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thesis entitled

A Method for Modeling the Electrical Variable Transmission

 

presented by

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A Method for Modeling the Electrical Variable Transmission
By

Stephen M. Bohan

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of
MASTER OF SCIENCE
Department of Electrical and Computer Engineering

2008

ABSTRACT

A Method for Modeling the Electrical Variable Transmission
By

Stephen M. Bohan

The increasing price of fuel and concern for the environment has lead to the
proliferation of hybrid electric vehicles. This has lead to the introduction of many
new electric machines to try to create the maximum efficiency for the hybrid vehicle.
In this thesis one such machine, the electric variable transmission, will be investigated.
A model will be developed for the electric variable transmission and that model will

be evaluated against a set of expected results to display its accuracy.

Copyright © by
Stephen M. Bohan
2008

To my Wife, Sarah

iv

ACKNOWLEDGMENTS

I would first like to show my appreciation to Dr. Elias Strangas, my thesis ad-
visor, for his guidance, patience, and support throughout this process. His technical
expertise and experience gave me the tools needed to complete this work and to guide
me in future endeavors. I also owe my gratitude to Dr. Gregory Wierzba and Dr.
Guoming Zhu for giving their time and energy to be a part of my committee.

This work would not have been completed without the help of the people at my
employer, BorgWarner, especially, my supervisors Mike Tuttle and Al Dove. I would
also like to express my appreciation to my coworkers for understanding my odd hours
and drowsiness during this process.

A special thank you is owed to all of my lab colleagues especially, Sinisa Jurkovic,
Carlos N ifio, and Wes Zanardelli.

Lastly and most importantly, I would like to express my deepest gratitude to my
wife, Sarah. Without her love, support and understanding, none of this work would

have been possible. It is to her that I dedicate this work.

 

TABLE OF CONTENTS

LIST OF TABLES ............................... ix
LIST OF FIGURES .............................. x
1 Introduction ................................. 1
2 Electric Variable Transmission ...................... 4
2.1 Introduction to the EVT ......................... 4
2.2 The Cascade System ........................... 4
2.3 The Dual Linked System ......................... 5
2.4 Concentric Implementation ........................ 9
2.5 The Electric Variable Transmission ................... 12
3 Analysis Methods .............................. 15
3. 1 Introduction ................................ 15
3.2 Resistances ................................ 16
3.3 Stator and Inner Rotor Self and Mutual Inductances ......... 17
3.4 Outer Rotor Self and Mutual Inductances ............... 20
3.5 Stator and Inner Rotor to Outer Rotor Inductances .......... 26
4 Simulation Setup and Results ...................... 28
4.1 FEA Setup ................................ 28
4.2 Simulation Geometry ........................... 28
4.3 Resistance Calculation Setup ....................... 30
4.4 Stator and Inner Rotor Setup ...................... 31
4.5 Outer Rotor Setup ............................ 33
5 Analysis Results ............................... 34
5.1 Resistance Results ............................ 34
5.2 Stator and Inner Rotor FEA Results .................. 35
5.3 Outer Rotor FEA Results ........................ 41
5.4 Outer Rotor to Stator and Inner Rotor FEA Results ......... 43
6 Model Development ............................ 50
6.1 Introduction ................................ 50
6.2 Ftequency Equations ........................... 50

vi

6.3 Voltage Equations ............................. 53

6.4 EVT Model ................................ 55
6.5 Torque Equations ............................. 55
7 Model Validation .............................. 58
7.1 Test 1 - Stator to Outer Rotor Linkage ................. 58
7.1.1 Setup ............................... 58
7.1.2 Results ............................... 59

7.2 Test 2 - Inner Rotor to Outer Rotor Linkage .............. 59
7.2.1 Setup ............................... 62
7.2.2 Results ............................... 62

7.3 Test 3 - Stator to Inner Rotor Linkage ................. 62
7.3.1 Setup ............................... 65
7.3.2 Results ............................... 65

7.4 Test 4 - Stator to Outer Rotor High Slip ................ 68
7.4.1 Setup ............................... 68
7.4.2 Results ............................... 68

7.5 Test 5 - Inner Rotor to Outer Rotor High Slip ............. 71
7.5.1 Setup ............................... 71
7.5.2 Results ............................... 71

7.6 Test 6 - Stator to Inner Rotor High Slip ................ 74
7.6.1 Setup ............................... 74
7.6.2 Results ............................... 74

7.7 Test 7 - Both Rotors in Motion ..................... 76
7.7.1 Setup ............................... 76
7.7.2 Results ............................... 76

7.8 Test 8 - Stator Torque Speed Curve ................... 79
7.8.1 Setup ............................... 79
7.8.2 Results ............................... 81

7.9 Test 9 - Inner Rotor Torque Speed Curve ................ 82
7.9.1 Setup ............................... 82
7.9.2 Results ............................... 82

8 Conclusions .................................. 84
Appendix A MATLAB Files ......................... 87
A.1 Resistancem ............................... 87
A2 SMI.SIR.m ................................ 89
A3 InductanceFromFile.m .......................... 92

vii

A.4 SMI-OR.m ................................. 94

A.5 Example Test M-Code - EVT_ODE_Ex1.m ............... 97
A6 Example Test M—Code - ODE_EVT1.m ................. 106
Appendix B Geometry ............................ 109
BIBLIOGRAPHY ............................... 113

viii

5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13

LIST OF TABLES

Winding and Rotor Bar Resistances ................... 34
Stator and Inner Rotor Voltages ..................... 35
Stator and Inner Rotor Inductances ................... 38
Stator Self and Mutual Inductances ................... 39
Inner Rotor Self and Mutual Inductances ................ 39
Stator to Inner Rotor Mutual Inductances ............... 40
Inner Rotor to Stator Mutual Inductances ............... 40
Magnetic Vector Potential of the Rotor Bars .............. 41
Rotor Region Magnetic Flux ....................... 42
Rotor Region Inductances ........................ 43
Outer Rotor Vector Potential ...................... 44
Outer Rotor Magnetic Flux ....................... 44
Mutual Inductance Stator and Inner Rotor to Outer Rotor ...... 47

ix

2.1
2.2
2.3
2.4
2.5

3.1
3.2

4.1
4.2
4.3

5.1
5.2
5.3
5.4
5.5
5.6

6.1
6.2

7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9

LIST OF FIGURES

Cascaded Dual Machine System ..................... 5
Dual Linked System ........................... 6
Power Distribution in a Dual Linked System .............. 8
Concentric Dual Machine Implementation ............... 9
The Electric Variable Transmission ................... 13
Outer Rotor regions 30 and Sb ...................... 21
Rotor region Sa in depth ......................... 24
Full EVT geometry ............................ 29
One quarter EVT geometry ....................... 30
Stator and Inner Rotor analysis circuit ................. 32
Sourced Stator - Voltage Graphs ..................... 36
Sourced Inner Rotor - Voltage Graphs ................. 37
Sourced Stator - Magnetic Vector Potential Graphs .......... 45
Sourced Inner Rotor - Magnetic Vector Potential Graphs ....... 46
Stator to Outer Rotor Inductances ................... 48
Inner Rotor to Outer Rotor Inductances ................ 49
EVT frequency definitions ........................ 52
Outer rotor equivalent circuit for outer rotor bars 1 and 2 ....... 54
Test 1 Machine Currents ......................... 60
Test 1 Machine Torques ......................... 61
Test 2 Machine Currents ......................... 63
Test 2 Machine Torques ......................... 64
Test 3 Machine Currents ......................... 66
Test 3 Machine Torques ......................... 67
Test 4 Machine Currents ......................... 69
Test 4 Machine Torques ......................... 70
Test 5 Machine Currents ......................... 72

7.10 Test 5 Machine Torques ......................... 73

7.11 Test 6 Machine Currents ......................... 75
7.12 Test 6 Machine Torques ......................... 77
7.13 Test 7 Machine Currents ......................... 78
7.14 Test 7 Machine Torques ......................... 80
7.15 Test 8 Stator to Outer Rotor Torque Speed curve ........... 81
7.16 Test 9 Inner Rotor to Outer Rotor Torque Speed curve ........ 83

xi

CHAPTER 1

Introduction

High fuel costs and concern for the environment have led many vehicle manufacturers
to introduce hybrid electric vehicles. In fact, Honda and Toyota have projected that
10% to 15% of the US. auto market will consist of hybrid electric vehicles by the year
2009[6]. Traditionally, hybrid electric vehicles have fallen into two categories: series
and parallel. Recently, two new classifications have been added. Series-parallel and
complex hybrids have been introduced to improve performance[2].

In a series hybrid, power is generated by an internal combustion engine which
is connected to a generator. The generator powers a DC bus which is used to feed
power to an electric motor. The electric motor then provides power to the wheels.
A large benefit of a series hybrid is that the electric motor can also be used for
regenerative braking to supply power back to the DC bus. Therefore, less power is
lost due to braking. One major drawback of a series hybrid is that all of the power
is transmitted through the electric components. Therefore, the electric components
must be sized for worst-case power draw. This results in large motors, generators,
and power electronics which can be very expensive and very heavy.

In a parallel hybrid, power can be delivered to the wheels either through an
electric motor or directly from the internal combustion engine. As in a series hybrid,

the motor can act as a generator during braking to reclaim power lost due to braking.

A parallel hybrid has an advantage over the series hybrid because it only requires
one electric machine, the electric motor. This results in fewer component costs and
weight which can hurt overall fuel economy. Another advantage is that all of the
power does not need to be transmitted through the electrical system. This results in
smaller power electronics and a smaller motor. Which will further add to the cost
and weight benefit over the series hybrid.

The disadvantage of a parallel hybrid in comparison to the series hybrid is that
the least efficient part of a hybrid vehicle system is usually the internal combustion
engine. In a series hybrid, all of the power from the internal combustion engine must
go through the electrical system. Therefore, the internal combustion engine can be
operated at peak efficiency for the majority of its duty cycle. This is not the case for
the parallel hybrid as the electrical system only provides a portion of the necessary
power.

It would be best to get the efficiency benefits of operating an internal combustion
engine at its peak efficiency as in the series hybrid but also benefit from the direct
mechanical power transmission of the parallel hybrid. This is where series-parallel
and complex hybrid systems come into play. To achieve this benefit, a machine must
transmit some of its power directly and some of it through power electronics. There
have been many proposals for a system which can do just that [1] [8].

This thesis will concentrate on one concept to meet this goal. Hoeijmakers and
Ferreira introduced a new electrical machine, the Electrical Variable Transmission,
(EVT)[4] which transmits power from the engine through both power electronics and
directly through a magnetic coupling to the drive shaft. The EVT is essentially
two concentrically arranged induction motors with a shared rotor. The shared rotor
is constructed thinner than what the combination of the two rotors would be and
therefore, introduces complexity into the machine. This complexity will be detailed

later in this paper.

In this thesis, a model will be developed for the electric variable transmission.
The concept of the EVT will be developed and evaluated. A method for finding
the machine parameters will be introduced and implemented using finite elements
analysis. The EVT model will then be developed. The model will be evaluated
against a set of expected results to display its accuracy. The model will be investigated
without operating with closed loop control. Control of the EVT is beyond the scope

of this thesis but, it is an opportunity for future work.

CHAPTER 2

Electric Variable Transmission

2.1 Introduction to the EVT

To help understand how the EVT works, a simple case will first be investigated and

complexity will be added to develop the concept.

2.2 The Cascade System

The system can first be thought of as two independent electrical machines with no
electric port. It will be comprised of a conventional motor and a specialized generator
linked only by a DC bus as seen in Figure 2.1. Electrical power from the first machine
is withdrawn from brushes on its rotor. The brushed rotor of this machine acts
similarly to how a stator would function in a traditional induction machine. The
exterior rotor acts as a rotor in a traditional induction machine.

Power enters this system as mechanical power applied to the input shaft of the
first machine

Pml = wmlel (21)

where wml is the rotational speed and Tml is the mechanical torque. For simplicity,

it is assumed that this system is lossless. After entering the system as mechanical

Pm1

    
 

 
   

 

Pm2

   

Tm1
wm1

Tm2
wm2

A /
_ Pe
——>

Figure 2.1. Cascaded Dual Machine System

 

 

 

 

 

 

 

 

 

 

 

 

 

 

power, the power is then converted to electrical power, Pe, by the first machine and
fed through a rectifier to create a DC voltage. This DC voltage is fed through an
inverter, which supplies electrical power to the second machine. This electrical power
is converted back to mechanical power, Pm2, by the second machine providing power

on the shaft of the second machine.
Pm2 = wm2Tm2 (2'2)

If this system is not lossless, as it is in reality, the numerous power conversions of

this method result in relatively low efficiency.

2.3 The Dual Linked System

The efficiency of the cascaded system could be greatly increased by directly trans-

mitting a portion of the input power to the shaft of the second machine. This could

 

 

 

 

Pe

 

 

 

 

 

 

 

 

 

 

Figure 2.2. Dual Linked System

be done by connecting the outer rotor of the first machine to the rotor of the second
machine, as seen in Figure 2.2.

If this new system is assumed to be lossless, the power transmitted through the
airgap of the first machine, P91, can be thought of as a function of two components.
The first component is converted to electrical power, Pa, and is a function of both shaft
speeds and the electromagnetic field torque, T91.The electromagnetic field torque of

machine one will be equal to the input torque.

Pa = (wm1 _ wm2)Tgl = (wm1 - wm2)Tm1 (2.3)

The electrical power is transmitted through the rectifier and inverter and then through

the air gap of the second machine, P92, as it was in the cascaded system.

The electrical power only accounts for part of the power transmitted to the second
machine. The rest of the power is transmitted directly through the air gap of the
first machine, Pd, through the mechanical linkage to the secondary shaft. This power
is due to the primary electromagnetic torque in the gap of the first machine and the

speed of the linked rotors.

Pd = wm2Tgl = wm2Tm1 (2'5)

The power transmitted through the system could be represented by summing the

electrical power and the power that is directly transmitted.

Pm1 = Fe + Pd = Pm2 (2-6)

As was the case with the cascade system, some power is transmitted through the
electrical components, P6, with relatively high losses. But now, a portion of the
power is also transmitted directly to the second machine, P91 = Pd, through the air
gap of the first machine. This power can be assumed to have relatively low losses.

Figure 2.3 depicts a situation in which the input torque, Tml, and speed, wm1,
are held constant. Since direct power transmission is generally more efficient then
electrical power transmission, efficiency will increase as wm2 approaches wm1. The
portion of the graph where wm2 is low is intentionally not represented because the
assumption of a lossless system is especially erroneous in this region.

Just as the power transmitted through the primary machine is comprised of two
components, the output torque of the machine also consists of two components. One

component, T91, is transmitted through the air gap of the first machine.

Tgl = Tml (2'7)

 

 

Pe

Figure 2.3. Power distribution with respect to speed under constant torque

The other component, T92, is transmitted through the air gap of the secondary

machine and is a result of the electric power transmitted through the system.

wm2 wm2

 

Equations (2.7) and (2.8) can be combined to show the relationship between the
input and output torque.

w
Tm2 = —’”le1 (2.9)

wm2
As it was in equation (2.9) Tm2 = Tm] when wm2 = wm1. At this point, all torque
is transmitted through T91 resulting in the most efficient torque transmission. When
the input speed is equal to the output speed, all of the power, and subsequently torque,
will be transmitted directly as Pd. This implies that all torque will be transmitted
directly through T91 and no torque will be transmitted though T 92. This should lead
to the highest efficiency as all of the power is transmitted through the path with the

fewest components.

 

//////////////////z‘;}”)})7/////////////// A

\\\\\\\\\\\\\ \\\\\\\\\\ \\

 

 

 

 

 

 

 

 

 

 

 

 

 

 

fl.

 

 

 

 

 

 

 

 

+Pe -

Figure 2.4. Concentric implementation of the dual linked system

2.4 Concentric Implementation

In order to save space in the implementation of the dual linked system, the two
machines could be arranged concentrically where the mechanically linked rotor is now
a shared rotor with two sets of windings as seen in Figure 2.4. Since the stator of the
first machine is not stationary it will be referred to as the inner rotor. Consequentially,
the shared rotor of the two machines will be referred to as the outer rotor.

So far, losses have been ignored except after the power flow has been developed.
In reality some key power losses should be considered while developing the power flow

through the machine. The power equations for this machine will be derived taking

into account key power losses.

The system is supplied by the shaft power of machine 1, Pm1 = wmlel. It is
then split into two separate parts. The first part is transmitted directly to the shared
rotor through the first air gap

Pdgl = wnggl (2.10)

where wgl is the rotational speed of the field in the inner air gap. The rotational speed
of the field in the air gap is supplied by the frequency of the inner rotor currents,

win, spinning at input speed, wm1, producing the following relationship.
“’91 = wirl + wm1 (2-11)

As it was in the dual linked system, the torque in the gap is equal to the input
torque, T 91 = Tml- The rotor losses of the first machine will now be considered. In
an induction machine, the rotor losses are a function of the slip in the machine. In
this case, the slip speed is the difference of the inner rotor frequency of machine 1,
wgl, and the rotor speed of machine 1. For the concentric implementation, the rotor
of machine 1 is mechanically linked to the rotor of machine 2. Since the mechanical
speed of machine 2 is the outer rotor speed, the outer rotor speed is equal to the
output speed, wm2

Worl = p(wgl _ wm2) (2°12)

where p is the number of pole pairs in machine 1. The outer rotor power losses due

to machine 1 can be estimated as
w l
Porl,losses = (I): Tm1 (2-13)

The power transmitted directly, Pd, is then found by subtracting the losses from

10

the power transmitted directly through the air gap
“1 1
Pd = (“’91 - —:—)Tm1 (2-14)

The remaining power flowing through the first machine is transmitted as electrical
power. As was the case with the direct power transfer, the power transmitted through
the inverter also suffers from machine losses. This time, the losses are first due to the
inner rotor of machine 1, Pir,losses- Since the inner rotor functions as a stator would
in a traditional induction machine, the winding losses can be treated the same as
they would be in a traditional stator. Stator winding losses are a function of machine
parameters and are independent of frequencies. As such, they will not be derived.
After removing inner rotor losses, the electrical power transmitted through the air

gap of machine and into the inner rotor will then be
Pir = P6 2 (”ml _ wgllel _ Pinlosses (2°15)

Pi, is supplied to the power electronics as P8 where it again suffers losses, Paws“.
The power is transmitted next through the stator of the second machine with losses

Ps,losses and into the air gap between the second machine and the outer rotor
P92 = Pe - Pe,losses - Ps,losses (2-16)
P92 can be related directly to the torque in the second air gap T92
P92 = wsTg2 (2.17)

where, ws is the frequency of the currents in the stator.

Before it is finally transmitted as output power, it suffers one more set of losses

11

from the outer rotor due to the second machine, Pal—2,103 se 3.

wslip,2

 

T92 (2.18)

Pm2,losses =

where

wslip,2 :—-_. p(wg2 _ wm2) (2'19)

In this case, the stator of the machine is stationary so wgg is found as

(.092 = ws (2.20)

where ws is defined as the stator frequency.
If this system was left as is, it would still show efficiency improvements over today’s
starter and alternator. But, the package is very large and heavy. To address these

problems, Hoeijmakers and Ferreira presented a new power conversion unit, the EVT.

2.5 The Electric Variable Transmission

If the restriction that the rotor slip frequencies, war, of both machines are kept equal is
imposed, the size of the yokes on the shared rotor can be greatly reduced. This is due
to the fact that in an induction machine, the torque developed is essentially a function
of the frequency of the rotor currents and the square of the flux density. If both
machines are operating with the same rotor current frequency, they can essentially
share the rotor currents.

The thinning of the yokes changes the electromagnetic behavior of the EVT. The
thin yokes cause the shared rotor of the machine to saturate much quicker then in the
concentric implementation. This causes an interesting interaction between the two
machines. As the shared rotor starts to saturate, a portion of the flux generated in

the inner rotor of the first machine will be passed directly to the stator of the second

12

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\§

§
\
\
\
\
x
x
\
\
§
.\
9.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.5. The Electric Variable Transmission

13

machine.

This interaction produces another torque component to consider when studying
the EVT. This torque, T r 3, adds another component to the air gap torque of machine
1.

T91 = Tml — Tm (2.21)

14

CHAPTER 3

Analysis Methods

3. 1 Introduction

In the previous section it was stated that the EVT was essentially two concentric
induction machines with a shared rotor. When the power flow through the machine
was developed, it was done so as a pair of traditional induction machines with some
added complexity due to the shared outer rotor. This method will be used to develop
the model for the EVT.

The first step is developing a set of equations that can be used to determine
the machine parameters. Once a set of equations has been established, a method
for finding the necessary values to solve those equations will be developed. In this
chapter the methods and equations for calculating electrical parameters for the EVT

using finite element analysis will be developed.

15

3.2 Resistances

The first machine parameter investigated will be the resistance. The resistance of a

wire can be found using the formula,
pl
R = — (3.1)

where p is the resistivity of the material, I is the length of the wire and A is the cross
sectional area of the wire. This equation can be applied directly to the rotor bars
since they can be treated as a single strand of wire.

Equation (3.1) can be modified slightly to take into account the number of wind-
ings found in each of the inner rotor and stator slots. If it is assumed that the slots
are completely filled with wire, then the cross sectional area of a single strand of wire

can be found by dividing the area of the slot by the number of windings.

 

_ Aslot
A _ N (3.2)

where Aslot is the cross sectional area of the slot and N is the number of windings
contained in the slot.

This is not the case. Stacking round wires on top of each other causes small
airgaps in the slot. A correction factor can be introduced to account for the small
airgaps. This factor will be referred to as the fill factor, 17. The total resistance in
the slot can then be found by multiplying the number of windings in the slot by the

resistance of one winding and dividing by the fill factor.

_ 2 pl
R _ N UAsiot (3'3)

16

3.3 Stator and Inner Rotor Self and Mutual In-

ductances

Since the EVT is essentially two concentrically arranged induction machines, anal-

ysis can be started using the voltage equations for a traditional induction machine.

Equations (3.4) through (3.9) are the voltage equations for a traditional induction

machine where the rotor can be represented by balanced three phase windings.

 

 

vas = rsias + d2?
vbs = Tsibs + 61—27%:
”as = Tsics + d_:t£s_
var = Trier + d221,.
vb, = nib, + 61—2-35
var = Trier + %r—

(3.7)

(3-8)

(3.9)

For the implementation to be used in this investigation, there will not be balanced

three phase outer rotor windings. Therefore, the self and mutual inductance of the

stator and inner rotor will first be discussed and the outer rotor will be ignored for

the moment. For the remainder of this section, it will be assumed that there are no

outer rotor bars. Equations (3.4) through (3.9) can be modified to represent the coils

in an EVT with no outer rotor bars.

. (D.
”as = Tszas + T?

. dA
”()3 = TSst + Eli?

17

(3.10)

(3.11)

 

dAcs

 

= ’ — 3.12

vcs rszcs + d t ( )
. d) -

”air = 7"irzair “l" '73:: (3°13)
. dA -

vbir‘ = Tirzbir + dz”. (3.14)
. dA '

var = Tirzcz'r + din. (3'15)

The flux linkage for a traditional machine would follow the form of Ads found in

equation (3.16).

Aas = Lasasias + Lasbsibs + Lascsics + Lasariar + Lasbribr + Lascricr (3.16)

Similar to the voltage equations, the flux linkage equations could be modified to
represent an EVT with no outer rotor bars. The flux linkages for the EVT without
outer rotor bars can be described in equations with the same form of Aas, found in

the following equation.

Aas = Lasasias + Lasbsibs + Lascsics + Lasairiair + Lasbz'rib'ir + Lasciricir (3'17)

If current is applied only to one coil, then the flux linkage equation (3.17) becomes
much simpler for each flux linkage. Since all of the other currents are zero, only one
term is left on the right side of equation (3.17). Equation (3.18) describes A5,, when

current is applied only to the phase B coil of the inner rotor.

)‘bir : Lbz'rbiribir (3'18)

If saturation is neglected, then the inductance can be assumed to be constant.

18

With this in mind, equation (3.18) can be inserted into equation (3.14) for Abrir-
. d .
var = Tierir + (’ftLbz'rbz’rzbt'r (3-19)

Equation (3.19) can now be solved for Lbirbz'r using phasor notation. This same
method could be used to find the self inductance of any inner rotor or stator coil.

Finding the mutual inductances in the inner rotor or stator is very similar to
finding the self inductance of one of the inner rotor or stator coils. The voltage
equations (3.10 - 3.15) along with the flux linkage equations of the form of equation
(3.17) would remain the same when looking for the mutual inductance in the inner
rotor or the stator. In fact, the mutual inductance between coils within one member
or between coils in different members can be found using the same method.

If current is applied to one coil in either the inner rotor or stator, equations of the
form of equation (3.17) can be found for all of the other coils in that member and in
the other members. The resulting equations would be simpler then equation (3.17)
since all but one of the currents would be zero. If a current was applied to the phase
b coil of the inner rotor, the flux linkage equations would be as follows and would

remain in this same form if current were applied to a different coil.

)‘as = Lasbz’ribir (3'20)
Abs = Lbsbz'ribir (321)
)‘CS = Lcsb-iribir (3°22)
Aaz'r = Lairbiribir (3'23)
Acir = Lcirbiribz'r (324)

As was done in order to find the self inductance, the flux linkage values found after

19

 

applying the current to a single coil could be substituted into the voltage equations
(3.10 - 3.15). The resulting equations could then be solved for the mutual induc-
tances using phasor notation. Continuing with the example from earlier, the voltage
equations necessary to find the mutual inductance between the phase b coil of the

inner rotor and all other coils would be as in equations (3.25) through (3.29).

d .

”as = Et'Lasbirzbt'r (3'25)
d .

vbs : EELbsb'irzbz'r (326)
d .

”CS = EELcsbirzbir (3'27)
d .

”air = d—tLairbierir (3°28)
d .

var = ELcirbirzbir (329)

This same method could be used to find the mutual inductance of any inner rotor or

stator coil with all other coils in the inner rotor and stator.

3.4 Outer Rotor Self and Mutual Inductances

The outer rotor bars need to be handled differently then the inner rotor and stator
coils. Since the outer rotor will consist of a set of cast aluminum rotor bars and not
three sinusoidal distributed coils, one rotor bar cannot be treated as one coil. This is
not the only issue concerning finding the inductances associated with the outer rotor.
The other problem comes in the fact that the rotor bars are not isolated from each
other. The rotor bar ends are all connected in the actual cast rotor.

To avoid these issues, the outer rotor must be investigated in a different manner.
The outer rotor can be looked at as a set of current carrying loops bounding closed

regions[7]. Figure 3.1 illustrates this concept.

20

 

Figure 3.1. The outer rotor with regions Sa and Sb

21

Closed loop Ca consists of a path down the center of rotor bars one and two and
the rotor rings at both ends of the rotor. Closed loop Ca bounds the surface Sa.
Similarly, closed loop Cb encloses surface Sb and is comprised of rotor bars 2 and 3
and the rotor rings. For the sake of this investigation, it is assumed that current is
uniformly distributed throughout the rotor bars and rotor rings. If a current I flows
through Ca, a magnetic field Ba will be created.

The magnetic flux (I) is related to the magnetic flux density B through a given

area bounded by a closed path C through the following equation
¢=fBas am)
3

The vector magnetic potential, A, can be used to find the inductances. The mag-
netic vector potential is defined such that the magnetic flux density, B, is expressed
as the curl of A.

B = V x A (3.31)

Equations (3.30) and (3.31) can be combined to find the relationship between

magnetic flux (I) and magnetic vector potential A.
¢=JVXAas an)
8

By applying Stoke’s theorem to equation (3.32) the equation can be further re-
duced to

c=¢AaL mm)
0

Equation (3.33) states that the line integral of A around any closed path is equal
to the total flux passing through the area enclosed by the path. If equation (3.33)
is applied to the closed path Ca the total flux passing through Sa can be found.

Similarly, equation (3.33) can be applied to any set of rotor bars creating a closed

22

path and used to calculate the flux passing through that path.

A coordinate system can be established to investigate the surface 50, as was de—
scribed in Figure 3.1. The y-axis, ya, will run down the height of the rotor bar. The
x—axis, ma, runs along the straight path connecting the centers of rotor bars one and
two. Figure 3.2, displays surface 50 in reference to this coordinate system.

If a current is applied such that the current in rotor bar one flows in the positive
ya direction and the current in rotor bar two flows in the negative ya direction, the
magnetic vector potential A will only have a ya component. In light of this decision,
A can be defined to have a line integral from 0 to h. This will simplify equation

(3.33) to a single integral in the :ra direction.
w
0

Ayl can be established as the magnetic vector potential from the point (0,0) to
(0,h) as is described in Figure 3.2. Similarly, Ayg can be established as the magnetic
vector potential from the point (111,0) to (w,h). Equation (3.34) can now be solved to

give the flux passing through the surface Sa.
<I>a = (Ayl — Ay2)w (3.35)

Equation (3.35) can be applied to all of the surfaces S as long it is adjusted for
the rotor bars involved.
The Biot-Savart Law states that the flux (I) in a magnetic material of constant

permeability will have linear relationship to current through the value inductance L.
(I) = LI (3.36)

This holds true for the case where there exists only one winding in the closed loop C.

23

(OJ!)
(

 

T c

ya (0,0)

i_,,a_.

 

 

 

 

 

 

/ A W
O 0 (w, h)
Ay1 Ay2
V
A S a Ca
0 0
(M40)

Figure 3.2. Region Sa enclosed by closed loop Ca

24

In the case of multiple windings equation (3.36) can be related to the flux linkage A
A = Mb (3.37)
where N is the number of turns in the closed loop C.
A = LI (3.38)

In this case there will only exist one winding so, equation (3.36) can be used to
find the inductance.

(I)
’3‘?

(3.39)

If it is assumed that the permeability of rotor bars does not change with I, the self
inductances of the closed loops C can be found by applying equation (3.39) directly.
For example, to find the self inductance of Ca equation (3.39) would be represented

as
<I>aa

L =
aa Ia

 

(3.40)

where Ia is the current in bars one and two.

The equation for finding the mutual inductance for two surfaces is very similar
to that of the self inductance. For example, the mutual inductance between surfaces
50 and Sb could be investigated. If a current Ia flows through the closed path Ca, a
magnetic field Ba will be created. Some of the magnetic flux due to Ba will link with
the surface Sb. This flux can be found by using an equation of the form of equation

(3.41) when current is applied to rotor bars one and two.

(Dab = (Ay2 — A((3)10 (3-41)

The mutual inductance can then be found as it was in the case of the self induc-

25

tance by dividing by the current applied to rotor bars one and two, Ia.

<r>
Lab = 7:9 (3.42)

3.5 Stator and Inner Rotor to Outer Rotor Induc-
tances

In Section 3.3 the outer rotor was excluded because it did not consist of three sinu-
soidally windings. Therefore, a different method than the one developed in Section
3.3 must be used to find the stator and inner rotor to outer rotor mutual inductances.
The method developed in the last section could be used instead. By applying current
to the stator and inner rotor, the flux due to those currents in the rotor bars could
be calculated by measuring the magnetic vector potential in specific locations in the
outer rotor.

Current can be applied to the stator one phase at a time and then to the inner rotor
one phase at a time. Then equations of the form of equation (3.35) can be applied
to find the flux passing through the outer rotor surfaces. Once the flux through the
surfaces is known, equations (3.37) and (3.38) can be combined to find the inductance

in the rotor loops caused by the stator and inner rotor windings.

L = _ (3.43)

For example, if a current were applied to phase A of the stator an equation of the
form of equation (3.41) could be used to find the flux in the region enclosed by Cb-

Equation (3.43) could then be modified as follows to find the mutual inductance

26

 

between phase a of the stator and the outer rotor region Sb

N (I)
Lasbor = 1::b0r ' (3.44)

Equation (3.44) could be modified to find any mutual inductance between the stator

or inner rotor and the outer rotor.

27

CHAPTER 4

Simulation Setup and Results

4.1 FEA Setup

In the last chapter, a set of equations was developed to find the machine parameters
necessary for the EVT model. In this chapter, a set of simulations to find the param-
eters necessary to solve those equations will be developed. Finite element analysis
will be used to simulate a given geometry and then the results of that simulation will

be analyzed to find the necessary parameters.

4.2 Simulation Geometry

A smaller overall geometry will be used then that presented in the Hoeijmakers and
Ferreira paper[4]. This will be done to help facilitate any future work using an actual
prototype part as a smaller machine should be less costly to have manufactured and
would be able to be operated with lower supply current. Since the actual dimensions
of the EVT are not known, a new geometry will be created. The overall machine
diameter will be set to 198.5 mm with an outer rotor thickness of 19 mm and 0.25
mm air gaps. The depth of the machine will be set at 235 mm. Appendix A.6 contains

detailed drawings of the machine geometry.

28

 

Figure 4.1. Full EVT geometry

The 4 pole induction machine will consist of 28 outer rotor bars, 36 inner rotor
slots and 48 stator slots. One quarter of the machine will be modeled to help reduce
simulation time. Figure 4.2 is a representation of the EVT geometry to be analyzed.

Since the purpose of this thesis is to create a linear model, the saturation effect
described in Section 2.5 will not be modeled. Instead, the effect will be simulated by
allowing a small amount of flux to link the stator to the inner rotor. The amount
of flux linking the stator and inner rotor is determined largely by the outer rotor
geometry. Originally, a geometry was chosen with 44 outer rotor bars instead of 28.
This geometry resulted in too much flux linking the stator to the inner rotor. The
outer rotor geometry was modified until a small amount flux linked the stator and
outer rotor.

The material selected for the stator and inner and outer rotors was iron. When

solving for the inductances, the material for the rotor bars and coils was selected as

29

 

Figure 4.2. The EVT geometry created to perform FEA analyis

vacuum.

The program chosen for the FEA analysis, Flux2D, only allows for one airgap
to be represented as a rotating airgap. Therefore, the rotating airgap will be set
according to Where current or voltage is applied. If power is applied to an inner rotor
coil, then the inner airgap will be set as rotating. If a stator coil is powered, then
the outer airgap will rotate. If power is applied to an outer rotor bar, then the outer
airgap is set as the rotating airgap. The non rotating airgap will be set as vacuum in

all these cases.

4.3 Resistance Calculation Setup

All of the resistances in the EVT can be found using equation (3.3). In order to
solve for the stator and inner rotor winding resistances and rotor bar resistances, the
physical characteristics outlined in equation (3.3) must be found. The overall physical
dimensions are the same as those defined in Section 4.2 and Appendix A.6.

Since the rotor bars are a single conductor, the number of turns is NOT = 1 and the

fill factor is 77 = 1. The resistivity is a material dependent value. Since the outer rotor

30

will be made out of aluminum, the resistivity will be set as par = 2.78 x 10—80m.
The length of the rotor bars is simply the depth of the machine, I = 235mm. The
area of the rotor bar can be found using geometry but, for simplicity, the FEA tool
will be used to find the area of the outer rotor bar.

The number of turns for the inner rotor windings will be established as Ni, =
44. Since the inner rotor slots will be filled with copper windings, the fill factor
will be set to 17 = 0.8 and the resistivity of the inner rotor windings will be set as
p,,. = 1.72 x 10-8Qm. The length of the inner rotor windings is the same as that
established for the outer rotor bars, I = 235mm. To determine the area of the inner
rotor slots we will again rely on the FEA tool as we did for the outer rotor bars.

The same number of windings will be used for the stator as were used for the inner
rotor, N3 = 44. The fill factor for the stator will be set to n = 0.8 as it was for the
inner rotor. The resistivity of the stator windings will be set as p3 = 1.72 x 10—89m
since the stator slots will be filled with copper windings. Again, the length of the
stator windings is the same as that established for the outer rotor bars, I 2 235mm.

To determine the area of the stator slots, the FEA tool will again be used.

4.4 Stator and Inner Rotor Setup

To find the currents and voltages needed to calculate the inductance due to the stator
and inner rotor windings a transient magnetic analysis will be run. A transient mag-
netic analysis was chosen because a steady state analysis would not have accounted
for the time varying signal needed as described in equations of the form of equation
(3.19). The geometry and machine parameters described in Section 4.2 will be used
in all of the analyses.

For the analyses, the coils in the circuit will be wired in a wye topology. A

resistance of 0.5575 0 will be added in series with each of the sets of coils to simulate

31

 

 

 

 

 

 

 

BTW—l—D—m
G rm E. [Er ar 59;—

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

w ' 4m I: ““—
WI *— rm_
«2' rm :- gal-w

 

 

l1

 

Figure 4.3. Circuit used to find self and mutual inductance of phase A of the stator.

the end winding resistance. Similarly, a 2.1 mH inductor will also be wired up in series
with the coils to simulate end inductance. One coil will be fed with 0.1 A current at
60 Hz. A low current was chosen so that the iron in the stator and both rotors would
not saturate. All of the other coil sets will have a large resistance, 40 k9, wired in
series with the coils to simulate an open circuit. The coil resistances will be set as
described in the previous section. A large parallel resistor, 20 k9, will be added to
each coil set to measure the voltage in each phase. Figure 4.3 is an example of the
type of circuit to be used. In this case, the current supply is connected to phase A
of the stator. For the simulations where current will be applied to other coils, the
current supply would be moved to the phase of interest and all other coils will have

a large resistance, 40 k9, set in series with them.

32

The rotor bars will be wired up as a squirrel cage. The end resistance of the
squirrel cage will be set at 40 k0 to isolate the rotor bars from each other. The end
inductance of the squirrel cage will be set at 4 nH.

In total, 6 simulations will be run; one each with one set of coils being fed in
either the inner rotor or the stator. The angular velocity in the rotating airgap will
be set to 0 rpm. The current source will be set to create a sinusoidal current with a

frequency of 60 Hz and a phase angle of 0°.

4.5 Outer Rotor Setup

In Section 3.4, equations (3.40) and (3.42) were derived for calculating the self and
mutual inductance of the outer rotor. In both equations there are no time varying
components. Since there are no time varying components, a magneto-static analysis
was performed on the EVT to find the values necessary to calculate the inductances
associated with the outer rotor.

The geometry described in Section 4.2 was used to simulate the EVT. A magneto-
static analysis does not require a circuit as was used in Section 4.4. Instead, current
is applied directly to regions where the current would flow. For the first analysis of
the EVT, the simulation will be run with current, 11 = 0.1A, applied to rotor bar 1
and the opposite current, 12 = —0.1A, to rotor bar 2. The current was kept low as it
was in the stator and inner rotor inductance simulations in order to avoid saturation
of the iron components. For the next simulation, positive current will be applied to
rotor bar 2 and negative current to rotor bar 3. Simulations will be repeated in this
manner until current is applied to rotor bar 7 and the opposite current to rotor bar
1. This will result in 7 total simulations in order to find all of the necessary magnetic

vector potential values.

33

CHAPTER 5

Analysis Results

5. 1 Resistance Results

In Section 4.3, most of the values necessary to calculate the resistances of all of the
windings and rotor bars in the EVT were established. The last value needed to
calculate the resistances was the area of each of the regions. Table 5.1 shows the
results for the area measurements from the FEA. It also shows the results of applying
these values and the values established in Section 4.3 to equation (3.3) to find the

resistances of the windings and rotor bars.

 

Region 1 Area (mmz) Fill Factor Resistance ((2)
Stator Winding 72 0.8 0.1359
Inner Rotor Winding 87.715 0.8 0.1115
Outer Rotor Bar 68.95 1 9.783x10"5

Table 5.1. Winding and Rotor Bar Resistances

34

 

 

Current Applied To
SA SB SC IRA IRB IRC
SA 20.83 8.505 8.544 0.732 0.360 0.363
SB 8.505 20.83 8.544 0.360 0.732 0.363
SC 8.544 8.544 20.82 0.360 0.360 0.738
IRA 0.834 0.417 0.404 9.667 3.964 3.931
IRB 0.417 0.834 0.404 3.964 9.667 3.931
IRC 0.410 0.410 0.836 3.931 3.931 9.663

 

 

 

 

 

 

 

 

 

Table 5.2. Stator and inner rotor voltages (V) when current was applied to a set of
windings.

5.2 Stator and Inner Rotor FEA Results

After the six simulations described in Section 4.4 were complete, the voltage across all
of the coils in the stator and inner rotor were measured. Figures 5.1 and 5.2 show the
voltages measured in all of the windings when current was applied to one phase of the
stator or inner rotor respectively. In order to perform the calculations described in
Section 3.3, the magnitude of all of the voltages must be found. This can be done by
performing an FFT on each voltage curve and then selecting the first harmonic value.
Table 5.2 contains the results of those measurements. These voltages will be used to
find the self and mutual inductances of the stator and inner rotor windings. Since a
current was applied directly to the windings, the current in all of the calculations will
be I = 01.4.

In Section 3.3, it was stated that if a voltage is applied to one set of windings in
the inner rotor or stator, equations of the form of equation (3.19) could be used to
calculate the self inductance of that set of windings. Equations of the form of equation
(3.19) can be represented using complex notation to illustrate the relationship between

the voltage and inductance.

an'r = Ibir(Rir + Xbir) (5-1)

35

Current Applied to Stator Phase A

0.15
0.10
0.05
0.00
-0.05
—0.10
—0.15

Voltage (V)
Current (A)

 

0.000 0.005 0.010 0.015
Time (3)

Current Applied to Stator Phase B

 

E E
E 8
0.000 0.005 0.010 0.015
Time (s)

Current Applied to Stator Phase C

 

 

30
9 20 2
I 10 I:
3 o s
s -10 '5
> -20 O
-30
0.000 0.005 0.010 0.015
Time (s)
—— Voltage_SA - - - Voltage_SB ------ Voltage_SC
— - - - Voltage_lRA - - - — Voltage_lRB — Voltage_lRC
—Applied Current

 

 

 

Figure 5.1. Stator and inner rotor voltages (V) measured when current was applied
to one stator phase.

36

Current Applied to Inner Rotor Phase A

 

15 0.15
’>" 10 0.10 2
I; 5 0.05 I:
g 0 0.00 E,
g -5 -0.05 5
> -10 -0.10 O

-15 -0.15

0.000 0.005 0.010 0.015

Time (5)

Current Applied to Inner Rotor Phase B

 

3
CD
0')
.9
T)
>
0.000 0.005 0.010 0.015
Time (5)

Current Applied to Inner Rotor Phase C

 

 

15 0.15
" 10 0.10 A
> <
I; 5 0.05 ::
8 o 0.00 5
g -5 -0.05 g
> -10 -0.10 O
-15 -0.15
0.000 0.005 0.010 0.015
Time (s)

— Voltage_SA — — — Voltage_SB ------ Voltage_SC

— - — - Voltage_lRA — - - — Voltage_lRB —Vo|tage_|RC

—AppIied Current

 

 

 

Figure 5.2. Stator and inner rotor voltages (V) measured when current was applied
to one inner rotor phase.

37

 

Current Applied To
SA SB SC IRA IRB IRC
SA 553 226 227 19.4 9.54 9.63
SB 226 553 227 9.54 19.4 9.63
SC 227 227 552 9.53 9.53 19.6
IRA 22.1 11.1 10.7 256 105 104
IRB 11.1 22.1 10.7 105 256 104
IRC 10.8 10.8 22.2 104 104 256

 

 

 

 

 

 

 

 

 

 

Table 5.3. Stator and inner rotor inductances (mH).

Since all of the voltage measurements made on the phases where current was
applied in Figures 5.1 and 5.2 are 90°out of phase with respect to the current, it can
be assumed that the resistance in equation (5.1) has a negligible effect. Therefore,
the reactance can be found by dividing the magnitude of the measured voltage by the

magnitude of the applied current.

sz'r = _ (5'2)

With the reactance calculated, the inductance can be found by dividing the reac-

tance by the frequency of the voltage.

___ Xbir
LU

Lbir (53)

Equations (5.2) and (5.3) could easily be modified for the case where current is
applied to any single stator or inner rotor phase.

Since equations of the form of equations (3.25 - 3.27) do not include resistance,
equations of the form of equations (5.2) and (5.3) can be modified to find the mutual
inductances in the stator and inner rotor. Following the example where current is
applied to phase B of the inner rotor, the mutual inductance between phase B of the

inner rotor and phase A in the stator could be found using equations (5.4) and (5.5).

38

 

ABC

 

 

A
B
C

553
226
227

 

 

226
553
227

 

227
227
552

 

Table 5.4. Stator self and mutual inductances (mH).

 

ABC

 

A
B
C

 

256
105
104

 

 

105
256
104

 

104
104
256

 

 

 

Table 5.5. Inner Rotor self and mutual inductances (mH).

Equations of this form could be used to find the mutual inductance between any set

of inner rotor or stator coils.

V
Xbiras = 1:78 (5'4)
1'
X .
th'ras = —b(::£§ (5-5)

Table 5.3 establishes the self and mutual inductances between stator and inner
rotor at a fixed position. This is appropriate for stator to stator mutual inductance
and the inner rotor to inner rotor mutual inductance as their relative position cannot
change. Tables 5.4 and 5.5 detail these values.

For the inductances between the two members, the inductance values need to be
found without respect to position. This can be done by dividing the values found in
Table 5.3 by the cosine of the angle between the phase where the voltage was applied

and where the current was measured. Table 5.6 and 5.7 detail the results of this

39

 

 

 

 

 

 

SA SB SC
IRA 19.4 -19.1 -19.3
IRB -19.1 19.4 -19.3
IRC -19.1 -19.1 19.6

 

Table 5.6. Stator to inner rotor mutual inductances (mH).

 

 

 

 

 

 

IRA IRB IRC
SA 22.1 -22.1 -21.5
SB -22.1 22.1 -21.4
SC -21.7 -21.7 22.2

 

 

 

Table 5.7. Inner rotor to stator mutual inductances (mH).

modification.

It will be useful to have a single value for the self and mutual inductances instead
of a table of values. The mutual inductance of the stator or inner rotor can be found
by first averaging the off-diagonal values of the stator and inner rotor self and mutual
inductance tables, Tables 5.4 and 5.5. The average values can then be multiplied
by 2, resulting in the mutual inductances. This differs from a traditional induction
motor because when the reactance was found, only the magnitude was found. If the
reactance had been found including the angle of the voltage, the final value would
need to be multiplied by -2.

L, = 453mH (5.6)

L,,. = 209mH (5.7)

The leakage inductance can be found by averaging the diagonal values of the
stator and inner rotor self and mutual inductances tables, Tables 5.4 and 5.5. And

then subtracting the mutual inductances found above, equations (5.6) and (5.7), from

40

 

 

 

 

 

 

 

 

 

 

Rotor Bar with positive current

1 2 3 4 5 6 7

1 18.2 1.79 0.446 0.344 0.453 1.79 18.3

2 -18.2 18.3 1.79 0.446 0.352 0.45 1.78

‘83 -1.78 -18.3 18.3 1.79 0.453 0.348 0.441
”24 -O.448 -179 -18.3 18.3 1.79 0.45 0.34
g5 -0.348 -0453 -179 -18.3 18.3 1.79 0.441
9:6 -0449 -0352 -0.446 -179 -18.3 18.3 1.78
7 -1.78 -0453 -0344 -0.446 -1.8 -18.3 18.3

 

 

Table 5.8. Magnetic vector potential (qu/ m) of rotor bars when current is applied
to two bars.

the resultant value.

L), = 100mH (5.8)

L1,, = 47.2mH (5.9)

Finally, the mutual inductance between the stator and inner rotor can be found
averaging the magnitudes of all of the mutual inductances found in Tables 5.6 and
5.7.

Lsz'r =2 Lirs = 20.6mH (5.10)

5.3 Outer Rotor FEA Results

After a total of 7 simulations were run, the magnetic potential was measured in the
center of each of the rotor bars. This was done to create regions Sa through 59 as was
described in Section 3.4. Table 5.8 contains the magnetic vector potential measured
in each of the rotor bars. The table is arranged such that the columns correspond
to the rotor bar where positive current was applied and the rows correspond to the

rotor bar where the magnetic vector potential was measured.

41

 

 

 

 

 

 

 

 

 

 

 

Rotor Region
1 2 3 4 5 6 7
A 0.5624 -0.2551 -0.0208 ~0.0016 0.0016 0.0207 0.2552
c, B -0.2537 0.5655 -0.2551 -0.0208 -0.0016 0.0016 0.0207
59,, C -0.0206 -0.2551 0.5655 -0.2551 -0.0207 -0.0016 0.0016
5?, D -0.0015 -0.0207 -0.2551 0.5655 -0.2551 -0.0207 -0.0016
‘5 E 0.0016 -0.0016 -0.0208 -0.2551 0.5655 -0.2551 -0.0207
g F 0.0206 0.0016 -0.0016 -0.0208 -0.2549 0.5655 -0.2552
G 0.2536 0.0207 0.0016 ~0.0016 -0.0208 -0.2551 0.5655

 

 

Table 5.9. Magnetic flux (pr) of regions enclosed by rotor bars when current is
applied to two bars.

The values in Table 5.8 can be used to calculate the flux passing through each
of the rotor regions 50 through 89. Equations of the form of equation (3.35) can
then be used to calculate the flux passing through the regions. Table 5.9 contains the
resulting magnetic flux from these calculations.

Finally, the self and mutual inductances of the rotor regions can be found by
dividing the magnetic flux in the region by the current applied to the rotor bars,
I = 01.4 as was described in equations (3.40) and (3.42). Table 5.10 contains the
inductances found by applying this method. The self inductances are listed along the
diagonal and mutual inductances are found throughout the rest of the table.

It would be useful to have a single value for the mutual inductance in the outer
rotor as well as the rotor leakage inductance. The mutual inductance can be found
by first finding the inductances in Table 5.10 regardless of position. This can be done
by dividing all of the values in the table by the relative angle to where the current
was applied. The mutual inductance of the outer rotor loops can then be found by

taking the average of the off diagonal values.

L0,. = 1.6791117 (5.11)

42

 

 

Rotor Region
1 2 3 4 5 6 7
A 5.6238 -2.5508 -0.2076 -0.0158 0.0156 0.2070 2.5523
g B -2.5369 5.6547 -2.5508 -0.2076 -0.0156 0.0158 0.2069
-b90C —0.2058 -2.5508 5.6547 -2.5508 -0.2066 -0.0158 0.0156
é D -0.0155 -0.2066 -2.5508 5.6547 -2.5508 -0.2070 -0.0156
‘5 E 00156 -0.0156 -0.2076 -2.5508 5.6547 -2.5508 -0.2069
g F 0.2056 0.0156 ~0.0158 -0.2076 -2.5493 5.6547 -2.5523
G 2.5369 0.2066 0.0158 -0.0158 -0.2081 -2.5508 5.6547

 

 

 

 

 

 

 

 

 

 

Table 5.10. Self and mutual inductance (,uH) of regions enclosed by two rotor bars.

The leakage inductance of the outer rotor loops can be found by taking the average
of the diagonal values of the inductance area found in the previous step and then

subtracting off the mutual inductance.

Llo, = 3.97211H (5.12)

5.4 Outer Rotor to Stator and Inner Rotor FEA

Results

To find the mutual inductances between the outer rotor and the stator and inner
rotor, the simulations described in Section 4.4 and used in Section 5.2 can be used.
The magnetic vector potential in the center of each rotor bar was measured for each of
the six simulations. Figures 5.3 and 5.4 shows the magnetic vector potential measured
in the center of each of the rotor bars when current was applied to a single phase of
the stator or inner rotor. In order to perform the calculations described in Section
3.5, the magnitude of the magnetic vector potentials must be found. To find these
values, each measurement point can be divided by the sine of the frequency of the

waveform multiplied by the time at each point. This should result in the instantaneous

43

 

 

Winding

SA SB so IRA IRB IRC

A 179.9 -1229 33.1 84.4 -58.6 15.5

a B 189.7 -557 100.2 88.5 -26.9 47.5
E00 143.9 11.8 165.3 67.5 5.6 77.8
5:; D 78.2 78.2 192.6 37.1 37.1 90.4
3 E 11.8 143.9 165.3 5.6 67.5 77.8
g F -55.8 189.7 100.2 -26.9 88.5 47.5
G -1229 179.9 33.0 -58.6 84.4 15.5

 

 

 

 

 

 

 

 

 

Table 5.11. Rotor bar vector potential (qu/ m) when current is applied to inner
rotor or stator windings.

 

 

 

 

 

 

Winding

SA SB 80 IRA IRB IRC

A —0.1525 .1.0375 -1.0380 -0.0639 -O.4889 04935
g B 0.7086 4.0434 -1.0056 0.3241 .0.5027 -0.4679
8,0 1.0155 10255 -04211 0.4703 -0.4863 -01952
5:; D 1.0254 —1.0154 0.4211 0.4863 -0.4703 0.1952
s E 1.0435 -0.7086 1.0055 0.5026 -0.3241 0.4679
g P 1.0374 0.1525 1.0381 0.4889 0.0638 0.4935
G 0.8801 0.8801 1.0212 0.3983 0.3982 0.4801

 

 

 

 

 

Table 5.12. Outer Rotor Magnetic Flux (aWb) when current is applied to the stator
and inner rotor.

magnitude of the signal. The resulting values can be averaged to find the average
magnitude of the signal. Table 5.11 contains the results of those measurements.

The values in Table 5.11 can be used to calculate the flux passing through each
of the rotor regions Sa through 39. This can be done by using the same method as
was used to find the outer rotor bar self and mutual inductances. Equations of the
form of equation (3.35) can be used to calculate the flux passing through the rotor
bar regions. Table 5.12 contains the flux resulting from that calculation.

The mutual inductance between the stator and inner rotor to the outer rotor

44

Current Applied to Stator Phase A

 

 

 

 

 

_ 3.0E-4 0.15
fig 2.0E-4 0.10 2
% 1.0E-4 0.05 r
a. 0.0E+0 0.00 8
§ -1.0E-4 -0.05 E
§-2.0E-4 -0.10 0
-3.0E-4 -0.15
0.000 0.005 0.010 0.015
Time (3)
Current Appplied to Stator Phase B
_ 3.0E-4 0.15
g 2.0E-4 0.10 2
% 1.0E-4 0.05 ::
CL 0.0E+0 0.00 5
g 4.054 -0.05 E
g —2.0E-4 -0.10 0
-3.0E-4 -0.15
0.000 0.005 0.010 0.015
Time (3)
Current Applied to Stator Phase C
3.0E-4
E 2.0E-4 ,4
E» 1.0E-4 §
8 0.015+0 5
g -1.0E-4 5
g 2054
-3.0E-4
0.000 0.005 0.010 0.015
Time (s)
Rotor Bar 1 - — — Rotor Bar 2 ------ Rotor Bar 3
— - — - Rotor Bar 4 - - - - Rotor Bar 5 — — - Rotor Bar 6
------ Rotor Bar 7 —Applied Current

 

 

Figure 5.3. Outer rotor magnetic vector potentials (Wb/m) measured when current
was applied to one stator phase.

45

Current Applied to Inner Rotor Phase A

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

     
 
  

 

 

 

 

 

 

 

 

1.0E-4 0.15
:0
E 5.0E-5 g:
8 0.05+0 5
.‘9 E
g 5.055 0
>
-1.0E-4
0.000 0.005 0.010 0.015
Time (3)
Current Applied to Stator Phase B
1.0E-4
E
E 5.0E-5 g
8 0.05+0 g,
.‘9 s
g 5.055 0
>
-1.0E-4
0.000 0.005 0.010 0.015
Time (3)
Current Applied to Inner Rotor Phase C
1.054 f“ 0.15
.713 -""’""'~‘;_.‘ -~ 0.10 A
g 5.0E'5 \ d_ 0.05 $
8 0 05+0 - 0.00 g
.9 ‘ ~ -0.05 5
° -5.0E-5 \-. I,“ 0
é’ V -- -0.10
4.054 . . "" . -0.15
0.000 0.005 0.010 0.015
Time (s)
Rotor Bar 1 - — — Rotor Bar 2 ------ Rotor Bar 3
— - - - Rotor Bar 4 - - - - Rotor Bar 5 - — - Rotor Bar 6
------ Rotor Bar 7 —Applied Current

 

 

 

Figure 5.4. Outer rotor magnetic vector potentials (Wb/m) measured when current
was applied to one inner rotor phase.

46

 

 

 

 

 

 

 

 

 

Winding

SA SB SC IRA IRB IRC

A -67.1 -456.5 -456.7 —28.1 -2150 -217.1
a B 311.8 -459.1 —442.5 142.6 -2211 -205.8
5,0 446.8 -451.2 -185.3 206.9 -213.9 -85.9
3;» D 451.2 -446.8 185.3 213.9 -206.9 85.9
3 E 459.2 -311.8 442.5 221.1 -1425 205.8
g F 456.5 67.1 456.8 215.1 28.1 217.1
G 387.3 387.3 449.4 175.2 175.2 211.2

 

 

Table 5.13. Mutual inductance (pH) between the stator or inner rotor and the outer
rotor regions.

regions can be found by dividing the magnetic flux in the region by the current
applied to the stator or inner rotor as was described in equation (3.44). Table 5.13
contains the inductances found by applying this method. It is only necessary to
calculate the inductance in one direction due to the linearity of the medium.

The mutual inductance between the stator and the outer rotor loops should be
reduced to single value. This can be done by finding the magnitude of the stator
to outer rotor mutual inductances curves formed by the inductances in Table 5.13.
The magnitude can be found by dividing the inductances in the table by the sine
of the angle between each rotor region and the sourced stator winding. Since the
inductances don’t have a perfect sinusoidal distribution, only those closest to the
sourced coil will be used to find the average inductance. The resulting sine waves
can be seen in Figure 5.5. After the magnitude is found for each mutual inductance
the overall mutual inductance between the stator and outer rotor can be found by
averaging the three resultant values.

L30. = 829.7/1H (5.13)

The mutual inductance between the inner rotor and the outer loops can be found

47

Inductance (H)

 

x 10-3 Stator to Outer Rotor Sinusoidial Approximations

 

T 1 I T I

 

 

 

csor
__Approxnmation of Law

—1—Approximation of Lbsor
_.._Approximation of Lcsor

 

 

 

 

 

 

l J l l l

 

A B C D E F G

Rotor Region

Figure 5.5. Stator to Outer Rotor Inductances

48

x 104 Inner Rotor to Outer Rotor Sinusoidial Approximations

 

 

4 T
I
3 _
2 ..
1 ’ Lairor
I‘biror
__ L .
curor

___Approximation of Lairor

Inductance (H)
o

——.—Approximation of Lbiror
_.._Approximation of Loiror

 

 

 

 

 

 

 

Rotor Region

Figure 5.6. Inner Rotor to Outer Rotor Inductances

using the same method as was used to find the mutual inductance between the stator

and the outer loops. Figure 5.6 shows the resulting sine waves.

Lira. = 37111H (5.14)

49

CHAPTER 6

Model Development

6. 1 Introduction

To create a model of the EVT, two sets of information are important: the frequencies
of the currents and voltages throughout the machine and the voltage equations. With
an understanding of both of these concepts it is possible to find the power distribution
throughout the EVT. In this chapter these concepts will be developed. This informa-
tion will then be used to develop the power flow throughout the EVT resulting in a
model of the EVT.

6.2 Frequency Equations

In a traditional induction machine, the mechanical speed of the device can be con-
trolled by the frequency of the currents applied to the stator. The EVT can be looked
at as two separate induction machines with two controlled frequencies, Luz-r and ws,
and two desired mechanical speeds, wm1 and wm2. To control the speeds in the EVT,
the frequencies throughout the machine should be established with respect to the
controlled frequencies and the desired speed.

To begin, the stator and outer rotor relationships will be developed. The stator

50

and outer rotor behave like the traditional stator and rotor in an induction machine.
The frequency of the magnetic field in the outer airgap will be the same as the
controlled frequency of the voltages in the stator. Since it has been established that
the frequencies in both airgaps of the EVT must be the same, the frequency in the

airgaps will be referred to as wf as seen in Figure 6.1.

Wf :ws (6.1)

In a traditional induction machine the frequencies of the voltages in the rotor will
be the difference between the frequency of the field in the gap and the mechanical
speed of the rotor. The same relationship holds true for the stator and outer rotor of

the EVT.

war = P(wf " wm2) = “’8 _ Wm2 (6-2)

The slip is defined as the ratio of the difference between the frequency of the
magnetic field in the airgap and the mechanical speed of the rotor divided by the
frequency of the field in the gap. The slip between the outer rotor and the stator of

the EVT will be the same as it was for an induction machine.

gamer = ‘j’fl = W (53)
Wf 013

Now that the frequencies associated with the outer airgap have been established,
the frequencies associated with the inner airgap can be established. The inner airgap
can be viewed such that the inner rotor would act like a traditional stator and the
outer rotor would, as it was in the case of the outer airgap, act like the rotor of
a traditional induction machine. There is one noticeable difference. As opposed to
the stator in a traditional induction machine which is stationary, the inner rotor is

spinning at a mechanical speed wm1, in units of radians per second. With this in

51

wm1

...r_,.. ._ .

\h
a
s
s

    
 

(Dr

(Us

 

 

%

Figure 6.1. EVT frequency definitions

mind, the frequency of the magnetic field in the inner airgap can be described in
terms of the mechanical speed of the inner rotor and the applied frequencies of the

voltages found in the inner rotor.

Wf = wir + Wml (6-4)

It would follow that the frequencies of the voltages found in the inner rotor would
be

wir = P(w f - wmll- (6-5)

In the EVT the frequencies for the magnetic field in the irmer and outer gaps
must be equal and therefore, the rotor frequency must be the same for both the inner
and outer airgaps. As such, the frequency of the currents and voltages in the outer

rotor can be described in terms of the mechanical speeds of the rotors and the applied

52

frequency of the voltages in the inner rotor.

war = wir " p(wm2 _' wm1) (6'6)

The slip is again described as the ratio of the difference between the frequency of
the field in the gap and the mechanical speed of the outer rotor. Since the inner rotor
is spinning at a mechanical speed, wml, the slip in the inner gap can be described as

war wir _ 17(me " wm1)

S. = _ = , 6.7
Inner Wf 011',- +Wml ( )

 

6.3 Voltage Equations

The voltage in the stator will consist of three parts. The first part will be due to
the currents flowing in the stator and the impedance of the stator windings. The
second component is due to the impedance caused by the mutual inductance between
the stator windings and the inner rotor windings. The final component is again due
to impedance caused by mutual inductance. This time the mutual inductance is
found between the stator and the outer rotor. If these components are combined, an

equation for the stator voltage will be created.
VS = (Rs + Xs)is + Xirsiir + Korsior (6.8)

The voltages in the inner rotor will have the same basic structure. There will be
one component due to the impedance of the rotor and two components due to the

impedance caused by the mutual inductance with the other two machine components.

Vir = (Rir + Xir)iir + XsiriS + Xoririor (6'9)

53

 

 

 

Figure 6.2. Outer rotor equivalent circuit for outer rotor bars 1 and 2

The outer rotor is a bit more complex. Since the outer rotor consists of rotor bars
and not windings, we need to account for the rotor bar ends not present in the stator
and inner rotor.

Figure 6.2 shows the equivalent circuit for the inner rotor bars one and two ignor-
ing, for now, the mutual inductances. The circuit can be solved for the current in the
outer rotor loop a composed of the rotor bar ends and rotor bars one and two. Using
Kirchoff’s voltage law, the voltage in rotor loop a, Vam, would be as follows.

2

Vaor = (Xaor + 7

2 . . .
Xend + §Rend + 2R0r)za — Rmzb — R0729 (6.10)

Equations of the form of equation (6.10) can be applied to any of the loops enclosed
by a pair of adjacent rotor bars. If the mutual inductances are no longer ignored,

equation (6.10) can then be written in vector notation

2 2 ,
7Rend1 + §XendI)10r

—Rorl(i:,',.1 + igrl) + XsoriS + Ximriir (6.11)

Vor = (Xor + 2R01'I +

54

 

Equation (6.11) can be modified to fit the form of equations (6.8) and (6.9).

_ 2 2 . . .
v... = (2Ror —- R321 — R0} + 5R.“ + Xor + gxendhor + xsorls + xirorntsn)

6.4 EVT Model

Everything needed to fully model the EVT is now known. The equations developed
in Section 6.3 can now be reevaluated taking into account the frequency equations
established in Section 6.2. The voltage equations will be established with the fre-
quencies described in terms of the controllable values, Lair and tag, and the desired

mechanical speeds, wm1 and wm2. Equations (6.13) through (6.16) represent the EVT

model.
Vs = (Rs + jwsLs)is + jwz’rLirsiir + jworLorsior (6‘13)
Vir = (Rir + jwirLir)iir + jwsLsiriS + ij’rLoririOI‘ (6'14)
__ 2 .
VOI' = (2R01‘ — Rgrl _ Ror1 + 7Rend + me‘LOI‘
2 . . . . . .
+§Jw0rLend)10r + stLsorls + JwirLirorlir (6'15)
where,
war = 0‘13 — Wm2 = Wir — 19(me — wmll (6-16)

6.5 Torque Equations

The energy stored in the coupling field of a linear rotational electromagnetic system

with J electrical inputs can be expressed as in equation (6.17).

. . 1 J J . .
Wf(21, ...,Zj,6) = E Z Z quZqu (6.17)

55

In the case of the EVT, there are three electrical inputs, the stator, the inner rotor
and the outer rotor. The EVT has three coupling fields: one between the stator and
outer rotor, one between the inner rotor and outer rotor and one between the stator
and outer rotor. The coupling field between the inner rotor and outer rotor will be
referred to as W91, the field between the stator and outer rotor as W92, and finally
the field between the stator and inner rotor as W”. Equation (6.17) can be solved

for the coupling field between the inner and outer rotors.

1 . . . . 1 . .
W91 = 5(1ir)T(Lir)lir + (lir)TLiror10T + 5(1W)T(L0T)10T (6'18)
The equations for the other two torque components, T92 and Trs, would be of the
same form as equation (6.18).
The electromagnetic torque of a rotational system at the kth mechanical input

can be described as in equation (6.19).

f) ___8W0(i3" 9") (6.19)

Tekfijygr) = (2 39¢

Since it was assumed that the machine is linear, the coupling field energy is equal

to the coenergy, Wf = WC. Equations of the type of equation (6.19) can now be used
to find the torque. In order to find the torque, We can be replaced with equation
(6.17 ) In equation (6.18) only the mutual inductances between the inner and outer
rotors are functions of Birm. Therefore, to solve for the torque, the self inductance

components of equation (6.18) can be ignored.

T91 = (g) mafia-mum (6.20)

The electromagnetic torque for the coupling fields W92 and Wrs would have similar

forms to that of equation (6.20). Solving for the partial derivative the torques would

56

become

P

Tgl = - (‘2‘) iirmiir{[“Cos(eiror)3in(62'ror) _ Sineirorlco‘SWz'rorHim‘} (621)

P

T92 = — <5) ismis{[—cos(esm)sin(650¢) — sin(esm)cos(630¢)]im}

A

P . . . .
Trs = _ (‘2‘) Lsir18{[‘COS(€sir)szn(6’sir) “ 31n<£sir)cos(gsir)llir}

where e is the relative angular position defined as

27r
0 3
6sir = _237_r_ 0
27r _ 277
T ‘3—
0 27r 471’ 611’
7— T T
_ 27r 87r 21r 47r
630'" — 3 — 2r " 2T 21
271' 207T 267r 327r
_3— 2T 71— ‘21—
0 27r 47r 67r
—7— T _7—
6' = _ 27r _ 87r __ 27r 477
nor ‘3— 2T 2T 2T
27r 207r 267r 327r
3— 2T '21— '21—

57

_'T

__ 327r
7T

47r

67r

_ 327r
7T

_47r
2T

__ 267r
7T

27r
2T

271'

__ 207r
71—

87r
2T

CHAPTER 7

Model Validation

To prove that the model developed actually represents a true machine, it must be
validated. Since an actual machine is not available the model will have to be tested
against a set of expected results. This section details a series of tests and their
results to verify the accuracy of the model. For all of the following tests, the machine
variables developed using the FEA process for the four pole machine presented earlier

will be used.

7 .1 Test 1 - Stator to Outer Rotor Linkage

This test treats the stator as a traditional stator and the outer rotor as a traditional
rotor. The inner rotor will be made an open circuit. The inner rotor will also be held
stationary for this test. This test shows the ability to pass energy from the stator to

the outer rotor.

7.1.1 Setup

To simulate this situation, the stator voltage will be set to V3 = 120V. Both the
inner and outer rotor applied voltages will be set to 0 V, Vir = Var = 0V. A large

resistance, 20 k9, will be added in series to each inner rotor phase to simulate an

58

open circuit. To simulate a stationary inner rotor, the speed of machine 1 will be set
to 0 rpm, wm1 = Orpm. To create a torque, a slip needs to be introduced between
the stator and outer rotor, Lush-p = 307‘pm. In order to have a synchronous speed of

1800 rpm, the outer rotor speed will be set to 1785 rpm, wm2 = 1785mm.

7.1.2 Results

In this case the stator currents settle out to sinusoidal signals at a frequency of 60
Hz. Since the inner rotor is an open circuit and stationary, the inner rotor currents
were also at 60 Hz but the amplitude was relatively small in comparison to the stator
and outer rotor currents. The inner rotor currents were 90 degrees out of phase with
respect to the stator currents. The outer rotor currents settled out to be sinusoidal
at 0.5 Hz.

There is torque between all three parts of the machine. The torque is positive in
the outer airgap and settles out to approximately 0.14 Nm. The inner airgap torque
is negative but is negligible in comparison to the outer airgap torque. There is a
relatively small torque between the inner rotor and stator in comparison to the outer

airgap torque.

7 .2 Test 2 - Inner Rotor to Outer Rotor Linkage

This test treats the inner rotor as a traditional stator and the outer rotor as a tra-
ditional rotor. The inner rotor will be held stationary for this test. The stator will
be made to act as an open circuit for this test. This tests the ability to pass energy

from the inner rotor to the outer rotor.

59

Stator Current - Phase A & B

 

 

 

 

 

 

 

 

 

I
IIIIIII
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As 2250

Outer Rotor Current - Phase A & B

 

 

 

 

 

 

    

 

 

0
.. _ \ a _ 2
m r
a la
n 7/7/
I 1 9
1
I 1 8
1
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0 O 0 O 01
4 2 9.. 4

As 2650

Time (sec)

 

 

 

 

 

 

 

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A3 2250

19.9 19.95 20
Time (sec)

19.85

19.8

Figure 7.1. Machine currents (A) measured during Test 1.

60

x 10-3 Torque Inner Airgap

 

1 l l T I

 

Torque (Nm)

 

 

 

 

 

 

 

 

Torque (Nm)

 

 

 

_3 L l 1 l
0 1 4
Time (sec)
x 104 Torque Stator and Inner Rotor
2 l I l l
A 0
E
é
0:9 -2
E'
o
'— 4
_6 l l l l
0 1 4
Time (sec)
Torque Outer Airgap
4
Time (sec)

Figure 7.2. Torque between the machine parts measured during Test 1.

61

7.2.1 Setup

For the validation of the linkage between the inner and outer rotors, the inner rotor
voltage will be set to 120V, Vi, = 120V. The stator and outer rotor will both have
no applied voltage, V; = Var = 0V. The stator will have large series resistances,
20 k0, added to each phase to simulate an open circuit. The inner rotor will again
not be allowed to rotate wm1 = Orpm. The slip frequency will again be set to
30 rpm, wslip = 307‘an and the outer rotor will be allowed to spin at 1785 rpm,

wm2 = 1785rpm.

7.2.2 Results

The stator currents are as they were expected to be, sinusoidal signals at 60 Hz with
a relatively low amplitude. The inner rotor currents are also 60 Hz sine waves but
they lead the stator currents by 90 degrees. The outer rotor currents are sinusoidal
and settled out to a frequency of 0.5 Hz.

There is torque between all three parts of the machine. The torque in the outer
airgap and the torque between the stator and in inner rotor are both negligible. The
torque is positive in the inner airgap with a steady state value of 0.13 Nm. These

results are all as expected.

7.3 Test 3 - Stator to Inner Rotor Linkage

The purpose of Test 3 is to check to make sure that energy is being transfered from
the stator to the inner rotor. The outer rotor will be held stationary for this test.
The stator will be treated as a traditional stator and the inner rotor a traditional

rotor.

62

 

 

 

 

 

 

 

 

 

 

0
2
5
1 9
9
............ 1
B r.......
co I’ll/{IV
A ...................
6 All: i
S ..............
a rim»
h nnnnnnnnnnnnn in:
P . ........ 9
_ 1 .......... L .
t 111111111 9
n iiiii 1
e .................
fl . .....
u ................
C ---..u
r IIIIIIIIIIII
o . ......
t till
a .............
t IIIII
S ............ _
I IIIIIIIIII 5
.I lllllllllllllll l B
----...... 9
.............. 1
--------..........v
«HHHH .........
a -------------U
0 uuuuuuuuuuuuuuuuu
1 Allllll
X -- ..... . _ w
2 1 O 1 _

g 22:6

Time (sec)
Outer Rotor Current - Phase A & B

 

20

 

 

 

 

 

9

1

7

 

15

 

 

-40

2v E250

Time (sec)
Inner Rotor Current - Phase A & B

 

 

 

 

 

 

 

20

 

 

ill/Inn, 5

............ -- - 9.

,.......- 9

- .............. 1

....... .. ......t 9

, ............... i 9.

I: ......... 1
.....V

--I------/... 5

............... - 8.

......... 9

.............. 1
llllilid

- -......... 8

_ iTr- 9

1 o 1 9....

E .550

Time (sec)

Figure 7.3. Machine currents (A) measured during Test 2.

63

Torque (Nm)

Torque (Nm)

Torque Inner Airgap

 

l I I

 

 

 

 

 

 

 

 

 

4
Time (sec)
x 104 Torque Stator and Inner Rotor
15 . x ‘17
10
5
0
-5 L l l
0 2 4
Time (sec)
x 10 Torque Outer Airgap
2 l I I
0
-2
.4
-6 l l 1
O 2 3 4
Time (sec)

Figure 7.4. Torque between the machine parts measured during Test 2.

64

 

 

 

7.3.1 Setup

To validate the transfer of power from the stator to the inner rotor the stator voltage
will be set to 120V, Vs = 120. The outer rotor will have a relatively large load applied
to it, 3 0. This value is much smaller than the resistance applied in the other tests.
This was done to decrease the time to solve using the ordinary differential equation
solver. The inner rotor will be made a short circuit by setting the applied voltage to
0V, V,-,. = 0. The outer rotor will not be allowed to rotate wm2 = Orpm. The inner
rotor will be allowed to spin at 1795 rpm wm1 = 1795mm. The slip will be set to 10,
Lush-p = 10. The slip value used for this test was set to a value less than the slip used
in the previous tests because the torque speed characteristics are slightly different
when the EVT is configured as it is in this test. Since this is not a normal operating
point, the model will have to be varied slightly. The frequency in the stator will be

set as

“’3 = Wml + wslip (7-1)

7 .3.2 Results

The stator currents are sinusoidal with a frequency of 60 Hz. The outer rotor currents
are also sinusoidal at 60 Hz but they are 90 degrees out of phase with the rotor
currents. The inner rotor currents are sinusoidal at a frequency of 0.1667 Hz.

The torque developed in the inner and outer airgap was extremely small. The
torque developed between the stator and inner rotor was larger than that developed
in either airgap but was still relatively small, 0.000942 Nm. This was as expected.
Some torque was developed between the stator and inner rotor displaying that there
is an interaction developed. The effect, in comparison to the torques developed in

Tests 1 and 2, is relatively small.

65

Stator Current - Phase A & B

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 0 0
1 1- 1
9
5 5
m 9. - 9.
llllll 9 B 9 B
-GHHV & &
............... A A 8
.............. w m
.IllV a a
........... .n h
an..- ) P ) P
I! llllllllllll ADIW - m —
ill. t
.......... - 9. be. m - 9. as. m 7
run... 9 e W 9 e W
iiiiiiiiiiiiii .m U .m u
......... a--. T m T pro
.nnil . m m
iiiiiiiiiiiii 0 0
IIII \\I\t R R
nun ..... .nlv mu 6
I II! t
............ nus m
II\\. 5 5 II
............... - 8. - 8.
\ll'l llllll 9 9
...Hqu
............... 5
Ann” ........
F 1.11-1 \i.\. a 8n n0. a _ 4
r3. 0 5 . . 2 1 0 1 2
- - 0 0 o- 0-
93 E950 20 E950

A3 E950

Time (sec)

Figure 7.5. Machine currents (A) measured during Test 3.

66

x 10'5 Torque Inner Airgap

 

2 I I I I

 

Torque (Nm)

 

 

 

I
a)

Time (sec)
x 10-3 Torque Stator and Inner Rotor

 

1.5 I I I I

 

Torque (Nm)

 

 

 

 

Time (sec)
x 10-3 Torque Outer Airgap

 

1.5 . w . T

 

Torque (Nm)

 

0 ‘ I I; I l

0 1 2 3 4
Time (sec)

Figure 7.6. Torque between the machine parts measured during Test 3.

67

 

7 .4 Test 4 - Stator to Outer Rotor High Slip

The purpose of this test is to show the relationship of slip and torque between the
stator and the outer rotor. In this case, much will be the same as Test 1. The
difference being that the slip will be increased. The increased slip should show an
increased torque between the stator and outer rotor then that which was measured

in Test 1.

7 .4.1 Setup

For this test, the stator voltage will be set to V3 = 120V. Both the inner and outer
rotor applied voltages will be set to zero, V2,. = V0,. = 0V. To simulate an open inner
rotor, a large resistance, 20 k9, will be added in series with each inner rotor phase.
To simulate a stationary inner rotor, the speed of machine 1 will be set to 0 rpm,
wm1 = Orpm. To create a greater torque than in Test 1 a greater slip needs to be
introduced between the stator and outer rotor. The slip will be set to twice what it
was in Test 1, cash-p = 60mm. To have a synchronous speed of 1800 rpm we will set

the outer rotor speed to 1770 rpm, wm2 = 1770mm.

7 .4.2 Results

In this case, the stator currents settles out to sinusoidal signals at a frequency of
60 Hz just as was in Test 1. The inner rotor currents were also at 60 Hz but the
magnitude of the currents was much lower then the stator and outer rotor currents.
The outer rotor currents settled out to be sinusoidal at 1 Hz with an amplitude of
about twice that of the currents in Test 1.

The torque between the stator and inner rotor and the torque in the inner airgap
are both negligible. The torque is positive in the outer airgap, 0.2754 Nm, and larger

then the torque found in Test 1 as was expected.

68

 

Stator Current - Phase A & B

 

 

 

 

 

 

 

\\

III

III

1!]

\\

III

\\I
III

 

 

0.5 r
0
0.5

9: E250

20

19.95

19.9
Time (sec)

Outer Rotor Current - Phase A & B

19.85

19.8

 

 

 

 

 

20

 

 

 

100

 

Time (sec)

 

 

 

 

 

 

 

 

 

I HHHV 5
1 ..................... 1 9
. ...... 9
B ....................... 1
& ...................
A ........... ....
e ....................
S H...
a ....................
Hm .............
. iiiiiiiiiiiiiiiiiiiiii .
.m ..................... 9.
e I lllllll I 9
n A ..................... 1
u iiiiiiiHIJ
C .....................
r ‘ IIIIIIIII
m ....................
R0 .uuilihuv
r- . ....................
e ................
n ............. ,
n ........ 5
I - ...................... I 8.
............. 9
................. 1
III/MU.
a ........
0 .................. ,
1 IIIIIII
X I--. ......... _. ...... _ %
2 1 0 1 .

Time (sec)

Figure 7.7. Machine currents (A) measured during Test 4.

69

-3 Torque Inner Airgap
15 I I l I

Torque (Nm)
UI

 

 

 

 

_5 1 I I 1
2 3
Time (sec)
4 Torque Stator and Inner Rotor

I I I

o
.1
h

x10

A—LN
0010

Torque (Nm)
0'!

 

O

 

 

I
001
Ar

#-

Time (sec)
Torque Outer Airgap

 

0.6

I I I

0.4

 

    

Torque (Nm)

.<'=
N

I l l l

' 0 1 2 3 4
Time (sec)

Figure 7.8. Torque between the machine parts measured during Test 4.

70

 

7 .5 Test 5 - Inner Rotor to Outer Rotor High Slip

This test is meant to Show that the torque between the inner and outer rotors varies
as the slip between those parts varies. This test will be a repeat of Test 2 with an

increased slip.

7.5.1 Setup

To show that the torque between the inner and outer rotors varies with the slip
between them, we will use many of the same values as we used in Test 2. Again,
the inner rotor voltage will be set to 120V, Vir = 120V. The stator will again have
series resistances of 20 k0 added to each phase. The inner rotor will again, not be
allowed to rotate, wm1 = Orpm. The difference between this test and Test 2 will be
that the slip frequency will be set to 60 rpm, Lush-p = 60mm. Therefore, to maintain
a synchronous speed of 1800 rpm the outer rotor will be allowed to spin at 1770 rpm,

wm2 = 1770mm.

7.5.2 Results

The inner rotor currents are as expected: sinusoidal signals with a frequency at 60 Hz.
The stator currents are also 60 Hz sine waves with an amplitude much less than the
inner and outer rotor currents. The outer rotor currents settle out to be sinusoidal
with a frequency of 1 Hz with an amplitude larger than that found in Test 2.

The torque in the inner airgap is positive with an amplitude of approximately 0.19
Nm. This torque value is greater than the torque found in Test 2. The other torque

values in the machine are both negligible. These results are all as expected.

71

 

 

 

 

 

Stator Current - Phase A & B

 

 

 

 

2V Eatso

1.. ”H”.
0 i .............
1 “III
X {J ......
4 2 0 2

19.95 20

19.9
Time (sec)

Outer Rotor Current - Phase A & B

19.85

 

 

 

 

 

 

 

 

 

A3 2250

Time (sec)
Inner Rotor Current - Phase A & B

 

20

 

 

 

 

 

 

19.95

19.9
Time (sec)

72

19.85

Figure 7.9. Machine currents (A) measured during Test 5.

 

 

2v €2.50

Torque Inner Airgap

 

S3
o:

I I I

.9
.rs

 

Torque (Nm)
0
N

 

 

 

 

 

 

 

 

 

 

 

 

0
'0.2 ' I 1
3 4
Time (sec)
x 10‘4 Torque Stator and Inner Rotor
15 . 1 I T
A 10
E
.2,
g 5
E
o
'— o
-5 4L ' l g
Time (sec)
x 10'3 Torque Outer Airgap
2 . . I .
A 0
E
E,
g -2
E'
o
'- 4
-6 l J l 1
0 1 4
Time (sec)

Figure 7.10. Torque between the machine parts measured during Test 5.

73

7 .6 Test 6 - Stator to Inner Rotor High Slip

To confirm that the torque between the stator and inner rotor varies with the slip
of those two components, Test 3 will be rerun this time with a slip that is greater
than that in Test 3. When complete, the results of this test can be compared to
the previous results to show the relationship of slip to torque between the stator and

inner rotor.

7.6.1 Setup

To show that the torque between the stator and inner rotor varies with the slip
between those components, all of the values used in Test 3 will again be used for
this test except for the slip speed. The stator voltage will be set to 120V, V3 = 120.
Both the inner and outer rotor applied voltages will be set to zero, Vir = V0,- : 0.
The outer rotor will again be made an open circuit by applying a relatively large
resistance, 30. The outer rotor will not be allowed to rotate, wm2 = Orpm. The inner
rotor will be allowed to spin at 1785 rpm wm1 = 1785mm. The slip will be set to 30
rpm, Lush-p = 30rpm. The frequencies in the model will be modified as they were in

Test 3.

7.6.2 Results

The stator currents are sinusoidal at 60 Hz as they were in Test 3. The outer rotor
currents are also sinusoidal at 60 Hz but they lag the stator currents. The inner rotor
currents are sinusoidal at 0.5 Hz as expected.

As was the case in Test 3, the torque developed in the inner and outer airgaps was
relatively small in comparison to the torque between the stator and inner rotor. The
torque developed between the stator and inner rotor was larger than that developed

in either airgap but was still relatively small, 0.00126 Nm. This value is larger than

74

Stator Current - Phase A & B

 

 

 

 

 

 

 

 

rnu

9.

...... 9
........... \\ i 9.
«mu... 9
I .HV 5

- ............. - 8.
.Iiii iiiiii 9
_ till- it... Ru.
5. o 5. 19.

0 no
2V E950

Time (sec)
Outer Rotor Current - Phase A & B

 

 

 

 

 

 

 

 

 

0
1
5
l 9.
l- ............ 9
Hmu.v
Aim“ iiiiiiii
i ................... 1 nuw
n ........ 9
....HIV
A ..............
IIIIIIIIII l4 5
i I ........ L Ru.
............... 9
-HHHIV
A.HH iiiiiiii
_ IIIIIII — IIIIIIIII _ 8.
2 1. O J 29

Time (sec)
Inner Rotor Current - Phase A & B

 

 

 

 

 

 

  

10

7
Time (sec)

 

 

o 0.9: 22:6

_
2
0..
n.u

-0.04

Figure 7.11. Machine currents (A) measured during Test 6.

75

that found in Test 3. This shows that near synchronous Speed the torque between
the stator and inner rotor will increase as the slip between those two components

increase.

7 .7 Test 7 - Both Rotors in Motion

The purpose of this test is to show that the relative motion of the inner rotor to outer
rotor is what causes the torque to be transfered, not the absolute speed. To show

this Test 2 will be repeated. This time, both rotors will be in motion.

7.7.1 Setup

All of the voltages used in Test 2 will be used for this test. The difference will be in
the speeds. The outer rotor will be allowed to rotate at 2785 rpm, wm2 = 2785rpm.
The inner rotor will be rotated at 1000 rpm, wm1 = 1000mm. This will result in the

same slip speed, Lush-p = 30, as used in Test 2.

7.7.2 Results

The currents in the inner rotor are sinusoidal at 60 Hz. They have relatively the
same magnitude and the same frequency as the inner rotor currents did in Test 2.
The currents in the outer rotor are sinusoidal with a frequency of 0.5 Hz. They have
relatively the same magnitude and frequency as the currents found in Test 2. The
stator currents have a slightly higher magnitude as those in Test 2 but the frequency
is now closer to 90 Hz. The magnitude of the stator currents is still much lower
than the inner and outer rotor currents. The slightly higher currents in all of the
components are due to the increased slip between the stator and inner rotor and the

stator and outer rotor. The effect is negligible but should be noted.

76

 

 

x 10-5 Torque Inner Airgap

 

r l I

 

 

 

 

 

 

 

 

 

 

 

 

4
Time (sec)
x 10-3 Torque Stator and Inner Rotor
’E‘
E 1
a:
D
E
a 0 ,
-1 I ' ' '
0 1 4
Time (sec)
x 10-3 Torque Outer Airgap
1.5 I I I I
’e‘
.2,
CD
:3
E
o
l-
4

 

Time (sec)

Figure 7.12. Torque between the machine parts measured during Test 6.

77

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

0 0
~ 111.. 2 ~ _ \ _ _ 2
s s IIIIIIIIII r pl
'3 ID IIIIII iuuv lm Inmfi
n III _
u rrrrrrrrrrrr J. n /
n ........ l n ,1/1
. ..... H II;
11111111111111. :11;
511111 rrrrrrrrrrrr v I 9
............. 5 1
I \III lllllllll I OU-
1111111. 9
nun iiiiiiiiii 1 B
B 111111111111”, &
& 1.... .......... A
A illuliuuu %
.r11 iiiiiiiiiii a 8
w 1 iiiiiiiiiiiiii 1n n J 1
a iiiiiiiiii 111 ) P
h 111111 iiiiiii m .
P 111 1111111111 9 S m
. - ........... n .( e
t 111 1111 9 e .l
n 11111111“ 1 r
m ............. i .m U
.I .r/ iiiiiii h C
U 1 111m”. .l 7
C .n ......... m - - 1
r iiiiiiiiiiiii 0
m ............ R
pHIl Ill! r
a llllll ”UV .5an
.r IIIIIIIIIIIII u
11 iiiiiiiiiii 111d % O
Anni- iiiiiii 9
iiiiiiiii 11 1111V 1 .I I 6
— IIIIIIIIIIIII 1
........--.. ...... -
A. ann11111 iiiiiii 11
0 rrrrrrrrrr 11 Mn. 1
1 IIIIIIIIIII 1”
X .1111 _ % _ _ I 5
4 2 0 2 41 0 0 0 0 01
. 4 2 Ad— 4

2v E950 9; E m ._ 5 0

Time (sec)
Inner Rotor Current - Phase A & B

 

 

 

 

 

 

 

 

 

0

2

IIIII-V 5

.............. - 9.

A iiiiiiii 9

lllllllllllllll 1

....... .1 UV
...... .. H... 9.
“1.111.111- 1 w
nummm ........

11111111111111» 5

.............. 1 8.

HH. 9

IIIIIIIIIIIIIIII 1

I IIIIIII1I» 8

_ 11.111 9.
1 0 1 —

Time (sec)

Figure 7.13. Machine currents (A) measured during Test 7.

78

The steady state torque in the inner airgap is positive and is relatively equivalent
to the torque found in Test 2. The steady state torque in the outer airgap is also
relatively the same value as found in Test 2. The torque between the stator and inner
rotor is relatively close to what it was in Test 2. This is the expected result as the

slip with respect to the stator has changed but the effect is negligible.

7.8 Test 8 - Stator Torque Speed Curve

For this test, the goal is to create the torque speed curve for the stator when the inner
rotor is an open circuit. This test is different from the previous tests as it is not as

much about verification of the model as about getting information from the model.

7.8.1 Setup

The voltage of the stator will be set to 120V, Vs = 120V. The inner and outer
rotor applied voltages will be set to zero, Vi, = Var = 0. Since we are looking for
a torque speed curve the slip and supply frequencies will be modified directly. The
stator frequency will be held constant at synchronous speed, ws = 27r60 rad/s. The
slip frequency will be varied from 100 rpm to 3600 rpm, Lash-p = lg?- to 27r60. For
this test, the inner rotor will be held stationary, wm1 = 0. The slip frequency will be
applied directly to the outer rotor, war = wslip. The inner rotor will be made an open
circuit by applying a series resistance of 20 k!) to each of the inner rotor phases.
The result will not be a true torque speed curve because the speed will be swept
from zero to twice synchronous speed throughout the test. To get a true torque speed
curve, the model would need to be allowed to hold at set speeds for a given time
period in order to reach steady state. The torque speed curve from this test will only

be useful to give basic characteristic data about the EVT.

79

Torque Inner Airgap

l I I

 

 

Torque (Nm)

 

 

 

 

 

 

 

 

 

 

4 5
Time (sec)
4 Torque Stator and Inner Rotor
x 10
20 1 1 . 1
15 -
E
E 10 -
OJ
3
E 5 —
o
[—
0
_5 I I I I
0 1 4 5
Time (sec)
5 x 10-3 Torque Outer Airgap
A 0 _
E
E
s -5 —
E"
o
I- _10 _
_15 l l l I
0 1 2 3 4 5
Time (sec)

Figure 7.14. Torque between the machine parts measured during Test 7.

80

n—

Torque Stator to Outer Rotor

0.4 I I1 f I fl I I

 

0.2 - -

 

0.1

 

Torque (Nm)
:3

 

 

 

 

 

500 1000 1500 2000 2500 3000 3500
mm (rpm)

Figure 7.15. Torque between the stator and outer rotor when the inner rotor is an
open circuit.

7.8.2 Results

The torque speed curve for the stator is presented in Figure 7.15. During startup, the
machine is unstable and then gains stability throughout the remainder of the curve.
After the synchronous speed of 1800 rpm, the machine stops acting as a motor and

starts acting as a generator.

81

7.9 Test 9 - Inner Rotor Torque Speed Curve

For this test, the goal is to create the torque speed curve for the inner rotor when
the stator is an open circuit. Again, the purpose of this test is to obtain information

from the model.

7.9.1 Setup

The voltage of the inner rotor will be set to 120V, Vs = 120V. The stator and outer
rotor applied voltages will be set to zero, V,,. = V0,— : 0. As was the case in the
last test, the supply and slip frequencies will be modified directly. The inner rotor
frequency will be held constant at synchronous speed, wir = 27r60 rad/s. The slip
frequency will be varied from 100 rpm to 3600 rpm, Lush-p = 10—31 to 27r60. For this
test, the inner rotor will be held stationary, wm1 = 0. The slip frequency will be
applied directly to the outer rotor, war = wslip. The stator will be made an open
circuit by applying a series resistance of 20 kfl to each of the inner rotor phases.

As was the case with the Test 8, the result of this test will not be a true torque
speed curve. The data found during this test is useful to gain basic characteristic

information about the EVT.

7 .9.2 Results

The torque speed curve for the inner rotor is presented in Figure 7.16. During startup,
the machine is unstable and then gains stability throughout the remainder of the
curve. After the synchronous speed of 1800 rpm, the machine stops acting as a motor

and starts acting as a generator.

82

, Torque Inner to Outer Rotor
0.2 I I I I I I I

 

0.15

I
l

0.1- _

I
l

 

0.05

O
1

Torque (Nm)

I

$5 - s5

.1 .0 o

01 .1. 01
I

-0.2 -

 

-0.25

I

 

 

 

 

500 1 000 1 500 2000 2500 3000 3500
wm2 (rpm)

Figure 7.16. Torque between the inner rotor and outer rotor when the stator is an
open circuit.

83

CHAPTER 8

Conclusions

The goal of this work was to create a model for the Electronic Variable Tfansmission.
This objective was met by first determining key motor parameters using finite ele-
ment analysis. After the methods were developed for determining key parameters, a
model was developed using techniques based on the model for a traditional induction
machine and adding in special considerations for the physical limitations of the EVT.

This model could be used to determine appropriate applications of an EVT. Dif—
ferent vehicle systems could be investigated to determine the effect of adding an EVT
style power transmission device. For example, an EVT style device could be designed
using the model developed in this document to replace the current on/ off style air
conditioning compressor clutches. The entire engine system could be simulated to de-
termine what if any power and subsequent fuel savings could be achieved. There are
many areas where an EVT style power transmission device could be applied including
modulating a water pump, replacing a mechanical fan drive, replacing a traditional
starter and alternator, or any number of power transmission devices.

Once an application of the EVT was determined, a model developed using the
method found in this work could be used to determine a proper control algorithm.
The control algorithm could be tuned for the specific application. If care is taken in

understanding the proper operating points for the given application, the individual

84

motor control for each part of the EVT could then be optimized.

Another application of the model development technique found in this work would
be to experiment with different motor types. Rather than using induction machines,
a permanent magnet style machine could be investigated. The method developed
for finding the model for the EVT could be used by, first finding key parameters
using FEA. Then the FEA results could be applied to a pair of traditional permanent
magnetic machines to determine the new model. Special considerations for the linkage
between the inner rotor and the stator could be taken as they were for the induction
machines in this work.

Future work could also include building a prototype machine and testing the
results of this work. The model could be verified for steady state operation. The

model could then be expanded, taking into consideration transient conditions.

85

APPENDICES

86

Appendix A MATLAB Files

A.1 Resistance.m

This file was used to calculate the resistances throughout the EVT.

% File: Resistance .m

% Purpose: Calculates the resistances for the stator,

% rotor and outer rotor
70 Author: S. Bohan
70 Date: 1—17—2007

global Rs Rir Rend Ror

%Coil End Resistances
Rs_end = 0.5575;
Rir-end = 0.5575;

% Resistivities

rho_s = 0.172e—7; % ohmm
rho_ir = 0.172e—7; % ohmrm
rho-or = 0.287e—7; % ohmm

%Stacking Factors

sf_s = 0.8;
sf_ir = 0.8;
sf_or = 1;

% Dimensions
length = 235e—3; % meters

area..s = 72e—6; % meters ‘2

87

inner

area_ir = 87.71e—6; % meters ‘2
area_or = 68.94e—6; 70 meters ‘2

% Number of winding turns

N..s = 44;
N_ir = 44;
N_or = 1;

% Number of slots per phase
3.3 = 4;

s_ir = 3;

% Slot Resistences

Rs_slot = N-s‘2*((rho_s*length)/(sf_s*area_s));
Rir_slot = N_ir“2*((rho_ir*length)/(sf_ir*area_ir));
Ror-bar = N_or“2*((rho_or*length)/(area-or*sf_or));

% Phase Resistance

Rs = s_s =1: Rs_slot + Rs_end; % ohm
Rir = s_ir =1: Rir_slot + Rir-end; % ohm
Ror = Ror_bar; % ohm

%E'nd Resistance (non calculated value)
Rend = 2.56—6; % ohm

88

A.2 SMI-SIR.m

This file, along with the ”InductanceFromFilem” file, were used to find the self and
mutual inductances associated with the stator and the mutual inductances between
the stator and the inner and outer rotors. These files were also used to find the self
and mutual inductances associated with the inner rotor and the mutual inductances

between the inner rotor and the stator and outer rotor.

% File: SMI.SIR.m

70 Purpose: Calculates the self and mutual inductances
70 between the stator and inner rotor

% Author: S. Bohan

% Date: 2—7—2008

Lsa = InductanceFromFile( ’Flux2dsDatasSets\SIA-Voltages . txt ’ );
st = InductanceFromFile( ’Flux2dsDataaSets\SIB-Voltages . txt ’ );
Lsc = InductanceFromFile( ’Flux2d..Data...Sets\SIC-Voltages . txt ’ );
Lira = InductanceFromFile( ’Flux2dsDatasSets\IRIA-Voltages . txt ’ );
Lirb = InductanceFromFile( ’Flux2d...Data..Sets\IRIB-Voltages . txt ’ );
Lirc = InductanceFromFile( ’Flux2dsDataaSets\IRIC_Voltages . txt ’ );

L_all = [Lsa’ st’ Lsc’ Lira’ Lirb’ Lirc ’];

%Find Ls and Lir
L3_ = L-a11(1:3,1:3);
Lir_ : L_all(4326,46),

70 Back out the angles for the mutual inductances between rotor
% and stator (Note: matlab notation mtria:(row,column)
for m: 1:3
theta_s = (m—1)*(2*pi/3);
for n = 4:6
theta_ir = (n-1)*(2*pi/3);
Lirs_((n—3),m) = L_all(n,m)/cos(theta_s — theta_ir);
Lsir_(m,(n—3)) = L_all(m,n)/cos(theta_ir — theta_s);

89

 

end
end
% Find the mutual inductance between the inner rotor and stator
% as an average

Lsir = mean(mean(abs ( [ Lsir- ; Lirs- ] ) ) );

% Find the Stator to Stator mutual inductance

Ls = (Ls_(1,2)+Ls-(1,3)+Ls_(2,1)+Ls_(2,3)+Ls_(3,1)+Ls_(3,2))/3;
%Find the Leakge Inductance

Lls = mean([Ls-(1,1) Ls_(2,2) Ls-(3,3)]) — Ls;

% Find the Inner Rotor to Inner Rotor Mutual Inductance

Lir = (Lir-(1,2)+Lir-(1,3)+Lir_(2,1)+Lir_(2,3)+Lir_(3 ,1)...
+Lir_(3,2))/3;

%Find the Leakge Inductance

Llir = mean([Lir_(1,1) Lir_(2,2) Lir_(3,3)]) — Lir;

% Note: End Inductance is a user entered value
Lsend = 2.1e—3;
Lirend = 2.1e—3;

Lsir2B-as = L_all(7:13,1)’.*(1./(sin((((l:7)*pi)/7)—(pi/6))));
Lasor = mean(Lsir2B_as(1:2));

Lsir2B_bs = L_all(7:13,2)’.*(1./(sin((((1:7)*pi)/7)-(5*pi/6))));
Lbsor = mean(Lsir2B_bs(5:7));

Lsir2B_cs = L-all(7:13,3)’.*(l./(sin((((1:7)*pi)/7)—(pi/2))));
Lcsor = mean(Lsir2B-cs(2:5));
% Find the average mutual inductance

Lsor = (Lasor+Lbsor+Lcsor)/3;

Lsir2B_air = L_a11(7:13,4)’.*(1./(sin((((1:7)*pi)/7)—(pi/6))));
Lairor = mean(LsirZB-air(1:2));

Lsir2B_bir = L-all(7:13,5)’.*(1./(sin((((1:7)*pi)/7)—(5*pi/6))));
Lbiror = mean(Lsir2B_bir (5:7));

90

Lsir2B_cir = L_all(7:13,6)’.*(1./(sin((((1:7)*pi)/7)—(pi/2))));
Lciror = mean(Lsir2B_cir(2:5));

% Find the average mutual inductance
Liror = (Lairor+Lbiror+Lciror)/3;

91

A.3 InductanceFromFile.m

This file, along with the ”SMI.SIR.m” file, were used to find the self and mutual
inductances associated with the stator and the mutual inductances between the
stator and the inner and outer rotors. The files also find the self and mutual
inductances in the inner rotor as well as between the inner rotor and the stator and

outer rotor.

% File: InductanceFromFile.m

% Purpose: Calculates the self and mutual inductances
70 between the stator and inner rotor for a
% single sourced coil

% Author: S. Bohan

% Date: 2—7—2008

function L = InductanceFromFile(fileName)
data: textread(fileName , ”, ’headerlines ’ , 4);

Time: data(:,l);

Voltages = cat(2, data(:,2), data(:,4), data(:,6),...
data(:,8), data(:,lO), data(:,12));

Current 2 data(:,14);

VectorPotential = cat(2, data(:,16), data(:,18), data(: ,20) ,
data(:,22), data(:,24), data(: ,26) ,...
data(:,28));

%Find the magnitudes of the voltages
V: fft(Voltages(1:90,:),90);

P_v = abs(V) / 45;

V_Mag = P_v(2,:);

%Find the magnitude of the current

I: fft(Current(1:90),90);
P_i = abs(I) / 45;

92

I_Mag = P_.i(2);

%Find the Inductances

X = (V_Mag./I_Mag)—(4*0.1359);
L = X ./ (2*pl*60);

%Find the magnitudes of the vector potentials

for j = 1:7
for i = 1:89
if i I: 45
VP(i,j) = VectorPotential(i,j)/sin(i*2*pi/90);
else
VP(i,j) =VP(i-1,J');
end
end
VP—MagU) = mean(VP(=aj));
end

% Define constants
w = 15.45e—3;
N = 44;

% Calculate the flux in the regions a—k
%(Note: matlab notation mtri$(row, column)
for n = 1:6

Phi_SIRZB(n) = (VP_Mag(n)—VP_Mag(n+1))*w;
end
Phi_SIR2B(n+1) = (VP_Mag(n+1)+VP_Mag(1))*w;

% Calculate the mutual inductances
for h = 1:7

Inductance-SIR2B(h) = 44*Phi_SIR2B(h)/I_Mag;
end

L = cat(l, L(:) , Inductance_SIR2B(:)) ’;

93

 

A.4 SMI_OR.m

This code reads in all of the data collected from the FEA data for the outer rotor self
and mutual inductances. It then finds the self and mutual inductances associated

with the outer rotor.

% File: SMI_0R.m

% Purpose: Calculates the self and mutual inductances in the
% outer rotor

% Author: S. Bohan

% Date: 10—21—2006

%Read in all of the Vector Potential values and populate an

% array

70 The measured potentials with current applied to 2 bars

% (1 ”coil”)

% Rows = Vector Potential

% Column # = bar where positive current was applied

70 Column #+1 = bar where negative current was applied

70 For Example the value in Row 4 Column 2 is the Vector Potential
70 of bar 4 with current applied to bar 2 and the opposite

% current to bar 3

VectorPotential_B2B (: ,1) =...

textread( ’F1ux2dsDatasSets\RB_12_reduced . txt ’, ” );
VectorPotentiaLB2B (: ,2) =...

textread( ’Flux2dsDatasSets\RB-23_reduced . txt ’, ”);
VectorPotentiaLBZB (: ,3) =...

textread( ’F1ux2d._.Data..Sets\RB_34_reduced.txt ’, ’ ’ );
VectorPotential_B2B (: ,4) =...

textread( ’Flux2dsDatasSets\RB_45_reduced . txt ’, ’ ’ );
VectorPotentiaLBZB (: ,5) =...

textread( ’Flux2dsDataaSets\RB_56_reduced . txt ’, ’ ’ );
VectorPotentiaLB2B (: ,6) =...

textread( ’Flux2dsDatasSets\RB_67_reduced . txt ’, ”);
VectorPotentiaLBZB (: ,7) = ...

textread( ’Flux2dsDatasSets\RB_71_reduced . txt ’ , ”);

94

70 Define constants
w = 15.45e—3;
I = 0.1;

70 Calculate the flux in the regions a—k
% (Note: matlab notation matria:(row, column)
for m = 1:7
for n = 1:6
Phi_B2B(n,m) = (VectorPotentiaLBZB(n,m)—...
VectorPotentiaLBZB(n+1,m))*w;
end
Phi_B2B(n+1,m) = (VectorPotentiaLBZB(n+1,m)+...
VectorPotentiaLBZB (1 ,m))*w;
end

% Calculate the inductances

for k = 1:7
for h = 1:7
Inductance_B2B(k, h) = Phi-B2B(k,h)/I;
end
end
offset_or=[ 0 2 4 6 —6 —4 —2,
—2 0 2 4 6 —6 —4,
-4 —2 0 2 4 6 —6,
—6 —4 —2 0 2 4 6,

2 4 6 —6 —4 —2 0;]*pi/7;

% Find the mutual inductance by calculating the offdiagonal
% inductances
for k = 1:7
for h = 1:7
M(k,h) = Inductance_B2B(k, h)/cos(offset-or(k,h));
end

end

Lor = sum(sum(abs (M—diag(diag (M)))))/(7”2 —7);

95

% Find the leakage inductance by calculating the diagonal
% inductance and subtracting from it the mutual inductance
Llor = mean(diag(M)) - Lor;

%Find the end inductance (Note: User entered value)
Lend = 4e—9; % H

96

 

A.5 Example Test M-Code - EVT_ODE_Ex1.m

Files of this type, along with files similar to ”ODE_EVT1.m”, were used to simulate
the EVT for the 9 model verification tests. All of the other files would be similar
with modifications made as were listed in chapter 7. These two files are examples of

the code used to simulate Test 1.

% Model Verfication for the EVT model

70 File: EVT_ODE_E$1.m

% Purpose: Simulates expirament 1 of the EVT model verfication
% Applies voltage to the stator and an open circuit to

% inner rotor.
% Author: S. Bohan
% Date: 1/19/08

%Clean up the workspace
clear all

% Define constants in case other files have modified them

i = Sqrt(-1);

971% Defined Values
global p Rs Rir Ror Lls Llir Llor Lsor Lsir Liror Vs Vir Vor wm2
global wm1 wslip Rend Lend Lor Lir Ls ws wir wor Lsend Lirend

 

 

= 4; % Unitless
P = 13/2; % Unitless

70 Get the current path
curr-dir = cd;

% Move up a directory
cd ..

cd (’EV'I‘PROPERTIE‘S ’)

% Call other m—files to calculate base resistances and

% inductances

97

 

Resistance;
SMI-SIR;
SMLOR;

70 Reset the path
cd(curr-dir);

Vs = 120;
Vir = 0;
Vor = 0;

W = (1785/60)*2*pi;
wm1 = (0/60)*2*pi;
wslip = (30/60)*2*pi;

ws = parwm2-I-wslip;

wir = p*(wm2—wm1)+wslip;

wor = wslip;

offset_s = [0 2 —2; —2 0 2; 2 —2 0]*pi/3;

offset-ir = [0 2 -4n —2 0 2; 2 —2 0]*pi/3;
offset-or =[ 0 2 4 6 -—6 —4 -2;

2 4 6 —6 -4 —2 0;]*pi/7;

offset_sor = offset_s(:,1)*[1 1 1 1 1 1 1]..
+ [1 1 1]’ * offset_or(1,:);

offset_iror = offset-ir(:,1)*[1 1 1 11 1 1]...
+[11 1]’ =1: offset_or(1,:);

offset-sir = offset-s(:,1)*[l 1 1]...
+ [1 1 1]’*offset_ir(1,:);

% Initial realtive positions

70 Outer rotor position with respect to stator position

98

evt_phi..ir = 0; % rad
evt-phi_s = 0;
evt-phi_or = 0;
t-step = 1/1000;
%Setup Initial condition vector
T = (0:19999)*t-step;

y0 = [0 0 0 0 0 0 0 0 0 0 0 0 0 evt_phi_s evt_phi_ir...
evt_phi_or (evt_phi_s—evt_.phi-ir )...
(evt_phi_s—evt-phi_or) (evt-phi-ir—evt-phi-or)];

[T, Y] = ode45(@ODE_EVT1, T,
is =Y(:,l:3);

iir =Y(:,4:6);

ior =Y(:,7:13);

phi_s =Y(:,14);

phi-ir =Y(:,15);

phi_or =Y(:,16);

theta-sir =Y(:,17)
theta-sor =Y(:,18)
theta_iror = Y(:,19

I

);

3’0);

Vs- = [sqrt(2)*Vs*sin(Y(:,14)+offset_s (1 ,1))...
sqrt(2)*Vs*sin(Y(:,14)+offset_s(2,1))...
sqrt(2)*Vs*sin(Y(:,14)+offset_s(3,1))];

Vir- = [sqrt(2)*Vir*sin(Y(:,15)+offset_s (1 ,1))...
sqrt(2)*Vir*sin(Y(:,15)+offset_s(2,1))...
sqrt(2)*Vir*sin(Y(:,15)+offset_s(3,1))];

Trs=—p*Lsir*((is (: ,1).*( iir (:,1)—0.5*iir(:,2) —0.5*iir (: ,3))...
+is (: ,2).*( iir(:,2)—0.5*iir(:,1)—0.5*iir (: ,3))...

+is(: ,3).*( iir(:,3)—-0.5*iir (: ,2)...

—0.5*iir (: ,1))).*sin(theta-sir)+...

(sqrt(3)/2)*(is(: ,1).*(iir(:,2)—iir (: ,3))
+is(: ,2).*(iir(:,3)—iir (: ,1))...
+is(: ,3).*(iir(:,1)—iir (: ,2)))...
.*cos(theta_sir));

99

'I‘ml =—p*Liror*(( iir (: ,1).*(cos(offset-iror (1,1))a1zior (: ,1)...
+cos(offset_iror(1,2))*ior(z ,2)...
+cos(offset-iror(1,3))*ior(z ,3)...
+cos(offset_iror(1,4))*ior(: ,4)...
+cos(offset_iror(1,5))akior(z ,5)...
+cos(offset-iror(1,6))*ior(: ,6)...
+cos(offset_iror(1,7))*ior(z ,7))...

+iir(z,2).*(cos(offset-iror(2,1))*ior(z ,1)...
+cos(offset-iror (2,2))*ior (: ,2)...
+cos(offset-iror(2,3))*ior(: ,3)...
+cos(offset-iror (2,4))*ior (: ,4)...
+cos(offset_iror(2,5))*ior(: ,5)...
+cos(offset-iror (2,6))*10r (: ,6)...
+cos(offset_iror(2,7))a1zior(: ,7))...

+iir (: ,3).*(cos(offset_iror (3,1))arior (: ,1)...
+cos(offset_iror(3,2))akior(: ,2)...
+cos(offset_iror (3,3))*ior (: ,3)...
+cos(offset-iror (3,4))*ior (: ,4)...
+cos(offset-iror(3,5))*ior(: ,5)...
+cos(offset_iror (3,6))*10r (: ,6)...
+cos(offset_iror (3,7))*ior(: ,7)))..
.*sin(theta-iror)+...

(iir(:,1).*(sin(offset_iror(1,1))*ior(z ,1)...
+sin(offset_iror(1,2))*ior(:
+sin(offset_iror(l,3))*ior(:
+sin(offset..iror(1,4)):kior(:
+sin(offset_iror(1,5))*ior(z
+sin(offset_iror(1,6))*ior(:
+sin(offset-iror(1,7))*ior(z

+iir(:,2).*(sin(offset-iror(2,1))*ior(z
+sin(offset-iror(2,2))*ior(z
+sin(offset-iror(2,3))*ior(z
+sin(offset-iror(2,4))*ior(:
+sin(offset-iror(2,5))*ior(z
+sin(offset_iror(2,6))*ior(:
+sin(offset_iror(2,7))akior(:
+iir(:,3).*(sin(offset-iror(3,1))*ior(:,
+sin(offset_iror(3,2))*ior(:
+sin(offset_iror(3,3))*ior(z
+sin(offset_iror(3,4))*ior(:

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+sin(offset_iror(3,5)):rior(: ,5)...
+sin(offset_iror(3,6))*ior(z ,6)...
+sin(offset-iror(3,7))akior(z ,7)))...
.*cos(theta_iror));

Trr12 =—p*Lsor*((is(:,1).*(cos(offset_iror(1,1))*ior(z ,1)...
+cos(offset_iror(1,2))*ior(:
+cos(offset_iror(1,3))*ior(z
+cos(offset_iror(1,4))*ior(:
+COS(OffSGt-lI'OI'(1,5))*101’(I
+cos(offset_iror(1,6))aior(z
+cos(offset-iror(1,7))*ior(z

+is(:,2).*(cos(offset_iror(2,1))*ior(: ,
+cos(offset_iror(2,2))*ior(:
+cos(offset_iror(2,3))*ior(z
+cos(offset_iror(2,4))z1cior(z
+cos(offset-iror(2,5))*ior(:
+cos(offset-iror(2,6))akior(z
+cos(offset_iror (2,7))*ior (: ,7

+is(:,3).*(cos(offset_iror(3,1))*ior(: ,
+cos(offset..iror(3,2))*ior(z
+cos(offset_iror(3,3))*ior(:
+cos(offset-iror(3,4))*ior(z
+cos(offset-iror(3,5))*ior(: ..
+cos(offset-iror(3,6))*ior(: ,6)...
+COS(OffS€t_iI‘OI‘(3,7))*10r(i
.*sin(theta-sor)+...

(is(:,1).*(sin(offset_iror(1,1))*ior(: ,1)...
+sin(offset_iror(1,2))*ior(: ,2)...
+sin(offset_iror(1,3))*ior(: ,3)...
+sin(offset_iror(1,4))akior(: ,4)...
+sin(offset-iror(1,5))*ior(z ,5)...
+sin(offset-iror(1,6))*ior(: ,6)...
+sin(offset_iror(1,7))*ior(z ,7))...

+is(:,2).*(sin(offset_iror(2,1))*ior(: ,1)...
+sin(offset_iror(2,2))akior(z ,2)...
+sin(offset_iror (2,3))*ior (: ,3)...
+sin(offset_iror (2,4))a: ior (: ,4)...
+sin(offset_iror(2,5))*ior(: ,5)...
+sin(offset_iror(2,6))*ior(: ,6)...
+sin(offset-iror(2,7))*ior(z ,7))...

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+is (: ,3).*(sin(offset-iror (3,1))*ior (: ,1)...
+sin(offset_iror (3,2)):rior (: ,2)...
+sin(offset-iror (3,3))*ior (: ,3)...
+sin(offset_iror (3,4))*ior (: ,4)...
+sin(offset-iror (3,5))*ior (: ,5)...
+sin(offset-iror (3,6))*ior (: ,6)...
+sin(offset_iror (3,7))*ior (: ,7)))...
.*cos(theta_sor));

% Create Time Based Figure

figurel '2 figure;

% Create Plot 1

axesl = axes(’Position’ ,[0.13 0.7093 0.775 0.2157] ,...
’Parent ’ ,figurel );

box(’on’);

hold(’all’);

xlabel( ’Time..(sec) ’ );

ylabel( ’Current..(A) ’);

title ( ’ StatorsCurrents—sPhasesAchaB ’ );

xlim([19.8 20]);

plotl = plot(T,is(:,1), ’Parent’, axesl);

plot2 = plot(T,is(:,2), ’Parent’, axesl);

set(plotl , ’Color ’ ,[0 0 0]);

set(plot2, ’Color’,[0.3137 0.3137 0.3137], ’LineStyle’,’—’);

% Create legend

legendl = legend(axes1,{ ’I_{as}’,’I_{bs}’});

70 Create Plot 2

axesZ = axes(’Position’ ,[0.13 0.4096 0.775 0.2157] ,...
’Parent ’,figure1);

box(’on’);

hold(’all ’);

x1abel( ’Time-(sec) ’ );

ylabel( ’Current..(A) ’ );

title ( ’Outer-Rotor..Current..—..Phase..A..&..B’ );

xlim([15 20]);

plot3 = plot (T, ior (: ,1), ’Parent ’ , axes2 );

plot4 = plot (T, ior (: ,2), ’Parent ’ , axesZ);

set(plot3 , ’Color ’ ,[0 0 0]);

102

set(plot4, ’Color’,[0.3137 0.3137 0.3137], ’LineStyle’,’—’);
70 Create legend
legend2 = legend(axes2,{ ’I-{aor}’,’I-{bor}’});

axes3 = axes(’Position’ ,[0.13 0.11 0.775 0.2157] ,...
’Parent’,figure1 );

box(’on’);

box(’on’);

hold(’all ’);

xlabel( ’Time..(sec) ’);

ylabel( ’Current..(A) ’ );

title ( ’Inner..RotorsCurrent..—..Phase..A..&..B’ );

x1im([19.8 20]);

plot5 = plot(T,iir(:,1), ’Parent’, axes3);

plot6 = plot(T,iir(:,2), ’Parent’, axes3);

set(plot5 , ’Color ’ ,[0 0 0]);

set(plot6, ’Color’,[0.3137 0.3137 0.3137], ’LineStyle’,’—’);

% Create legend

legend3 = legend(axes3,{’I_{air}’,’I_{bir}’});

figure3 2 figure;

% Create Plot 4

axes4 = axes(’Position ’ ,[0.13 0.7093 0.775 0.2157] ,...
’Parent ’ , figure3 );

box(’on’);

hold( ’all ’);

title ( ’TorqueaInnersAirgap ’ );

xlabel( ’Time..(sec) ’);

ylabel( ’Torque..(Nm) ’);

xlim([O 5]);

plot7 = plot(T,Tm1(: ,1), ’Parent ’, axes4);

% Create Plot 5

axes5 = axes(’Position’ ,[0.13 0.4096 0.775 0.2157] ,...
’Parent ’ , figure3 );

box(’on’);

hold(’all ’);

xlabel( ’Time..(sec)’);

ylabel( ’Torque...(Nm) ’);

103

title ( ’TorquesStator..andUInnerHRotor ’ );
xlim([0 5]);
plot8 = plot(T,Trs(: ,1) , ’Parent’, axes5);

% Create Plot 6

axes6 = axes(’Position’ ,[0.13 0.11 0.775 0.2157] ,...
’Parent ’ ,figure3 );

box(’on’);

hold( ’all ’);

title ( ’TorquesOutersAirgap ’ );

xlabe1( ’Time..(sec) ’ );

ylabel( ’Torque...(Nm) ’ );

xlim([O 5]);

plot9 = plot(T,Tm2(: ,1) , ’Parent ’ , axesfi);

%Create Frequency Based figure
figure2 2 figure;

f = 1000*(0z200)/1000;

% Create Plot 7

axes7 = axes(’Position ’ ,[0.13 0.7106 0.3347 0.211] ,...
’Parent’,figure2);

box(’on’);

hold(’all ’);

title ( ’Frequencyscontentsofsias ’)

xlabel( ’frequency...(Hz) ’)

%Find major frequency components

F = fft(is (: ,1) ,1000);

Pff = F.* conj (F)/1000;

plot10 = plot(f,Pff(1:201), ’Parent’, axes7);

% Create Plot 8

axes8 = axes( ’Position ’ ,[0.5703 0.7106 0.3347 0.211] ,...
’Parent ’ , figure2 );

box(’on’);

hold(’all ’);

title ( ’Frequencyscontentsofsiaor ’)

xlabel( ’frequency...(Hz) ’)

%Find major frequency components

104

F = fft(ior(:,1),1000);
Pff = F.=I= conj (F)/1000;
plotll = plot(f,Pff(1:201), ’Parent’, axes8);

% Create Plot 9

axesQ = axes(’Position’ ,[0.13 0.4109 0.3347 0.211] ,...
’Parent ’ , figure2 );

box(’on’);

hold(’all ’);

title ( ’Frequencyscontentsofsiair ’);

xlabel( ’frequency...(Hz) ’)

%Find major frequency components

F = fft(iir (: ,1) ,1000);

Pff = F.* conj(F)/1000;

plot12 = plot(f,Pff(1:201), ’Parent’, axesQ);

% Create Plot 10

axes10 = axes(’Position’ ,[0.5703 0.4109 0.3347 0.211] ,...
’Parent’,figure2);

box(’on’);

hold(’all ’);

title ( ’Frequencyscontent..ofsVas ’)

xlabel( ’frequency..(Hz) ’)

%Find major frequency components

F = fft(Vs_(: ,1) ,1000);

Pff = F.* conj(F)/1000;

plot13 = plot(f,Pff(1:201), ’Parent’, axes10);

70 Create Plot 12

axesl2 = axes(’Position’ ,[0.5703 0.11 0.3255 0.2157] ,...
’Parent’,figure2);

box(’on’);

hold(’all ’);

title ( ’Frequencyscontent..ofsVair ’)

xlabel( ’frequency..(Hz) ’)

%Find major frequency components

F = fft(Vir_(:,1),1000);

Pff = F.* conj(F)/1000;

plot15 = plot(f,Pff(1:201), ’Parent’, axesl2);

105

A.6 Example Test M-Code - ODE_EVT1.m

Files of this type, along with files similar to ”EVT_ODE_Ex1.m”, were used to
simulate the EVT for the 9 model verification tests. All of the other files would
be similar with modifications made as were listed in chapter 7. These two files are

examples of the code used to simulate Test 1.

function dy = ODE_EVT1(t ,y)
dy=zeros(19 ,1);

global p Rs Rir Ror Lls Llir Llor Lsor Lsir Liror Vs Vir Vor wm2
global wm1 wslip Rend Lend Lor Lir Ls ws wir wor Lsend Lirend

offset_s = [0 2 —2; —2 0 2; 2 —2 0]*pi/3;
offset..ir = [0 2 —2; —2 0 2; 2 -2 0]*pi/3;
offset_or = [ 0 2 4 6 —6 —4 —2;

2 4 6 —6 —4 —2 O;]*pi/7;
offset_sor = offset_s(:,1)*[1 1 1 1 1 1 1]...
+ [1 1 1]’ * offset-or(1,:);

offset_iror = offset_ir(:,1)*[1 1 1 11 1 1]..
+[11 1]’ * Offset_0r(1,:);

offset_sir = offset_s(:,1)*[1 1 1]...
+ [1 1 1]’*offset_ir(1,:);

VS_ 2 sqrt(2)*Vs*sin(y(14)+offset_s (: ,1));
Vir- = sqrt(2)*Vir*sin(y(15)+offset_ir (: ,1));
Vor- = zeros(7, 1);

V- = [Vs_;Vir_;Vor_];

106

% Setup the resitence matrices
Rs- = Rs =1: eye(3);

R11_ = (Rir+20e3)*eye(3);

Ror_p = Ror =1: eye(7);

Ror-b = circshift(Ror_p,[0,-1]);
Ror_.f = circshift(Ror_p,[0 ,1));
Rend- (2*Rend/7) * eye(7);
Ror- = 2*Ror-p-Ror-b—Ror-f+Rend_;

Ls- = Ls * cos(offset-s)+(Lls+Lsend)*eye(3);

Lir- = Lir =1: cos(offset_ir)+(Llir+Lirend)*eye(3);
Lend_= ((2*Lend/7)+Llor) * eye(7);

Lor- = Lend_+Lor*cos(offset_or);

Lsir- = Lsir .* [cos(y(17)+offset_sir)l;

3

Lirs- = Lsir- ;

dLsir- = —Lsir*(cos(offset_sir)*sin(y(17))...
+sin(offset_sir)*cos(y(17)));
dLirs- = dLsir- ’;

Lsor- = Lsor .* [cos(y(18)+offset_sor )];

9

Lors- = Lsor- ;

dLsor- = —Lsor*(cos(offset-sor)*sin(y(18))...
+sin(offset-sor)*cos(y(18)));
dLors- = dLsor- ’;

Liror- = Liror .* [cos(y(19)+offset_iror )];
Lorir- = Liror- ’;
dLiror- = —Liror*(cos(offset_iror)*sin(y(19))...

+sin(offset_iror)*COS(y(19)));
dLorir_ = dLiror- ’;

A = [Rs_ dLsir_*p*wm1 dLsor_*p*wm2;...

dLirs-*p*wm1 Rir- dLiror_*p*(wm2—wml);...
dLors_*p*wm2 dLorir_*p*(wm2—wm1) Ror-];

107

B = [Ls_ Lsir- Lsor_;...
Lirs- Lir- Liror-;...

Lors- Lorir- Lor- ];

dy(1:13)=inv(B)*(V- — A*y(1:13));

dy(14) = ws; %dphi_s
dy(15) = wir; %dphi-ir
dy(16) = wor; %dphi_or

dy(17) = p*wm1;
dy(18) = p*WIn.2;
dy(l9) = p*(wm2-—wm1);

108

Appendix B Geometry

 

 
     

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111

BIBLIOGRAPHY

112

BIBLIOGRAPHY

[1] T. Backstrom, C. Sadarangani and S. Ostlund. Integrated Energy Transducer for
Hybrid Electric Vehicles. Eighth International Conference on Electrical Machines
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[2] CC. Chan. The State of the Art of Electric, Hybrid, and Fuel Cell Vehicles.
Proceedings of the IEEE. pages 704—718, April 2007.

[3] D.K. Cheng. Field and Wave Electromagnetics. Addison-Wesley Publishing Com-
pany, 1992.

[4] M.J. Hoeijmakers and J .A. Ferreira. The Electric Variable Transmission. IEEE
Industry Applications Conference, pages 2770—2777, October 2004.

[5] RC. Krause, O. Wasynczuk, and SD. Sudhoff. Analysis of Electric Machinery
and Drive Systems. IEEE Press, 2002.

[6] J .M. Miller. Hybrid Electric Vehicle Propulsion System Architectures of the e—
CVT Type. IEEE Transactions on Power Electronics, pages 756—767, May 2006.

[7] AR. Mufioz and TA. Lipo. Complex Vector Model of the Squirrel-Cage Induction
Machine Including Instantaneous Rotor Bar Currents. IEEE Transactions on
Industry Applications, pages 1332—1340, November/ December 1999.

[8] E. Nordlund and C. Sadarangani. Four-quadrent Energy Transducer for Hybrid
Electric Vehicles. Record of the Industry Applications Conference, 2002. 37th IAS
Annual Meeting. Conference, pages 390—397, 2002.

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