Degree project in
Mechanical Dynamics
of a Sensorless PMSynRel Drive
Yingbei Yu
Mechanical dynamics of a sensorless
PMSynRel drive
by
Yingbei Yu
Master Thesis
Royal Institute of Technology
School of Electrical Engineering
Dept. Electrical Energy Conversion
Stockholm 2013
Mechanical dynamics of a sensorless PMSynRel drive YINGBEI YU
c
YINGBEI YU, 2013.
School of Electrical Engineering
Department of Electrical Machines and Power Electronics Kungliga Tekniska H¨ogskolan
Abstract
Hybrid electric vehicle (HEV) concept has, combining conventional internal combustion engines and electric drives, gained more and more interest due to its environmental friendly features. A PMSynRel based electric drive is considered as a good option due to its high torque density and high efficiency. To reduce the overall cost of HEVs, the position resolvers can be replaced by Hall-sensors or using sensorless control. However, the dynamics of such electric drives may be degraded. The main objective of this MSc project is to develop torque dynamics of such electric drives when operating with/without a position sensor. The developed torque dynamic can be used to analyze the limits of hall senor/sensorless strategy when, e.g. anti-oscillation control is required. The torque dynamic is presented as a matrix based transfer function extracted from the speed responses and torque responses using Identification Tool Box in Matlab. Firstly, the transfer function was derived by means of simulations in both time and frequency domains. Secondly, similar procedures were applied to extract the transfer functions based on the experimental results.
Keywords: Bode plot, Hall-effect sensor, Matlab Identification Toolbox, MIMO, PMSynRel,
sensorless control, transfer function.
Sammanfattning
Elektriska hybridfordon, där en konventionell förbränningsmotor kombineras med ett elektriskt drivsystem, uppmärksammas mer och mer på grund av de miljömässiga fördelarna. En eldrift baserad på en permanentmagnetiserad synkron reluktansmaskin (PMSynRel) är ett bra alternativ tack vara den höga momenttätheten och den höga verkningsgraden. För att minska systemkostnaden kan positionsgivarenen (resolver) ersättas med Hall-givare eller att motorn styrs med, så kalla, sensorlös reglering. En nackdel med dessa alternativ är att den mekaniska dynamiken kan försämras. Huvudmålet med detta examensarbete är att studera hur momentdynamiken kan kvantifieras med och utan en positionsgivare. De framtagna modellerna kan användas till att utvärdera huruvida tex svängningar i fordonets drivlina kan dämpas ut med hjälp av det elektriska drivsystemet i de fall då positionen mäts med en Hall-givare eller skattas via den sensorlösa algoritmen. I detta arbete modelleras momentdynamiken med hjälp av en matrisbaserad överföringsmatris var element identifierats i en simuleringsmodell implementerad i Matlab/Simulink. I examensarbetets sista skede jämfördes den modellerade dynamiken med tidiga experimentella försök i en laborativ försöksrigg.
Nyckelord: Bodediagram, Hall-givare, MIMO, PMSynRel, sensorlös reglering,
Acknowledgements
In the first place, I would like to thank Phd student Shuang Zhao and my supervisor and examiner Dr. Oskar Wallmark for providing me with thorough technical explanation and guidance through the whole thesis work. I’ve learned a lot during each discussion and meeting for this thesis work. Especially thanks to Shuang Zhao, he was very patient to help me with any questions I had related to this project.
I am also very grateful to Mats Leksell for introducing me to this valuable opportu-nity to work on this project at the Department of Electrical Energy Conversion.
I also want to thank the people who work at this department at KTH for assisting me in different questions and problems.
In addition, I’d like to appreciate my manager Vidar Grimelind working at FM-CTechnologies AS for the permission on my temporary study leave. Thanks for his kind-ness, generousness and understanding.
Finally, I would like to express my deepest gratitude to my parents for all the sup-ports and love, not only during the period I am abroad, but through all my life. They are the greatest parents in the world! Last but not least, I would like to thank my husband Xu Yuan for all the encouragements, understandings and love.
Yingbei Yu
Contents
Abstract v
Acknowledgements vii
Contents ix
1 Introduction 1
1.1 Background and Objectives . . . 1
1.2 Motivation of Hybrid Electric Vehicles . . . 1
1.3 Configurations of HEVs . . . 2
1.3.1 Series HEV Configuration . . . 2
1.3.2 Parallel HEV Configuration . . . 2
1.4 PMSynRel machine in HEV applications . . . 2
1.5 Control of PMSynRel in HEV applications . . . 4
1.6 Outline of Thesis . . . 4
2 PMSynRel Control 5 2.1 Field Oriented Control . . . 5
2.2 Resolver and Hall-effect sensor . . . 6
2.2.1 Resolver . . . 6
2.2.2 The Hall-Effect sensor . . . 6
2.3 The Sensorless Control . . . 8
3 Simulation Models 13 3.1 Matlab/Simulink . . . 13
3.1.1 Basic Simulink Model . . . 13
3.1.2 Advanced Simulink Model/Flux map model . . . 14
4 Implementation of rotor position detection in Simulink 15 4.1 Modeling of rotatory position sensors . . . 15
4.1.1 Resolver modeling . . . 15
Contents
4.2 Modeling of sensorless control . . . 17
4.2.1 Signal Injection . . . 17
4.3 Result of the Hall-effect sensor and sensorless control . . . 18
5 Transfer Function 23
5.1 Set input and output . . . 23
5.2 Identification and evaluation process . . . 24
5.2.1 Transfer functions identification and evaluation process in time domain 24
5.2.2 Transfer functions identification and evaluation process in frequency domain 26
6 Experimental Result 43
6.1 Experiment . . . 43
7 Conclusion and future work 47
7.1 Summary . . . 47
7.2 Future work . . . 48
7.2.1 Sensorless control and Hall-effect sensors . . . 48
7.2.2 Torque dynamics in frequency domain from the experiment . . . 48
A Laboratory Setup 49
Chapter 1
Introduction
This chapter briefly introduces the background and the outline of this thesis. The moti-vation, configuration and principle of hybrid electric vehicles is also presented in this chapter.
1.1
Background and Objectives
HEV applications have gained more and more interest due to its environmental friendly features. A PMSynRel based electric drive is considered as good option due to its high torque density and high efficiency. The main objectives of this thesis work is to develop torque dynamics of such electric drives when operating without a position sensor. The developed torque dynamic can be used to analyze the limits of sensorless strategy when, e.g. anti oscillation control is required.
1.2
Motivation of Hybrid Electric Vehicles
As the concerns of environmental degradation are growing, many countries have made na-tional plans to significantly reduce the oil consumption. More and more strict regulations are pronounced by governments, which force manufactures to find alternatives to replace conventional vehicles. Besides the environmental feature, relatively low operational costs is another main motivation of the HEV technology.
Chapter 1. Introduction
1.3
Configurations of HEVs
HEVs are typically classified into two basic configurations, series or parallel.
1.3.1
Series HEV Configuration
A series HEV configuration is shown in Figure 1.1 where there is no physical connection between the ICE and transmission. The generator converts the mechanical power (deliv-ered from ICE) to electrical power which can be used either for charging the battery or for propelling the vehicle. The advantages of this configuration is listed as follows: 1. Since the ICE is only used to charge the battery, the operating point can be optimized to achieve a high efficiency.
2. Since electrical machine is used to propel the wheels, which can operate in a wide speed range, the multi-gear transmission can be removed from the power train.
However, this configuration requires several energy conversions which lead to a low over-all efficiency. Furthermore, this HEV configuration requires a large electrical machine, therefore, all drivetrain components have to be designed to match the peak power of the electrical machine.
1.3.2
Parallel HEV Configuration
A parallel HEV configuration is shown in Figure 1.2, where the propulsion power is pro-vided by the ICE and/or electrical machine. The major advantage of the parallel HEV configuration compared to the series HEV configuration is that the electrical machine can be used to convert the mechanical power to the electrical power. Thus, the generator can be eliminated. Since the vehicle is propelled by electric-machine and ICE(provided the battery is never be depleted), both of them can be downsized.
1.4
PMSynRel machine in HEV applications
1.4. PMSynRel machine in HEV applications dƌĂŶƐŵŝƐƐŝŽŶ
Figure 1.1: Series-HEV configuration: the arrows indicate the possible directions of en-ergy flow. dƌĂŶƐŵŝƐƐŝŽŶ ! "# ! $ %"& '(# #!) * +(", %$
Chapter 1. Introduction
known as the permanent-magnet assisted synchronous reluctance (PMSynRel) machines. The permanent magnet in the rotor provides additional permanent torque and therefore increases the torque density. The field-weakening capability of PMSynRel machines is better than PMSMs due to less permanent-magnet.
1.5
Control of PMSynRel in HEV applications
Field oriented control (FOC) is commonly used to control PMSynRel machines to achieve high performance. However, knowledge of the rotor position is required for correct opera-tion. Mechanical resolvers are often mounted on the rotor shaft to detect the rotor posiopera-tion. However, this additional components and their associated cabling may degrade the reli-ability of the overall system and increase the cost. One solution to remove the resolvers is to implement position estimation technology (known as sensorless control). Therefore, reduction of the cost and an improved reliability can be achieved.
1.6
Outline of Thesis
The outline of the thesis is summarized as:
Chapter 2: The theory of Field Oriented Control (FOC), Resolver, Hall-effect sensor and
strategies of different types of sensorless control are briefly introduced.
Chapter 3: Matlab Simulink is introduced in this chapter. Two Simulink models for the
PMSynRel are briefly discussed.
Chapter 4: Different rotor position detection methods are simulated in Simulink and
studied. PLL bandwidth is adjusted to see the impacts on different rotor position detec-tion methods. Harmonic order is also taken into consideradetec-tion for modeling the Hall-effect Sensor.
Chapter 5: The torque dynamics is presented as a matrix based transfer function extracted
from the spped responses and torque responses using Identification Tool Box in Matlab. The transfer function was derived in both time and frequency domain. Evaluations and the result of the torque dynamics are also included in this chapter.
Chapter 6: The experimental set-up in the lab are stated in this chapter, the experimental
results are listed and studied.
Chapter 7: This chapter summarizes the conclusion of the work and provides the future
Chapter 2
PMSynRel Control
In this chapter, the field oriented control strategy is briefly introduced. The resolver and the Hall-effect sensor are studied and presented. Sensorless control strategies for high-speed and low-high-speed position estimation are also introduced in this chapter.
2.1
Field Oriented Control
Field Oriented Control (FOC), also known as vector control,is to control the stator cur-rents represented by vector. The three phase quantities, e.g. the curcur-rents, are measured
and converted intoα − β system by applying Clarke transformation as follows [5]:
iα = ia (2.1) iβ = 1 √ 3ia+ 2 √ 3ib (2.2)
Next, the quantities in theα − β frame are transformed to the d-q system by applying the
Park transformation provided by the knowledge of the rotor position. The d-q reference frame rotates synchronously with the stator flux, and the d-axis is defined as it is aligned to the rotor flux. The Park transformation is shown as:
id= iαcos θ + iβsin θ (2.3)
iq = −iαsin θ + iβcos θ (2.4)
where θ is the rotor position. Then the id and iq are controlled to follow the references
iref
d (flux reference) andirefq (torque reference). The differences between the measured
cur-rents and the references are forced to zero by the PI regulators. The output of the PI
regulators areuref
d and urefq , which are transformed to the abc reference frame using the
Chapter 2. PMSynRel Control as: uref α = u ref d cos θ − u ref q sin θ (2.5) urefβ = u ref d sin θ + u ref q cos θ (2.6)
2.2
Resolver and Hall-effect sensor
As shown in(2.3)(2.4) and (2.5),(2.6), the rotor position is required to perform FOC. The rotor position can be measured by rotary position sensors. Rotary position sensors can be divided into two groups, absolute and incremental. Absolute sensors can detect the current position of the shaft at any given time, while the incremental sensors can only indicate the motion of the shaft. The rotary position sensors can be mounted either on the shaft, partial-through-shaft or end-of-shaft and the position readings can be either axially or radially [3].
In this work, resolver and Hall-Effect sensors are modeled and studied as examples of the absolute and incremental rotary position sensors, respectively, are studied and modeled. The impacts of both sensors are illustrated and compared in Chapter 4.
2.2.1
Resolver
Resolver is a very common type of rotary position sensor used in PMSynRel. It is usu-ally mounted on the rotor shaft and provides rotor angle information required in FOC. The most common type of resolver may consider as as a small electrical motor having both stator and rotor. The wire winding configuration inside of the resolver, illustrate in Figure 2.1, makes it different than a normal motor. The resolver consists of two stator windings and one rotor winding. The rotor winding refer as the excitation winding will be applied an excitation signal on and the signal will be induced to the stator windings.
Assume the excitation signal is:Uref = E sin(ω
extt) with a frequency ofωextand
ampli-tude E. As shown in Figure 2.1, two stator windings are configured at 90 degrees from
each other, the induced excitation signal can be expressed as:Usin = E sin(ωextt) sin(θ),
Ucos= E sin(ωextt) cos(θ), where θ is the actual rotor position and can be determined by
dividingUsinandUcos, (the value oftan(θ) will be given) [3] [9].
The resolver is an analogue device which requires demodulation to achieve digital sig-nals. Resolver-to-Digital Conversion (RDC) is used for demodulating and generating the excitation signals for resolvers [3].
2.2.2
The Hall-Effect sensor
2.2. Resolver and Hall-effect sensor
-./
.0
Figure 2.1: Working principle of resolver.
be generated which is perpendicular to the magnetic field and varies with the change-rate of the magnetic field. A Hall-effect sensor measures magnetic field strength, and it used to measure speed as well (e.g.the angular speed of turning shaft), when place the sensor besides the moving magnet (e.g rotor). The sensor will be triggered and a pulse will be produced once the rotor magnet passes. In the meanwhile, the pulse signals will be fed into a counter and the counter will count the number of pulses obtained in a specific time interval. Furthermore, by integration of the angular speed the electrical angular position can be found.
The machine system normally consists of three Hall-effect sensors and are placed axially outside the rotor with 120 electrical degree apart. According to this arrangement, the
Hall-effect sensors can only provide the position information every60◦(electrical degree) [3].
Chapter 2. PMSynRel Control
2.3
The Sensorless Control
By using absolute rotatory sensors, required rotor position to FOC can be obtained. Un-fortunately, several drawbacks to have sensors in the system need to be concerned. First of all, the drive train has a limited space. Therefore, when designing the motor, the factor of the physical size needs to be considered. In addition, the reliability of the sensors re-lies on several respects, such as vibrations, dirt and disturbance. Cost is another issue for manufactures to premeditate. To reduce the cost and to improve the reliability, many rotor position estimation methods (or sensorless control) have been proposed to remove the ro-tary sensors. All the presented methods utilize the machine itself as a sensor and observe the rotor position from the electrical quantities (voltages or currents). In principle, there are two main sensorless control approaches. One is to use the back electromotive force (EMF) estimation. This method has high accuracy from the medium to high speed range, but may fail at the low or zero speed since the back EMF is gradually reduced when speed decreases. The other one, known as signal injection method, injects high frequency sig-nals to the stator voltage, so that the rotor position can be detected from the corresponding results of the interaction of the high frequency signal and the rotor saliency [11]. Theoret-ically, signal injection method can be used for all speeds, including standstill. However, this method might introduce additional noise, torque ripple and losses [12]and therefore, only considered at low-speed range.
Back EMF method
The back emf is defined from the PMSynRel model which is given by a set of equations [6]: ud = Rsid+ Ld did dt − ωrLqiq (2.7) uq = Rsiq+ Lq diq dt + ωrLdid+ ef (2.8)
Where, the back emfef isef = ωψpm.udanduq,idandiqare the stator voltages and
cur-rents in the rotor reference frame, respectively, and can be measured.R is the resistance,
Ld,Lq are the machines inductances. ψpm is the flux generated by permanent magnets.
Thoses are all approximately known quantities. Therefore, ωr, electrical rotor speed can
be solved from the motor equations. Using the estimated rotor coordinates, (2.7) and (2.8)
are transformed into the estimated rotor reference frame by applying a position error ˜θr.
equa-2.3. The Sensorless Control tions [4]: ude = Rside − ˆωr(idecos ˜θr(Ld− Lq) + iqe(Lqcos 2˜ θ + Ldsin 2 ˜ θr)) − ωrψmsin ˜θr (2.9) uqe = Rsiqe− ˆωr(−ide(Lqsin 2˜ θr+ Ldcos 2 ˜ θr) − iqecos ˜θr(Ld− Lq)) + ωrψmsin ˜θr (2.10)
where ˜θr=θr− ˆθris an unknown value while motor parameters are precisely known. ”de”
and ”qe” denote the quantity is in the estimated rotor reference frame. The estimator can estimate the steady state voltage by the following expression:
ˆ
ude = ˆRside− ˆωrLˆqiqe (2.11)
ˆ
uqe= ˆRsiqe+ ˆωrLˆdiqe + ˆωrψˆm (2.12)
The voltage errors for the d,q axes can be obtained by subtracting(2.11)(2.12) from (2.9)(2.10):
ude− ˆude = ˜ude = −ˆωrsin ˜θr((Ld− Lq)(idecos ˜θr+ iqesin ˜θr) − ωrψmsin ˜θr (2.13)
uqe− ˆuqe = ˜uqe= −ˆωrsin ˜θr((Ld− Lq)(idecos ˜θr− iqesin ˜θr) − ωrψmcos(˜θr− 1)
(2.14)
Assuming no parameter errors and the machine is non-salient(△L = 0), ˜ude can be
sim-plified as:
ude− ˆude = ˜ude = −ωrψmsin ˜θr (2.15)
Therefore, the position error used for the back-EMF estimator is:
sin ˜θr= −
˜
ude
ωrψm
(2.16) according to(2.16), it is not difficult to find out once the motor speed is low or at standstill,
˜
θr is significant. In another word, the back-emf method might fail in such conditions.
Signal Injection Method
The main drawback of back-emf estimator is the instability when operating at low and zero speed. Therefore, in the low speed range, high frequency signal injection methods are studied and presented. The injected voltages can be either rotating or pulsating to ex-tract the information of rotor position [7]. A transient state of the machine is created when injecting a high frequency voltage. The resultant currents contain the rotor position
infor-mation provided by the inductance variations (Ld 6= Lq). Figure 2.2 describes the general
idea of signal injection method.
Chapter 2. PMSynRel Control
vector signal uc = vccos(ωct) + j0 is applied to the estimated ˆd-direction with the
am-plitude of injected voltage vc and frequencyωc, Rs is negligible compared to ωcL, since
the injection frequency is in the kHZ region. Therefore, the PMSynRel can be modeled as pure inductive load:
ucde≈ Ld dicde dt − ωrLqicqe (2.17) ucqe≈ Lq dicqe dt + ωrLdicde+ ef (2.18)
here,ef can be omitted since it is rather small.
Equations (2.17) and (2.18) elaborates that theq axis current also oscillates even withoutˆ
any injected signal when the rotor rotates. ( ˆd and ˆq axis are coupled) This should be
avoided since when the position error is zero, the goal is to have current only in the ˆd
axis oscillate. This problem can be solved by adding a suitable signal in theq axis whichˆ
makesicqe = 0. Now zero icqeis substituded into (2.17) and (2.18),which gives:
icde = vcsin(ωct) ωcLd (2.19) ucde = vccos(ωct) (2.20) ucqe= vcsin(ωct) ωr ωc (2.21)
ucqe is assumed to 0, since the study only focuses at the low speed region (ωc ≫ ωr).
Therefore, the high frequency flux in the estimated reference frames can be expressed as:
ψcde=
vcsin(ωct)
ωc
(2.22)
ψcqe= 0. (2.23)
When transformψdandψq from rotor reference frame toα, β stationary reference frame
with the frequency of the injected voltage,ωc, following equations can be obtained:
ψcα = (L0+ L2cos 2θr)icα+ L2sin 2θricβ (2.24)
ψcβ = L2sin 2θricα+ (L0 − L2cos 2θr)icβ (2.25)
whereiαβ are the measured current vector of the injection frequency components, L0 =
Ld+Lq
2 andL2 = Ld−Lq
2 . Equation (2.22) and (2.23) can be transformed in to the
station-ary reference frame and together with (2.24) and (2.25), the measured current vector in stationary reference frame can be expressed as:
2.3. The Sensorless Control afterward, equations (2.26) (2.27) are transformed to the estimated rotor reference frame as: ıcde = sin(ωct) vc 2ωc(− Ld+ Lq LdLq − Ld− Lq LdLq cos(2˜θr) (2.28) ıcqe= − sin(ωct) vc 2ωc (Ld− Lq LdLq sin 2˜θr) (2.29)
Now the error information ˜θris contained into the current component of estimated q-axis,
the high frequency from the injected voltage can be extracted by applying a band pass filter. Then, followed by the demodulation process to get a pulsating DC signal. In the end the error signal will be filtered by a low pass filter and the result is the input signal for phase locked loop (PLL).
For rotating voltage vector injection method, instead of pulsating signal, the injected volt-age is a rotating voltvolt-age vector and inject in the stator reference frame. In this thesis work, pulsating voltage vector injection is mainly studied.
Phase locked loop structure
The phase locked loop structure is to extract the rotor position. The algorithm of a PLL observer is as follows [13]:
˙ˆωr = k1ε (2.30)
˙ˆθr = ˆωr+ k2ε (2.31)
where ωˆr is the estimated rotor speed, ˆθr is the estimated rotor position,k1, k2 are
esti-mator gains.ε, is the error signal can be obtained from using either back-EMF estimate
or using signal injection method. The detail information have been given in the first two parts of this section. Generally, for both two methods, the error signal information is in
the form of: ε = ˜θr. The control method of PLL can be explained as follows. Assume
the angular error signal is small, therefore ε can be expressed as: ε ≈ ˜θr. In this case, if
θr > ˆθr, thenk1 > 0 provides ωr will increase. The system will continually follow and
update ωˆr as long as ˜θ 6= 0. Moreover, as stated in equation (2.31), ˆθ is updated as the
integral of speed estimate together with the correction of error signal,k2ε.
The characteristic polynomial,c(p), indicating the error dynamics [13], is found as
c(p) = p2
+ k2p + k1. (2.32)
To obtain a well damped system, the poles are placed at p = −ρ, where ρ is a positive
constant, provided by selectingk1, k2as [13]:
k1 = ρ
2
Chapter 2. PMSynRel Control 12 345673859 :;985<7
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Simulation Models
In this chapter, the simulation environment including the simulation tools and methods are introduced. Two different simulation models for the vector controlled PMSynRel are briefly presented.
3.1
Matlab/Simulink
To study the impacts of different rotatory position sensors and the sensorless control strat-egy, a simulation model is implemented in Matlab/Simulink.
3.1.1
Basic Simulink Model
As a starting point, a simple simulation model, shown in Fig.3.1 , was implemented in Matlab/Simulink. As seen, this model consists of a current controller (FOC) and a PM-SynRel model. The impacts of discretisation and PWM are disregarded which means the voltage references commanded by the current controller are directly sent to the PMSynRel model. The PMSynRel machine is modeled as follows:
ψd = Ldid+ ψpm (3.1) ψq = Lqiq (3.2) ψd = 1 s(ud− Rid+ ωψq) (3.3) ψq = 1 s(uq− Riq+ ωψd) (3.4) id = (ψd− ψpm)/Ld (3.5) iq = psiq/Lq (3.6)
How-Chapter 3. Simulation Models J K L M NOPQPRPK SPTUVK WX Y MR RPK NY Z K NR Z XXPR [\]Y^ _``WSaUK PbUK PWR ]M NL M NWK c \PPcdWSefUVK WX
Figure 3.1: Basic Simulation Model.
ever, those effects are exists in the real life, which means in order to obtain a precise machine model those phenomena have to be taken into account.
3.1.2
Advanced Simulink Model/Flux map model
Chapter 4
Implementation of rotor position
detection in Simulink
In Chapter 2, different rotor position detection methods of the PMSynRel have been discussed theoretically. Modeling of these methods in Simulink environment is presented in this Chapter. Hall-effect sensor and sensorless control are mainly studied.
4.1
Modeling of rotatory position sensors
4.1.1
Resolver modeling
As described in chapter 2.2.1, the resolver provides the absolute rotor position at a fairly high resolution. If the measurement noises are disregarded the rotor position given by resolvers can be assumed the same as the real rotor position. Therefore, for modeling the
resolver in Simulink, the real rotor position, θr, obtained from the PMSynRel model is
directly used as an input to the current controller (FOC).
4.1.2
Hall-effect sensor modeling
The Hall Effect sensor provides the rotor position every 60◦. Figure 4.1 shows a
com-parison of the electrical angle measured by the Hall-effect sensorθhall and the real rotor
positionθr.
As can be seen from Figure 4.1, the difference between the real rotor position and the measurements from the Hall-effect sensor are significant. In another word, if the measured
rotor position from hall sensor,θhall, is given to the current controller as an input without
Chapter 4. Implementation of rotor position detection in Simulink 0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 5 6 θr θh a ll (r ad ) t (s)
Figure 4.1: Comparison of θhall and θr:θhall is plotted by dashed green line and θr is
plotted by solid blue line.
Hall-effect sensor measurement. This equation is stated as following [?].
Fdifth =
(−1)k
3k sin(k(ˆθr+
π
6) × 6) (4.1)
This equation demonstrates at kth harmonic order, the way to calculate the difference (
Fth
dif) between the Hall-effect sensor measurement and real rotor position, where ˆθr is the
estimated rotor position estimated in PLL. In theory, if k is infinite, and ˆθr is as accurate
as the real rotor position, the sum ofFth
dif is exactly the same as the difference between the
rotor position detected by Hall-effect sensor and the real rotor position. The comparison
between θhall − θr and P Fdifth is illustrated in Figure 4.2. The Hall-effect sensor
mea-surements can be optimized by using (4.1). The sum of (4.1) with all harmonic orders is
subtracted from the original measurements θhallto obtain the optimized Hall-effect rotor
position signal. The equation is expressed as :
θopth = θhall−
X
Fth
dif (4.2)
where θopth can be seen as the optimized Hall-effect sensor signal. The optimized rotor
position from Hall-effect sensor is compared with the real rotor position in Figure 4.3.
Compared to Figure 4.1, Figure 4.3 indicates much better consistency betweenθhall and
θr. However, some noises exists every 60 electrical degree which is caused by the
resolu-tion of the Hall-effect sensor and this can be improved with PLL estimator. The optimized
Hall-effect sensor rotor position can be used for calculating the error signalε, to input to
4.2. Modeling of sensorless control 0 0.1 0.2 0.3 0.4 0.5 0.6 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 θr (r ad ) t (s)
Figure 4.2: Comparison ofθhall− θrandP Fdifth :θhall− θris plotted by blue line,P Fdifth
is plotted by green line
part of section 2.3. Figure 4.4 presents the comparison θr, θhall and θhopt with PLL
es-timation. In this Figure,θhopt with PLL estimation gives a comparably best ”follow-up”
result. Even though, this method is not recommended since the performance will not be stable, especially while, for example, with fast and frequent speed variations, Hall-effect sensor can not respond fast enough to keep up with the changes.
4.2
Modeling of sensorless control
4.2.1
Signal Injection
Chapter 4. Implementation of rotor position detection in Simulink 0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 5 6 θr (r ad ) t (s)
Figure 4.3: Comparison of real rotor position and the optimized hall sensor rotor position
.θopth is plotted in green line,θris plotted in blue line
4.3
Result of the Hall-effect sensor and sensorless control
˜
ωr is the difference between the real rotor speedωr and the estimated speedω. The elec-ˆ
trical angleθris the integration ofωr. With the implementations of different control
meth-ods into the Simulink model, either by comparing the simulation result of ω˜r or ˜θr is a
good reference to observe the impacts of implementing different rotor position detection models. Hall-effect sensor and sensorless control (based on pulsating signal injection) are
implemented in Simulink. They are compared with respect to ω˜r. The PLL bandwidthρ
is an important parameter to the system by using different control methods.
In this section, only the Hall-effect sensor and the sensorless control(based on the pulsat-ing voltage vector injection method) is compared due to the high resolution of resolver ensure the the resolver gives absolutely a better performance.
Sensorless Control
By giving a step torque reference and varying the PLL bandwidth ρ, how ρ impacts the
simulation results are observed in Figure 4.6. Although Figure 4.6 illustrates the lower
ρ, the better ˜ωr is obtained (ω˜r approaching to 0), this does not mean it has the absolute
positive impact for the whole system. PLL requires more responding time with the small ρ. Meanwhile, when ρ is big, the system will be more sensitive which means more noises
4.3. Result of the Hall-effect sensor and sensorless control 0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 5 6 θr (r ad ) t (s)
Figure 4.4: Comparison ofθr, θhall and θhoptwith PLL θr : blue line, θhall : green slash
line ,θPLL
hall : red dot line
Hall-effect sensor
In the Hall-effect sensor simulation, the same step torque reference as used in signal
injection modeling is given.Both ρ and harmonic orders, k, varies sequently to observe
the impacts from different parameters. As shown in Figure 4.7, When different ρ are
applied to the system,ω˜r is getting smaller whileρ becomes small. Again this does not
mean it has better influences to the system. Some pulses are appeared in this figure, which
is caused by (4.1). ˆθr estimated from PLL is using in this equation, which might cause
some periodic errors. Onceθris substituted in this equation, the pulses vanish.
When different harmonic order numbers, k, applies to the Hall-effect sensor modeling,
different results ofω˜r can be obtained. Ideally, high harmonic order number gives more
accurate result, if uses real electrical angle θr as the input to (4.1). However, it is not
realistic, since the real angle is not known. Instead, estimated angle ˆθr is used in (4.1)
to subtract the certain harmonics. Therefore, if the position error is significant, and k is
large,P Fth
dif might not accurate which impacts the result ofω˜r at the same time. From
Figure 4.8, when ρ=60, k = 9, 7, 5, 3, 1 is applied. Better results are found when k=3
and 5. Anyhow, it is difficult to predict the value of k to achieve the smallestω˜r, and to
summarize how does the value of k vary with different PLL parameters applied to the
Chapter 4. Implementation of rotor position detection in Simulink g hij kl mn o p mqr st uv wim xhipwu yzvh{wvm||}w~hztk q n llmq pvqtpvlt uuml 000000000000000000000000000000000000000000000000000000000000000000000000н 00000000000000 yl ltlzhipwuklt{ mzvh{wvm|o}w~hz 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 y~vlwqv g hijwzz huvml 00 00 00 00 00 00 00 00000000 00000000 m{t| nuwvhtp 00 00 00 00 00 00 00000000 00000000 twzz huvml 00 00 00 00 00 00 00 00000000 00000000 00000000 ltqmzzm| ml ltlzhipwu 00 00 00 00 00 00 00 00000000 00000000 00000000000000000000000000000000000000000000000000 nv nvlt v tl tzhvhtp 00000000000000000000000000000000000000000000000000
Figure 4.5: Logic diagram for voltage vector injection method
1.5 2 2.5 3 3.5 4 4.5 5 −1 0 1 1.5 2 2.5 3 3.5 4 4.5 5 −1 0 1 1.5 2 2.5 3 3.5 4 4.5 5 −1 0 1 t (s),ρ=100 t (s),ρ=80 ˆω (r ad /s ) ˆω (r ad /s ) ˆω (r ad /s ) t (s),ρ=10
4.3. Result of the Hall-effect sensor and sensorless control 0.5 1 1.5 2 2.5 −50 5 10 ˜ωr ρ=100 0.5 1 1.5 2 2.5 3 −10 0 10 ˜ωr ρ=70 0.5 1 1.5 2 2.5 3 −10 0 10 ρ=50 ˜ωr 0.5 1 1.5 2 2.5 3 −10 0 10 t(s) ˜ωr ρ=30
Chapter 4. Implementation of rotor position detection in Simulink 0.5 1 1.5 2 2.5 3 −2 0 2 ˜ωr (r a d / s) harmonic order=9 0.5 1 1.5 2 2.5 3 −2 0 2 ˜ωr (r a d / s) harmonic order=7 0.5 1 1.5 2 2.5 3 −2 0 2 ˜ωr (r a d / s) harmonic order=5 0.5 1 1.5 2 2.5 3 −2 0 2 ˜ωr (r a d / s) harmonic order=3 0.5 1 1.5 2 2.5 3 −2 0 2 t(s) ˜ωr (r a d / s)
harmonic order =1 (without Fourier)
Chapter 5
Transfer Function
Transfer functions are used to capture the dynamics of systems. For the PMSynRel model, anti-oscillation controller can be designed based on the identified torque dynamic (trans-fer functions). The trans(trans-fer functions identified in this chapter are based on the simulation results in Matlab environment. The same methodology can be used in real application to identify the PMSynRel system. The PMSynRel model is seen as a MIMO system and the transfer function is derived from the simulation results by using System Identification Toolbox. The comparison of results from the transfer function and the full simulation will be conducted. All the simulations in this chapter are based on the basic Simulink model introduced in Chapter 3.
5.1
Set input and output
In HEVs applications, the PMSynRel machine is torque controlled. And the position is
required in FOC control which results the speedωr. Torque reference has impact on both
torque output and ωˆr. Likewise, speed reference has certain impact on torque output as
well. The system input and output of transfer function can be presented as below: Input:
U1: Torque reference
U2: Speed reference
Output:
Y1: Output torque
Y2: Output speed (estimated speed)
The modeling system can be defined as:
Y1(s) = G11U1(s) + G12(s)U2(s) (5.1)
Chapter 5. Transfer Function
whereG11, G12, G21andG22are the transfer functions derived from different simulations.
G11is the transfer function betweenTref and the output of torque,Tout.G12is the transfer
function fromωref toTout.G21represents the relationship betweenTref andωˆr. Last,G22
defines the transfer function of ωref andωˆr. The layout of the whole system is shown in
Figure 5.1.
Figure 5.1: Transfer function system layout
5.2
Identification and evaluation process
Matlab Identification Toolbox is used to derive the transfer functionsG11, G12, G21, G22.
And this toolbox can be used to analyze the data in both frequency domain and time domain. In this thesis work, both methods are studied. The sequence of how to identify and evaluate the transfer functions for both time domain and frequency domain can be described in Figure 5.2.
5.2.1
Transfer functions identification and evaluation process in time
domain
In this section, procedures described in Figure 5.2 are presented in details for time domain data. Certain simulations with the selected input reference signals need to be conducted for the input to the identification toolbox. To model the time domain data, step reference signals are selected as the reference signals. To obtain each transfer functions, different reference signals are defined in Table 5.1. Then, based on the simulation results (full simulation), Matlab Identification tool box is used to derive the transfer functions. The general procedure for using Matlab Identification tool box is briefly introduced as below,
5.2. Identification and evaluation process
Figure 5.2: Identification and evaluation process for transfer functions
”work space variable”. ”Process models” is the estimation method to use for identifying
G11. The procedure showing in Figure 5.3 can only identify one transfer function at a
time, repeat work is required for identifying the rest of the transfer functions, i.e. G12,
G21andG22.
Afterwards, onceG11,G12,G21andG22are defined, the transfer function matrixG(s) =
(G11(s) G12(s)
G21(s) G22(s)
can be determined. Therefore,G(s) can be modeled in Simulink with
reference signalsTref,ωref by using equations (5.1) and (5.2). The output signalsTout,ωˆr
can be compared with the full simulation by using the same input reference signals. The results of the output signals between the full simulation and the transfer function matrix
simulation are shown in Figure 5.4 and 5.5. Figure 5.4 compares the result ofTout with
step reference on both Tref and ωref. The step reference signals are already defined in
Table 5.1.Tout of the transfer function matrix model tracks the full simulation model in
the steady state.(when the torque reach 5 Nm). The consistency at the transient state is
poor.(at step time 1.2s). In the same way,ωˆr is compared in Figure 5.5. The consistency
Chapter 5. Transfer Function
Table 5.1: Reference signals in time domain
Tref(U1) Nm ωref(U2) (Hz) Tout(Y1) (Nm) ωˆr(Y2) (Hz)
G11(s) step signal (0-5)
at step time 1.2s
2 (constant) output value –
G12(s) 5 (constant) step signal (0-2)at
step time 1.2s
output value –
G21(s) Step signal (0-5)
at step time 1.2s
2 (constant) – output value
G22(s) 5 (constant) step signal (0-2)at
step time 1.2s
– output value
5.2.2
Transfer functions identification and evaluation process in
fre-quency domain
The transfer function matrix, G(s) derived within the time domain can only fit the time
domain input reference (step reference, constant reference etc). However, if frequency domain input references are applied in the full simulation models(sinusoidal references), G(s) must be derived again in frequency domain from Matlab Identification Toolbox. Similar as section 5.2.1, reference input need to be identified to the full simulation model.
Table 5.2 contains the information of selected reference input. Here Amp=1 is used for
the reference inputs. Unlike in section 5.2.1, reference inputs and output can not be used directly at frequency domain in Matlab Identification Toolbox. Instead, amplitude (gain) and phase shift of the input reference and output are used as the input information in Mat-lab Identification Toolbox.
Bode plot
With the amplitude (gain) and phase shift information provided, Bode plots can be gen-erated. Bode plots are a very useful way to present the gain and phase of a system as a function of frequency [1]. It is a usually a combination of a magnitude plot and a phase plot. The Bode magnitude plot is expressing the gain of the transfer function, and the Bode phase plot expresses the frequency response phase shift [2]. Bode plots are an ef-ficient evidence to evaluate if the system have lost a lot of gain or have a lot of changes on phase shift with a certain frequency range. Several simulations with different cut-off frequencies of low-pass, high-pass filters and different bandwidth of the PLL estimator have been conducted to optimize the PMSynRel model which is evaluated by Bode plots. The PMSynRel model considered in this chapter is the basic PMSynRel for simplicity. Furthermore, reference input signal is given at low speed range which is from 0-40 Hz. The speed at 25 Hz is set as reference checking point point. The gain and phase shift is
compared at 25 Hz with different settings of the cutoff frequencies (high-pass filter ωnh,
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Figure 5.3: Procedure for using Matlab Identification tool box in time domain
Hz and 94.25 rad/s, and the Bode plots obtained in Figure 5.6. This figure shows one ex-ample for the Bode lots at the initial values, it is found that when the speed is at reference checking point (25 Hz), the gain ratio is only 0.2655, too low to the system. To increase
the gain, several combinations of ωnh, ωnl and ρ are adjusted and the results are shown
in Table 5.3. The magnitude and the phase shift is highlighted at the reference checking point (25 Hz).
Table 5.3 indicates that the PLL estimator bandwidth has a great impact on the gain
but rather small impact on the phase shift. Whenρ is increased from 15 ×2π rad/s to 20
×2π rad/s, the value of magnitude gain increases nearly 3 time, from 0.26 to 0.68. It also increases the phase shift, though not as much as the gain. The Bode plots in Figure 5.7
shows the change of the gain and phase shift whenρ is increasing from 15 ×2π rad/s to
20 ×2π rad/s with applying the same cut-off frequencies. When ρ=30 ×2π rad/s, with
the same cut-off frequencies, the sensorless control becomes unstable, but with adjusting
Chapter 5. Transfer Function 1.16 1.18 1.2 1.22 1.24 0 1 2 3 4 5 6 7 Full Simulation TF t (s) T o rq u e (N m )
Figure 5.4: Comparison of Tout in full simulation model and transfer function matrix
model
For this system, the maximum PLL bandwidth can be reached is 35×2π rad/s, otherwise,
the system is unstable. Figure 5.8 presents the magnitude change while vary the cut-off
frequency of low and high pass filter with keepingρ unchanged. The carrier frequency of
the injection signal has a great impact on the phase shift. However, the frequency of the
carrier signal is limited in the lab which means to keepωc to 500×2π is more reasonable.
ωnh=5×2π rad/s, ωnl=70×2π rad/s and ρ=30×2π rad/s are selected to the simulation
based on the Bode plot evaluations. The Bode plots corresponding to G11, G12, G21and
G22 are plotted in Figure 5.9 and 5.10 to derive the transfer functions. From Figure 5.9,
it shows the impact from Tref to ωˆr is rather small which can be assumed the impact is
zero.
Results and Evaluations
5.2. Identification and evaluation process 0.5 1 1.5 2 2.5 0 2 4 6 8 10 12 14 16 Full Simulation TF t (s) ˆω (r ad /s )
Figure 5.5: Comparison ofωˆrin full simulation model and transfer function matrix model
estimation are listed: 1. select ”Frequency Domain” in ”Import Data” section with ”Fre-quency Function (Amp/phase) as ”Data Format for Signals”. 2. Evaluate the results by clicking ” Frequency resp” instead of ”Model output”.
By using the Matlab Identification tool box, the system can be identified as:
Chapter 5. Transfer Function
Table 5.2: Reference signals in frequency domain
Tref(U1) Nm ωref(U2) (Hz) Tout(Y1) (Nm) ωˆr(Y2) (Hz)
G11(s) Amp · cos(ωt) ω
range:0-40Hz in-creasing rate: 0.5 Hz
2 (constant) output value –
G12(s) 5 (constant) Amp · sin(ωt) ω
range:0-40Hz in-creasing rate: 0.5 Hz output value – G21(s) Amp · cos(ωt) ω range:0-40Hz in-creasing rate: 0.5 Hz
2 (constant) – Measured value
G22(s) 5 (constant) Amp · sin(ωt) ω
range:0-40Hz in-creasing rate: 0.5 Hz – Measured value G22(s) = Gωrefωˆr (5.7) = KpT wwh× (1 + T zwwh× (s)) 1 + (2 × (Zetawwh) × (T wwwh)) × (s) + (T wwwh× (s))2× (1 + T p3wwh× (s) (5.8) Kpwwh = 1.0026, T wT wh = 0.0051808, ZetaT wh = 0.45176 T zT wh = −0.0015445 and T p3T wh= 0.0050678.
The estimation result of G12 is quite poor by using ”process model” estimation method.
Therefore, another simulation method ”Linear Parametric Models” from identification
tool box is selected. And the transfer function ofG12can be identified as:
G12(s) = GωrefTout = arx[datasource, orders,’Focus’,’Simulation’,’InitialState’,’Estimate’]
(5.9)
Again, as described in section 5.2.1, G(s) can be modeled in Simulink, and the
out-put can be compared with the full simulation results when the same inout-put references are given to the full simulation model. The comparing results for each transfer function
G11, G12, G21, G22 are shown in Figure 5.11. Further more, various simulations are
car-ried out with different reference input signals to evaluateG(s) in this stage, and the details
5.2. Identification and evaluation process 0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 X: 25 Y: 0.2655 Hz gain (−) 0 5 10 15 20 25 30 35 40 −300 −200 −100 0 Hz degree
Figure 5.6: bode plot for compareωref toω at ωˆ nh=5 Hz,ωnl=40 Hz,ρ =94.25 rad/s
Table 5.3:ωref-ˆω at sinusoidal input signal is 25 Hz
ωc (rad/s) ωnh(rad/s) ωnl(rad/s) ρ (rad/s) Magnitude gain (–) Phase shift (◦)
500×2π 5×2π 40×2π 15×2π 0.2655 -217.4
500×2π 5×2π 40×2π 20×2π 0.6861 -208.5
500×2π 10×2π 40×2π 20×2π 0.6601 -208.1
500×2π 25×2π 40×2π 20×2π 0.5543 -212.3
500×2π 25×2π 50×2π 20×2π 0.4745 -187.4
500×2π 5×2π 15×2π 30×2π out of control out of control
1000×2π 40×2π 80×2π 30×2π 0.7672 -81.06
1000×2π 50×2π 100×2π 30×2π 0.7294 -83.16
1000×2π 30×2π 70×2π 30×2π 0.794 -79.2
Chapter 5. Transfer Function 0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 0 5 10 15 20 25 30 35 40 −300 −200 −100 0 Hz degree ρ=15 ρ=20
5.2. Identification and evaluation process 0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 0 5 10 15 20 25 30 35 40 −300 −200 −100 0 Hz degree ωnh=15Hz, ωnl=90Hz ωnh=15Hz, ωnl=70Hz ωnh=5Hz, ωnl=70Hz
Figure 5.8: bode plot for ρ =30×2π rad/s, under ωnh=5×2π,15×2π rad/s ωnl
Chapter 5. Transfer Function 0 10 20 30 40 0 0.5 1 1.5 0 10 20 30 40 −200 −150 −100 −50 0 Hz degree 0 10 20 30 40 0 0.01 0.02 0.03 0 10 20 30 40 −400 −300 −200 −100 0 ω-ˆω ω-T
Figure 5.9: bode plot for ω-ˆω and ω-T ρ=30×2π rad/s,ωnh=5×2π rad/s, ωnl =70×2π
rad/s 0 10 20 30 40 0.97 0.98 0.99 1 1.01 0 10 20 30 40 −15 −10 −5 0 Hz 0 10 20 30 40 0 2 4 6 x 10 −4 0 10 20 30 40 −400 −300 −200 −100 0 Hz Tref-T Tref-ˆω
Figure 5.10: bode plot for Tref-T and Tref-ˆω ρ=30 ×2π rad/s,ωnh=5×2π rad/s, ωnl
5.2. Identification and evaluation process 0.05 0.1 0.15 0.2 0.25 0.3 −1 −0.5 0 0.5 1 G11: Tref-T Full Simulation TF 0.05 0.06 0.07 0.08 12 12.5 13 G21: Tref-ˆω Full Simulation TF 0.1 0.2 0.3 0.4 4.95 5 5.05 G12: ωref-T 0.05 0.1 0.15 0.2 0.25 0.3 −1 −0.5 0 0.5 1 G22: ωref-ˆω Full Simulation TF Full Simulation TF T o rq u e (N m ) T o rq u e (N m ) t (s) t (s) t (s) t (s) ˆω (r ad /s ) ˆω (r ad /s ) Figure 5.11: Comparison ofG11, G12, G21, G22
Table 5.4: Simulations on evaluating the transfer functions
Tref(U1) Nm ωref(U2) (Hz) Tout(Y1) (Nm) ωˆr(Y2) (Hz)
Case 1 sinusoidal signal:Amp ·
cos(ωt)
sinusoidal signal:Amp ·
sin(ωt)
output result output result
1 ω=5 Hz ω=5 Hz Fig 5.9 Fig 5.12 2 ω=15 Hz ω=15 Hz Fig 5.13 Fig 5.13 3 ω=25 Hz ω=25 Hz Fig 5.14 Fig 5.14 4 ω=40 Hz ω=40 Hz Fig 5.15 Fig 5.15 5 ω=50 Hz ω=50 Hz Fig 5.16 Fig 5.16 6 ω=60 Hz ω=60 Hz Fig 5.17 Fig 5.17
Case 2 Step signal: 0-5 Nm Step signal:0-10 rad/s output result output result
1 step time: 0.1s step time: 0.1s Fig 5.18 Fig 5.18
2 step time: 0.2s step time: 0.1s Fig 5.19 Fig 5.19
3 step time: 0.1s step time: 0.2s Fig 5.20 Fig 5.20
Case 3 Step signal/sinusoidal
signal:0-5
Nm/Amp · cos(ωt)
Step signal/sinusoidal
signal:0-10
rad/s/Amp · sin(ωt)
output result output result
1 sinusoidal signal: ω =
5 rad/s
step signal: step time 0.1s
Fig 5.21 Fig 5.21
2 step signal: step time
0.1s
sinusoidal signal: ω =
5 rad/s
Chapter 5. Transfer Function 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −2 −1 0 1 2 Full Simulation TF 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −2 −1 0 1 2 Full Simulation TF T o rq u e (N m ) t (s) t (s) ˆω (r ad /s )
Figure 5.12: Transfer function result for Torque andωˆr atω=5 Hz, case 1.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −2 −1 0 1 2 Full Simulation TF 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −2 −1 0 1 2 Full Simulation TF T o rq u e (N m ) t (s) t (s) ˆω (r ad /s )
5.2. Identification and evaluation process 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −1 −0.5 0 0.5 1 Full Simulation TF 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −1 −0.5 0 0.5 1 Full Simulation TF T o rq u e (N m ) t (s) t (s) ˆω (r ad /s )
Chapter 5. Transfer Function 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −1 −0.5 0 0.5 1 Full Simulation TF 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −1 −0.5 0 0.5 1 Full Simulation TF T o rq u e (N m ) t (s) t (s) ˆω (r ad /s )
Figure 5.15: Transfer function result for Torque andωˆratω=40 Hz, case 1.
In case 1 of Table 5.4, sinusoidal reference signals are given for both torque ref-erence and speed refref-erence. Figure 5.12 to Figure 5.7 present the results of case 1, the output results from transfer function matrix model have agreed with the results from full
simulation model when the frequency ω of the sinusoidal reference signals is low. The
transfer functions are derived whenω is from 0-40 Hz. Even though, the output results of
transfer function matrix model can still follow the full simulation when ω is 50 Hz. The
deviations of the magnitude and phase is obvious whenω reaches 60 Hz.
Step reference is applied on both torque and speed reference in case 2 of Table 5.4, as shown in Figure 5.18, the agreement with the output from transfer function matrix model and the full simulation model is nice. However, from Figure 5.19 and 5.20, when the step time of two step references is inconsistent (0.1s step time on one reference input and 0.2 s step time on the other), the output results from the transfer function matrix model can not follow the transient which exists in the full simulation which is caused by another step reference signal. Simulation of using step reference to one input reference and sinusoidal reference to the other input applied in case 3. From Figure 5.21, the output from transfer function matrix model matches the output from the full simulation mode. Anyhow, the output results between transfer function matrix model and full simulation model can not
5.2. Identification and evaluation process 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −1 −0.5 0 0.5 1 Full Simulation TF 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −0.5 0 0.5 Full Simulation TF T o rq u e (N m ) t (s) t (s) ˆω (r ad /s )
Figure 5.16: Transfer function result for Torque andωˆratω=50 Hz, case 1.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −1 −0.5 0 0.5 1 Full Simulation TF 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −0.2 0 0.2 0.4 0.6 Full Simulation TF T o rq u e (N m ) t (s) t (s) ˆω (r ad /s )
Chapter 5. Transfer Function 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0 2 4 6 Full Simulation TF 0.05 0.1 0.15 0 5 10 Full Simulation TF T o rq u e (N m ) t (s) t (s) ˆω (r ad /s )
Figure 5.18: Transfer function result for Torque andωˆr, case 2.
0.05 0.1 0.15 0.2 0.25 0.3 0 2 4 6 Full Simulation TF 0.05 0.1 0.15 0.2 0.25 0.3 0 5 10 Full Simulation TF T o rq u e (N m ) t (s) t (s) ˆω (r ad /s )
5.2. Identification and evaluation process 0.05 0.1 0.15 0.2 0.25 0.3 0 2 4 6 Full Simulation TF 0.05 0.1 0.15 0.2 0.25 0.3 0 5 10 Full Simulation TF T o rq u e (N m ) t (s) t (s) ˆω (r ad /s )
Figure 5.20: Transfer function result for Torque andωˆr, case 2.
0.05 0.1 0.15 0.2 −1 0 1 Full Simulation TF 0.05 0.1 0.15 0.2 0 5 10 Full Simulation TF T o rq u e (N m ) t (s) t (s) ˆω (r ad /s )
Chapter 5. Transfer Function 0.05 0.1 0.15 0.2 0.25 0.3 0 2 4 6 Full Simulation TF 0.05 0.1 0.15 0.2 0.25 0.3 −1 0 1 Full Simulation TF T o rq u e (N m ) t (s) t (s) ˆω (r ad /s )
Chapter 6
Experimental Result
From Chapter 5, transfer functions are possible to be estimated in Simulink program which initialize the thoughts to get the transfer function in the real case to design the controller to the system. Measurements are taken in the laboratory and recorded as inputs to Matlab identification Tool box. And the transfer functions obtained from the laboratory will be compared and analyzed from the one stated in the last chapter. The experimental set up and information is presented in Appendix A.
6.1
Experiment
The laboratory set-up is described in Appendix A where a load machine is driven by the PMSynRel via a coupling torque meter. Ideally in Figure 6.1, the output torque from
PMSynRelT1 shall be equal toTs andTL. This is indeed valid at steady state operation.
However, due to the resonance induced by the finite stiffness kt, Ts does not follow T1
during transients. This requires a closer investigation of the effect of the practical set-up.
The torque measurement Ts is plotted in Figure 6.2. It is obvious that a large amount
of harmonics present in this measurement and it does indicate the right torque dynamics. Naturally, Fourier analysis is applied to this measurement and three dominant harmonics are found, as shown in Figure 6.3. The Three dominant harmonic orders are identified at 48 Hz, 240 Hz and 500 Hz (1 Hz is used as a reference frequency for Fourier analysis).
The 48 harmonics is identified to be the slot harmonics which is speed dependent due to the fact that the electrical angle is set to 2 Hz, therefore the mechanical angle is 4 Hz since the pole pair number is 2. And the harmonic at 48 Hz is the 12th harmonic order of the mechanical angle. This harmonic varies with different electrical speed. The 240 harmonics is identified to be a speed independent resonance and this harmonic is not changing while the speed changes. The 500 harmonics is clearly the torque ripple caused by the injection frequency. Based on the above investigation, it is practically very
difficult to identify a transfer function based onTsdue to the speed dependent harmonics.
Chapter 6. Experimental Result ŽƵƉůŝŶŐĂŶĚƚŽƌƋƵĞ ŵĞƚĞƌ >ŽĂĚ WD^LJŶZĞů dІ T c T k L T s T 1 eq J Jeq2 WD^LJŶZĞů >ŽĂĚŵĂĐŚŝŶĞ
Figure 6.1: Laboratory set-up and physical coupling
withT1.iq,P M is plotted in Figure 6.4. And the low-pass filterediq,P M is plotted in Figure
6.1. Experiment
Figure 6.2: Torque measurementsTs
Chapter 6. Experimental Result
Figure 6.4:Tsandiq,P M
Chapter 7
Conclusion and future work
The conclusion from this thesis is made in this chapter. Some suggestions for future re-search related to the thesis are also stated.
7.1
Summary
The first half of the thesis illustrates and compares different methodologies of acquiring rotor position information which are essential for the realization of field oriented control: • Resolver
• Hall-effect sensor
• Sensorless (based on Signal Injection)
Full scale simulations have been performed taking into account nonlinearity of the motor, discretized sampling, and the PWM switching. Therefore, the simulations shall be closest possible theoretical results to reality. The main discovery from the simulation work is that the PLL bandwidth have quite significant impact to the error of estimation of the rotor position. The value of PLL bandwidth can neither be too high nor too low. High PLL bandwidth causes additional sensitivity to the system while low value increases system responding time. Experience is that the bandwidth has to be customized. For Hall-effect sensor, other than PLL bandwidth, the harmonic order number introduced in Chapter 4 has a great impact on the error estimation. Theoretically, the larger k the better result. However, the theory is based on the rotor information is known and can be used as an input to equation (4.1). In reality, only estimated rotor position can be used for (4.1) cal-culation, large number of k might cause the accumulation of error information. However, it is difficult to summarize which k or PLL bandwidth is the most suitable for each spe-cific systems. Experience is that simulations need to be run to obtain the most appropriate values.
Chapter 7. Conclusion and future work
”Best Fit” for each simulated input/output. Identification Tool box can be used to derive the transfer functions from both frequency domain data and time domain data. The trans-fer function matrix derived based on the frequency domain data is also applicable to the
time domain system in the study. In this work, the accuracy of the transfer functionsG11
(FromTref toTout) andG22 (Fromωref toωˆr) are the highest since those input and
out-put are directly effected. Good agreement is achieved onG12which means Identification
Tool Box have a great possibility to derived the cross effected objects. The result of G21
is poor. Although based on the Bode plots, the impacts from Tref is rather small, it has
certain impacts in the full simulations.
However, it has to be noted that the simulated data is based on a Basic Model (sen-sorless), which could be quite ”different” from a laboratory set-up. It is doubted that this difference is so large that it makes a transfer function impossible to identify applying the same methodology to the measured input/output data. In the laboratory work, measure-ments of torque output are taken in time domain and found out the slot harmonic is the dominated harmonic which cannot be decoupled in time domain. In this case, frequency domain should be considered for the experimental measurements.
7.2
Future work
7.2.1
Sensorless control and Hall-effect sensors
Research may be done to summarize if there is any general rules to identify the range of PLL bandwidth with different system set up. This might save a lot of work for others and
it can also be a guidance to the other for selecting a efficientρ in the system.
7.2.2
Torque dynamics in frequency domain from the experiment
Appendix A
Laboratory Setup
The laboratory setup is shown in Fig.A.1. The PMSynRel is connected to a servo ma-chine via a torque meter. A torque meter is mounted on the interconnected shaft to record the torque data. The resolver mounted on the servo machine is used to obtain rotor posi-tion informaposi-tion for both controllers. Both PMSynRel and SERVO drive are powered by voltage source inverters (VSIs). The sensorless control method is implemented for PM-SynRel. All control algorithms are implemented in C-code in dSPACE system (DS1005).
Figure A.1: Experimental Setup
Slave boards used in this dSPACE system are listed in in Table A.1.Asymmetrical PWM is considered in this work and the sampling frequency and switching frequency are 10 kHz and 5 kHz respectively on the PWM used in the VSIs.
Appendix A. Laboratory Setup
Table A.1: Configuration of the hardware of system setup.
Board number Function
DS1005 Control board
DS5101 PWM generation
DS4001 32 digital I/O
DS2001 5 analog inputs
Table A.2: Nominal values of the PMSynRel
Value Unit
Winding type Y-Connection
Number of pole pairs 2
-Rated current 30 A
Rated frequency 50 Hz
Rated power 21 kW
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