent Magnet ......................................................................................... 25

2.5 Air Gap Flux Tubes ..................................................................................................26

ALGEBRAIC SYSTEM OF MAGNETIC CIRCUIT EQUATIONS ....................... 33

3.1 Stator Teeth Mesh Equations ...................................................................................33

iv

Page

3.2 Nodal Analysis at Stator Tooth Tip Nodes ..............................................................34

3.3 Mesh/Nodal Analysis at Claw Section Nodes .........................................................35

3.4 Relating Loop Flux to Phase Flux ............................................................................37

3.5 Mesh Analysis of Rotor ...........................................................................................37

3.6 Overall Algebraic System ........................................................................................39

3.7 Solution of Nonlinear Algebraic System .................................................................40

MODELS FOR DIFFERENT LOADING CONDITIONS ....................................... 42

4.1 Passive Rectifier State Model ..................................................................................42

4.2 Open Circuit Model ..................................................................................................49

4.3 Active Rectifier Model .............................................................................................51

VALIDATION OF MODELS ................................................................................... 55

5.1 Passive Rectifier State Model Validation ................................................................55

5.2 Open-Circuit Model Validation ...............................................................................58

5.3 Active Rectifier Model Validation ...........................................................................59

CONCLUSION .......................................................................................................... 60

LIST OF REFERENCES .................................................................................................. 61

APPENDICES

A. AIRGAP PERMEANCE EXPRESSIONS ............................................................. 62

A.1. Triangular Region 1 .................................................................................................63

A.2. Rectangular Region ..................................................................................................64

A.3. Triangular Region 2 .................................................................................................65

A.4. Chamfer Region .......................................................................................................66

B. MATRIX EXPANSION ......................................................................................... 78

v

LIST OF TABLES

Table .............................................................................................................................. Page

2.1: Variable Identifiers and Descriptions, Stator Quantities. ............................................ 9

2.2: Variable Identifiers and Descriptions, Rotor Quantities ............................................ 15

2.3: Variable Identifiers and Descriptions, Air Gap Quantities ........................................ 28

vi

LIST OF FIGURES

Figure ............................................................................................................................. Page

Figure 2.1: Example rotor of a claw-pole machine, without field coil. .............................. 4

Figure 2.2: Simple cylindrical flux tube. ............................................................................ 5

Figure 2.3: Flux tube with non-uniform cross-sectional area. ............................................ 6

Figure 2.4: Flux tube with non-uniform length. ................................................................. 7

Figure 2.5: Generic stator MEC. ......................................................................................... 9

Figure 2.6: Illustration of yoke flux tube. ......................................................................... 10

Figure 2.7: Stator tooth. .................................................................................................... 11

Figure 2.8: Coffin-shaped slot leakage permeance. .......................................................... 12

Figure 2.9: Rotor Section. ................................................................................................. 14

Figure 2.10: Physical Rotor. ............................................................................................. 14

Figure 2.11: Shaft and core flux tubes. ............................................................................. 16

Figure 2.12: End disk of rotor. .......................................................................................... 17

Figure 2.13: End piece, or “knuckle,” of rotor. ................................................................ 18

Figure 2.14: Detail of rotor claws. .................................................................................... 20

Figure 2.15: Rotor field leakage path detail. ..................................................................... 22

Figure 2.16: Detail of the claw-claw leakage permeance. ................................................ 25

Figure 2.17: Permanent magnet in between claws. ........................................................... 26

vii

Figure Page

Figure 2.18: Formation of airgap flux tubes. .................................................................... 27

Figure 2.19: Two cases for determining airgap length over the chamfer region. ............. 30

Figure 3.1: MEC structure detailing the discretization of the airgap and stator into respective flux tubes. ..................................................................................... 34

Figure 3.2: Discretization of claws into sections. ............................................................. 36

Figure 3.3: MEC structure detailing discretization of rotor into respective flux tubes. ... 38

Figure 4.1: Passive rectifier connection to machine. ........................................................ 43

Figure 4.2: Lower diode voltage. ...................................................................................... 44

Figure 4.3: Solution procedure block diagram. ................................................................ 49

Figure 4.4: Active rectifier connection to machine. ......................................................... 52

Figure 5.1: Comparison of MEC and hardware AC quantities for rectified alternator with resistive load. ..........................................................................................56

Figure 5.2: Comparison of MEC and hardware AC quantities for rectified alternator with resistive load. ..........................................................................................57

Figure 5.3: Comparison between open-circuit line-line voltage prediction of MEC, FEA models and with experimental data. ......................................................58

Figure 5.4: Torque vs speed profiles for FEA and MEC. ................................................. 59

Appendix Figure

Figure A.1: Formation of airgap flux tubes. ..................................................................... 62

Figure A.2: Two cases for determining flux tubes over chamfer region. ......................... 66

viii

ABSTRACT

Horvath, Daniel C. M.S.E.C.E., Purdue University, May 2016. Mixed Mesh/Nodal Magnetic Equivalent Circuit Modeling of a Six-Phase Claw-Pole Automotive Alternator. Major Professor: Steven Pekarek.

Claw-pole, or Lundell alternators, are used as the charging source in most

commercial ground vehicles. Increasingly, manufacturers of these machines are being

required to meet strict acoustic noise and vibration specifications. This has led to

alternator designs with more than the traditional three phases. Increasing phase count

above three has been shown to reduce torque ripple, which is an acknowledged source of

the acoustic noise.

In this research, a magnetic equivalent circuit is used to first establish a model of

the nonlinear relationship between stator and field winding current and magnetic flux in a

claw-pole machine. The flux/current relationship is coupled with a state model to predict

the electrical dynamics of the machine connected to a passive rectifier/battery.

Subsequently, two variants of the model are derived to facilitate design optimization

under alternative excitation strategies. In one, the inputs are selected as the stator currents

in order to consider the performance of the machine connected to an active rectifier. In

another, the model is structured to compute open-circuit performance, which facilitates

relatively quick evaluation of model accuracy. Validation of the models is performed

using both FEA and laboratory experiments.

1

INTRODUCTION

Magnetic equivalent circuits (MECs) have been employed by many researchers to

model the relationship between magnetic flux and current in electromagnetic systems

such as electric machines, transformers and inductors [1] ,[2]. Magnetic circuits are

analogous to electric circuits where voltage, current, resistance and conductance are the

respective counterparts of magneto-motive force (MMF), magnetic flux, reluctance and

permeance. The solution of MECs can be accomplished with the plethora of techniques

developed for electrical circuit analysis. Specifically, mesh analysis, based on Kirchoff’s

Voltage Law (KVL), and nodal analysis, based on Kirchoffs Current Law (KCL), are two

very common solution techniques. Once an MEC is established, the question is often of

which circuit analysis technique should be applied in order to minimize computational

effort.

For linear circuits, there is little advantage to using mesh over nodal analysis.

Using one method may yield a system with fewer equations, but for most problems the

difference in unknowns is insignificant. When analyzing nonlinear magnetic systems,

researchers have noted a significant difference in mesh versus nodal analysis. Derbas et al

have noted that for nonlinear MECs a mesh analysis reduces the number of iterations

required to solve the nonlinear system using a Newton-Raphson method [3]. It was

further shown that for strong nonlinearities caused by magnetic saturation, a nodal-based

solution will often fail to converge whereas a mesh-based solution will converge.

2

It is relatively easy to apply MEC analysis to stationary magnetic systems.

However, modeling electric machinery with MECs can be challenging since the circuit

structure can depend on the position of the rotor. Specifically, in the case in which mesh-

based solution techniques are applied, the circuit components representing the airgap will

tend to infinite values as stator/rotor structures (i.e. teeth) come into and out of alignment.

As a result, one must eliminate these elements and establish new KVL loops with the

remaining non-infinite components. Researchers have developed algorithms to automate

the loop construction process [4]. However, the algorithms require one to categorize all

potential overlap positions, which is a challenge for claw-pole machines. One does not

experience this issue in nodal-based solution techniques. However, since machines tend

to operate in saturation, issues of convergence are often encountered.

In this research, an alternative solution technique is provided in which mesh

analysis is used in all magnetically nonlinear flux tubes while nodal analysis is used to

solve for quantities in the airgap. This has the potential to use the advantage of each

solution technique. This research builds upon that presented in [5], in which permeance

expressions for all flux tubes were developed for a nodal MEC model of a three-phase

claw-pole alternator with a delta-connected stator. In addition to the mixed mesh/nodal

system a second focus is to explore new model configurations including six-phase

machines with wye-connected stator and permanent magnets on the rotor. Validation of

the models that are proposed is performed using both FEA models and experimental data

from commercially-available alternators.

3

GEOMETRY OF LUNDELL ALTERNATOR AND RELUCTANCE/PERMEANCE DERIVATION

The claw-pole machine will require a more complex geometrical description than

most machines encountered because its magnetic structure must be considered in three

dimensions. In this chapter, a focus is to first introduce the structure of the stator/rotor.

Subsequently, the circuit structure and circuit elements are defined and parameterized

which is based on the derivation in [5], however modifications are made where necessary

for the six-phase, P pole machine.

2.1 Claw Pole Geometry Introduction and Construction

The claw-pole machine is a wound rotor synchronous machine (WRSM) with a

stator structure that is similar to that used on most salient-pole machines. In most

automotive applications, the stator winding is unity slot/pole/phase, single-layer that

allows the stator to be constructed in an automated and cost-effective manner.

The rotor of a claw-pole alternator is this machine’s defining feature. The name

“claw-pole” comes from the shape of the rotor pole faces, which resemble claws and as

can be seen in Figure 2.1. The vast majority of machines encountered in the literature can

be sufficiently analyzed in two dimensions because they have a consistent cross section

when traveling along the rotational axis. Due to the complex shape of the rotor, claw-pole

machines do not have a consistent cross section and so the traditional two-dimensional

analysis techniques are not sufficient to capture the detail of the magnetic flux paths.

4

Figure 2.1: Example rotor of a claw-pole machine, without field coil.

The rotor is constructed using two halves each with a set of tapered “claws.” Each half of

the rotor can be quickly constructed using an automated forging process. The main source

of rotor MMF is a single bobbin-wound concentrated coil which occupies the toroidal

cavity beneath the claws radially, with the coil’s axis coinciding with the rotational axis.

The coil can be wound on a bobbin as a separate step before being placed between the

claws. These simple components of the rotor are significantly different from the rotors of

salient-pole WRSMs which are constructed using lamination stacks, individual field coils

per pole, and damper bars. The minimal cost of construction of the claw-pole machine is

an important factor that has led to its widespread use for power generation in ground

vehicles.

5

2.2 Flux Tube Fundamentals

Magnetic Equivalent Circuit theory is based upon the flux tube which is defined as

a volume of a material or space which has magnetic flux enter (in a normal direction) at a

single surface of equal magnetic potential and leave (in a normal direction) the flux tube

volume only at a second surface of equal magnetic potential. The flux tube is analogous

to an electrical resistor. Figure 2.2 shows a simple representative cylindrical flux tube

example.

Figure 2.2: Simple cylindrical flux tube.

The flux tube parametric property is its reluctance R , which is defined as the

ratio of magnetic potential across its terminals to the flux flowing through it

1 2f fRφ−

= (2.1)

Similar to resistance, the reluctance can be defined only in terms of its dimensions and

material properties. In particular, reluctance is proportional to its length tl and inversely

proportional to its cross-sectional area tA . It is expressed as

t tt

t

lRA

ν= (2.2)

where tν is the reluctivity. It is a popular choice to instead express the relationship in

terms of the material’s magnetic permeability tµ which is the reciprocal of reluctivity

6

t

t t

lRAµ

= (2.3)

For many situations it is useful to work with a quantity that is the reciprocal of reluctance,

called the permeance

1PR

= (2.4)

t t

t

APl

µ= (2.5)

which is analogous to conductance in electrical circuits.

Figure 2.3: Flux tube with non-uniform cross-sectional area.

Flux tubes may have non-uniform cross-sectional area, as shown in Figure 2.3. If

encountered, the entire flux tube can be thought of as a collection of smaller reluctances

in series, each with its own constant cross-sectional area and length. The differential

reluctance can be expressed in terms of the differential length dx and area as a function

of position as in [2],

( ) ( )

dxdRx A xµ

= (2.6)

7

In order to determine the total reluctance of the non-uniform flux tube, the differential

reluctance is integrated which represents the series combination.

( ) ( )

dxRx A xµ

= ∫ (2.7)

Flux tubes with non-uniform length between the equipotential surfaces are often

encountered, as shown in Figure 2.4. In this case it is desired to work with differential

permeances. Each differential permeance is assumed to have constant cross-sectional area,

and the length is a function of position.

( ),

dAdPl x yµ

= (2.8)

The total permeance between the equipotential planes can then be thought of as the

parallel combination of the differential permeances. For permeances, like conductances in

electrical circuits, this amounts to a summation or rather an integration in this instance

( )( ),,A A

x y dAP dP

l x yµ

= =∫ ∫ (2.9)

Figure 2.4: Flux tube with non-uniform length.

8

As a general rule, in order to determine reluctance, an analytical expression is required

for the cross-sectional area, and for permeance an analytical expression is required for the

flux tube length.

In order to establish a magnetic potential difference across a flux tube, free

current must flow in a nearby conductor. It can be shown that conductors carrying current

can be thought of as an “MMF source,” by using Ampere’s law

enclosedH dl i⋅ =∫

(2.10)

where H

is the magnetic field intensity and enclosedi is the current enclosed by the closed

path. Windings are often composed of a collection of coils with each coil being

composed of a series of turns. If the closed path encloses all N conductors, then

Ampere’s law may be expressed as

H dl Ni⋅ =∫

(2.11)

The MMF source F in the magnetic circuit is defined as

F Ni≡ (2.12)

2.3 Stator Flux Tubes

A generic machine’s stator can be modeled using a simple magnetic circuit as

shown in Figure 2.5. Therein, a reluctance or permeance can represent each yoke, tooth

and slot leakage flux tube, while the conductors in the stator slots form a complex system

of MMF sources. The location of the MMF sources is subject to a discussion which is

detailed in section 3.1.

9

Figure 2.5: Generic stator MEC.

A list of the variables used in the derivation of stator elements is provided in Table 1

Table 2.1: Variable Identifiers and Descriptions, Stator Quantities.

Description

DBS stator yoke depth

GLS Stator stack length

SFL Stacking factor

ID Stator inner diameter

LT Length of tooth shank

OD Stator outside diameter

SL Number of stator slots

SLTINS Height of conductor above bottom of slot

STTW Stator tooth tip width

STW Stator tooth width at shank

tµ Permeability of stator tooth material

yµ Permeability of stator yoke material

0µ Permeability of free space

10

2.3.1 Yoke Flux Tube

Each yoke can be thought of as a pie-section of a toroid, in which the flux travels in the

tangential direction as shown in Figure 2.6. Since length of the flux paths is not constant,

an expression for reluctance is obtained by integrating differential permeances. Taking

the angular span as 2SLπ

, the length of each differential permeance is then 2 rSLπ

. The

differential area isGLS SFL dr⋅ ⋅ , and the permeance of the yoke can be expressed using

cylindrical coordinates by integrating along the radial direction.

Figure 2.6: Illustration of yoke flux tube.

2

2

2

OD

yy

OD DBS

P drr

SL

GLS SFLµπ

−

⋅= ∫ (2.13)

The final expression for the permeance of the stator yoke is

ln2 2

yy

GLS SFL SL ODPOD DBS

µπ⋅ ⋅ = −

(2.14)

where the magnetic permeability yµ is the product of the relative permeability of the

yoke material and the permeability of free space

11

, 0y y rµ µ µ= (2.15)

The reluctance of the yoke flux tube is then obtained using the reciprocal relationship

(2.4).

2.3.2 Tooth Flux Tube

Figure 2.7: Stator tooth.

Stator teeth, shown in Figure 2.7, provide a low-reluctance path from the yoke to

the airgap. The mean path taken by the flux includes 12

DBS for a total length

12

2OD ID DBSl − −

= . Neglecting the tooth tip, the cross-sectional area of the tooth is

GLS FSTW S L⋅ ⋅ . Using (2.3), the stator tooth reluctance is expressed as

2T

T

OD ID DBSTW SFL

SRGLSµ ⋅ ⋅

− −= (2.16)

2.3.3 Slot Leakage Flux Tube

The spaces between the stator teeth form the slots, which are cavities in which

windings are placed to produce stator MMF. Though the majority of the flux produced by

the MMF flows through the yoke and tooth, some will take a path across the stator slot

tangentially. This is referred to as leakage flux, and is modeled by placing a permeance

across the tooth tip potentials which span the slot. The leakage permeance for the coffin-

shaped slot shown in Figure 2.8 can be expressed as [6]

Figure 2.8: Coffin-shaped slot leakage permeance.

13

2 4

0

22

2 2 1 1 0log log ...2 1 1 1 0 0 0

2 1 2 2 3log4 43

2 21 1

TL

H B H B HB B B B B B B

B B BP GLS BS BS BSH

BS B BBS BS

µ

+ + + − − − − − = ⋅ − ⋅ −

(2.17)

where, in terms of the set of variables being used here,

3 0 1 22 2

OD IDH DBS SLTINS H H H = − − − + + +

(2.18)

( )21 0 12

IDB H H STWSLπ = + + −

(2.19)

( )22 0 1 22

IDB H H H STWSLπ = + + + −

(2.20)

( )2 0 1 2 32

IDBS H H H H STWSLπ = + + + + −

(2.21)

Many different tooth/slot shape combinations are encountered in practice. An example of

rectangular teeth with coffin-shaped slots has been presented here, but in reality the

shapes will vary considerably. The specific alternator used for the majority of the

validation had rather peculiar tooth/slot shape and so custom expressions were derived in

place of (2.16) and (2.17).

2.4 Rotor Flux Tubes

A simplified view of the rotor is shown in Figure 2.9 where the flux paths are

illustrated as well as the orientation of the field winding. This differs from most

synchronous machines where the rotor flux is primarily oriented in the radial and

14

tangential directions, but in the claw-pole rotor, the flux is concentrated at the axis of the

machine in the axial direction. Flux tubes and their reluctance or permeance will now be

established for the paths shown in Figure 2.9. A list of variables used in rotor derivations

is provided in Table 2.2.

Figure 2.9: Rotor Section.

Figure 2.10: Physical Rotor.

15

2.4.1 Shaft and Core Flux Tubes

Table 2.2: Variable Identifiers and Descriptions, Rotor Quantities

Description

CL core length

COILH field coil height

CSN section number

DC core diameter

DD end disk diameter

G1 main airgap

G3 gap between claw sections

GLP length of rotor pole

HPR rotor tooth height at root

HPT rotor tooth height at tip

PITCH angle of pitch in of side of claw

RID inside diameter of claw

RISE angle of rise of claw underside

COD Field coil plastic slot outer diameter

ROD outside diameter of claw

RP number of poles in machine

SD shaft diameter

SPC sections per claw

TAPER angle of claw taper from base to tip

TED end piece thickness

WPR rotor tooth width at root

WPT rotor tooth width at tip

β1 half of angular width of narrow end of claw section

β2 half of angular width of wide end of claw section

µc rotor claw permeability

µcor rotor core permeability

16

Table 2.2 Continued

µd rotor end disk permeability

µend rotor end piece permeability

µsha rotor shaft permeability

µ0 permeability of free space

Figure 2.11 shows the simple cylindrical flux tubes for the rotor’s shaft and core.

The expressions for these flux tubes are relatively straightforward. The area of the shaft is

222

SDRP

π

and the length is CL TED+ . A coefficient of 2

RP is needed in the cross-

sectional area to account for the fact that the MEC is only a model of a single pole pair of

the machine. The extra term TED in the length is to account for the extra length that the

flux travels into the end disks on either side, which is a result of applying the mean path

approximation.

( )SHA 2

cor

CL TEDRPR2 SDμ π

2

+=

(2.22)

Figure 2.11: Shaft and core flux tubes.

17

The core is a cylindrical shell of a flux tube where the machine shaft resides in the cavity

in its center. The cross-sectional area is 2 22

2 2DC SD

RPπ

− and the length is the

same as the shaft flux tube. The reluctance is then

COR 2 2

o cor

RP CL+TEDR2 DC SDμ μ π

2 2

= −

(2.23)

2.4.2 End Disk Flux Tube

Figure 2.12: End disk of rotor.

In the shaft and core, the flux is in the axial direction. In the end disk shown in

Figure 2.12, the flux changes direction and then travels radially toward the stator. Though

the transition in direction is not truly abrupt, for modeling purposes an abrupt transition is

assumed. Looking into the disk radially the cross-sectional area is 4πA(r) r TEDRP

= ⋅ ⋅ .

With a differential length dr , the end disk differential reluctance is

RPπ

18

4d

DPdrdRr TEDµ π

=⋅

(2.24)

The total reluctance is obtained by integrating (2.24) from the inner to outer radius,

2

4

4

DD

SD

d DTEDP drR

rπµ= ∫ (2.25)

ln 24D

d

P DDRTED SDπ µ

=⋅⋅

(2.26)

2.4.3 End Piece Flux Tube

The intermediate path from the end disk to the claws is referred to as the end

piece. If the pole faces are the claws, then the end pieces can be thought of as the

knuckles, as shown in Figure 2.13.

Figure 2.13: End piece, or “knuckle,” of rotor.

The flux tube representing the knuckle is modeled using two series reluctances

which are perpendicular. Using a mean path approximation argument, the flux first

19

travels a distance 1 ( )2

ROD DD HPR− − radially from the end disk and then travels

axially toward the claw a distance 12

TED . The radial flux tube has cross-sectional area

TED WPR⋅ and the axial flux tube has cross-sectional area WPR HPR⋅ .

22 END END

ENDROD DD HPR TEDR

TED WPR WPR HPRµ µ− −

= +⋅ ⋅

(2.27)

2.4.4 Claw Section Flux Tube

The claws are a rather complex portion of the MEC for several reasons. Not only

are the claws difficult to define geometrically, but also the condition that flux must only

enter or exit an equipotential surfaces is violated as the flux leaves the pole face and

traverses the airgap. In order to model the effect of flux leaving the pole face, the length

of the claw is divided into several sections, described by the parameter SPC , the

“sections per claw.” Throughout the derivation of the MEC 4SPC = is used. At each

claw section junction, a node is placed in the MEC where an airgap flux tube is

connected which provides a path to traverse the airgap. The parameters for claw width

and claw height are assumed to be parameters of a particular claw section rather than the

full length of the claw.

As shown in the side view in Figure 2.14, the claw can be thought of as a

cylindrical shell whose inner radius changes linearly along the axial direction, described

by the parameter RISE . In addition to the underside radius changing, there is a tapering

of the angular span of the claw, as shown in the top view of Figure 2.14. This is described

by the parameter TAPER .

20

Figure 2.14: Detail of rotor claws.

The intermediate parameter TAPER can be expressed in terms of the geometrical

descriptors in Table 2-2 as

1tan2

WPR WPTTAPERGLP

− − = ⋅ (2.28)

Using a similar geometrical argument, the intermediate parameter RISE can be

expressed as

1tan HPR HPTRISEGLP

− − =

(2.29)

Noticing that the cross-sectional area of the claw section flux tube varies along the length

of the machine, the approach taken is to use differential reluctances as before

( )c

dzdRA zµ

= (2.30)

21

where the cross-section area of the claw as a function of position z down the axis of the

machine can be thought of as the difference in area of an outer and inner pie-section,

( ) ( ) (z)outer innerA z A z A= − , as detailed in the back view of Figure 2.14. The areas of the

pie-sections are functions of the angular span and radius. The radius of the inner pie-

section as well as the angular span of each section are also functions of z . First, ( )outerA z

can be expressed in terms of a circular area and a proportion of angular span to the whole

circle,

2

( ( ))2outer

RODA z zβππ

=

(2.31)

where ( )zβ can be expressed as the linear transition from 2β to 1β over the distance

/GLP SPC

2( )z m zβ β= ⋅ + (2.32)

and ( )1 2 SPCm

GLPβ β− ⋅

= , the slope of the linear transition. A similar approach is taken

for the inner pie-section, but it must be considered that the inner radius changes with

distance down the z axis. The claw section index CSN is included in the expression to

account for distance in the z direction that is taken up by claw sections preceding the

claw section of interest

( ) ( ) ( )2

1( ) 2

2inner

CSN GLP RIDA z m z z TAN RISESPC

β − ⋅

= ⋅ + ⋅ + ⋅ + (2.33)

where m is the same slope as before. Combining (2.30) – (2.33) yields

22

( ) ( ) ( )22 1

22 2o c

dzdRCSN GLPROD RIDm z z TAN RISE

SPCµ µ β

= − ⋅ ⋅ + ⋅ − + ⋅ +

(2.34)

In the MATLAB code implementation, (2.34) is integrated numerically to obtain the final

reluctance.

2.4.5 Rotor Leakage Flux Tubes

Some of the flux through the core and shaft does not go to the claws/airgap and

instead takes a path back through the space occupied by the field winding beneath the

claws. This path is herein referred to as the field leakage path. The geometry of the field

leakage path is shown in Figure 2.15.

Figure 2.15: Rotor field leakage path detail.

The field leakage path is modeled by two parallel paths. Flux that travels through

the winding is partially-linked flux and must be treated with an energy-based approach of

determining leakage permeances as detailed in [7]. Specifically, the energy is first

determined using the volume integral

23

212 V

E H dvµ= ∫ (2.35)

Subsequently another expression for energy

2 212

E N i P= (2.36)

is equated to (2.35) and then the permeance P determined. To do so, Ampere’s law is

applied along the core and back through the field winding, the same path the flux takes in

Figure 2.15. The H

field in the magnetic material is neglected as the MMF drop is

expected to be small there, and the current enclosed is then a function of radius. When

2 2CID CIDr COILH≤ ≤ + ,

2( ) CLfd

z

CIDTRC i rH r

COILH

⋅ − ⋅ = (2.37)

and when 2

CIDr COILH≥ + , the winding is fully-linked and ( ) CLz fdH r TRC i⋅ = ⋅ .

Substituting (2.37) into (2.35),

2

1 22

fdCIDTRC i r

CLE drCL COILH RP

π ⋅ − =

⋅

⋅

∫ (2.38)

where RP is used to represent the pole count so as to avoid mistaking it for permeance.

Evaluating the integral over 2

CIDr = to2

CID COILH+ yields

( )2 021 2 32 3fd

COILHE TRC i CID COILHCL RP

πµ ⋅= +

⋅ ⋅ (2.39)

Equating (2.39) to (2.36) and cancelling terms,

24

( )0 2 33coil

COILHP CID COILHCL RP

πµ ⋅= +

⋅ ⋅ (2.40)

The flux also occupies the space between the field coil and the underside of the claws,

where the permeance is that of a cylindrical shell

( )( )22

0AIR

RID CID 2COILHπ μP2 RP CL

− +⋅=

⋅ (2.41)

Then the parallel combination of these two permeances is

FLD COIL AIRP P P= + (2.42)

Magnetic flux will also leak from the claw on one rotor half to the adjacent claw

on the other rotor half, and it will do this on both sides. To model this path, it is assumed

that the radial component of the permeance cross-section uses the average of the claw

underside radius as denoted by _under rad in Figure 2.16.

1_2 2 2

RID RODunder rad HPT = + − (2.43)

( )1GLPh

COS TAPER= (2.44)

Substituting (2.44) into into (2.5) and noting the cross-sectional area is the product of 1h

and _2

R radOD under−

, the leakage permeance from claw to claw is

( )

0 _2

3CL

RODGLP under radP

COS TAPER SPC G

µ ⋅ ⋅ − =

⋅ ⋅ (2.45)

where 3G is the distance from claw to claw perpendicular to the claw edge, which can be

expressed as

25

( ) ( )WPR WPT2π RODG3 COS TAPERRP 2 2

− = − ⋅

(2.46)

Figure 2.16: Detail of the claw-claw leakage permeance.

2.4.6 Permanent Magnet

A second source of rotor MMF is utilized in some alternators in order to reduce

the claw-to-claw leakage flux. The direction is critical in that the magnets are oriented to

oppose the leakage flux that would occur with positive field current. If permanent

magnets are used in the alternator, the permanent magnet’s MEC in Figure 2.17 replaces

the leakage permeance (2.45).

0PM

PMH PMLPPMW SPC

µ ⋅ ⋅=

⋅ (2.47)

PM PMPMC PMW P= ⋅ ⋅Φ (2.48)

26

Figure 2.17: Permanent magnet in between claws.

2.5 Air Gap Flux Tubes

To model the complex collection of airgap flux tubes, it is assumed that a flux tube

is formed between the rotor and stator tooth only when there is some amount of angular

overlap between the two, and if there is not an angular overlap logic must be used to

determine if the rotor pole face is near the stator tooth, in which case a fringing flux tube

is described. If the rotor pole face is not near the stator tooth, then the airgap flux tube’s

permeance is zero. More specifically, when any portion of the claw is within the angular

range of the tooth tip’s edges, there is a direct path of length 1G , and on the approach to

angular overlap there is a more complex fringing path. An additional feature of the

airgap model is the representation of a chamfer along the tapered edge of the claw which

is a common feature in existing alternators as a means to control electrical and torque

harmonics and thus audible noise. A claw face with a trailing edge chamfer is shown in

27

Figure 2.18 where Region 4 represents the chamfer. This chamfer introduces an

additional airgap that is non-constant and must be dealt with accordingly. As described

earlier, the claw face area is divided into sections down the length of the machine’s axis

described by the parameter SPC . Table 2-3 lists the parameters used to describe the

airgap flux tubes along with descriptions of each variable.

φ

θφs

r

r

−αα

Tooth

GLPRe

gion

1 Region 2 Region 3Region 4

CW

z

l( )φr

β1β2 −β1 −β2

Figure 2.18: Formation of airgap flux tubes.

In Figure 2.18 the variable sφ is an angular location on the stator relative to a

reference axis that is fixed on the stator. Similarly, rφ is the angular location on the rotor

relative to an axis that is fixed on the rotor. For a chosen angular location, the two angles

can be related by

rr sφ θφ= − (2.49)

where rθ is the angle between the axis fixed on the stator and the axis fixed on the rotor.

28

Table 2.3: Variable Identifiers and Descriptions, Air Gap Quantities

Description

G1 main airgap length

GLP length of rotor pole

ID inside radius of stator stack

CD chamfer depth

CW chamfer width

β1 half of angular width of narrow end of claw section

β2 half of angular width of wide end of claw section

µ0 permeability of free space

θr mechanical position of rotor

From Figure 2.18 the angular location of the tooth tips in rotor angles are

( )r rα θφ α− = − − (2.50)

( )r rα θφ α= − (2.51)

The length of the overlapping edge can be expressed as

( ) 2r r

1 2 1 2

β GLPGLPβ β β β

l φ φ ⋅= −

− − (2.52)

When the tooth tip’s edge ( )α− begins entering the Region 1, i.e. when 1 ( ) 2β βr αφ −≤ ≤

and ( ) 2βr αφ ≥ , then the area of overlap is expressed as

( )2

r

β

r rα θ

A ID dl φ φ− −

= ∫ (2.53)

To consider the next two cases, it is noted that for some designs the total width of the

tooth tip can exceed the width of Region 1, or the opposite can occur – the width of

29

Region 1 can exceed the width of the stator tooth tip. When Region 1 spans the entirety

of the tooth width, or when 1 ( ) 2β βr αφ −≤ ≤ and 1 ( ) 2β βr αφ≤ ≤ , the area of overlap is

( )r

r

α θ

r rα θ

A ID dl φ φ−

− −

= ∫ (2.54)

When the tooth tip spans the entirety of Region 1, or when ( ) 1βr αφ − ≤ and ( ) 2βr αφ ≥ , the

area of overlap can be expressed as

( )2

1

β

r rβ

A ID dl φ φ= ∫ (2.55)

The next case that must be considered is for when the tooth is transitioning out of Region

1. This is described by the inequalities ( ) 1r αφ β− ≤ and 1 ( ) 2r αβ βφ≤ ≤ . In this situation, the

area of overlap is expressed as

( ) ( )( )r

1

α θ

r r 1 r 1 2 r2 1β

ID GLPA ID d β α θ α β 2β θ2 β β

l φ φ−

= = − + + − −−∫ (2.56)

A similar analysis can be applied to the remaining regions 2 and 3. The details and results

are provided in Appendix A.

Now the chamfer region with non-constant airgap is analyzed which is Region 4

in Figure 2.18. The parameter CW describes the angular width of the chamfer along the

tip and base of the tooth. Two cases arise when dealing with the chamfer, as shown in

Figure 2.19. On the left side of Figure 2.19, the corner point of the claw tip, described by

the angle 1β− , when projected onto the base is outside the angular span CW at the base

of the claw. The other case is illustrated on the right side of Figure 2.19 where the

projection of the corner point is within the angular span CW at the base of the claw. More

compactly, the inequality for the left side of Figure 2.19 is 1 2 CWβ β− ≥ − + and the

30

inequality is 1 2 CWβ β− ≤ − + for the right side. The chamfer analysis is further broken

down into three regions as shown in Figure 2.19.

GLPGLPCWCW

zz

φrφr

1 1

2 2

3 3

z1

z2 z2

z1

−β2−β1 −β2−β1

Figure 2.19: Two cases for determining airgap length over the chamfer region.

The airgap length in the chamfer region is varying with respect to position, therefore the

total permeance over the region can be determined by integrating differential permeances

where airgap length is a function of position. Using cylindrical coordinates, the

permeance of the chamfer region is

( )

rchamfer 0

r

dzdIDP μ2 g , z

φφ

= ∫∫ (2.57)

where ( , )rg zφ is the airgap length as a function of position. The chamfer is described

geometrically as a plane which contains the points ( ,, )g zφ : ( )21, ,0G CWβ− + ,

( )11, ,G CW GLPβ− + and ( )11 , ,G CD GLPβ+ − . An expression for g is then

( ) ( )2 1 2r r

CD CDCDg ,z z CD G1CW GLP CW CW

β β βφ φ− ⋅

= − + + −⋅

(2.58)

An expression for the line 1z which is the edge of the pole face where 1g G= is

( ) ( )1 r r 22 1

GLPz CWφ φ ββ β

= + −−

(2.59)

31

and 2z where 1g G CD= + is

( ) ( )22

2 r r1

GLPz φ φ ββ β

= +−

(2.60)

These z lines are used to bound the integration (2.57). Again with a case by case

treatment, the various expressions for airgap permeance can be expressed. When the

stator tooth is transitioning into Region 1 for both 1 2 CWβ β− ≥ − + and 1 2 CWβ β− +≤ −

(both left and right sides of Figure 2.19), the airgap permeance of the chamfer region is

( )

1

r 1

CW GLPr

chamfer 0rα θ z

dzdIDP μ2 g ,z

β φφ

− +

− −

= ∫ ∫ (2.61)

As before, in some designs the stator tooth width can entirely span the region or the

region can entirely span the tooth. When the tooth width is larger than the width of

Region 1, or 1 rβ α θ− ≥ − − and 1 rCWβ α θ− + ≤ −

( )

1 1

1 CW GLPr

chamfer 0rz

dzdIDP μ2 g ,z

β

β

φφ

− +

−

= ∫ ∫ (2.62)

When the tooth width is smaller than the width of Region 1, or

1 1r CWβ α θ β≤ − ≤ − +− − and 1 1r CWβ α θ β− ≤ ≤ − +− ,

( )

1

GLPr

chamfer 0rz

dzdIDP μ2 g ,z

r

r

α θ

α θ

φφ

+

− +

= ∫ ∫ (2.63)

When the tooth is transitioning out of Region 1, the limits of integration are again

changed accordingly and the airgap permeance is

( )

1 1

GLPr

chamfer 0rz

dzdIDP μ2 g ,z

rα θ

β

φφ−

+

= ∫ ∫ (2.64)

32

The same analysis techniques can be used to determine the airgap permeance over the

chamfer for Regions 2 and 3 for both 1 2 CWβ β− ≥ − + and 1 2 CWβ β− +≤ − (both left

and right sides of Figure 2.19). The results of all integrations in this section are

summarized in Appendix A.

A simple means of approximating fringing effects has been used in the MEC

whereby the airgap 1G is lengthened by Carter’s coefficient and the tooth width is

lengthened to account for fringing. Carter’s coefficient is

4 1 ln 1

4 1

ss sts

ssst

G ww wc

wG

ππ

+=

+ + ⋅

⋅ (2.65)

where 0ssw b= is the width of the slot, 0b is an input dimension of the machine and

0stID bwSL

π= − is the tooth tip width. The modified airgap 1 1G c Gs= ⋅′ is used in place of

1G in all airgap permeance expressions. To compensate for the lengthened airgap, a

modified tooth width is also used so that the fringing path begins as soon as the edge of

the rotor passes into the halfway-point of the slot width. Specifically the tooth tip width

α replaced by 'SLπα = in the airgap permeance expressions.

33

ALGEBRAIC SYSTEM OF MAGNETIC CIRCUIT EQUATIONS

Using the expressions for each reluctance or permeance, a system of equations is

formed and a solution method is presented. Within this research, nodal analysis is

employed in the airgap, where permeances will become zero-valued, and mesh analysis is

used elsewhere in the machine. Herein, a loop flux emanates from a node which is

nonlinear material and terminates when it encounters an air path.

3.1 Stator Teeth Mesh Equations

Referring to Figure 3.1, traversing the loop of mesh flux 1LΦ , the mesh equation

for the first stator slot can be expressed as

1 1 2 2 1 2 1 12 2 1 2 1( ) ( ) 0T L L T L L L y T TR R R f f F F− + − + + − + −Φ Φ Φ Φ =Φ (3.1)

The other 11 loop equations for the remaining teeth are similar. The MMF sources in

teeth 1 and 2 can be related to the total current in the slot by

2 1asTC i F F⋅ = − (3.2)

The turns count is the variable TC and is common to all windings. Similar equations can

be written for the remaining slots. In matrix form, the entire set of stator mesh equations

is

(12,12) (12,1) (12,12) (12,1) (12,6) (6,1)LT T LL L IP abcxyzs+ + = 0A f A A iΦ (3.3)

All matrices are expanded in Appendix A.

34

Figure 3.1: MEC structure detailing the discretization of the airgap and stator into respective flux tubes.

3.2 Nodal Analysis at Stator Tooth Tip Nodes

Summing all flux entering the node at stator tooth tip 1, including connections to all

the claw section nodes (10 claw section nodes when 4SPC = ), though most of these

permeances will be zero-valued for a given rotor position indicating there is not a path for

them at this rotor position, ones can use KCL to express

6 1 2 1 12 1

10 1 1 10 9 1 1 9 8 1 1 8 7 1 1 7 6 1 1 6

5 1 1 5 4 1 1 4 3 1 1 3 2 1 1 2 1 1 1 1

) )) ) ) ) )

) ) ) )

( (( ( (

)( (

( ( ( ( (0

L L T T TL T T TL

C T T C C T T C C T T C C T T C C T T C

C T T C C T T C C T T C C T T C C T T C

f P f Pf

f ff f f f ff f f f

P f P f P f P f Pf P f P f P f P f Pf

+ ++ + + + ++ +

Φ − Φ − −− − − − −

− − − −+ + +=

− (3.4)

The remaining 12 stator tooth tips have an identical form. Placing all of them together

yields a matrix expression,

35

(12,12) (12,1) (12,10) (10,1) TL(12,12) (12,1)TT T TC C L+ + = 0A f A f A Φ (3.5)

For the purposes of determining magnitude of flux density in each tooth, which is

required information for determining reluctance, the flux in each tooth is the difference of

the two loop fluxes present in a particular tooth, since they are opposite in direction. For

example, the flux density in tooth 1 is

1 1 12T L LΦ = Φ − Φ (3.6)

and the remaining teeth are similar.

3.3 Mesh/Nodal Analysis at Claw Section Nodes

Since the rotor’s interaction with the airgap is at the nodes between each claw

section, nodal analysis is employed. As an example, claw nodes 1 and 2 are similar in

form,

13 1 1 1 1 2 1 2 1 3 1 3 1 4 1 4 1

5 1 5 1 6 1 6 1 7 1 7 1 8 1 8 1

9 1 9 1 10 1 10 1 11 1 11 1 12 1 12 1

10

( ) P ( ) P ( ) P ( ) P( ) P ( ) P ( ) P ( ) P( ) P ( ) P ( ) P ( ) P2(

CS T C T C T C T C T C T C T C T C

T C T C T C T C T C T C T C T C


C

f f f f f f f ff f f f f f f ff f f f f f f ff f

+ − + − + − + −

+ − + − + − + −

+ − + − + − + −

+

Φ

− 1 1,10)C CL

PM

P= Φ−

(3.7)

having only one mesh flux as they are on the tips of the claws (most narrow portions).

The inner claw section nodes have an additional mesh flux, for example, the claw section

node at potential 3Cf can be expressed as

35 13 1 3 1 3 2 3 2 3 3 3 3 3 4 3 4 3

5 3 5 3 6 3 6 3 7 3 7 3 8 3 8 3

9 3 9 3 10 3 10 3 11 3 11 3 12 3 12 3

( ) P ( ) P ( ) P ( ) P( ) P ( ) P ( ) P ( ) P( ) P ( ) P ( ) P ( ) P2(

CS CS T C T C T C T C T C T C T C T C



f f f f f f f ff f f f f f f ff f f f f f f f

Φ − Φ + − + − + − + −

+ − + − + − + −

+ − + − + − + −

+ 8 3 3,8)C C CL

PM

f Pf= Φ−

− (3.8)

36

The remaining nodes take on a similar form, although the nodes at potentials 9Cf and

10Cf have an additional ENDΦ term from the knuckles. In matrix form, the set of 10

(when 4SPC = ) equations is

(10,12) (12,1) (10,10) (10,1) (10,1) (1,1) (10,8) (8,1) (10,17) (17,1)CT T CC C CE END CS CS PM+ + + = CA f A f A A bΦ Φ (3.9)

where PMC is a connection matrix that relates the magnetic circuit equations to system

inputs in the vector b , such as permanent magnet flux sources. If permanent magnets are

present between the claws, the leakage permeances CLP are replaced with (2.47).

Figure 3.2: Discretization of claws into sections.

In addition to the set of nodal equations for the claw section nodes, a set of mesh

equations is needed to solve for the mesh fluxes 1,3CSΦ – 8,10CSΦ . Referring to Figure 3.2,

for example, the flux in the claw section between nodes at potentials 1Cf and 3Cf is

1,3 1,3 3 1( ) 0CS CS C CR f f− − =Φ (3.10)

In matrix form the set of mesh fluxes can be expressed as

10Cf

CLP

8Cf

6Cf

4Cf

2Cf9Cf

7Cf

5Cf

3Cf

1Cf

7,9CSR

5,7CSR

3,5CSR

1,3CSR8,10CSR

6,8CSR

4,6CSR

2,4CSR1,3CSΦ

3,5CSΦ

5,7CSΦ

7,9CSΦ

9Cf

7Cf

5Cf

3Cf

1Cf

2,4CSΦ

4,6CSΦ

6,8CSΦ

8,10CSΦ

37

FS(8,10) (10,1) (8,8) (8,1)C FD CS+ = 0A f A Φ (3.11)

3.4 Relating Loop Flux to Phase Flux

The state variables of the system are the flux linkages of each winding. Phase flux

linkage is related to the phase flux by a pole count factor 2P

and a turns ratio for an

integer slot/pole/phase winding scenario. The tooth fluxes constituting a phase flux are

the sum of the teeth which are spanned by the respective winding. Subsequently, since

the a phase conductors span teeth 2-7, the phase flux is related to stator tooth fluxes by

, 2 3 4 5 6 7as p T T T T T TΦ Φ Φ Φ= + + + Φ +Φ Φ+ (3.12)

where the p denotes the flux in a pole pair. Using the relationship between tooth and

mesh flux, as in (3.6), for example in the a-phase

, 7 1as p L LΦ Φ Φ= − (3.13)

In matrix form,

(6,12) (12,1) (6,17) (17,1)FF L F=A C bΦ (3.14)

where the phase fluxes are members of the input vector b .

3.5 Mesh Analysis of Rotor

Mesh analysis is employed in the rotor of the alternator. This is accomplished using

KVL loops in each of the three circuital paths shown in Figure 3.3. The end disk DR is

divided into two series reluctances, an inner and outer disk, denoted DIR and DOR

respectively. Correspondingly, the limits of integration in (2.25) are adjusted and the

38

integral divided into two parts. The inner disk is integrated from 4

SD to

4SD DC+

and

the outer is integrated from 4

SD DC+ to

2DD

.

Figure 3.3: MEC structure detailing discretization of rotor into respective flux tubes.

9 102 ( ) 0END END END FD FD C CR R f f+ − + −Φ Φ =Φ (3.15)

( ) (2 )FD DO FD END FD FD SHA COR fdTRC iR R RΦ + Φ − Φ Φ = ⋅− Φ+ (3.16)

( )2 0SHA DI SHA RHA SHA FD CORR R RΦ + Φ + Φ − Φ = (3.17)

In matrix form, the three mesh equations describing the rotor can be placed into the

tableau

[ ]0

0

c

ENDRFC REND RFD RS fd

FD

SHA

f

A A A A TRC i

= ⋅

ΦΦΦ

(3.18)

ENDΦ

FDΦ

SHAΦ

9Cf 10Cf

SHAR

CORR

DIR

DOR

ENDR

FDR

39

3.6 Overall Algebraic System

Each subset of magnetic circuit equations can be combined into an overall system

of the form =Ax Cb where

(12,10) (12,1) (12,1) (12,12) (12,8) (12,6) (12,1)

(12,10) (12,1) (12,1) (12,12) (12,8) (12,6) (12,1)

(12,12)

(

(10,12) (10,1) (10,1) (10,12) (10,8) (10,6) (10,1)

12,12)

(10,10)

LT

TT

CC

LL IP

TC TL

CT CE CS=

0 0 0 0 00 0 0 0 0

0 0 0 00

A A AA A AA A A A

A(8,12) (8,10) (8,1) (8,1) (8,12) (8,8) (8,6) (8,1)

(6,12) (6,10) (6,1) (6,1) (6,12) (6,8) (6,6) (6,1)

(3,12) (3,10) (3,1) (3,1) (3,12) (3,8) (3,6) (3,1)

FS FD

FF

RFC REND RS RFD

0 0 0 0 00 0 0 0 0 0 00 0 0 0

A AA

A A A A

(3.19)

(1,12) (1,10) (1,1) (1,1) (1,12) (1,8) (1,6) (1,1)

T

T C END S L CS abcxyzs fdiΦ Φ ix = f f Φ Φ (3.20)

(17,12) (17,12) PM(17,10) (17,8) (17,6) (17,3)

T

F RFD = 0 0 0C C C C (3.21)

(1,10) (1,6) (1,1)

T

PM abcxyzs fd Φ b = Φ Φ (3.22)

When solving circuits in terms of node potentials, the potential is always relative

to the user-selected reference node. In this research, the reference node was chosen to be

stator tooth tip 1, which has potential 1Tf . To implement this, the equation corresponding

to the row which solves for 1Tf is deleted, and the variable is removed from (3.20).

When developing the system of equations for a wye-connected stator, it was noted

that one must account for the physical constraint that the sum of the currents in each wye

is zero. Therefore, the third is redundant, and the stator teeth mesh equations and the

relationship between loop and phase flux was altered to eliminate csi and zsi from the

system. This modification affects matrices IPA , FFA and FC . Details are provided in

40

Appendix A for the expanded matrices, including changes for wye and delta-connected

stators.

3.7 Solution of Nonlinear Algebraic System

From the MEC algebraic system, it is desired to solve for the vector x in

=Ax Cb . At first glance, the answer is simply 1−=x A Cb but the matrix A is a function

of x . More specifically, in each nonlinear flux tube, the relative permeability is a

function of the average flux density in each flux tube. The well-known Newton-Raphson

method is used to solve the nonlinear system of equations

) ( )( = −A xx xf Cb (3.23)

for the roots x . The Newton-Raphson iterator is

( ) ( ) ( ) ( )11 )( ( )i i i i−+ − =x x J x f x (3.24)

where ( ) )( iJ x is the Jacobian matrix of f and the superscript i denotes the iteration

number. In this research ( )J x is approximated as ( )A x , as was done in [3], for two

reasons. First, the nodal system which used this approximation in [3] converged in more

cases than with the full analytical Jacobian matrix. Second, less computation is needed to

express the approximation than the full Jacobian matrix, speeding up the time to solution.

As verification, a numerically-computed full Jacobian matrix was tested and compared to

(( ) )= AJ xx , and the numerically obtained Jacobian matrix did not improve

convergence. To further ensure convergence, the Newton-Raphson method is relaxed by

the constant α , 0 1α≤ ≤ and the iterator becomes

( ) ( ) ( ) ( )11 ( ) )(i i i iα−+ = − x x J x f x (3.25)

41

To initiate the Newton-Raphson method, initial values of phase and field flux linkages

are required which are user-selected. From the flux linkages, magnetic flux values are

determined and fed into the MEC algebraic system as the input vector b (3.22). The

Newton-Raphson method is then iterated until error criteria are met. Specifically, since

the members of x are MMF, current and flux as can be seen from (3.20), three error

tolerances are checked corresponding to the three types of variables Fe , eΦ and Ie

where

( ) ( ) ( ) ( )1 1

2 2

i i i iF r aF F F Fe x x K x x K+ += − − + − (3.26)

( ) ( ) ( ) ( )1 1

2 2r ai i i ie x x K x x K+

Φ Φ Φ+

Φ Φ= − − + − (3.27)

( ) ( ) ( ) ( )1 1

2 2

i i i iI I r I I aIe x x K x x K+ += − − + − (3.28)

This is similar to the nonlinear solver in [7]. The Newton-Raphson solver stops iterating

when each respective error is less than zero. rK and aK are small constants that describe

the relative and absolute error tolerance respectively. In other words, the iteration stops

when the solution difference between successive iterations is small relative to the

magnitude of the variables and relative to the absolute error.

42

MODELS FOR DIFFERENT LOADING CONDITIONS

In this chapter, several model structures are developed to enable coupling of

common loads to the MEC algebraic system. First a passive rectifier state model is

presented, next a model for stator open-circuit conditions is presented and then a

simplified way of simulating an active rectifier is presented without the need to model

individual switch states.

4.1 Passive Rectifier State Model

For the six phase machine being modeled, the first three phase set is labeled abc ,

which is connected in a wye configuration, and the second three-phase set is labelled

xyz and is also in a wye configuration, but it is displaced physically in the stator by 30°

electrically, or by one slot. The neutral points of the wye terminations are not connected.

Figure 4.1 illustrates the machine connection to the rectifier. Faraday’s law is used as the

starting point for the state model derivation as in [8]

abcxyzs s abcxyzs abcxyzsr p= +v i λ (4.1)

fd fd fdv r i= (4.2)

where dpdt

= .

43

Figure 4.1: Passive rectifier connection to machine.

Rearranging (4.1) – (4.2) and expanding,

as as s asv rp iλ = − (4.3)

bs bs s bsv rp iλ = − (4.4)

cs cs s csv rp iλ = − (4.5)

xs xs s xsv rp iλ = − (4.6)

ys ys s ysv rp iλ = − (4.7)

zs zs s zsv rp iλ = − (4.8)

fd fd fd fdv rp iλ = − (4.9)

fdv

Load

aibi

ci

dci

dcv

+

−

bsv

asv

csv

+

−

+

− +

−

+

−

+

−

+−xsvysv

zsv

xsi

ysizsi

g

g

agv+

−

bgv+

−

cgv+

−

xgv+

−

ygv+

−

zgv+

−

n

n

44

r rpθ ω= (4.10)

State models typically take the form = +x Ax Bu where x is the vector of state

variables and u is the vector of system inputs. Since the right hand sides of (4.3) –(4.10)

are not in terms of the variables being differentiated on the left-hand side, the model is in

a “quasi-state” model form. The quasi-state model could consist of (4.3) – (4.10) , but as

will be shown, the state model must be solved in the arbitrary reference frame in order to

avoid making an assumption about the stator voltages.

After the Newton-Raphson iteration has converged to a solution, the stator

currents, members of x (3.20), are input into the rectifier model, which determines

terminal voltages on the stator. The diode bridge model is illustrated in Figure 4.2. The

diode voltage as a function of diode current ( )D Dv i is modeled using the Shockley diode

equation which is solved for voltage, namely

1 n( ) l 1DD D

iv iD Dβ α

= +

(4.11)

where the diode parameters Dα and Dβ are selected for the particular diode being

modeled.

vag

vdc/2

v + vdc d

ialε−ε-vd

Figure 4.2: Lower diode voltage.

45

When the phase current Xsi is greater than , the upper diode is conducting and

the lower diode voltage ( )Xg dc D Xsv v v i= + , where [ ]X abcxyz∈ . If the phase current is in

the range [ ],− , a simple transitioning between the upper diode in full conduction and

the lower diode in full conduction is assumed, where the lower diode voltage is the linear

interpolant of the points ( )(, )Dv− and ( )(, )dc Dv v+ . When the phase current is less

than − , the lower diode is fully conducting and its voltage Xgv is ( )D Xsv i− . Once all

the lower diode voltages are established, a relationship to the phase voltages can be

derived. The analysis is performed on the first three phase set abc and the same analysis

is applied to the second three phase set xyz . Referring to Figure 4.1, KVL loops can be

made through the lower diode, phase winding and across potential ngv .

0as ag ngv v v− + = (4.12)

0bs bg ngv v v− + = (4.13)

0cs cg ngv v v− + = (4.14)

Summing (4.12)-(4.14) yields

( ) ( )1 13 3ag bg cg as bs cng sv v v v v v v+ + − + += (4.15)

At this point, it is common to make the assumption that the sum of the phase voltages

will be zero in each wye connection, however back-emf harmonics exist in machines with

unity slot/pole/phase windings and this causes the instantaneous sum of the phase

voltages in each three phase set to be nonzero. Substituting (4.15) into (4.12) - (4.14) and

rearranging yields

46

( ) ( )1 13 3as as bs cs ag ag bg cgv v v v v v v v− = + +−+ + (4.16)

( ) ( )1 13 3bs as bs cs bg ag bg cgv v v v v v v v− = + +−+ + (4.17)

( ) ( )1 13 3cs as bs cs cg ag bg cgv v v v v v v v− = + +−+ + (4.18)

Representing (4.16) - (4.18) in matrix form,

abcs abcg=Av Av (4.19)

where

2 1 1

1 1 2 13

1 1 2

− − = − − − −

A (4.20)

Substituting the subset abc of (4.1) into (4.19),

s as as

s bs bs abcg

s cs cs

r i pr i pr i p

λλλ

+ + = +

A Av (4.21)

abcs abcg s abcsp v r= −A A Aiλ (4.22)

In order to use this relationship in a state model, it is necessary to solve for each time-

derivative of flux linkage. Since A is not invertible, this cannot be done directly. When a

reference frame transformation is applied to (4.22), the differential equation then

becomes

[ ]( ) [ ]( )1 10 0s qd s abcg s s qd sp r− −= −A K Av A K iλ (4.23)

where sK is the arbitrary reference frame transformation matrix.

47

( )

( )

2 23 3

2 2

cos cos cos

2 sin sin sin3

1 1 12 2 2

3 3s

π πθ θ θ

π πθ θ θ

=

− +

+

−K (4.24)

For the second three phase set the analysis is similar except that the transformation matrix

takes on a 30° phase shift. Multiplying (4.23) by sK yields

[ ]( ) [ ]( )1 10 0s s qd s s abcg s s s qd sp r− −= −K A K A A K iK v Kλ (4.25)

Expanding the left-hand side of (4.25) yields

[ ]( ) [ ] ( ) [ ]( )1 1 10 0 0s s qd s s s qd s s abcg s s s qd sp p r− − −+ = −K A K K A K K K A K iAvλ λ (4.26)

where

[ ]( )10 1 01 0 00 0 0

s sp ω− = −

K A K (4.27)

[ ] 11 0 00 1 00 0 0

s s−

=

K A K (4.28)

Substituting (4.27) and (4.28) into (4.26), the time-derivatives of flux-linkages can be

solved for in the arbitrary reference frame

1 0 00 1 0

dsqs qss abcg s

qsds ds

ip r

iλλ

ωλλ

= −

− +

K Av (4.29)

The 2 3× matrix multiplying the voltage term extracts the upper two rows from [ ]sK A .

Electrical dynamics are simulated by integrating the quasi-state model derivatives in

(4.29) and (4.9) using the Forward Euler method. Mechanical dynamics are not modeled;

48

the model simulates constant rotor speed and the rotor position is updated using (4.10)

and the Forward Euler method. With (4.29), the Forward Euler update equations for qsλ

and dsλ are

( ) ( ) ( )1k k kqs qs qsp tλ λ λ+ = + ∆ (4.30)

( ) ( ) ( )1k k kds ds dsp tλ λ λ+ = + ∆ (4.31)

where the superscript k denotes the time step in the simulation. Since the bottom row of

(4.28) is composed of zeros, there is no explicit expression for 0spλ and it is not included

in the state model. From (4.30) and (4.31) the inverse reference frame transformation can

be applied.

( ) [ ]

( )

( )

( )

1

11 1

0

kqs

k kabcs s ds

ks

λ

λ

λ

+

−+ +

=

Kλ (4.32)

The input to the MEC (3.22) requires updated phase flux values which are obtained from

(4.32)

( ) ( )1 11k kabcs abcsTC

+ += λΦ (4.33)

Note that an approximation is made in (4.32) where 0sλ is obtained from the solution to

the Newton-Raphson method from time step k

( ) ( ) ( ) ( )( )013

k k k ks as bs csTCλ = Φ + Φ + Φ (4.34)

With updated phase fluxes, the next simulation time step is computed. The block diagram

in Figure 4.3 illustrates the overall solution procedure. Simulation results are presented in

the next chapter. The simulation output is compared to data from lab experiments.

49

Figure 4.3: Solution procedure block diagram.

4.2 Open Circuit Model

Open circuit modeling of the machine does not require the solution of a state

model, but solves a similar nonlinear MEC algebraic system as before using the Newton-

Raphson method. Again the overall system is described by some =Ax Cb . In open-circuit

conditions the stator current is zero and thus the algebraic system of equations established

in section 3.6 still holds, but some modifications must be made. Terms involving stator

current are removed as well as the set of equations relating loop to phase flux (3.14). This

corresponds to column 7 and row 5 being removed from (3.19).

MEC/Newton-Raphson

,TC TRC÷ ,TC TRC÷

,abcxyzs fdΦΦ

Rectifier Model

,abcxyzs fdi i

abcxyzgv

(4.29)

Forward Euler Integration

0qd spλ

sK

0qd sλ

[ ] 1s

−K

,abcxyzs fdλλ Initial Guess

0qd sλabcxyzsλ

abcxyzsifdi

0qd si

abcxyzsλ

50

(12,10) (12,1) (12,1) (12,12) (12,8) (12,1)

(12,10) (12,1) (12,1) (12,12

(12,12)

(12,12)

(10

) (12,8) (12,1)

(10,12) (10,1) (10,1) (10,12) (10,8) (10,1)

(8,12)

,

(8,10) (8,1

1

)

0)

(

LL

TC TL

CT

LT

TT

CE CS

FS

CC=

0 0 0 0 00 0 0 0

0 0 00 0 0

A AA A AA A A AA

A 8,1) (8,12) (8,8) (8,1)

(3,12) (3,10) (3,1) (3,1) (3,12) (3,8) (3,1)

FD

RFC REND RS RFD

0 00 0 0

AA A A A

(4.35)

Since stator currents no longer need to be solved for, they are removed from the solution

vector. Additionally, the flux through the field winding is added to the solution vector.

(1,12) (1,10) (1,1) (1,1) (1,12) (1,8) (1,1)

T

T C END S L CS fd Φ Φ Φx = f f Φ Φ (4.36)

The matrix (3.21) is now

(11,12) (11,12) (11,10) (11,8) (11,3)

T

PM RFD = 0 0 C 0C C (4.37)

The input to the system is now field current, the sole excitation of the system, thus (3.22)

becomes

(1,10) (1,1)

T

PM fd= i b Φ (4.38)

as the phase fluxes abcxyzsΦ are no longer driven by state of the system. When the

Newton-Raphson solver reaches a solution, the mesh flux vector LΦ is extracted from x ,

then the relationship between phase flux and mesh flux in the stator teeth (3.14) is applied

to yield phase flux abcxyzsΦ . Multiplying the phase flux by the stator turn count TC will

yield a set of stator flux linkages. The phase voltages are then

abcxyzs abcxyzsp=v λ (4.39)

which are computed by numerically differentiating the flux linkages, using the average of

forward and backward derivatives. Simulation results are presented in the next chapter.

The simulation output is compared to FEA predictions as well as data from lab

experiments. Matrices are expanded in Appendix B.

51

4.3 Active Rectifier Model

Automotive alternators have historically been used as generators which are

connected to passive rectifiers to power electrical loads and charge the on-board battery.

In recent years the automotive industry has manufactured vehicles with increasing use of

fuel-saving technology such as start-stop. A variant of the start-stop technology is

alternator-based start/stop, where an alternator is connected to an active rectifier in order

to provide starting torque to an internal combustion engine. In this section, an alternative

method of using the MEC to predict motoring torque is shown. The MEC is used to

predict the maximum achievable torque vs speed characteristic subject to maximum

current and bus voltage constraints. Figure 4.4 shows the circuit topology of the

machine/rectifier connection.

The MEC model can again be rearranged so that its inputs are field and stator

currents. It is similar in structure to the open circuit system. The system matrix A is

given by (4.35) and the solution vector x is given by (4.36). Now (4.37) becomes

(12,6)

(10,12)

(17,12) PM(17,10) (17,8) (17,3)

(1,12)

IP

T

TRFD

= −

00 C 0

0

C A C (4.40)

52

Figure 4.4: Active rectifier connection to machine.

Stator currents are added to the input vector and it becomes

(1,10) (1,6) (1,1)

T

PM abcxyzs fd= i b iΦ (4.41)

Electromagnetic torque is expressed in terms of quantities that are readily

obtained from the MEC. Analytical expressions for the airgap permeances derived in

Chapter 2 can be differentiated with respect to rθ and the electromagnetic torque can

then be directly calculated by [9]

( )2

1

12

nj

jej

r

r

dPT F

dθ

θ=

= ∑ (4.42)

Where n is the number of airgap elements.

fdv

Load

aibi

ci

dci

dcv

+

−

bsv

asv

csv

+

−

+

− +

−

+

−

+

−

+−xsvysv

zsv

xsi

ysizsi

53

In order to identify a maximum performance envelope for the machine, a genetic

algorithm (GA) optimization is employed to maximize torque over a three variable search

space r rdq fs dsi i i subject to maximum current and voltage constraints. Specifically, at

each rotor speed of interest, a GA performs the single-objective optimization subject to

the operating constraints

( ) ( )2 2

,2r rqs s Rateds dIi i ≤ ⋅+ (4.43)

( ) ( )2 2 13

r rqs ds dcv v v≤+ (4.44)

In order to determine the voltages so that constraint (4.44) may be checked, first the

phase currents in physical variables are given by

( ) ( )

3 3

3

cos sin2 2cos sin

2 2co sin3

s

rqs

r r

abcs r r rds

r r

ii

θ θπ πθ θ

π πθ θ

= − −

+ +

i (4.45)

cos sin6 6

2 2cos si3 3

n6 6

2 2cos sin6 63 3

r r

xyz

rqsrds

s r r

r r

ii

π πθ θ

π π π πθ θ

π π π πθ θ

− − = − − − −

+ − + −

i (4.46)

Similar to the previous section, the phase fluxes can be obtained from the stator teeth

mesh flux values. Then flux linkages are obtained for each phase. The flux linkages are

numerically differentiated using the average of the forward and backward numerical

54

derivatives. The phase voltages are then obtained by (4.1). The phase voltages are

transformed to the rotor reference frame by

0r

abcr

qd s s s=v K v (4.47)

Note that individual switch states are not simulated. Whether the modulation strategy for

the active rectifier is Space Vector Modulation, Sin-Triangle with third harmonic

injection or a current control method, sinusoidal currents result in the stator while (4.44)

is valid. The fitness function of the GA is simply electromagnetic torque (4.42). The

MEC was solved for several rotor positions spanning one slot pitch and then the torque

and voltages were averaged in order to include slotting effects. The maximal torque vs

rotor speed predicted by the optimization are presented in the next chapter. Matrices are

expanded in Appendix B.

55

VALIDATION OF MODELS

In order to validate the MEC models, three-dimensional finite element models

were created that accurately model the geometrical features and electrical output of a

commercially-available claw-pole alternator. One such FEA model was used to quantify

the flux linkage in the unloaded stator winding with varied rotor position which then

allowed the calculation of open-circuit voltages. In addition to the open-circuit FEA

model, the commercially-available alternator was tested in laboratory experiments where

it was affixed to a dynamometer test stand and loaded according to the test.

5.1 Passive Rectifier State Model Validation

Figure 5.1 and Figure 5.2 show a comparison between the prediction capability of

the MEC model and a laboratory test done on the commercially-available alternator. A

resistive load of .2 Ω was connected to the output of the alternator, the rotor was set to a

speed of 1750 RPM and had a field voltage applied such that rated field current was

achieved at 5 A. The stator current shows agreement in the harmonic content, but the

amplitudes of the harmonics don’t appear to match very well. The fundamental

component of stator current predicted by the MEC has a 4.7% difference from the

experimental data. However, when comparing RMS values, the difference increases to

10.4%. It has been observed that the harmonics are sensitive to parameters such as the

modified tooth width α′ used in the fringing airgap permeance calculation, as well as

56

claw-claw leakage permeance CLP . In addition, the amplitude of the stator currents are

sensitive to the diode parameters Dα and Dβ .

Figure 5.1: Comparison of MEC and hardware AC quantities for rectified alternator with resistive load.

The line-line voltage waveforms differ on the slope, but have a similarity in

magnitude near the peak. The chattering that is present in the slope of the MEC line-line

voltage is believed to be due to the diode model. Specifically, the transition region

between full lower diode conduction to full upper diode conduction is an abrupt change

57

in the voltage, which can cause oscillation in the state, accompanied by voltage

oscillation.

In Figure 5.2, the various DC quantities of interest are compared between the

MEC and lab test data. Field current matches well, but the DC output current has

approximately a 3.9 A difference. Inaccuracy in the stator current harmonics causes

significant discrepancy in the output current as the harmonics are rectified. There is

disagreement in both the amplitude and the harmonics of the DC voltage waveform.

Figure 5.2: Comparison of MEC and hardware AC quantities for rectified alternator with resistive load.

58

The disagreement in amplitude can be explained by the error in DC current for a given

resistance, but the disagreement in harmonics is believed to be due to the switching

events of an electronic load used in the experimental setup.

5.2 Open-Circuit Model Validation

Figure 5.3 illustrates the comparison between open circuit predictions of the MEC,

FEA and lab test data. In this comparison, the conditions were the same as for the

rectified testing (rotor speed 1750 rpm, field current 5A) except that the stator windings

have been disconnected from the rectifier and the line-line voltage is measured. The

amplitudes of all three waveforms in Figure 5.3 show agreement, however the MEC

predicts slightly higher amplitude.

Figure 5.3: Comparison between open-circuit line-line voltage prediction of MEC, FEA models and with experimental data.

59

Specifically, the fundamental component of the waveform predicted by the MEC

is 29.7 V whereas the FEA and Lab Test waveforms both predict a fundamental

component of 25.5 V. This corresponds to a 20% difference.

5.3 Active Rectifier Model Validation

In order to validate the torque prediction of the MEC, a comparison was done

between an FEA model and the MEC model for the same current magnitude, current

angle, field current and rotor speed. The normalized torque comparison is shown in

Figure 5.4. The GA coupled with the MEC was then run, as detailed in section 4.3, in

order to explore whether a higher torque value could be achieved while staying within the

machine current rating and obeying bus voltage limits. From Figure 5.4 higher torque is

indeed achievable than was shown for the FEA model. The torque has been normalized to

the maximum torque predicted by the FEA model.

Figure 5.4: Torque vs speed profiles for FEA and MEC.

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

rpm

Nor

mal

ized

Tor

que

FEAMECGA/MEC

60

CONCLUSION

In this research, the MEC model originally derived in [5] was recast as a mixed

mesh/nodal system in order to explore potential numerical improvements that were

expected based on well-performing mesh-analysis-based MEC models in [3] and [10].

Additional models and features were added and validated such as active rectification,

wye connection, permanent magnets between rotor claws and a higher phase count.

The MEC predictions show reasonable agreement with FEA and hardware testing,

in both harmonic content as well as amplitude. The error for loaded conditions is

acceptable, but the error can be larger in the open circuit voltage. The error is acceptable

since the computation time of MEC simulation is several orders of magnitude less than

FEA.

Structuring the circuit equations in a particular mixed mesh/nodal arrangement

was done with the intent that the Newton-Raphson method would have robust

convergence properties, and would not require the use of a user-tuned relaxation constant.

Despite this, the system matrix was ill-conditioned which caused poor convergence

properties and did not rid the problem of the relaxation constant. Further exploration of

nonlinear solution methods in future work is warranted in order to attain consistent

convergence with a minimal iteration count.

LIST OF REFERENCES

61

LIST OF REFERENCES

[1] C. J. Carpenter, “Magnetic equivalent circuits,” Proc. Inst. Electr. Eng., vol. 115, no. 10, pp. 1503–1511, Oct. 1968.

[2] V. Ostović, Dynamics of Saturated Electric Machines. New York, NY: Springer New York, 1989.

[3] H. W. Derbas, J. M. Williams, A. C. Koenig, and S. D. Pekarek, “A Comparison of Nodal- and Mesh-Based Magnetic Equivalent Circuit Models,” IEEE Trans. Energy Convers., vol. 24, no. 2, pp. 388–396, Jun. 2009.

[4] M. L. Bash, J. M. Williams, and S. D. Pekarek, “Incorporating Motion in Mesh-Based Magnetic Equivalent Circuits,” IEEE Trans. Energy Convers., vol. 25, no. 2, pp. 329–338, Jun. 2010.

[5] J. Williams, “Derivation of a magnetic equivalent circuit model for analysis and design of claw-pole alternator based automotive charging systems,” Masters Theses, Jan. 2001.

[6] “9780974547022: Introduction to AC Machine Design - AbeBooks - Thomas A. Lipo: 0974547026.” [Online]. Available: http://www.abebooks.com/9780974547022/Introduction-AC-Machine-Design-Thomas-0974547026/plp. [Accessed: 04-Apr-2016].

[7] S. D. Sudhoff, Ed., “Magnetics and Magnetic Equivalent Circuits,” in Power Magnetic Devices, Hoboken, New Jersey: John Wiley & Sons, Inc., 2014, pp. 45–112.

[8] P. Krause, O. Wasynczuk, S. Sudhoff, and S. Pekarek, Eds., “Distributed Windings in ac Machinery,” in Analysis of Electric Machinery and Drive Systems, Hoboken, NJ, USA: John Wiley & Sons, Inc., 2013, pp. 53–85.

[9] S. D. Sudhoff, Ed., “Force and Torque,” in Power Magnetic Devices, Hoboken, New Jersey: John Wiley & Sons, Inc., 2014, pp. 133–153.

[10] M. L. Bash and S. D. Pekarek, “Modeling of Salient-Pole Wound-Rotor Synchronous Machines for Population-Based Design,” IEEE Trans. Energy Convers., vol. 26, no. 2, pp. 381–392, Jun. 2011.

APPENDICES

62

A. AIRGAP PERMEANCE EXPRESSIONS

Airgap permeance derivations for Regions 1-4 are considered one by one. Areas of

overlap based on conditional statements are listed. To compute the airgap permeance in

Regions 1-3, the various area expressions can be substituted into

0

1RegionX

RegionX

AP

Gµ

= (A.1)

Explicit permeance expressions are derived for Region 4 because of the non-constant

airgap length.

φ

θφs

r

r

−αα

Tooth

GLP

Regi

on 1 Region 2 Region 3

Region 4

CW

z

l( )φr

β1β2 −β1 −β2

Figure A.1: Formation of airgap flux tubes.

Repeated for convenience,

rr sφ θφ= − (A.2)

( )r rα θφ α− = − − (A.3)

63

( )r rα θφ α= − (A.4)

A.1. Triangular Region 1

( ) 2r r

1 2 1 2


l φ φ ⋅= −

− − (A.5)

Tooth edge ( α− ) transitioning into Region 1, or when ( ) ( )1 ( ) 2 ( ) 2r rα αβ φ β φ β−≤ ≤ ∧ ≥ ,

( ) ( )2

r

β2

r r r 21

12α θ

ID GLPA ID l d α θ β4 β βRegion φ φ

− −

= = + +−∫ (A.6)

Tooth spans Region 1 or Region 1 spans tooth,

( )

( )1

r

2

r 1 ( ) 2 1 ( ) 2

1

( ) 2 ( )

α θ

r r 1

rID l d , ( ) ( )2

A

ID l d , ( ) ( )2

r

r r

Region

r r

α θ

α α

β

α αβ

φ φ β φ β β φ β

φ φ θ β θ β

−

−

−

− −

≤ ≤ ∧ ≤ ≤

= ≥ ∧ ≤

∫

∫

(A.7)

( )( ) ( ) ( )

( )

2( ) ( ) 2

2 1

1

2 1( ) 2 ( 1

1 2 1

)

GLP ID,

AGLP ID

, ( ) ( )4

rr r

Region

r r

α α

α α

α β θβ φ β β φ β

β β

β βθ β θ β

−

−

⋅ +≤ ≤ ∧ ≤ ≤ −=

⋅ − ≥ ∧ ≤

(A.8)

Tooth transitioning out of Region 1, or when ( ) ( )( ) 1 ( )2 1r rα αβ φ β φ β−≥ ≥ ∧ ≤ ,

( )1

1 r rID l d2

r

RegionAα θ

β

φ φ−

= ∫ (A.9)

64

( ) ( ) ( )2 2

1 2 1 2124Region r

GLP IDA β β α θ ββ β

⋅ = − ++− − (A.10)

A.2. Rectangular Region

Area calculations consist of the product of arc lengths and l GLP= , no integration

necessary.

Tooth Transitioning into Region 2, or when ( ) ( )1 ( ) 1 ( ) 2r rα αβ φ β φ β−− ∧≤ ≤ ≥ ,

( )2 12Region rIA D GLP β α θ⋅

+= + (A.11)


( ) ( )

( ) ( )

1 ( ) 1 1 ( ) 1

2

1 ( ) 1 ( ) 1

,

,

r r

Region

r r

GLP ID

GLP

CW CW

I CD

A

W

α α

α α

α β φ β β φ β

β φ β φ β

−

−

⋅ ⋅ − ≤ ≤ ∧ − ≤ ≤ + +=

⋅ ⋅ ≥ ∧ ≤ − +

(A.12)

Tooth Transitioning out of Region 2, or when

( ) ( )1 2 1r rCW CWβ α θ β α θ β≥ − ≥ − − ≤+ ∧ − + ,

( )2 12Region rIA GLPD α θ β⋅

+−= (A.13)

65

A.3. Triangular Region 2

( ) ( )2r r

1 2 1 2


CWl φ φ

− ⋅= − −

− − (A.14)

Tooth transitioning into Region 3, or when

( ) ( )2 ( ) 1 ( ) 1r rCCW CWWα αβ φ β φ β−≤ ≤ − + ≥− + ∧ + ,

( ) ( ) ( ) ( )1

r

-

r r2

2 23 2 1

1α2

θ

ID ID GLPA d2 4 β βReg r

W

i

C

on l CWβ

φ φ β β φ β+

− −

= = − − + − −⋅

∫ (A.15)


( ) ( ) ( )

( ) ( ) ( )1

2

r 2 ( ) 1 2 ( ) 1

3

( ) 1 ( )

r

2r

ID d ,2

A

ID d ,2

r

r

r r

RegioCW

n

r rCW

r

CW Cl CW CW

l CW C

W

W

α θ

α αα θ

β

α αβ

φ φ β φ β β φ β

φ φ φ β φ β

−

−−

+

+

−

−

−−

− + ≤ ≤ − + ≤ ≤ − +

≥ − ≤ −

∧ − += + ∧ +

∫

∫ (A.16)

( )

( ) ( )

( )

( ) ( )

22 1

2 ( ) 1 2 ( ) 1

3

2 1

( ) 1 ( ) 2

ID ,

AID GLP ,

4

r

r r

Region

r r

GLP CW

CW CW

CW CW

CW CWα α

α α

α θ ββ β

β φ β β φ β

β β

φ β φ β

−

−

− + − − + ∧ − +=

⋅ ⋅

≤ ≤ − + ≤ ≤ − +

⋅

≥

− + ∧ +

− ≤ −

(A.17)

Transitioning out of Region 3, or when

( ) ( )1 ( ) 2 ( ) 2r rCW CW CWα αβ φ β φ β−− + + ≤ −∧− +≥ ≥ ,

66

( )2

23 2

2 1

ID( )2

r

RegioCW

n r r rIDA l d G CWLPα θ

β

φ φ α θ ββ β

−

− +

= = +− −⋅−∫ (A.18)

A.4. Chamfer Region

The pole face is subdivided further into sub-regions of the chamfer region, as

shown in Figure A.2. Each sub-region is considered on a case-by-case basis, for both the

left and right possibilities in Figure A.2, i.e. 2 1CW β β≤ − and 2 1CW β β≥ − .

GLPGLPCWCW

zz

φrφr

1 1

2 2

3 3

z1

z2 z2

z1

−β2−β1 −β2−β1

Figure A.2: Two cases for determining flux tubes over chamfer region.

( ) ( )2 1 2r r

CD CDCDg ,z z CD G1CW GLP CW CW

β β βφ φ− ⋅

= − + + −⋅

(A.19)

( ) ( )1 r r 22 1

GLPz CWφ φ ββ β

= + −−

(A.20)

( ) ( )22

2 r r1

GLPz φ φ ββ β

= +−

(A.21)

Each case is considered first for the left possibility in Figure A.2, or when

2 1CW β β≤ − . For a tooth transitioning into sub-region 1, or when

( ) ( )1 ( ) 1 ( ) 1r rCW CWα αβ φ β φ β−≤ ≤ − ≥ −+ +− ∧ ,

67

( )

r 1

1 CW GLPr

0rα θ z

dz dIDμ2 g ,zchamferP

β φφ

− +

− −

= ∫ ∫ (A.22)

( )lnchamfer a b c dP = − (A.23)

where

( )0 2

2 1

CWGLPaCD

µβ β

=−

⋅

1CD +CD CW+CW g0+CD +CD rb β α θ= − ⋅ ⋅ ⋅ ⋅ ⋅

CW g0

bc =⋅

( )1CD + +CWrd α θ β= − −

For a tooth fully inside sub-region 1, or when

( ) ( )1 ( ) 1 1 ( ) 1r rCW CWα αβ φ β β φ β−− + ∧ ≤ ≤ −−− +≤ ≤ ,

( )

r

r 1

α θ GLPr

0rα θ z

dz dIDμ2 g ,zchamferP φ

φ

−

− −

= ∫ ∫ (A.24)

[ ]log( ) log( ) 2 log(CW/C ) 11Dchamfer GP a b c d e α= − −⋅ ⋅ ⋅ −⋅ (A.25)

where

0

2 1

GLP CW( - ) CD

a µβ β ⋅

=⋅

1( / 1 )rb CW CD G CWα θ β ⋅= + − + +

1+ - +CW/CD 1+CWrc Gα θ β= ⋅

1( )1/rd CW CD G CWα θ β ⋅= − + − + +

68

When sub-region 1 is fully-spanned by the tooth, or when

( ) ( )( ) 1 ( ) 1r rCWα αφ β φ β−+≥ − ≤ −∧ ,

( )

1

1

1

GLPr

0rz

dz dIDμ2 g ,zchamfe

C

r

W

Pβ

β

φφ

− +

−

= ∫ ∫ (A.26)

2

02

2 1

1( 1) ln( )CD 1chamfer

GLP CW G CDP CD G CDG

µβ β

⋅ + + − − = (A.27)

When tooth is transitioning out of sub-region 1, or when

( ) ( )1 ( ) 1 ( ) 1r rCWα αβ φ β φ β−− + ∧≤ ≤ − ≤ −

( )

1

( )

1

GLPr

0rz

dz dIDμ2 g ,z

r

chamferPαφ

β

φφ−

= ∫ ∫ (A.28)

( ln( ) ln( 1) ln( ) )chamfer b b c CW G d dP a e= − + ⋅ + + (A.29)

Where

02

2 1( )GLP CW

CDa µ

β β=

−⋅

1 1rb CD CD CD CW G CD CWα θ β= − + − + ⋅ + ⋅

( )1rc CD θ β α−= −

( )1CW Gd CD= +

( )1re CD α θ β+= − −

69

For the right possibility in Figure A.2, or when 2 1CW β β≥ − , each angular location is

considered again on a case-by-case basis. For a tooth transitioning into sub-region 1, or

when ( ) ( )1 ( ) 1 ( ) 1r rCW CWα αβ φ β φ β−≤ ≤ − ≥ −+ +− ∧ , then (A.22) and (A.23) apply again.

When the tooth is fully inside sub-region 1, or when

( ) ( )2 ( ) 1 2 ( ) 1r rCCW W WCW Cα αβ φ β β φ β−− + ∧ −≤ ≤ − + ≤ ≤ − ++ , then (A.24) and

(A.25) apply again.


( ) ( )1 ( ) ( ) 2r rCW CWα αβ φ φ β−− ≤ −∧≤ ++ ,

( )

1

1

2

GLPr

0rz

dz dIDμ2 g ,zchamf

CW

CWerP

β

β

φφ

− +

− +

= ∫ ∫ (A.30)

02

2 1( )1lnchamfer

CW Ga ba

GLP CWPh

µβ β

= − + ⋅ ⋅

− (A.31)

where

1 2CD -CD -CW G1a β β= ⋅ ⋅ ⋅

2 1-b CD CDβ β= +⋅ ⋅


( ) ( )2 ( ) 1 ( ) 2r rCW CWCW α αβ φ β φ β−≤ ≤ − + ≤ −∧ +− +

( )

12

GLPr

0rz

dz dIDμ2 g ,z

r

chamfeW

rC

Pα θ

β

φφ− +

−

= ∫ ∫ (A.32)

[ ]02

2 1(ln( ) ln( 1) l

)n( )chamfer

GLP CWP a a b CW G cCD

cµβ β

= − + +⋅

⋅−

(A.33)

1CD -CD +CD -CW G1-CD CWra α θ β⋅ ⋅ ⋅ ⋅ ⋅=

70

2- -rb CD CD CD CD CWθ α β= +⋅ ⋅ ⋅ ⋅

1 2-CD +CW G1+CDc β β⋅ ⋅ ⋅=

Sub-region 2 is now considered. First, when 2 1CW β β≤ − , when the tooth is

transitioning into sub-region 2, or when ( ) ( )( ) 2 (1 ) 1r rCWα αβ φ β φ β−∧ +− ≤ − ≤ ≤ − ,

( )

1

1 2zr

0rz

dz dIDμ2 g ,z

r

cham

CW

ferPβ

α θ

φφ

−

− −

+

= ∫ ∫ (A.34)

01

2 1

CW GLP CD( + )ln 1+( )CD G1chamfer rP µ α β θβ β

⋅ = − − (A.35)


( ) ( )2 ( ) 1 2 ( ) 1r rCW CWα αβ φ β β φ β−− ≤ ≤ − − ≤+ ≤ −+ ∧ ,

( )

1

2zr

0rz

dz dIDμ2 g ,z

r

r

chamferPα θ

α θ

φφ

−

− −

= ∫ ∫ (A.36)

0

2 1

GLP CW G1ln( ) 1+CD

2CD GchamferP αµ

β β = −

⋅⋅

⋅ (A.37)


( ) ( )( ) 2 (1 ) 1r rCWα αβ φ β φ β−∧ +− ≤ − ≤ ≤ − ,

( )

1

1 2

2

zr

0rz

dz dIDμ2 g ,zcha r

Cmfe

W

Pβ

β

φφ−

−

+

= ∫ ∫ (A.38)

02 1

2 1

CW GLP CD( CW)ln 1+( )CD G1chamferP µ β ββ β

= − −

⋅

− (A.39)

When transitioning out of sub-region 2, or when

( ) ( )2 ( ) 1 ( ) 2r rCW CWα αβ φ β φ β−≤ ≤ − ≤ −− + ∧ + ,

71

( )

1

2

2

zr

0rz

dz dIDμ2 g ,z

r

chamf rW

eC

Pα θ

β

φφ+

−

−

= ∫ ∫ (A.40)

01

2 1

CW GLP CD( )ln 1+( )CD G1chamfer rP CWµ α θ ββ β

⋅ = + − − − (A.41)

Now, when 2 1CW β β≥ − , when the tooth is transitioning into sub-region 2, or when

( ) ( )( ) 2 1 ( ) 2r rCW CWα αφ β β φ β−∧ ≤ ≤+ +≥ − ,

( )

2r

0r0

dz dIDμ2 g ,z

r

GLP

cham

CW

ferPβ

α θ

φφ

−

− −

+

= ∫ ∫ (A.42)

[ ]02

2 1

ln(- ) * 1*ln( * 1) ln( ) - l ( )(

n)

CW GLP a a CW G CCD

D G b b c cµβ β

⋅+ +

− (A.43)

where

4 - 3- 1a CD b CD b CW G⋅ ⋅ ⋅=

1- 1rb CD CD CD CW CD CW Gα θ β⋅ ⋅ ⋅ +⋅+ ⋅= +

2- 1rc CD CD CD CW CD CW Gα θ β⋅ ⋅ +⋅+ ⋅= +


( ) ( )1 ( ) 2 1 ( ) 2r rCW CWα αβ φ β β φ β−− ≤ ≤ − + − ≤ ≤ − +∧ ,

( )

r0

r0

dz dIDμ2 g ,z

r

r

GLP

chamferPα θ

α θ

φφ

−

− −

= ∫ ∫ (A.44)

( ) ( )0

2 1

GLP CW( )CD

ln ln ln( ) ln( )chamfer a a b b c d dP cµβ β

⋅+ + −

−= (A.45)

where

72

11

ra CW GCWCD

α θ β ⋅− + − + +=

11

rCW GCW

CDb α θ β ⋅

+ += − +

21

rc CW GCWCD

α θ β ⋅− + − + +=

2G1

rCWd CW

CDα θ β ⋅

= + − + +

When sub-region 2 is fully-spanned by tooth, or when ( ) ( )( ) 2 ( ) 1r rCWα αφ β φ β−+≥ − ≤ −∧

( )

2

1

r0

r0

dz dIDμ2 g ,z

CW GLP

chamferPβ

β

φφ

− +

−

= ∫ ∫ (A.46)

[ ]02

2 1

ln( ) ln(b) ln( ) lC n( )W G( )CD

LPchamfer a a b c cP dµ

β β⋅

− + + +−

= (A.47)

Where

2 1 1CD CD CW Ga β β− ⋅ + ⋅ − ⋅=

1b CW G= ⋅

( 1 )CW Gc CD⋅ +=

1 21d CD CW G CD CD CWβ β= − ⋅ − ⋅ + ⋅ − ⋅


( ) ( )1 ( ) 2 ( ) 1r rCWα αβ φ β φ β−+ ∧− ≤ ≤ − ≤ − ,

( )

1

r0

r0

dz dIDμ2 g ,z

r GLP

chamferPα θ

β

φφ

−

−

= ∫ ∫ (A.48)

73

( ) [ ]0

22 1

ln( ) ln( ) ln( ) ln( )chamferGLP a aCWPCD

b b c c d dµβ β

⋅− + + − +

−= (A.49)

where

2 1 1CD CD CW G CDa CWβ β⋅ ⋅ − ⋅ − ⋅= −

( 1 )CW Gb CD⋅ +=

1 1rCD CD Cc D CW CD CW Gα θ β⋅ − ⋅ − ⋅ + ⋅ − ⋅=

2 1rCD CD CD CW CD W Gd Cα θ β− ⋅ + ⋅ + ⋅ − ⋅ + ⋅=

Sub-region 3 is now considered. First, the case when 2 1CW β β≤ − is considered for all

angular location possibilities. When the tooth is transitioning into sub-region 3, or when

( ) ( )2 ( ) 2 2 ( )r rCW CWα αβ φ β β φ− ≤ ≤ − − + ≤∧+ ,

( )

2 2

( )

r0

r0

dz dIDμ2 g ,z

r

z

cha

C

mfe

W

rPα

β

φ

φφ

−

− +

= ∫ ∫ (A.50)

( )( ) ( )02

2 1

l( )

n 1 lnchamferCW GLPP a

Dc

Cb dµ

β β =

⋅+ + −

(A.51)

where

2 ra CD CD CW CD CDβ α θ⋅ ⋅ ⋅− + + ⋅= +

2

CW G1+CW CDCD +CD -CD +CD CW+CW G1r

bα θ β

⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅

=

1c CW G= ⋅

2

CW G1CD +CD -CD +CD CW+CW G1r

dα θ β

⋅⋅ ⋅ ⋅ ⋅

=

74


( ) ( )2 ( ) 2 2 ( ) 2r rCW CWα αβ φ β β φ β−− ≤ ≤ − + − ≤ ≤ − +∧ ,

( )

( ) 2

( )

r0

r0

dz dIDμ2 g ,z

r

r

z

chamferPα

α

φ

φ

φφ

−

= ∫ ∫ (A.52)

[ ]0

2 1

GLP CW 2 (ln( ) 1) - ln( ) ln( )( ) CDchamferP CD a a b bµ αβ β

= + +⋅ ⋅− ⋅

(A.53)

where

2- - -1r

CWa CWCD G

α θ β+⋅

=

2- - - -1r

CWb CWCD G

α θ β= +⋅


( ) ( )2 ( ) ( ) 2r rCW α αβ φ φ β−+ ∧− ≤ ≤ − ,

( )

2 2

2

r0

r0

dz dIDμ2 g ,z

z

cha f

W

m er

C

Pβ

β

φφ

− +

−

= ∫ ∫ (A.54)

[ ]2

02

2 1

1(ln( 1) ln( 1 ))*( )chamfer

CW GLP G CW G CW G CW CD CWD

PCµ

β β⋅

⋅ − ⋅ + ⋅ +=−

(A.55)

When the tooth is transitioning out of sub-region 3, or when

( ) ( )2 ( ) 2 ( ) 2r rCWα αβ φ β φ β−+ ∧− ≤ ≤ − ≤ − ,

( )

( ) 2

2

r0

r0

dz dIDμ2 g ,z

r z

chamferPαφ

β

φφ−

= ∫ ∫ (A.56)

[ ]022

2 1( )ln( ) ( - )chamfer r

CW GLPP a b DCD

Cµ α β θβ β

⋅⋅

−= + + (A.57)

where

75

2-CD +CD -CD +CD CW+G1 CWra α θ β⋅ ⋅ ⋅ ⋅ ⋅=

2CD (- + - +CW)+CW G1CW G1+CW CD

rb α θ β⋅ ⋅⋅ ⋅

=

Now, the case when 2 1CW β β≥ − is considered for all angular location possibilities,

again for sub-region 3. When the tooth is transitioning into sub-region 3, or when

( ) ( )2 ( ) 1 1 ( )r rα αβ φ β β φ∧− ≤ ≤ − − ≤ ,

( )

1 2

( )

r0

r0

dz dIDμ2 g ,z

r

z

chamferPα

β

φ

φφ

−

−

= ∫ ∫ (A.58)

[ ]02 ln(b) ln( ) - ln( )

( 3 4)chamfer b b CCW GLPP a c c d

Dd eµ ⋅

−= + + (A.59)

where

1- CD+CD +CD ra β α θ⋅ ⋅ ⋅=

1b CW G CW CD= +⋅ ⋅

1 21c CW CW G CD CD CWβ β ⋅= − +⋅ ⋅+⋅

21rd CD CD CW G CD CD CWα θ β⋅ ⋅ ⋅ ⋅+ − + ⋅= +

1re CD CD CDα θ β⋅ ⋅ − ⋅= +


( ) ( )2 ( ) 1 2 ( ) 1r rα αβ φ β β φ β−− ≤ ≤ − − ≤ ≤ −∧ ,

( )

( ) 2

( )

r0

r0

dz dIDμ2 g ,z

r

r

z

chamferPα

α

φ

φ

φφ

−

= ∫ ∫ (A.60)

[ ]0

2 1

2 (1 ln( )) - ln( ) ln( )( )chamfer

GLP CWP CD a a b bCD

µ αβ β

= + +−

⋅ ⋅⋅

(A.61)

76

where

2- - -1r

CWa CWCD G

α θ β+⋅

=

2- - - -1r

CWb CWCD G

α θ β= +⋅

When the sub-region 3 is fully-spanned by the tooth, or when

( ) ( )1 ( ) ( ) 2r rα αβ φ φ β−− ∧≤ ≤ − ,

( )

1 2

2

r0

r0

dz dIDμ2 g ,z

z

chamferPβ

β

φφ

−

−

= ∫ ∫ (A.62)

02

2 1

ln( )

CW GLP baCD

dc

µβ β

⋅+ −

(A.63)

where

2 1- CD+CW G1+CD CW+ CDa β β⋅= ⋅ ⋅ ⋅

CW G1+CW CDb = ⋅ ⋅

2 1- CD+CW G1+CD CW+ CDc β β⋅= ⋅ ⋅ ⋅

2-CD -CD*b3d β=

When the tooth is transitioning out of sub-region 3, or when

( ) ( )2 ( ) 1 ( ) 2r rα αβ φ β φ β−− ≤ ≤ ≤ −∧− ,

( )

( ) 2

2

r0

r0

dz dIDμ2 g ,z

r z

chamferPαφ

β

φφ−

= ∫ ∫ (A.64)

[ ]02

2 1(ln( ) ln(

))chamfer

CW GLPP a b c cCD

dµβ β

+⋅

−= + (A.65)

where

77

2CD -CD + CD-CW G1-CD CWra α θ β⋅ ⋅ ⋅ ⋅ ⋅=

CW G1+CW CDb = ⋅ ⋅

2-CD +CD - CD+CW G1+CD CWrc α θ β⋅ ⋅ ⋅ ⋅ ⋅=

2CD -CD +CDrd β θ α⋅= ⋅ ⋅

78

B. MATRIX EXPANSION

The matrices for the system of MEC equations are expanded here. Where applicable, a

matrix may appear as several different versions depending on which model it is used in or

which stator phase connection is used.

(12,12)

1 1 0 00 1 1 0

0 1 1 00 1 1 0

0 1 1 00 1 1 0

0 1 1 00 1 1 0

0 1 1 00 1 1 0

0 0 0 1 11 0 0 1

LT

− − − − −

− = −

− − − − −

A (2.1)

(12,10) , 1 12, 1 10TC TiCjP i j= = = A (2.2)

(10,12) (12,10)CT TCT=A A (2.3)

(12,12) (12,12)TLTTL =A A (2.4)

79

(3,10)

0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

FDL FDL

RFC

P PA

− =

(2.5)

(3,1)2 0

2

T

END FDL FDLREND

FDL COR DO FDL

R R RR R R R

+ −= − − −

A (2.6)

(3,1)20

2

T

COR SHA COR DIRS

COR DO FDL COR

R R R RR R R R

− + += − − −

A (2.7)

(3,1) 0 02

T

RFDCOR DO FDL

TRCR R R

−= − − −

A (2.8)

(3,7)

0 0 10 0 10 0 1

RFD

=

C (2.9)

80

80

(12,12)

122 1

1 2

232 3

2 3

343 4

3 4

454 5

4 5

565 6

5 6

676 7

6 7

787

7

0 0

0 0 0

0 0

0 0

0 0

0 0

0

LL

YT T

T T

YT T

T T

YT T

T T

YT T

T T

YT T

T T

YT T

T T

YT

T T

A

RR R

R R

RR R

R R

RR R

R R

RR R

R R

RR R

R R

RR R

R R

RR

R R

=

+ − − +

+− −

+

+− −

+

+− −

+

+− −

+

+− −

+

+−

+

88

898 9

8 9

9109 10

9 10

101110 11

10 11

111211 12

11 12

1211 12

12 1

0

0 0

0 0

0 0

0 0 0

0 0

T

YT T

T T

YT T

T T

YT T

T T

YT T

T T

YT T

T T

R

RR R

R R

RR R

R R

RR R

R R

RR R

R R

RR R

R R

− + − − +

+− −

+

+− −

+

+

− −+

+−

+ −

(2.10)

81

81

1

1

1

1

1

2

3

4

5

(12,12

1

)

20 0

20 0 0

20 0

20 0

20 0

20

c

c

c

c

c

N

i

N

i

N

TL

TL TLT Ci

TL

TL TLT Ci

TL

TL TLT C

ii

TL

TL TLT Ci

TL

TL TLT Ci

N

i

N

i

TL

TL

TT

PP P

P

PP P

P

PP P

P

PP P

P

PP P

P

PP

P

=

=

=

=

=

− −

− −

− −

− −

− −

−

−

=

∑

∑

∑

∑

∑

A6

7

8

9

10

11

1

1

1

1

1

1

0

20 0

20 0

20 0

20 0

20 0 0

c

c

c

c

c

c

TLT Ci

TL

TL TLT Ci

TL

TL TLT Ci

TL

TL

N

i

N

i

N

i

N

i

N

TLT Ci

TL

TL TLT Ci

TL

TLT Ci

i

N

i

P

PP P

P

PP P

P

PP P

P

PP P

P

PP

P

=

=

=

=

=

=

− −

− −

− −

− −

− −

∑

∑

∑

∑

∑

∑

11

2

20 0 c

TL

TL

TL TN

i

LT Ci

P

PP P

P=

− −

∑

(2.11)

82

82

1

110

1101

29

292

38

383

47

474

56

(10,

1

1

)

1

10

20 0 0 0 0 0 0 0 2

20 0 0 0 0 0 0 2 0

20 0 0 0 0 0 2 0 0

20 0 0 0 0 2 0 0 0

20 0 0 0

t

t

t

t

N

i

N

i

N

CL

CLTiC

CL

CLTiC

CL

CLTiC

CL

CLTiC

CL

Ti

C

i

i

C

N

PP

P

PP

P

PP

P

PP

P

P

P

=

=

=

=

− −

− −

− −

− −

−

−

=

∑

∑

∑

∑

A 1

1

1

1

565

56

566

47

477

38

388

29

299

1

2 0 0 0 0

20 0 0 0 2 0 0 0 0

20 0 0 2 0 0 0 0 0

20 0 2 0 0 0 0 0 0

20 2 0 0 0 0 0 0 0

t

t

t

t

t

N

i

N

i

N

CLC

CL

CLTiC

CL

CLTiC

CL

CLTiC

CL

CL

i

N

i

T

N

iCi

P

PP

P

PP

P

PP

P

PP

P

=

=

=

=

=

− −

− −

− −

− −

∑

∑

∑

∑

∑

110

11

1

010

22 0 0 0 0 0 0 0 0 tN

CL

CLTiC

i

PP

P=

− −

∑ (2.12)

83

For wye connection,

(12,4)

1 0 0 00 0 1 01 1 0 00 0 1 10 1 0 00 0 0 11 0 0 0

0 0 1 01 1 0 0

0 0 1 10 1 0 00 0 0 1

IP TC

= ⋅ −

− − − − − − −

A (2.13)

(4,12)

1 0 1 0 0 0 1 0 1 0 0 00 0 1 0 1 0 0 0 1 0 1 00 1 0 1 0 0 0 1 0 1 0 00 0 0 1 0 1 0 0 0 1 0 1

FF

− − − − = − − − − −

A (2.14)

(4,7)

1 0 0 0 0 0 00 1 0 0 0 0 00 0 0 1 0 0 00 0 0 0 1 0 0

F

=

C (2.15)

84

For delta connection,

(12,6)

1 0 0 0 0 00 0 0 1 0 00 0 1 0 0 00 0 0 0 0 10 1 0 0 0 00 0 0 0 1 01 0 0 0 0 0

0 0 0 1 0 00 0 1 0 0 00 0 0 0 0 10 1 0 0 0 00 0 0 0 1 0

IP TC

− − = ⋅ −

− − −

A (2.16)

(6,12)

1 0 0 0 0 0 1 0 0 0 0 00 0 0 0 1 0 0 0 0 0 1 00 0 1 0 0 0 0 0 1 0 0 00 1 0 0 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 0 0 0 10 0 0 1 0 0 0 0 0 1 0 0

FF

− − −

= − −

−

A (2.17)

(4,7)

1 0 0 0 0 0 00 1 0 0 0 0 00 0 1 0 0 0 00 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 0 1 0

F

=

C (2.18)

85

(12,8)

1 0 0 0 0 0 0 00 1 0 0 0 0 0 01 0 1 0 0 0 0 0

0 1 0 1 0 0 0 00 0 1 0 1 0 0 00 0 0 1 0 1 0 00 0 0 0 1 0 1 00 0 0 0 0 1 0 10 0 0 0 0 0 1 00 0 0 0 0 0 0 1

CS

− − − −

= −

− −

A (2.19)

[ ](10,1) 0 0 1 1 TCE = −A (2.20)

(8,10)

1 0 1 0 00 1 0 1 0

0 1 0 1 00 1 0 1 0

0 1 0 1 00 1 0 1 0

0 1 0 1 00 0 1 0 1

FS

− − − − = −

− −

−

A (2.21)

13

24

35

46(8,8)

57

68

79

810

0 00 0

0 00 0

0 00 0

0 00 0

C

CS

CS

CSFD

CS

CS

CS

CS

RR

RR

RR

RR

=

A (2.22)

Documents

Mixed mesh/nodal magnetic equivalent circuit modeling of a