© 2019 - Shivang

BRUSHLESS DOUBLY-FED RELUCTANCE MACHINE DRIVE FORTURBO-ELECTRIC DISTRIBUTED PROPULSION SYSTEMS

BY

- SHIVANG

THESIS

Submitted in partial fulfillment of the requirementsfor the degree of Master of Science in Electrical and Computer Engineering

in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2019

Urbana, Illinois

Adviser:

Assistant Professor Arijit Banerjee

ABSTRACT

Turbo-electric distributed propulsion systems are considered to be a crit-

ical enabler for low-carbon emission in the aircraft industry. A brushless

doubly-fed reluctance machine (BDFRM) is an attractive option to drive the

distributed propeller fans for these megawatt-scale turbo-electric propulsion

systems due to use of a partially-rated power converter, reduced maintenance,

and absence of permanent magnets. This thesis investigates the torque pro-

duction in BDFRM, discusses machine modeling and drive control architec-

ture, and reports on an initial sizing of a 1.5 MW motor. However, the

BDFRM has inherently poor torque density because of machine saturation,

even at low current-density, that offsets all the benefits. This thesis proposes

an approach to maximize the torque density by finding appropriate electri-

cal excitations on the two stator windings for a given machine dimension

while remaining within flux- and current-density limits. A single-objective

optimization problem is formulated. The obtained results prove that while

designs with equal electrical loadings on both stators, and an initial current

phase offset of π/2 between the two stators, may seem a good design ap-

proach, they are far from optimal. Our optimized solution establishes that

the phase offset of 2π/3 provides maximum torque capability for an identical

dimension. This procedure is validated using FEA simulations. Operating

with this design also leads to higher machine efficiency and better power fac-

tor on the secondary stator, thus reducing the converter rating. Finally, the

thesis concludes with a summary of findings and suggestions for future work

in this field.

ii

To my family, for their constant support.

And to the many friends, new and old, I have made along the way.

iii

ACKNOWLEDGMENTS

First and foremost, I would like to acknowledge my family who always have

been there, continuously supporting my education. Being so far away from

them has not always been easy but they have continued to have their faith

in me, motivate me for further success and always believe in me.

I would like to express my sincere gratitude to my supervisor Prof. Arijit

Banerjee for his patience, motivation and immense knowledge. The support,

guidance and academic freedom he provided allowed me to develop skills that

will surely help me become a successful engineer, leader, and mentor. His

careful and precious guidance was extremely valuable for my study. I was

also lucky to be a Teaching Assistant with him in ECE 464. Watching him

teach in the class so passionately helped me enhance my teaching skills.

I want to express my gratitude to Raymond Beach at NASA Glenn Re-

search Center for giving me the opportunity to work on such a cool project

in my first year of MS. The Grainger Center for Electric Machinery and Elec-

tromechanics has always been there for us to provide the financial support

we needed.

To the awesome power group, thank you for being great co-workers and

friends. The power group here at Illinois is something very unique and will

always hold a special place in my heart. You guys made sure that I stick

around for a couple of more years and do my PhD here at Illinois.

I perceive the opportunity to do my MS at UIUC as a big milestone in

my career development. I will strive to use skills and knowledge gained here

in the best possible way, and I will continue to work on improving them, in

order to attain my desired career objectives.

iv

TABLE OF CONTENTS

CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . 1

CHAPTER 2 STEADY-STATE ANALYSIS . . . . . . . . . . . . . . 62.1 Air-gap flux density distribution with different rotor structures 72.2 Machine inductances . . . . . . . . . . . . . . . . . . . . . . . 10

CHAPTER 3 DYNAMIC MODEL . . . . . . . . . . . . . . . . . . . 123.1 Model in ABC framework . . . . . . . . . . . . . . . . . . . . 133.2 α - β transformation . . . . . . . . . . . . . . . . . . . . . . . 153.3 d-q model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Torque computation . . . . . . . . . . . . . . . . . . . . . . . 193.5 Speed computation . . . . . . . . . . . . . . . . . . . . . . . . 21

CHAPTER 4 DRIVE CONTROL ARCHITECTURE . . . . . . . . . 224.1 Stator flux estimation . . . . . . . . . . . . . . . . . . . . . . . 224.2 Current controllers design . . . . . . . . . . . . . . . . . . . . 234.3 Speed controller . . . . . . . . . . . . . . . . . . . . . . . . . . 264.4 Reactive power controller . . . . . . . . . . . . . . . . . . . . . 26

CHAPTER 5 ELECTROMAGNETIC DESIGN METHODOLOGY . 285.1 D2l sizing derivation . . . . . . . . . . . . . . . . . . . . . . . 285.2 Dl sizing derivation . . . . . . . . . . . . . . . . . . . . . . . . 325.3 Example design of a 1.5 MW BDFRM . . . . . . . . . . . . . 33

CHAPTER 6 TORQUE DENSITY IMPROVEMENT . . . . . . . . 366.1 Optimization problem formulation . . . . . . . . . . . . . . . . 386.2 Impact of optimization on torque density . . . . . . . . . . . . 396.3 Impact of optimization on power converter size . . . . . . . . . 426.4 Impact of optimization on grid and converter power factor . . 46

CHAPTER 7 FINITE ELEMENT ANALYSIS AND RESULTS . . . 50

CHAPTER 8 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . 558.1 Summary and design insights . . . . . . . . . . . . . . . . . . 558.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

v

APPENDIX A ELECTROMAGNETIC TORQUE PRODUCEDBY BDFRM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

APPENDIX B DETERMINATION OF KP AND KI BASED ONFIRST-ORDER SYSTEM . . . . . . . . . . . . . . . . . . . . . . . 60

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

vi

CHAPTER 1

INTRODUCTION

The U.S. aviation industry produces 11% of total transportation-related

emissions domestically, and about 3% of the global CO2 emissions annu-

ally are produced by planes. Substantial improvements in air transportation

technologies continue to increase the efficiency of air transportation by mov-

ing passengers and cargo over the same distance with less fuel consumed and,

hence, fewer carbon emissions. The International Civil Aviation Organiza-

tion anticipates a 3x increase in the global international aviation emissions

by 2050 [1]. This exponential increase in the air travel has prompted a signif-

icant effort to mitigate the impact of commercial aviation on climate change.

At the request of the National Aeronautics and Space Administration

(NASA), the National Academies of Sciences, Engineering, and Medicine

convened a committee to develop a national research agenda for reducing

CO2 emissions from commercial aviation, focusing on single-aisle and twin-

aisle large commercial aircraft. They proposed four high-priority approaches

that could reduce CO2 emissions and could be implemented during the next

10 to 30 years [2]:

� Advances in aircraft–propulsion integration,

� Improvements in gas turbine engines,

� Development of turbo-electric distributed propulsion (TeDP) systems,

and

� Advances in sustainable alternative jet fuels.

This thesis focuses on the development of turbo-electric propulsion systems.

These propulsion systems employ gas turbines to drive electric generators,

which deliver power to several motors driving a set of distributed propeller

fans. The propeller thrust is regulated by controlling the individual motor

1

Figure 1.1: Turbo-electric propulsion system architecture [3].

speeds. Figure 1.1 shows this architecture. The electrical power system

decouples the turbine and propeller speed, such that both operate at the

speeds required for their efficient operation. NASA's N3-X is based on this

fully turbo-electric architecture. This architecture is estimated to reduce fuel

requirements by 70 percent as compared to a twin-aisle aircraft [3].

Most prospective turbo-electric system architectures have focused on using

a single-port motor, such as a permanent magnet synchronous motor, to

drive the propeller fans [4, 5]. This requires full-power rated power-electronic

converters to decouple the electric generator speed from the motor. The need

to handle megawatts of power makes these converters massive, complex, and

bulky. Figure 1.2 shows the weight distribution of various components for a

30,000 hp turbo-electric system [6], with power converters occupying a big

chunk of this share.

In contrast, NASA Convergent Aeronautics Solutions proposed the use of

Figure 1.2: Weight distribution of various components in a TeDP system [6].

2

a doubly-fed machine (DFM), both as a generator and a motor, by adopting

a high-voltage, wide-range variable frequency ac power distribution architec-

ture [7]. This architecture reduces the power-electronic converter requirement

by 85% as compared to the one with single-port machines — an advantage

highly relevant in megawatt-class aircraft applications. Figure 1.3 shows the

reduction in power converter size using a dual-port machine. DFMs require

brushes and slip rings to deliver power to rotating windings which leads to

poor reliability and frequent maintenance. This disadvantage creates a fun-

damental bottleneck with the proposed architecture.

A brushless doubly-fed reluctance machine (BDFRM) eliminates this bot-

tleneck by eliminating brushes and slip-rings and offers exciting trade-offs

from an electromechanical energy conversion system perspective [8, 9, 10].

From a machine perspective, BDFRM does not require permanent magnet,

thus increasing reliability and fault-tolerant capability [11]. From the power-

electronic drive perspective, a significant reduction in weight, size, and cost

is achieved as compared to conventional single-port machine [4]. A wide

speed range can also be accomplished by using a switched power converter

architecture without sacrificing the benefits of reduced power-electronic re-

quirements [12]. Figure 1.4 shows the proposed architecture for driving a

single propeller fan.

(a) Power converter processes full rated power Ps with a single-port machine

(b) Power converter processes partial power Pr with a dual-port machine

Figure 1.3: Reduction in the size of power converter using a dual-portmachine.

3

Figure 1.4: BDFRM based AC power architecture [8].

This thesis contributions are:

� The conventional D2l volumetric sizing equation and an unconventional

Dl sizing equations are derived, which are useful to get an initial esti-

mate of the motor size.

� BDFRM have inherently poor torque density because of under-utilized

machine magnetics in terms of achievable peak air-gap flux density.

This thesis takes an approach to maximize the torque density by finding

appropriate electrical excitations on the two stator windings for a given

machine dimension while remaining within flux- and current-density

limits.

� A design methodology for the power architecture of a BDFRM drive

is presented. The size of the drive-end power converter is computed

in terms of the converter current and voltage rating. Control of the

drive-end converter using field oriented control is also explained.

Chapter 2 discusses the production of torque in BDFRM and shows the

differences in air-gap flux density in a conventional uniform air-gap machine

and a non-uniform air-gap BDFRM. In order to develop the necessary control

algorithms, a machine model is essential. BDFRM is modeled in Chapter 3

using generalized machine model. The control drive architecture using field

oriented control approach is presented in Chapter 4. The basic D2l and Dl

sizing equations are derived in Chapter 5, which are used to design a 1.5 MW

BDFRM, capable of driving a propeller fan at 4000 rpm. The conventional

approach to maximizing torque density in a BDFRM has been to create two

orthogonal magnetic fields in the rotor reference frame using the primary and

4

secondary stator windings, similar to a uniform air gap sinusoidal machine

[8, 13]. In this design approach, the allowable current density is implicitly

restricted by the flux density limit rather than independent thermal design

considerations. Fundamentally, this unconventional limit on the allowable

current density is due to the non-sinusoidal air-gap flux density distribution

in a BDFRM, which comprises several spatial harmonics.

Chapter 6 opens up the design space by allowing the angle between the

two magnetic fields to be varied to increase the current density towards the

thermal-consideration-driven constraint. Intuitively, this angle is similar to

the definition of torque angle in a synchronous machine. An optimization

problem is formulated to maximize the output torque while choosing the

electrical loadings on both stators as design variables. Operating flux density,

current density, grid voltage, and machine dimensions are used to set the

design constraints. The machine torque capability is enhanced by 75% for an

identical machine dimension [14] while simultaneously improving the power

factor of the secondary stator, and hence reducing the power converter size.

The overall efficiency also increases with the proposed design. The simulation

results as carried out using ANSYS Maxwell are presented in Chapter 7 and

decisively prove the benefits proposed in theory. The conclusions and future

work are provided in Chapter 8.

5

CHAPTER 2

STEADY-STATE ANALYSIS

A brushless doubly-fed machine (BDFM) works on the principle of air-gap

flux modulation. The machine has two stator windings, commonly known

as primary or grid winding and secondary or control winding, with different

pole numbers. An induction-based or a reluctance-based rotor modulates

the air-gap field to couple the two stator windings [15]. These two classes

of machines have been denoted as brushless doubly-fed induction machine

(BDFIM) and brushless doubly-fed reluctance machine (BDFRM). The lat-

ter, with a radially laminated reluctance rotor with flux guides, will be con-

sidered here since it exhibits fewer magnetic losses [16]. Figure 2.1 shows the

drive architecture. The primary stator is directly connected to the ac sup-

ply, while the secondary stator is connected through a bi-directional power

converter.

The presence of two stator fields of different pole numbers, different fre-

quencies and possibly different sequences, and the resulting rotor fields pro-

duced by their interactions, make the BDFRM analysis comparatively more

Figure 2.1: Architecture of a brushless doubly-fed reluctance machine drive[12].

6

complex than that of induction and synchronous motors. The following as-

sumptions are made throughout this work:

� The magnetic core material has infinite permeability.

� Both the stator windings are excited by three-phase balanced sinusoidal

currents.

� Both the stator windings are also uniformly distributed in space so that

their winding function can be represented by sinusoidal functions.

� Slotting effects are neglected.

Section 2.1 discusses the air-gap flux density in BDFRM with both salient

pole and more commonly used radially laminated ducted rotor and shows the

differences in the spatial domain. The harmonic content in the air-gap flux

density is what makes BDFRM so different from other conventional machines

and is essential to understand and design the machine accordingly. The

machine inductances required for modeling and characterizing the machine

are computed in the following section.

2.1 Air-gap flux density distribution with different

rotor structures

This section explores the spatial frequency distribution of air-gap flux den-

sity with three different rotor structures: uniform air-gap, salient-pole, and

ducted rotor with a common stator winding. Rotor geometry plays a signif-

icant role in BDFRM design. Although the salient pole rotor is useful for

understanding the BDFRM operating principles, its performance is inferior

to an axially and radially laminated rotor [17, 18]. A radially laminated ro-

tor with flux guides, known as the ducted rotor, will be considered since it

exhibits lower magnetic losses than an axially laminated rotor [16].

A uniform air-gap machine with p poles excited using balanced three-phase

sinusoidal currents of amplitude I and frequency ω results in an air-gap flux

7

density given by

Buniform(t, φ) = Bmaxcos(ωt− p

2φ), where (2.1)

Bmax =µ0

gMmax =

3

2

µ0

g(4

π

kwN

p)I (2.2)

where g is the uniform air gap, kw denotes the winding factor and N repre-

sents the total number of series turns per phase. The flux density contains

one spatial frequency at p poles. For example, Fig. 2.2(a) shows the spatial

frequency distribution for an 8-pole stator winding. A BDFRM with a salient

(a) Uniform air-gap with rotorhaving 8 poles (b) Salient-pole rotor having 6 poles

(c) Ducted rotor having 6 poles

Figure 2.2: Spatial frequency distribution in the air-gap flux density due toan 8-pole stator winding using different rotor structures. The flux density is

normalized with Bmax =µ0

gMmax.

8

pole rotor of pr pole introduces a non-uniform air gap g, modeled as

g−1(φ, t) =1

G(1 + cospr(φ− wrmt)) (2.3)

where φ is the mechanical air-gap angle, wrm is the rotor mechanical speed

and G is the mean air gap. The rotor modulation creates two side-bands in

the air-gap flux density along with the fundamental component by exciting

the primary stator having pp poles [19]. The air-gap flux density distribution

is given by

Bsalient = Bmaxcos(ωpt−pp2φ)

+Bmax

2cos[(ωp − prωrm)t− (

pp2− pr)φ)]

+Bmax

2cos[(ωp + prωrm)t− (

pp2

+ pr)φ)]

(2.4)

The secondary stator is constructed either with pr −pp2

pole-pair or pr +pp2

pole-pair to couple with the primary stator. For example, Fig 2.2(b)

shows the spatial frequency distribution for an 8-pole primary stator and

6-pole salient rotor. The secondary stator must be either 4 or 20 to produce

steady torque. The uncoupled side-band contributes only to the leakage

flux, resulting in poor efficiency, torque density and power factor compared

to uniform air-gap machines.

In practice, a radially-laminated ducted rotor having pr poles improves the

efficiency and torque density compared to salient-pole and axially-laminated

rotors [20]. One such radially-laminated ducted rotor structure is shown in

Fig. 2.3. Assuming the iron is infinitely permeable, all the flux entering one

Figure 2.3: Flux paths in ducted rotor [13].

9

of the rotor ducts at an air-gap angle of φ exits from the same duct at φf

[13], where

φf = φ+2π

pr− 2mod(φ,

2π

pr) (2.5)

The air-gap flux density due to the primary stator winding having pp poles

is

Bp(φ) =µ0Mp

g

1

2[cos(

pp2φ)− cos(

pp2φf )] (2.6)

Inherent non-linearity in φf introduces spatial harmonics in the air-gap flux

density. The flux density distribution is expanded as a Fourier series to give

Bp(φ) =∞∑i=1

Bpicos(iφ) =µ0Mp

g

∞∑i=1

Cpicos(iφ) (2.7)

where Cpi are the Fourier coefficients. Figure 2.2(c) shows the spatial fre-

quency distribution of the flux density for an 8-pole primary stator and a

6-pole ducted rotor. Comparison of Fig. 2.2(b) and 2.2(c) shows that in a

ducted-rotor, additional harmonic poles are generated that do not obey the

general formulation given by (2.4), which is valid for salient-pole rotors. For

the current example with 8-pole primary stator and a 6-pole rotor, the most

dominant sideband appears at 4 poles. Thus having a secondary stator of 4

poles will lead to maximum torque capability.

2.2 Machine inductances

This section computes the self and mutual inductances of the primary and

secondary stator windings. These inductances are used to model the machine

and compute the base-line torque produced by the interaction between the

primary and secondary stator in Chapter 3.

Using (2.7) and (2.2), the air-gap flux density with the primary stator

A-phase excited is given by

BpA =µ0

g

( 4

π

kwNp

pp

)Ip

∞∑i=1

Cpicos(iφ) (2.8)

Considering the winding function of primary stator A-phase wpa to be sinu-

10

soidal, the self flux linkage ϕpp is calculated as

ϕpA,pA = rl

∫ 2π

0

BpAwpAdφ (2.9)

=µ0rl

g

( 4

π

Np

pp

)2Ip

∫ 2π

0

∞∑i=1

Cpicos(iφ)cos(pp2φ)dφ (2.10)

where r and l are the air-gap radius and stack length, respectively. The

above integral is non-zero for only i = pp/2 simplifying to

ϕpA,pA =µ0rl

g

( 4

π

Np

2pp

)2(πCpp)Ip (2.11)

Including the leakage, the primary stator A-phase self-inductance is given by

LpA,pA =ϕpA,pAIp

+ Lplk =16µ0rl

πg

(Np

pp

)2Cpp︸ ︷︷ ︸

Lpp

+Lplk (2.12)

The mutual flux linkage between the primary A and primary B phase is given

by [21]

ϕpA,pB = rl

∫ 2π

0

BpAwpBdφ (2.13)

=µ0rl

g

( 4

π

Np

pp

)2Ip

∫ 2π

0

∞∑i=1

Cpicos(iφ)cos(pp2φ− 2π/3)dφ, (2.14)

=µ0rl

g

( 4

π

Np

pp

)2(πCpp)Ipcos(2π/3) (2.15)

The mutual inductance between the primary A and primary B phase is there-

fore given by

LpA,pB =ϕpA,pBIp

= −1

2

16µ0rl

πg

(Np

pp

)2Cpp = −1

2Lpp (2.16)

The secondary self and mutual inductances are computed similarly. The

mutual inductance between primary and secondary stator phase is a function

of rotor position, and its maximum value is expressed as

Lpsmax =16µ0rl

πg

(NpNs

ppps

)Cps (2.17)

11

CHAPTER 3

DYNAMIC MODEL

This chapter discusses a generalized machine model for BDFRM, which will

be used to control the motor and to ascertain the size of the power electronic

converter. A conventional approach to modeling a machine incorporates

computation of variables such as currents, torque and speed for a given set of

input voltages and load torque. A similar approach is used for a BDFRM, as

shown in Fig. 3.1. For a given set of primary stator voltages vpABC, secondary

stator voltages vsABCand load torque τL, the primary stator currents ipABC

,

secondary stator currents isABC, electromagnetic torque τe and the mechanical

speed of the machine ωrm need to be computed. A generalized machine model

looks like that shown in Fig. 3.2. The first step is to compute both the stator

fluxes from the voltages fed to the machine. Thus a relation for flux in terms

of voltages is required. The second step is to calculate currents from fluxes

followed by the electromagnetic torque. Last, the speed of the machine is

calculated using the mechanical equation of the machine.

While the machine can be modeled in the three-phase or ABC domain, it

is difficult to control the machine in this domain. An easier way to control a

machine is to make all the quantities dc, which can be achieved by transform-

ing all the variables to a reference frame rotating at the same speed as the

variables. Thus before moving on to Step 1 of calculating of fluxes, all the

variables should be transformed into their respective reference frames. The

objective of this chapter is to develop the BDFRM model in these rotating

Figure 3.1: BDFRM model showing inputs and outputs.

12

Figure 3.2: Flow-chart of machine modeling using generalized machinetheory.

reference frames and derive all the relations, as shown in Fig. 3.2.

3.1 Model in ABC framework

The stator voltage equations for both primary and secondary stator windings

of BDFRM are expressed as:

vpABC= RpipABC

+ pϕpABC(3.1)

vsABC= RsisABC

+ pϕsABC(3.2)

where all the matrices are given by

vpABC=

vpAvpBvpC

; Rp =

Rp 0 0

0 Rp 0

0 0 Rp

; ipABC=

ipAipBipC

; ϕpABC=

ϕpAϕpBϕpC

vsABC=

vsAvsBvsC

; Rs =

Rs 0 0

0 Rs 0

0 0 Rs

; isABC=

isAisBisC

; ϕsABC=

ϕsAϕsBϕsC

The flux linkages for both primary and secondary stator windings can be

written as:

ϕpABC= L[p,ABC−p,ABC]ipABC

+ L[p,ABC−s,ABC]isABC(3.3)

ϕsABC= L[s,ABC−p,ABC]ipABC

+ L[s,ABC−s,ABC]isABC(3.4)

13

where the inductance matrices are given by:

L[p,ABC−p,ABC] =

LpA,pA LpA,pB LpA,pC

LpB,pA LpB,pB LpB,pC

LpC,pA LpC,pB LpC,pC

(3.5)

=

Lpp + Lplk −Lpp

2−Lpp

2

−Lpp2

Lpp + Lplk −Lpp2

−Lpp2

−Lpp2

+ Lplk Lpp + Lplk

(3.6)

L[s,ABC−s,ABC] =

LsA,sA LsA,sB LsA,sC

LsB,sA LsB,sB LsB,sC

LsC,sA LsC,sB LsC,sC

(3.7)

=

Lss + Lslk −Lss

2−Lss

2

−Lss2

Lss + Lplk −Lss2

−Lss2

−Lss2

+ Lslk Lss + Lslk

(3.8)

L[p,ABC−s,ABC] =

LpA,sA LpA,sB LpA,sC

LpB,sA LpB,sC LpB,sC

LpC,sA LpC,sB LpC,sC

= L[s,ABC−p,ABC] (3.9)

=

cos(θr) cos(θr − 2π/3) cos(θr + 2π/3)

cos(θr − 2π/3) cos(θr + 2π/3) cos(θr)

cos(θr + 2π/3) cos(θr) cos(θr − 2π/3)

Lpsmax

(3.10)

where θr is the rotor electrical angle and the inductance Lpp, Lss and Lpsmax

values as derived in the Chapter 2 are given by:

Lpp =16µ0rl

πg

(Np

pp

)2Cpp (3.11)

Lss =16µ0rl

πg

(Ns

ps

)2Css (3.12)

Lpsmax =16µ0rl

πg

(NpNs

ppps

)Cps (3.13)

14

3.2 α - β transformation

The next step is to convert the three phase equations into α-β reference frame,

or express the equations using space vector approach. The space vector for

the primary winding is constructed from the three phases as follows:

vp =2

3(vpA + avpB + a2vpC )

ip =2

3(ipA + aipB + a2ipC )

ϕp

=2

3(ϕpA + aϕpB + a2ϕpC )

(3.14)

Similarly for the secondary winding:

vs =2

3(vsA + avsB + a2vsC )

is =2

3(isA + aisB + a2isC )

ϕs

=2

3(ϕsA + aϕsB + a2ϕsC )

(3.15)

where a = ej2π/3. As

cos(θ) =ejθ + e−jθ

2

cos(θ − 2π

3) =

a2ejθ + ae−jθ

2

cos(θ +2π

3) =

aejθ + a2e−jθ

2

(3.16)

the inductance matrix L[p,ABC−s,ABC] (= L[s,ABC−p,ABC]) derived in Section

3.1 can be converted to

L[p,ABC−s,ABC] =Lpsmax

2

1 a2 a

a2 a 1

a 1 a2

ejθr +Lpsmax

2

1 a a2

a a2 1

a2 1 a

e−jθrExpanding ϕpA from (3.3),

ϕpA = (Lpp + Lplk)ipA −1

2LppipB −

1

2LppipC

+1

2Lpsmax [(isA + a2isB + aisC )ejθr + (isA + aisB + a2isC )e−jθr ]

(3.17)

15

For the first part, since there are no zero sequence currents, ipA+ipB +ipC = 0,

i.e. ipB + ipC = −ipA so that the first term becomes:

(Lpp + Lplk)ipA −1

2Lpp(ipB + ipC ) = (

3

2Lpp + Lplk)ipA (3.18)

For the second part, using

(isA + aisB + a2isC ) =3

2is

(isA + a2isB + aisC ) =3

2i∗s

(3.19)

in (3.17) to get

[(isA + a2isB + aisC )ejθr + (isA + aisB + a2isC )e−jθr ] =3

2[i∗se

jθr + ise−jθr ]

Combining both and using them in (3.17):

ϕpA = (3

2Lpp + Lplk)ipA +

3

2

1

2Lpsmax [i∗se

jθr + ise−jθr ] (3.20)

The same procedure is implemented for ϕpB and ϕpC to get

ϕpB = (3

2Lpp + Lplk)ipB +

3

2

1

2Lpsmax [a2i∗se

jθr + aise−jθr ] (3.21)

ϕpC = (3

2Lpp + Lplk)ipC +

3

2

1

2Lpsmax [ai∗se

jθr + a2ise−jθr ] (3.22)

Using these expressions of ϕpA , ϕpB and ϕpC in (3.14):

ϕp

= (3

2Lpp + Lplk)ip +

3

2Lpsmaxi

∗sejθr (3.23)

Similarly for the secondary stator winding:

ϕs

= (3

2Lss + Lslk)is +

3

2Lpsmaxi

∗pejθr (3.24)

16

Since L[p,ABC−s,ABC] = L[s,ABC−p,ABC], a similar expression is seen for both

the stator windings. Assuming

3

2Lpp + Lplk = Lp

3

2Lss + Lslk = Ls

3

2Lpsmax = Lps

(3.25)

the following equation is obtained:

ϕp

= Lpip︸︷︷︸ϕpp

+Lpsi∗sejθr︸ ︷︷ ︸

ϕps

(3.26)

The role of ejθr in (3.26) is to convert the frequency from secondary to pri-

mary. The self flux linkage ϕpp

and mutual flux linkage ϕps

are given by

ϕpp

= Lpip = LpIpejωpt; and (3.27)

ϕps

= Lpsi∗sejθr = LpsIse

−j(ωst+δ)ejωrt = LpsIsej(ωpt−δ) (3.28)

Both the flux linkage space vectors ϕpp and ϕps rotate with the same angular

velocity ωp, the latter lagging by δ. The net primary stator flux is given by

ϕp

= ϕpej(ωpt−θ0); where (3.29)

tanθ0 =ϕpssinδ

ϕpp + ϕpscosδ=

LpsIssinδ

LpIp + LpsIscosδ(3.30)

Similarly the secondary stator flux is given by

ϕs

= Lsis︸︷︷︸ϕss

+Lpsi∗pejθr︸ ︷︷ ︸

ϕsp

(3.31)

Using (3.1), (3.2), (3.14) and (3.15), the voltage equations are written as:

vp = Rpip + pϕp

(3.32)

vs = Rsis + pϕs

(3.33)

17

3.3 d-q model

As the primary and secondary windings are fed with different frequency ex-

citation currents, two different reference frames are required to obtain the

BDFRM d-q model [22]. The two d-q frames for BDFRM are, however, not

independent of each other. Aligning the d-axis for primary stator along pri-

mary stator flux also fixes the d-axis for secondary stator, although both the

stators move at different speeds. Multiply (3.26) by e−j(ωpt−θ0) to get:

ϕpr

= Lpipr + Lpsi∗sr; where (3.34)

ipr = ipe−j(ωpt−θ0) = Ipe

jθ0 = Ipcosθ0︸ ︷︷ ︸ipd

+j Ipsinθ0︸ ︷︷ ︸ipq

; and (3.35)

isr = ise−j(ωst+θ0) = Ise

j(δ−θ0) = Iscos(δ − θ0)︸ ︷︷ ︸isd

+j Issin(δ − θ0)︸ ︷︷ ︸isq

(3.36)

The primary stator flux initial phase angle θ0 in (3.36) can be substituted

using (3.30) to get:

δ = arctan( isqisd

)+ arctan

(Lps

√i2sd + i2sqsinδ

Lp√i2pd + i2pq + Lps

√i2sd + i2sqcosδ

) (3.37)

Similarly the secondary stator flux is calculated as

ϕsr

= Lsisr + Lpsi∗pr (3.38)

The d-q equations for the BDFRM are developed from (3.34) and (3.38) by

breaking the equations into real and imaginary parts, and then writing these

components as separate equations. This leads to four equations, two for the

d-axis and two for the q-axis:

ϕp = Lpipd + Lpsisd (3.39)

0 = Lpipq − Lpsisq (3.40)

ϕsd = Lsisd + Lpsipd (3.41)

ϕsq = Lsisq − Lpsipq (3.42)

18

since ϕpd = ϕp and ϕpq = 0 in primary stator flux reference frame. To get the

primary stator voltage equation in the primary stator flux reference frame,

multiply (3.32) by e−j(ωpt−θ0) to get:

vpr = Rpipr + pϕpr− ϕ

ppe−j(ωpt−θ0) (3.43)

= Rpipr + pϕpr

+ jωpϕpr (3.44)

Similarly the secondary stator voltage is calculated as

vsr = Rsisr + pϕsr

+ jωsϕsr (3.45)

The d-q voltage equations for the BDFRM are developed from (3.44) and

(3.45) by breaking the equations into real and imaginary parts, and then

writing these components as separate equations.

vpd = Rpipd + pϕp (3.46)

vpq = Rpipq + ωpϕp (3.47)

vsd = Rsisd + pϕsd − ωsϕsq (3.48)

vsq = Rsisq + pϕsq + ωsϕsd (3.49)

3.4 Torque computation

Similar to a conventional machine, the torque for this machine is derived

using virtual work function. The co-energy function for an electro-mechanical

system is given by:

Eco =1

2[i]T [L][i] (3.50)

where the [i] and [L] matrices are given by

[i]T = [ipA , ipB , ipC , isA , isB , isC ] (3.51)

[L] =

[L[p,ABC−p,ABC] L[p,ABC−s,ABC]

L[s,ABC−p,ABC] L[s,ABC−s,ABC]

]Since all the matrices L[p,ABC−p,ABC], L[s,ABC−s,ABC], L[p,ABC−s,ABC] and L[s,ABC−p,ABC]

are 3*3 matrices, [L] is a 6*6 matrix. From this co-energy function, torque

19

can be derived since

τe =∂Eco∂θrm

∣∣∣i=constant

= pr∂Eco∂θr

∣∣∣i=constant

(3.52)

Using the value of co-energy from (3.50),

τe =1

2pr[i]

T[ dLdθr

][i] (3.53)

Computation ofdL

dθrand torque τe is shown in Appendix A.1 and leads to

τe =1

2pr[i]

T[ dLdθr

][i] = j

3

4prLps[i

∗pri∗sr − iprisr] (3.54)

The expression derived above for torque is in terms of primary and secondary

winding currents. Alternatively, this can also be written in terms of primary

stator flux and secondary stator currents as:

τe =j3

4prLps[

1

Lp(ϕ∗

pr− Lpsisr)i∗sr −

1

Lp(ϕ

pr− Lpsi∗sr)isr]

τe =j3

4prLpsLp

[ϕ∗pri∗sr − Lpsi2sr − ϕprisr + Lpsi

2sr]

τe =j3

4prLpsLp

[ϕ∗pri∗sr − ϕprisr]

Using ϕpr

= ϕpd + jϕpq and isr = isd + jisq in the above expression:

τe =j3

4prLpsLp

[(ϕpd − jϕpq)(isd − jisq)− (ϕpd + jϕpq)(isd + jisq)]

τe =3

2prLpsLp

[ϕpdisq + ϕpqisd]

(3.55)

Aligning the d-q frame with that of primary flux vector makes ϕpq = 0,

simplifying the equation to:

τe =3

2prLpsLp

[ϕpdisq] =3

2prLpsLp

[ϕpisq] (3.56)

Since the primary winding is connected to the grid, ϕp is constant and thus

by controlling secondary winding q-axis current isq, the desired torque can

be achieved.

20

3.5 Speed computation

The final step is to compute the rotor speed. The mechanical rotor speed

ωrm is calculated using

τe − τL = Bωrm + Jdωrmdt

(3.57)

where

τe: Electromagnetic torque produced by the machine

τL: Load torque

B: Damping coefficient

J : Machine inertia constant

ωrm: Rotor mechanical speed

To get the rotor electrical speed ωr, multiply both sides of (3.57) by pr to

get

pr(τe − τL) = Bprωrm + Jprdωrmdt

pr(τe − τL) = Bωr + Jdωrdt

(3.58)

21

CHAPTER 4

DRIVE CONTROL ARCHITECTURE

Various control methodologies have been investigated for the BDFRM, in-

cluding scalar control [23], [24], field-oriented control (FOC) [24, 25, 26], and

direct torque control (DTC) [27]. The conventional FOC approach, as shown

in Fig. 4.1, has been used for this study. The machine d-q model developed

in Chapter 3 will be used to develop the necessary control of secondary stator

through a drive end converter. Since the machine model is based on primary

stator flux reference frame, the calculation of the magnitude and phase angle

of primary stator flux is done first. This flux angle along with the rotor an-

gle is used to transform secondary stator variables into d-q reference frame,

required by the two current controllers. The design of both the secondary

stator current controllers is discussed next followed by the speed and reactive

power controllers.

4.1 Stator flux estimation

The primary stator flux ϕp along with the flux angle θp (with respect to

primary stator A-phase axis) and flux frequency wp are estimated using the

measured values of ac source voltage vp and current ip connected to three

phases of the primary stator. The computed primary stator flux angle is

used to transform the variables to primary stator flux reference frame. Using

(3.32), the primary stator flux space vector is given by

ϕp

=

∫(vp −Rpip)dt (4.1)

For practical implementation, the integrator is replaced by a low-pass filter

with a suitable time-constant to avoid problems with measurement offset and

drift in the integral computations [28]. The cutoff frequency of these low-pass

22

Figure 4.1: Field-oriented control architecture for BDFRM.

filters must be well below the signal frequency. From the space phasor ϕp,

both the flux magnitude ϕp and flux angle θp are calculated. The primary

stator flux frequency wp is computed by integrating the flux angle θp. The

flux frequency wp is used in the feed-forward terms by both the secondary

stator current controllers.

4.2 Current controllers design

The secondary stator-side inverter control requires a feedback loop to control

the d-axis and q-axis secondary stator currents. The control of q-axis sec-

ondary stator current followed by the controller for d-axis secondary stator

current is derived in this section. Using (3.40) and (3.42), ipq is eliminated

to get

ϕsq = (Ls −L2ps

Lp)isq (4.2)

23

Figure 4.2: Block diagram for the design of q-axis current controller: Kp

and Ki are chosen based on the first-order plant model.

Using this value of ϕsq in the dynamic equation of (3.49):

vsq = Rsisq +dϕsqdt− (wp − wr)ϕsd (4.3)

= Rsisq + (Ls −L2ps

Lp)disqdt− (wp − wr)ϕsd (4.4)

Converting into frequency or s-domain:

isq(s) =vsq(s) + (wp − wr)ϕsd(s)

rs + s(Ls −L2ps

Lp)

(4.5)

The drive vsq is provided by the inverter connected to the secondary stator

windings. The desired v∗sq is computed as the output of a proportional integral

(PI) controller added to a speed voltage term that cancels the cross-coupling

effect of ϕsd in (4.5). Using the BDFRM machine model, ϕsd is indirectly

estimated as:

ϕsd = Lsisd+Lpsisd = Lsisd+Lps(ϕp − Lpsisd

Lp) = (Ls−

L2ps

Lp)isd+

LpsLp

ϕp (4.6)

The overall q-axis controller along with the machine dynamics based on (4.5)

is shown in Fig. 4.2 where RE and LE denote the equivalent resistance and

inductance used to design the first-order plant of Fig. 4.2 and FF denotes

the feed-forward term. These quantities are given by:

RE = Rs

LE = Ls −L2ps

Lp

FF = ϕsd(wp − wr)

(4.7)

24

Proportional and integral gains (Kp and Ki) of the controller are chosen to

ensure loop stability, desired transient response, and acceptable steady-state

error. The determination of Kp and Ki based on the first-order closed loop

system is derived in Appendix A.2

The design of the d-axis current controller is similar. Using (3.39) and

(3.41),

ϕsd = (Ls −L2ps

Lp)isd +

LpsLp

ϕp (4.8)

Using this value of ϕsd equation in (3.48),

vsd = Rsisd+dϕsddt

+(wp−wr)ϕsq = Rsisd+d

dt[(Ls−

L2ps

Lp)isd+

LpsLp

ϕp]+(wp−wr)ϕsq(4.9)

Using (4.2) to equate ϕsq in (4.9),

vsd = Rsisd +d

dt[(Ls −

L2ps

Lp)isd +

LpsLp

ϕp] + (wp − wr)(Ls −L2ps

Lp)isq (4.10)

vsd = Rsisd + (Ls −L2ps

Lp)disddt

+LpsLp

dϕpdt

+ (wp − wr)(Ls −L2ps

Lp)isq (4.11)

Also the rate of change of primary fluxdϕpdt

is expressed using (3.46) and

(3.39) as:

dϕpdt

= vpd −Rpipd

= vpd −Rp(ϕp − Lpsisd

Lp)

(4.12)

Using this value in (4.11),

vsd = (Rs+Rp

L2ps

L2p

)isd+(Ls−L2ps

Lp)disddt

+LpsLp

vpd−RpLpsL2p

ϕp+(wp−wr)(Ls−L2ps

Lp)isq

(4.13)

The equivalent inductance and resistance of the machine dynamics are ap-

propriately changed in Fig. 4.2 using (4.13) to design Kp and Ki for the PI

25

controller by making

RE = Rs +Rp

L2ps

L2p

LE = Ls −L2ps

Lp

FF =LpsLp

vpd −RpLpsL2p

ϕp + (wp − wr)(Ls −L2ps

Lp)isq

(4.14)

4.3 Speed controller

Using the rotor speed equation derived earlier in (3.58),

wr =pr(τe − τL)

B + sJ=

τe − τLB

pr+ s

J

pr

(4.15)

Since the drive torque is related to q-axis secondary stator current by (3.56)

τe =3

2prLpsLp

ϕpisq (4.16)

the reference q-axis secondary stator current is given by

i∗sq =τ ∗e

3

2prLpsLp

ϕp

=2

3

LpprLps

τ ∗e1

ϕp(4.17)

The overall control loop including both the speed and q-axis current controller

is determined and is shown in Fig. 4.3. The bandwidth of the speed controller

is chosen to be much smaller than that of the q-axis current controller. This

is a design choice that will determine the final performance of the propulsion

drive.

4.4 Reactive power controller

While the motor torque is controlled by the converter q-axis current, the

converter d-axis current is commanded to control the flow of primary stator

reactive power to and from the ac grid. The objective is to maintain the

26

Figure 4.3: The speed-control forms an outer control loop with the q-axissecondary stator current control inside it.

maximum possible power factor on the ac grid connected to the primary

stator. The primary stator reactive power is given by [19]:

Qp =3

2(vpqipd − vpdipq) =

3

2wpϕp(

ϕp − xpsisdxp

) (4.18)

This can be made zero, thus having a unity power factor on the ac grid, by

having

i∗sd =ϕpxps

(4.19)

This calculated reference d-axis current must also be less than the maximum

allowable value given by

i∗sdmax=√I2srated − i∗sq

2 (4.20)

The reference d-axis secondary stator current is hence given by

i∗sd = min(ϕpxps

,√I2srated − i∗sq

2) (4.21)

Just like the speed controller, the bandwidth of the reactive power controller

is chosen such that it is around one tenth that of the d-axis current controller.

27

CHAPTER 5

ELECTROMAGNETIC DESIGNMETHODOLOGY

This chapter discusses the electromagnetic design methodology of a BDFRM

for a TeDP system. The volumetric D2l sizing equation is derived. As the

design of BDFRM is unconventional, the design equation D2l is subsequently

transformed to a Dl sizing equation. The machine specifications and design

constraints are presented in Section 5.3.

5.1 D2l sizing derivation

The torque produced by a machine is proportional to the product of elec-

trical and magnetic loadings. The electrical loading is limited by the i2R

loss in the conductor, effectiveness of the cooling media, and the allowable

temperature rise in the insulating material. The magnetic loading is limited

by the saturation point of the material, and hysteresis and eddy current loss.

Therefore, the torque has an upper bound for a given amount of copper and

iron.

The basic design equations of a BDFRM having an ideal circular ducted

rotor are presented in [13]. The induced peak phase voltage (Ep) and peak

phase current (Ip) in the primary stator winding are given by

Ep = (lD)kwpNp2wppp

Bpp (5.1)

Ip =πD

6kwpNp

Ap (5.2)

where Bpp and Ap represent peak flux density and peak electric loading of

the primary stator winding with pp poles, respectively. Using (5.1) and (5.2),

28

output power from the primary stator is calculated by

Pp =3

2EpIpcosφpη = π(D2l)

wppp

ApBpp

2cosφpη (5.3)

Similarly, the output power from the secondary stator is

Ps = π(D2l)wsps

AsBps

2cosφsη = π(D2l)

(αwp)

ps

AsBs

2cosφsη (5.4)

where α = ws/wp is the frequency ratio of the secondary stator current to

the primary current. The mechanical output power of the machine is

Prated = PP + Ps = π(D2l)wp2

[ApBp

ppcosφp + α

AsBs

pscosφs

]η (5.5)

The secondary stator parameters Bs, As and cosφs are expressed in terms

of primary stator parameters Bp, Ap and cosφp to simplify (5.5). The peak

MMF produced by the primary stator winding Mp, when excited by current

of amplitude Ip, is given by

Mp =D

ppAp =

6

π

NpKwp

ppIp (5.6)

As derived in (2.7), the air-gap flux density due to this primary stator MMF

consists of several spatial harmonics, and is given by:

Bp(φ) =∞∑i=1

Bpicos(iφ) =µ0Mp

g

∞∑i=1

Cpiksatcos(iφ) (5.7)

where ksat accounts for the MMF drop in the iron. The spatial harmonic at

pp poles and ps poles are given by

Bpp = Cppksatµ0Mp

g(5.8)

Bsp = Cspksatµ0Mp

g(5.9)

Similarly, the peak MMF produced by the secondary stator winding Ms, and

the air-gap flux density produced by the secondary stator winding, when

excited by current of amplitude Is, with an initial phase offset δ, are given

29

by

Ms =D

psAs =

6

π

NsKws

psIs (5.10)

Bs(φ) =∞∑i=1

Bsicos(iφ− δ) =µ0Ms

g

∞∑i=1

Csiksatcos(iφ− δ) (5.11)

The spatial harmonics at pp poles and ps poles are given by

Bps = Cpsksatµ0Ms

g(5.12)

Bss = Cssksatµ0Ms

g(5.13)

The term δ in Bs is due to the phase difference in the excitation currents

between primary and secondary stator currents. δ = π/2 ensures a maximum

electromagnetic torque [29]. Therefore, peak magnitude of the primary stator

flux density (spatial harmonic pp poles) and secondary stator flux density

(spatial harmonic ps poles) are given by

Bpp =√Bpp

2 +Bps2 =

µ0ksatg

√(CppMp)2 + (CpsMs)2 (5.14)

Bps =√Bss

2 +Bsp2 =

µ0ksatg

√(CssMs)2 + (CspMp)2 (5.15)

As and Ms are also expressed in terms of Ap and Mp, respectively, using

(5.2), (5.6) and (5.10), as

As =kws

kwp

Ns

Np

IsIpAp (5.16)

Ms =ppps

AsApMp (5.17)

Reference [13] also discusses the computation of primary and secondary wind-

ing power factors as

tanφp =pspp

CppCps

ApAs

(5.18)

tanφs =ppps

CssCsp

AsAp

(5.19)

The extent of coupling between the two stator windings, denoted by the cou-

pling factors, plays a major role in sizing of the BDFRM. These depend upon

30

the pole combination of both the stators and the rotor. A pole combination

of pp = 8, ps = 4 and pr = 6 is chosen. Using these pole values, the coupling

factors are

Cpp = 0.6034, Csp = Cps = 0.4135, Css = 0.2935 (5.20)

For the initial machine design, the primary and secondary stator currents

and turns per phase are considered equal. Using Eqs. (5.16) and (5.17),

Ap ≈ As (5.21)

Ms ≈ 2Mp (5.22)

Using the coupling factors (Eq. (5.20)) and peak MMF (Eq. (5.22)) in Eqs.

(5.14) and (5.15), peak magnitude of the primary stator flux density Bpp and

secondary stator flux density Bps are given by

Bpp = 1.0237ksatµ0Mp

g(5.23)

Bps = 0.717ksatµ0Mp

g= 0.7Bpp (5.24)

The shear stress for both windings is

σp =BppAp

2(5.25)

σs =BpsAs

2= 0.7σp (5.26)

σnet = σp + σs = 1.7σp (5.27)

The net or rated power for the chosen pole combination is computed by

substituting (5.16) – (5.27) in (5.5) as

Prated = 0.125π(1 + α)wpσpcosφpη(D2l) = 0.88π2

60Nratedσnetcosφpη(D2l)

(5.28)

where Nrated is the rated rotor speed. The D2l equation for the selected pole

combination is given by

D2l =Prated

0.88π2

60Nratedσnetcosφpη

(5.29)

31

The D2l equation in (5.29) can be used as an initial sizing estimate of BD-

FRM, similar to a conventional machine. The shear stress is estimated to

be around 20-35 kPa for aerospace applications [30]. However, the machine

shear stress given by Eq. (5.27) should be re-evaluated for BDFRM to ensure

that such high shear stress does not force the iron into saturation. While the

D2l sizing equation is very useful for initial sizing of conventional machines,

this is not meaningful for BDFRM due to the dependency of magnetic and

electrical loadings. Therefore, a more useful Dl sizing equation is derived in

Section 5.2 and is used for initial sizing in Section 5.3.

5.2 Dl sizing derivation

Shear stress is computed in terms of design parameters and Bairgappeak , which

is chosen according to the core material saturation flux density. Using (2.5)

and (2.6), the air-gap flux density due to primary and secondary stator for

the chosen pole-combination is given by

Bp(φ) =µ0Mp

2g

[cos4φ− cos4

(φ+

π

3− 2mod(φ,

π

3))]

(5.30)

Bs(φ) =µ0Ms

2g

[cos(

2φ+π

2

)− cos

(2(φ+

π

3− 2mod(φ,

π

3))

+π

2

)](5.31)

Figure 5.1 shows both the normalized flux densities Bp, Bs along with the

net flux density in the air-gap with Ms = 2Mp, as discussed earlier in Section

5.1. The peak air-gap flux density, as observed from this figure is

Bairgappeak = 3.45µ0Mp

2g= 1.725

µ0Mp

g(5.32)

Using (5.6), (5.32) and (5.27), the net shear stress is expressed as

σnet = 1.7σp = 1.7BppAp

2= 2.34

ksatµ0

g

DB2airgappeak

(5.33)

32

Figure 5.1: Air-gap flux density (normalized with µ0Mp/2g) due to primaryand secondary stator with equal current amplitude and number ofturns/phase (Ap = As).

Substituting this value of σnet in Eq. (5.29), the D2l equation is converted

to the Dl equation as

Dl =Prated

2.06π2

60Nrated

ksatµ0

gB2airgappeak

cosφpη

(5.34)

This Dl equation leads to a more practical design for the initial sizing of a

BDFRM as compared to the one using the D2l equation. While the sizing

equation derivation was shown for a 6-8-4 pole combination of rotor, pri-

mary and secondary stator, a similar process can be adopted for various pole

combinations for a BDFRM having a ducted reluctance rotor.

5.3 Example design of a 1.5 MW BDFRM

The objective is to design a 1.5 MW motor capable of driving a propeller

fan at 4000 rpm using the Dl sizing equation derived in the previous section.

The rotor poles should be minimized to achieve high rotor speed, and small

volume of the machine, for the provided ac grid frequency. The minimum

rotor pole number with sufficient coupling between the two stator windings,

6 (pp = 8, ps = 4) is chosen. The ac grid maximum frequency is chosen

as 1000 Hz as proposed in [7]. This allows a high synchronous speed of

33

Table 5.1: Design specifications of BDFRM

Parameter Value

Shaft output power 1.5 MWPropeller fan speed 4000 rpm

Primary stator (grid) maximum frequency 1000 HzPrimary stator poles 8

Secondary stator poles 4Rotor poles 6

Max. ratio of frequency of secondary to primary stator currents (α) 0.1Maximum synchronous speed 10000 rpm

Maximum rotor speed 11000 rpm

10,000 rpm with 6 rotor poles and hence reduces the machine size. Using

power electronics of 10% rated power, the maximum rotor speed achievable

is 11000 rpm in the super-synchronous mode. Table 5.1 shows the design

specifications of the BDFRM.

Hiperco-50 steel is selected to construct both the stator and rotor since

it has a high saturation flux-density of about 2.1 T. The rotor diameter is

chosen to limit the rotor tip-speed to 0.8 Mach (280 m/s) [31]. The stack-

length is calculated using the Dl equation in (5.34), with ksat = 0.5. Two

slots per pole per phase for the primary winding were chosen, thus giving a

total of 48 stator slots. The number of rotor ducts is taken as 66 duct rotor

to minimize the harmonic effects and torque ripple [16]. Yoke depth and

tooth width are chosen to limit the flux-density below the saturation limit

of Hiperco-50. Table 5.2 shows these basic sizing parameters.

The next step is to evaluate the primary stator voltage and current ratings.

For a high-speed motor, the number of turns per phase for both windings is

minimized to reduce the required stator voltage. The number of turns per

pole is taken as 1 for the primary and 2 for the secondary stator winding.

Table 5.2: Basic sizing parameters

Parameter Value

Air-gap diameter 486 mmStator outer diameter 640 mm

Stack length 1000 mmAir-gap 1 mm

Number of stator slots 48Number of rotor ducts 66

Slot depth 20 mm

34

This gives a total of 8 turns per phase for both stator windings. Equations

(5.21) and (5.32) are then used to compute the current rating for both stator

windings. Neglecting the resistances, (5.1) and (5.23) are used to determine

the primary stator voltage rating. Table 5.3 shows the calculated motor

ratings following the FEA validation of analytical design equations. The

conventional approach to designing BDFRM leads to low shear stress and

low torque density. Chapter 6 discusses an approach to improving the torque

density in this machine.

Table 5.3: Motor ratings

Parameter Value

Primary stator phase voltage 4 kV peakPrimary and secondary stator phase current 300 A peakPrimary and secondary stator turns/phase 8

Primary winding power factor 0.8Net Shear stress 4.6 kPa

Efficiency 97.6%

35

CHAPTER 6

TORQUE DENSITY IMPROVEMENT

The torque density of the designed 1.5 MW BDFRM is poor, as seen in

Chapter 5. The conventional approach of having equal electric loading on

both stator windings is not appropriate since the stators have different pole

numbers. As will be seen in this chapter, the choice of initial phase offset

δ being π/2 also is not an optimum value for maximum torque production.

An optimization problem is formulated to maximize torque density, with the

two electric loadings and the phase offset δ as the design variables. The

flux density, current density, machine dimensions and primary stator voltage

rating are chosen as constraints.

In order to express air-gap flux density due to primary stator Bp in terms

of its peak specific electric loading Ap, the relationship between Mp and Ap

is used. This leads to the following equation using (2.6):

Bp(φ) =1

2

[cos

pp2φ− cos

pp2φf

]µ0DApgpp

(6.1)

Similarly, the secondary stator winding with ps poles excited using currents

of amplitude Is and an initial phase shift δ results in an air-gap flux density

given by

Bs(φ) =1

2

[cos(ps

2φ− δ

)− cos

(ps2φf − δ

)]µ0DAsgps

(6.2)

Torque in the synchronous operation mode is given by

Te =µ0lD

3

2g

prppps

CpsApAssinδ (6.3)

The peak primary current density is expressed in terms of peak electrical

loadings as:

Jp =τsApfαAslot

(6.4)

36

Figure 6.1: Voltage induced in primary stator (Ep) is the vector sum of Eppand Eps.

where τs is the slot pitch, f is the fill factor, Aslot is the total slot area and

α is the slot area fraction available for the primary stator. The α is equal to

Ap/(Ap + As) to minimize the total stator copper loss [32].

The peak induced voltage in stator i due to excitation of j stator winding

is given by [13]

Eij =2µ0lD

2

g

kwiNi

pipjwiAjCij (6.5)

where Cij denotes the coupling between ith and jth stator windings, given

by the Fourier series expansion of (6.1) and (6.2). Figure 6.1 shows the net

induced voltage in the primary stator (Ep), which is the vector sum of Epp

(emf due to primary stator itself) and Eps (emf due to secondary stator),

both given by (6.5). As shown in Fig. 6.1, current Ip lags the voltage Ep by

θp which is computed using sine law:

Eppθp + δ − π/2

=Eps

π/2− θp(6.6)

This yields the primary stator power factor as

cosθp = sin( mp

1 +mp

δ)

where mp =CpsCpp

ppps

AsAp

(6.7)

The secondary stator power factor is also derived similarly. With the objec-

tive and the constraint functions expressed in terms of design variables, the

optimization problem is constructed next using an example design presented

in Chapter 5.

37

6.1 Optimization problem formulation

The objective is to maximize torque under the flux and current-density con-

straints without modifying the machine's physical dimensions. The optimiza-

tion problem is set up as [14]:

maximizeAp,As, δ

Te = keApAssinδ; ke =µ0lD

3

2g

prppps

Cps

subject to |∑i=p,s

Ji| ≤ Jmax,

|Bi(φ)| ≤ Bmax; i = p, s,

|Bnet(φ)| = |∑i=p,s

Bi(φ)| ≤ Bmax

(6.8)

The conventional FEA optimization solver is time consuming. The motor's

properties are examined here analytically so that the optimization is solved

using a time-saving tool like MATLAB. Instead of applying the flux den-

sity constraints in the iron core, these constraints are applied only along the

air gap. The maximum flux density in stator tooth and rotor ducts is ap-

proximately 1.7 times the maximum air-gap flux density Bmax [33]. Bmax is

chosen based on the material saturation flux density. Even at the air gap,

the optimization problem is quite challenging to solve due to the time- and

space-varying nature of flux density. Therefore, constraints are applied only

at t = 0 s since the amplitude of Bnet waveform remains nearly constant, i.e.,

independent of rotor position [29]. To avoid the space variation along the air

gap, critical φ values are identified by carefully observing the nature of Bnet

in Fig. 6.2, instead of satisfying the constraints for all air-gap angles φ. Due

to the chosen pole combination, the air-gap flux density is symmetric about

π air-gap angle. The critical φ air-gap angles are φ1 = π/3, φ2 = 2π/3, and

φ3, which lies between 2π/3 and π. The first two, φ1 and φ2, exist because

of the discontinuity in the rotor structure. The last one, φ3, exists between

2π/3 and π because it is the only region where both Bp and Bs are in phase.

Using (6.1) and (6.2), the net air-gap flux density in this region is

Bnet(φ) = kAp[cos(4φ) + cos(4φ+

π

3)]

+ 2kAs[cos(2φ− δ) + cos(2φ− π

3+ δ)

] (6.9)

38

Figure 6.2: Air-gap flux density waveforms using the design in [8]. Thestators have equal electrical loading with δ = π/2.

where k = µ0D/16g. The value of angle φ3 giving the minima or maxima of

Bnet is obtained by solving ∂Bnet/∂φ = 0, leading to the following quadratic

equation:

2Apcosπ

6sin2γ − Ascos

(π6− δ)sinγ − Apcos

π

6= 0,

where γ = 2φ− π

6

(6.10)

The critical air-gap angles for which the net flux-density conditions need to

be satisfied in (6.8) have been expressed in terms of design variables. The

peak current density (Jmax) is chosen as 15 A/mm2 to allow indirect oil

cooling [34].

6.2 Impact of optimization on torque density

This section discusses optimization results obtained after using the fmincon

solver in MATLAB. Figure 6.3 shows the variation of primary stator voltage

rating after applying optimization. An upper limit of 4.5 kV, the optimum

value at δ = π/2, is placed on the primary stator voltage to make a fair

comparison between the various designs. Figure 6.4(a) shows both variables,

primary and secondary stator electrical loadings, with varying δ after apply-

39

Figure 6.3: Primary stator voltage rating after optimization for various δ.

ing optimization given by (6.8). Output torque for the optimum design with

various δ and corresponding power factor of both windings, given by (6.3)

and (6.7), respectively, are shown in Fig. 6.4(b). This clearly indicates that

δ = 2π/3 is a better operating point than δ = π/2, in terms of both torque

and power factor. With an equal primary stator voltage rating, a design with

δ = 2π/3 has the capability of producing 75% more torque than the design

with δ = π/2.

This brings up the question of improvement in torque density with design

(a) Current distribution for statorwindings (Np, Ns = 8) to achieveoptimized design at various δ.

(b) Torque capability andcorresponding power factor of bothstators after using optimization.

Figure 6.4: Optimization results indicate that δ = 2π/3 allows a higherelectrical loading, thus leading to higher torque and power factor than thecommonly used δ = π/2.

40

Figure 6.5: Air-gap flux density waveforms with equal electrical loading onboth stators with δ = 2π/3.

at δ = 2π/3. Figure 6.5 illustrates the air-gap flux density with δ = 2π/3 and

equal electric loading on both the stator windings. Comparing Fig. 6.5 with

Fig. 6.2, the major difference lies in the air-gap region between 2π/3 and

π. The flux densities produced by both the primary and secondary stators

are in phase and hence add up with δ = π/2 in this region. However, with

δ = 2π/3, the flux density produced by secondary stator goes to zero in this

region. This implies that the primary stator current can be increased to

achieve the same net air-gap flux density while enhancing the torque density.

Phase offset δ = 2π/3 is not a universal optimum operating point for all

BDFRM designs. As shown in Fig. 6.6, the torque maximum is achieved at

different δ for various limits on primary stator voltage. Since the air-gap flux

density patterns depend on the chosen stators and rotor pole combination,

δ = 2π/3 design may not be best suited for BDFRMs having different pole

configurations. While the pole combination of two stators and rotor chosen

in this thesis is the most widely adopted one, a similar optimization strategy

may be used for other pole combinations.

41

Figure 6.6: Torque capability at various grid voltage levels shows thatδ = 2π/3 is not always an optimum design point to maximize torquedensity.

6.3 Impact of optimization on power converter size

This section discusses the influence of stator electrical loadings and initial

current phase offset δ on the size of the power converter connected to the

secondary stator. The steady-state machine model as derived in Chapter

3 is required for power converter sizing. Since the drive-end converter is

connected to the secondary stator directly, the drive-end converter current

and voltage ratings are same as those of the secondary stator.

To compute the converter current rating, the limits on d and q-axis primary

and secondary stator currents are calculated to ensure that the machine

remains within its rated condition. Based on the limits, a converter current

rating is chosen for both the designs with δ = π/2 and δ = 2π/3 such that the

example BDFRM is utilized to its maximum torque capability. The converter

voltage rating is computed next, and it is shown that the ratio of the power

converter size to output shaft power is much lower for the proposed design

of δ = 2π/3. The steady-state machine equations in the primary stator flux

orientation as summarized in (3.39) - (3.49) will be used in this section.

6.3.1 Converter current rating

The first step in computing the converter current rating is to decide the op-

erating region of the power converter, such that both primary and secondary

42

stator currents remain within bounds, while yielding the demanded torque.

Neglecting the voltage drop across the primary stator resistance, the primary

stator flux is estimated as

ϕp =Vpwp

(6.11)

To remain within primary and secondary stator currents limits,

i2pd + i2pq ≤ I2prated (6.12)

i2sd + i2sq ≤ I2srated (6.13)

(a) δ = π/2

(b) δ = 2π/3

Figure 6.7: Due to a higher permissible current limit on the primary statorwith δ = 2π/3, the allowable operating region is extended for δ = 2π/3.

43

The primary stator currents in (6.12) are substituted with the secondary

currents using (3.39) and (3.40):(ϕp − Lpsisd

Lp

)2

+

(LpsLp

isq

)2

≤ I2prated (6.14)

Both current constraints, given by Eqs. (6.13) and (6.14), are shown in Fig.

6.7. The overlapping region occupied by these curves, ABCD for δ = π/2,

as shown in Fig. 6.7(a), and PQRS for δ = 2π/3, as shown in Fig. 6.7(b),

is the area for safe operation. Phase offset δ = 2π/3 allows a much higher

primary stator current rating Iprated . This increase in current rating extends

both the major and minor axes of the ellipse given by (6.14) for δ = 2π/3,

extending the desired operating region as compared to δ = π/2.

The torque produced by BDFRM is given by (3.56). Since the motor

dimensions and primary stator voltage and frequency are kept identical, there

is no change in the motor's inductances and primary stator flux for the two

designs. The differentiating factor is only isq. The maximum possible positive

torque is obtained at maximum allowable isq, represented by C for δ = π/2,

as shown in Fig. 6.7(a), and R for δ = 2π/3, as shown in Fig. 6.7(b).

Changing the design from δ = π/2 to δ = 2π/3 allows an increase of 75% in

isq, and hence a 75% increase in rated torque. The converter current rating is

equal to Israted for both the designs as both points C and R lie on the ellipse

given by (6.13).

6.3.2 Converter voltage rating

As seen from (3.48) and (3.49), the secondary stator/converter voltage re-

quirement (vs =√v2sd + v2sq) for an ideal BDFRM (no resistance, no leakages

and negligible magnetizing current) at various rotor speeds for a fixed pri-

mary stator frequency is shown in Fig. 6.8(a). The voltage requirement near

synchronous speed Np (ws = 0) is lower than at zero and high speeds. The

converter voltage rating is therefore dependent on the speed range needed

across the synchronous speed. Ideally no power (only losses) is required to

be transferred via power electronics at this synchronous speed. Because the

synchronous speed can itself be changed to vary rotor speed using a variable

frequency ac grid, no power needs to be handled by the power electronics.

44

(a) Converter voltage requirementincreases significantly below and abovesynchronous speed.

(b) By varying grid frequency in discretesteps along with a ± 10% variation inrotor speed due the converter, thevoltage rating of converter can bereduced significantly.

Figure 6.8: Converter voltage requirement for the adopted architecture withrotor speed for an ideal BDFRM.

However, there may exist certain circumstances where some amount of power

electronics must be utilized such as (a) dynamic operation when the propeller

speed would be varied without the ac grid frequency altering instantly or (b)

the set of distributed propellers running at different speeds. The converter

rating is decided by the speed difference between the motors, whose maxi-

mum value is considered to be 20% for the current studies, i.e., the operating

speed of a motor can vary between 0.9Np and 1.1Np for a given synchronous

speed Np, as shown in Fig. 6.8(a). Figure 6.8(b) shows the converter voltage

requirement when the primary stator frequency (wp) is varied in discrete steps

of 0.1Nmax, 0.3Nmax, 0.5Nmax, 0.7Nmax and 0.9Nmax along with using power

electronics to extend the operating speed to ±10% across each synchronous

speed to achieve complete control over the entire speed range. The converter

voltage rating is computed using (3.48) and (3.49) at the maximum operat-

ing speed Nmax. Nmax is equal to 1.1Npmax , where Npmax is the maximum

synchronous speed and is considered as 10000 rpm for both designs.

Table 6.1 summarizes the power converter size required for δ = π/2 and

δ = 2π/3. This shows that for the same output shaft power, the power

converter size required for the δ = 2π/3 design is much lower than that for

the conventional design with δ = π/2. As will be shown in Section 6.4,

this reduction in converter size occurs due to a significant increase in the

secondary stator power factor.

45

Table 6.1: Power converter specifications

Drive Specifications δ = π/2 δ = 2π/3

Converter current rating 289 A 331 AConverter voltage rating 625 V 628 VConverter power rating 271 kVA 312 kVA

Rated torque 1.62 kN-m 2.83 kN-mRated shaft power 1.7 MW 2.96 MW

Ratio of converter powerrating to rated shaft power 0.16 0.105

6.4 Impact of optimization on grid and converter

power factor

The primary stator power factor has an impact on the quality of power being

delivered from the grid, while the secondary stator power factor affects the

required size of the power converter. Thus, this section discusses the impact

on both stator power factors with the proposed δ being 2π/3 .

While the torque is controlled by the secondary stator q-axis current, the

d -axis current is commanded to control the flow of primary stator reactive

power to and from the ac mains. The primary stator reactive power is given

by [19]:

Qp =3

2(vpqipd − vpdipq) =

3

2wpϕp(

ϕp − LpsisdLp

) (6.15)

This can be made zero by having

isd =ϕpLps

=Vp

wpLps(6.16)

which implies a unity power factor operation of the primary stator. This

locus of unity power factor (UPF) is shown in Fig. 6.9. Since it lies outside

the allowable operating region for both designs, a unity power factor is not

possible at all speeds and torques. In fact, the best possible power factor on

the primary stator is obtained on the locus given by:

isd = isdmax =√I2srated − i2sq (6.17)

While the d -axis current is evaluated from (6.17) to maximize the primary

stator power factor, another common strategy of maximizing torque per to-

tal amperes (MTPTA) can be adopted. For singly-fed machines such as a

46

(a) δ = π/2

(b) δ = 2π/3

Figure 6.9: None of the designs allow us to have unity power factor on grid,but δ = 2π/3 allows us to operate on MTPTA.

PMSM, the converter d -axis current is used as zero for MTPTA. For doubly-

fed machines such as BDFRM, the solution is a bit more challenging since

there are two stator currents, both being magnetically coupled. The torque

per total amperes is given by

τeIp + Is

=

3

2prLpsLp

ϕpisq√i2pd + i2pq +

√i2sd + i2sq

(6.18)

The primary stator currents, ipd and ipq, in (6.18) are replaced by secondary

stator currents using (3.39) and (3.40). For a given isq, the value of isd for

achieving the MTPTA is obtained using MATLAB. The locus of isd over the

entire allowable operating region is shown in Fig. 6.9.

47

As seen from Fig. 6.4(b), there is a significant improvement of secondary

stator power factor at the rated conditions when δ changes from π/2 to 2π/3.

However this improvement is true at all operating points and not merely at

rated torque. The real and reactive power flowing into the secondary stator

are given by:

Ps =3

2(vsdisd + vsqiisq) (6.19)

Qs =3

2(vsqisd − vsdisq) (6.20)

Both the real and reactive power flowing from the secondary stator at the

maximum operating speed wrmax are presented in Fig. 6.10 at various torque

conditions. This shows a significant improvement of power factor in the

secondary stator at all operating conditions when δ changes from π/2 to

2π/3. At rated torque, there is a considerable increase in the active power

being handled by the converter at δ = 2π/3 due to much higher rated torque.

Nevertheless, a significant improvement in the secondary stator power factor

decreases the reactive power handled by the converter at δ = 2π/3. The

power converter size remains almost constant for both the designs as seen

earlier in Section 6.3.

48

(a) δ = π/2

(b) δ = 2π/3

Figure 6.10: δ = 2π/3 allows a higher isq, thus a higher rated torque, and ahigher power factor on the secondary stator.

49

CHAPTER 7

FINITE ELEMENT ANALYSIS ANDRESULTS

FEA simulations using ANSYS Maxwell are used to compare the BDFRM

performance between δ = 2π/3 and the conventional case of δ = π/2. As

shown in Fig. 7.1, the maximum flux density at rated conditions is approxi-

mately 2.2 T, the same as the saturation flux density of the material used in

the stator teeth and rotor ducts for both designs. The current and induced

voltage in the A-phase of the primary stator are also shown in Fig. 7.2 when

the motor is running at its natural synchronous speed of 10,000 rpm while

being connected to an ac grid operating at 1000 Hz. The stator current am-

plitudes used in the simulation are obtained using the optimization procedure

discussed in Section 6.2 with a constraint on peak primary stator voltage.

Figure 7.3 compares the torque profile in one mechanical cycle between both

designs. Using δ = π/2, the average torque observed is 1.36 kN-m and is 17%

less than the analytical value. Using δ = 2π/3, the average torque observed is

2.38 kN-m and is 15% less than the analytical value. This shows an increase

of 75% in the torque density of BDFRM with the proposed architecture.

(a) δ = π/2 (b) δ = 2π/3

Figure 7.1: Flux-density plot shows that the material used is not saturatedin either of the designs.

50

(a) δ = π/2(b) δ = 2π/3

Figure 7.2: In the primary stator A-phase, induced voltage was keptconstant during the optimization while the current for δ = 2π/3 design ismuch higher as shown.

Differences between the torque obtained from FEA and the analytical

method are attributed to saturation in the iron ribs, stator slotting effects,

and presence of other air-gap flux density harmonics due to the distributed

stator windings. The iron-ribs are essential for providing mechanical strength

and maintaining the rotor structure at high speeds. Rotor structure can be

further optimized to reduce the torque ripple.

Increasing the primary stator current increases copper losses substantially,

but since core losses are the major contributor to the total losses, there is

only a minor increase in the net loss. In fact, there is an increase in the

(a) δ = π/2 (b) δ = 2π/3

Figure 7.3: There is a 75% increase in torque with δ = 2π/3 when machineis rotating at its natural synchronous speed of 10,000 rpm.

51

Table 7.1: Performance comparison

Specifications δ = π/2 δ = 2π/3

Electrical loading (Ap, As) 10186 A/m, 9086 A/m 17605 A/m, 10469 A/mPower factor (cosθp, cosθs) 0.77, 0.63 0.8, 0.9

Torque 1.36 kN-m 2.38 kN-mCopper losses 1.14 kW 2.35 kW

Core losses 57 kW 70 kWEfficiency 96.1% 97.2%

overall efficiency with the proposed design. Table 7.1 shows the comparisons

between the two optimized designs with δ = π/2 and δ = 2π/3.

A 1.5 MW BDFRM for a rated speed of 10,000 rpm is designed with

δ = 2π/3 and optimum stator currents using (6.8). An air gap of 2 mm

was selected as a feasible value considering cooling requirements. The motor

dimensions and weight are listed in Table 7.2. Active material weight is

obtained from FEA software, where all the inactive material including shaft,

insulation material, and motor housing are neglected for this study. A rib-

width of 3 mm is selected to provide adequate mechanical integrity after

conducting the rotor stress analysis.

The motor current and voltage ratings, and its performance at rated con-

ditions, are listed in Table 7.3. Since the motor is running at such a high

speed, it is necessary to include windage loss in the efficiency computation.

The windage loss is predicted using the model in [35]. Figure 7.4 shows the

2-D cross sectional area of the designed 1.5 MW motor. Figure 7.5 shows

the torque profile for one mechanical rotation cycle. The torque ripple is

approximately 17%.

Table 7.2: BDFRM dimensions

Parameter Value

Air-gap diameter 420 mmStator outer diameter 575 mm

Stack length 400 mmShaft diameter 120 mm

Air-gap 2 mmNumber of stator slots 48Number of rotor ducts 66

Slot depth 25 mmActive weight 671 kg

52

Table 7.3: Motor rated parameters and performance

Parameter Value

Primary stator phase voltage 1.65 kV peakPrimary stator phase current 1120 A peak

Secondary stator phase current 640 A peakPrimary stator maximum frequency 1000 Hz

Primary winding power factor 0.73Core loss 23 kW

Copper loss 4.5 kWWindage loss 12.2 kW

Efficiency 97.42%

Figure 7.4: 2-D cross-section of the 1.5 MW motor as designed usingANSYS Maxwell.

53

Figure 7.5: Torque profile of the designed motor running at rated operatingconditions of 10,000 rpm.

54

CHAPTER 8

CONCLUSIONS

8.1 Summary and design insights

This thesis has addressed a wide range of topics within the scope of electric

machine development for turbo-electric aircraft propulsion systems. With the

ever increasing demand for air travel, it is essential to explore more energy-

efficient means for deployment in the commercial aviation industry. This

motivates the use of electric machines to drive the propeller fans, hence de-

coupling the speed of turbine and propeller. Brushless doubly-fed reluctance

machines (BDFRM) are seen as an attractive choice to drive the distributed

propeller fans as they use partially rated power converter, have brushless op-

eration, and low rotor losses. This thesis explores a BDFRM by computing

the machine and power converter size for the rated operation.

The motor developed after the initial sizing, and following the conventional

approach of having equal electrical loadings on both the stator windings, has

poor torque density. This thesis presents an approach to enhance torque

density. A single objective function along with multiple constraints is formu-

lated in terms of three design variables: electrical loadings of primary and

secondary stators, and phase offset in the secondary stator current (δ), which

plays a role similar to the torque angle in a synchronous motor. Optimization

results reveal that the most commonly used δ = π/2 is not an optimum op-

erating point in terms of torque. Phase offset δ = 2π/3, along with suitable

electrical loadings, has the capability to increase the torque by 75%. There is

also an improvement in efficiency and power factor. 2-D FEA using ANSYS

Maxwell was also carried out to validate the optimization results.

55

8.2 Future work

The following future work will be done during the author's PhD program:

� The rotor is modeled having infinite number of ducts for ease of analyt-

ical formulation. The rotor will be modeled considering finite widths of

iron ducts and air-barriers using reluctance network. This can enhance

the rotor modeling, leading to a more accurate prediction of torque

density improvement.

� The copper and core losses are computed using FEA. A framework

for the loss computation will be developed. Considering the amount

of flux-density harmonics in the material in this type of machine, the

modeling of core loss will be challenging but needs to be looked at for

maximizing efficiency.

� A switched drive architecture similar to the one presented in [36] will be

investigated. A motor in an electrical aircraft operates under various

operating conditions. The maximum torque or the rated conditions are

required during take-off, which lasts only a few minutes. The motor

needs to be operated at a significantly lower torque and speed at cruise.

Since the cruise conditions last for several hours, these conditions de-

mand the motor to be efficient at operating points far away from rated

ones. Designing the motor to operate efficiently at low speeds while

delivering the required torque at high speeds is the key for such appli-

cations and can probably be achieved using switched drive architecture.

� Since the motor is running at speeds close to 10,000 rpm, the mechan-

ical integrity of the rotor needs to be investigated. A trade-off study

between the output torque and the stress for different rotor iron rib

widths will be done to ensure the motor is capable of delivering suffi-

cient torque while maintaining mechanical integrity.

� A 5-hp prototype motor will be designed, constructed, tested and val-

idated along with developing necessary drive control algorithms.

56

APPENDIX A

ELECTROMAGNETIC TORQUEPRODUCED BY BDFRM

As seen in (3.53), output torque is given by

τe =1

2pr[i]

T[ dLdθr

][i] (A.1)

where the inductance varying matrixdL

dθris given by

[ dLdθr

]=

[dL[p,ABC−p,ABC]

dθr

] [dL[p,ABC−s,ABC]

dθr

][dL[s,ABC−p,ABC]

dθr

] [dL[s,ABC−s,ABC]

dθr

] (A.2)

Since the matrices L[p,ABC−p,ABC] and L[s,ABC−s,ABC] are constant,[dL[p,ABC−p,ABC]

dθr

]=[dL[p,ABC−p,ABC]

dθr

]= 0. Since L[p,ABC−s,ABC] is known,

thedL[p,ABC−s,ABC]

dθrmatrix is computed as:

dL[p,ABC−s,ABC]

dθr= j

Lpsm2

1 a2 a

a2 a 1

a 1 a2

ejθr − jLpsm2

1 a a2

a a2 1

a2 1 a

e−jθr

Since L[p,ABC−s,ABC] = L[s,ABC−p,ABC],dL[p,ABC−s,ABC]

dθr=dL[s,ABC−p,ABC]

dθr.

57

The inductance matrixdL

dθris written as:

jLpsm

2

0 0 0 1 a2 a

0 0 0 a2 a 1

0 0 0 a 1 a2

1 a2 a 0 0 0

a2 a 1 0 0 0

a 1 a2 0 0 0

ejθr−jLpsm

2

0 0 0 1 a a2

0 0 0 a a2 1

0 0 0 a2 1 a

1 a a2 0 0 0

a a2 1 0 0 0

a2 1 a 0 0 0

e−jθr

the inductance matrix[ dLdθr

][i] is given by:

[ dLdθr

][i] = j

Lpsm2

isA + a2isB + aisCa2isA + aisB + isCaisA + isB + a2isCipA + a2ipB + aipCa2ipA + aipB + ipCaipA + ipB + a2ipC

ejθr − jLpsm

2

isA + aisB + a2isCaisA + a2isB + isCa2isA + isB + aisCipA + aipB + a2ipCaipA + a2ipB + ipCa2ipA + ipB + aipC

e−jθr

= j3

2

Lpsm2

i∗sr

a2i∗sr

ai∗sr

i∗pr

a2i∗pr

ai∗pr

ejθr − j 3

2

Lpsm2

isr]

aisr

a2isr

ipr

aipr

a2ipr

e−jθr

58

Using (3.53), the net electromagnetic torque is given by:

τe =1

2pr[i]

T[ dLdθr

][i]

=j3

4prLpsm

2

[ipA ipB ipC isA isB isC

]

i∗sr

a2i∗sr

ai∗sr

i∗pr

a2i∗pr

ai∗pr

ejθr

−j 3

4prLpsm

2

[ipA ipB ipC isA isB isC

]

isr

aisr

a2isr

ipr

aipr

a2ipr

e−jθr

τe =j3

4prLpsm

2[(ipA + a2ipB + aipC )i∗sr + (isA + a2isB + aisC )i∗pr]e

jθr

−j 3

4prLpsm

2[(ipA + aipB + a2ipC )isr + (isA + aisB + a2isC )ipr]e

−jθr

=j3

4prLpsm

2[(

3

2i∗pr)i

∗sr + (

3

2i∗sr)i

∗pr]e

jθr − j 3

4prLpsm

2[(

3

2ipr)isr + (

3

2isr)ipr]e

−jθr

=j3

4prLps[i

∗pri∗sr − iprisr]

=j3

4prLps[i

∗pri∗sr − iprisr]

59

APPENDIX B

DETERMINATION OF KP AND KI

BASED ON FIRST-ORDER SYSTEM

As shown in Fig. B.1, the closed loop transfer function with a unity feedback

is given by [37]:

G(s) =

sKp +Ki

s(sLE +RE)

1 +sKp +Ki

s(sLE +RE)

=sKp +Ki

LEs2 + (RE +Kp)s+Ki

(B.1)

where Kp and Ki are the proportional and integral gains of the controller,

respectively, and LE and RE are the equivalent inductance and resistance of

the plants, respectively. Since the objective is to get a first-order system dy-

namics with a time-constant τ , the transfer function in (B.1) can be equated

to:

G(s) =sKp +Ki

LEs2 + (RE +Kp)s+Ki

=1

sτ + 1(B.2)

which yields the proportional gain Kp and the integral gain Ki as:

Kp =LEτ

(B.3)

Ki =RE

τ(B.4)

Figure B.1: Block diagram for the design of PI controller: Kp and Ki arechosen based on the first-order plant model.

60

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