SENSITIVITY OF ELECTRICAL DRIVE PERFORMANCE TO CONTROLLER
PARAMETER ERROR
By
Andrew Stephen Babel

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Electrical Engineering – Master of Science
2013

ABSTRACT
SENSITIVITY OF ELECTRICAL DRIVE PERFORMANCE TO
CONTROLLER PARAMETER ERROR
By
Andrew Stephen Babel
Saturation, cross-saturation, heating of the windings and magnets are nonlinear effects that
can affect the measured machine parameters and performance. Because the machine parameters are used in the controller, inaccuracies in parameters will be reflected in the final
controller. With parameter error, the controller will not operate as designed. Most studies
justify the work of improving controllers and detecting faults by arguing that poor modeling
and faults have a deleterious effect upon performance. They fail to completely define the
effect on performance. The effect that controller modeling errors have upon performance is
described for poor characterization, some minor faults, and the failure to include unmodeled
dynamics. The sensitivity of performance to characterization is determined because of the
vital nature of accurate parameter determination. Performance effects due to controller parameter error, saturation and other unmodeled dynamics are calculated to justify controller
improvements, detailed characterization techniques and fault detection.

ACKNOWLEDGMENTS

I would like to express my deepest gratitude to my advisor, Professor Elias Strangas, whose
assistance, advice, and professional guidance have helped greatly in my professional pursuits
and development. Without his guidance this thesis would not have been possible.
Additionally, the work performed would not have been possible without the support staff,
including, but not limited to: Brian Wright, Gregg Mulder and Roxanne Peacock.
My friends and colleagues have been instrumental in completing my thesis. Their diverse
experiences and input have greatly enriched my experience. Their assistance, time, advice,
and constant collaboration made this work possible. I would like to thank Jorge G. Cintr´no
Rivera and Shanelle Foster for their close collaboration and time, as well as: Reemon Haddad,
Cristian Lopez-Martinez, Eduardo Montalvo-Ortiz, Dr. Muhammad Shahid Nazrulla, Feng
Niu, Dr. Carlos Nino-Baron, Arslan Qaiser, Dr. Abdul Rahman Tariq, Muhammad Jawad
Zaheer, and Dr. Sajjad Zaidi.
My wife Cassandra has given me the crucial love and support I needed throughout this
experience. My grandmother Fern Keen has, as always, given me invaluable support, as have
my parents, Larry and Janet, and siblings Christina and Matthew.

iii

TABLE OF CONTENTS

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

vi

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

vii

KEY TO SYMBOLS AND ABBREVIATIONS . . . . . . .
Chapter 1

. . . . . . . .

xi

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter 2 Background and Theory . . . . . . . . . . . . . . . . . . .
2.1 Machine Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Machine Model Considering Saturation and Cross-Saturation
2.1.2 Machine Parameter Determination . . . . . . . . . . . . . .
2.1.3 Machine Control Development . . . . . . . . . . . . . . . . .
2.1.3.1 Torque Control Development . . . . . . . . . . . .
2.1.3.2 Matlab Controller Extraction . . . . . . . . . . . .
2.2 Parametric Error . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Sources of Parametric Error . . . . . . . . . . . . . . . . . .
2.2.1.1 Characterization Errors . . . . . . . . . . . . . . .
2.2.1.2 Nonlinear Steel Characteristics . . . . . . . . . . .
2.2.1.3 Eccentricity . . . . . . . . . . . . . . . . . . . . . .
2.2.1.4 Uniform Demagnetization . . . . . . . . . . . . . .
2.2.1.5 Heating . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Effect of Parametric Error Upon Performance . . . . . . . .
2.2.3 Quantification of Parametric Error . . . . . . . . . . . . . .
2.3 Sensitivity Analysis Procedure . . . . . . . . . . . . . . . . . . . . .

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4
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9
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25
25
26
26
27
30
30
40
42

Chapter 3 Experimental and Simulation Results . . . . . .
3.1 Experimental Characterization Procedure . . . . . . . . . .
3.1.1 Characterization Procedure . . . . . . . . . . . . .
3.1.1.1 Open Circuit Testing . . . . . . . . . . . .
3.1.1.2 Phase Resistance . . . . . . . . . . . . . .
3.1.1.3 Flux linkages . . . . . . . . . . . . . . . .
3.1.2 Flux Linkage Data processing . . . . . . . . . . . .
3.2 SPM Machine Current Controller Sensitivity . . . . . . . .
3.2.1 Method of Characterizing SPM and IPM machines
3.3 IPM Machine 1 Torque Controller Sensitivity Using FEA .
3.3.1 Finite Element Analysis for Characterization . . . .
3.3.2 Finite Element Model . . . . . . . . . . . . . . . .
3.3.3 Finite Element Simulations . . . . . . . . . . . . .
3.4 IPM Machine 2 Experimental and Simulation Results . . .
3.4.1 Flux Linkages . . . . . . . . . . . . . . . . . . . . .

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44
44
45
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46
47
51
53
53
57
57
57
60
65
67

iv

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3.4.2
3.4.3
Chapter 4

Controller Development . . . . . . . . . . . . . . . . . . . . . . . . .
Sensitivity Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68
72

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

BIBLIOGRAPHY

. . . . . . . . . . . . .

v

. . . . . . . . . . . . . . . . .

82

LIST OF TABLES

Table 2.1

Operating point quantities . . . . . . . . . . . . . . . . . . . . . . .

31

Table 3.1

Characteristics of surface mounted permanent magnet machine . . .

54

Table 3.2

Table showing the various output metrics with respect to varying
parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Table showing the various dynamic response metrics with respect to
varying parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Table showing the sensitivity of output and dynamic response metrics
with respect to varying parameters . . . . . . . . . . . . . . . . . . .

55

Table showing the experimentally determined sensitivity to parametric error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

Table 3.6

Fig. 3.9 color legend

58

Table 3.7

Mesh point values for Fig. 3.10

. . . . . . . . . . . . . . . . . . . .

60

Table 3.8

Characteristics of interior permanent magnet machine 1 . . . . . . .

61

Table 3.9

Table showing the finite element determined sensitivity to parametric
error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

Table showing the experimentally determined sensitivity to parametric error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

Table 3.3

Table 3.4

Table 3.5

Table 3.10

. . . . . . . . . . . . . . . . . . . . . . . . . .

vi

LIST OF FIGURES

Figure 2.1

Machine rotor styles . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

Figure 2.2

Three and two phase machine axes . . . . . . . . . . . . . . . . . . .

7

Figure 2.3

Stator and rotor oriented machine axes . . . . . . . . . . . . . . . .

7

Figure 2.4

B-H curve for a particular material at different current values . . . .

10

Figure 2.5

Speed controller for machine being characterized with current controller in the inner loop . . . . . . . . . . . . . . . . . . . . . . . . .

14

Voltage and current limit circles for the nonsalient case. For interpretation of the references to color in this and all other figures, the
reader is referred to the electronic version of this thesis. . . . . . .

17

Figure 2.7

Maximum torque per current lines . . . . . . . . . . . . . . . . . . .

18

Figure 2.8

Voltage and current limit circles for the salient case . . . . . . . . .

20

Figure 2.9

Development of lookup table and use during operation . . . . . . . .

21

Figure 2.10

Calculation points depending on speed, direct and quadrature axis
current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

Figure 2.11

Optimal choice of ids demonstrated for a non-salient machine . . . .

23

Figure 2.12

Optimal choice of ids demonstrated for a salient machine . . . . . .

24

Figure 2.13

Inductance and flux curves showing the effect of saturation in iq . .

27

Figure 2.14

Eccentric IPM machine model . . . . . . . . . . . . . . . . . . . . .

28

Figure 2.15

Flux curve for an eccentric and concentric machine (Based on a
graphic in [3]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

Flux curve for demagnetized machine (Based on a graphic in [11]) .

29

Figure 2.6

Figure 2.16

vii

Figure 2.17

Change in the direct axis developed current with changing inductances 33

Figure 2.18

Change in the direct axis developed current with changing resistance
and flux linkage due to the magnets . . . . . . . . . . . . . . . . . .

34

Change in the quadrature axis developed current with changing inductances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

Change in the quadrature axis developed current with changing resistance and flux linkage due to the magnets . . . . . . . . . . . . .

35

Figure 2.21

Change in the torque with changing flux linkage due to the magnets

37

Figure 2.22

Change in the torque with changing direct and quadrature axis inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

Change in the torque with changing direct and quadrature axis flux
linkages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

Figure 2.24

Plots for the changes in voltage ellipse due to parameter changes . .

40

Figure 2.25

Flowchart of the procedure to determine sensitivity set

43

Figure 3.1

The angle offset plot used to first find angle offset, subtract this from
the angle to achieve the angle aligned to the positive back-EMF peak

46

Figure 3.2

DC test for phase resistance . . . . . . . . . . . . . . . . . . . . . .

47

Figure 3.3

Current vector range applied during characterization . . . . . . . . .

48

Figure 3.4

Vector diagram including fictitious rotor magnet . . . . . . . . . . .

49

Figure 3.5

Sinusoids for the phase A quantities . . . . . . . . . . . . . . . . . .

50

Figure 3.6

Demonstration of filtering to produce clearer waveforms . . . . . . .

52

Figure 3.7

Plotting the phase A voltage with back-EMF to find d- and q- axis
values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

Physical test setup for SPM machine characterization with inverters,
machines and sensors . . . . . . . . . . . . . . . . . . . . . . . . . .

54

Figure 2.19

Figure 2.20

Figure 2.23

Figure 3.8

viii

. . . . . . .

Figure 3.9

View of the partial Flux2D geometry finite element model . . . . . .

59

Figure 3.10

View of the partial Flux2D geometry finite element model with mesh

59

Figure 3.11

Circuit of the FEM . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

Figure 3.12

Current dependency of the direct axis flux linkage . . . . . . . . . .

62

Figure 3.13

Current dependency of the quadrature axis flux linkage . . . . . . .

63

Figure 3.14

All torque and speed curves . . . . . . . . . . . . . . . . . . . . . . .

64

Figure 3.15

Physical test setup for IPM machine characterization with inverters,
machines and sensors . . . . . . . . . . . . . . . . . . . . . . . . . .

67

Figure 3.16

Current dependent direct axis flux linkage . . . . . . . . . . . . . . .

68

Figure 3.17

Current dependent quadrature axis flux linkage . . . . . . . . . . . .

68

Figure 3.18

Direct axis flux linkage from simplified method from Jorge Cintr´no
Rivera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Figure 3.19

Quadrature axis flux linkage from simplified method from Jorge Cintr´no
Rivera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Figure 3.20

Torque-speed curve for the nominal controller . . . . . . . . . . . . .

Figure 3.21

Torque-speed curve for simplified characterization flux linkage controller 70

Figure 3.22

Operating point dependent current magnitude command

. . . . . .

71

Figure 3.23

Operating point dependent direct axis current command . . . . . . .

71

Figure 3.24

Operating point dependent current magnitude command from simplified characterization . . . . . . . . . . . . . . . . . . . . . . . . .

72

Operating point dependent direct axis current command from simplified characterization . . . . . . . . . . . . . . . . . . . . . . . . .

73

Figure 3.25

Figure 3.26

70

Direct axis torque sensitivity with respect to operating point (∆λd→T ) 75

ix

Figure 3.27

Quadrature axis torque sensitivity with respect to operating point
(∆λq→T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

76

KEY TO SYMBOLS AND ABBREVIATIONS

Symbol

Meaning

Units

α

Temperature coefficient of resistance

αd

Direct axis inductance error coefficient

αf

Flux linkage due to the magnets error coefficient

αf d

Direct axis flux linkage error coefficient

αf q

Quadrature axis flux linkage error coefficient

αq

Quadrature axis inductance error coefficient

αr

Resistance error coefficient

αs

Nonsalient inductance error coefficient

χ

Saliency factor of machine

δ

Current command angle

V

∆ξ→P

Sensitivity measure from parameter ξ to performance measure P

V

eabc,s

EMF induced in the machine phases

V

iabc,s

Stator currents in the stator frame of reference

A

i∗
dq0,s

Stator current commands in the rotor oriented frame of reference

A

idq0,s

Stator current in the rotor oriented frame of reference

A

Is

Stator current magnitude

A

isc

Short circuit current of motor

A

Ix%

Value of x percentage of rated machine current

A

Ki

Gain from inverter command to output

Lxx

Self inductance of phase X

1
◦C

H

xi

Symbol

Meaning

Units

Lxy

Mutual inductance given by phases X and Y

H

Ld

Direct axis inductance

H

Lq

Quadrature axis inductance

H

ωe

Electrical speed of machine

rad
sec

ωs

Operating speed of machine

rad
sec

ωb

Base speed of machine

rad
sec

Ω

Mechanical speed of machine

RPM

P

Number of poles in machine

poles

λA

Phase A flux of machine

V ·s

λaf

Flux linkage due to magnets

V ·s

λdq0,s

Rotor oriented flux linkages

V ·s

Rs

Phase resistance of machine

Ω

Rs0

Nominal phase resistance of machine

Ω

T

Torque of machine

Tabc

Frame of reference transformation matrix

θr

Electrical rotor position

Ti

Inverter delay/switching period

sec

T0

Nominal temperature

◦C

Ts

Current operating temperature

◦C

vabc,s

Stator voltage in the stator frame of reference

V

vdq0,s

Stator voltage in the rotor oriented frame of reference

V

∗
vdq0,s

Stator voltage commands in the rotor oriented frame of reference

V

N ·m

degrees

xii

Symbol

Meaning

Units

Vl−l,RM S

Line to line RMS voltage

VRM S

Vx%

Value of x percentage of rated machine voltage

xabc,s

Generalized stator oriented quantities

xdq0,s

Generalized rotor oriented quantities

ξ

Machine parameter in question

xiii

V

Chapter 1
Introduction
It is oft said [9] that electrical machines use greater than forty percent of the world’s electricity. This is obvious considering the amount of devices which use them, from washing
machines and pumps in the home, traction drives in hybrid and electric vehicles on the road,
and serving as electromagnetic actuators in a multitude of industrial applications. With this
in mind, any improvement upon the performance of these drives is welcomed. In years past,
induction machines served many of these purposes, and would merely be plugged into the
wall and allowed to reach steady-state levels and this was sufficient. In modern times the
large transient currents, and therefore losses are no longer acceptable. Permanent magnet
motors are a fixture of modern drives, as they are more power dense and have smaller losses,
as there are no rotor windings. In PM drives, controlling both the flux and torque directly
has replaced the blind application of wall voltage and doing this has reduced transient losses.
This is accomplished by controlling the current in and out of alignment with the magnet
current. Doing this, the machine torque can be precisely controlled at any speed with lesser
transient loss than the past.
The use of inverters to apply specific currents to the electrical machine has allowed for the
precise control of machine flux and torque, moving past the uncontrolled operation of motors.
Two types of controllers exist for the precise control of torque and flux of a permanent magnet
machine. The first is a method which directly controls the torque and flux of the machine

1

without controlling the current in [12]. While effective, the goal is maximizing precision,
thus current control is used. This rotor oriented method applies currents to control the
direct and quadrature axis current and thus the machine torque and flux. The torque is
chosen to match that of the load and to allow for the machine to accelerate as desired.
The flux is chosen based upon the speed of the machine [5]. This current control method
achieves a desired torque at any given speed by the application of specific currents. These
models are however dependent upon the machine model inductance, resistance, and the flux
linkage due to the magnets. Thus, to have confidence in the performance of the model
and therefore control of machine torque at any speed, the individual parameters must be
known accurately. Parameter identification can be done analytically, experimentally, or
with numerical modeling tools, or experimentally. The goal is to have the parameters be as
accurate as possible.
Discussion of the dependence of high performance controllers upon the accuracy of parameters is often not very clear, and involves nonspecific statements. Papers discussing
characterization such as [7] cite this as a motivation for their improved characterization
method. It is well understood that in order to create a controller some rough parameters
must be known, in order to model the machine performance and operation. This begs the
question: How will performance be affected if the parameters contain some level of inaccuracy? In closed loop PI control, parametric error will have a limited effect upon the
steady-state performance, as the integrator portion of the control will drive the error to
zero. In the open loop portions of the control, as in the torque and current relationship,
these errors will result in the inaccurate development of torque. Currently, this is avoided
by the use of accurate characterization techniques or online parameter estimation as in [8].
To simplify the controller, the former is often chosen over the latter.
2

In order to improve controllers, the characterization must be as accurate as possible, but
if one were to take into account all effects which may be encountered during operation, the
characterization process would be much too cumbersome. In order to choose which effects
and thus parameters to characterize most accurately, it should be known which parameter
is most vital to accurate performance. In order to find this, parametric error is introduced
into SPM and IPM machine controllers and the performance changes are noted. Measuring
this effect in a standardized way can establish sensitivities for this particular machine and
allow the control design to be optimized around those parameters which are most important
in the operation of the machine. It is this procedure and simple relationship which is used
to optimize characterization procedures and establishes how more accurate characterizations
affect the high performance control of these machines. This thesis discusses a method which
measures the effect of parametric error upon output performance. This is done analytically,
with finite element analysis and experiments.

3

Chapter 2
Background and Theory

2.1

Machine Modeling

Permanent magnet synchronous machines are valuable for their performance; the magnets
used are small, and have a high magnetic flux. Additionally, the fact that the coercivity of the
Neodymium Iron Boron and Samarium Cobalt magnets is so high prevents demagnetization
at low currents. This allows for the design of small machines with a high of power, thus
allowing for small kW-sized machines. A great deal of this size reduction can be attributed
to reasons in [6]: the lack of rotor conduction and thus ohmic losses; permanent magnet
machines typically have neither field windings nor rotor bars. While the cost of these motors
is high due to the magnet cost, the power density compensates.
The machines are typical in their stator design, but the rotor design can vary. In these
machines, magnets can either be attached in to the surface of the rotor, or embedded in the
rotor as in Fig. 2.1 (a) and (b) respectively. Other machine designs are not investigated and
are not included. In the case of the magnets being embedded in the rotor, the inductance is
not uniform along the angle of the machine. Machine axes are defined as from the center of
the rotor to its outermost edge. There are two main axes: the axis which coincides with the
center of the magnet flux and the axis which coincides with the point between the positive
and negative fluxes. For reference, note that the positive flux is defined to be the ‘north’

4

q-axis

q-axis

d-axis

d-axis

(a) SPM

(b) IPM
Figure 2.1: Machine rotor styles

pole or the area in which the flux lines exit the rotor. In the case of the IPM and SPM rotors
drawn here, the d-axis coincides with the middle of the magnet. Through the magnet, a flux
line along this path would have a large reluctance, since the permeability of the magnet is
that of air, which is much smaller than that of the steel which makes up the rest of the rotor.

The machine can be modeled by the phase equations in (2.1). Note that the e{a,b,c}s
quantities correspond to the emfback-EMF induced in the machine by the magnets. In the
IPM machine discussed previously, the difference in permeability of the magnet in comparison
with the steel path give rise to the problem of nonuniform inductance. Thus, it makes more
sense to develop the equations of the machine in a new two phase frame of reference. This
two phase frame of reference is fictitious and is defined to be the axes which align with
the magnets and perpendicular to it, so that the inductance in each of these two phases is

5

constant in all cases. This reference frame is defined as the rotor frame of reference.














vas 



vbs 



vcs

=










Rs

0

0

Rs

0

0





0   ias 







 i  +
0   bs 




ics
Rs

d
dt















Laa Lab Lac   ias 
Lba Lbb

Lca Lcb




 

 

 

Lbc   ibs  + 
 

 

 

ics
Lcc



eas 



ebs 



ecs

(2.1)

The rotor oriented frame of reference for an interior permanent magnet machine is not aligned
to the stator, or fixed axes, but the moving, or rotor axes. In particular, it is oriented with
the rotor magnet flux axis. The end goal is to have the system be aligned such that the
horizontal axis aligns with the middle of the magnet flux. This axis is designated the direct
axis. The vertical axis is then set to be ninety degrees ahead of this direct axis, and should be
ninety degrees ahead of the maximum positive magnet flux point and ninety degrees behind
the maximum negative flux point. Note the axes given in Fig. 2.2, and shown on an example
machine in Fig. 2.3. The α and β axes are the horizontal and vertical axes in the stator
frame of reference. These do not change position as the machine moves, and are aligned to
the machine such that the horizontal axis is in line with the flux developed by the machine
with only a positive phase A current. The β axis is merely defined as ninety degrees ahead
of this.
Note that the north pole of the magnet is not in alignment with the α axis, due to the
rotation of the rotor. The d-axis is always aligned with the rotor pole of the magnet to
maintain alignment with the positive flux of the magnet. Thus, if a stator current were
applied in alignment with the d-axis, flux would be applied in a direction which strengthens
the flux in the machine, while applying a negative d-axis current would weaken the flux.
Further, applying a quadrature axis current would induce torque in the machine with a
relationship to the rotor oriented axes as in equation (2.2). The value of operating the
6

β-axis
c-axis
d-axis
q-axis
θr

a-axis
α-axis

b-axis

Figure 2.2: Three and two phase machine axes

-axis
A+

!-axis
C-

B-

q-axis
B+

C+

d-axis
A-

A-

C+

B+

B-

CA+

Figure 2.3: Stator and rotor oriented machine axes

7

machine in this coordinate system can be seen by the simplicity of the torque equation.

3
T = P [λaf + (Ld − Lq )ids ]iqs
4

(2.2)

The final issue remaining is the application of rotor oriented axis current. This can be
done with knowledge of the proper rotor oriented quantities desired, and then applying a
transformation (2.4) to achieve the appropriate stator oriented quantities.




xa 
 
 
 
x 
 b
 
 
xc


Tabc =

cos(θr )



sin(θr )




1/2



=



xq 
−1  
 
 
xq 
Tabc
 
 
 
x0

cos(θr −

2π )
3

(2.3)

cos(θr +

sin(θr − 2π ) sin(θr +
3
1/2

1/2


2π )
3 


2π )

3 



(2.4)

With this understanding, the machine is modeled by equations in this rotor frame of
reference as in (2.5)-(2.8).

dλ
vd = ids Rs + d − ωe λq
dt
di
. . . = ids Rs + Ld ds − ωe Lq iqs
dt
dλq
vq = iqs Rs +
+ ω e λd
dt
diqs
+ ωe λaf + ωe Ld ids
. . . = iqs Rs + Lq
dt

(2.5)
(2.6)
(2.7)
(2.8)

The machine equations relate the current and voltage with the resistances, inductances and

8

the flux linkage due to the magnets. The resistance Rs is simply the resistance of the phase,
Ld is the inductance of the path through the direct axis, Lq is the inductance of the path
through the quadrature axis, and the λaf is the flux which links the stator due to the
magnets.

2.1.1

Machine Model Considering Saturation and Cross-Saturation

At first glance, the machine model appears to assume that the parameters are constant. In
reality, these parameters vary greatly in different circumstances. The resistance is the most
obvious, depending upon the temperature of the machine at any given point as well as the
frequency applied to the machine due to the skin effect. The flux linkage due to the magnets
can change slightly due to saturation of the iron of the machine as well as temperature.
The main components that change are the inductances. The Ld can change slightly due to
saturation and cross-saturation, but because in both machine cases discussed in section 2.1,
the equivalent airgap is large, thus the current required to saturate the machine is much
greater than the operating current of the machine. In the case of Lq , because the magnets
are embedded, the airgap can become very small. Because of this, when current is applied,
much more flux is developed. Because a great deal more flux is developed, the machine
will saturate rapidly. This saturation caused by the high level of flux, especially in an IPM
machine, is a typical source of variation in the parameters. The typical quadrature axis flux
curve shows saturation at large current quantities in Fig. 2.4.

Beyond the effect of mere

saturation, cross saturation can lead to a reduction in flux in that axis. This is especially
true for the quadrature axis with large quadrature axis currents and direct axis currents.
This curve in Fig. 2.4 also shows decreasing quadrature axis flux with respect to direct axis
current. The saturation changes which affect the machine model can be represented by a
9

q-axis flux (T)
d-axis current (A)

q-axis current (A)

Figure 2.4: B-H curve for a particular material at different current values
machine model which depends on both axis currents as in equations (2.9)-(2.12).

dλd (ids , iqs )
− ωe λq (ids , iqs )
dt
di
. . . = ids Rs + Ld (ids , iqs ) ds − ωe Lq (ids , iqs )iqs
dt
dλq (ids , iqs )
+ ωe λd (ids , iqs )
vq = iqs Rs +
dt
diqs
. . . = iqs Rs + Lq (ids , iqs )
+ ωe λaf + ωe Ld (ids , iqs )ids
dt
vd = ids Rs +

2.1.2

(2.9)
(2.10)
(2.11)
(2.12)

Machine Parameter Determination

In order to design a controller, the machine parameters must be known. These parameters
can be determined in a variety of ways, with varying levels of accuracy. The simplest methods
are motor standstill techniques, which apply a voltage or current to a machine and measure

10

the rise time to determine the inductance and steady state relation to determine resistance.
The flux linkage due to the magnets as well as d- and q- axis flux linkage with applied current
must however be calculated by rotating the machine at a constant speed.
The simplest parameter to determine is the resistance as the machine. The measurement
does not require the rotor to be moving, and this requires no other parameters. The resistance
is measured as in [4] by applying a current to phase A, with the return current coming
through the parallel combination of phases B and C. Apply a current for both the rated
machine current and 150% the rated machine current. At each current measure the voltage
and apply the relationship in equation (2.13).

Ra =

2 V150% − V100%
3 I150% − I100%

(2.13)

The flux linkage due to the magnets is also an easy parameter to measure accurately,
without requiring any other machine parameters. The back-EMF is the voltage induced by
the flux linkage of the magnets. As such, if the machine is open circuited, and the voltage is
measured while spinning the machine at constant speed, the flux linkage due to the magnets
can be determined. If the line-to-line voltage and speed is precisely applied, the relationship
in equation (2.16) is derived from the machine equations with current zero in equations (2.14)
and (2.15).

vd = 0

(2.14)

vq = λaf ωe

(2.15)

λaf =

Vl−l,rms
√

Ω πP√ 3
60 2

11

(2.16)

The resistance and flux linkage due to the magnets are the simplest parameters to calculate due to the simplicity of these tests. The inductance has a similar simple method which
is found from a short circuit test on the machine operating at or above the base speed. The
machine equations in the short circuit test simplify such that the only current is due to the
flux linkage of the magnets, and that this current is in the direct axis. While the machine
terminals are shorted and the RMS phase current is measured, the direct axis inductance
can be determined as in equation (2.17) if the flux linkage due to the magnets is known, as
the induced current is limited by the inductance Ld .

Ld =

λaf
√

isc 2

(2.17)

This direct axis inductance is an excellent measure of inductance if the machine has surface
mounted magnets. In this case, the inductances are typically equal in both axes and saturation has little effect due to the large airgap. The assumption of no saturation is insufficient
with the IPM machines due to the fact that it measures only the direct axis inductance
at one point, at a high current level in the machine. In an IPM motor, this would mean
calculating the direct axis inductance, and not having information on either the quadrature
axis inductance or the effects of both currents on this quadrature axis inductance value. The
aforementioned tests solve this by applying a voltage at standstill and measuring the rise
time; the rise time method is the classical inductance determination tool. In order to take
into account all current combinations and actual operating conditions, the most sensible
characterization method is that of the steady state dynamic characterization. The main
method used for inductance calculation has the machine spin at a constant speed while a
constant current vector is applied to the machine. With knowledge of the resistance and flux

12

linkage due to the magnets, the voltage is measured and the flux linkages then calculated.
This flux linkage depends on both currents and can thus be used for controller development
with parameters which will match machine performance during operation like in [7].
Having the machine moving at constant speed with a constant current vector will remove
the derivative terms for the currents from equations (2.5)-(2.8), and thus of fluxes in equation
(2.18).

vd = ids Rs − ωe λq

(2.18)

vq = iqs Rs + ωe λd
With the resistance known, the flux depending on both currents can be found from equation
(2.19).
v −ids R
λq (ids , iqs ) = d −ωe s

λd (ids , iqs ) =

vq −iqs Rs
ωe

(2.19)

Now that accurate machine parameters are known the controller depending on these
parameters can be developed.

2.1.3

Machine Control Development

During operation, there are two quantities of importance for proper torque and current
development: current commands to generate the torque command and voltage commands
to generate the current commands. In order to precisely develop torque, current must be
applied to the machine in the correct axes with knowledge of the parameters. This torque
relationship was given in equation (2.2). Thus, the desired phase current must be quickly and

13

accurately applied to the machine. All speed current controller systems take the form of Fig.
2.5. In most real world systems, this is done through the use of an inverter applying voltage
to the machine terminals and allowing the current to build through use of PI controllers.
With perfect machine parameter knowledge, the voltage required for an exact current can

‒

Iqs*

‒

INV

‒

T*

SVM

PI

[Tabc]-1

ω*

Ids*

Current
Controller

VDC

Torque
Controller

be found as in equation (2.20).

[Tabc]

M

Θ

d/dt

ω

Figure 2.5: Speed controller for machine being characterized with current controller in the
inner loop

L

q
∗
vqs = Rs iqs + T K (i∗ − Ki iqs ) + ωr (λaf + Ld ids )
i i qs
∗
vds

= Rs ids +

Ld
∗
Ti Ki (ids

(2.20)

− Ki ids ) + ωr (Lq iqs )

On first glance it is clear that there is no integral action in this controller. In fact, with
accurate parameters no integral action is needed! In any realized controller, any error in
current by application of this voltage would come from errors in the parameters. For this
reason, this proportional controller is used to measure the effect of errors in parameters
upon the output performance discussed later. This error in performance is the control effort
typically applied to the system by this integral action. This is a fairly straightforward method
to control the current in either the salient or nonsalient machine case.
14

2.1.3.1

Torque Control Development

With the machine model the current can be easily applied with the controller previously
discussed. While the current controller can be applied to any PM machine, salient or non
salient, the torque controller requires more consideration. Initially, only the current commands are needed in order to control torque in an electrical machine. By applying a specific
current vector, a specific torque can be expected given the machine parameters and equation
(2.2). In the case of a non-salient rotor, the inductances are the same in the torque equation
(2.21); the only torque in this case is due to the magnets. Therefore, controlling iq can allow
for precise torque control. Thus, in this nonsalient case, the torque can be simply controlled
by the modified torque equation which includes the fact that the inductances are the same.
This equation is solved for the quadrature axis inductance of interest given a torque command in equation (2.23). The trajectory of this current is only along the vertical, quadrature
axis as in Fig. 2.6. Quadrature axis current can be applied until the magnitude of this is
equal to the maximum current magnitude of the motor.

3
T = P [λaf + (Ls − Ls → 0)ids ]iqs
4
3
... = P λaf iqs
4
4 T
iq =
3 P λaf

(2.21)
(2.22)
(2.23)

The next main portion of the controller is important when the machine is operating above
base speed. In this operating condition, the voltage limit is reached, and thus the operating
point is on the circle defined by equation (2.24) and shown by the voltage limit circle in Fig.
2.6, therefore the voltage induced by the magnets must be reduced as base speed is exceeded.
To do this, the airgap flux must be reduced. An analytical result is found, that will result
15

in the amount of negative field weakening current to be applied to the machine in equations
(2.26) and (2.27).

(ids + isc

)2

+ i2
qs

=

... =

ids = −isc +
iqs =

i2 (
sc

λmax
s
Ls

2

Vmax
ωe Ls

ωs
ωs 2
) + ( )2 − 1 (iqs )2
ωb
ωb

4 T
3 P λaf

(2.24)
(2.25)

(2.26)
(2.27)

In the case of the controller development, since the inductance is dependent upon the
applied current, the currents corresponding to a specific torque and speed command are
found computationally. The inductances with respect to he currents must be used. For
below the base speed, the relationship between the torque and quadrature axis current is
found. Above base speed, for each speed, the resulting torques and voltages are found with
the machine model for every current combination. There will be a direct axis current level at
which the torque is maximum, and this is the value of direct axis current which will be used
for operation at that given speed. Other speeds will exceed the voltage limit while other
exceed the current limit. The various limits on the machine are illustrated in Fig. 2.6. Fig.
2.6 shows the current limiting circle, defined merely by the maximum allowable current. The
figure also shows the voltage circle, noting that as speed increases, the voltage circle shrinks,
as seen by the two different voltage limit circles at ω1 and ω2 . The circle shrinking can be
seen by the speed dependence in equation (2.24). During operation the current applied to
the machine must remain within both the current and voltage limit circles. Beyond this,
16

Quadrature Axis Current (A)
Voltage Limit
at 1

Non-Salient
Machine Trajectory

Vmax
1 Ls

Current Limit

Voltage Limit
at 2 > 1

I max
Vmax
2 Ls

-isc

Direct Axis Current (A)

Figure 2.6: Voltage and current limit circles for the nonsalient case. For interpretation of
the references to color in this and all other figures, the reader is referred to the electronic
version of this thesis.
there exist two cases for machine type. Some machines have the center of the voltage circle,
the short circuit current, in the current limit circle. It is this case the thesis discusses since
the machine of interest has a short circuit current within the current limit circle.
This trajectory is derived from the ratio of torque to current. For each speed and torque
point the minimum stator current is what determines the current and the trajectory. The
current requirement that corresponds to a certain Torque/Speed point is added to the controller lookup table as the ideal current condition at that Torque/Speed operating point.
This is the point appended to the specified maximum torque per ampere trajectory for each
speed. The current trajectories are shown in Fig. 2.7, and can also be found analytically
17

Quadrature Axis Current (A)

Non-Salient
Machine Trajectory

Salient Machine Trajectory

Direct Axis Current (A)

Figure 2.7: Maximum torque per current lines
using the Torque/Current relationship in equation (2.28), which results in (2.32), given some
machine information.
If the current limit is reached before the voltage limit, the trajectory must continue along
the current limit line as in equation (2.33).

T
Is

= 3 P λaf + (Ld − Lq )ids iqs /Is
4

(2.28)

2χ
T = iq 1 − isc ids

(2.29)

L −L
χ= q d
2L

(2.30)

d

λ

af
isc = L
d

ids = isc −
8χ
iqs =

i2
s

(is )2
2

− i2
ds
18

(2.31)
2
+ isc
8χ

(2.32)

i2 = i2 + i2
s
qs
ds

(2.33)

A controller designed like this suffices only below the base speed. Above the base speed,
the voltage limit is reached, and the machine must be operated at the maximum voltage,
or constant flux ellipse. This ellipse (which is for the ideal case of constant inductances)
is defined by (2.36) and shown in Fig. 2.8. An increasing speed translates to an ellipse of
smaller radius. Using the parameters which are dependent upon the machine current, the
voltage and torque for all current and speed combinations are found. Once the commanded
torque has been reached, the constant torque line in equation (2.29) is followed by setting
the torque constant and finding the least current possible for that torque at a given speed;
this defines a line to travel from beyond the base speed until the maximum current line
in equation (2.33) is reached. When the constant torque or maintaining a path along the
maximum current circle is not possible, the maximum torque per flux line in equations (2.34)
and (2.35) is maintained and followed to the horizontal axis as the speed increases.

( Vmax )2
ωe L

isc
−
ids = −isc + (1 + 2χ)
8χ
iqs =

d

2

isc
+ (1 + 2χ)
8χ

1
Vmax 2
) − (κ + ids )2
(
1 + 2χ ωe Ld

(ids + isc )2 + (iqs )2 (2χ + 1)2 = ( Vmax )2
ωe L d

2

(2.34)
(2.35)

(2.36)

These methods result in motor controllers considering of all parameters. These controllers
are used in the operation of the machine.

19

Quadrature Axis Current (A)

Voltage Limit
at 1

Vmax

Voltage Limit
at 2 > 1

1

L d (2 +1)
Vmax

2

L d (2 +1)

Vmax
2

Ld
-isc

Direct Axis Current (A)

Vmax
1

Ld
I max

Current Limit

Figure 2.8: Voltage and current limit circles for the salient case

20

Preparation (LUT Creation)
ID, IQ

VDC

Ω

Generate Voltage and Torque
Save Optimal Points
(one per T/Ω/VDC pair)
(ID,IQ)=f(T,Ω,VDC)
Operation
T*

VDC

Ω

(ID,IQ)=f(T,Ω,VDC)

Figure 2.9: Development of lookup table and use during operation
2.1.3.2

Matlab Controller Extraction

In order to design the complete machine controller as it depends upon the current dependent
flux linkages a simple analytical method will not suffice in a simple way. This is due to the
fact that the flux changes, so the paths will not be as smooth as assumed. Initially, the flux
linkages with their dependence upon the machine currents should be found. In addition to
these flux linkages, the limits of the machine, including the DC link voltage, phase current
limit, phase resistance, number of poles, speed range and current range should be found.
The controller development and utilization are two steps in the process of the development
of a working controller, as shown in Fig. 2.9. The first step of preparation begins with: The
direct and quadrature axis voltages as well as torque are calculated for each theoretical point.
The result of this will be a three dimensional matrix with respect to both currents and speed
as in Fig. 2.10.
From this, only the possible combinations are allowed. While the current in either axis
21

ed

Quadrature Axis Current

e
Sp

Negative Direct Axis Current

Figure 2.10: Calculation points depending on speed, direct and quadrature axis current
can vary from zero to the maximum allowable current, the magnitude if both axes had the
maximum present exceeds the machine limit; the points exceeding these limits are removed.
The same applies for voltage, where the voltage points which exceed the level the machine
can operate at are removed.
Once this is completed, the optimal operating points will be extracted. In the case of
the SPM machine, the direct axis current will be zero for operation below base speed, but
constant above base speed. If the direct axis current is too low, the machine voltage limits
will be exceeded; if too high, there will be wasted flux weakening, and thus the voltage will
be too low. Therefore for both operating regions, finding the maximum torque per unit
current will give the optimal voltage point, which will happen to be increasing below base
speed and hold to the voltage limit above base speed. Such a line for an SPM machine is

22

Quadrature Axis Current

Negative Direct Axis Current

Figure 2.11: Optimal choice of ids demonstrated for a non-salient machine
given in Fig. 2.11 and is found by incrementing the quadrature axis current and finding
for a certain speed and quadrature axis current the direct axis value at which the torque is
greatest and current lowest. This method is likewise followed for the IPM machine shown in
Fig. 2.12 where even below base speed this calculation results in nonzero direct axis current.
Once this is complete, the only remaining points will be the determination of realizable,
optimal points for that current and speed combination. The points can then be assembled
into a lookup table. All these points now give us a relationship in equation (2.37), while for
torque control we want the inverse relationship. Finding this relationship requires mapping
the relationship in the other direction. In order to do this the tables are traversed and for
each torque, speed point the least current point, or nearest to the rated voltage is found and
added to the set of final operating points. It is this relationship of direct and quadrature

23

Quadrature Axis Current

Negative Direct Axis Current

Figure 2.12: Optimal choice of ids demonstrated for a salient machine
axis currents dependent upon torque and speed which is the lookup table.














ID 



ID 




→

Ω












VD 



VQ 




(2.37)

T

The only step remaining is the resampling and interpolation of the table to achieve a
uniform table with the desired number of points, as is shown in the ‘Operation’ step in Fig.
2.9. The DSP can then interpolate these values, so an analytical relationship is not needed,
and this lookup table will suffice.

2.2

Parametric Error

Parametric error receives the brunt of the blame for controller performance degradation. This
is due to the fact the variation in parameters will change the performance of the machine.
24

In the case of the torque controller for instance, an error in the parameters can result in a
different current command than desired. This will then produce a torque either higher or
lower than expected. It is the degree of this ‘lower’ or ‘higher’ that is to be quantified with
respect to various parameters.

2.2.1

Sources of Parametric Error

2.2.1.1

Characterization Errors

The first and main source of parametric error is the failure to characterize the machine
accurately. The main source of this is the assumption that the machine parameters from the
manufacturer are true for all machines produced. Typically when a machine is given to the
user, the parameters are also given. It is these rough parameters that are mostly used. With
a multitude of variations including manufacturing, operating conditions, etc. the parameters
can vary from those given by the manufacturer; parameter error is present in situations where
each machine is not characterized individually. In addition to the lack of characterization
and complete trust in manufacturer parameters, the system used to characterize by the
manufacturer may have errors. These errors can be a result of misalignment, sensor, or
computational characterization error. The most common characterization error results form
the variation in resistance, which is used to calculate the flux linkages; the flux linkages will be
incorrect in this case. It is therefore important to do as accurate a characterization as possible
to have a controller with achieves precise performance. Having parameters dependent upon
as many parameters as possible helps, but typically current dependence will suffice for the
salient machine case.

25

2.2.1.2

Nonlinear Steel Characteristics

A problem source in the case of the salient machine is the fact that the iron saturates. In the
non-salient machine, as the airgap is large, the flux developed for a certain current is not very
large compared to a machine with a smaller airgap. An IPM machine has a smaller airgap,
given that the magnets are embedded in the rotor. Because this airgap is smaller, the flux
produced can be quite high. As the level of flux increases, the machine steel saturates. This
saturation will reduce the effective inductance of the machine and thus affect the controller
greatly. If the control design does not take into account the saturation of the steel, the
controller will have a great deal of error. The quadrature axis, having a greater inductance
than the direct axis, is most susceptible to this effect. The direct axis is weakly affected
by flux, but the current in this axis can cause further saturation in the quadrature axis,
especially at higher currents. Thus, the characterization should include a variety of currents
in both the direct and quadrature axis to have inductances which depend on these, and thus
saturation and cross-saturation. Recall that Fig. 2.4 shows the effect that current can have
upon the flux developed and thus the inductance. The inductance depending on this flux is
shown in Fig. 2.13.

2.2.1.3

Eccentricity

One case of inductance changing is from a combination of a manufacturing variation and
saturation of the machine iron. Eccentricity faults manifest themselves in a rotor being
shifted from the concentric position, resulting in the airgap no longer being uniform. This is
very possible due to manufacturing as it is very difficult to center the rotor. In the eccentric
case, the machine can be divided into two parts, the upper and lower circuit as in Fig. 2.14.
The flux developed in the direct axis for instance in this case is greater on the upper circuit
26

q-axis Inductance (H)

q-axis flux (T)

q-axis current (A)

Figure 2.13: Inductance and flux curves showing the effect of saturation in iq
until the point at which saturation takes effect as in Fig. 2.15, the average of the upper
and lower fluxes will be lower than that of the nominal machine. Thus, the flux developed
for the same current in an eccentric machine is lower; the lower flux for a certain current
corresponds to a lower inductance. From this, it is seen that eccentricity can be a source
of parametric error with respect to the inductances. A similar inductance change happens
in the quadrature axis inductance. Thus, this is one more reason to accurately know the
parameters of the machine and the effects of parametric error.

2.2.1.4

Uniform Demagnetization

Uniform demagnetization is a phenomenon where the magnets of the machine all lose the
same amount of remnant flux. This is a problem for torque development, which is heavily
dependent upon the flux linkage due to the magnets. This phenomenon is not uncommon, as
the rare-earth based magnets can become demagnetized at high, but possible temperatures.
27

q-axis
d-axis

Figure 2.14: Eccentric IPM machine model

d-axis flux (T)
Upper Magnetic Circuit

λaf

Lower Magnetic Circuit
Healthy Machine

d-axis current (A)

Figure 2.15: Flux curve for an eccentric and concentric machine (Based on a graphic in [3])

28

B

20
Celsius

100
Celsius

H

Figure 2.16: Flux curve for demagnetized machine (Based on a graphic in [11])
This is due to the fact that the magnetic hysteresis curve is shifted during high temperatures,
because the coercivity is reduced.
Note the plots of flux on the flux density and field intensity axes in Fig. 2.16 with the
intrinsic and normal curves of the flux for both 20 ◦ C and 100 ◦ C. If at 20 ◦ C the operating
point on the normal curve is the light green dot. If the magnet temperature rises to 100
◦C

the new operating point is given by the hollow circle with a red outline. Because this

operating point is below the knee of this curve, operating point does not return to the original
operating point, but a new operating point. This new operating point is given by the dark
blue dot, and the remnant flux of the magnet reduced to the intersection of the dashed
line with the flux density axis. This is therefore important, as the remnant flux density is
reduced, which will affect control performance; for this reason we seek to quantify the effect
of demagnetization upon controller performance.

29

2.2.1.5

Heating

It is well known that as the windings of a motor are heated, the resistance of these windings
change. This is a common occurrence during prolonged operation, as is common in these
machines. The resistance changes with the relationship given in equation (2.13) from [10].
This simple relation is the motivation for studying resistance errors in the control of machines.

Rs = Rs0 + αRs0 (Ts − T0 )

(2.38)

This relationship is found from Rs0 , the resistance at an initial temperature T0 ,. Further,
−
α is the temperature coefficient for the material which is copper (2.92e − 3 ◦ C ) and Ts is the

temperature of concern.

2.2.2

Effect of Parametric Error Upon Performance

An analytical method is used to predict the variation of performance with respect to changing
parameters from [5]. This method is expanded upon to include all parameters and include
operational analysis at all speeds. The analysis performed will concern five sets of equations:
the current controller equations (2.39)-(2.42), the nonsalient torque equations (2.43)-(2.44),
the salient torque equations (2.45)-(2.46), the flux dependent torque equation in equations
(2.47)-(2.48), the nonsalient voltage circle equations (2.49)-(2.50), and the salient voltage
ellipse equation (2.51)-(2.52). Note in these, the quantities with errors present are denoted
by their primed values. α is a fraction used to show variation in a parameter. The X’ is
the value of incorrect X parameter assumed. αd , αq , αr , αs , αf , αf d , and αf q represent
the d-inductance, q-inductance, resistance, stator inductance in the nonsalient case, flux
linkage due to the magnets, direct axis flux, and quadrature axis flux fraction errors. These
30

Table 2.1: Operating point quantities
Parameter
ωe
Ld
Lq
λd
λq
Id
Iq
Ti
Rs
λaf

Description
Value
Electrical Motor Speed
167.6 rad
sec
Direct Axis Industance
48.7(mH)
Quadrature Axis Inductance
86(mH)
Direct Axis Flux Linkage
0.55(V · s)
Quadrature Axis Flux Linkage
1.021(V · s)
Direct Axis Current
−6.604(A)
Quadrature Axis Current
11.87(A)
Inverter switching period
100(µsec)
Phase Resistance
1.4(Ω)
Flux Linkage due to the Magnets 0.87(V · s)

values are allowed to range anywhere from 0 to 2, where 1 represents accurate parametric
knowledge. For this analysis, the third machine used in this paper with an actual operating
point and experimental data. The parameters from this are given in Table (2.1).
In the case of the current controllers, with multiple parameters, a graphical approach is
taken to analyze the parametric sensitivity. The equations in (2.39)-(2.42) show the voltage
command equations with respect to their actual parameters, commands and α error fraction.
In order to do this analysis, these equations are solved for their actual current value. Assuming that the controller parameters remain constant, and the machine parameters change,
the changing voltage command with respect to parameters can be found. This is of interest
because the machine parameter change could cause voltages to be applied which are out of
the machine voltage operating range. Even when within the limits, voltage command errors
will translate to errors in the developed current. In Fig. 2.17, it can be seen that the direct
axis inductance has a significant effect, while quadrature only has a significant effect at low
values of quadrature axis inductance. Fig. 2.18 shows that resistance has little effect on the
direct axis voltage command, and that the flux linkage due to the magnets clearly has no

31

effect because the equation does not depend upon the flux linkage to to the magnets. Fig.
2.19 shows that the direct axis inductance does not have a large effect, which the quadrature
axis inductance does. Finally, Fig. 2.20 shows the flux linkage due to the magnets has a
significant effect while resistance less so. Note that the flux linkage due to the magnets has a
lesser effect than normal as the machine is heavily loaded, and as such the voltage is mostly
determined by the resistance and inductance.
L′
q
(i∗ − Ki iqs ) + ωr (λ′ + L′ ids )
af
d
Ti Ki qs
αq Lq ∗
... = αr Rs iqs +
(i − Ki iqs ) + ωr (αf λaf + αd Ld ids )
Ti Ki qs
L′
′
∗′
vds = Rs ids + d (i∗ − Ki ids ) + ωr (L′ iqs )
q
Ti Ki ds
α L
... = αr Rs ids + d d (i∗ − Ki ids ) + ωr (αq Lq iqs )
Ti Ki ds
′

′
∗
vqs = Rs iqs +

(2.39)
(2.40)
(2.41)
(2.42)

In the nonsalient torque equation as given with the parameter scaling factor in equation
(2.44), the effect of the changing flux linkage quantity can be investigated. As αf is reduced
below 1, or as the flux linkage due to the magnets is reduced, the torque expected is reduced.
Thus, for the maintenance of the desired torque, more current would need to be applied. This
in effect lowers the torque limit, as the machine rated current remains constant. Similarly,
if this αf is increased to 2, the torque expected is doubled for the same current. The actual
torque developed in an open loop torque controller will be half, as we apply only half of the
current of interest to get the previously desired torque. This in effect lowers the maximum

32

10
0

′

id /id (dB)

−10
−20
αq
−30
−40
−50
−60
0

0.5

1
αd

1.5

2

Figure 2.17: Change in the direct axis developed current with changing inductances

33

0.03

0.02

id′/id (dB)

0.01

0

−0.01

−0.02

−0.03
0

0.5

1
αr

1.5

2

Figure 2.18: Change in the direct axis developed current with changing resistance and flux
linkage due to the magnets
10
5

iq′/iq (dB)

0
−5
αq
−10
−15
−20
−25
0

0.5

1
αd

1.5

2

Figure 2.19: Change in the quadrature axis developed current with changing inductances
34

0.2
αr

0.15
0.1

iq′/iq (dB)

0.05
0
−0.05
−0.1
−0.15
−0.2
0

0.5

1
αf

1.5

2

Figure 2.20: Change in the quadrature axis developed current with changing resistance and
flux linkage due to the magnets

35

torque, as the controller will apply half of the current actually needed to give a given torque.

3
T = P λ′ iqs
4 af
3
... = P αf λaf iqs
4

(2.43)
(2.44)

In the salient case, the torque produced is due to both the magnets and the saliency. The
magnet torque is affected in the same manner as the nonsalient case as αf is changed in
Fig. 2.21. The new analysis here is dealing with the change in direct and quadrature axis
inductances. Clearly, both the direct and quadrature axis inductances have a large effect
upon the torque as in Fig. 2.22. This makes their accurate determination during operating
important.

3
T = P [λ′ + (L′ − L′ )ids ]iqs
q
af
d
4
3
... = P [αf λaf + (αd Ld − αq Lq )ids ]iqs
4

(2.45)
(2.46)

Rather than building a controller with inductances, creating one with flux linkages has its
advantages. These advantages are apparent when one tries to find the inductance in either
axis with little current present. The inductance will be quite large with a small current
measuring error, as the flux derived in the characterization section 2.1.2 with equation (2.19)
is divided by this current. Thus, looking at sensitivity of the torque equation with respect to
the flux dependant torque equation will be of some use. These equations are given in (2.47)
and (2.48). The effect that errors in the known fluxes can be seen in Fig. 2.23, where both
flux errors have a large effect upon the developed torque. The large effect can be understood

36

6
4

T′/T (dB)

2
0
−2
−4
−6
−8
0

0.5

1
αf

1.5

2

Figure 2.21: Change in the torque with changing flux linkage due to the magnets

6
4
2

′

T /T (dB)

0
−2
αq
−4
−6
−8
−10
−12
0

0.5

1
αd

1.5

2

Figure 2.22: Change in the torque with changing direct and quadrature axis inductance
37

in that changing the flux indicates an error in the inductance/current product.

3
T = P λ′ iqs − λ′ ids
q
d
4
3
... = P αf d λd iq − αf q λq id
4

(2.47)
(2.48)

60

40

αfq

0

′

T /T (−)

20

−20

−40

−60
0

0.5

1
αfd

1.5

2

Figure 2.23: Change in the torque with changing direct and quadrature axis flux linkages

In the case of the nonsalient voltage limit circle equations in (2.49) and (2.50), the effect
of inductance error upon the controller is seen in the change in the radius of the voltage
circle. If αs is 0.5, the radius of the voltage circle is doubled, which allows the controller to
operate outside of its actual voltage limits. This can destroy the machine or prevent normal
operation, and thus care should be taken to allow for DC link voltage margin that allows for

38

some change in the inductance while keeping below the voltage limits. If αs is 2, the voltage
circle will shrink to half of its normal value, and merely artificially limit the operating range
of the machine.

(ids + isc )2 + i2 =
qs
... =

Vmax
ωe L′
s

(2.49)

Vmax
ωe αs Ls

(2.50)

(ids + isc )2

(iqs )2
+ V
= 12
Vmax 2
max )2
)
(
(
ωe L ′
ωe L ′
q
d

(ids + isc )2
( ωVmax )2
e αd Ld

+

(iqs )2
( ωVmaxq )2
e αq L

= 12

(2.51)

(2.52)

In the case of the salient voltage limit ellipse equation simplified to show the radii in
(2.52), the effect of errors in both inductances upon the controller is seen in the change in
the radii of the voltage ellipse in Fig. 2.24. If αd is 0.5, the horizontal radius of the ellipse
will double, and again assume operation is possible outside of the actual voltage constrains.
The machine will either not operate, or in the extreme case, be damaged. The same is for
an αd of 0.5, which would double the vertical radius, and allow for operation outside the
machine voltage ratings. Again in the salient case, care should be taken to allow to allow for
a DC link voltage margin slightly higher than what the controller is designed for to allow for
an unmodeled decreased inductance value. If either αd or αq were to double, the range would
merely be reduced. The reduction in operating range reduces the utility of the machine, and
thus the inductances should be known accurately for effective machine use. Further, this
exemplifies the need for having a model including saturation during controller development.

39

Quadrature Axis Current (A)
Voltage Limit
at
Vmax
L q (2"+1)

Voltage Limit
at !q = 2

0.5 Vmax
L q (2"+1)
0.5Vmax
Ld
Vmax
Ld

-isc

Direct Axis Current (A)

I max
Current Limit

Voltage Limit
at !d = 2

Figure 2.24: Plots for the changes in voltage ellipse due to parameter changes

2.2.3

Quantification of Parametric Error

A standardized technique must be found in order to make the relationship between parametric error and performance error clear. The parameters and performance measures have
been clearly defined as tools for controller operation evaluation. Comparing the ‘output’, the
variation in performance with the ‘input’ variation in the parameter which led to this error
is the clearest measure possible. In order to experimentally determine the effect of parametric error, an error is induced in the controller while the machine remains unchanged. This
simple error is given in [1] by the relationship in equation (2.53). The parameter quantity is

40

given by ξ and the error induced in it is δξ.

ξerror = ξ(1 − error% )
100%

(2.53)

Once the error has been introduced and the controller has been modified with the erroneous parameters, the machine is operated from standstill to an operating point and the
output performance is measured. The performance measure is given as P and the change
in performance is δP . The ratio of the normalized change in performance to the normalized
change in the parameter is defined as the sensitivity in equation (2.56).

∆ξ→P =

δP
P
δξ
ξ

(2.54)

δP = Pactual − Perror

(2.55)

δξ = ξactual − ξerror

(2.56)

This relationship gives a simple, quick, standardized way of measuring the effect that an
error in a parameter has upon the output performance.
In the case of the dynamic performance of a machine, numerous dynamic performance
measures can be used. These parameters represent the operation of the machine in the
transient as the machine transitions to the steady-state. The parameter and its error are
again used, but to distinguish the dynamic performance from the power performance, S and
its difference from the original δS are used. This new relationship gives the effect of a change
in a parameter upon the output performance of operation, such as a change in inductance’s
effect on the time to reach the steady-state torque. This relationship is given in equation

41

(2.59).
δS

S
∆ξ→S = δξ

(2.57)

ξ

δS = Sactual − Serror

(2.58)
(2.59)

2.3

Sensitivity Analysis Procedure

As the goal of this thesis is to find the effect that multiple parameter errors have, the process
of computing the sensitivity with performance measurements of a machine or controller
with erroneous parameters must be repeated multiple times. This process of computing
the sensitivity from an erroneous controller is defined as a sensitivity set. A set results in a
sensitivity from a specific parameter error to a specific performance outcome. This sensitivity
set involves a process of characterization, controller development, performance measurement,
and sensitivity calculation. This procedure is standardized and repeated multiple times to
result in an array of sensitivity measures; various parameters and performance measures are
used. Fig. 2.25 shows this procedure for one such sensitivity set.
The sensitivity procedure is summarized in Fig. 2.25. The first step is to define the
machine model and the typically fixed parameters which are used in the machine model.
The second step is then to extract these parameters using a characterization procedure.
Then one controller with accurate parameters and another with parameters which have
errors introduced in them is used to control the machine. The performance is measured and
compared by calculating the sensitivity.

42

Machine Model

Characterization

Introduce Parametric Error
Develop Nominal Controller
Develop Skewed Controller

Apply to the Machine
Performance

Sensitivity Calculation

Figure 2.25: Flowchart of the procedure to determine sensitivity set

43

Chapter 3
Experimental and Simulation Results

3.1

Experimental Characterization Procedure

The foremost goal of experimental and FEA is to verify assumptions made in the analysis.
The analysis discussed shows that it is possible to evaluate parametric sensitivity. The FEM
verifies that the analysis holds with an actual machine model. The experimental work proves
both the analysis and FEA, demonstrating that the work is practical. FEA and Experimental
work are all done to verify the predicted results.
Overall, for data acquisition, the voltages, currents, torque, speed, and position are
recorded. The appropriate quantities are then converted from their line-to-line values in the
case of voltage to the phase voltage in equation (3.1). The phase voltages are found, then
transformed to the rotor frame of reference by the relationship in equation (3.2), the opposite
of the transformation discussed in section 2.1 with the transformation matrix of equation
(2.4).
Va = 2 Vab + 1 Vbc
3
3
Vb = −Vab + Va
Vc = −Vbc + Vb

44

(3.1)






r
xqs 




 r 
x 
 ds 




x0

=



xas 






Tabc  xbs 






xcs

3.1.1

Characterization Procedure

3.1.1.1

(3.2)

Open Circuit Testing

The back-EMF is found by spinning the machine which receives a speed command, having
removed the inverter connections from the second machine, or the one being characterized,
and only measuring its terminal voltage. In order to characterize it, the machine must be
appropriately oriented to apply current aligned with the flux, or of any angle beyond this.
The rotor alignment is similar to [7], oriented to the positive peak of the back-EMF. In Fig.
3.1 the alignment is displayed with respect to the mechanical position as well as the measured
back-EMF. This is from the acquired data and is for clarity. Thus, with the characterized
machine being open-circuited, the machine equations simplify to (2.14) and (2.15), and are
repeated in equation set (3.1). The voltage can then be transformed into the rotor frame of
reference. Once in the rotor frame of reference, the d-component of the back-EMF should
be zero, whereas the q component contains the constant value which is the back-EMF if the
machine is properly aligned.
For the measurement of this λaf , the rotor must be spun stably at a constant velocity
while the voltage is measured. From this voltage the BEMF is determined with the equations

45

Normalized Units (Vn), (degrees n)

1.004

Back EMF
Unaligned Angle
Aligned Angle

1.002
1
0.998
0.996
0.994
0.992
0.99
0.988

0.986
0.0247 0.0248 0.0249 0.025 0.0251 0.0252 0.0253 0.0254
Time (sec)
Figure 3.1: The angle offset plot used to first find angle offset, subtract this from the angle
to achieve the angle aligned to the positive back-EMF peak

in (3.3).
vqs = ωr λaf

(3.3)

vds = 0
In addition to determining the alignment with respect to rotor position and back-EMF
from these voltage measurements, the number pole pairs can be determined by plotting
the phase voltage. The number of complete back-EMF cycles in one mechanical revolution
defines the number of pole pairs.

3.1.1.2

Phase Resistance

In order to determine the phase resistance, an elementary DC test is used which is performed
with the setup in Fig. 3.2 including two phases of the machine, a voltage transducer, and
a DC supply with controllable current and voltage limit. From this test, a voltage and
46

current will be derived. Using these two quantities, the resistance can be calculated with the
equation in (2.13). Note that the resistance measured is twice that of the phase resistance.

CT

CS
Figure 3.2: DC test for phase resistance

3.1.1.3

Flux linkages

In this section, the flux linkages for all possible currents are determined. The machine
being characterized must be connected to the inverter, and without flux linkage knowledge,
a simple controller must be developed. This controller discussed earlier will take the form of
Fig. 2.5.
In this, a direct and quadrature axis current command is given, and if the gains of the PI
controllers are chosen properly, the machine will attain these in reasonable time. A current
angle and magnitude are defined and applied, resulting in the direct and quadrature axis
currents as in equation (3.4).
ids = |Is | cos δ

(3.4)

iqs = |Is | sin δ
In characterizing, commands for as many operating points are given as should be expected
during operation. Typically this means giving current magnitude commands ranging from
47

q -axis
I

d-axis

Figure 3.3: Current vector range applied during characterization

no current to the maximum current rating of the machine. For each of these magnitudes,
the angle should be varied from the point of maximum torque and no flux weakening, to
the point of maximum flux weakening and no torque. An example of this is given for the
entire second quadrant in Fig. 3.3. Knowing the magnitude variation is straightforward,
but knowing the angle variation is not. Recall the voltage and flux relationship in equation
(3.5).
dλ

A
VA = dt

(3.5)

The back-EMF is the voltage induced by the magnet flux. For alignment, the back-EMF
can be visualized as a cosine wave with zero phase. The vector is represented in Fig. 3.4
diagram showing the magnetic axes with a fictitious one pole pair rotor. This means that
the integral of a cosine wave will result in the flux linkage by phase A. In order to weaken
this field a negative sine must be applied for phase A. From the defined transformation, an
angle of δ = 0 for any magnitude value gives a positive direct axis current, and a sine wave
as the voltage command. This means that an angle of δ = 0 results in the maximal flux
strengthening. Thus, to operate in the flux weakening range with positive torque which the
48

machine will most likely be operated in, the angle δ should range from 90 ◦ through 180 ◦ .
A plot showing the relationship between phase A quantities with repect to rotor position is
given in Fig. 3.5.
q-axis
b-axis

Θr

S

d-axis

N
a-axis

c-axis

Figure 3.4: Vector diagram including fictitious rotor magnet

λA = VA dx

(3.6)

Thus to gather the flux linkage data, a large number of magnitudes and angles should be
used spanning the range of operation. For each, the magnitude, angle, torque, speed, as well
as the voltage and current waveforms were recorded. Using the transformation to the rotor
frame of reference, the rotor aligned values are found, and should be, like the speed, constant.
This information will be used to find the flux linkages if operation at constant speed and
current is maintained. This is necessary as the machine model simplifies to equation (3.7),

49

Current (A), Voltage (V)

Back−EMF (V)
Magnet Flux (V⋅ s)
Maximum FW Current (A)
Magnet Torque Current (A)

0

1

2

3
4
Angle (rad)

5

6

7

Figure 3.5: Sinusoids for the phase A quantities

repeated from section 2.1.2.
λq =
λd =

−vds +ids Rs
ωe
vqs −iqs Rs
ωe

(3.7)

The flux linkages are found from this, and this procedure should be repeated for every
operating point of interest, so that that the flux linkages are found from magnitudes ranging
from zero to |Imax | and from angles ranging from 90◦ to 180◦ . It was shown before that the
flux linkage is found with integration of the voltage, which would of course have the problems
introduced by integration; in this circumstance as the flux linkages are calculated without
integration these problems are avoided. The fluxes can then be used to find the inductances
and transient inductances. If the data is noisy, some post processing may be helpful.

50

3.1.2

Flux Linkage Data processing

For each of the voltage and current data, a simple filter can remove switching noise and
other harmonics, so that the fundamental can be derived, being the main torque producing
component. A third order Butterworth filter is used to filter the data in this paper. Using
MATLAB, a digital filter is created, which is used to extract the fundamental from all of
the noisy waveforms. As the f iltf ilt() function is used in MATLAB, zero phase delay is
achieved, but the magnitude attenuation must be determined using the bode function as
the signal is attenuated by the square of the filter magnitude at the appropriate frequency.
Note that the f iltf ilt() function determines the phase delay for the resulting frequencies,
and uses this to compensate and thus achieves zero phase delay. In order to determine the
exact frequency, the Fast Fourier Transform is performed on the filtered waveform, and this
frequency is used to determine the magnitude attenuation. Using this method, the waveform
in Fig. 3.6b can be achieved from Fig. 3.6a.
Once the fundamental can be derived, the d- and q- axis components can be derived
in one of two ways: the Park’s transformation or measurement at specific positions on the
waveform. The Park’s transform will give a constant direct and quadrature axis value, as
it calculates at every angle. The phase A, B and C values as well as position are read
and transformed with equation (3.2). This is one method, but another can be performed
by simply comparing the phase A voltage or current with the position. In Fig. 3.7, the
back-EMF compared to the phase angle. The value when position is zero degrees gives the
direct axis quantity, while the value at position of ninety degrees gives the quadrature axis
quantity. It is clear that this will not give you the direct and quadrature axis values at all
times, just those that coincide with the angle at 0 ◦ or 90 ◦ .

51

Voltage (V)

10
0
−10

Voltage (V)

0

0.05

0.1
Time (sec)
(a) Unfiltered Back EMF

0.15

10
0
−10
0

0.05

Time (sec)

0.1

0.15

(b) Filtered Back EMF

Normalized Units (Vn),(degrees n)

Figure 3.6: Demonstration of filtering to produce clearer waveforms

Back EMF
Electrical Angle

1.5
Quadrature Axis Value

1
0.5
0
Direct Axis Value

−0.5
−1
0.132 0.134 0.136 0.138 0.14 0.142 0.144 0.146
Time (sec)

Figure 3.7: Plotting the phase A voltage with back-EMF to find d- and q- axis values

52

3.2
3.2.1

SPM Machine Current Controller Sensitivity
Method of Characterizing SPM and IPM machines

Note that in the model, the main parameters which must be found are the phase resistance,
the flux linkage constant due to the magnets, and the flux linkages at each operating point.
In order to do this characterization for the SPM machine a characterizing system which
utilizes two identical machines, each controlled by an inverter is used. In one inverter, a
DSP, the Texas Instruments fixed point DSP TMS320F2810 is used to control the speed
of one machine, with a simple PI controller system, from commands given through the
Controller Area Network by an external computer controlled by a Labview interface. Another
inverter containing the Texas Instruments floating point DSP TMS320F28335, which controls
a machine mechanically attached to the shaft of the other, takes current commands, including
magnitude and angle from a computer, through the controller area network.
For each machine, two line-to-line voltages and two phase currents are measured for
data acquisition by voltage sensors and current transducers (CT), respectively. Position is
read with an encoder from both inverters, which is used to measure position and speed. A
torque sensor, which connects to the shafts of both machines, measures the torque applied
between the machines. This data acquisition is performed by a Labview developed data
acquisition system, which reads inputs through two National Instruments Data Acquisition
(DAQ) cards. The general machine system is shown in Fig. 3.8, including all sensors.
The sensitivity measure in section 2.2.3 is applied to the machine in simulation, found to
have the parameters as in Table (3.1). Note that the inductance is not current dependant,
but chosen to be constant due to a low degree of saturation. The speed is not the rated
machine value, but the machine operating condition for the tests performed. Note that
53

CAN BUS

Inverter

Inverter

PC
CT
Θ

CT
M

CT

Torque
Sensor

CT
M

Θ

PMSM

PMSM

Figure 3.8: Physical test setup for SPM machine characterization with inverters, machines
and sensors

only one inductance is given. This is due to the fact that the machine has surface mounted
magnets, which should then not result in a large of inductance variation by axis.
Table 3.1: Characteristics of surface mounted permanent magnet machine
27.5 (mΩ)
Resistance of One Phase
30
Number of Poles
0.0862 (V · s) Flux Linkage due to Rotor Magnets
3.585 (mH)
Total Inductance
rad
80 sec
Electrical speed of machine

Now that the sensitivity measurements have been defined, simulations will be performed
while varying each parameter. For each variation of parameter, the output performance is
measured: dynamic, power, efficiency, and torque performance measurements. The first table
shows the performance data: the torque, power and efficiency readings. The second shows
the effect of parametric errors on dynamic performance data including rise time, settling
time and steady state error. Finally, the last table shows the sensitivity, calculated with the
method discussed in the last section. Note that in each instance an error is introduced into

54

one of the machine parameters in the controller. For instance in Table (3.2) the first line is
the result of no error, whereas the second line is the result of halving the phase resistance.
Only one parameter is given an error at each test, and each is halved in the respective test.
Table 3.2: Table showing the various output metrics with respect to varying parameters
P
Nom.
Rs
Ls
λaf

Error (%)
0%
50%
50%
50%

T orque (N · m)
97.776
97.77
97.57
97.6

P ower (W )
4991.88
4991.69
4981.88
4982.62

η (%)
97.94%
97.93%
97.93%
97.94%

Table 3.3: Table showing the various dynamic response metrics with respect to varying
parameters
S
Nom.
Rs
Ls
λaf

Error (%)
0%
50%
50%
50%

Tr (sec)
Ts (sec)
ess (%)
2.71e − 4 0.1003259
0%
1.27e − 4 0.1003258 −0.00383%
2.64e − 4
0.1003
0%
2.6444e − 4 0.1003311 −0.18033%

Table 3.4: Table showing the sensitivity of output and dynamic response metrics with respect
to varying parameters
∆ξ→S,P
Rs
Ls
T (N · m) 70.09e − 6 4.09e − 3
P (W )
0.076e-3
4e-3
η (−)
1.81e-5
9.22e-5
Tr (sec)
1.06
4.7e-2
Ts (sec)
2e-6
3.5e-4
xss (A)
7.67e-5
0

λaf
3.62e − 3
3.71e-3
6.2e-5
4.63e-2
1.05e-4
3.61e-3

From the data shown, some observations are made as to the sensitivity effects of various
machine parameters. The torque is most affected by errors in the inductance as well as flux
55

linkage due to the magnets, being significantly larger than the other two. This is because the
torque depends heavily upon the inductances and flux linkage due to magnets. Resistance
is shown to have a much lesser effect upon the output. The power is most affected by the
inductance since this affects the calculated voltage command and an almost negligible amount
by the phase resistance. The rise time had a large sensitivity with respect to phase resistance
since the voltage command is reduced for any speed whereas the other metrics are much less
significant. In the case of settling time, only the phase resistance does not have a large effect,
the others all had a fairly similar effect. Finally, the steady state error is affected by the flux
linkage due to the magnets and inductance, so for instances where accurate current control
is desired, these should be the most accurately determined parameters.
Beyond the simulations, an experiment was performed to prove this performance data.
In order to gather data, the test machine is spun with another machine at constant speed
while a 5 A quadrature axis current was applied to the test machine. The quadrature axis
current command was then abruptly changed to 10 A, while the data acquisition measures
all operating data. This was then used to gather the same data which was found during
simulation. This experiment was repeated with the introduction of parametric error in the
parameters during simulation. As with the simulation, the experimental data is used to
generate sensitivity data. This data was given in Table (3.5). From the experimental data
the flux linkage due to the magnets, much like the simulation, seems to have a large effect
upon the output. This is also seen more so with the inductance variation, the sensitivity
being higher especially in the case of steadystate error and other transient responses. The
resistance has the least effect, which is also seen in the simulated data.

56

Table 3.5: Table showing the experimentally determined sensitivity to parametric error
∆ξ→S,P
T (N · m)
P (W )
η (−)
Tr (sec)
Ts (sec)
xss (A)

3.3

Rs
0.705
0.785
0.536
0.750
0.028
0.002

Ls
0.610
0.695
0.664
1.563
1.008
1.031

λaf
2.201
1.811
1.289
0.100
1.285
1.521

IPM Machine 1 Torque Controller Sensitivity Using FEA

3.3.1

Finite Element Analysis for Characterization

An IPM machine was modeled with the Flux2D software package, for use in the simulation
of these machines. The designed finite element model is verified by conducting a back-EMF
test as well as applying known currents to check whether the predicted torque is achieved. In
addition to performance fidelity with the actual machine, the finite element model software
allows us to apply currents and voltages by use of the controllers and measure all performance
parameters.

3.3.2

Finite Element Model

A finite element model is sought to first prove the concepts of sensitivity, as it best approximates machine behavior outside of experimental work. The finite element model was
constructed and given a mesh using the Flux2D finite element analysis software package.
The simulation parameters are discussed here. Note that the general finite element model
design is given in Fig. 3.9, where the colors define certain regions.
57

Table 3.6: Fig. 3.9 color legend
FEA Color Legend
Other
Infinite Box Black
Magnets Magnets
Red
A+
Yellow
ABlack
B+
Green
Windings
BRed
C+
Cyan
CBlue
Stator
Fuchsia
Iron
Rotor
Cyan
Rotor
Black
Air
Stator
Blue
Stator
Blue
Airgap
Rotor
Black

Note that in Table (3.6) the colors for the various regions in the FEM are given. As some
are repeated, take note that the coils are the small rectangular regions within the mode, the
large center black circle is the rotor air, the large black on the outside is the infinite box.
The blue area just within the infinite box is the air surrounding the stator. The large cyan
area with the magnets embedded is the rotor iron. The fuchsia area inside of the stator air
is the stator iron.
The mesh and geometry for this finite element model is seen in Fig. 3.10. It is difficult
to see the details of the mesh, so the mesh point information is given by Table (3.7).
A circuit must be coupled to this model, and from the winding diagrams the circuit
in Fig. 3.11 corresponds to the machine. In Flux2D, the coil conductor lumped element
defines the current through the appropriate coil conductor of the geometry. In this, each
positive circuit element corresponds to the outward facing conductor in the geometry, while
the negative circuit element corresponds to the inward facing conductor. Note that also the

58

Figure 3.9: View of the partial Flux2D geometry finite element model

Figure 3.10: View of the partial Flux2D geometry finite element model with mesh
phase resistance and leakage inductance are included, determined from the Speed software
package. Additionally, three AC voltage sources are shown, while in simulation either voltage
or current may be used.

59

Table 3.7: Mesh point values for Fig. 3.10
FEA Mesh Values
Inner Rotor Iron
Large
Outer Stator Iron
Magnets
Medium
Radial Outermost Slot
Radial Innermost Magnet
Small
Other Slot
Airgap
Airgap
Small Airgap Airgap Midpoint

Lla

AC Coil A+ Coil AAC Coil B+ Coil BAC Coil C+ Coil C-

15 (mm)
15 (mm)
3 (mm)
3 (mm)
0.5 (mm)
0.5 (mm)
0.2 (mm)
0.1 (mm)

RA
RB

Llb
Llc

RC
Figure 3.11: Circuit of the FEM

3.3.3

Finite Element Simulations

Using this circuit along with the FEM, certain scenarios are used for simulation of a healthy
machine mimicking the performed experiments. The characterization imposes a constant
speed of 2449 RPM, while applying a set DC quantity of either direct or quadrature axis
current with a current source, and again measures all quantities. This is conducted for a
time from 0 to 0.16 seconds in steps of 300 µsec and applies current from 0 to 350 A in steps
of 35 A. The performance simulation applies currents to achieve a specified torque from the
torque/speed to current lookup table while spinning the machine at 2449 RPM in order to
generate a torque of 180 N · m. In experimentation: 166.6 N · m is achieved, thus a torque

60

error of 7.4% exists in simulation.)
The sensitivity measure in section 2.2.3 was applied to the IPM machine in the finite
element analysis software package Flux2D, with the parameters in Table (3.8). Note that only
the quadrature axis inductance is current dependent, as the self saturation of the quadrature
axis is quite high. The flux with varying current was found and the inductance from this was
used to calculate a family of torque/speed curves. The speed was not the rated machine value,
but the machine operating condition for the tests performed. A torque/speed point taken
from the torque/speed curve was used and the performance at this point was measured. This
measurement was compared to the parametrically correct controller for torque and efficiency;
this was used to determine the sensitivity.
Table 3.8: Characteristics of interior permanent magnet machine 1
29 (mΩ)
12
0.078 (V · s)
0.84 − 0.0016 (mH)
0.243 (mH)
1539 rad
sec

Resistance of One Phase
Number of Poles
Flux Linkage due to Rotor Magnets
Quadrature Axis Inductance
Direct Axis Inductance
Electrical speed of machine

The flux was found as discussed for the nominal machine to be used as a starting point.
The flux curves depending on both the direct and quadrature axis current are shown in
Figs. 3.12 and 3.13. Note that the quadrature axis flux is heavily dependent upon the
current, especially at the high values of quadrature axis current, where the flux saturates
heavily. Some cross saturation is evident from the flux changing for different values of direct
axis current. Also note that the direct axis flux and inductance is much less effected by
saturation, but some effect can be seen. In the nominal controller, the quadrature inductance
is approximated by a linear function which best represents the inductance found from these
61

0.07
IQ=0(A)
IQ=−35(A)
IQ=−70(A)
IQ=−105(A)
IQ=−140(A)
IQ=−175(A)
IQ=−210(A)
IQ=−245(A)
IQ=−280(A)
IQ=−315(A)
IQ=−350(A)

0.06
0.05

λd (V ⋅ s)

0.04
0.03
0.02
0.01
0
−0.01
−0.02
−350

−300

−250 −200 −150 −100
Direct Axis Current (A)

−50

0

Figure 3.12: Current dependency of the direct axis flux linkage
flux curves. The direct axis inductance is assumed to be constant.
The parameters were used to find torque/speed controllers for the machine. This controller is represented in the form of torque/speed curves which have the current values needed
for each operating point. The relationship between torque/speed and D/Q current was found
and used to operate the machine. The parameters had error introduced into them, the controller recalculated, and operated at this point once more. The torque speed plots are shown
in Fig. 3.14. The torque speed curves change only due to the change in the known machine parameters. Changing the parameters will only change the model and controller for
the model, not the actual machine. Neglecting saturation, the maximum torque below base
speed is high, due to the assumption that high current will not saturate the quadrature axis,
and thus the saliency torque will always remain at its highest level. Like in previous results,
the resistance does not change the controller much at all. As was seen before, an error in
λaf has a large effect upon the torque; when this is assumed to be half, the magnetic torque
expected is halved. The torque below base speed is not halved because a great deal of torque
62

0.12
ID=0(A)
ID=−35(A)
ID=−70(A)
ID=−105(A)
ID=−140(A)
ID=−175(A)
ID=−210(A)
ID=−245(A)
ID=−280(A)
ID=−315(A)
ID=−350(A)

0.1

λq (V ⋅ s)

0.08
0.06
0.04
0.02
0
−0.02
−0.04
0

50

100
150
200
250
Quadrature Axis Current (A)

300

350

Figure 3.13: Current dependency of the quadrature axis flux linkage
in an IPM machine comes from saliency. The base speed increases because the voltage the
magnets induce in the stator is assumed to be halved. The torque expected when there is
error in Lq is halved because the saliency torque is then assumed to be much less. Typically,
the quadrature axis inductance is twice that of the direct axis inductance; when the quadrature axis is halved, the saliency torque is assumed to be gone. The base speed increases
in part because a halving of the inductance reduces the voltage induced in the direct axis
when quadrature current is present. This means that field weakening is not required until
higher direct and quadrature axis currents are needed. If the direct axis inductance is halved,
the torque increases compared to the nominal because more saliency torque is assumed to
be developed. The maximum speed decrease can be seen in the maximum speed equation
(3.10), assuming all of the current applied contributes to field weakening. When the direct
axis inductance is reduced, so is the maximum speed attainable with maximum current.

63

800
Nominal
No Saturation
Resistance error
λaf error
LQ error
LD error

700

Torque (N ⋅ m)

600
500
400
300
200
100
0
0

5000

10000

15000

Speed (RPM)

Figure 3.14: All torque and speed curves
2
2
2
vs = vds + vqs

(3.8)

... = (ids Rs )2 + (ωmax λaf + ωmax Ld ids )2

(3.9)

2
2
vs − i2 Rs
ds
ωmax = 2
λaf + L2 i2 + 2λaf Ld ids
d ds

(3.10)

With the parametric sensitivity procedure, the sensitivity is calculated for the effect of
various parameters upon output performance and shown in Table (3.9). The results agree
with that previously seen in the nonsalient machine. The inductance here has a larger
effect, especially the quadrature inductance, as this has an effect on saliency torque. The
torque change is reflected in both the power and efficiency, too. The resistance has a small
effect as before as does the direct axis inductance. The flux linkage due to the magnets is
seen to have a large effect, too as this has a large effect upon the magnet torque. A new
phenomenon is seen in this case compared to the nonsalient machine case. The sensitivity of

64

direct axis current to quadrature axis inductance is very high. This is due to the dependence
of the operating point on the level of saliency from equation (2.32). With an error in this
quadrature axis inductance, the saliency is assumed to be reduced and thus the saliency
torque. With less dependence on saliency torque, direct axis current is not important for
torque, only flux control.
Table 3.9: Table showing the finite element determined sensitivity to parametric error
∆ξ→P
T (N · m)
η (−)
Pmech (W )
Pelec (W )
ID (A)
IQ (A)

3.4

λaf
0.8765
0.1226
0.8765
1.0646
2.300
0.1739

Ld
0.1887
0.0049
0.1887
0.1932
0.3500
0.0217

Lq
1.5715
0.1785
1.5715
1.9198
2.600
0.6957

Rs
0.0410
0.0292
0.0410
0.0713
0.1500
0.2391

IPM Machine 2 Experimental and Simulation Results

Accurate performance measurement and controllers rely on accurate characterization methods. However, characterization costs time and money, and as such the minimization of effort
in characterization is important for motor and drive system manufacturers. In order to evaluate the sensitivity of performance to characterization, two characterization procedures with
different levels of measurement accuracy are performed in this section. The first is a detailed
characterization which measures high frequency values of the machine quantities during operation. The second uses the quantities the inverter observes during operation. The simple
sensitivity measure from section 2.2.3 is used to determine the sensitivity to characterization
65

by measuring torque controller performance at two points, 800 and 1500 RPM 30, 40 and
50 N · m. Note that in the analysis, no error is introduced in the parameters. The entire
process from characterization to sensitivity calculation is performed in the search for the
most accurate results. With this analysis, the time which must be spent on characterization
can be minimized, by not characterizing the lowest sensitivity quantities accurately.
Another characterization was performed on a small test IPM machine. The machine being characterized is a 10kW Samarium Cobalt magnet interior permanent magnet machine.
A setup is used – shown in Fig. 3.15 – to find the sensitivity for this IPM machine. This
setup consists of a DC machine rated to operate at torques exceeding the IPM machine
torque and speeds of 2650 RPM, above the base speed of the IPM machine being characterized. The DC machine is driven by a thyristor drive system. The DC machine acts as
a generator while the IPM machine is driven as a motor during characterization. Using
the same sensors as the SPM machine setup, a torque sensor, high frequency current, and
voltage sensor is used to gather data for machine characterization. The IPM machine is
driven by an APS inverter with the Texas Instruments floating point DSP TMS320F28335.
The machine is given commands like the SPM machine setup. The inverter, CAN setup and
CAN acquisition was done by Jorge Cintr´n-Rivera. Data is gathered in two ways. The first
o
way data is gathered is by the use of a high frequency 1.25 GSamples scope to record the
Second
data form the high frequency voltage and current sensors. The incremental encoder zeroing
pulse, which signifies the completion of one mechanical rotation, is used to determine the
relationship between the voltage, current, and the position of the machine rotor by acting
as the trigger. This relationship between electrical quantities and rotor position is required
for characterization in the two phase frame of reference. A second simplified method of
characterization [2] is implemented to see how sensitive the final controller operation is to
66

the machine characterization method. This uses data gathered from voltage, current and
position sensors on the inverter and then transmitted through CAN to the Labview interface
program, where they are recorded on a computer.
C

PC

Inverter

DC
Drive
CT

CT

M

M

IPM

DCM

Figure 3.15: Physical test setup for IPM machine characterization with inverters, machines
and sensors

3.4.1

Flux Linkages

The flux linkages were found from the methods discussed in sections 3.2.1. The flux linkages
were found from the scope characterization method shown in Fig. 3.8 to be as in Figs. 3.16
and 3.17. These will then be used to calculate the controller.
Similarly the flux linkages were found from the simplified characterization setup shown
in Fig. 3.15. The flux linkages are shown in Figs. 3.18 and 3.19.

67

λd
25

0.1

0.3

0.5

0.5

|I| (A)

0.1

0.3

0.7

15

0

0.2

0.4

0.6

20

−0.1

0.4

0.8
0.6

10

0.3

0.5

0.7

0.4

0.6

0.8

5

0.2

0.7

0.8
0
90

100

110

120

130 140 150
Angle (degrees)

160

0.5
0.6
0.7
0.8
170 180

Figure 3.16: Current dependent direct axis flux linkage
λq

0.8

25

1

0.8

|I| (A)

1

6

10

0.

0.8

0.4

0.6
5

0
90

0.2

1

15

0.4

1.2

1.2

0.6

20

0.4
0.2

0.6
0.4
0.2
100

110

120

130 140 150
Angle (degrees)

0.2

160

170

180

Figure 3.17: Current dependent quadrature axis flux linkage

3.4.2

Controller Development

The controller was extracted such that all operating points are within the maximum torque
curve of the torque/speed in Fig. 3.20. With this information, the lookup tables are extracted
68

λd

−0

0

25

0.1

0.2

0.3

0.4

0.5

.1

20

|I| (A)

0.6

15

0
0.5

4

0.7

0.6

10

0.3

0.

0.1

0.2

0.4

0.5

0.3
0.5

5

0.6
0.

0.6

7

0
90

100

110

120

130 140 150
Angle (degrees)

160

170

180

Figure 3.18: Direct axis flux linkage from simplified method from Jorge Cintr´n-Rivera
o

0.6

1

1.2

0.4

1.2

0.8

λq
25

20

15

1

0.8
0.6

0.4

|I| (A)

1

10

0.8

0.8
0.6

0.6

5

0.4

0.4
100

110

0.2

0.2

0.2
0
90

0.4

120

130 140 150
Angle (degrees)

160

170

180

Figure 3.19: Quadrature axis flux linkage from simplified method from Jorge Cintr´n-Rivera
o
for use in the controller. The interpolation step of controller development smooths the
controller, the controller is given in Figs. 3.22 and 3.23. Similarly the fluxes from the
simplified characterization are used to generate a controller in Figs. 3.24 and 3.25.

69

80
70

Torque (N ⋅ m)

60
50
40
30
20
10
0

1000

2000

3000
4000
Speed (RPM)

5000

6000

Figure 3.20: Torque-speed curve for the nominal controller
70
60

Torque (N ⋅ m)

50
40
30
20
10
0
0

1000

2000

3000
4000
Speed (RPM)

5000

6000

Figure 3.21: Torque-speed curve for simplified characterization flux linkage controller

70

|IS| (A)

70
60 20
20

15

40
15

Torque (N ⋅ m)

50

30

20

10
10

20

15

20

5
5

10
0
0

1000

2000

15

10

3000
4000
Speed (RPM)

5000

6000

Figure 3.22: Operating point dependent current magnitude command
ID (A)

70
−15

60
−10

40
−5

0
−1

30

−20

−1

−5

Torque (N ⋅ m)

0
−2 5
−1

50

5

20

−20

−10

10
0 0
0

−15

−5

1000

2000

3000
4000
Speed (RPM)

5000

6000

Figure 3.23: Operating point dependent direct axis current command

71

|IS| (A)

70

25
25

60
50
20

Torque (N ⋅ m)

20

40
30

15

10

20

10

15

20
5

5

10
15

10
0
0

1000

2000

3000
4000
Speed (RPM)

5000

6000

Figure 3.24: Operating point dependent current magnitude command from simplified characterization

3.4.3

Sensitivity Results

Once the controllers are extracted, various performance tests were completed with the machine test setup. The tests of interest here are intended to cover the range of operating points
below and above base speed with a range of torques. The machine was operated at both
800 RPM and 1500 RPM speeds held constant while changing the torque command from
30 N · m to 40 N · m in the 800 RPM case and 30 N · m to 50 N · m in the 1500 RPM case.
During these tests, the torque, voltage, current, speed and current commands are measured
in real time. The torque was calculated based upon estimated flux and current to compensate for poor torque sensor accuracy. This was used to calculate the power and efficiency
measures. These quantities are then used with the sensitivity measure from section 2.2.3 to
calculate the sensitivity. The sensitivity was calculated with respect to the fluxes with the

72

ID (A)

70

−15

60
−15

−10

0
−2

40
0
−1

Torque (N ⋅ m)

50

30

−1

−20

5

−5

20

−10

10

−15

−5

0 0
0

1000

0
2000

3000
4000
Speed (RPM)

5000

6000

Figure 3.25: Operating point dependent direct axis current command from simplified characterization
results in Table (3.10).
While no error was artificially imposed in the system, small changes occur and this is
what was measured. For this reason, the sensitivity measure for torque and power is high
with respect to the quadrature axis flux. This is because the magnet torque is most affected
by the quadrature axis flux. In most cases, the sensitivity of the output torque and power
performance with respect to direct axis inductance is small due to the lesser dependence
of these upon saliency torque. The efficiency sensitivity is very low. This is because the
existence of flux error changes the performance through the current command at a certain
torque and speed operating point. The new operating point will be different, but the losses
for that point will be on the same magnitude in any case. The direct and quadrature axis
currents greatly affect the machine controller. The large sensitivity of these quantities result

73

from the dependence of the controller upon the flux linkages. The controller was derived
from the maximum torque per ampere line, which coincides with the maximum operating
voltage above base speed. If the fluxes are different, the voltage induced will also change.
This means that the maximum voltage predicted changes. From this, the current command
derived changes due to the change in voltage.
Table 3.10: Table showing the experimentally determined sensitivity to parametric error
800 (RPM)/30(N · m)
800 (RPM)/40(N · m)
1500 (RPM)/30(N · m)
1500 (RPM)/50(N · m)
-

∆ξ→P
λd
λq
λd
λq
λd
λq
λd
λq

T
0.034
0.176
0.061
0.269
0.061
0.269
0.622
0.782

Pm
0.036
0.187
0.688
0.602
0.061
0.27
0.62
0.782

Pe
0.067
0.346
0.58
0.507
0.052
0.23
0.583
0.583

η
Id
Iq
0.1
2.492 0.375
0.524 13.01 1.96
0.122 1.069 0.101
0.107 0.9361 0.088
0.112 0.382 0.253
0.49
1.68 1.113
0.048 0.448 0.107
0.061 0.563 0.135

To make the relationship between the torque and errors in flux linkage clear, simulations
were performed by applying both the simplified and detailed controllers from Figs. 3.22 and
3.23 as well as 3.24 and 3.25, respectively, to the detailed machine model given in Figs. 3.16
and 3.17. This results in two sets of torques and two sets of fluxes, from which the controllers
are derived. These can be used with equation (2.56) to find the sensitivity over the entire
operating range of the drive. From this, it is clear that the developed torque is sensitive to
λd in the field weakening region, as the currents are determined in field weakening based
upon the predicted λd . Further, note that the sensitivity of torque to λq is high in the high
torque low speed region, as the torque estimate is not as reliant upon the field weakening
calculation, and thus λd , but upon the magnet flux linkage, and thus λq .

74

60
5

50

0

−1

−10

5

−2

Torque (Nm)

40

−10

30

−4

0

5

5
10

20

20
35
50

20

20

35

5
5

0
0

500

1000

1500
2000
2500
Speed (RPM)

3000

3500

4000

Figure 3.26: Direct axis torque sensitivity with respect to operating point (∆λd→T )

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60
15
5

50

15

5
−5

30

−1

−5

Torque (Nm)

40

5

−5

20

5

−5

10

5
0
0

500

1000

1500

2000
2500
Speed (RPM)

3000

3500

4000

Figure 3.27: Quadrature axis torque sensitivity with respect to operating point (∆λq→T )

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Chapter 4
Conclusion
This thesis details the effect of error in parameters upon the output performance and sensitivity. Knowing the effect that errors in the parameters have upon the output performance
can at least allow for more accurate determination of the most important parameters. A
method was also reviewed from literature to accurately determine parameters, including the
exact setups used. It then details a controller from literature, which measured performance
with an experimental test, finite element simulation, and an analytical model simulation.
The controller parameters then have error induced to observe the change in performance.
This effect of error in each parameter is compared, using various metrics for three machines,
to determine which parameter has the greatest effect upon control. Another test used this
sensitivity measure to determine the sensitivity of machine performance to characterization.
A sensitivity measurement is used to compare the output sensitivity to error in various
parameters.
For the SPM machine, torque is found to be most affected by the quadrature axis inductances as well as flux linkage due to the magnets. Thus, for areas where torque accuracy is
required, these parameters should be most accurately determined. The settling time is most
affected by the inductances and flux linkage due to the magnets, so if a fast transient response is desired the inductances and flux linkage due to the magnets should be determined
very closely to the actual value. Finally, the steady state error can be found to be most

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affected by the flux linkage due to the magnets, so where high current accuracy is required
on the output, these quantities should be accurately found. As the current controller does
not vary with the machine topology, the current sensitivity is generalized to any machine.
For the first IPM machine, the flux linkage due to the magnets was also seen to affect
the torque and power performance. The resistance was not found to be of great importance
where machine performance is concerned. The results differ from the salient machine in
that the controller development is heavily dependent upon the axis inductances. This is
because the saliency is vital in the development of torque. Thus, when designing controllers,
the characterization should accurately determine the inductances. The change in these
parameters were also seen to heavily affect the expected operating range of the machine.
The second IPM machine was used to determine the sensitivity of performance to the
entire characterization process without any introduced errors. Characterization error was
found by a process of characterizing the IPM machine with two different methods. Once
more, the SPM machine and FEA from the first IPM machine results were verified. Even
while characterizing with the best effort, the resultant parameters from the two characterization methods differed. The sensitivity of the controller to flux error was shown. It was
seen that generally the quadrature flux axis has the largest effect on performance. This is
not true for all operating points; some operating points have a greater sensitivity to the flux
parameter error. The sensitivity of torque to λd is highest in the field weakening region.
And further, the sensitivity of torque to λq is high in the high torque low speed region.
The importance of using accurate models, characterization methods, and accounting for
faults is demonstrated by showing that when these are lacking, errors occur in performance.
Torque error was clearly a result of changes in λd , especially in field weakening. This error
can easily occur, as rotor eccentricity, heating of the magnets, and uniform demagnetization
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cause changes in the direct axis flux. Torque error also resulted from λq , especially where
field weakening was not present, where magnet torque dominates. The quadrature axis flux
is also important to know well, as saturation reduces the expected flux at high currents.
Overall, errors in the flux, a direct result of poor modeling, poor characterization, or faults
have a detrimental effect on performance.
In this thesis parameter dependent controllers are discussed. Parametric error causes the
voltage applied for a given current command to be incorrect. Since the torque controller
depends upon the parameters to determine the correct current operating point, the current
operating point will be incorrect with parameter error. By developing a controller which
estimates the parameters online, the controller can then be changed. In order to account for
the change in parameters while maintaining robust operation, a controller implemented in
tandem with the online parameter estimation would result in a controller not only robust to
disturbance, but also to parametric error. A controller which can operate a PMSM or IPM
machine through a combination of adaptive and robust control schemes and accomplishes
a minimization of the sensitivity quantity would succeed in attaining a higher degree of
performance.
These results guide the development of high performance controllers, as the output performance accuracy can be enhanced by accurate determination of the parameters which
induce a larger performance error. To clarify, the flux linkage due to the magnets could be
accurately known for all currents and temperatures if accurate magnetic torque performance
is desired. By understanding this analysis, the results of applications requiring PMSMs and
IPM machines can be improved. With more accurate controllers, smaller machines can be
used since a “margin of error” is not required with increased efficiency; the machine does
not need to be overdesigned. The controller response can also be improved as dependence
79

on integral action is lessened, and thus are transient losses. This information can be used to
guide the development of a controller and demonstrate whether a controller can effectively
compensate for parametric error if sensitivity is reduced.

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