CONTROL METHODS FOR SMOOTH OPERATION OF PERMANENT MAGNET SYNCHRONOUS AC MOTORS
Greg Heins
A thesis for the degree of Doctor of Philosophy at the School of Engineering and Logistics Charles Darwin University, Australia
Submitted on March 31, 2008. ii Declaration
I hereby declare that the work herein, now submitted as a thesis for the degree of Doctor of Philosophy of the Charles Darwin University, is the result of my own inves- tigations, and all references to ideas and work of other researchers have been specifically acknowledged. I hereby certify that the work embodied in this thesis has not already been accepted in substance for any degree, and is not being currently submitted in candidature for any other degree. iv Abstract
Many applications require a motor capable of providing a smooth torque output. The use of control methods to ensure smooth operation is attractive because it minimises restrictions on motor design and manufacture. Programmed reference current wave- form (PRCW) methods are a commonly proposed control method as they have the ability to work at a range of speeds and load torques, however their experimental implementation to date has been inconclusive. This research compares three published PRCW methods with a standard sinusoidal current waveform. For the test motor considered, all PRCW methods tested are able to reduce the RMS pulsating torque to approximately 3−4% down from the 8−9% created when using a sinusoidal current. While this is a substantial reduction, no method was clearly superior. One method however, the time domain method, produced slightly better results. Further reduction in pulsating torque requires a greater accuracy in motor and controller parameter estimation. To achieve this further reduction, this thesis develops a ‘pulsating torque decoupling’ (PTD) method where the parameters are determined from the pulsating torque itself. The use of this technique allows calibration of the critical parameters and produces a further reduction of the pulsating torque to approximately 1%. As with the uncal- ibrated results, the difference between the different PRCW methods is minimal, with the time domain method producing only slightly less pulsating torque. These results suggest PRCW methods can be effective in producing a smooth torque output if the relevant parameters are estimated to a suitable accuracy. The use of the PTD method presented in this thesis can achieve this required accuracy.
v vi List of publications
As of the March 31, 2008, the following papers with contributions from the results presented in this thesis have been published, or are under review:
[36] Greg Heins, Friso de Boer, Jeroen Wouters, and Roel Bruns Experimental com- parison of reference current waveform techniques for pulsating torque minimiza- tion in PMAC motors. In International Electric Machines and Drives Conference (IEMDC ’07), volume 1, pages 1031–1035, Antalya, Turkey, 2007.
[35] Greg Heins, Friso De Boer, and Sina Vafi. Characterisation of the mechanical motor parameters for a PMSM using induced torque harmonics. In Australian Universities Power Electronics Conference, Perth, 2007.
[34] Greg Heins and Friso De Boer. Modeling of a synchronous permanent magnet motor to determine reference current waveforms. In Australian Universities Power Electronics Conference, Brisbane, 2004.
vii viii Acknowledgments
This thesis would not have happened without the guidance and patience of Friso de Boer and Nic Hannekum. It would certainly not be finished yet without the assistance of Charles Darwin University staff members: David Van Munster, Mark Thiele and Jim Mitroy and Eindhoven University of Technology students: Roel Bruns, Jeroen Wouters, Pieter Poels and Erik Grassens. I am also grateful for the support of L’institut Francais de Mecanique Avancee students: David Ahounou, Pascal Magnan, Gabriel Caroux, Ben Errard, Nick Ferriere and Florian Barnet, Ecole Superieure d’Ingenieurs student: Vincent Lafont and Charles Darwin University students: Jasveer Saini, James Canning and Charles Gammon. Towards the end, somewhere between listening to Up all night by The Waifs and the soundtrack to Mission Impossible, things inevitably went somewhat pear shaped. I was very grateful for the love and support from Kelly Mashford, my folks: Terry and Diana, and my sister Karen.
ix x .
To Gaz. I have always been a fan of enthusiasm. You have big mobs of it.
xi xii Contents
Declaration iii
Abstract v
List of publications vii
Acknowledgments ix
Table of Contents xvi
List of Figures xx
List of Tables xxi
List of Symbols xxviii
Abbreviations xxix
1 Introduction 1 1.1 Background ...... 2 1.2 Research goal ...... 7 1.3 Research approach ...... 7 1.4 Chapter overview ...... 8
2 Review: Programmed reference current waveform methods 9 2.1 PMAC mathematical model ...... 9 2.2 Fundamental control problem ...... 11 2.3 PRCW methods ...... 12
xiii CONTENTS
2.4 Goals and constraints ...... 13 2.5 Frequency domain method (FDM) ...... 13 2.6 Time domain method (TDM) ...... 19 2.7 ‘Park-like’ method (PLM) ...... 22 2.8 Experimental implementation ...... 24 2.9 Implementation challenges ...... 25 2.10 Summary of PRCW methods ...... 28
3 Review: determination of motor parameters 31 3.1 Back EMF ...... 31 3.2 Cogging torque ...... 33 3.3 Current ...... 39 3.4 Rotor Position ...... 40 3.5 Sensitivity analysis ...... 40 3.6 Summary of methods for determination of motor parameters ...... 41
4 Theoretical comparison of reviewed methods 43 4.1 Methodology ...... 43 4.2 Parameter and constraint variation ...... 44 4.3 Performance with different back EMFs (without cogging torque) .... 44 4.4 Comparison of cogging torque compensation ...... 49 4.5 Upper frequency limit constraint ...... 52 4.6 Summary of theoretical comparison of PRCW methods ...... 53
5 Pulsating torque decoupling approach to motor parameter determi- nation 55 5.1 Motor model overview ...... 56 5.2 Decoupling of pulsating torque: determination of current imbalance and cogging torque ...... 58 5.3 Modifications if only dynamic torque measurement is available ..... 62 5.4 System gain determination ...... 64 5.5 Sensitivity analysis ...... 66 5.6 Summary of PTD approach to motor parameter determination ..... 67
xiv CONTENTS
6 Experimental setup 69
6.1 Motor description ...... 69
6.2 Review of past implementation problems ...... 70
6.3 Mechanical design ...... 70
6.4 Current control design ...... 75
6.5 Data acquisition ...... 80
6.6 Experimental setup summary ...... 83
7 Results 85
7.1 Validity of assumptions ...... 85
7.2 Current controller ...... 91
7.3 Eddy current brake testing ...... 95
7.4 Parameter determination ...... 95
7.5 Parameter determination - PTD method ...... 103
7.6 Comparison of PRCW methods ...... 109
7.7 Sensitivity analysis ...... 113
7.8 Summary of results ...... 115
8 Discussion 119
8.1 Experimental setup ...... 119
8.2 Parameter estimation ...... 121
8.3 Comparison of PRCW methods ...... 122
9 Conclusion 127
9.1 Further work ...... 129
References 131
A Hardware design 141
A.1 Hardware overview ...... 141
A.2 Mechanical drawings ...... 142
A.3 Electrical Design ...... 154
xv CONTENTS
B Additional Calculations 159 B.1 Design and sizing of eddy current brake ...... 159 B.2 Modal analysis of experimental setup ...... 164
C Additional results 165 C.1 Theoretical sensitivity analysis ...... 165 C.2 Pulsating torque for PRCW methods ...... 166 C.3 Pulsating torque comparison within methods ...... 171
xvi List of Figures
2.1 Fundamental PMAC torque control scheme ...... 12 2.2 PRCW control scheme ...... 12
3.1 FBD for motor ...... 36 3.2 FBD for rotor ...... 37 3.3 FBD for stator ...... 37
4.1 Current and torque output for sinusoidal back EMF ...... 45 4.2 Current and torque output for trapezoidal back EMF ...... 46 4.3 Performance of different methods if one phase has a magnitude variation 48 4.4 Performance of different methods if one phase has a phase variation .. 49 4.5 Ability of each method to compensate 1Nm of cogging torque at different harmonics ...... 50 4.6 Comparison of currents with and without a star connection constraint . 51 4.7 Ability of each method to compensate 1Nm of cogging torque at different harmonics ...... 52
5.1 Block diagram of parameters to be determined ...... 57 5.2 Block diagram of parameters to be determined (simplified) ...... 58 5.3 Block diagram of parameters to be determined (simplified) ...... 62
5.4 ~α, β~, ~τcog variation leading to 1% RMS ~τmeas ...... 67
6.1 Stylised motor assembly cross section - full detail see A.2.2 ...... 73 6.2 CDU Experimental Setup (Magnet portion of the eddy current brake has been removed for clarity) ...... 75
xvii LIST OF FIGURES
6.3 Full bridge drive topology ...... 76 6.4 Star connected drive topology ...... 76 6.5 SimulinkTMmodel of current controller ...... 79 6.6 Screen shot of LabviewTMvirtual instrument ...... 83
7.1 External drive for measurement of back EMF ...... 86 7.2 Raw back EMF over the angular velocity range ...... 87 7.3 Mean normalised back EMF ...... 88 7.4 Percentage error of normalised back EMF over the angular velocity range 89 7.5 Proportionality of torque to current ...... 90 7.6 Bode plot of Plant ...... 92 7.7 Simulated step response of system ...... 92 7.8 Current controller performance ...... 94 7.9 Current error ...... 95 7.10 Harmonic content of back EMF ...... 98 7.11 Error from truncation of back EMF ...... 99 7.12 Dynamically measured cogging torque (one mechanical revolution) ... 100 7.13 Dynamically measured cogging torque (one electrical revolution) .... 101 7.14 Dynamically measured cogging torque error (one mechanical revolution) 101 7.15 Dynamically measured cogging torque error (one electrical revolution) . 102 7.16 Measured cogging torque harmonics ...... 103 7.17 System transfer function estimation ...... 104 7.18 System transfer function estimation (no multiple of 8 harmonics) .... 105 7.19 Decoupling the pulsating torque (time domain) - (‘rest’ is the remaining pulsating torque for which the source is unknown) ...... 107 7.20 Decoupling the pulsating torque (frequency domain) - (‘rest’ is the re- maining pulsating torque for which the source is unknown) ...... 107 7.21 Determined cogging torque for different set-points ...... 108 7.22 Cogging torque error for different set-points ...... 108 7.23 Comparison of PRCW methods (* refers to the use of the star connection constraint) ...... 110 7.24 Uncalibrated sin wave method ...... 111
xviii LIST OF FIGURES
7.25 TDM - without star connection constraint ...... 112 7.26 Comparison of PRCW current waveforms (* refers to the use of the star connection constraint) ...... 113 7.27 RMS current increase from PRCW methods (* refers to the use of the star connection constraint) ...... 114 7.28 Sensitivity of pulsating torque to a α error ...... 115 7.29 Sensitivity of pulsating torque to a β error ...... 116
∗ 7.30 Sensitivity of pulsating torque to a ~τcog error ...... 116 7.31 Sensitivity of pulsating torque to a error ...... 117
8.1 Pulsating torque due to current controller error ...... 124
A.1 CDU motor test assembly ...... 142 A.2 Overview of electrical components ...... 154 A.3 Overview of current inverter ...... 155 A.4 Closeup of gate drive and MOSFET module ...... 155 A.5 Current sensor board with LEM LTS25NP ...... 156 A.6 Labview interface board ...... 156 A.7 DSP interface board ...... 157
B.1 Assembled eddy current brake ...... 160 B.2 Expected torque from eddy-current brake ...... 163 B.3 Results of modal analysis ...... 164
C.1 Sensitivity of pulsating torque to parameter variation ...... 166 C.2 FDM - without star constraint ...... 167 C.3 FDM - with star constraint ...... 167 C.4 TDM - without star constraint ...... 168 C.5 TDM - with star constraint ...... 168 C.6 SIN ...... 169 C.7 FDM - without star constraint ...... 169 C.8 FDM - with star constraint ...... 170 C.9 TDM - with star constraint ...... 170
xix LIST OF FIGURES
C.10 FDM (frequency domain) ...... 171 C.11 TDM (frequency domain) ...... 172
xx List of Tables
2.1 Back EMF and current harmonic combinations responsible for torque harmonics (including star connection constraints) ...... 17
4.1 Performance of PRCW methods with a sinusoidal back EMF ...... 45 4.2 Performance of PRCW methods with a trapezoidal back EMF ..... 47 4.3 Performance of PRCW methods with a modified trapezoidal back EMF 51
7.1 Induced scaling and offset errors and the found compensating values .. 109 7.2 Average total RMS current usage increase over operating range ..... 113
xxi LIST OF TABLES
xxii List of Symbols
= overall system gain independent of ~α, β~ and ~τcog,
DSP Ki = DSP integral gain,
Ko = system gain,
Kp = controller gain,
DSP Kp = DSP controller gain,
λ = Lagrange multiplier,
νo = 63% rise time, op = current inverter offset for phase p,
∗ op = current inverter supply offset for phase p,
ω = mechanical angular velocity (rad/sec),
τo = delay time,
θ = mechanical rotor position (rad),
Θ = total number of encoder states,
θe = electrical rotor position (rad),
Ti = integral time constant, ts = sample time, up = torque transducer gain for phase p,
xxiii LIST OF SYMBOLS
∗ up = estimated torque transducer gain for phase p, wp = current inverter gain for phase p,
∗ wp = estimated current inverter gain for phase p, m = harmonic number of torque series, n = harmonic number of back EMF and current series,
P = number of pole pairs,
αp = current scaling error in phase p,
~α = 3 × 1 vector of current scaling error,
GAA(ω) = input auto spectrum,
GBB(ω) = output auto spectrum,
βp = current offset error in phase p,
β~ = 3 × 1 vector of current offset error,
C = Lagrange cost constraint,
γ2(ω) = coherence,
C1,θ = Lagrange cost constraint at angle θ,
∗ = matrix multiplication operator,
GAB(ω) = cross spectrum,
C2,θ = Lagrange cost constraint at angle θ,
Eθ,p = back EMF for phase p at angle θ (V s/rad),
Eθ,p = back EMF for phase p at encoder point θ (V s/rad),
÷ = element-wise division operator,
• = element-wise multiplication operator,
xxiv LIST OF SYMBOLS f = Lagrange minimization function at,
A(ω) = frequency input function, fθ = Lagrange minimisation function at angle θ,
B(ω) = frequency output function,
H(ω) = frequency response function,
H1(ω) = alternate frequency response function,
Icog = Θ × 3 matrix of the current required to compensate for torque (A),
Iˆn = current Fourier series coefficients,
I~τref = Θ × 3 matrix of the current required to achieve the reference torque (A),
I = Θ × 3 matrix of the current (A)(NOT the identity matrix),
Iref = Θ × 3 matrix of the reference current (A),
ref iθ,p = reference current for phase p at angle θ (A),
iθe,a = current in phase a at electrical angle θe (A), iθ,a , iθ,b , iθ,c = currents in three phase reference frame at at angle θ (A), iθ,d , iθ,q , iθ,0 = currents in rotating reference frame at at angle θ (A), iθ,p = current in phase p at angle θ (A),
~ Iˆ = ? × 1 vector of current coefficients,
~ip = Θ × 1 vector of the current in phase p (A),
Kˆn = normalised back EMF Fourier series coefficients,
K = Θ × 3 matrix of the back EMF (V s/rad),
Kˆ = ?×? matrix of the back EMF coefficients,
kθe,a = normalised back EMF for phase a at electrical angle θe (V s/rad),
xxv LIST OF SYMBOLS
kθ,a , kθ,b , kθ,c = normalised back EMFs in three phase reference frame at at angle θ (V s/rad), kθ,d , kθ,q , kθ,0 = normalised back EMFs in rotating reference frame at at angle θ (V s/rad), kθ,p = normalised back EMF for phase p at angle θ (V s/rad),
~kp = Θ × 1 vector of the normalised back EMF of phase p (V s/rad),
λ1,θ = Lagrange multiplier at angle θ,
λ2,θ = Lagrange multiplier at angle θ,
~λ = Θ × 1 vector of Lagrange multipliers,
L = limit chosen for the truncation of the back EMF series,
M = limit chosen for the torque harmonics to be canceled,
N = limit chosen for the truncation of the current series,
~ 1Θ×1 = Θ × 1 vector of ones,
Npole pair = number of pole pairs,
Nslot = number of stator slots,
~o = 3 × 1 vector of the current inverter offset,
~o∗ = 3 × 1 vector of the estimated current inverter offset,
1 = Θ × 3 matrix of ones,
+ = pseudo-inverse operator,
Q = Lagrange minimization objective,
Qθ = Lagrange minimization objective at angle θ,
τaverage = average torque output,
∗ ~τcog = Θ × 1 vector of the cogging torque estimate (Nm),
xxvi LIST OF SYMBOLS
cog τ = cogging torque at electrical angle θe (Nm), θe
cog τθ = cogging torque at angle θ (Nm),
cog∗ τθ = estimate of cogging torque at angle θ (Nm),
~τcog = Θ × 1 vector of the cogging torque (Nm),
em T = Θ × 3 matrix of the electromagnetic torque (Nm),
em τ = electromagnetic torque at electrical angle θe (Nm), θe
em τ = electromagnetic torque produced by phase a at electrical angle θe (Nm), θe,a
em τ = electromagnetic torque produced by phase p at electrical angle θe (Nm), θe,p
em τθ = electromagnetic torque at angle θ (Nm),
~τem = Θ × 1 vector of the electro-magnetic torque (Nm),
~τem,p = Θ × 1 vector of the electromagnetic torque produced by phase p (Nm),
Tˆm = total electromagnetic torque Fourier series coefficients,
Tˆm,a = electromagnetic torque Fourier series coefficients for phase a,
~ 13×1 = 3 × 1 vector of ones,
m τθ = motor torque at angle θ (Nm),
~τm = Θ × 1 vector of the motor torque (Nm),
meas τθ = measured torque at angle θ (Nm),
∗ ~τpulsating = Θ × 1 vector of the pulsating torque estimate (Nm),
~τpulsating = Θ × 1 vector of the pulsating torque (Nm),
~τmeas = Θ × 1 vector of the measured torque (Nm),
ref τ = reference torque at electrical angle θe (Nm), θe
ref τθ = reference torque at angle θ (Nm),
xxvii LIST OF SYMBOLS
~τref = Θ × 1 vector of the constant reference torque (Nm),
~ Tˆ = ? × 1 vector of torque coefficients,
~u = 3 × 1 vector of the torque transducer gain,
~u∗ = 3 × 1 vector of the estimated torque transducer gain,
~w = 3 × 1 vector of the current inverter gain,
~w∗ = 3 × 1 vector of the estimated current inverter gain,
∗ X = Θ × 6 matrix of torque and back EMF estimate,
X = Θ × 6 matrix of torque and back EMF,
~y = 9 × 1 vector of scaling and offset errors,
~0 = Θ × 1 vector of zeros,
~z = Θ × 1 vector of residuals,
xxviii Abbreviations
BLDC Brushless DC machines
EMF electro-motive force
FBD free body diagram FDM frequency domain method FEA finite element analysis
MC numerically controlled MCM ‘minimum current’ method
PLM Park-like method PMAC permanent magnet synchronous AC PRCW programmed reference current waveform PTD pulsating torque decoupling
RMS root mean squared
TDM time domain method
xxix Abbreviations
xxx Chapter 1
Introduction
Smooth electric motor torque output is desirable in many applications. Traditionally, for permanent magnet synchronous AC (PMAC) motors this smooth torque has been achieved using carefully designed motors that are manufactured to close tolerances[40].
The potential exists for a well designed control system to avoid these design and manufacture restrictions. One of the most popular control schemes proposed to achieve this goal is the use of programmed reference current waveforms (PRCW). These wave- forms are determined from prior knowledge of the motor parameters.
To date however, the experimental implementation of these methods has been inconclusive, mainly due to challenges in acquiring the prior knowledge of the motor parameters to a significant accuracy.
The goal of this research is to determine if, for motors requiring a smooth torque output, PRCW methods can be serious contenders to replace the current technology of careful motor design and manufacture.
To focus the discussion, smooth operation is defined and an overview of various motor types is provided. Justification for using control methods to achieve smooth operation is given and this research is focused on PRCW control methods. The tasks to be completed are listed along with an overview of the content of the following chapters.
1 CHAPTER 1. INTRODUCTION
1.1 Background
1.1.1 Smooth operation
Motors with a smooth output torque are essential for applications that require precise tracking. These applications include arc welding, laser cutting, numerically controlled (NC) machining and antenna tracking [1], [24]. The existence of pulsating torque can have a negative impact on processes. One such example is the impact of pulsating torque on the surface finish when using rotary machine tools [40]. Pulsating torque also has the potential to excite resonances in the mechanical drive-train of a system along with the production of acoustic noise [37], [78].
1.1.2 Synchronous alternating current machines
Electric machines are classified as either alternating current (AC) or direct current (DC) machines. AC machines can be further classified as asynchronous (induction), or synchronous.
• DC machines are simple to control, however they require regular maintenance due to the sliding contact of their brushes.
• AC asynchronous machines (induction machines) are the most widely used mo- tors due to their low cost, however they are less efficient and unable to be con- trolled as accurately as other types of machines.
• AC synchronous machines offer a level of control comparable with DC machines while avoiding the maintenance issues associated with brushes. Control however, is more difficult because of their requirement for a variable frequency drive with accurate position feedback.
• One category of AC synchronous machines with trapezoidal back EMF and rectangular current excitation are often referred to as Brushless DC machines (BLDC), because of the similar output characteristics.
2 1.1. BACKGROUND
Three-phase permanent magnet AC synchronous (PMAC) machines
AC synchronous machines can be further differentiated by the way that the rotating (rotor) magnetic field is generated. This can either be by electromagnets or by perma- nent magnets. Machines that use electromagnets can create a higher power to weight ratio, however they are less efficient due to the losses associated with the current re- quired in the field windings. The use of electromagnets also requires brushes and slip rings to transfer the field current to the rotor. Synchronous PMAC machines are also classified by the number of independent windings, or phases. To achieve smooth torque production with evenly spaced phase 360o windings phase offset = number of phases , at least 3 phases are required. The focus of this thesis is on the analysis and testing of three-phase synchronous permanent magnet AC machines. For the remainder of this thesis, the abbreviation ‘PMAC’ will refer this style of machines.
Rationale of studying PMAC machines
PMAC machines avoid the maintenance issues associated with DC motors and are superior to other AC motors in terms of controllability and efficiency. With the intro- duction of high performance rare-earth magnets, PMAC machines have been capable of improved dynamic performance and higher efficiency [25]. Due to the need for sophisticated controllers and in some cases, the cost of rare-earth magnets, PMAC machines have been a more expensive option. In the last few decades however, the price of power electronic components and rare earth magnets has decreased [62]. For smaller motor sizes (up 10 - 15kW) PMAC machines are increasingly the machine of choice for servo drives and vehicle applications [25].
1.1.3 Pulsating torque
PMAC machines generate torque by the interaction between the magnetic flux from the permanent magnets in the rotor and the magnetic flux from the electromagnets in the stator (stationary part of the motor). The term pulsating torque refers to any periodic variation in the torque output of a motor. It is created in two ways: cogging torque and torque ripple. Ambiguity exists
3 CHAPTER 1. INTRODUCTION for the description of pulsating torque mechanisms. However, this thesis will use the terms defined by Jahns and Soong in their 1996 paper: Pulsating torque minimisation techniques for permanent magnet AC motor drives - a review [40].
Cogging torque
In a PMAC machine, the electromagnets that make up the stator usually have steel cores. Copper windings fill the space between consecutive steel cores. Hanselman [31] [p111] describes cogging torque as “the torque created when the rotor permanent magnets attempt to align themselves with the maximum amount of ferromagnetic material”, i.e. the permanent magnets on the rotor have a much greater attraction to the steel cores than to the copper windings. Even without any current flowing in the motor, torque needs to be applied to ‘cog’ from where the magnets align to one set of cores to where the magnets align to the next set of cores.
Torque ripple
The interaction between the magnetic flux from the rotor and stator can be charac- terised by the shape of the voltage induced in each phase winding by motor motion (back EMF). Torque is created in PMAC machines through the interaction of the back EMF waveform in each phase and the current waveform in that phase. For a three phase machine, if the back EMF from the rotor is purely sinusoidal and the current in each of the three phases is also purely sinusoidal then the torque will be constant. Alternatively, a trapezoidal back EMF excited by a square wave current will also result in a constant torque. Torque ripple is the term applied to any periodic variation in torque created when either of these waveforms deviate from ideal waveforms.
1.1.4 Control methods for smooth operation
Torque ripple is minimised by optimising the interaction between the phase currents and the phase back EMF waveforms. This optimisation can be achieved by either mechanical means (altering the back EMF shape) or by electrical means (altering the current waveform) [40].
4 1.1. BACKGROUND
There are two major disadvantages of motor modification methods: design trade- offs and the accurate manufacturing process required.
Design trade-offs
Sometimes the modification of motor design to reduce cogging torque can have a nega- tive impact on torque ripple. In their 2002 paper Analysis of torque ripple in a BDCM [79](p1293), Zeroug, Boukais, and Sahraoui describe this effect for a brushless DC ma- chine. Efforts to reduce either cogging torque or torque ripple can also have a negative effect on average torque [29], [6].
Manufacturing tolerances
Jahns and Soong [40] noted that: “techniques which require a high accuracy of as- sembly, magnetisation, magnet placement or dimensions may prove to be impractical for low-cost, high volume production.” If pulsating torque is to be minimised by me- chanical means, high accuracy manufacture is required, limiting the practicality for low-cost, high volume production.
Control method benefits
In contrast to the motor design approach, the control based approach allows motors to be designed for maximum average torque and minimum manufacturing cost. Ripples can be removed that were created by a design that maximises average torque or by manufacturing inaccuracies. A control based approach can use feedback information from a torque transducer, however the associated cost and complexity is usually prohibitive. This research will focus on reducing pulsating torque using only current and position feedback informa- tion.
1.1.5 Programmed reference current waveform PRCW methods
Jahns and Soong [40], characterised control methods for minimising pulsating torque into five categories:
1. commutation torque minimisation;
5 CHAPTER 1. INTRODUCTION
2. speed loop disturbance rejection;
3. high speed current regulator saturation;
4. estimators and observers; and
5. programmed current waveform control.
Of these categories, commutation torque minimisation is only relevant to motors with a trapezoidal back EMF driven by a square wave current (120o electrical), high speed current regulator saturation is only relevant for high speed operation and speed loop disturbance rejection is only relevant for low speed operation. Most estimator and observer methods require a very high resolution position signal to be effective at low speed [59],[20]. They can also have difficulty coping with load fluctuations unless adaptive control is used [14]. Some research has been done on this problem by using adaptive control [29], [76], [52]. Control systems benefit most from adaptive control when parameters are time varying. This research only considers the implementation of programmed current wave- form methods under a set of operating conditions where the parameters are time invari- ant. Under different operating conditions, parameters may vary and adaptive control may be beneficial. This is a subject for further work (see section 9.1). Of the methods suggested, only PRCW control has the potential to work at all speeds and work independently of the applied load. As such, research focuses on these methods.
1.1.6 Previous implementation of PRCW methods
Despite the potential of PRCW methods in theory, Jahns and Soong [40] commented that: “experimental verification of the proposed harmonic injection techniques is gen- erally weak”. Since 1996, further work has been done on PRCW methods however no clear experimental verification has been published. In 2004, Bianchi and Cervaro [6] suggested that “The suppression of the torque ripple of SPM machines is a problem that is not completely solved”. The major challenges to the experimental implementation of these methods are:
6 1.2. RESEARCH GOAL
1. the determination of motor parameters to a suitable accuracy;
2. accurate torque measurement;
3. the provision of a smooth load to the motor; and
4. the presence of mechanical resonances in the experimental setup.
1.2 Research goal
In view of the limited experimental implementation and verification of PRCW methods, this research will use a motor with an inherently high pulsating torque and a well designed experimental setup to compare the different PRCW methods. In doing so, it will be determined if any of the methods can achieve a smooth torque comparable to that of a motor specifically designed for that goal. The rationale behind the use of a motor with high pulsating torque is to consider the extreme case. Methods that can reduce the pulsating torque of such a motor to an acceptable level should be applicable for machines with lower initial pulsating torque.
1.3 Research approach
Completion of the research goal involved the following tasks:
1. A literature review of:
• published PRCW methods for minimising pulsating torque; and
• existing methods for determination of motor parameters.
2. A theoretical comparison of the published methods to determine the conditions under which each is the preferred method;
3. A theoretical evaluation of alternative motor parameter determination methods;
4. Design of an experimental setup that overcomes the challenges described in sec- tion 1.1.6; and
5. Experimental comparison of:
7 CHAPTER 1. INTRODUCTION
• motor parameter determination techniques;
• different PRCW methods; and
• motor parameter sensitivities.
1.4 Chapter overview
To explain the tasks outlined above, chapter 2 discusses published PRCW methods and chapter 3 describes the parameter determination which is critical to the success of these methods. Chapter 4 theoretically compares the published PRCW methods based on various back EMF and cogging torque motor parameters. Chapter 5 proposes a ‘pulsating torque decoupling’ (PTD) approach for accurate motor parameter determination and theoretically analyses that method. Chapter 6 discusses the challenges and solutions involved in creating the experimen- tal setup and chapter 7 reports the results of the experiments. Results are discussed in chapter 8 and conclusions are made in chapter 9. The drawings and calculations for the detailed design of the test rig and computer controlled power supply are presented in the appendices along with additional results.
8 Chapter 2
Review: Programmed reference current waveform methods
The introduction presented the problem of pulsating torque in PMAC motors and focused this research on programmed reference current waveform (PRCW) methods. This chapter reviews past research into these control methods to provide a background for the theoretical comparison discussion in chapter 4, and for the experimental setup design described in chapter 6.
A mathematical model for a PMAC motor is presented and the general control problem is considered. Three published PRCW methods are discussed. To allow later comparison, each method has been explained in a common nomenclature which may differ slightly from the original published nomenclature.
The last section of the chapter reviews the experimental implementation of PRCW methods and discusses the challenges previous researchers have highlighted from their experimental work.
2.1 PMAC mathematical model
Control of PMAC motors relies on a mathematical model. This model is derived from a relationship between mechanical and electrical power and is based on several assumptions.
9 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS
2.1.1 Assumptions and Constraints
1. Back EMF is proportional to angular velocity, and independent of current: Maxwell’s equation states that the back EMF across a phase is proportional to the speed of the machine [38].
2. Torque produced is proportional to phase currents: The Lorentz force equation states that the instantaneous torque produced by a particular phase winding is proportional to the phase currents [38].
3. DC bus voltage: For the purposes of this comparison, speed limits imposed by a finite voltage source are not considered.
2.1.2 General Model Equation
For a three phase synchronous motor, electrical power is given by the product of the current and voltage. Mechanical power is given by the product of the torque and the speed. Assuming an efficiency of 100%, the relationship between the mechanical and electrical power is given by [31](eq 8.1):
em X τθ ω = iθ,pEθ,p (2.1) p=a,b,c where:
θ = mechanical rotor position (rad) em τθ = electromagnetic torque at angle θ (Nm) ω = mechanical angular velocity (rad/sec)
iθ,p = current in phase p at angle θ (A)
Eθ,p = back EMF for phase p at encoder point θ (V s/rad).
This equation is simplified by assuming the back EMF is proportional to angular velocity, allowing En to be defined by a normalised waveform scaled by the mechanical Eθ,p rotor speed. ( ω = kθ,p). Equation 2.1 becomes: em X τθ = iθ,pkθ,p (2.2) p=a,b,c where:
10 2.2. FUNDAMENTAL CONTROL PROBLEM
kθ,p = normalised back EMF for phase p at angle θ (V s/rad).
If cogging torque is considered then the total torque is:
m em cog τθ = τθ + τθ (2.3) X cog = iθ,pkθ,p + τθ p=a,b,c
where:
m τθ = motor torque at angle θ (Nm) cog τθ = cogging torque at angle θ (Nm)
It should be noted that this equation is based on the mechanical position and mechanical angular velocity, however some of the methods below are based on the electrical position and angular velocity. In a complete mechanical revolution of the motor, there will be an electrical revolution for each pair of poles. Therefore:
θe = P θ (2.4)
where:
θe = electrical rotor position (rad) θ = mechanical rotor position (rad) P = number of pole pairs
2.2 Fundamental control problem
The most basic form of the control problem considered has three inputs to the con- troller, torque reference, current feedback and position feedback. The output of the controller is the EMF applied to the ends of each of the three phases. This control scheme is shown in figure 2.1.
11 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS
Figure 2.1: Fundamental PMAC torque control scheme
2.3 PRCW methods
PRCW methods split the fundamental control problem into two parts. A current reference generator uses predetermined information about the back EMF and cogging torque to determine the current for a particular position. The current controller ensures that the actual current follows the current commanded by the current reference block. This scheme is shown in figure 2.2.
Figure 2.2: PRCW control scheme
Previous researchers have taken a number of different approaches, of which the most popular are:
1. Frequency domain method (FDM) where the normalised back EMF (k) is defined as a Fourier series which leads to a definition of the current reference (i) as a
12 2.4. GOALS AND CONSTRAINTS
Fourier series [43], [38], [32], [10], [9] and [6].
2. Time domain method (TDM) where k is defined as an array of values for each encoder value θ. This array is then used to determine i as an array [75].
3. ‘Park-like’ method (PLM) where a modification of field oriented control is used to determine i in a rotating reference frame [30], [46], [59],[12], [11] and [58].
2.4 Goals and constraints
The primary goal of each of these methods is to minimise pulsating torque. This may include the requirement to suppress cogging torque. For the FDM and the TDM there is an additional constraint to minimise RMS current. Depending on the topology of the motor there may be an additional ‘star connection’ constraint to ensure that the sum of all currents at any time is zero.
2.5 Frequency domain method (FDM)
This analysis follows that presented by Hung and Ding [38]. It is similar to that presented by Le-Huy [43], Hanselman [32] and Bianchi and Cervaro [6]. Chapman, Sudhoff and Whitcomb [10], [9] describe this method as the ‘minimum current no ripple’ method. They present a further addition to this method which they describe as the ‘minimum current minimum ripple’ method [10], [9]. This addition allows the designer to place more emphasis on either the ripple minimisation constraint or the current minimisation constraint. This modification however, adds considerable complexity to the analysis and studies have shown [10], [34] that there is negligible benefit to be gained by allowing more torque ripple in an attempt to further minimise the current.
2.5.1 Assumptions
This method requires several further assumptions to those outlined in section 2.1.1:
1. All phases have the same back EMF and current waveform shapes and are 120o out of phase.
13 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS
2. The back EMF waveform is identical for each electrical revolution (allowing the
analysis to be done in terms of θe instead of θ.
2.5.2 Parameter descriptions
Back EMF
For this method, the back EMF is converted into an exponential Fourier series (trigono- metric Fourier series in [10]). It should be noted that this analysis is done in electrical degrees rather than mechanical degrees.
∞ X ˆ jnθe kθe,a = Kne (2.5) n=−∞ where:
kθe,a = normalised back EMF for phase a at electrical angle θe (V s/rad)
θe = electrical rotor position (rad) n = harmonic number of back EMF and current series
Kˆn = normalised back EMF Fourier series coefficients
Further:
1 Z 2π ˆ −jnθe Kn = (kθe,a)e (2.6) 2π θe=0 ˆ ˆ It is worth noting that since kθe,a is real, Kn and K−n are complex conjugates.
Current
It is assumed that the current and electromagnetic torque can also be represented as a Fourier series.
∞ X ˆ jnθe iθe,a = Ine (2.7) n=−∞ where:
iθe,a = current in phase a at electrical angle θe (A)
Iˆn = current Fourier series coefficients
14 2.5. FREQUENCY DOMAIN METHOD (FDM)
k and i can be found by replacing θ with (θ − 2π ). k and i can be θe,b θe,b e e 3 θe,c θe,c 2π found by replacing θe with (θe + 3 ).
Electromagnetic torque in each phase
It is also assumed that the electromagnetic torque produced by each phase can be represented by a Fourier series. For phase a, the electromagnetic torque is then given by:
∞ em X jnθe τ = Tˆm,ae (2.8) θe,a m=−∞ where:
em τ = electromagnetic torque produced by phase a at electrical angle θe θe,a (Nm)
Tˆm,a = electromagnetic torque Fourier series coefficients for phase a m = harmonic number of torque series
The combination of the current and back EMF harmonics can be modeled as a convolution sum so:
∞ X Tˆm,a = Kˆm−nIˆn (2.9) n=−∞
Total electromagnetic torque
The total electo-magnetic torque produced will be a sum of the three phase torques.
∞ 2π 2π em X jmθe jm(θe− ) jm(θe+ ) τ = Tˆm,ae + Tˆm,be 3 + Tˆm,ce 3 (2.10) θe m=−∞ From the assumption that the back EMF and current harmonics are the same for all phases: Tˆm,a = Tˆm,b = Tˆm,c, equation 2.10 can be rewritten as:
∞ 2π 2π em X jmθe jm(θe− ) jm(θe+ ) τ = Tˆm,a e + e 3 + e 3 (2.11) θe m=−∞ Which can be simplified to:
15 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS
∞ em X 2πm jmθe τ = Tˆm,a 1 + 2cos( ) e (2.12) θe 3 m=−∞ 2π 2π jm(− 3 ) jm(+ 3 ) 2πm By noting that : 1 + e + e = 1 + 2cos( 3 ), equations 2.12 and 2.9 can be combined to determine an expression for the coeffients of the total torque harmonics:
∞ X 2πm Tˆ = Kˆ Iˆ 1 + 2cos( ) (2.13) m m−n n 3 n=−∞ Now:
2πm 1 + 2 cos = 3 n = ±3, ±6, ±9 ... (2.14) 3 2πm 1 + 2 cos = 0 n 6= ±3, ±6, ±9 ... (2.15) 3 So:
∞ X Tˆm = 3 Kˆm−nIˆn n = ±3, ±6, ±9 ... (2.16) n=−∞ As expected, consideration of equations 2.9 and 2.16 suggests that the total torque is three times the torque produced by each phase.
2.5.3 Torque ripple and RMS current minimisation
The relationship between back EMF, current and generated torque harmonics can be seen clearly in Table 2.1 which is a modification of Table 1 presented by Favre, Cardoletti and Jufer [23]. The ‘0’ harmonics are those responsible for average torque creation. Those marked with an x would create torque harmonics, however a star connection constraint ensures that there are no current harmonics that are multiples of three. It is worth noting not only which harmonics will combine to create torque ripple, but conversely, which harmonics must be present in order to cancel out particular cogging torque harmonics. For example if the back EMF is purely sinusoidal, the fifth and seventh current harmonics are required to cancel out a sixth harmonic in the cogging torque.
16 2.5. FREQUENCY DOMAIN METHOD (FDM)
Table 2.1: Back EMF and current harmonic combinations responsible for torque har- monics (including star connection constraints)
Kˆn 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 0 3 3 6 6 9 9 12 12 15 15 18 2 3 0 6 3 9 6 12 9 15 12 18 15 3 x x x x x 4 3 6 0 9 3 12 6 15 9 18 12 21 5 6 3 9 0 12 3 15 6 18 9 21 12 6 x x x x x 7 6 9 3 12 0 15 3 18 6 21 9 24 8 9 6 12 3 15 0 18 3 21 6 24 9
Iˆn 9 x x x x x 10 9 12 6 15 3 18 0 21 3 24 6 27 11 12 9 15 6 18 3 21 0 24 3 27 6 12 x x x x x 13 12 15 9 18 6 21 3 24 0 27 3 30 14 15 12 18 9 21 6 24 3 27 0 30 3 15 x x x x x 16 15 18 12 21 9 24 6 27 3 30 0 33 17 18 15 21 12 24 9 27 6 30 3 33 0
17 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS
Limits of series
So far the limits of the Fourier series have been considered infinite. In practice, a limit needs to be chosen for each of the series. To ensure that equation 2.16 is not under-constrained, the limit of the current harmonics (N) must be greater or equal to the limit of the back EMF harmonics (L). N can be equal to L as long as L is not a multiple of three. The largest torque harmonic that can be cancelled is equal to the sum of the current and back EMF harmonics (M = N + L). This is particularly important to consider when cogging torque harmonics need to be cancelled.
Matrix representation
Equation 2.16 can be represented in matrix form by:
~ ~ Kˆ Iˆ = Tˆ (2.17)
where:
Kˆ = matrix of the back EMF coefficients ~ Iˆ = vector of current coefficients ~ Tˆ = vector of torque coefficients
As long as the system of equations defined in equation 2.17 is over-constrained, an additional constraint to ensure the current is minimised can be added. By using the minimum norm solution, the optimal current can be defined as:
~ˆ ˆ T ˆ ˆ T −1 ~ˆ Ioptimal = K (KK ) T (2.18)
2.5.4 Cogging torque suppression
~ If cogging torque suppression is required, the first element of Tˆ should be the average torque required and the remaining elements should be the negative of the cogging torque Fourier coefficients. In this way the torque ripple should cancel out the cogging torque.
18 2.6. TIME DOMAIN METHOD (TDM)
2.5.5 Star connection constraint
In the frequency domain, the addition of a star connection constraint requires that there are no current harmonics that are multiples of three. If this is required, the rows of Kˆ that correspond to these harmonics should be removed.
2.6 Time domain method (TDM)
2.6.1 Parameter descriptions
This method follows the analysis presented by Wu and Chapman [75]. The back EMF is defined as an array of values for each discrete value of θ. This array is then used directly in calculations to define the reference current.
2.6.2 Torque ripple and RMS current minimisation
The optimisation is done using the method of Lagrange multipliers where the objective Q is defined and subjected to a cost constraint C. The optimal solution is found by differentiating the resulting function with respect to i. Objective to be minimised (RMS current):
X 1 Q = (i )2 (2.19) θ 2 θ,p p=a,b,c em Cost constraint: deviation of electromagnetic torque from reference torque (τθ - ref τθ ):
X ref C1,θ = kθ,piθ,p − τθ (2.20) p=a,b,c Overall function:
fθ = Qθ + λ1,θC1,θ (2.21)
so:
X 1 X f = (i )2 + λ k i − τ ref (2.22) θ 2 θ,p 1,θ θ,p θ,p θ p=a,b,c p=a,b,c
19 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS
∂fθ This function will be minimised with respect to iθ,p when = 0. Taking this ∂iθ,p partial derivative of equation 2.22, expressions are found for optimal current:
iθ,p = −λ1,θkθ,p (2.23)
λ1,θ , however is still unknown. To find the optimal value of λ1,θ requires setting ∂f θ = 0, which gives: ∂λ1,θ
ref X τθ = kθ,piθ,p (2.24) p=a,b,c This implies that the cost function is satisfied.
To find an expression for λ1,θ that is independent of iθ,p , equation 2.23 needs to be substituted into equation 2.24, to give:
ref X 2 τθ = λ1,θ (kθ,p) (2.25) p=a,b,c
which when rearranged to solve for λ1,θ gives:
ref −τθ λ1,θ = P 2 (2.26) p=a,b,c(kθ,p) Now equation 2.26 can be substituted into equations 2.23 to find the optimal cur- rents:
! ref kθ,p iθ,p = τθ P 2 (2.27) p=a,b,c(kθ,p)
2.6.3 Cogging torque suppression
ref If cogging torque suppression is required, τθ should be defined as the average torque required minus the cogging torque. As in the FDM, this should ensure that the torque ripple will cancel out the cogging torque.
2.6.4 Star connection constraint
If a star connection constraint is to be considered, it can be written in matrix form as:
20 2.6. TIME DOMAIN METHOD (TDM)
X iθ,p = 0 (2.28) p=a,b,c This equation can be formulated as a Lagrange cost constraint:
X C2,θ = iθ,p (2.29) p=a,b,c Now there are two constraints in the Lagrange function which becomes:
X 1 X X f = (i )2 + λ k i − τ ref + λ i (2.30) θ 2 θ,p 1,θ θ,p θ,p θ 2,θ θ,p p=a,b,c p=a,b,c p=a,b,c
∂fθ As before, the function will be minimised with respect to iθ,p when = 0. Tak- ∂iθ,p ing this partial derivative of equation 2.30, expressions are found for optimal current:
X iθ,p = (−λ1,θkθ,p − λ2,θ) (2.31) p=a,b,c
∂fθ To find the optimal value of λ2,θ requires setting = 0, which gives: ∂λ2,θ
X iθ,p = 0 (2.32) p=a,b,c Again, this implies that the cost function is minimised. Substituting equation 2.31 into equation 2.24 gives:
ref X τθ = kθ,p (−λ1,θkθ,p − λ2,θ) (2.33) p=a,b,c
ref X 2 X τθ = −λ1,θ kθ,p − λ2,θ kθ,p (2.34) p=a,b,c p=a,b,c
and substituting equation 2.31 into equation 2.32 gives:
X (−λ1,θkθ,p − λ2,θ) = 0 (2.35) p=a,b,c X −λ1,θ kθ,p − 3λ2,θ = 0 (2.36) p=a,b,c
21 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS
λ1,θ and λ2,θ can be found from equations 2.33 and 2.35 to give:
−3τ ref λ = θ (2.37) 1,θ P 2 P 2 3 p=a,b,c kθ,p − ( p=a,b,c kθ,p)
ref P τ kθ,p λ = θ p=a,b,c (2.38) 2,θ P 2 P 2 3 p=a,b,c kθ,p − ( p=a,b,c kθ,p) Which can be substituted back into equation 2.31 to find an expression for the optimal current:
P ! 3kθ,p − kθ,p i = τ ref p=a,b,c (2.39) θ,p θ P 2 P 2 3 p=a,b,c kθ,p − ( p=a,b,c kθ,p)
2.7 ‘Park-like’ method (PLM)
The PLM is a variation on field oriented control where the current is described with reference to a rotating coordinate system. This discussion will follow the most recent implementation of this technique which was done by Park et al. in 2001 [58]. A similar technique was described by Grenier et al. [30] Lu et al. [46] and Chen and Sekiguchi [12][11].
2.7.1 Parameter descriptions
In this method, all the parameters are converted from their three phase coordinates (a, b, c) to a rotating reference frame (d, q).
kθ,d kθ,a kθ,q = C kθ,b (2.40) kθ,0 kθ,c
where:
2π 2π sin(θ) sin(θ − 3 ) sin(θ + 3 ) 2 C = 2π 2π (2.41) cos(θ) cos(θ − 3 ) cos(θ + 3 ) 3 0.5 0.5 0.5
22 2.7. ‘PARK-LIKE’ METHOD (PLM)
2.7.2 Torque ripple minimisation
In a rotating reference frame, the general torque equation stated in equation 2.2 can be rewritten as:
3 τ em = (k i + k i + k i ) (2.42) θ 2 θ,d θ,d θ,q θ,q θ,0 θ,0
The torque produced by id and i0 is assumed to be zero. Substituting this into equation 2.42 and rearranging gives the following expression for the iq current required to provided the desired reference torque:
em 2 τθ iθ,q = (2.43) 3 kθ,q
Obviously this will cause a problem if kθ,q is zero, however as long as kθ,a, kθ,b and kθ,c do not deviate drastically from sinusoids then this will not be the case. If it is desired to know the currents in the three phase reference frame then the current can be converted back by multiplying the d, q reference currents by C−1:
iθ,a iθ,d −1 iθ,b = C iθ,q (2.44) iθ,c iθ,0
Unlike the other two methods, this method does not include a current minimisation constraint. Adding this constraint is not possible as an additional constraint was already added (namely assuming id is zero).
2.7.3 Cogging torque suppression
Like the other two methods, cogging torque suppression is done by defining a reference electromagnetic torque that is the negative of the cogging torque.
2.7.4 Star connection constraint
The PLM ensures that i0 = 0 so there will always be a star connection constraint.
23 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS
2.8 Experimental implementation
As suggested in the introduction, experimental implementation of PRCW methods is limited. This is partly due to a lack of a quantitative measure of the remaining pulsating torque. If the torque is actually measured, most authors resort to presenting graphical results and commenting that the pulsating torque has been ‘significantly reduced’. This section will consider the goal for pulsating torque, and discuss the challenges faced by previous PRCW method researchers when implementing their methods ex- perimentally.
2.8.1 Pulsating torque goal
The goal for ‘smooth’ operation of a PMAC motor is not well defined. Jahns and Soong [40] stated that for motor modification methods such as skewing, about 1% of rated torque is the lowest achievable in practice. Grcar et al. [29] suggested that for high performance drives, a torque pulsation under 1-2% is typically considered as the desired objective. Liu et al. [45] discussed a power steering application with a peak-peak pulsating torque requirement of less than 2-5%. Specifications of commercially available ‘high performance’ motors also vary con- siderably. Trust Automation [3] claims a 0.3% ripple torque for their SE700 motors. For their BL series motors, Malivor [48] do not quote a pulsating torque however pro- vide data on reluctance torque (cogging torque) as 3.5-6%. Parker Motion [55] (p 154) quotes 5% ripple peak-peak for their ‘Dynaserv’ system and ThinGap [73] quotes 0.045% ripple. That value however, is not based on measurements but rather on the harmonic distortion of the back EMF. Additional confusion is created by multiple methods for calculating the percentage pulsating torque. Often the method is not quoted. When the method is quoted, it is usually the ratio of peak-peak torque to rated torque. Peak-peak torque however, can vary significantly between trails as it is only based on two data points. To avoid that issue, for this research, the RMS torque is used as suggested by Gieras and Wing [25] (p245). However, rather than divide the RMS torque by the average torque as they did, division is done by the maximum torque. The justification for this
24 2.9. IMPLEMENTATION CHALLENGES is that at low torque set points, the use of average torque creates distortion. This is because, for the motor used, a large proportion of the pulsating torque is related to cogging which does not scale with average torque.
RMS Terror Tpulsating = (2.45) Tmax While the RMS gives a lower pulsating torque than the peak-peak value, a rough comparison is to consider the relationship for a pure sine wave where the peak to peak √ value is 2 2 times larger than the RMS value. Assuming the figures given above for ‘high performance drives’ are peak-peak values then a RMS value of 1% would fit somewhere in the range of motors surveyed and so is the goal of this reaserch.
2.9 Implementation challenges
The most significant challenge facing the implementation of PCRW methods is the determination of motor parameters to a suitable accuracy [40]. This is discussed in detail in chapter 3. The other challenges, as outlined in the introduction are:
1. accurate torque measurement;
2. the provision of a smooth load to the motor, and
3. the presence of mechanical resonances in the experimental setup.
2.9.1 Torque measurement
In the absence of a suitable torque sensor, many PRCW method experimental set- ups rely on recreating the torque from the measured currents and back EMF wave- forms. Unfortunately, this neglects any cogging torque or pulsating torque created from parameter variations in the windings. These factors have the potential to have a significant impact on the effectiveness of programmed current waveform methods. When the torque is measured, one potential issue is achieving resolution. Li et al. [69] highlighted a problem with traditional methods of testing for torque ripple. A torque sensor with a large enough range to deal with the maximum load will not necessarily have the resolution to measure the torque ripple.
25 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS
Another issue with torque measurement is adequate bandwidth. All the sensors used for PRCW research have been either strain gauge or surface acoustic wave tech- nology. The surface acoustic wave technology sensor used by Wu and Chapman [75] had a bandwidth up to 1 kHz. Aghili, Buehler and Hollerbach [2] used a Himmelstein strain gauge sensor which can have a bandwidth somewhere between 200 − 1000Hz depending on the signal type [17] and the signal conditioner used [18]. With band- widths in this range, the speed of the motor is limited if higher harmonics are to be measured.
Beccue et al. [4] suggest a piezoelectric polymer for use in PMAC motors in pref- erence to the the other methods discussed. Their justification however, was cost on the assumption that this type of sensor would be installed in all motor installations. As PRCW methods seek to avoid a torque sensor in all installations and only use a torque sensor for initial calibration and validation, cost is a less significant issue.
2.9.2 Load application
To apply a load, previous PRCW method researchers have either used: DC motors, hysteresis brakes, eddy current brakes, hydraulic motors or friction brakes.
DC generators
DC generators provide a load by operating as a generator and turning kinetic energy into electrical energy. The electrical energy then needs to be either fed back into the power supply for the test motor or dissipated in a resistor bank. The torque applied can be readily controlled by adjusting the current flowing out of the generator. Some issues can arise however, when this method is used in a setup for measuring pulsating torque. Qian, Panda and Xu [64] described how the pulsating torque from the DC generator interfered with the measurement of the pulsating torque from the ‘test’ motor. This can be minimised with careful choice of load generator however, the commutations will always create some pulsating torque.
26 2.9. IMPLEMENTATION CHALLENGES
Hysteresis
Hysteresis is a property of magnetic materials where the flux density is a function of previous field intensity across the material [31]. In a hysteresis brake, a disk made of magnetic material rotates between a series of magnetic poles. Poles previously induced in the disk interact with the stationary magnetic poles and torque is transmitted until the disk begins to slip. Once slipping, the poles in the disk move as the material is magnetised and demagnetised [16], [72] and [68]. Energy is dissipated due to the hysteresis of the material. Load torque is adjusted by varying the strength of the applied magnetic field. Hysteresis motors have the benefit that the torque applied is independent of speed so torque can be applied down to zero speed. This means however, that if speed is to be regulated then a control system is required to do so. Wu and Chapman [75] found that the controller from the commercially available hysteresis brake was not able to keep a constant low speed. Another problem associated with hysteresis brakes is the potential for residual cogging torque to be created if the brake is incorrectly shut down [47](p53). As hysteresis brakes are unable to drive the motor, another drive source is required to determine back EMF. Any pulsating torque from the drive motor is not an issue however, because the back EMF is normalised with velocity.
Eddy current
In an eddy current brake a highly conducting disk (such as copper or aluminium) rotates in a magnetic field. The magnetic field induces eddy currents in the disk. These eddy currents in turn create a magnetic field that opposes the original magnetic field. As the current produced is a function of the velocity the torque produced is a function of motor speed [68], [71]. This proportionality of torque to speed, limits the applicability of eddy current brakes as they cannot be used for zero speed testing (‘locked rotor’). Another issue is that due to their inherent damping they have a large mechanical time constant so are unsuitable for high frequency dynamic testing. They do however have the benefit that the torque applied is completely smooth.
27 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS
Because eddy current brakes are unable to create motion, like hysteresis brakes, another drive motor is required when measuring the back EMF.
Hydraulic motors
Hydraulic motors can be used to apply a load in a similar way to a DC motor [2]. Instead of controlling the torque by the amount of current, the pressure is regulated. As with DC motors, if the motor is not chosen carefully it can create pulsating torque. Due to the peripheral equipment required, hydraulic motors are usually more complex and expensive than other options.
Prony friction brake
Another device used for research into PRCW methods is a ‘Prony’ brake test device which was used in [22]. This is a rudimentary device made of a drum immersed in cooling water with a belt running over it. Accurate control of this device is difficult and the authors noted that there was a significant once per revolution component due to the braking mechanism.
2.9.3 Mechanical resonance problems
In their 2005 paper [75], Wu and Chapman suggest that mechanical resonances in the experimental setup obscured the fluctuating torque that they were attempting to measure. Unless measures are taken to avoid this problem, other experimental setups would suffer from the same problem. The use of flexible couplings to connect in-line torque transducers is a particular concern as their flexibility creates additional system dynamics that are difficult to model.
2.10 Summary of PRCW methods
Three PRCW methods have been presented: the time domain method (TDM) the frequency domain method (FDM) and the Park-like method (PLM), each claiming to minimise pulsating torque. The FDM and the TDM also attempt to minimise RMS current. The performance of each of these methods is compared for various back EMF and cogging torque waveforms in chapter 4.
28 2.10. SUMMARY OF PRCW METHODS
Methods of describing pulsating torque were discussed and levels of torque in in- dustrial drives considered. This allowed a goal for smooth operation to be defined. The challenges facing successful experimental implementation were presented. The largest challenge facing all of these methods is the accuracy of the determination of the motor parameters. Published methods for determination of these parameters are discussed in chapter 3.
29 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS
30 Chapter 3
Review: determination of motor parameters
Common to all of the programmed current waveform methods discussed in chapter 2 is the need for accurate information about motor parameters. If the back EMF, cogging torque, current flow and rotor position are known for a motor then it is theoretically possible to eliminate pulsating torque. Problems are created if these parameters are determined inaccurately. This chapter will review existing work done on the determination of motor param- eters to ensure that the experimental methods implemented are based on the most accurate motor information available. Consideration will be given to determination by calculation and by measurement. Explanation will also be given on how the variation of each property will affect the pulsating torque. The final part of the chapter will discuss the importance of sensitivity analysis and review relevant publications.
3.1 Back EMF
Back EMF is the induced EMF created by the interaction of rotor and stator magnetic fields when the motor is turned. It is normally measured with the phases open-circuit while the motor is being rotated by an external drive. Though measurement is reason- ably straightforward, a brief discussion is provided as to the likely harmonics present,
31 CHAPTER 3. REVIEW: DETERMINATION OF MOTOR PARAMETERS so that the credibility of the measured results can be checked.
3.1.1 Analytical determination
Generally the back EMF in a PMAC is a shape that is somewhere between a sinusoid and trapezoid [40]. Waveforms of this shape will be half wave symmetric [31] [p194], which implies the waveform in the second half of the period is the negative of the waveform in the first half of the period. Faraday’s law defines that the back EMF is the derivative with respect to angle of the flux linkage. The flux linkage itself is a function of air-gap flux density, wind- ing inductance, winding resistance, the number of slots, the number of magnets and geometrical factors. In his book, Brushless Permanent Magnet Motor Design [31], Hanselman provides a series of predicted back EMF shapes for various motor configu- rations. These can be used for a rough guide to check against experimental results.
3.1.2 Finite element analysis (FEA)
Much work has been done in determining the back EMF using finite element analysis mostly as part of the design process for new motors. As the motor design is beyond the scope of this research these methods will not be discussed in detail. Of interest however, is an analysis done by Patterson [61] on a very similar motor to the one used for the experimental part of this research. Results suggested an error of 1% between the back EMF calculated from FEA and the measured back EMF. Determination of back EMF using finite element analysis, while potentially accu- rate, usually assumes that the back EMF will be identical for each electrical revolution. Unless specifically catered for, manufacturing variation will lead to errors in the deter- mined waveform.
3.1.3 Measurement
Measurement of back EMF is generally straightforward. In their book Design of Brush- less Permanent-Magnet Motors [41](p11-2) Hendershot and Miller describe it as: ‘per- haps the simplest and most useful test which can be performed on a brushless DC motor’. This statement can be extended to other types of PMAC mtors. An external
32 3.2. COGGING TORQUE drive is needed to provide rotation while the open circuit voltage is measured. As the final goal is the speed normalised back EMF, it is also critical that an accurate mea- surement of velocity is obtained. This is particularly important if there is any speed fluctuation during the measurement resulting from cogging torque.
3.1.4 Effect of error on pulsating torque
As long as the assumption holds that the back emf is proportional to velocity (see section 2.1.1), the shape of the back emf will remain constant. That allows the back EMF error to be divided into offset and scaling errors. In a similar way to the current errors discussed in section 3.3.2, an offset error will lead to a pulsating torque harmonic at the fundamental electrical frequency and a scaling error will lead to a pulsating harmonic at twice the electrical frequency.
Temperature
The research published about PRCW methods rarely considers the effects of tem- perature. Mattavelli, Tubiana and Zigliotto [49] suggested that the back EMF could possibly vary with motor temperature, but not significantly. However, the magnitude of the back EMF is a function of the magnetic flux [31] [eq 7.46], which is in turn a function of the remanence [31] [eq 2.21], which is dependant on temperature [31] [eq 2.20]. This dependence on temperature is often published for particular magnets. One company, Oemag, has data showing that for NdFeB magnets of the type used in this research, the remanence falls by about 0.12%/oC [57]. As the motor heats up to operating temperature, a significant reduction in back EMF could be expected.
3.2 Cogging torque
The formal definition of cogging torque is: ‘pulsating torque components generated by the interaction of the rotor magnetic flux and angular variations in the stator magnetic reluctance.’ [40]. For the purposes of PRCW methods, it is the component of pulsating torque that is independent of excitation current.
33 CHAPTER 3. REVIEW: DETERMINATION OF MOTOR PARAMETERS
Any current reference waveform method that seeks to remove pulsating torque must compensate for the cogging torque and so needs an accurate, time invariant description of this torque. In contrast to the back EMF, experimental determination of cogging torque is much more difficult, mainly because of the difficulty in decoupling the cogging toque from either the torque ripple or from dynamic effects resulting from the speed variation of the rotor.
3.2.1 Analytical determination
As with back EMF, it is worth considering which harmonics will be present in the cog- ging torque to provide a guide to the validity of any measurements. The fundamental cogging torque harmonic will be: [25][p247]
Nslot fcogging = (3.1) Npole pair where:
Nslot = number of stator slots
Npole pair = number of pole pairs
Detailed analytical calculations can be done to determine the actual waveform. The cogging waveform depends on the tooth Fourier series coefficients which are a function of the magnetic field distribution around each tooth, the air gap length and the size of the slot opening between teeth [31](p210). This detailed analysis is beyond the scope of this thesis.
3.2.2 FEA
As with back EMF, much work has been done on the calculation of cogging torque by FEA. It is generally done as an aid to motor design and so is beyond the scope of this research. Some research, such as that presented by Islam, Mir and Sebastian [39] and Shaotang, Namuduri and Mir [13] is relevant to this research because they use FEA to consider the effect of motor construction variation on pulsating torque.
34 3.2. COGGING TORQUE
3.2.3 Experimental
Experimental methods for determining cogging torque can be classified into static, quasi-static and dynamic methods.
Static
Some static measurements of cogging torque are only designed to determine the ab- solute magnitude. For instance, a method is described by Caricchi et al. [7] where a measurement was taken of the force applied to a lever arm sufficient to move the rotor from one equilibrium state to the next. The static method presented by Chandler [8] allows the determination of the actual waveform. The rotor was mounted in a rotary ‘dividing head’ from a milling machine. The stator was held in place by a beam with a strain gauge attached. By rotating the dividing head the force on the strain gauge could be noted. Problems were noted with this method however. For adequate sensitivity, the strain gauge was placed on a flexible beam. This flexibility made accurate measurements difficult, particularly when the motor was crossing from unstable position from one tooth to the next.
Quasi-static
A quasi-static method is described by Aghili, Buehler and Hollerbach [2] where the motor velocity is kept sufficiently low (1o/s), to ensure that the inertial torque does not interfere with the measurement. Their setup is driven by a hydraulic motor with the pressure set sufficiently high to ensure that the angular speed remained constant regardless of the test motor torque.
Dynamic
Most motor measurement equipment is configured for measurements while the motor is rotating. As such, it is attractive to attempt cogging torque measurement while the motor is rotating. This form of measurement is discussed by Holtz and Springob [37] and by Bianchi and Bolognani [5]. As an in-line torque sensor was used, both authors pointed out the need for keeping rotor speed constant during the measurements.
35 CHAPTER 3. REVIEW: DETERMINATION OF MOTOR PARAMETERS
Figure 3.1: FBD for motor
When measuring cogging torque dynamically, the location of the torque sensor is important. Figure 3.1 shows a free body diagram (FBD) for the motor. The two possibilities for the location of a torque sensor are:
1. In-line torque sensor which will measure the load torque in the shaft.
2. Reaction torque sensor which will measure the reaction torque on the back of the stator.
In the case of the in-line torque sensor, if a FBD is taken of just the rotor (see figure 3.2) then the sum of the moments in the axis of rotation gives:
Tmotor = Tload + Jα + bω (3.2)
On the other hand, with a reaction torque sensor (see FBD in figure 3.3) the sum of moments in the axis of rotation gives (assuming that the reaction torque sensor is suitably stiff):
Tmotor = Treaction (3.3)
Care must be taken when using an in-line torque sensor, as compensation has to be made for inertial torques due to acceleration. This issue is avoided by the use of a reaction torque sensor.
36 3.2. COGGING TORQUE
Figure 3.2: FBD for rotor
Figure 3.3: FBD for stator
37 CHAPTER 3. REVIEW: DETERMINATION OF MOTOR PARAMETERS
Many other published works ([60], [80], [42], [39], [77] and [44]) present data for ‘experimentally measured’ cogging torque but have limited information as to how these measurements were performed.
3.2.4 Effect of error on pulsating torque
As the pulsating torque is the sum of the torque ripple and the cogging torque, the pulsating torque will increase by the error in the estimate of the cogging toque. Errors can be classified into: manufacturing (magnet placement, eccentricity and material property variation), and operating point dependent (temperature and torque set point dependent). This classification is important as theoretical determination will not usu- ally account for manufacturing errors, whereas direct measurement will measure these errors. Variation that is operating point dependent will be difficult to determine by either method.
Material properties
Cogging torque is a function of the magnetic interaction between the permanent mag- nets and the steel in the stator. As such, any variation in the magnetic properties of either of these materials will cause variation in cogging torque. Morcos, Brown and Campbell [54] suggested that poor uniformity of the magnets can lead to high cogging torque.
Manufacturing tolerances
Islam, Mir and Sebastian [39] report that if a magnet is misplaced from its “perfect” position by 1 mechanical degree (in a 6 pole 27 slot motor), then the magnitude of the cogging torque can be increased by over three times. Obviously this result will vary with motor topology but the fact that accurate manufacturing is critical is still worth noting. It is suggested by Hartman and Lorimer [33] that other common defects are non-concentric stators and rotors (equivalent to misalignment in an axial flux motor such as that used for this research).
38 3.3. CURRENT
Temperature
As with the effect of temperature on back EMF, there has been very little published about the effect of temperature on cogging torque. Grcar et al. [29] claimed that the waveform of the cogging torque varies with the operating conditions (temperature), however they did not report the the extent of this variation. It was discussed earlier the the back EMF is a function of the magnetic flux so is likely to be temperature dependant. The cogging torque is also related to the magnetic flux [31] [eq 3.37], so will also be effected by as the motor warms up to operating temperature.
3.2.5 Variation with torque set-point
Although usually defined as being independent of current flow, Gieras and Wing [25][p247] suggest that cogging torque can be affected by saturation effects associ- ated with high current flow . Dai, Keyhani and Sebastian [19] discuss how an uneven tooth flux density can cause a current related component of the ‘cogging’ torque.
3.3 Current
Unlike the back EMF and cogging torque, the optimal current is calculated and so does not need to be ‘determined’. However, as current is used as the feedback variable in the controller, its measurement is critical. Any measurement errors, along with any errors in the controller itself will lead to pulsating torque.
3.3.1 Measurement
Current measurement is normally done by measuring the voltage across a shunt or by a Hall-Effect device. Chapter 6 contains a detailed discussion of current sensors including justification of the sensors chosen for this research.
3.3.2 Effect of error on pulsating torque
As long as the current waveform is within the bandwidth of the sensor, current mea- surement error is normally considered as a combination of an offset and a gain error
39 CHAPTER 3. REVIEW: DETERMINATION OF MOTOR PARAMETERS
(i.e. the assumption is made that the output remains linear). Analysis of current error has been discussed in several publications [65], [15], however the most detailed analysis is reported by Chen, Namuduri and Mir [13].
Offset error
The addition of an offset into the torque equation (equation 2.2) induces a torque ripple at the fundamental electrical frequency. Chen, Namuduri and Mir calculated that in the worst case, a 1% error in offset could lead to a 4% error in torque ripple.
Scaling error
The addition of a scaling error to the torque equation (equation 2.2) leads to a torque ripple of twice the fundamental electrical frequency. Chen, Namuduri and Mir sug- gested that in the worst case, a 1% scaling error between sensors in different phases could lead to a 2.3% torque ripple.
3.4 Rotor Position
3.4.1 Measurement
Normally rotor position is either measured by an encoder, where the position signal is a digital signal, or by a resolver where the signal is a sinusoidally varying analog signal. Detailed discussion of position sensors is presented in chapter 6.
3.4.2 Effect of error on pulsating torque
Analysis by Chen, Namuduri and Mir [13] suggests that for a low resolution encoder (10 electrical degrees/ count in their setup) the induced torque ripple can be up to 5%. This emphasises the need for a high resolution encoder when implementing PRCW methods.
3.5 Sensitivity analysis
Jahns and Soong [40] noted that there was minimal work published on sensitivity anal- ysis for PRCW methods. Since then there has not been much further work done on the
40 3.6. SUMMARY OF METHODS FOR DETERMINATION OF MOTOR PARAMETERS subject. Grcar et al. [28] stated that “Since motor parameters can considerably vary under a wide range of operating conditions (temperature, saturation, load variations), a sensitivity analysis must be included as a part of the design for every particular drive”. They do not however, report the results of their analysis.
3.6 Summary of methods for determination of motor parameters
This chapter discussed several methods for the determination of motor properties. In most cases, to account for manufacturing errors, experimental methods are preferred to theoretical methods. Theoretical methods are important as a guide to the credibility of the experimental results.
41 CHAPTER 3. REVIEW: DETERMINATION OF MOTOR PARAMETERS
42 Chapter 4
Theoretical comparison of reviewed methods
Chapter 2 discussed the mechanisms for the creation of pulsating torque, presented a mathematical model of PMAC machines and reviewed the published methods for pulsating torque minimisation.
This chapter theoretically compares the reviewed methods when applied to a gen- eralised motor model with a variety of back EMF and cogging torque waveforms.
Initially, pulsating torque minimisation is considered without cogging torque using various back EMF waveforms. Some of these variations break the assumptions imposed by some methods to determine the sensitivity of each method to those assumptions.
Next, consideration is given to the ability of each method to compensate for cogging torque harmonics.
4.1 Methodology
In this chapter, all calculations are done using the mathematical computer package MatlabTM. Optimal currents are calculated using one of the PRCW methods. By assuming that the current controller exactly follows the reference current, the torque produced can be determined using equation 2.3 for a given back EMF and cogging torque.
43 CHAPTER 4. THEORETICAL COMPARISON OF REVIEWED METHODS
4.2 Parameter and constraint variation
4.2.1 Baseline for comparison
Generally the goals of reference current waveforms methods are minimum pulsating torque using minimum current. One baseline for comparison was a pure sine wave (SIN) as that is what is most widely used. The other baseline was what is described in [10] as ‘minimum current’ method (MCM). This mode demonstrates that minimum current is used if the current is the same shape as the back EMF. The amount of current used is quoted as the percentage increase required over the MCM. The pulsating torque is quoted as the RMS variation as a percentage of full scale.
4.2.2 Scope of variation
The performance of the methods is primarily affected by the back EMF and cogging torque waveforms. It is also affected by whether a star connection constraint is applied.
4.3 Performance with different back EMFs (without cogging torque)
Generally the shape of the back EMF in a PMAC motor varies between sinusoidal and trapezoidal [40]. There are infinite possible variations between these extremes, so this analysis will only consider one sinusoidal waveform and one trapezoidal waveform. If a method appears superior for both cases, then it will be assumed it is superior for a back EMF waveform that is between a sinusoid and a trapezoid.
Another variation that may occur is that the back EMFs from each phase are unbalanced. This may occur in one of two ways, a magnitude variation or a phase variation. To test the magnitude variation, phase B was scaled to an error of ± 1%, 2% and 3% of the value of the other two phases and the effect on torque ripple was noted. To test the phase variation, phase B was shifted by 1, 2 and 3 degrees and the torque ripple noted.
44 4.3. PERFORMANCE WITH DIFFERENT BACK EMFS (WITHOUT COGGING TORQUE)
4.3.1 Sinusoidal back EMF
As a baseline, the three methods were compared for a sinusoidal back EMF. As ex- pected, all three determined that the best current shape would be a sinusoid and all were capable of completely removing pulsating torque. In figure 4.1 all the lines are on top of one another as all methods give the same results. Table 4.1) shows that all methods have no pulsating torque and use the same amount of current.
p Current − Phase A
3 FDM 2 TDM
A PLM SIN 1 MCM
0 0 20 40 60 80 100 120 140 160 180
Torque output
5.05 FDM TDM 5 PLM Nm SIN 4.95 MCM
4.9 0 20 40 60 80 100 120 140 160 180 electrical angle (degrees)
Figure 4.1: Current and torque output for sinusoidal back EMF
FDM TDM PLM SIN MCM % Pulsating Torque 0.0 0.0 0.0 0.0 0.0 % Current Increase 0.0 0.0 0.0 0.0 0.0
Table 4.1: Performance of PRCW methods with a sinusoidal back EMF
45 CHAPTER 4. THEORETICAL COMPARISON OF REVIEWED METHODS
4.3.2 Trapezoidal back EMF - adding odd harmonics
At the other end of the range of possible back EMFs is a trapezoidal back EMF. The trapezoidal shape assumed was a linear rise 0o to 30o, flat top, linear fall from 150o to 180o. For this analysis, an approximation of a trapezoidal back EMF was achieved by adding the appropriate odd harmonics up to a certain limit. That limit was determined by the number of harmonics required to reduce the truncation error between an ‘ideal’ trapezoid and the approximation to a RMS variation of less than 1%. This required harmonics up to the 17th.
As with the sinusoidal back EMF, each of the methods was able to eliminate pulsat- ing torque with a small increase in current (see figure 4.2 and Table 4.2). The increase in current required for the TMD and the PLM is 2.7% and the FDM is slightly higher at 3.5% above that required for the MCM.
Current − Phase A
3 FDM 2 TDM
A PLM SIN 1 MCM
0 0 20 40 60 80 100 120 140 160 180 Torque output 6 FDM 5.5 TDM PLM Nm 5 SIN MCM 4.5 0 20 40 60 80 100 120 140 160 180 electrical angle (degrees)
Figure 4.2: Current and torque output for trapezoidal back EMF
46 4.3. PERFORMANCE WITH DIFFERENT BACK EMFS (WITHOUT COGGING TORQUE)
FDM TDM PLM SIN MCM % Pulsating Torque 0.0 0.0 0.0 4.3 11.5 % Current Increase 3.6 2.7 2.7 2.5 0.0
Table 4.2: Performance of PRCW methods with a trapezoidal back EMF
4.3.3 Variation between phases
As noted in section 2.5.1, the FDM assumes that each of the back EMFs and currents are identical and spaced 120 electrical degrees apart. To test each of the methods sensitivity to violation of this assumption tests were done varying the magnitude and the phase of phase ’B’ relative to the others and the effect on pulsating torque noted.
Magnitude variation
It can be seen in figure 4.3 that while the TDM and the PLM were able to cope with this variation, the output from the FDM was affected. The effect on pulsating torque however is relatively minor, with a variation of 3% only creating a 0.5% RMS variation in the torque.
47 CHAPTER 4. THEORETICAL COMPARISON OF REVIEWED METHODS
Average torque variation 101 FDM
% 100 TDM PLM 99 97 98 99 100 101 102 103 Pulsating torque
0.4 FDM
% 0.2 TDM 0 PLM −0.2 97 98 99 100 101 102 103 Current requirement increase 6 FDM
% 4 TDM PLM 2 97 98 99 100 101 102 103 Phase B scaling (%)
Figure 4.3: Performance of different methods if one phase has a magnitude variation
Phase variation
Figure 4.4 shows that as with the magnitude variation, the TMD and the PLM still can eliminate pulsating torque if the back EMF has an offset variation. The impact of this change is has a much greater affect on the output of the FDM with an offset of 3 degrees creating a 6% RMS variation in the torque.
48 4.4. COMPARISON OF COGGING TORQUE COMPENSATION
Average torque variation 101 100 FDM
% 99 TDM 98 PLM −3 −2 −1 0 1 2 3 Pulsating torque 6 4 FDM
% 2 TDM 0 PLM −2 −3 −2 −1 0 1 2 3 Current requirement increase 10 FDM
% 5 TDM PLM 0 −3 −2 −1 0 1 2 3 Phase B offset (degrees)
Figure 4.4: Performance of different methods if one phase has a phase variation
4.4 Comparison of cogging torque compensation
To determine the capability of each method to cancel out cogging torque, they were tested with a simulated cogging torque containing only one harmonic. Figure 4.5 considers the pulsating torque output from each method for a cogging torque containing a harmonic from 1 to 40. Table 2.1 suggested that the FDM is not capable of compensating cogging torque harmonics that are not a multiple of three. Figure 4.5 clearly demonstrates that this was the case in the simulations. As the frequency to be compensated increased, the current required by the FDM increased considerably. This current increase was due to the small size of the higher back EMF harmonics. For example, consideration of table 2.1, shows that when using the FDM, a 30th harmonic can only be compensated by a combination of a 13th and 17th or 14th and 16th current and back EMF harmonics. As the trapezoidal back EMF only contained odd harmonics, compensation of the 30th harmonic depended on the the 13th and 17th back EMF harmonics. As these were
49 CHAPTER 4. THEORETICAL COMPARISON OF REVIEWED METHODS small, the matching current harmonics had to be very large. In addition, because the back EMF only had harmonics up to 17, the FDM could only compensate up to the 33rd harmonic. Both the TDM and the PLM could successfully compensate all harmonics with minimal increase in current.
Pulsating torque 15
10 FDM TDM 5 PLM % pulsating torque 0 0 5 10 15 20 25 30 35 40
4 x 10 Current requirement 10
FDM 5 TDM PLM
% current increase 0 0 5 10 15 20 25 30 35 40 cogging torque harmonic
Figure 4.5: Ability of each method to compensate 1Nm of cogging torque at different harmonics
4.4.1 Star connection constraint
The comparisons described so far were done with a star connection constraint, however the test motor is configured to allow the star constraint to be removed. This allowed a study of how the methods performed without a star constraint. Though there is no benefit from removing this constraint if the back EMF was sinusoidal, figure 4.6 and Table 4.3 show that the FDM and TDM benefited from the removal of this constraint by lowering the additional current required to 0.7%. There
50 4.4. COMPARISON OF COGGING TORQUE COMPENSATION is no change in the PLM as it always includes a star connection constraint.
FDM: Star connected currents FDM: Independent currents
A 2 2 B C 0 0 Total
−2 −2
0 100 200 300 0 100 200 300 TDM: Star connected currents TDM: Independent currents
2 2
0 0
−2 −2
0 100 200 300 0 100 200 300 PLM: Star connected currents PLM: Independent currents
2 2
0 0
−2 −2
0 100 200 300 0 100 200 300 Position (degrees) Position (degrees)
Figure 4.6: Comparison of currents with and without a star connection constraint
FDM TDM PLM % Current Increase (with star constraint) 3.6 2.7 2.7 % Current Increase (without star constraint) 0.7 0.7 2.7
Table 4.3: Performance of PRCW methods with a modified trapezoidal back EMF
51 CHAPTER 4. THEORETICAL COMPARISON OF REVIEWED METHODS
4.5 Upper frequency limit constraint
The analysis presented so far, suggests that the TDM is superior. It can always eliminate pulsating torque using the same or less current than the other methods. One consideration that could limit the effectiveness of the TDM is that an upper frequency limit is not prescribed for the desired current. The existence of high frequencies in the TDM current could be an argument for using the FDM. The prescribed frequency limit of the FDM limits the current controller bandwidth required. A current controller with a higher bandwidth usually requires a higher switching frequency resulting in increased switching losses.
To check the effect of current controller bandwidth on the TDM and the PLM, the frequency spectrum of all methods was determined for each of the comparisons described.
Figure 4.7 shows the current harmonics above the specified harmonic limit for the trapezoidal back EMF. The 17th harmonic shown is the highest in the frequency domain method. Though the TDM has a 19th harmonic, its magnitude is only about 0.1% of the fundamental, so is insignificant. If all harmonics above the harmonic limit are removed from the TDM method before the torque is calculated, there is still no noticeable pulsating torque.
Current harmonics
0.1 FDM TDM 0.05 PLM % of fundamental 0 16 18 20 22 24 26 28 30 32 34 harmonic number
Figure 4.7: Ability of each method to compensate 1Nm of cogging torque at different harmonics
52 4.6. SUMMARY OF THEORETICAL COMPARISON OF PRCW METHODS
4.6 Summary of theoretical comparison of PRCW methods
Comparisons between different PRCW methods for various back EMF waveforms and cogging torques show that regardless of the shape of the waveforms, the time domain method (TDM) is always the best method. This validates the approach of only consid- ering sinusoidal and trapezoidal back EMF shapes. As the TDM is superior for both these extreme back EMF shapes, it is reasonable to expect it will be superior for an intermediate waveform. If implementation is to be done in a rotating reference frame (such as space vector modulation) then the use of the Park-like method (PLM) may be beneficial. The frequency domain method (FDM), while unlikely to produce the best results, is useful for providing insight into pulsating torque production. A possible concern that the TDM’s success might require the injection of higher harmonics than the FDM (potentially beyond the bandwidth of a current controller) was checked and found to be unfounded.
53 CHAPTER 4. THEORETICAL COMPARISON OF REVIEWED METHODS
54 Chapter 5
Pulsating torque decoupling approach to motor parameter determination
Chapter 3 discussed the importance of obtaining accurate information on motor prop- erties to ensure minimum pulsating torque when using PRCW methods.
The task of parameter determination is a considerable challenge as any measure- ment usually involves a number of conversions or scaling factors. Previous implementa- tions of PRCW methods have either used datasheet values for this scaling or separate calibrations of each individual scaling factor. The limited success of these implemen- tations suggests that a more accurate method is required for parameter estimation. This chapter presents an approach to parameter determination using pulsating torque decoupling (PTD).
The proposed method uses a best guess at parameter values for an initial motor trial. Any inaccuracies in these parameter values will cause pulsating torque. By decoupling this resulting pulsating torque into components related motor parameters, inaccuracies can be quantified and compensated for in future operation. This process can be viewed as a method for calibrating the sensors and is far simpler than attempting calibration of each sensor individually.
Discussion begins by restating the equation responsible for torque creation in block
55 CHAPTER 5. PULSATING TORQUE DECOUPLING APPROACH TO MOTOR PARAMETER DETERMINATION diagram form and highlighting sources of pulsating torque. The PTD method for decoupling the pulsating torque into the parameter errors responsible for its creation is then presented.
The special case where the torque sensor only measures dynamic torque is discussed. It is demonstrated that in that situation, the overall system gain is the only additional information required. A determination method for this gain is presented.
5.1 Motor model overview
Figure 5.1 is a block diagram of the experimental implementation of the general motor equation as expressed in equation 2.3.
To simplify this analysis, and allow the electrical transfer function to be replaced by a simple gain, it is assumed that the current controller is effective in following the reference current. The validity of this assumption is verified in section 7.2.3. This gain for each motor phase is designated wp, where p = a, b, c. Usually, in a practical implementation, there will also be an offset (op) associated with the current control hardware. This offset is due to the output of the current sensor being 2.5V for 0A.
This model assumes that the transfer function between motor torque and measured torque is only a gain (up). If there is a mechanical resonance in the system or if an indirect measurement is used (such as an observer) then this assumption may not hold.
For correct compensation, estimates of these three parameters are required: current ∗ ∗ control hardware gain estimate (wp), current control hardware offset estimate (op) and ∗ torque sensor gain estimate (up). Though an initial estimate is usually still required, this method allows compensation for errors in this initial estimate. The dotted black line in Figure 5.1 denotes the extremities of the hardware. The gain and offset blocks shown outside this box are software compensation for the hardware gains and offsets inside.
56 5.1. MOTOR MODEL OVERVIEW
Figure 5.1: Block diagram of parameters to be determined
5.1.1 ‘Known’ parameters
The PTD approach requires that some parameters are assumed to be accurate (shown in white in figure 5.1). Experiments (see chapter 7) determined that the back EMF ref ref (kθ,p) was accurate. The reference current (iθ,p ) and the torque measured (τθ ) are also known.
5.1.2 Block diagram simplification
Pulsating torque is created when one of the parameters in Figure 5.1 is inaccurately de- ∗ termined (for example, wp 6= wp). To assist the analysis, the parameters and estimates can be replaced by other parameters which represent the estimation error.
wpup αp = ∗ ∗ (5.1) wpup wpup ∗ βp = ∗ (op − op) (5.2) up cog∗ up cog τθ = ∗ τθ (5.3) up (5.4)
With these new variables, Figure 5.1 can be modified as shown in Figure 5.2.
57 CHAPTER 5. PULSATING TORQUE DECOUPLING APPROACH TO MOTOR PARAMETER DETERMINATION
Figure 5.2: Block diagram of parameters to be determined (simplified)
5.2 Decoupling of pulsating torque: determination of current imbalance and cogging torque
If the assumptions made are valid for a particular motor, and if the ‘ideal’ currents have been correctly calculated by one of the methods presented in chapter 4, pulsating torque will only come from an error in either:
1. cogging torque; and/or
2. an unbalance in the current, caused by an offset or gain error in the current sensors.
These are shown in pink in Figures 5.1 and 5.2, and their determination will be the focus of this section.
By decoupling the pulsating torque into the components created from each of these errors, it is possible to determine where the errors lie and compensate accordingly. To do this, it is important to note that the cogging torque will be independent of current input.
The cogging torque is redefined as the residual resulting from a least squares min- imisation matching the electro-magnetic torque to the measured torque.
58 5.2. DECOUPLING OF PULSATING TORQUE: DETERMINATION OF CURRENT IMBALANCE AND COGGING TORQUE
5.2.1 General formula including scaling errors, offset errors and cogging torque
Rewriting Equation 2.3 with the current offset and scaling factors as shown in Figure 5.2:
meas X cog∗ τθ = (iθ,pαp + βp) kθ,p + τθ (5.5) p=a,b,c where:
meas τθ = measured torque at angle θ (Nm)
iθ,p = current in phase p at angle θ (A)
αp= current scaling error in phase p
βp= current offset error in phase p
kθ,p = normalised back EMF for phase p at angle θ (V s/rad) meas τθ = measured torque at angle θ (Nm)
or in matrix notation:
~ ∗ ~τmeas = (Iref •K)~α + (K)β + ~τcog (5.6)
where:
~τmeas = Θ × 1 vector of the measured torque (Nm)
Iref = Θ × 3 matrix of the reference current (A) ~α = 3 × 1 vector of current scaling error β~ = 3 × 1 vector of current offset error
K = Θ × 3 matrix of the back EMF (V s/rad) ∗ ~τcog = Θ × 1 vector of the cogging torque estimate (Nm) • = element-wise multiplication operator
if we concatenate the matrices to let:
X = I•KK (5.7)
59 CHAPTER 5. PULSATING TORQUE DECOUPLING APPROACH TO MOTOR PARAMETER DETERMINATION
and concatenate the vectors to let: ~α ~y = (5.8) β~
then:
∗ ~τmeas = X~y + ~τcog (5.9)
∗ where ~y and ~τcog are unknown. We previously defined cogging torque as the residual from a least squares min- imisation matching the electro-magnetic torque to the measured torque. The Moore- Penrose pseudo inverse is a convenient way of conducting a least squares minimisation ∗ [53][chapter 5, p13]. By using this inverse and assuming that ~τcog will be the residual, ~y can be found.
+ ~y = X ~τmeas (5.10)
where:
+= pseudo-inverse operator
∗ The residual (~τcog) can then be found by rearranging equation 5.9:
∗ ~τcog = ~τmeas − X~y (5.11)
∗ 5.2.2 Determination of ~y and ~τcog over operating range
The vector ~y is attributed to errors in the current sensor system, so regardless of speed ∗ and torque set-point it should remain constant. The cogging torque (~τcog) should also ∗ be independent of operating point. This method will only be valid if ~y and ~τcog are independent of speed and torque. One method to ensure that the same ~y is determined for all operating points is to combine all tests at different operating points into one long X. This matrix will have 6 columns and the number of rows will be Φ times the number of different operating points considered. Though this gives only one ~y for all trials, it does give a different
60 5.2. DECOUPLING OF PULSATING TORQUE: DETERMINATION OF CURRENT IMBALANCE AND COGGING TORQUE
∗ ~τcog for every trial. The validity of this method will be determined by the error ∗ between the determined residuals (~τcog).
~ 5.2.3 Compensation for ~α, β and ~τcog
~ ∗ Once ~α, β and ~τcog have been determined from an uncompensated set of measurements over the operating range, they can be used to pre-compensate Iref to cancel their effect for future operation.
First Iref is split up into the current required to achieve the reference torque (I~τref ) ∗ and the current required to compensate for ~τcog (Icog):
Iref = I~τref − Icog (5.12)
where:
I~τref = Θ × 3 matrix of the current required to achieve the reference torque(A)
Icog = Θ × 3 matrix of the current required to compensate for cogging torque (A)
A PRCW method is used to ensure that:
I~τref •K = ~τref (5.13)
and:
∗ Icog•K = ~τcog (5.14)
the reference current can be improved by compensating with the previously deter- mined ~α and β~
~ ~ I~τref − Icog − 1Θ×1 ∗β = (5.15) Iref ~ 1Θ×1 ∗~α where:
61 CHAPTER 5. PULSATING TORQUE DECOUPLING APPROACH TO MOTOR PARAMETER DETERMINATION
~ 1Θ×1 = Θ × 1 vector of ones
If this modified reference current is used in equation 5.6:
~ ~T I~τref − Icog − 1Θ×1 ∗β ~τ = • ~α + ( )β~ + ~τ ∗ (5.16) meas ~ T K K cog 1Θ×1 ∗~α T ~ ~ ~ ∗ = I~τref − Icog − 1Θ×1 ∗β •K + (K)β + ~τcog (5.17)