UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Modelling and control of the LLC resonant converter Cheng, Brian Cheak Shing 2012

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
24-ubc_2013_spring_cheng_brian.pdf [ 2.04MB ]
Metadata
JSON: 24-1.0073435.json
JSON-LD: 24-1.0073435-ld.json
RDF/XML (Pretty): 24-1.0073435-rdf.xml
RDF/JSON: 24-1.0073435-rdf.json
Turtle: 24-1.0073435-turtle.txt
N-Triples: 24-1.0073435-rdf-ntriples.txt
Original Record: 24-1.0073435-source.json
Full Text
24-1.0073435-fulltext.txt
Citation
24-1.0073435.ris

Full Text

MODELLING AND CONTROL OF THE LLC RESONANT CONVERTER  by  Brian Cheak Shing Cheng  B.Sc., Queen's University, 2010  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF  THE REQUIREMENT FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  in  The Faculty of Graduate Studies  (Electrical & Computer Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA  (Vancouver)  December 2012  c Brian Cheak Shing Cheng, 2012  Abstract To achieve certain objectives and specications such as output voltage regulation, any power electronics converter must be coupled with a feedback control system. Therefore, a topic of considerable interest is the design and implementation of control systems for the LLC resonant converter. Additionally, with the current trend of smaller, more cost eective and reliable digital signal processors, the implementation of digital feedback control systems has garnered plenty of interest from academia as well as industry. Therefore, the scope of this thesis is to develop a digital control algorithm for the LLC resonant converter. For output voltage regulation, the LLC resonant converter varies its switching frequency to manipulate the voltage gain observed at the output. Thus, the plant of the control system is represented by the small signal control-to-output transfer function, and is given by  P (s) =  Vo f .  The diculty in designing compensators for the LLC resonant converter is the lack of known transfer functions which describe the dynamics of the control-to-output transfer function. Thus, the main contribution of this thesis is a novel derivation of the small signal control-to-output transfer function. The derivation model proposes that the inclusion of the third and fth harmonic frequencies, in addition to the fundamental frequency, is required to fully capture the dynamics of the LLC resonant converter. Additionally, the eect of higher order sideband frequencies is also considered, and included in the model. In this thesis, a detailed analysis of the control-to-output transfer function is presented, and based on the results, a digital compensator was implemented in MATLAB  R  . The compensator's  functionality was then veried in simulation. A comparison of the derivation model and the prototype model (based on bench measurements) showed that the derivation model is a good approximation of the true system dynamics.  It was  therefore concluded that both the bench measurement model and the derivation model could be used to design a z-domain digital compensator for a digital negative feedback control system. By using the derivation model, the main advantages are reduced computational power and the requirement for a physical prototype model is diminished.  ii  Table of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Table of Contents List of Tables  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  List of Figures  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  List of Symbols  List of Abbreviations  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  List of SI Units and Prexes  xi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xiii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xiv  Acknowledgments Dedication .  ii  1 Introduction  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.1  Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.2  Motivation  2  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2 LLC Resonant Converter 2.1  2.2  2.3  2.4  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3  H-bridge Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  6  2.1.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  7  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  8  Zero Voltage Switching  Resonant Tank  ωr1 ,ωr2  2.2.1  Resonant Frequencies  2.2.2  High Frequency Isolation Transformer  9  . . . . . . . . . . . . . . . . . . . . . .  9  Rectier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  10  2.3.1  Synchronous Rectication . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  11  Theory of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  14  2.4.1  Region 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  15  2.4.2  Region 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  15  3 Control of LLC Resonant Converter  3.1  . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . .  18  3.0.3  Variable Frequency Control . . . . . . . . . . . . . . . . . . . . . . . . . . . .  18  3.0.4  Pulse Width Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  19  Digital Control 3.1.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Eects of Sampling Frequency  . . . . . . . . . . . . . . . . . . . . . . . . . .  19 20  iii  3.2  3.3  3.1.2  Design of Digital Control Systems  3.1.3  2-Pole-2-Zero Compensator  . . . . . . . . . . . . . . . . . . . . . . . .  20  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  21  Compensator Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  22  3.2.1  Root Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  22  3.2.2  Bode Diagram  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  22  Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  23  3.3.1  Bode Diagram  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  23  3.3.2  Nyquist Plot  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  23  4 Implementation and Verication . 4.1  Design Methodology  25  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  25  4.1.1  Overview  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  25  4.1.2  Bench Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  26  4.1.3  Compensator Design Results  . . . . . . . . . . . . . . . . . . . . . . . . . . .  27  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  30  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  34  4.1.4 4.2  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Verication in PSIM  Texas Instruments  R  DSP  R  5 Derivation of Control-to-Output Transfer Function . 5.1  Overview  5.2  Transfer Function Derivation  5.3  . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  36 36  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  36  5.2.1  Frequency Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  37  5.2.2  Square Wave Approximation  38  5.2.3  Bessel Functions of the First Kind  . . . . . . . . . . . . . . . . . . . . . . . .  39  5.2.4  Amplitude Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  41  5.2.5  Results of Frequency and Amplitude Modulation . . . . . . . . . . . . . . . .  45  5.2.6  Isolation Transformer and Rectication Stage . . . . . . . . . . . . . . . . . .  47  5.2.7  Results of Analysis of the Derivation Model . . . . . . . . . . . . . . . . . . .  47  5.2.8  Phase Response of Control-to-Output Transfer Function . . . . . . . . . . . .  48  . . . . . . . . . . . . . . . . . . . . . . . . . . .  Verication of Derivation Model in PSIM  R  . . . . . . . . . . . . . . . . . . . . . . .  50  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  54  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  54  6 Conclusions and Future Work 6.1  Summary  6.2  Future Work  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  55  References .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  56  Appendices  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  59  Appendix A: Observation of Stability in Continuous and Discrete-time domains . . . . . . A1: Continuous-time domain  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  59 59  iv  A2: Discrete-time domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B: PSIM  R  simulation schematic . . . . . . . . . . . . . . . . . . . . . . . . . . .  Appendix C: MATLAB  R  derivation model code . . . . . . . . . . . . . . . . . . . . . . . .  60 61 62  v  List of Tables 1  Results of prototype model  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  32  2  Results of derivation model  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  53  vi  List of Figures 1  A direct link between DC source(s) and load(s) . . . . . . . . . . . . . . . . . . . . .  1  2  Resonant tank circuits of resonant converter topologies . . . . . . . . . . . . . . . . .  3  3  LCC resonant tank circuit schematic . . . . . . . . . . . . . . . . . . . . . . . . . . .  4  4  Comparison of LCC and LLC DC gain characteristic with varying Q-factors . . . . .  5  5  LLC resonant converter circuit schematic  . . . . . . . . . . . . . . . . . . . . . . . .  5  6  H-bridge inverter circuit schematic  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  7  7  Resonant tank circuit schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  8  8  Rectier circuit modelled as  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  10  9  SR phase delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  12  Rac R  10  Synchronous rectication PSIM  model . . . . . . . . . . . . . . . . . . . . . . . . .  13  11  Regions 1, 2, and 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  14  12  Operation of LLC resonant converter in region 1  . . . . . . . . . . . . . . . . . . . .  16  13  Operation of LLC resonant converter in region 2  . . . . . . . . . . . . . . . . . . . .  17  14  Block diagram of negative feedback loop . . . . . . . . . . . . . . . . . . . . . . . . .  18  15  Block diagram of digital negative feedback loop . . . . . . . . . . . . . . . . . . . . .  19  16  2 pole 2 zero DSP implementation  22  17  Bench test result of DC gain characteristic for Region 1  18  Plot of relative frequency response of prototype model under dierent loading conditions 27  19  Prototype model frequency response data in MATLAB  20  Open-loop Bode plot of prototype frequency response data . . . . . . . . . . . . . . .  29  21  MATLAB  22  PSIM  R  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  R  . . . . . . . . . . . . . . . .  26  SISOTOOL GUI environment 28  . . . . . . .  30  circuit schematic of LLC resonant converter . . . . . . . . . . . . . . . . . .  step response of closed-loop system using prototype model  31  controller schematic of LLC resonant converter  23  R PSIM R  . . . . . . . . . . . . . . . .  31  24  Error voltage of prototype model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  31  25  Output voltage ripple of prototype model  . . . . . . . . . . . . . . . . . . . . . . . .  32  26  PSIM  closed-loop response to step load change using prototype model . . . . . . .  33  27  Voltage loop ow diagram for DSP implementation . . . . . . . . . . . . . . . . . . .  35  28  Analysis road map  37  R  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Rac  29  Tank lter circuit schematic with equivalent resistance  30  Comparison of the signicance between fundamental, third and fth harmonics  . . .  46  31  Comparison of the fundamental component, with and without sideband frequencies .  46  32  Comparison of prototype frequency response data and derivation model (magnitude)  48  33  Comparison of derivation model and curve t model (magnitude) . . . . . . . . . . .  49  34  Comparison of prototype frequency response data and curve t model (phase) . . . .  50  35  MATLAB  50  R  . . . . . . . . . . . . . .  step response of closed-loop system using derivation model  . . . . . . .  43  vii  36  Error voltage of derivation model  37  Output voltage ripple of derivation model  38  PSIM  A-1  Unit circle in the z-domain  B-1  PSIM  B-2  PSIM  R R R  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  51  . . . . . . . . . . . . . . . . . . . . . . . .  51  closed-loop response to step load change using prototype model . . . . . . .  52  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  closed loop circuit schematic of LLC resonant converter open loop circuit schematic of LLC resonant converter  60  . . . . . . . . . . .  61  . . . . . . . . . . . .  61  viii  List of Symbols an  nth th  coecient of 2P2Z transfer function denominator  bn  n  Co  output capacitor  Cp  parallel resonant capacitor  Cr  series resonant capacitor  E  energy of carrier and sideband frequency signals  e[k − n]  nth  fs  switching frequency  fsample  sampling frequency  fmax  maximum bandwidth frequency  Jn (β)  Bessel function of the rst kind  KV CO  VCO gain constant  LM  magnetizing inductor  Lr N  series resonant inductor  Nprimary  number of turns on primary side of transformer  Nsecondary  number of turns on secondary side of transformer  pi  ith  Psw  power loss of semiconductor switch device  Q  ratio between the characteristic impedance and the output load  r  reference input  Rac  equivalent AC resistance  Rc  output capacitor equivalent series resistance  Ro  output resistance  Rr  series resistance  T  sampling time  Tof f  semiconductor switch device turn o time  coecient of 2P2Z transfer function numerator  previous error term  transformer turns ratio  pole  Ton  semiconductor switch device turn on time  u[k − n]  nth  Vctrl  control voltage  Vds  drain-source voltage  Vg  input voltage  Vo  output voltage  Vp  peak voltage  previous term of compensator output  ix  Vs,1  fundamental component voltage expression  Vs,3  third harmonic voltage expression  Vs,5  fth harmonic voltage expression  zi  ith  β  modulation index    amplitude of small signal perturbation  Γ(x) ωc  gamma function  ωm  angular modulation frequency  ωr  angular resonant frequency  ω̂  voltage controlled oscillator free-running angular frequency  =  equals  6=  not equal to  ≈  approximately equal to  ±  plus and minus  zero  angular carrier frequency  x  List of Abbreviations 2P2Z  2-pole-2-zero  AC  alternating current  ADC  analog-to-digital converter  AM  amplitude modulation  DAC  digital-to-analog converter  CLA  Control Law Accelerator  CPU  central processing unit  DC  direct current  DSP  digital signal processor  TM  EMI  electromagnetic interference  FM  frequency modulation  GUI  graphical user interface  LCC  inductor-capacitor-capacitor  LLC  inductor-inductor-capacitor  LPF  low pass lter  MOSFET  metal oxide semiconductor eld eect transistor  P  proportional control  PI  proportional, integral control  PID  proportional, integral, derivative control  PWM  pulse-width modulation  RHPZ  right hand plane zero  SISO  single input single output  SMPS  switched-mode power supply  SR  synchronous rectier  UPS  uninterruptible power supply  VCO  voltage-controlled oscillator  ZVS  zero-voltage switching  ZCS  zero-current switching  ZOH  zero-order hold  xi  List of SI Units and Prexes A  amperes  dB  decibel  Hz  hertz  s  seconds  V  volts  ◦  degrees  p  pico(10  n  −12  )  −9  nano (10  )  −6  µ  micro (10  m  milli (10  k  kilo (10 )  M  Mega (10 )  G  Giga (10 )  T  12  −3  )  )  3  6  9  Tera (10  )  xii  Acknowledgments I would rst like to express my sincere and utmost gratitude to my research supervisor, Dr. William Dunford, for his patience and guidance throughout my studies at UBC. His support and mentorship in both my academic and professional endeavors has been invaluable, and I am indebted to him for his help over the past two years.  Secondly, I would like to thank my friends and colleagues at the UBC Electric Power and Energy Systems Group. My time at UBC has been much more memorable because of the people I have met here. In particular, I'd like to thank Rahul Baliga, Justin Wang, and William Wang for the many technical and non-technical discussions that have been had. Additionally, I would like to thank my former classmates and friends Colin Clark and Kyle Ingraham for their technical advice and friendship.  I would also like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) and Alpha Technologies Ltd. for their generous nancial support of this research project. I must give my extended gratitude to Mr. Victor Goncalves for giving me the opportunity to complete my internship at Alpha Technologies Ltd., as well as the many people I was fortunate enough to collaborate with at Alpha Technologies Ltd. The technical expertise I received over the course of this work was indispensable. Additionally, I would also like to thank Mr. Brian Bella of the Faculty of Graduate Studies at UBC for his assistance with the application to the Industrial Postgraduate Scholarship program.  Last but not least, I must thank my family for their unconditional support throughout the years. Growing up, they have been my source of inspiration, and have provided me with exemplary examples of the person I one day hope to become. To each of them, I owe my deepest gratitude. I would like to take this opportunity to individually thank each of these people who have made it possible for me to get to where I am today. My grandmother; my parents Tom and Daisy, my uncles and aunts Patrick and Janet; Millie and Gilbert, Cora and Joe; and Juno and Alex. A special `thank you' also needs to extended to my cousins Jerey, Steven, Gibson, Jackie and Megan, who have all become like my own brothers and sisters.  Finally, I would like to acknowledge and thank my grandfather, whose memory remains strong with us to this very day.  xiii  Dedicated to my parents  xiv  1 Introduction 1.1 Background Switched-mode power supplies (SMPS) are now found in many dierent industrial applications and their function can vary from high power electric vehicles to low power biomedical devices and equipment. There is particular interest in applying SMPS to medium and high power applications for uninterruptible power supplies (UPS) designed for telecommunications-grade applications. The next generation of SMPS aim to achieve high eciency, high reliability, high power density, as well as low cost. As an example, in renewable energy applications, due to constraints in cost and the physical limitations of energy storage, the benets of a well designed SMPS are immediate. The application of DC-DC converters has recently become an important area of SMPS design as the emergence of distributed generation and battery-based systems continues to grow. Additionally, as the number of DC loads increases [1], using local DC power sources becomes more of a sensible solution, and further encourages the development of DC-DC conversion technology. For some applications, DC-DC converters are particularly useful, as it provides a direct interface between energy storage elements, which are typically DC voltage sources, and DC loads.  By  reducing the number of intermediate energy conversion processes, the overall eciency of the system can be increased, while also potentially minimizing the cost of the system.  To meet safety and  protection requirements, DC-DC converters can also be implemented with galvanic isolation between the input source and output load.  AC Input  ≈/=  =/=  VS DC Input  =/=  Figure 1: A direct link between DC source(s) and load(s)  A particular form of DC-DC converter is the resonant converter. In literature, resonant converters have been thoroughly studied and it has been shown that they can oer many benets in performance, size, and cost [2]. For example, resonant converters are able to achieve low switching losses through the use of soft-switching techniques, and are able to be operated at greater switching  1  frequencies than other comparable converters. The ability to operate at higher switching frequencies has the superior advantage of increasing eciency, as well as decreasing the size of the discrete components, notably inductors and capacitors, within the hardware. This is in comparison to pulse-width modulation (PWM) converters, where the turn-on and turn-o losses of the switching devices at high switching frequencies can be high enough to prohibit operation of the converter, even when soft-switching techniques are used [3]. Moreover, PWM converters utilizing high switching frequency operation can cause disturbances such as electromagnetic interference (EMI) and suer from the eects of parasitic impedances. However, under proper design, it is possible for resonant converters to utilize the leakage inductances of the circuit as part of the resonant tank circuit. It was also found that certain resonant converters are able to operate with low EMI [4].  Because of these advantages, resonant converters with  switching frequencies in the range of MHz are conceivable [2][5]. Some typical resonant topologies include the series resonant, parallel resonant, and the seriesparallel resonant converters.  1.2 Motivation Because of the demand for resonant conversion, methods on how to design eective feedback control systems for resonant converters becomes a topic of considerable interest. To make a converter valuable for practical system applications, and to achieve specications such as output voltage regulation, it is necessary to adopt some form of feedback control. Furthermore, with the advancements in digital signal processors, digital control techniques have become a feasible option. The application of digital controllers has allowed for more exible designs when compared to analog controllers, and allows for much greater reliability and system integration. Consequently, this thesis will be centered around designing and implementing a digital control system for a resonant power electronics converter topology to be used in a medium power application. Chapter 2 will discuss in detail a resonant converter topology, followed by discussion on implementing a digitally controlled negative feedback control loop for output voltage regulation in Chapters 3 and 4. Lastly, a novel mathematical model of the small signal control-to-output transfer function is presented in Chapter 5.  2  2 LLC Resonant Converter Resonant power converters contain L-C networks, or resonant tanks, whose voltage and current waveforms vary sinusoidally during one or more subintervals of each switching period [6]. Three well-known resonant topologies include the series resonant, parallel resonant, and series-parallel resonant. Although there are peculiar dierences in each of these topologies, the essential operation is the same: a square pulse of voltage or current is generated, and applied to the resonant tank circuit. Energy circulating within the resonant tank will then either be fully supplied to the output load, or be dissipated within the tank circuit [2]. As documented in [7], the series and parallel resonant topologies shown in Figure 2 have several limiting factors which make them the non-ideal choice for practical applications.  Lr  For the series  resonant converter topology, light load operation requires a very wide range of switching frequencies in order to retain output voltage regulation.  It was also observed that for high input voltage  conditions, the series resonant converter suers from high conduction losses, and the switching  Lr  network transistors experience high turn-o current. Vsquare(t) V(t) Cp Compared to the series resonant converter, the parallel resonant converter topology does not require a wide range of switching frequencies to maintain output voltage regulation. However, at  Cp  V(t) input voltage conditions, the parallel resonant converter shows worse conduction losses, and high higher turn-o currents.  Lr  Lr Cr  Lr Vsquare(t)  Vsquare(t)  Cr  V(t)  Cp  Vsquare(t)  Cp  V(t)  V(t)  (a) Series resonant converter  (b) Parallel resonant converter  Figure 2: Resonant tank circuits of resonant converter Ltopologies r  Cr One possible solution to overcome the deciencies found in the above two converters is the seriesparallel or LCC (inductor-capacitor-capacitor) converter shown in Figure 3. Since it is known that  V(t) the operation around the resonant frequencyVhas square(t)the greatest eciency, it is desired to operate the converter around this operating point [7].  3  Lr  Lr Cr  Vsquare(t)  Vsquare(t)  V(t)  Cp  Lr Cr Figure 3: LCC resonant tank circuit schematic  Figure 4a plots the DC gain characteristic of the LCC resonant converter with dierent values of the variable  Q,  and it can be seen in Figure 4a that more than one resonant frequency exists,  depending on value of  Vsquare(t)  Q. Q  is dened as the ratio between the characteristic impedance and the  output load and can be given by Equation 1 [8].  q Q= where  R  Lr Cr  (1)  R  is dened as the value of the output load resistance.  Furthermore, from the DC gain characteristic of Figure 4, it is understood that the segments of the DC gain characteristic with a positive gradient are regions intended for zero-current switching (ZCS) operation [7]. Given that the designed converter is to use metal-oxide eld eect transistors (MOSFET) as the semiconductor switching device, the desired region of operation should therefore be on the negative gradient, a region intended for zero-voltage switching (ZVS) operation [7]. This is because the preferred soft-switching mechanism of MOSFET devices is ZVS. As outlined in [6], ZVS mitigates the switching loss otherwise caused by diode recovery charge and semiconductor capacitance often found in MOSFET switching devices. Since it is known that operation on the negative gradient of the DC gain characteristic is desired, the characteristics of this operating region should be observed. As an example, it can be seen in Figure 4a that to achieve a gain value of 1.0 for increasing values of  Q,  the range of the ratio of the  fs switching frequency to the resonant frequency fr varies from 0.7 to 1.2. Therefore, it can be noted that a fairly large range of switching frequencies is required to maintain a gain value of 1.0 when the output load is changing.  4  Lr  Lr Rr  Rr  Cr  V(t)  V(t)  LM  Cr  LM  Vsquare(t)  Rac  (a) LCC  V(t)  (b) LLC  Figure 4: Comparison of LCC and LLC DC gain characteristic with varying Q-factors  DC With this understanding of the LCC converter characteristics, the LLC Vsquare (t) V(t) Co (inductor-inductorRo Vo(t)  capacitor) resonant converter shown in Figure 5 becomes a potential solution. Essentially the dual of the LCC converter, the DC gain characteristics of Figure 4a are reversed in the LLC resonant converter and are shown in Figure 4b.  Lr Rr  Cr  DC  Co  LM  Ro Vo(t)  Figure 5: LLC resonant converter circuit schematic  b0  e(k)  +  +  u(k)  From Figure 4b, it can be seen that the operation at around the resonant frequency, the operation is in a region such that the DC gain characteristic has a negative gradient and therefore, ZVS  -1 region. capabilities can be achieved in this  z  -1 Furthermore, by observing the DC characteristic of  z  Figure 4b, and what was noted about the switching frequency range of the LCC, it can be seen that in the LLC resonant converter, the gain value of 1.0 can be achieved for all loading conditions  b  +  within a very narrow range of switching1 frequencies.  +  -a1  An additional benet of the LLC DC gain characteristic is that the resonant frequency of Figure 4b has now shifted to a higher switching frequency in comparison to the equivalent ZVS region resonant frequency of Figure 4a, and thus the potential for improvements in eciency and  z-1  z-1  b2  -a2  5  Rac Vo(t)  power density are greatly improved. Some other advantages of the LLC resonant converter over other resonant topologies is its ability to maintain ZVS characteristics under light load conditions, and low electromagnetic interference (EMI) [4]. Since it has now been determined that the LLC resonant converter has many favorable features for DC-DC conversion applications, Sections 2.1 - 2.4 will discuss each stage of the LLC resonant converter, as well as the theory of operation.  2.1 H-bridge Inverter The rst stage of the LLC resonant circuit is the H-bridge inverter. It is composed of four triodemode MOSFET transistors which are used to invert the input DC voltage to an AC sinusoidal waveform. The use of MOSFET switches are preferred since they have high input impedance and can operate at very fast switching speeds [9]. And as previously discussed, for eciency purposes, the LLC resonant converter utilizes ZVS to eliminate the switching losses of the MOSFETs. Finally, the frequency at which the switches are turned on and o will determine the frequency to be applied to the resonant tank. To be more precise, the H-bridge generates a square wave with frequency  fs  equal to the fre-  quency of the MOSFET switching. This quasi-square wave is established by switching on diagonally placed switches at the same time to generate the high and low values of the square-wave waveform. It can also be noted that to prevent a shoot through condition, a small dead band time can be included between the turn-on and turn-o times of the diagonal switches. Finally, the quasi-square wave output can be mathematically characterized by Equation 2 and the basic circuit conguration of the H-bridge inverter can be given by Figure 6.  vsquare (t) =  ∞ X  4Vg sin(n2πfc t) nπ n=1,3,5,...  (2)  6  V(t)  V(t)  Rac  LM  DC  Vsquare(t)  V(t)  Figure 6: H-bridge inverter circuit schematic  Lr  2.1.1 Zero Voltage Switching  Rr  Cr  Zero voltage switching (ZVS) is the preferred soft-switching mechanism for MOSFET devices, as it mitigates the switching loss caused by diode recovery charge and semiconductor output capacitance [6]. In [10], switching loss is dened as the simultaneous overlap of voltage and current in power  DC  LM  MOSFET switches. It is shown in [10] that by allowing the drain-to-source voltage switch turns on, the switching power loss is made to be zero. between  Vds  and the current  ids ,  Vds  to reach zero before the  However, when there is overlap  a non-zero loss can be observed. To allow  Vds  to reach zero, the  internal capacitance of the MOSFET must be discharged by reversing the direction of the current ow through the MOSFET. In general, ZVS occurs when the switching network is presented with an inductive load, and hence, the switch voltage zero crossings lead the zero crossings of the switch current [6].  In the  case of the LLC resonant converter, ZVS operation of the H-bridge inverter switching devices is achieved and maintained by the presence of the magnetizing inductance [3][11].  b0  +  An additional benet of ZVS is the reduction of electromagnetic interference (EMI) typically  e(k)  associated with switching device capacitances [6].  z-1 7  b1  +  2.2 Resonant Tank The resonant tank circuit is the chief constituent of the LLC resonant converter, and is comprised of series resonant inductor  Lr ,  series resonant capacitor  Cr ,  and a parallel resonant inductor  LM .  There are numerous possible congurations in which the tank components can be arranged, but the most frequently used arrangement found in literature is the connection of inductor  LM  Lr  and  Cr  in series and  in parallel to the load. This arrangement is identical to the series resonant topology  with the addition of inductor  LM .  For a more complete model of the resonant tank, a series resistor  can also be included.  Lr Rr  Cr  LM  Vsquare(t)  Rac  V(t)  Figure 7: Resonant tank circuit schematic  The no-load transfer function of the LLC resonant tank circuit shown Figure 7 is given by Equation 3  H(s) = V(t)  C  s2 (LM Cr ) s2 Cr (LM + Lr ) + sCr Rr + 1  (3)  Rac V (t)  R  o Well-known relationships between impedance and frequency are given by Equation 4. o o Vo(t)  ZL = jωL (4)  ZC =  1 jωC  From these fundamental equations, it is shown that the impedance of the resonant tank can be tuned according to the frequency applied to the tank circuit. It can then be said that by varying  Lr  the impedance of the resonant tank, that the voltage gain seen at the output will dier, depending  8  LM  Co  Ro Vo(t)  on the frequency applied to the resonant tank.  It is then concluded that this voltage gain will  determine the attainable output voltage value, and thus is the method in which output voltage regulation can be achieved with the LLC resonant converter.  2.2.1 Resonant Frequencies ωr1 ,ωr2 From the values of  ωr2 .  and  LM ,  two important operating conditions can be identied:  ωr1  and  These operating points are dened as the resonant frequencies, and are dened by the applied  loading condition.  LM  Lr , Cr ,  From Figure 7, it can be seen that at the no-load condition, the inductance  is seen by the tank circuit as a passive load and thus, the resonant frequency can be given by  Equation 5.  ωr2 = p  1  (5)  (Lr + LM )Cr  In the case where a nominal load is applied, the load seen by the resonant tank is eectively the large output capacitance  Co  in parallel with inductance  LM .  Thus, the inductor  LM  is bypassed  by the eective AC short circuit and the resonant frequency can be given by Equation 6.  ωr1 = √  1 Lr Cr  (6)  In this thesis, operation is focused around the resonant frequency given by Equation 6, as this operating point allows for greater eciency, and allows for regulation using only a narrow range of switching frequencies.  2.2.2 High Frequency Isolation Transformer Following the resonant tank circuit is a high frequency isolation transformer that can be used to either buck or boost the sinusoidal voltage to the secondary side of the converter. The transformer also serves the dual purpose of providing galvanic isolation, such that no direct current ows between the input and output. A rather remarkable feature of the transformer is that the magnetizing and leakage inductances can be used as part of the tank circuit. For instance, the magnetizing inductance of the transformer can be used as or part of the parallel resonant inductance  LM ,  therefore potentially reducing the  number of additional discrete components required. Similarly, the leakage inductance can be made a part of the series inductor  Lr ,  depending on the design of the transformer parameters. This so-  called integrated magnetic can therefore be designed to serve the purpose of potentially increasing the converter's power density. The transformer ratio between the primary and secondary sides is given by the transformer turns ratio and shown by Equation 7.  9  N=  Nsecondary Nprimary  (7)  2.3 Rectier  L  r The last stage of the LLC resonant converter is the full bridge rectier with capacitor output lter, r r which transforms the AC waveform to DC output. Similar to what was found in [6] and [12], the  R  C  full bridge rectier with capacitor lter is modelled as resistor  Rac .  The relationship between  Rac  and the output load is given by  V(t)  LM  Vsquare(t)  Rac Rac  Rac =  LM  VI 8Vo 8 = 2 = 2 Ro Iac π Io π  V(t) (8)  can then be reected onto the primary side by multiplying Equation 8 by the transformer  turns ratio given by Equation 7.  are(t)  V(t)  Co  Rac Vo(t)  Ro Vo(t)  Figure 8: Rectier circuit modelled as  Lr Rr  Cr  Rac  From [6], the nal DC value of the output voltage and the output voltage ripple can also be  determined by the following relationships.  Vo,dc = Vp (1 −  1 ) 2fs Co Ro  C  (9)  R  o o V (t) For the case of an M ideal full bridge rectier that has negligible ripple,o the switching frequency  L  fs  and output capacitance  Co  are large such that the DC output voltage can be approximated by  10  b0  u(k)  Equation 10.  Vo,dc ≈ Vp where  Vp  is the peak value of the AC input voltage,  (10)  V (t).  2.3.1 Synchronous Rectication To further increase the eciency of the LLC resonant converter, a synchronous rectication (SR) network can be implemented in lieu of the full bridge diode rectier. In a synchronous rectier, the diodes are replaced with MOSFET devices, and when current ow is detected on the secondary side, the MOSFETs are turned on to allow current ow to the load. This is a much more ecient strategy in comparison to the diode rectier, as the voltage drop across the diode is eliminated. Additionally, unlike diodes, which are only able to provide unidirectional ow of power, a SR network makes bidirectional power ow between the input and output feasible. The disadvantage of implementing SR with the LLC resonant converter is the increased complexity. Computer simulations of Figure B-1 found in Appendix B show the presence of a phase delay between the voltage and current when the switching frequency is away from the resonant frequency. The results of the simulation are shown in Figure 9. Because of this, it is dicult to determine the timing of the SR turn-on and turn-o. Usually, additional detection circuitry is required on the secondary side to determine when there is current ow through the rectier, and to determine the control signal to the gates of the SR MOSFETs. Furthermore, since MOSFETs do not have the ability for automatic reverse current blocking, the timing of the turn-o is also important, such that shoot through conditions are avoided. Some literature has been produced on the control of the SR gate drive signals in [13][14][15]. It is known that the critical event for triggering the SR gate drive signal is the detection of current in the secondary side. It was outlined in [15], that there are two main methods in which this can be accomplished. The rst is to directly sense the current using a current transformer. Although this may be the simplest method of detection, the use of a current transformer introduces limitations of the power density as well as the maximum achievable switching frequency.  Additionally, the  increased series inductance is detrimental to current commutation in the synchronous rectication switch network [15]. Another possible method is to detect the drain-source voltage, switches. The sensed value of and turn-o time of the SRs.  Vds  Vds  of the synchronous rectier  is then processed by control circuits to determine the turn-on  This approach can be relatively easily veried in simulation, but  the diculty of this method is determining a method in which  Vds  can be accurately measured.  Because there exists a package inductance from the MOSFETs in the SR, the measured value of  11  C:\Users\Brian\Documents\PSIM\difference_eq_CL_May17.smv Date: 05:07PM 09/27/12 Vtank  Itra  150  100  50  0  -50  -100  -150  C:\Users\Brian\Documents\PSIM\difference_eq_CL_May17.smv (a) Voltage and current waveforms in phase when Vtank  fs = fr1  Date: 05:08PM 09/27/12  Itra  60  40  20  0  -20  -40  -60  (b) Voltage and current waveforms out of phase when  fs 6= fr1  Figure 9: SR phase delay  12  Vds  becomes highly deviated if the package inductance is not properly considered. If the physically  sensed value is far deviated from the true value of  Vds ,  a false trigger of the SR circuits may occur,  and the potential for shoot through condition increases. Some literature proposed by [13][14], have shown a number of methods to improve the accuracy of sensing  Vds .  This is at the expense of circuit complexity, as several additional compensating  components are required. A computer simulation using a current-sensing method has been proposed and is shown in Figure 10. Vout  V  Iload  A Vmos1  Vmos3  Vmos1  A  Imos1  Isec  V Vmos1 Vmos3  Imos3 Vmos3  V Vmos3  Vmos1  Figure 10: Synchronous rectication PSIM  R  model  Title Designed by Revision  Page 1 of 2  13  2.4 Theory of Operation The fundamental operation of the LLC resonant converter is based on applying a voltage with frequency  fs  to the resonant tank to vary the impedance of the tank circuit, thus controlling the  achievable gain seen at the output. From Figure 11, it can be seen that as the switching frequency is increased, the output gain is decreasing. This implies that as the switching frequency is increased, the impedance of the resonant tank is increased, and therefore, a larger voltage drop is observed across the resonant tank circuit components, and less voltage is transferred to the load. The reverse is true as well, such that as the switching frequency decreases, the DC gain increases, implying that the voltage drop across the tank circuit has decreased, and the output voltage gain can be increased. In the following sections, the operation of the LLC resonant converter is more closely considered, and a single switching interval is analyzed. There are three main regions of operation, commonly described as Region 1, 2, and 3. Operation in Region 1, 2, and 3 is determined by the location of the operating point on the DC gain curve shown in Figure 11. Region 1 and 2 are located on the negative gradient of the DC gain curve, and Region 3 is located on the positive gradient. Region 1 is the set of switching frequencies greater than the resonant frequency, while Region 2 is the set of switching frequencies below the resonant frequency.  Figure 11: Regions 1, 2, and 3  Depending on the design specications and requirements that are desired, operation in any of these regions is possible. As it is preferred to switch the MOSFET devices under ZVS conditions, the focus will be on operation in Regions 1 and 2, which is graphically shown in Figure 11 as the negative gradients of the DC gain curves.  14  2.4.1 Region 1 In Region 1, the magnetizing inductance  LM  from the transformer does not resonate with the other  tank circuit components, and is viewed as a passive load by the resonating series inductor and series capacitor. Because there is always this passive load, it is possible for the LLC resonant converter to operate at no load without having to force the switching frequency to very high levels. The passive load also ensures ZVS is achieved for all loading conditions [7]. Region 1 operates at the resonant frequency given by Equation 6.  2.4.2 Region 2 In Region 2, there are two distinct operating modes.  In the rst time sub-interval, the series  inductor and series capacitor resonate together while the magnetizing inductance is clamped to the output voltage, and has operation similar to that of Region 1. Therefore, the resonance only occurs between  Lr  and  Cr .  The current in the resonant tank  increase until the magnetizing current  LM  starts to resonate with  resonating with the  Lr  and  Lr  and  Cr ,  IM  Cr ,  and  Ir  Ir  then begins to increase and continues to  are the same. When the two currents are equal,  and the second sub-interval begins. Since  LM  is now also  the resonant frequency is given by Equation 5.  In Region 2, since multiple resonant frequencies are observed over one switching period, the LLC resonant converter is considered to be a multi-resonant converter. Figure 13 shows the plots of the LLC resonant converter operating in Region 2. Operation in the second sub-interval begins during the time interval in which the output current is equal to zero. Figures 12 and 13 show the plots of the gate-source drive signal the magnetizing current  ILM ,  the capacitor voltage  VCr  Vgs ,  the resonant current  and the output current  Io .  Ir ,  Figure B-2 in  Appendix B shows the labelled circuit schematic used to obtain the plots of Figures 12 and 13.  15  C:\Users\Brian\Documents\PSIM\difference_eq_CL_May17.smv Date: 05:22PM 09/27/12 Vgs1 1 0.8 0.6 0.4 0.2 0 I_r  I_LM  100  50  0  -50  -100 V_Cr 40  20  0  -20  -40 I_o 20 15 10 5 0 -5 0.00055  0.00056  0.00057  0.00058  0.00059  0.0006  Time (s)  Figure 12: Operation of LLC resonant converter in region 1 16  C:\Users\Brian\Documents\PSIM\difference_eq_CL_May17.smv Date: 05:23PM 09/27/12 Vgs1 1 0.8 0.6 0.4 0.2 0 I_r  I_LM  200  100  0  -100  -200 V_Cr 100  50  0  -50  -100 I_o 50 40 30 20 10 0 -10 0.00055  0.00056  0.00057  0.00058  0.00059  0.0006  Time (s)  Figure 13: Operation of LLC resonant converter in region 2 17  3 Control of LLC Resonant Converter When correctly applied, the application of control theory and feedback can eliminate steady state errors, moderate system sensitivity to parameter changes and disturbances, modify the gain or phase of the system over a desired frequency range, and make unstable systems stable. For this thesis, the objective is to design a voltage control loop to regulate the output voltage, according to a predened reference voltage. In other words, regardless of disturbances to the system, and more specically, the load; the output voltage is to remain at a constant value. Consequently, a negative feedback loop was designed to achieve these specications.  Disturbances  r  Reference Input  + -  e  Compensation Network G(s)  Actuator (VCO)  Switching Converter P(s)  Vo  y  Sensor Gain  Figure 14: Block diagram of negative feedback loop  From Figure 14, it can be seen that the function of the negative feedback control loop is to determine an input to the plant, such that the desired output behavior can be obtained. Specically, for the LLC resonant converter, the objective is to generate a set of gate driving signals of frequency  f  r  Reference Inputfor the primary side switch network. s  Figure 14 shows that by sensing the output y and comparing it to reference input r , the comDigital e u Actuator Digital-to-Analog Plant Compensator (VCO) signals based on Converter P(s) pensator network G(s) can generate the gate drive the given error signal e. The D(s)  + -  error signal is the y dierence between the  y  and  r,  and in a functional control loop, tends to zero  after some time period.  3.0.3 Variable Frequency Control  Analog-to-Digital Converter  Sensor Gain  Under normal operating conditions, the LLC resonant converter uses variable frequency control to regulate the output voltage. This type of control requires a gate drive signal that has constant duty cycle, but varying frequency.  18  To implement variable frequency control, an actuator that can produce a variable frequency signal is required and can be realized by a voltage controlled oscillator (VCO). The VCO is an electronic circuit designed to produce an oscillation frequency based on the control voltage  Vctrl  and can be implemented as an analog circuit or with a digital signal processor. The relationship between the input signal  Vctrl  and the output frequency signal  ωo  is shown in Equation 11.  ωo = ω̂ − KV CO Vctrl In Equation 11,  ω̂  (11)  is the free-running frequency of the VCO, and  KV CO  is the gain of the voltage  controlled oscillator.  3.0.4 Pulse Width Modulation Disturbances  Pulse-width modulation (PWM) is a xed frequency control method, and modies the duty cycle of the pulses to regulate the output voltage. It has been suggested in the literature [16][17] that for r  Reference Input  light and no load conditions that it may be more eective to control the LLC resonant converter  + -  e  V  Compensation  Switching work, Converter by using PWM rather than variable frequency control.Actuator However, in this it will be oassumed Network (VCO)  G(s)  P(s)  that the converter always operates under nominal loading conditions, and therefore only requires y  the use of the variable frequency control method.  3.1 Digital Control  Sensor Gain  Digital controllers have many advantages over analog designs. Digital controllers can be designed to be more robust, and can be easily manipulated for optimal control performance. Additionally, the recent decreases in the cost and size of programmable micro-controllers and digital signal processors (DSP) has made digital control a viable option for power electronics applications.  Reference Input  r  + -  e  Digital Compensator D(s)  u  Actuator (VCO)  Digital-to-Analog Converter  Analog-to-Digital Converter  Sensor Gain  Plant P(s)  y  Figure 15: Block diagram of digital negative feedback loop  19  3.1.1 Eects of Sampling Frequency An essential consideration that is relevant to digital systems is the principle of sampling, a non-zero timed event to capture the continuous-time data. Sampling is required to convert continuous-time data to discrete-time for processing in the digital signal processor and is physically realized with an analog-to-digital (A/D) converter. The aim of the A/D converter is to accept measured signals from the output of the power electronic converter and then convert these signals into an electrical voltage level that can be read by the DSP. Conversely, once the processing has been completed in the DSP, a digital-to-analog (D/A) converter is used to produce a physical signal to be read by the power electronic converter. A crucial factor in digital design is the sampling rate at which the continuous-time signal is sampled. Ideally, the sampling rate is innite such that the discrete-time system is equivalent to the continuous-time model.  Unfortunately, since this is not a realistic solution, the alternative  solution is to apply the Nyquist rate shown in Equation 12.  fsample ≥ 2fmax  (12)  The Nyquist rate states that the sampling frequency must be at least twice the maximum bandwidth in the system. By satisfying this criterion, aliasing and signal distortion in the reconstructed signal can be avoided. It can be added that, in fact, it is recommended in [6] to make the sampling frequency ten times the maximum bandwidth.  3.1.2 Design of Digital Control Systems Digital controllers can be implemented using two primary methods: emulation and direct digital design. Digital controllers designed through emulation method use the continuous-time plant model, and obtain the compensator model in continuous-time. The continuous-time s-domain model is then converted to a discrete-time z-domain compensator by applying mathematical transformations. Numerous methods to transform continuous-time transfer functions to discrete-time transfer functions exists.  Some typical methods include the Tustin method, zero-order hold, rst-order  hold, etc. From [18], it was found that for the most accurate models, the rst-order hold is the most accurate discretization method, but at the expense of computational complexity.  A more  general method that is typically used is the zero-order hold discretization as it provides a balance of accuracy and computational eciency. The advantage of the emulation method is that well-developed and familiar continuous-time design methods can be applied. This method produces reasonably accurate results given that the sampling rate is very fast. The main disadvantage of the emulation method is that after mapping  20  the compensator from continuous to discrete-time, it is possible that the control system can no longer achieve the same performance characteristics as in the continuous-time domain model [18]. The other method which can be used to design a digital compensator is by using the direct digital design methodology. Compensators implemented using the direct digital design technique usually have signicantly better performance when implemented on digital signal processors. This method of design immediately begins with a plant model already in discrete-time, and then the compensator is designed based o the digitized plant model in the z-domain [19]. Thus, the probability of obtaining unexpected controller response and dynamics are mitigated when the controller is implemented using a DSP.  3.1.3 2-Pole-2-Zero Compensator Irrespective of which compensator design method is used, the end goal is to design a compensator  G(s)  to supply the correct signals to maintain output voltage regulation.  Loop compensation is  achieved by the placement of additional poles and zeros to the feedback system, in conjunction with the system's natural poles and zeros. The system loop gain can then be shaped to the desired performance and stability characteristics by the new poles and zeros introduced to the system. An example of compensator in the digital domain is the 2-pole-2-zero compensator (2P2Z). The 2-pole-2-zero compensator was selected as the compensator of choice as it can be simply implemented in the z-domain as a dierence equation. Its transfer function is given by Equation 13.  H2p2z (z) =  bo + b1 z −1 + b2 z −2 1 + a1 z −1 + a2 z −2  (13)  Determining the roots of the quadratic numerator and denominator of Equation 13 gives the location of the zeros and poles of the compensator respectively. Compared to P, PI, or PID controllers, the 2P2Z compensator allows for ve degrees of freedom, and allows for the use of complex poles and zeros [20]. In fact, the P, PI, and PID controllers are simply particular cases of the 2P2Z. For example, the PID controller uses the assumption that the coecients  a1 = −1  and  a2 = 0. HP ID (z) =  bo + b1 z −1 + b2 z −2 1 − z −1  (14)  Furthermore, for implementation in the digital signal processor, the 2P2Z compensator transfer function can be easily re-written in the form of a dierence equation.  u(k) = bo e(k) + b1 e(k − 1) + b2 e(k − 2) − a1 u(k − 1) − a2 u(k − 2)  (15)  A functional block diagram of the 2P2Z is shown in Figure 16.  21  DC  Co  LM  b0  e(k)  +  Ro Vo(t)  +  u(k)  z-1  z-1  b1  +  +  -a1  z-1  z-1 b2  -a2  Figure 16: 2 pole 2 zero DSP implementation  3.2 Compensator Design Two popular methods in which a compensator can be designed for a feedback system are the Bode diagram and root locus.  3.2.1 Root Locus The root locus method is a plot of all system poles of the closed loop transfer function as some parameter of the system is varied.  This design method relocates the closed-loop poles to meet  performance specications such as overshoot, rise and settling times. This method is ensures that acceptable transient response characteristics are reached, as well as robust compensator design [21]. A disadvantage of the root locus method is that the system under test must be able to be approximated as a second order system.  3.2.2 Bode Diagram The Bode diagram displays the magnitude and phase response of a feedback system as two separate plots with respect to the logarithmic frequency [22]. The Bode diagram is useful since the multiplication of transfer functions simply become problems of summation. The phase plot is generally  ◦  given in degrees ( ) and the magnitude plot is given in decibels (dB) scale. The conversion of magnitude to magnitude in decibels is given by Equation 16.  XdB = 20log10 |X|  (16)  22  The performance of compensators designed using the Bode diagram method are based on the bandwidth, gain margin and phase margin [21]. In the Bode diagram, a pole is represented by a 20 dB per decade decay, and a zero is a 20 dB per decade increase.  3.3 Stability To verify that the designed compensator and plant form a stable system, the stability of the closed loop feedback system can be studied in either the continuous-time domain or the discrete-time domain. The dierences in characteristic between the continuous-time and discrete-time domains are discussed in Appendix A. In this thesis, the stability of the control system will be observed using the Bode and Nyquist diagrams.  3.3.1 Bode Diagram The stability of the feedback system can be determined from the Bode diagram by observing the gain and phase margins. crossover frequency  fc .  ◦  The phase margin is dened as the phase dierence from -180  at the  The crossover frequency is dened as the instance at which the magnitude  plot is at unity. The phase margin can be calculated using Equation 17.  P.M. = 180 + 6 P (jωcrossover )  (17)  The gain margin is the point on the magnitude plot that corresponds to the point on the phase  ◦  plot at which the phase crosses the -180  axis.  Typically, the target phase margin is between 45  ◦  ◦   60 , and can be negotiated depending on the  requirements for the transient settling time and the stability. For the gain margin, it is generally accepted that good gain margin is greater than 9 dB [23].  3.3.2 Nyquist Plot Alternatively, the stability of the system can be observed using the Nyquist plot.  The Nyquist  plot displays the frequency response as a single plot in the complex plane, and is a graphical representation of the loop transfer function as  jω  traverses the contour map [24].  Stability in the Nyquist plot can be observed by applying the Nyquist stability criterion. This criterion is determined by examination of the enclosure around the critical point  (−1, 0).  For closed  loop stability, the Nyquist plot must encircle the critical point once for each right hand pole, in a direction that is opposite to the contour map [24].  23  It would be appropriate and useful to relate the Nyquist stability criterion to the Bode diagram magnitude and phase plots [25]. There are two main relationships that can be made:  •  The unit circle of the Nyquist plot is equivalent to the 0 dB line of the magnitude plot.  •  The negative real axis of the Nyquist plot is equivalent to the -180  ◦  phase line of the phase  plot.  24  4 Implementation and Verication In this thesis, the plant process is given by the ratio of the output voltage to the switching frequency, and is expressed in Equation 18.  Since it was determined that the direct digital design method  gives superior results, it is the selected method for the design of the compensator.  P (s) =  Vo fs  (18)  Consequently, the rst step is to obtain a description of the plant process model, in either continuous or discrete time, which can be discretized so that the compensator may be directly designed in the discrete-time z-domain. However, since the mathematical relationship described by Equation 18 is not well-dened in the literature, one proposed solution is to measure the frequency response of the LLC resonant converter using a network analyzer. The frequency response can be obtained by probing the output voltage, and the switching frequency while the switching frequency is undergoing small signal perturbations around a designated operating point. This methodology has the main advantage of capturing the true behavior and dynamics of the LLC resonant converter, as well as any additional behaviors that appear due to parasitic components. This measurement method provides the most realistic representation of the plant process model, however it assumes a prototype model has already been designed and is functional.  4.1 Design Methodology 4.1.1 Overview The approach taken to design the digital controller was to rst use a laboratory bench prototype model and network analyzer to obtain the frequency response data of the plant process model. Dierent loading conditions were then applied to the prototype model to observe the changes in the frequency response. Additionally, bench data of the DC gain characteristics under dierent loading conditions were obtained to compare with the theoretical models presented in Chapter 2. The obtained frequency response data of the physical plant model was then imported into the MATLAB MATLAB  R  .  R  environment, and based on the obtained data, a compensator was designed in  To verify that the compensator design is satisfactory, the designed compensator  coecients can be extracted from MATLAB  R  , and then exported to a software more suitable for  power electronics simulation. Finally, an example implementation using a Texas Instruments  R  -based digital signal processor  (DSP) is given.  25  4.1.2 Bench Test Results 1.03  1.02  1.01  1  Voltage Gain (V/V)  0.99  0.35A 0.98A  0.98  2.3A 2.92A  0.97  3.6A 4.6A  0.96  0.95  0.94  0.93 100  105  110  115  120  125  130  135  140  145  Switching Frequency (kHz)  Figure 17: Bench test result of DC gain characteristic for Region 1  Figure 17 shows the DC voltage gain characteristic of the prototype model when operated above the resonant frequency. From these results, it is evident that for light loading conditions, the LLC resonant converter begins to display non-linear characteristics, which suggests a separate control loop may be necessary for light load operation. For loading conditions approaching the nominal load, it appears that the prototype model follows the theoretical DC gain characteristics as given by Figure 4b. Figure 17 also shows that the boundary condition between nominal and light loading conditions for this particular converter lies around 2.3 A. Given that the resonant frequency is the quiescent operating point, Figure 18 shows the frequency response of the prototype model with dierent loading conditions.  Figure 18 plots the  magnitude of the ratio between the output voltage and control voltage in decibels (dB) with respect to the relative frequency in Hertz (Hz).  From these results, it was determined that the  frequency response of the plant model also has a large dependency on the loading conditions. Since there may be frequent changes to the load, the compensator design should be based on the plant process model that represents the worst-case scenario. For this thesis, it will be assumed that the converter is operated under nominal operating conditions, and thus, the worst-case is dened as the loading condition such that the phase margin is minimum, as this is the condition that is  26  Bode Diagram -10 1A 2.3 A 3A 3.6 A 4.6 A  -15 -20  Magnitude (dB)  -25 -30 -35 -40 -45 -50 -55 -60 180  Phase (deg)  90  0  -90  -180 10  2  3  10 Frequency (Hz)  10  4  Figure 18: Plot of relative frequency response of prototype model under dierent loading conditions  most likely to be overcome by problems of instability. In Figure 18, this scenario is represented by the curve `4.6 A.' It can be noted that the frequency response generally follows the form of a low-pass lter (LPF), with the addition of a high frequency pole-zero combination. Because this system appears to be greater than second order, design of the compensator using the Bode diagram technique is preferred over other design methods.  4.1.3 Compensator Design Results MATLAB  R  has several useful tools for digital controller development. Firstly, built-in functions  such as `c2d ' make it straightforward to convert continuous-time s-domain models to discrete-time z-domain models.  Secondly, the Control System Toolbox features the SISO Design Tool which  27  allows for simple, visual design of control systems.  Root Locus Editor for Open Loop 1 (OL1) 1  Open-Loop Bode Editor for Open Loop 1 (OL1) -10  2.5e4 3e4  0.8  2e4  -15 0.1 1.5e4  3.5e4  -20  0.2 4e4  1e4  0.4 0.5  0.4  0.6 0.7 4.5e4  0.9  -30 -35 -40 -45 -50 G.M.: 77.5 dB -55 Freq: 5e+004 Hz Stable loop  5e4 5e4  -60 180 P.M.: Inf 135 Freq: NaN  -0.2 4.5e4  5e3 90  -0.4  -0.6  -0.8  4e4  1e4  3.5e4  1.5e4 3e4  -1 -1  -0.5  45 0 -45 -90 -135  2e4 2.5e4 0 Real Axis  Phase (deg)  Imag Axis  0  5e3  0.8  0.2  Magnitude (dB)  -25  0.3  0.6  0.5  -180 1  10  2  Figure 19: Prototype model frequency response data in MATLAB  3  10 Frequency (Hz)  R  10  4  SISOTOOL GUI environment  In this case, the frequency response data of the plot `4.6 A,' a continuous-time domain function, is transformed into a discrete-time z-domain frequency response plot with the `c2d ' command and an appropriate sampling time. With the z-domain frequency response in hand, the `sisotool ' command opens the SISO Design GUI for interactive compensator design, and allows for controller design using root locus, Bode diagram, and Nichols techniques. Using the Bode diagram design editor, the `sisotool ' GUI environment allows the user to manually add and position the poles and zeros of the compensator according to the specications required for the control system [26]. The bandwidth, gain and phase margins, as well as the closed-loop step response can then be easily obtained from the GUI environment to verify the stability and performance of the designed compensator model.  28  Figure 20 shows the response of the open-loop system with the designed digital compensator.  ◦  The open loop response appears to be stable and has a gain and phase margin of 53.8 dB and 60 respectively.  Open-Loop Bode Editor for Open Loop 1 (OL1) 20  Magnitude (dB)  10  0  -10  -20  -30  G.M.: 53.8 dB Freq: 5e+004 Hz Stable loop  -40  Phase (deg)  -90  -135  P.M.: 60 deg Freq: 481 Hz -180 10  2  10 Frequency (Hz)  3  10  4  Figure 20: Open-loop Bode plot of prototype frequency response data  Given that a step change to the control voltage input is applied to the system, the closed-loop step response of the system is given by Figure 21, and shows a rise time of approximately 1 ms, and after 2.5 ms, the steady state error is reduced to zero. The overshoot of the step response is approximately 10%. From the results of Figure 20 and Figure 21, it appears that a stable compensator with appropriate gain and phase margin has been designed. Since these results appear to be reasonable, the designed compensator coecients can be exported to the PSIM  R  simulation model for verication.  29  Step Response 1.4  1.2  1  Amplitude  0.8  0.6  0.4  0.2  0  0  0.5  1  1.5  2 Time (sec)  Figure 21: MATLAB  R  2.5  3  3.5 x 10  -3  step response of closed-loop system using prototype model  4.1.4 Verication in PSIM R The negative feedback control loop of Figure 15 was implemented and simulated using Powersim's PSIM  R  simulation software, a software specically designed for power electronics simulation [27].  The simulation le includes a model of the LLC resonant converter in a negative feedback loop conguration. Additionally, to more closely model the eects of the DSP circuit, the sample-andhold, and quantization eects of the analog-to-digital converter are also included in the model. As seen in [28], the A/D converter can be approximately modelled as an ideal switch switching at frequency  fsample  and a zero-order hold (ZOH) block. It will be assumed that the sampling rate  is high enough such that the sampling can be considered ideal. Additionally, a quantization block been included to simulate the quantization error during typical A/D conversion [27]. The complete circuit schematic of the PSIM  R  simulation model is shown in Figure 22, and the  controller components are shown in Figure 23. The model of the 2P2Z compensator is implemented in the dierence equation format according to Figure 16, and the VCO actuator is designed according to Equation 11. In the simulation, the reference set point value of the voltage control loop was set to be 190 V. Figure 24 shows that the steady-state value of the error voltage approaches zero, and conrms that the controller meets the output regulation requirement.  30  C:\Users\Brian\Documents\PSIM\LOW2HI_FINALplay.smv Date: 11:49PM 10/09/12 Verror 3 2.5  Vout  V  Vmos1  V Vgs1  1.5  Vds1  Vmos3 MOS1  MOS3  Vmos1 MOS2  MOS4  Vc  Vgs3  Q1  V  Q3  Ires  Ip  A  1  Vg  Iload  A  2 Vgs1  A  A Isec  Vtank  Vm  Vsec  Vgs3 0.5  V  Vds3  Vgs3  0 -0.5  Q2  Vmos3  Vgs1 Q4  Isw  Imos2  Figure 22: PSIM  R  circuit schematic of LLC resonant converter  60  f_s  Verror  V  40  V r  H(z)  K  ZOH  K  K  sin  Vgs1  K  2P2Z_Compensator  Vgs3 Title  V_ref  Designed by  20  Revision  Page 1 of 1  Voltage Controlled Oscillator Error Signal Generator  0  -20  Figure 23: PSIM  R  controller schematic of LLC resonant converter  Verror 0.14 0.12 0.1 Title  0.08  Designed by Revision  Page 1 of 1  0.06 0.04 0.02 0 0.005  0.01  0.015  0.02  0.025  0.03  Time (s)  Figure 24: Error voltage of prototype model  31  Imos2 60  40 A study of the eects of a steady-state load step change are shown in Figure 26. A step load change in the simulation is applied at t = 15ms, and from the results of Figure 26, it has been veried  20  that the designed control system that has been implemented is functional and is able to maintain output voltage regulation even when undergoing changing loading conditions. switching frequency 0  fs  As expected, the  increases when the load was decreased. Figure 26 also shows the response  of the current through the resonant tank, and the switching frequency. For interest, the ripple voltage of the output voltage was also observed and was found to be  -20  approximately 1.4 mV and is shown in Figure 25.  Vout 190.0165  190.016  190.0155  190.015  190.0145  190.014 0.0295  0.02952 Time (s)  0.02954  0.02956  Figure 25: Output voltage ripple of prototype model  The results of the simulation are summarized in Table 1. Based on the nal results, it can be concluded that a sucient digital controller has been designed to achieve output regulation. Output voltage value:  190 V  Steady-state error:  0 V  Output voltage ripple  1.4 mV  % overshoot:  10%  Rise-time:  1 ms  Settling-time:  2.5 ms  Table 1: Results of prototype model  32  C:\Users\Brian\Documents\PSIM\LOW2HI_FINALplay.smv Date: 10:43PM 09/27/12 Vout  190  185  180  Ires 200  100  0  -100  -200  f_s 300K  200K  100K  0K 0  Figure 26: PSIM  0.01  R  0.02 Time (s)  0.03  0.04  closed-loop response to step load change using prototype model  33  4.2 Texas Instruments R DSP To implement the voltage control loop digitally, a DSP is required. Texas Instruments  R  (TI) has  a large portfolio of micro-controllers and digital signal processors available for use. In this thesis, the TMS320F28035 Piccolo was selected as the ideal digital signal processor as it has the capability to use both the on-board central processing unit (CPU) as well as the  TM (CLA) platform. The CLA has the advantage of minimizing  additional Control Law Accelerator  the processing latency of regular DSPs since it can execute time-critical control algorithms in parallel with the main CPU [29]. The voltage control loop can be implemented onto the CLA partition of the TMS320F28035 DSP using assembly language programming. To assist with the programming related to the CLA, the controlSUITE  TM package oered by TI provides many example code snippets for use.  Figure 27 shows an algorithm that can be used to implement the voltage control loop with the CLA.  34  VOLTAGE LOOP MCU DESIGN FOR UPS LLC Voltage Setpoint  Setpoint, max output voltage, 2p2z coefficients  2P2Z Digital Compensator  Voltage Controlled Oscillator  LLC Resonant Converter  Initialize Code  YES  Overvoltage Protection  NO  Calculate Error Voltage  Voltag  1) Initialize Code: · Define voltage setpoint, max output voltage · Define 2p2z coefficients · Initialize CLA processor · Disable primary PWM  Output Voltage ADC Read-in  Vo > Vomax  Voltage Divider  Disable PWM  End Process 2P2Z Compensator  Control Voltage Limiter  Voltage Controlled Oscillator  2) ADC Read: · Read output voltage (find average value?) · If (Voltage Value > Vomax) à OVP triggered and PWM disabled. 3) Voltage Control Loop (in CLA): · Calculate error Verror = (Vsetpoint – Vo) · Send error value to 2p2z compensator à 2p2z outputs VCTRL · Limit VCTRL between 6V – 8V. · Send VCTRL to VCO àVCO outputs fswitching 4) PWM Update/Enable: · Update bit.TBPRD in “Main” code. · Enable primary PWMs. 5) Restart: · Read new output voltage value · Loop steps 2 – 5.  Update PWM switching frequency/period  Enable Primary side PWM  Figure 27: Voltage loop ow diagram for DSP implementation 35  5 Derivation of Control-to-Output Transfer Function 5.1 Overview The control-to-output transfer function of the LLC resonant converter represents the ratio of the output voltage  Vo to small signal variations in the switching frequency fs .  The diculty in designing  compensators for the LLC resonant converter lies in the fact that there are few known examples in literature of the control-to-output transfer function, which in this case represents the plant process model of the control loop. Modelling the small signal characteristics of the LLC resonant converter is particularly dicult because many averaging techniques, such as state space models cannot be used. In [7] and [30], it is noted that unlike PWM converters, the control-to-output transfer of the converter cannot be obtained by the state space averaging methods, due to the way energy is processed in the LLC resonant converter. Currently, most known applications ([30][31]) rely on bench measurements and frequency response data to help determine the plant process model for the design of the compensation network. In practice, the transfer function can be obtained by using a network analyzer to measure the frequency response of the plant process model, and then using the data to design the compensator. This methodology has the inherent disadvantage of requiring a pre-built functioning prototype model. A proposed control-to-output transfer function that does not use frequency response data was presented in [32]. It was proposed that the control-to-output transfer function could be approximated as a third order polynomial. However, for the transfer function to be computed, variables such as the damping factor and the beat frequency of the converter need to be known. Additionally, a third order polynomial equation would not be able to predict the high frequency pole-zero shown in Figure 32. Finally, similar to the method proposed in [7], computer simulation can also be used.  The  diculty in creating a simulation model that can accurately model the true component and parasitic models may be dicult. With digital controllers, computer simulation is further complicated since the modelling of the eects of the DSP hardware can only be approximated. Additionally, as noted in [7], the computer simulation method requires extensive computing power.  5.2 Transfer Function Derivation In the proceeding sections, a novel model of the small signal characteristics of the control-to-output transfer for the LLC resonant converter will be presented. A MATLAB  R  -based software program  has been developed to determine the frequency response of the derivation results, and will be compared to the frequency response data obtained in Chapter 4.  36  DC Input  V x H(jω) Full Bridge Square Power Full Bridge Resonant Tank V x H(j(ω+ωm)) Rectifier with Wave Transformercontrol regulate the output As per Chapter 2, resonant converters that utilize variable frequency Converter H(jω) V x H(j(ω-ωm)) Capacitor Filter Vs(t) voltage by supplying sinusoids of dierent frequencies to the input of the resonant tank. The eect of this is to alter the tank impedance, and thus control the voltage drop across the resonant tank. Generating the appropriate sinusoid is accomplished by varying the switching frequency of the  Voltage Controlled primary side switch network.  Compensator  Oscillator  In order to develop a method to calculate the small signal control-to-output transfer function of the LLC resonant converter, techniques borrowed from communications theory can be used. More specically, the operation and analysis of the converter can be described by using analogies similar to that of frequency modulation (FM), as well as amplitude modulation (AM). The generalized block diagram shown in Figure 28 shows the steps of the analysis that will be taken for the development of the derivation model.  Square wave Voltage  Frequency modulation  ωc ωc +ωm ωc -ωm  Amplitude modulation H(jω)  Resonant Tank Voltage Output  Figure 28: Analysis road map  5.2.1 Frequency Modulation Recall that the general equation for a sinusoid is given by  v(t) = A cos (φ(t))  (19)  Frequency modulation is dened as a deviation in frequency from the carrier signal frequency. Mathematically, this deviation in frequency can be expressed as the addition of an additional cosine term to the carrier frequency [33]. The radial frequency of a signal undergoing frequency modulation can be expressed as  ωs (t) = nωc t + ∆ω cos(ωm t) In the above equation, and  ωm ,  (20)  ∆ω is dened as the amplitude of the deviation from the carrier frequency,  the modulating frequency, is dened as the rate of carrier deviation.  To arrive at an equation that can be substituted into Equation 19, the following relationship between the angular velocity  ωs (t)  and the angle  φ(t)  is used.  37  DC Output  Z φ(t) =  ωs (t)dt (21)  φ(t) = nωc t +  ∆ω sin(ωm t) ωm  In the scope of this thesis, it is known that the deviation is actuated by a voltage controlled oscillator and  ∆ω  can be substituted by  ∆ω = 2πKV CO  where  KV CO  (22)  is the gain of the voltage controlled oscillator, and the amplitude of the applied  small signal perturbation is given by the constant  .  Therefore, Equation 21 can be expressed as    KV CO  sin(ωm t) φ(t) = 2π nfc t + ωm To simplify Equation 23, the modulation index  β=  β  (23)  can be substituted and is dened as  2πKV CO  ωm  (24)  5.2.2 Square Wave Approximation The aforementioned mathematical relationships give a description of the eects of frequency modulation on the carrier frequency. However, prior to applying the above equations for use in analysis, the form of the original unmodulated signal must be determined. The aperiodic square wave waveform is supplied from a DC voltage source and is produced as a result of switching complementary pairs of eld eect transistor (FET) switches in the primary side FET bridge. Mathematically, a generic square wave signal undergoing frequency modulation is given as  Vs (t) =  ∞ X  4Vg sin(n2πfc t + β sin(ωm t)) nπ n=1,3,5,...  (25)  In [7], it is acknowledged that analysis including harmonics up to the fth harmonic may be useful in modelling the control-to-output transfer function of the LLC resonant converter. Computer simulations recorded in [7] show that by including the harmonic content, the accuracy of the models are improved.  38  The model proposed in this thesis includes the third and fth harmonics, in addition to the fundamental frequency of the square wave waveform. By including the third and fth harmonics in the model, a more accurate understanding of the true dynamics of the converter can be observed. Additionally, the eects of including supplementary sideband frequencies (for the case where does not meet the condition  |β| << 1)  |β|  are studied and included in the new model.  Equation 25 is therefore dened as  4Vg sin (ωc t + β sin(ωm t)) π 4Vg + sin (3ωc t + β sin(ωm t)) 3π 4Vg + sin (5ωc t + β sin(ωm t)) 5π  Vs (t) =  For simplicity, each individual harmonic component is redened as  (26)  Vs1 (t), Vs3 (t),  and  Vs5 (t).  5.2.3 Bessel Functions of the First Kind To gain better insight to the eects of frequency modulation, Bessel functions are used to determine the amplitudes of the carrier and resulting sideband frequencies. Expanding Equation 26 by using Bessel functions has the advantage of determining the locations of the modulated frequencies in the frequency spectrum, as well as giving the amplitude of each frequency component. As an example,  Vs1 (t)  can be expanded using Bessel functions and is given  by  Vs1 (t) = Jo (β)  4Vg 4Vg sin(ωc t) ± J1 (β) sin ((ωc ± ωm )t) π π (27)  ±J2 (β) Jn (β)  4Vg 4Vg sin ((ωc ± 2ωm )t) ± ... ± Jn (β) sin ((ωc ± nωm )t) π π  is denoted as a Bessel function of the rst kind and can be calculated by  ∞ X  (−1)m Jn (β) = m!Γ(m + n + 1) m=0 In Equation 28, the variable  n  is the  nth   2m+n β 2  sideband frequency, and  (28)  β  is the modulation index.  In lieu of Equation 28, there are existing tables developed in [34], which can be used to determine corresponding values of  Jn (β)  given the value of  β.  39  In [35], there is an assumption that the value of  |β|  is much less than one. This is normally a  useful assumption as the following simplications of the Bessel functions become valid.  J0 (β) ≈ 1 (29)  βn Jn (β) ≈ n!2n As can be seen in Equation 29, for values of  β  much less than one, higher order sideband  frequencies will be approximately equal to zero. However, since the assumption that  β  is much less than one is not always valid, the use of  Equation 29 is quite restrictive. Therefore, in this thesis, the assumption of than one is not used. Instead, the values of  |β|  |β|  being much less  are determined and calculated in MATLAB  R  with  the function `besselj'. This function is based on the tables found in [34]. Since the  |β|  value is no longer assumed to be small, the higher order values of  signicant. Therefore, the  n  th  Jn (β)  become  highest sideband to be considered remains to be determined.  It is known that the number of sidebands for the fundamental and harmonic components are determined by the value of the modulation index  β.  Theoretically, under frequency modulation,  there are an innite number of sideband frequencies, and therefore, a nite number of sideband frequencies must be selected in a way such that the majority of the signal's energy is captured. According to Equation 24, drawn that as  ωm  β  is inversely proportional to  ωm .  From this, the conclusion can be  increases, the modulation index becomes small, and the total number of sideband  frequencies is reduced. Therefore, it can be said that the "worst-case" is the scenario when such that the value of be located at  1000π  |β|  ωm  is at its smallest value,  becomes large. For the purpose of this thesis, the value of  rad/s. By setting  ωm  to  1000π  rad/s, a corresponding value of  β  ωm  is taken to  is determined,  and the number of sidebands required can also be determined. To verify that there are a sucient number of sideband frequencies, the sideband frequencies that account for 99% of the signal energy (in the pre-determined worst case) are considered. The energy of a carrier and sideband signals is determined by  E = Jo (β)2 +  x X  2(Jn (β))2  (30)  n=1 To determine how many sidebands are needed to have 99% of the signal energy, an increasing number of sidebands  x  is added to the carrier frequency amplitude until the value of  E  is equal to  0.99. It was determined that by including three pairs of symmetrical sideband frequencies to the fun-  40  damental, third and fth harmonic components, that 99% of the signal energy would be accounted for the case where  ωm  is greater than  1000π  Vs1 (t) = Jo (β)  4Vg 4Vg sin(ωc t) ± J1 (β) sin ((ωc ± ωm )t) π π  rad/s. Thus, Equation 27 can be written as  (31)  ±J2 (β)  4Vg 4Vg sin ((ωc ± 2ωm )t) ± J3 (β) sin ((ωc ± 3ωm )t) π π  5.2.4 Amplitude Modulation Each individual frequency component is then passed through and ltered by the resonant tank circuit at their respective frequencies.  This is equivalent to amplitude modulation (AM) and is  mathematically equivalent to multiplying each frequency component with the resonant tank transfer function at the corresponding frequency.  The resulting signal represents a voltage, that will be  eventually applied to the secondary side of the converter. As an example, the carrier, upper and lower sideband frequencies of the third harmonic undergoing amplitude modulation are given by Equations 32 - 38. Similar equations can be determined for the fundamental and fth harmonic components.  4Vg sin(3ωc t) × H(3j(ωc )) 3π  (32)  4Vg sin ((3ωc + ωm )t) × H(j(3ωc + ωm )) 3π  (33)  Vs3,carrier = J0 (β) ×  Vs3,upper1 = J1 (β) ×  4Vg sin ((3ωc − ωm )t) × H(j(3ωc − ωm )) 3π  (34)  4Vg sin ((3ωc + 2ωm )t) × H(j(3ωc + 2ωm )) 3π  (35)  Vs3,lower1 = −J1 (β) ×  Vs3,upper2 = J2 (β) ×  Vs3,lower2 = −J2 (β) ×  Vs3,upper3 = J3 (β) ×  4Vg sin ((3ωc − 2ωm )t) × H(j(3ωc − 2ωm )) 3π  4Vg sin ((3ωc + 3ωm )t) × H(j(3ωc + 3ωm )) 3π  Vs3,lower3 = −J3 (β) ×  4Vg sin ((3ωc − 3ωm )t) × H(j(3ωc − 3ωm )) 3π  (36)  (37)  (38)  41  The Euler formula is useful in this case, and is given by  sin(θ) =  ejθ − e−jθ j2  (39)  Applying Equation 39 to the trigonometric term of the resonant tank outputs, the above equations can be rewritten. As an example, the equations of the carrier and sideband frequencies for the third harmonic are shown in Equations 40 - 46. The equations for the fundamental and fth harmonic components can be determined similarly.  Vs3,carrier =  (41)  (Bl e−jωm t )ej3ωc t − (Bl e−jωm t )∗ e−j3ωc t j2  (42)  (Bu,2 ej2ωm t )ej3ωc t − (Bu,2 ej2ωm t )∗ e−j3ωc t j2  (43)  (Bl,2 e−j2ωm t )ej3ωc t − (Bl,2 e−j2ωm t )∗ e−j3ωc t j2  (44)  (Bu,3 ej3ωm t )ej3ωc t − (Bu,3 ej3ωm t )∗ e−j3ωc t j2  (45)  (Bl,3 e−j3ωm t )ej3ωc t − (Bl,3 e−j3ωm t )∗ e−j3ωc t j2  (46)  Vs3,upper2 =  Vs3,lower2 =  Vs3,upper3 =  Vs3,lower3 = Where  (40)  (Bu ej3ωm t )ejωc t − (Bu ejωm t )∗ e−j3ωc t j2  Vs3,upper =  Vs3,lower =  Bo ej3ωc t − Bo∗ e−j3ωc t j2  Bo , Bu , Bl , Bu,2 , Bl,2 , Bu,3 , Bl,3  are dened by Equations 47 - 53  4Vg H(j3ωc ) π  (47)  Bu = J1 (β) ×  4Vg H(j(3ωc + ωm )) 3π  (48)  Bl = −J1 (β) ×  4Vg H(j(3ωc − ωm )) 3π  (49)  Bu,2 = J2 (β) ×  4Vg H(j(3ωc + 2ωm )) 3π  (50)  Bo = J0 (β) ×  42  The resonant tank circuit  Bl,2 = −J2 (β) ×  4Vg H(j(3ωc − 2ωm )) 3π  Bu,3 = J3 (β) ×  4Vg H(j(3ωc + 3ωm )) 3π  Bl,3 = −J3 (β) ×  4Vg H(j(3ωc − 3ωm )) 3π  (51)  (52)  (53)  H(s) is given by Figure 29 and the s-domain transfer function is given  by Equation 54.  Lr Rr  Cr  V(t)  Rac  LM  Figure 29: Tank lter circuit schematic with equivalent resistance  H(s) =  s3 Lr Cr LM + s2 (Cr Rac LM  V(t)  Rac  s2 Cr Rac LM + LM Cr Rr + Lr Cr Rac ) + s(LM + Rr Cr Rac ) + Rac  In the above equations, the frequency domain variable  DC  s  is taken such that  is the quiescent operating point. This is done in order to work in  Vsquare(t) terms of  s = jωm ,  (54)  and  jωc  the relative frequency,  V(t)  rather than the absolute frequency. The individual mathematical representations of the fundamental, third and fth harmonic voltage outputs from the resonant tank have now been determined. The total output voltage of the resonant tank is then the sum of the harmonic output equations.  The complete output voltage  equation can therefore be written as  v(t) = ||A|| sin(ωc t + 6 A) + ||B|| sin(3ωc t + 6 B) + ||C|| sin(5ωc t + 6 C) Continuing with the previous given example, the vector  B  is therefore dened as  Rr  (55)  Lr  Cr 43  DC  LM  B = Bo + Bu ejωm t + Bl e−jωm t + Bu,2 ej2ωm t + Bl,2 e−j2ωm t + Bu,3 ej3ωm t + Bl,3 e−j3ωm t  (56)  Equation 55 represents the output voltage as a sinusoid having time-varying amplitude and also time-varying phase. Since the goal of this analysis is to determine the change in output voltage due to the eect of small variations to the switching frequency, the magnitude of Equation 55 must be evaluated. As was discussed, the eect of passing frequency signals through the resonant tank is the same as applying amplitude modulation (AM). Therefore, similar to AM analysis techniques, the envelope of the waveform should be extracted to study the change in the output voltage [33]. From AM analysis, it is known that the magnitude of the coecients  ||A||, ||B||, and ||C|| contain  the envelope of the signal. The square of the magnitude can rstly be determined by multiplying the vector with its complex conjugate. E.g. for  B,  the magnitude can be determined by  ||B||2 = BB ∗  (57)  To make use of some mathematical simplications, it is assumed that only the dominant DC components are relevant. If this is true, then the following approximation can be made.  p Thus, the magnitudes of  A, B,  and  C  1 1+ξ ≈1+ ξ 2  (58)  are determined to be  ||A|| = ||Ao || + ||Al || ||Ao A∗l + Au A∗o + Al A∗l,2 + Au,2 A∗u || sin(ωm t + 6 (Ao A∗l + Au A∗o + Al A∗l,2 + Au,2 A∗u )) ||Ao || ||Ao A∗l,2 + Au A∗l + Al A∗l,3 + Au,2 A∗o + Au,3 A∗u || + sin(3ωm t+ 6 (Ao A∗l,2 +Au A∗l +Al A∗l,3 +Au,2 A∗o +Au,3 A∗u )) ||Ao || ||Ao A∗l,3 + Au A∗l,2 + Au,2 A∗l + Au,3 A∗o || + sin(5ωm t + 6 (Ao A∗l,3 + Au A∗l,2 + Au,2 A∗l + Au,3 A∗o )) (59) ||Ao || +  44  ||B|| = ||Bo || + ||Bl || ∗ ||Bo Bl∗ + Bu Bo∗ + Bl Bl,2 + Bu,2 Bu∗ || ∗ sin(ωm t + 6 (Bo Bl∗ + Bu Bo∗ + Bl Bl,2 + Bu,2 Bu∗ )) ||Bo || ∗ ∗ ||Bo Bl,2 + Bu Bl∗ + Bl Bl,3 + Bu,2 Bo∗ + Bu,3 Bu∗ || ∗ ∗ sin(3ωm t+6 (Bo Bl,2 +Bu Bl∗ +Bl Bl,3 +Bu,2 Bo∗ +Bu,3 Bu∗ )) + ||Bo || ∗ ∗ ||Bo Bl,3 + Bu Bl,2 + Bu,2 Bl∗ + Bu,3 Bo∗ || ∗ ∗ + Bu Bl,2 + Bu,2 Bl∗ + Bu,3 Bo∗ )) (60) + sin(5ωm t + 6 (Bo Bl,3 ||Bo ||  +  ||C|| = ||Co || + ||Cl || ∗ ||Co Cl∗ + Cu Co∗ + Cl Cl,2 + Cu,2 Cu∗ || ∗ sin(ωm t + 6 (Co Cl∗ + Cu Co∗ + Cl Cl,2 + Cu,2 Cu∗ )) ||Co || ∗ ∗ + Cu,2 Co∗ + Cu,3 Cu∗ || + Cu Cl∗ + Cl Cl,3 ||Co Cl,2 ∗ ∗ + sin(3ωm t+ 6 (Co Cl,2 +Cu Cl∗ +Cl Cl,3 +Cu,2 Co∗ +Cu,3 Cu∗ )) ||Co || ∗ ∗ ||Co Cl,3 + Cu Cl,2 + Cu,2 Cl∗ + Cu,3 Co∗ || ∗ sin(5ωm t + 6 (Co Cl,3 + + Cu A∗l,2 + Cu,2 Cl∗ + Cu,3 Co∗ )) (61) ||Co ||  +  In Equations 59, 60, and 61, it is shown that each harmonic component has a DC and a small signal cosine term. It is known that the envelope function is represented by the amplitude of the small signal term, and this is the value of output voltage that is to be observed in the control-tooutput transfer function.  Vo f , the magnitude of the envelope voltage ex∆ω pression is divided by the small signal variation in frequency represented by 2π , as dened by Therefore, to get the transfer function  P (s) =  Equation 22.  5.2.5 Results of Frequency and Amplitude Modulation Figure 30 shows the frequency response of each harmonic component and shows the contribution of each harmonic after being ltered by the resonant tank circuit. The results of Figure 30 conrm the third and fth harmonic components have signicant contribution to the transfer function, and therefore it has been shown that they may not be neglected in analysis. The result of the summation of the fundamental, third and fth harmonic components is also shown in Figure 30. For comparison, Figure 31 is a plot of the fundamental harmonic component and shows the eect of including and not including the sideband frequencies in the derivation model. From the results of Figure 31, it can be seen that the sideband frequencies have a signicant eect on the resulting magnitude plot.  45  Figure 30: Comparison of the signicance between fundamental, third and fth harmonics  Figure 31: Comparison of the fundamental component, with and without sideband frequencies  46  5.2.6 Isolation Transformer and Rectication Stage The next stage of the LLC resonant topology is the high frequency isolation transformer, located after the resonant stage. To model the eect of the transformer on the control-to-output transfer function, the transformer turns ratio can be multiplied to the magnitude of Equation 55 in order to get an equation of the voltage seen on the secondary side of the converter.  vsecondary (t) =  N1 × v(t) N2  (62)  The frequency response data of Figure 18 shows that a pole-zero combination appears at high frequencies. It was determined that the cause of the zero is a result of the output capacitor equivalent series resistance, as well as the presence of a right hand plane zeros (RHPZ). In [36], it is stated that the RHPZ is a result of the inductor current not being able to instantaneously change, and is a function of the inductance and load [37]. The ESR zero is located at a xed frequency, and is given by Equation 63.  ωesr =  1 C o Rc  (63)  The ESR and RHPZ zeros are compensated by high frequency poles which were determined to be contributed by system delays [38]. A system delay in the s-domain can be given by  Hdelay (s) = e−sT  (64)  To evaluate Equation 64, the third order Padé approximation is used and the transfer function of the high frequency pole is given by Equation 65 [39].  e−sT ≈  60 − 24sT + 3(sT )2 60 + 36sT + 9(sT )2 + (sT )3  (65)  By combining all of the above eects presented with the results of Figure 30, the analysis nally gives the magnitude plot of the derivation model of the control-to-output transfer function. The results of the analysis can be compared to the frequency response data of Chapter 4 and are shown in Figure 32.  5.2.7 Results of Analysis of the Derivation Model Figure 32 compares the results of the derivation model and the frequency response data of the prototype model. From the results, it can be seen that the derivation model is a good approximation of the frequency response data, and therefore conrms the proposed model is adequate for modelling the small signal control-to-output transfer function. Therefore, it has been shown that by including  47  Figure 32: Comparison of prototype frequency response data and derivation model (magnitude)  harmonics up to the fth harmonic, as well as sideband frequencies in the derivation model, a reasonably accurate model of the small signal control-to-output transfer function can be achieved. From Figure 32, it is observed that at the lower frequencies, the error is largest, and there is approximately a maximum of 5 dB dierence between the frequency response data and the derivation model.  This non-conformity can be attributed to have not included enough of the  sideband frequencies in the model. As was discussed in Section 5.2.3, the derivation model only considers the sideband frequencies that appear under the condition  ωm = 1000π  rad/s. The equivalent value of this  ωm  in frequency  is at 500 Hz, and from Figure 32, it can be seen that this is approximately the point at which the discrepancies appear. Therefore, if the chosen value of  ωm  was smaller, more sideband frequencies  would have been added to the model, and the accuracy of the derivation model would improve for the low frequency region. The importance of the sideband frequencies is further shown in Figure 32, such that for frequencies greater than 2 kHz, the derivation model gives a very close approximation of the magnitude response of the bench prototype model.  5.2.8 Phase Response of Control-to-Output Transfer Function To gain an understanding of the phase response of the control-to-output transfer function, the asymptotic Bode tracing technique of [40] can be applied to the magnitude response. This method approximates the s-domain transfer function of frequency response plots by using knowledge of the behaviors of simple poles and zeros.  48  Qm |(s − zi )| |H(s)| = k Qni=1 |(s − pi )| i=1 6  H(s) =  m X 6  (s − zi ) −  i=1  n X 6  (s − pi )  (66)  (67)  i=1  By extracting an approximate s-domain transfer function equation of the magnitude response, the corresponding phase response can be obtained. The accuracy of the plot can be increased simply by increasing the number of poles and zeros used to model the magnitude response. The curve t result is shown in Figure 33 and is compared to the frequency response phase data.  Figure 33: Comparison of derivation model and curve t model (magnitude)  Figure 33 shows a reasonable curve tting of the derivation model frequency response.  As  previously stated, the accuracy of the curve t can be adjusted by increasing the number of poles and zeros used during the curve tting process. There are several alternative methods in which a curve t of the frequency response can be obtained. However, compared to asymptotic Bode tracing, many of the analytical methods presented in literature such as [41] [42], are cumbersome and unmanageable for normal use, and methods that use software tools are costly to acquire and oer little to no exibility in terms of pole/zero placement [30]. Figure 34 compares the results of the phase plot from the derivation model and is compared with the frequency response data of Chapter 4. The results appear to be a good quality approximation of the obtained data. However, it can be noted that the curve t model presents a more pessimistic view of the phase response, and was unable to capture phase lead characteristic between 2 and 4 kHz.  49  Figure 34: Comparison of prototype frequency response data and curve t model (phase)  5.3 Verication of Derivation Model in PSIM R To verify that the derivation model can be used as an approximation of the control-to-output transfer function, a compensator based on the derivation model results was designed in MATLAB  R  . The  results are shown in Figures 35 - 38, and are summarized in Table 2.  Step Response 1.4  1.2  1  Amplitude  0.8  0.6  0.4  0.2  0  0  0.5  1  1.5 Time (sec)  Figure 35: MATLAB  R  2  2.5 x 10  -3  step response of closed-loop system using derivation model  50  6 -100  4 -200  2  Verror  0.12 Ip 0.1 200 0.08 0.06 100 0.04 0 0.02 0 -100 0.01  0.02  0.03  0.04  Time (s) -200  Figure 36: Error voltage of derivation model  Vout 190.0175  190.017  190.0165  190.016  190.0155  190.015 0.025  0.02502 Time (s)  0.02504  Figure 37: Output voltage ripple of derivation model  The results of the digital negative feedback control system designed by using the derivation model are summarized in Table 2. It was found that the results of Table 2 are comparable to those of Table 1 found in Chapter 4. Therefore, it can be concluded that the derivation model can be used for digital compensator design, and is able to produce similar results to a compensator designed based on physical prototype data.  51  C:\Users\Brian\Documents\PSIM\LOW2HI_FINALplay.smv Date: 06:55PM 10/02/12 Vout 192  190  188  186  184  182  Ires 200  100  0  -100  -200  f_s  200K  0K  -200K  -400K  -600K 0  Figure 38: PSIM  0.01  R  0.02 Time (s)  0.03  0.04  closed-loop response to step load change using prototype model  52  Output voltage value:  190 V  Steady-state error:  0 V  Output voltage ripple  1.5 mV  % overshoot:  10%  Rise-time:  0.9 ms  Settling-time:  2 ms  Table 2: Results of derivation model  The merit of these results is the conrmation that the small signal control-to-output transfer function obtained by the derivation model can also be used to determine a digital compensator in lieu of using prototype frequency response data. The rst advantage of the derivation model method is that only the circuit parameters of the proposed LLC resonant converter are required, and no physical prototype model is needed to obtain the frequency response data.  Secondly, compared  to computer simulation software which can also be used to obtain the plant transfer function, the derivation model requires signicantly less computational power, and is considerably faster in terms of simulation run-time.  53  6 Conclusions and Future Work 6.1 Summary In this thesis, the LLC resonant converter topology and the variable frequency control method for output voltage regulation was discussed in detail. In Chapter 2, it was determined that the LLC resonant converter is an excellent choice for a power electronic converter topology to be used in isolated DC-DC conversion applications.  The  LLC resonant converter was found to feature high eciency, high power density, and the ability to operate over a wide range of loading conditions. The LLC resonant converter also has many other advantageous features including low electromagnetic interference, and soft-switching capabilities. Chapter 3 discusses the application of a controller to the LLC resonant converter.  It was  determined that the preferred control method under nominal loading conditions is variable frequency control, a method which varies the switching frequency of the converter to regulate the output voltage.  The 2-pole-2-zero compensator was introduced as a possible suitor for a digital control  system, as it can be easily implemented within digital signal processors by using dierence equations. The stability of continuous and discrete-time control systems was also discussed and the application of Bode diagrams and Nyquist plots for control system design was explored. It was then shown in Chapter 4 that by using a combination of laboratory bench measurements, and computer simulation, that a digital compensator could be designed to very tightly regulate the output voltage. An algorithm was then presented to implement the voltage control loop in a digital signal processor. In Chapter 5, a novel derivation of the small signal control-to-output transfer function was completed. The model includes the use of the fundamental, third and fth harmonics, and showed that the higher order harmonics have a signicant contribution to the overall transfer function, and it should be necessary to include them to capture the true dynamics of the control-to-output transfer function. The new model also includes higher order sideband frequencies and it was conrmed that the higher order sideband frequencies are required for modelling the converter for the case(s) where the magnitude of the modulation index  β  does not meet the condition  |β| << 1.  It was then veried that the small signal control-to-output transfer function developed in the derivation model could used to design a digital compensator for a digital negative feedback control system.  The simulation results show that an acceptable closed-loop step response and output  voltage regulation was achieved, and had comparable results to the results of Chapter 4. Finally, it was concluded that the results of Chapter 5 were comparable to the results of Chapter 4 and that the small signal control-to-output transfer function could be obtained using the derivation model. The advantages of using the derivation model are reduced computational complexity and the diminished requirement for a physical prototype model.  54  6.2 Future Work There are several areas which can be explored further to improve the modelling of the LLC resonant converter control-to-output transfer function. Firstly, development of the model for the low frequency range can be investigated, and can be achieved by decreasing the value of  β  is large.  ωm , and therefore, introducing the case where modulation index  To calculate large values of  function coecient  Jn (β)  β,  a dierent method to calculate and obtain the Bessel  may be required.  Secondly, a more in depth method to describe the rectier model is also needed.  As there  are certain loading conditions which cause the LLC resonant converter to go into discontinuous conduction mode, the linearizion of the rectier circuit may not be valid for all scenarios. Thirdly, another topic of interest would be to determine a practical method in which synchronous rectication can be achieved. In addition to an increase in the eciency of the converter, there is also potential of having bidirectional power ow between the input and output. In the case where a battery is used to provide DC power at the input, this means the battery could also be charged by the load without any additional conversion processes. Fourthly, it was also observed that there was a signicant eect on the control-to-output transfer function from the output capacitance and its equivalent series resistance. Therefore, a self-tuning controller has potential to be of great value for the design of controllers for the LLC resonant converter. It may be able to oset errors in the modelling process that are caused by the tolerances, the eects of aging capacitance, or the variation in capacitance from operation in environments with non-ideal temperatures. Lastly, there have also been several improvements in switching device technology since the beginning of this work. Silicon carbide-based transistors and diodes have been shown many signicant advantages including lower conduction losses, higher frequency operation, as well as higher operable temperatures.  55  References [1] D. Becker and B. Sonnenberg, Dc microgrids in buildings and data centers, in Telecommunications Energy Conference (INTELEC), 2011 IEEE 33rd International, 2011.  [2] B. Mammano, Resonant mode converter topologies, pp. 12, 2001. Texas Instruments application note. [3] F. Canales, P. Barbosa, and F. C. Lee, A wide input voltage and load output variations xed-frequency zvs dc/dc llc resonant converter for high-power applications, in Industry Applications Conference, 37th IAS Annual Meeting, 2002.  [4] G. Pledl, M. Tauer, and D. Buecherl, Theory of operation, design procedure and simulation of a bidirectional llc resonant converter for vehicular applications, in Vehicle Power and Propulsion Conference (VPPC), 2010 IEEE, 2010.  [5] D. J. Tschirhart and P. K. Jain, Digital control of resonant converters at ultra high frequency, in Telecommunications Energy Conference (INTELEC), 32nd International, 2010. [6] R. W. Erickson and D. Maksimovic, Fundamentals of Power Electronics. Springer; 2nd edition, 2001. [7] B. Yang, Topology investigation of front end DC/DC converter for distributed power system. PhD thesis, Virginia Polytechnic Institute and State University, 2003. [8] B. Lu, W. Liu, Y. Liang, F. C. Lee, and J. D. van Wyk, Optimal design methodology for llc resonant converter, in Applied Power Electronics Conference and Exposition, 2006. APEC '06. Twenty-First Annual IEEE, 2006.  [9] R. Locher, Introduction to power mosfets and their applications, p. 1, 1998. [10] Y.-F. Liu and W. Eberle, Recent developments in switching mode power supply technologies, IEEE Canadian Review, vol. 61, pp. 913, 2009.  [11] D. Fu, Topology Investigation and System Optimization of Resonant Converters. PhD thesis, Virginia Polytechnic Institute and State University, 2010. [12] H. Choi, Analysis and design of llc resonant converter with integrated transformer, in Applied Power Electronics Conference, APEC 2007 - Twenty Second Annual IEEE, 2007.  [13] D. Fu, Y. Liu, F. C. Lee, and M. Xu, A novel driving scheme for synchronous rectiers in llc resonant converters, Power Electronics, IEEE Transactions on, vol. 24, pp. 13211329, 2009.  56  [14] D. Wang, L. Jia, Y.-F. Liu, and P. Sen, An improved driving method for synchronous rectier using drain-source voltage sensing, in Applied Power Electronics Conference and Exposition (APEC), 2011 Twenty-Sixth Annual IEEE, 2011.  [15] Y. Liu, High eciency optimization of llc resonant converter for wide load range, Master's thesis, Virginia Polytechnic Institute and State University, 2007. [16] Y. Ye, C. Yan, J. Zeng, and J. Ying, A novel light load solution for llc series resonant converter, in Telecommunications Energy Conference, 2007. INTELEC 2007. 29th International, 2007. [17] W.-J. Lee, S.-W. Choi, C.-E. Kim, and G.-W. Moon, A new pwm-controlled quasi-resonant converter for a high eciency pdp sustaining power module, Power Electronics, IEEE Transactions on, vol. 23, pp. 17821790, 2008.  [18] R. Poley, Digital control theory seminar; student manual. Texas Instruments, 2012. [19] G. F. Franklin, J. D. Powell, and M. L. Workman, Digital Control of Dynamic Systems. Addison-Wesley, 1998. [20] B. Akin and D. Chang, Optimal digital power control using llc resonant converters, April 2012. [21] M. Gopal, Control Systems: Principles and Design. Tata McGraw-Hill Publishing Company, 2008. [22] S. Engelberg, A Mathematical Introduction to Control Theory. Imperial College Press, 2005. [23] G. F. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback Control of Dynamic Systems. Addison-Wesley Long Man Publishing Co., 1993. [24] R. Poley, Control theory seminar; student manual. Texas Instruments, 2012. [25] S. M. Shinners, Modern Control System Theory and Design. John Wiley & Sons, 1998. [26] MATLAB, version 7.11.0.584 (R2010b). Natick, Massachusetts: The MathWorks Inc., 2010. [27] Powersim, PSIM User Manual. www.powersimtech.com. [28] N. Macia, Modeling and Control of Dynamic Systems. Thomson Delmar Learning, 2005. [29] S. Chon, What it takes to do ecient and cost-eective real-time control with a single microcontroller: The c2000 advantage, p. 3, January 2011.  57  [30] P. Spanik, P. Drgona, and M. Frivaldsky, Simulation based method for mathematical model design of llc power converter, [31] H. Huang, Feedback loop design of an llc resonant power converter, Texas Insturments Application Report, 2010.  [32] J. Jang, M. Joung, S. Choi, Y. Choi, and B. Choi, Current mode control for llc series resonant dc-to-tc converters, in Applied Power Electronics Conference and Exposition (APEC), 2011 Twenty-Sixth Annual IEEE, 2011.  [33] T. G. Thomas and S. C. Sekhar, Communication Theory. Tata McGraw-Hill Publishing Company, 2006. [34] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. National Burea of Standards Applied Mathematics Series,  December 1972. [35] R. W. Erickson, Fm/am modeling of the envelope transfer function. ECEN 5817 course notes. [36] C. P. Basso, Switch-Mode Power Supply SPICE Cookbook. McGraw-Hill Prof Med/Tech, 2001. [37] R. Zaitsu, Voltage mode boost converter small signal control loop analysis using the tps61030, Texas Instruments Application Report, pp. 28, 2009.  [38] R. Petkov and G. Anguelov, Current mode control of frequency controlled resonant converters, in Telecommunications Energy Conference, 1998. INTELEC. Twentieth International, 1998. [39] M.Vajta, Some remarks on pade approximations, in 3rd TEMPUS-INTCOM Symposium, 2000. [40] J. Marti, Wave propagation in frequency dependent lines: The fd-line model, 2004. EECE 560 course notes. [41] W. S. Cleveland, S. J. Devlin, and E. Grosse, Regression by local tting: Methods, properties, and computational algorithms, Journal Of Econometrics, vol. 37, pp. 87114, 1988. [42] G. Ferreres and J. Biannic, Frequency domain curve tting: a generalized approach, tech. rep., ONERA/DCSD: Systems Control and Flight Dynamics Department, 2001.  58  Appendices Appendix A: Observation of Stability in Continuous and Discrete-time domains A1: Continuous-time domain To determine the stability and the dynamic characteristics of the closed loop feedback system of Figure 14, the transfer function of Figure 14 can be described by Equation A-1. Assuming that the actuator has a gain equal to 1,  T (s) =  G(s)P (s) 1 + G(s)P (s)  (A-1)  In the continuous-time domain, the stability of the feedback system is observed on the s-domain plane. The stability can be determined by evaluating the denominator of Equation A-1, also known as the characteristic equation. Solving for the roots of the characteristic equation gives the poles of the system, which are indicators of the system's stability. A system that contains only poles on the left hand side of the  jω  axis of the s-domain plane are  considered to be stable, while systems with poles on the right hand side of the Marginally stable systems have poles on the  jω  jω  axis are unstable.  axis.  59  A2: Discrete-time domain In the discrete-time domain, the stability of the feedback system is observed with respect to the z-domain unit circle. Stable systems have all poles located inside of the unit circle, while unstable systems have poles located outside of the unit circle. Finally, marginally stable systems have poles located on the unit circle.  Figure A-1: Unit circle in the z-domain  60  Appendix B: PSIM R simulation schematics Vout Iload  V  A Vgs1  Vmos1  Vmos3 MOS1  V Vds1  Vgs1  Vc  Vgs3  Q1  MOS3  V  Q3  Ires  Ip  A  A  A  Vout_digital  Isec Vm  Vtank  V  Vsec  ZOH  V Vgs3  V  Vds3  Vgs3  Vmos3  Vgs1  Q2  Vmos1 MOS2  MOS4  Q4  Isw  Vmos1 Verror  f_s  V  V  Imos1  ZOH  K  Error Vctrl  V Vmos1  r  K  sin  K  Vgs1  K  Vmos3  Vgs3  Imos3  V Vmos3  V_ref  Figure B-1: PSIM  R  closed loop circuit schematic of LLC resonant converter  Vout  V  I_o  A  A  Vgs1 Title  V Vds1  Vgs1  Designed by  V_Cr  Vgs3  Q1  Revision  V  Page 1 of 1  A  Q3  I_r  Ip  A  A  Isec  Vin Vm  Vtank  I_LM Vgs3  V Vds3  Vgs3  Vgs1  Q2  Q4  f  Isw  V r  K  ZOH  K  sin  Vgs1  K  Vgs3  Figure B-2: PSIM  R  open loop circuit schematic of LLC resonant converter  Title Designed by Revision  Iload  Page 1 of 1  61  Vout  Appendix C: MATLAB R derivation model code 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45  clear a l l close a l l clc Vg = 48; Kvco = 32779; Tratio = 15/4; delta_w = 2 ∗ pi ∗ (Kvco) ∗ 0.02; Lm = (80e −6) ∗ ((4/15) ^2) ; Lr = (10e −6)/((15/4) ^2) ; Cr = 9 ∗ 270e − 9; R = 1; Rr = R; Ro = 170/4.6; Rac = ((8 ∗ Tratio ^2)/ pi ^2) ∗Ro; res_freq = 1/( sqrt (Lr ∗ Cr) ∗ 2 ∗ pi ) w_so = 2 ∗ pi ∗ res_freq ; V_s1 = (4/ pi ) ∗Vg; V_s3 = (4/(3 ∗ pi ) ) ∗Vg; V_s5 = (4/(5 ∗ pi ) ) ∗Vg; Rc = 0.159; C = 470e − 6; ESRzero = t f ( [ (C∗ Rc) 1 ] , [ 1 ] ) ; [ mag_ESRzero phase_ESRzero ] = bode (ESRzero , logspace (1 ,6 ,200) ) ; mag_ESRzero = squeeze (mag_ESRzero) ; phase_ESRzero = squeeze (phase_ESRzero) ; fz = 3100; RHPzero = t f ([ − 1/(2 ∗ pi ∗ fz ) 1 ] , [ 1 ] ) ; [mag_RHPzero phase_RHPzero ] = bode (RHPzero , logspace (1 ,6 ,200) ) ; mag_RHPzero = squeeze (mag_RHPzero) ; phase_RHPzero = squeeze (phase_RHPzero) ; T = 1/8000; ZOHpole = t f ([3 ∗ T^2 −24∗T 60] ,[T^3 9 ∗T^2 36 ∗T 60]) ; [ mag_ZOHpole phase_ZOHpole ] = bode (ZOHpole , logspace (1 ,6 ,200) ) ; mag_ZOHpole = squeeze (mag_ZOHpole) ; phase_ZOHpole = squeeze (phase_ZOHpole) ; %%%%%%%%fundamental frequency 62  46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80  a b c d e  = Cr ∗ Rac ∗Lm; = Lr ∗ Cr ∗Lm; = (Cr ∗ Rac ∗Lm)+(Lm∗ Cr ∗ Rr)+(Lr ∗ Cr ∗ Rac) ; = Lm+(Rr ∗ Cr ∗ Rac) ; = Rac ;  %with Rload H_jwso = t f ( [ a ∗ ( i ∗ w_so) ^2] ,[ b ∗ ( i ∗ w_so)^3+c ∗ ( i ∗ w_so)^2+d ∗ ( i ∗ w_so)+e ] ) ; H_jwsoConj = t f ( [ a ∗( − i ∗ w_so) ^2] ,[ b∗( − i ∗ w_so)^3+c ∗( − i ∗ w_so)^2+d∗( − i ∗ w_so)+e ] ) ; %with Rload H_jwsoPluswm = t f ( [ a a ∗ 2 ∗ i ∗ w_so −a ∗ w_so^2] ,[ b (b ∗ 3 ∗ i ∗ w_so+c ) (−b ∗ 3 ∗ w_so^2+c ∗ 2 ∗ i ∗ w_so +d) (e−c ∗ w_so^2−b ∗ i ∗ w_so^3+d ∗ i ∗ w_so) ] ) ; H_jwsoPluswmConj = t f ( [ a a ∗ 2 ∗ i ∗ w_so −a ∗ w_so^2] ,[ − b (−b ∗ 3 ∗ i ∗ w_so+c ) (b ∗ 3 ∗ w_so^2+c ∗ 2 ∗ i ∗ w_so−d) (b ∗ i ∗ w_so^3− c ∗ w_so^2−d ∗ i ∗ w_so+e ) ] ) ; %with Rload H_jwsoMinuswm = t f ( [ a −2∗ i ∗ w_so ∗ a −a ∗ w_so^2] ,[ − b (b ∗ 3 ∗ i ∗ w_so+c ) (b ∗ 3 ∗ w_so^2− c ∗ 2 ∗ i ∗ w_so−d) (e−b ∗ j ∗ w_so^3− c ∗ w_so^2+d ∗ i ∗ w_so) ] ) ; H_jwsoMinuswmConj = t f ( [ a −2∗ i ∗ w_so ∗ a −a ∗ w_so^2] ,[ b (−b ∗ 3 ∗ i ∗ w_so+c ) (−b ∗ 3 ∗ w_so^2− c ∗ 2 ∗ i ∗ w_so+d) (e−c ∗ w_so^2+b ∗ i ∗ w_so^3−d ∗ i ∗ w_so) ] ) ; %with Rload H_jwsoPlus2wm = t f ([4 ∗ a 4 ∗ a ∗ i ∗ w_so −a ∗ w_so^2] ,[8 ∗ b (12 ∗ b ∗ i ∗ w_so+4∗ c ) (−b ∗ 6 ∗ w_so^2+4∗ i ∗ c ∗ w_so+2∗d) ( e+d ∗ i ∗ w_so−b ∗ i ∗ w_so^3− c ∗ w_so^2) ] ) ; H_jwsoPlus2wmConj = t f ([4 ∗ a 4 ∗ a ∗ i ∗ w_so −a ∗ w_so^2] ,[ − 8 ∗ b ( −12∗ b ∗ i ∗ w_so+4∗ c ) (b ∗ 6 ∗ w_so ^2+4∗ i ∗ w_so ∗ c −2∗d) (b ∗ i ∗ w_so^3− c ∗ w_so^2−d ∗ i ∗ w_so+e ) ] ) ; %with Rload H_jwsoMinus2wm = t f ([4 ∗ a −4∗ a ∗ i ∗ w_so −a ∗ w_so^2] ,[ − 8 ∗ b (12 ∗ b ∗ i ∗ w_so+4∗ c ) (6 ∗ b ∗ w_so ^2−4∗ c ∗ i ∗ w_so−2∗d) ( e+d ∗ i ∗ w_so−b ∗ j ∗ w_so^3− c ∗ w_so^2) ] ) ; H_jwsoMinus2wmConj = t f ([4 ∗ a −4∗ a ∗ i ∗ w_so −a ∗ w_so^2] ,[8 ∗ b ( −12∗ b ∗ i ∗ w_so+4∗ c ) ( −6∗ b ∗ w_so^2−4∗ c ∗ i ∗ w_so+2∗d) (e− i ∗ d ∗ w_so−c ∗ w_so^2+b ∗ i ∗ w_so^3) ] ) ; %wth Rload H_jwsoPlus3wm = t f ([9 ∗ a 6 ∗ a ∗ i ∗ w_so −a ∗ w_so^2] ,[27 ∗ b (27 ∗ b ∗ i ∗ w_so+9∗ c ) ( −9∗ b ∗ w_so ^2+6∗ c ∗ i ∗ w_so+3∗d) ( e+d ∗ i ∗ w_so−c ∗ w_so^2−b ∗ i ∗ w_so^3) ] ) ; H_jwsoPlus3wmConj = t f ([9 ∗ a 6 ∗ a ∗ i ∗ w_so −a ∗ w_so^2] ,[ − 27 ∗ b ( −27∗ b ∗ i ∗ w_so+9∗ c ) (9 ∗ b ∗ w_so^2+6∗ c ∗ i ∗ w_so−3∗d) (e− i ∗ d ∗ w_so−c ∗ w_so^2+b ∗ i ∗ w_so^3) ] ) ; %with Rload H_jwsoMinus3wm = t f ([9 ∗ a −6∗ a ∗ i ∗ w_so −a ∗ w_so^2] ,[ − 27 ∗ b (27 ∗ b ∗ i ∗ w_so+9∗ c ) (9 ∗ b ∗ w_so ^2−6∗ c ∗ i ∗ w_so−3∗d) ( e+d ∗ i ∗ w_so−c ∗ w_so^2−b ∗ i ∗ w_so^3) ] ) ; H_jwsoMinus3wmConj = t f ([9 ∗ a −6∗ a ∗ i ∗ w_so −a ∗ w_so^2] ,[27 ∗ b ( −27∗ b ∗ i ∗ w_so+9∗ c ) ( −9∗ b ∗ w_so^2−6∗ c ∗ i ∗ w_so+3∗d) (e−d ∗ i ∗ w_so−c ∗ w_so^2+b ∗ i ∗ w_so^3) ] ) ; beta = t f ( [ delta_w ] ,[ − 1 ∗ i 0]) ; 63  81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125  [ mag_beta phase_beta wout ] = bode ( beta , logspace (1 ,6 ,200) ) ; mag_beta = squeeze (mag_beta) ; phase_beta = squeeze ( phase_beta ) ; %fundamental , sideband 0 alpha = 0; J0_1 =b e s s e l j ( alpha , mag_beta) ; [mag_H_jwso phase_H_jwso ] = bode (H_jwso , logspace (1 ,6 ,200) ) ; mag_H_jwso = squeeze (mag_H_jwso) ; phase_H_jwso = squeeze (phase_H_jwso) ; [ mag_H_jwsoConj phase_H_jwsoConj ] = bode (H_jwsoConj , logspace (1 ,6 ,200) ) ; mag_H_jwsoConj = squeeze (mag_H_jwsoConj) ; phase_H_jwsoConj = squeeze (phase_H_jwsoConj) ; Ao_mag = V_s1∗ J0_1. ∗ mag_H_jwso; AoConj_mag = V_s1∗ J0_1. ∗ mag_H_jwsoConj ; %fundamental sideband 1 alpha1_1 = 1; J1_1 =b e s s e l j ( alpha1_1 , mag_beta) ; %plus [mag_H_jwsoPluswm phase_H_jwsoPluswm ] = bode (H_jwsoPluswm , logspace (1 ,6 ,200) ) ; mag_H_jwsoPluswm = squeeze (mag_H_jwsoPluswm) ; phase_H_jwsoPluswm = squeeze (phase_H_jwsoPluswm) ; [ mag_H_jwsoPluswmConj phase_H_jwsoPluswmConj ] = bode (H_jwsoPluswmConj , logspace (1 ,6 ,200) ) ; mag_H_jwsoPluswmConj = squeeze (mag_H_jwsoPluswmConj) ; phase_H_jwsoPluswmConj = squeeze (phase_H_jwsoPluswmConj) ; Au_mag = V_s1∗ J1_1. ∗ mag_H_jwsoPluswm; AuConj_mag = V_s1∗ J1_1. ∗ mag_H_jwsoPluswmConj ; %minus [mag_H_jwsoMinuswm phase_H_jwsoMinuswm ] = bode (H_jwsoMinuswm , logspace (1 ,6 ,200) ) ; mag_H_jwsoMinuswm = squeeze (mag_H_jwsoMinuswm) ; phase_H_jwsoMinuswm = squeeze (phase_H_jwsoMinuswm) ; [ mag_H_jwsoMinuswmConj phase_H_jwsoMinuswmConj ] = bode (H_jwsoMinuswmConj , logspace (1 ,6 ,200) ) ; mag_H_jwsoMinuswmConj = squeeze (mag_H_jwsoMinuswmConj) ; phase_H_jwsoMinuswmConj = squeeze (phase_H_jwsoMinuswmConj) ; Al_mag = −V_s1∗ J1_1. ∗ mag_H_jwsoMinuswm; 64  126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169  AlConj_mag = −V_s1∗ J1_1. ∗ mag_H_jwsoMinuswmConj ; %fundamental , sideband 2 alpha2 = 2; J2_1 =b e s s e l j ( alpha2 , mag_beta) ; %plus2wm [ mag_H_jwsoPlus2wm phase_H_jwsoPlus2wm ] = bode (H_jwsoPlus2wm , logspace (1 ,6 ,200) ) ; mag_H_jwsoPlus2wm = squeeze (mag_H_jwsoPlus2wm) ; phase_H_jwsoPlus2wm = squeeze (phase_H_jwsoPlus2wm) ; [ mag_H_jwsoPlus2wmConj phase_H_jwsoPlus2wmConj ] = bode (H_jwsoPlus2wmConj , logspace (1 ,6 ,200) ) ; mag_H_jwsoPlus2wmConj = squeeze (mag_H_jwsoPlus2wmConj) ; phase_H_jwsoPlus2wmConj = squeeze (phase_H_jwsoPlus2wmConj) ; Au2_mag = V_s1∗ J2_1. ∗ mag_H_jwsoPlus2wm ; Au2Conj_mag = V_s1∗ J2_1. ∗ mag_H_jwsoPlus2wmConj ; %minus2wm [mag_H_jwsoMinus2wm phase_H_jwsoMinus2wm ] = bode (H_jwsoMinus2wm , logspace (1 ,6 ,200) ) ; mag_H_jwsoMinus2wm = squeeze (mag_H_jwsoMinus2wm) ; phase_H_jwsoMinus2wm = squeeze (phase_H_jwsoMinus2wm) ; [ mag_H_jwsoMinus2wmConj phase_H_jwsoMinus2wmConj ] = bode (H_jwsoMinus2wmConj , logspace (1 ,6 ,200) ) ; mag_H_jwsoMinus2wmConj = squeeze (mag_H_jwsoMinus2wmConj) ; phase_H_jwsoMinus2wmConj = squeeze (phase_H_jwsoMinus2wmConj) ; Al2_mag = V_s1∗ J2_1. ∗ mag_H_jwsoMinus2wm; Al2Conj_mag = V_s1∗ J2_1. ∗ mag_H_jwsoMinus2wmConj ; %fundamental , sideband3 alpha3 = 3; J3_1 =b e s s e l j ( alpha3 , mag_beta) ; %plus3wm [ mag_H_jwsoPlus3wm phase_H_jwsoPlus3wm ] = bode (H_jwsoPlus3wm , logspace (1 ,6 ,200) ) ; mag_H_jwsoPlus3wm = squeeze (mag_H_jwsoPlus3wm) ; phase_H_jwsoPlus3wm = squeeze (phase_H_jwsoPlus3wm) ; [ mag_H_jwsoPlus3wmConj phase_H_jwsoPlus3wmConj ] = bode (H_jwsoPlus3wmConj , logspace (1 ,6 ,200) ) ; mag_H_jwsoPlus3wmConj = squeeze (mag_H_jwsoPlus3wmConj) ; phase_H_jwsoPlus3wmConj = squeeze (phase_H_jwsoPlus3wmConj) ; Au3_mag = V_s1∗ J3_1. ∗ mag_H_jwsoPlus3wm ; 65  170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206  Au3Conj_mag = V_s1∗ J3_1. ∗ mag_H_jwsoPlus3wmConj ; %minus3wm [mag_H_jwsoMinus3wm phase_H_jwsoMinus3wm ] = bode (H_jwsoMinus3wm , logspace (1 ,6 ,200) ) ; mag_H_jwsoMinus3wm = squeeze (mag_H_jwsoMinus3wm) ; phase_H_jwsoMinus3wm = squeeze (phase_H_jwsoMinus3wm) ; [ mag_H_jwsoMinus3wmConj phase_H_jwsoMinus3wmConj ] = bode (H_jwsoMinus3wmConj , logspace (1 ,6 ,200) ) ; mag_H_jwsoMinus3wmConj = squeeze (mag_H_jwsoMinus3wmConj) ; phase_H_jwsoMinus3wmConj = squeeze (phase_H_jwsoMinus3wmConj) ; Al3_mag = V_s1∗ J3_1. ∗ mag_H_jwsoMinus3wm; Al3Conj_mag = V_s1∗ J3_1. ∗ mag_H_jwsoMinus3wmConj ; a1 = (Ao_mag. ∗ AlConj_mag)+(Au_mag. ∗ AoConj_mag)+(Al_mag. ∗ Al2Conj_mag)+(Au2_mag. ∗ AuConj_mag) ; a2 = (Ao_mag. ∗ Al2Conj_mag)+(Au_mag. ∗ AlConj_mag)+(Al_mag. ∗ Al3Conj_mag)+(Au2_mag. ∗ AoConj_mag)+(Au3_mag. ∗ AuConj_mag) ; a3 = (Ao_mag. ∗ Al3Conj_mag)+(Au_mag. ∗ Al2Conj_mag)+(Au2_mag. ∗ AlConj_mag)+(Au3_mag. ∗ AoConj_mag) ; fundamentalOutput = ( sqrt ((1/3) ∗ ( a1.^2+a2.^2+a3 .^2) ) ) ./(Ao_mag) ; fundamentalOutput = ((1/ delta_w ) ∗ fundamentalOutput ) ; %%%%%%%%third harmonic %with Rload H_3jwso = t f ( [ a ∗ (3 ∗ i ∗ w_so) ^2] ,[ b ∗ (3 ∗ i ∗ w_so)^3+c ∗ (3 ∗ i ∗ w_so)^2+d ∗ (3 ∗ i ∗ w_so)+e ] ) ; H_3jwsoConj = t f ( [ a ∗( −3∗ i ∗ w_so) ^2] ,[ b ∗( −3∗ i ∗ w_so)^3+c ∗( −3∗ i ∗ w_so)^2+d ∗( −3∗ i ∗ w_so)+e ]) ; %with Rload H_3jwsoPluswm = t f ( [ a a ∗ 6 ∗ i ∗ w_so −a ∗ 9 ∗ w_so^2] ,[ b (b ∗ 9 ∗ i ∗ w_so+c ) (−b ∗ 27 ∗ w_so^2+c ∗ 6 ∗ i ∗ w_so+d) (e −9∗ c ∗ w_so^2 −27∗b ∗ i ∗ w_so^3+3∗d ∗ i ∗ w_so) ] ) ; H_3jwsoPluswmConj = t f ( [ a a ∗ 6 ∗ i ∗ w_so −a ∗ 9 ∗ w_so^2] ,[ − b (−b ∗ 9 ∗ i ∗ w_so+c ) (b ∗ 27 ∗ w_so^2+c ∗ 6 ∗ i ∗ w_so−d) (27 ∗ b ∗ i ∗ w_so^3− c ∗ 9 ∗ w_so^2 −3∗ d ∗ i ∗ w_so+e ) ] ) ; %with Rload H_3jwsoMinuswm = t f ( [ a −6∗ i ∗ w_so ∗ a −9∗ a ∗ w_so^2] ,[ − b (b ∗ 9 ∗ i ∗ w_so+c ) (b ∗ 27 ∗ w_so^2− c ∗ 6 ∗ i ∗ w_so−d) (e −27∗b ∗ j ∗ w_so^3−9∗ c ∗ w_so^2+3∗d ∗ i ∗ w_so) ] ) ; H_3jwsoMinuswmConj = t f ( [ a −6∗ i ∗ w_so ∗ a −9∗ a ∗ w_so^2] ,[ b (−b ∗ 9 ∗ i ∗ w_so+c ) (−b ∗ 27 ∗ w_so ^2− c ∗ 6 ∗ i ∗ w_so+d) (e−c ∗ 9 ∗ w_so^2+27∗ b ∗ i ∗ w_so^3−3∗d ∗ i ∗ w_so) ] ) ; %with Rload H_3jwsoPlus2wm = t f ([4 ∗ a 12 ∗ a ∗ i ∗ w_so −9∗ a ∗ w_so^2] ,[8 ∗ b (36 ∗ b ∗ i ∗ w_so+4∗ c ) (−b ∗ 54 ∗ w_so^2+12∗ i ∗ c ∗ w_so+2∗d) ( e+3∗d ∗ i ∗ w_so−27∗b ∗ i ∗ w_so^3−9∗ c ∗ w_so^2) ] ) ; 66  207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245  H_3jwsoPlus2wmConj = t f ([4 ∗ a 12 ∗ a ∗ i ∗ w_so −9∗ a ∗ w_so^2] ,[ − 8 ∗ b ( −36∗ b ∗ i ∗ w_so+4∗ c ) (b ∗ 54 ∗ w_so^2+12 ∗ i ∗ w_so ∗ c −2∗d) (27 ∗ b ∗ i ∗ w_so^3 −9∗ c ∗ w_so^2 −3∗ d ∗ i ∗ w_so+e ) ] ) ; %with Rload H_3jwsoMinus2wm = t f ([4 ∗ a −12∗ a ∗ i ∗ w_so −9∗ a ∗ w_so^2] ,[ − 8 ∗ b (36 ∗ b ∗ i ∗ w_so+4∗ c ) (54 ∗ b ∗ w_so^2 −12∗ c ∗ i ∗ w_so−2∗d) ( e+3∗d ∗ i ∗ w_so−27∗b ∗ j ∗ w_so^3−9∗ c ∗ w_so^2) ] ) ; H_3jwsoMinus2wmConj = t f ([4 ∗ a −12∗ a ∗ i ∗ w_so −9∗ a ∗ w_so^2] ,[8 ∗ b ( −36∗ b ∗ i ∗ w_so+4∗ c ) ( −54∗ b ∗ w_so^2 −12∗ c ∗ i ∗ w_so+2∗d) (e −3∗ i ∗ d ∗ w_so−9∗ c ∗ w_so^2+27∗ b ∗ i ∗ w_so^3) ] ) ; %wth Rload H_3jwsoPlus3wm = t f ([9 ∗ a 18 ∗ a ∗ i ∗ w_so −9∗ a ∗ w_so^2] ,[27 ∗ b (81 ∗ b ∗ i ∗ w_so+9∗ c ) ( −81∗ b ∗ w_so^2+18∗ c ∗ i ∗ w_so+3∗d) ( e+3∗d ∗ i ∗ w_so−9∗ c ∗ w_so^2 −27∗b ∗ i ∗ w_so^3) ] ) ; H_3jwsoPlus3wmConj = t f ([9 ∗ a 18 ∗ a ∗ i ∗ w_so −9∗ a ∗ w_so^2] ,[ − 27 ∗ b ( −81∗ b ∗ i ∗ w_so+9∗ c ) (81 ∗ b ∗ w_so^2+18∗ c ∗ i ∗ w_so−3∗d) (e −3∗ i ∗ d ∗ w_so−9∗ c ∗ w_so^2+27∗ b ∗ i ∗ w_so^3) ] ) ; %with Rload H_3jwsoMinus3wm = t f ([9 ∗ a −18∗ a ∗ i ∗ w_so −9∗ a ∗ w_so^2] ,[ − 27 ∗ b (81 ∗ b ∗ i ∗ w_so+9∗ c ) (81 ∗ b ∗ w_so^2 −18∗ c ∗ i ∗ w_so−3∗d) ( e+3∗d ∗ i ∗ w_so−9∗ c ∗ w_so^2 −27∗b ∗ i ∗ w_so^3) ] ) ; H_3jwsoMinus3wmConj = t f ([9 ∗ a −18∗ a ∗ i ∗ w_so −9∗ a ∗ w_so^2] ,[27 ∗ b ( −81∗ b ∗ i ∗ w_so+9∗ c ) ( −81∗ b ∗ w_so^2 −18∗ c ∗ i ∗ w_so+3∗d) (e −3∗d ∗ i ∗ w_so−9∗ c ∗ w_so^2+27∗ b ∗ i ∗ w_so^3) ] ) ; %third Harmonic , sideband 0 alpha0_3 = 0; J0_3 = b e s s e l j ( alpha0_3 , mag_beta) ; [mag_H_3jwso phase_H_3jwso ] = bode (H_3jwso , logspace (1 ,6 ,200) ) ; mag_H_3jwso = squeeze (mag_H_3jwso) ; phase_H_3jwso = squeeze (phase_H_3jwso) ; [ mag_H_3jwsoConj phase_H_3jwsoConj ] = bode (H_3jwsoConj , logspace (1 ,6 ,200) ) ; mag_H_3jwsoConj = squeeze (mag_H_3jwsoConj) ; phase_H_3jwsoConj = squeeze (phase_H_3jwsoConj) ; Bo_mag = V_s3∗ J0_3. ∗ mag_H_3jwso; BoConj_mag = V_s3∗ J0_3. ∗ mag_H_3jwsoConj ; %thirdHarmonic , sideband1 alpha1_3 = 1; J1_3 =b e s s e l j ( alpha1_3 , mag_beta) ; %3 plus [ mag_H_3jwsoPluswm phase_H_3jwsoPluswm ] = bode (H_3jwsoPluswm , logspace (1 ,6 ,200) ) ; mag_H_3jwsoPluswm = squeeze (mag_H_3jwsoPluswm) ; phase_H_3jwsoPluswm = squeeze (phase_H_3jwsoPluswm) ; [ mag_H_3jwsoPluswmConj phase_H_3jwsoPluswmConj ] = bode (H_3jwsoPluswmConj , logspace (1 ,6 ,200) ) ; 67  246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287  mag_H_3jwsoPluswmConj = squeeze (mag_H_3jwsoPluswmConj) ; phase_H_3jwsoPluswmConj = squeeze (phase_H_3jwsoPluswmConj) ; Bu_mag = V_s3∗ J1_3. ∗ mag_H_3jwsoPluswm ; BuConj_mag = V_s3∗ J1_3. ∗ mag_H_3jwsoPluswmConj ; %3minus [mag_H_3jwsoMinuswm phase_H_3jwsoMinuswm ] = bode (H_3jwsoMinuswm , logspace (1 ,6 ,200) ) ; mag_H_3jwsoMinuswm = squeeze (mag_H_3jwsoMinuswm) ; phase_H_3jwsoMinuswm = squeeze (phase_H_3jwsoMinuswm) ; [ mag_H_3jwsoMinuswmConj phase_H_3jwsoMinuswmConj ] = bode (H_3jwsoMinuswmConj , logspace (1 ,6 ,200) ) ; mag_H_3jwsoMinuswmConj = squeeze (mag_H_3jwsoMinuswmConj) ; phase_H_3jwsoMinuswmConj = squeeze (phase_H_3jwsoMinuswmConj) ; Bl_mag = −V_s3∗ J1_3. ∗ mag_H_3jwsoMinuswm; BlConj_mag = −V_s3∗ J1_3. ∗ mag_H_3jwsoMinuswmConj ; %thirdHarmonic , sideband2 alpha2_3 = 2; J2_3 =b e s s e l j ( alpha2_3 , mag_beta) ; %3 plus2 [ mag_H_3jwsoPlus2wm phase_H_3jwsoPlus2wm ] = bode (H_3jwsoPlus2wm , logspace (1 ,6 ,200) ) ; mag_H_3jwsoPlus2wm = squeeze (mag_H_3jwsoPlus2wm) ; phase_H_3jwsoPlus2wm = squeeze (phase_H_3jwsoPlus2wm) ; [ mag_H_3jwsoPlus2wmConj phase_H_3jwsoPlus2wmConj ] = bode (H_3jwsoPlus2wmConj , logspace (1 ,6 ,200) ) ; mag_H_3jwsoPlus2wmConj = squeeze (mag_H_3jwsoPlus2wmConj) ; phase_H_3jwsoPlus2wmConj = squeeze (phase_H_3jwsoPlus2wmConj) ; Bu2_mag = V_s3∗ J2_3. ∗ mag_H_3jwsoPlus2wm ; Bu2Conj_mag = V_s3∗ J2_3. ∗ mag_H_3jwsoPlus2wmConj ; %3minus2 [mag_H_3jwsoMinus2wm phase_H_3jwsoMinus2wm ] = bode (H_3jwsoMinus2wm , logspace (1 ,6 ,200) ) ; mag_H_3jwsoMinus2wm = squeeze (mag_H_3jwsoMinus2wm) ; phase_H_3jwsoMinus2wm = squeeze (phase_H_3jwsoMinus2wm) ; [ mag_H_3jwsoMinus2wmConj phase_H_3jwsoMinus2wmConj ] = bode (H_3jwsoMinus2wmConj , logspace (1 ,6 ,200) ) ; mag_H_3jwsoMinus2wmConj = squeeze (mag_H_3jwsoMinus2wmConj) ; phase_H_3jwsoMinus2wmConj = squeeze (phase_H_3jwsoMinus2wmConj) ;  288  68  289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329  Bl2_mag = −V_s3∗ J2_3. ∗ mag_H_3jwsoMinus2wm; Bl2Conj_mag = −V_s3∗ J2_3. ∗ mag_H_3jwsoMinus2wmConj ; %thridHarmonic , sideband3 alpha3_3 = 3; J3_3 =b e s s e l j ( alpha3_3 , mag_beta) ; %3 plus3 [ mag_H_3jwsoPlus3wm phase_H_3jwsoPlus3wm ] = bode (H_3jwsoPlus3wm , logspace (1 ,6 ,200) ) ; mag_H_3jwsoPlus3wm = squeeze (mag_H_3jwsoPlus3wm) ; phase_H_3jwsoPlus3wm = squeeze (phase_H_3jwsoPlus3wm) ; [ mag_H_3jwsoPlus3wmConj phase_H_3jwsoPlus3wmConj ] = bode (H_3jwsoPlus3wmConj , logspace (1 ,6 ,200) ) ; mag_H_3jwsoPlus3wmConj = squeeze (mag_H_3jwsoPlus3wmConj) ; phase_H_3jwsoPlus3wmConj = squeeze (phase_H_3jwsoPlus3wmConj) ; Bu3_mag = V_s3∗ J3_3. ∗ mag_H_3jwsoPlus3wm ; Bu3Conj_mag = V_s3∗ J3_3. ∗ mag_H_3jwsoPlus3wmConj ; %3minus3 [mag_H_3jwsoMinus3wm phase_H_3jwsoMinus3wm ] = bode (H_3jwsoMinus3wm , logspace (1 ,6 ,200) ) ; mag_H_3jwsoMinus3wm = squeeze (mag_H_3jwsoMinus3wm) ; phase_H_3jwsoMinus3wm = squeeze (phase_H_3jwsoMinus3wm) ; [ mag_H_3jwsoMinus3wmConj phase_H_3jwsoMinus3wmConj ] = bode (H_3jwsoMinus3wmConj , logspace (1 ,6 ,200) ) ; mag_H_3jwsoMinus3wmConj = squeeze (mag_H_3jwsoMinus3wmConj) ; phase_H_3jwsoMinus3wmConj = squeeze (phase_H_3jwsoMinus3wmConj) ; Bl3_mag = −V_s3∗ J3_3. ∗ mag_H_3jwsoMinus3wm; Bl3Conj_mag = −V_s3∗ J3_3. ∗ mag_H_3jwsoMinus3wmConj ; b1 = (Bo_mag. ∗ BlConj_mag)+(Bu_mag. ∗ BoConj_mag)+(Bl_mag. ∗ Bl2Conj_mag)+(Bu2_mag. ∗ BuConj_mag) ; b2 = (Bo_mag. ∗ Bl2Conj_mag)+(Bu_mag. ∗ BlConj_mag)+(Bl_mag. ∗ Bl3Conj_mag)+(Bu2_mag. ∗ BoConj_mag)+(Bu3_mag. ∗ BuConj_mag) ; b3 = (Bo_mag. ∗ Bl3Conj_mag)+(Bu_mag. ∗ Bl2Conj_mag)+(Bu2_mag. ∗ BlConj_mag)+(Bu3_mag. ∗ BoConj_mag) ; thirdHarmonicOutput = ( sqrt ((1/3) ∗ (b1.^2+b2.^2+b3 .^2) ) ) ./(Bo_mag) ; thirdHarmonicOutput = ((1/ delta_w ) ∗ thirdHarmonicOutput ) ; %%%%%%%%f i f t h harmonic %with Rload 69  330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363  H_5jwso = t f ( [ a ∗ (5 ∗ i ∗ w_so) ^2] ,[ b ∗ (5 ∗ i ∗ w_so)^3+c ∗ (5 ∗ i ∗ w_so)^2+d ∗ (5 ∗ i ∗ w_so)+e ] ) ; H_5jwsoConj = t f ( [ a ∗( −5∗ i ∗ w_so) ^2] ,[ b ∗( −5∗ i ∗ w_so)^3+c ∗( −5∗ i ∗ w_so)^2+d ∗( −5∗ i ∗ w_so)+e ]) ; %with Rload H_5jwsoPluswm = t f ( [ a a ∗ 10 ∗ 1 i ∗ w_so −a ∗ 25 ∗ w_so^2] ,[ b (b ∗ 15 ∗ i ∗ w_so+c ) (−b ∗ 75 ∗ w_so^2+c ∗ 10 ∗ i ∗ w_so+d) (e −25∗ c ∗ w_so^2 − 125 ∗ b ∗ i ∗ w_so^3+5∗ d ∗ i ∗ w_so) ] ) ; H_5jwsoPluswmConj = t f ( [ a a ∗ 10 ∗ i ∗ w_so −a ∗ 25 ∗ w_so^2] ,[ − b (−b ∗ 15 ∗ i ∗ w_so+c ) (b ∗ 75 ∗ w_so ^2+c ∗ 10 ∗ i ∗ w_so−d) (125 ∗ b ∗ i ∗ w_so^3− c ∗ 25 ∗ w_so^2−5∗d ∗ i ∗ w_so+e ) ] ) ; %with Rload H_5jwsoMinuswm = t f ( [ a −10∗ i ∗ w_so ∗ a −25∗ a ∗ w_so^2] ,[ − b (b ∗ 15 ∗ i ∗ w_so+c ) (b ∗ 75 ∗ w_so^2− c ∗ 10 ∗ i ∗ w_so−d) (e − 125 ∗ b ∗ j ∗ w_so^3 −25∗ c ∗ w_so^2+5∗ d ∗ i ∗ w_so) ] ) ; H_5jwsoMinuswmConj = t f ( [ a −10∗ i ∗ w_so ∗ a −25∗ a ∗ w_so^2] ,[ b (−b ∗ 15 ∗ i ∗ w_so+c ) (−b ∗ 75 ∗ w_so^2− c ∗ 10 ∗ i ∗ w_so+d) (e−c ∗ 25 ∗ w_so^2+125∗ b ∗ i ∗ w_so^3−5∗d ∗ i ∗ w_so) ] ) ; %with Rload H_5jwsoPlus2wm = t f ([4 ∗ a 20 ∗ a ∗ i ∗ w_so −25∗ a ∗ w_so^2] ,[8 ∗ b (60 ∗ b ∗ i ∗ w_so+4∗ c ) (−b ∗ 150 ∗ w_so^2+20∗ i ∗ c ∗ w_so+2∗d) ( e+5∗d ∗ i ∗ w_so−125∗ b ∗ i ∗ w_so^3 −25∗ c ∗ w_so^2) ] ) ; H_5jwsoPlus2wmConj = t f ([4 ∗ a 20 ∗ a ∗ i ∗ w_so −25∗ a ∗ w_so^2] ,[ − 8 ∗ b ( −60∗ b ∗ i ∗ w_so+4∗ c ) (b ∗ 150 ∗ w_so^2+20 ∗ i ∗ w_so ∗ c −2∗d) (125 ∗ b ∗ i ∗ w_so^3 −25∗ c ∗ w_so^2 −5∗ d ∗ i ∗ w_so+e ) ] ) ; %with Rload H_5jwsoMinus2wm = t f ([4 ∗ a −20∗ a ∗ i ∗ w_so −25∗ a ∗ w_so^2] ,[ − 8 ∗ b (60 ∗ b ∗ i ∗ w_so+4∗ c ) (150 ∗ b ∗ w_so^2 −20∗ c ∗ i ∗ w_so−2∗d) ( e+5∗d ∗ i ∗ w_so−125∗ b ∗ j ∗ w_so^3 −25∗ c ∗ w_so^2) ] ) ; H_5jwsoMinus2wmConj = t f ([4 ∗ a −20∗ a ∗ i ∗ w_so −25∗ a ∗ w_so^2] ,[8 ∗ b ( −60∗ b ∗ i ∗ w_so+4∗ c ) ( − 150 ∗ b ∗ w_so^2 −20∗ c ∗ i ∗ w_so+2∗d) (e −5∗ i ∗ d ∗ w_so−25∗ c ∗ w_so^2+125∗ b ∗ i ∗ w_so^3) ] ) ; %wth Rload H_5jwsoPlus3wm = t f ([9 ∗ a 30 ∗ a ∗ i ∗ w_so −25∗ a ∗ w_so^2] ,[27 ∗ b (135 ∗ b ∗ i ∗ w_so+9∗ c ) ( − 225 ∗ b ∗ w_so^2+30∗ c ∗ i ∗ w_so+3∗d) ( e+5∗d ∗ i ∗ w_so−25∗ c ∗ w_so^2 −125∗ b ∗ i ∗ w_so^3) ] ) ; H_5jwsoPlus3wmConj = t f ([9 ∗ a 30 ∗ a ∗ i ∗ w_so −25∗ a ∗ w_so^2] ,[ − 27 ∗ b ( − 135 ∗ b ∗ i ∗ w_so+9∗ c ) (225 ∗ b ∗ w_so^2+30∗ c ∗ i ∗ w_so−3∗d) (e −5∗ i ∗ d ∗ w_so−25∗ c ∗ w_so^2+125∗ b ∗ i ∗ w_so^3) ] ) ; %with Rload H_5jwsoMinus3wm = t f ([9 ∗ a −30∗ a ∗ i ∗ w_so −25∗ a ∗ w_so^2] ,[ − 27 ∗ b (135 ∗ b ∗ i ∗ w_so+9∗ c ) (225 ∗ b ∗ w_so^2 −30∗ c ∗ i ∗ w_so−3∗d) ( e+5∗d ∗ i ∗ w_so−25∗ c ∗ w_so^2 −125∗ b ∗ i ∗ w_so^3) ] ) ; H_5jwsoMinus3wmConj = t f ([9 ∗ a −30∗ a ∗ i ∗ w_so −25∗ a ∗ w_so^2] ,[27 ∗ b ( − 135 ∗ b ∗ i ∗ w_so+9∗ c ) ( − 225 ∗ b ∗ w_so^2 −30∗ c ∗ i ∗ w_so+3∗d) (e −5∗d ∗ i ∗ w_so−25∗ c ∗ w_so^2+125∗ b ∗ i ∗ w_so^3) ] ) ; %fifthHarmonic , sideband0 alpha0_5 = 0; J0_5 = b e s s e l j ( alpha0_5 , mag_beta) ; [mag_H_5jwso phase_H_5jwso ] = bode (H_5jwso , logspace (1 ,6 ,200) ) ; mag_H_5jwso = squeeze (mag_H_5jwso) ; phase_H_5jwso = squeeze (phase_H_5jwso) ; 70  364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407  [ mag_H_5jwsoConj phase_H_5jwsoConj ] = bode (H_5jwsoConj , logspace (1 ,6 ,200) ) ; mag_H_5jwsoConj = squeeze (mag_H_5jwsoConj) ; phase_H_5jwsoConj = squeeze (phase_H_5jwsoConj) ; Co_mag = V_s5∗ J0_5. ∗ mag_H_5jwso; CoConj_mag = V_s5∗ J0_5. ∗ mag_H_5jwsoConj ; %fifthHarmonic , sideband1 alpha1_5 = 1; J1_5 =b e s s e l j ( alpha1_5 , mag_beta) ; %5 plus [ mag_H_5jwsoPluswm phase_H_5jwsoPluswm ] = bode (H_5jwsoPluswm , logspace (1 ,6 ,200) ) ; mag_H_5jwsoPluswm = squeeze (mag_H_5jwsoPluswm) ; phase_H_5jwsoPluswm = squeeze (phase_H_5jwsoPluswm) ; [ mag_H_5jwsoPluswmConj phase_H_5jwsoPluswmConj ] = bode (H_5jwsoPluswmConj , logspace (1 ,6 ,200) ) ; mag_H_5jwsoPluswmConj = squeeze (mag_H_5jwsoPluswmConj) ; phase_H_5jwsoPluswmConj = squeeze (phase_H_5jwsoPluswmConj) ; Cu_mag = V_s5∗ J1_5. ∗ mag_H_5jwsoPluswm ; CuConj_mag = V_s5∗ J1_5. ∗ mag_H_5jwsoPluswmConj ; %5minus [mag_H_5jwsoMinuswm phase_H_5jwsoMinuswm ] = bode (H_5jwsoMinuswm , logspace (1 ,6 ,200) ) ; mag_H_5jwsoMinuswm = squeeze (mag_H_5jwsoMinuswm) ; phase_H_5jwsoMinuswm = squeeze (phase_H_5jwsoMinuswm) ; [ mag_H_5jwsoMinuswmConj phase_H_5jwsoMinuswmConj ] = bode (H_5jwsoMinuswmConj , logspace (1 ,6 ,200) ) ; mag_H_5jwsoMinuswmConj = squeeze (mag_H_5jwsoMinuswmConj) ; phase_H_5jwsoMinuswmConj = squeeze (phase_H_5jwsoMinuswmConj) ; Cl_mag = −V_s5∗ J1_5. ∗ mag_H_5jwsoMinuswm; ClConj_mag = −V_s5∗ J1_5. ∗ mag_H_5jwsoMinuswmConj ; %fifthHarmonic , sideband2 alpha2_5 = 2; J2_5 =b e s s e l j ( alpha2_5 , mag_beta) ; %5 plus2 [ mag_H_5jwsoPlus2wm phase_H_5jwsoPlus2wm ] = bode (H_5jwsoPlus2wm , logspace (1 ,6 ,200) ) ; mag_H_5jwsoPlus2wm = squeeze (mag_H_5jwsoPlus2wm) ; phase_H_5jwsoPlus2wm = squeeze (phase_H_5jwsoPlus2wm) ;  408  71  409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449  [ mag_H_5jwsoPlus2wmConj phase_H_5jwsoPlus2wmConj ] = bode (H_5jwsoPlus2wmConj , logspace (1 ,6 ,200) ) ; mag_H_5jwsoPlus2wmConj = squeeze (mag_H_5jwsoPlus2wmConj) ; phase_H_5jwsoPlus2wmConj = squeeze (phase_H_5jwsoPlus2wmConj) ; Cu2_mag = V_s5∗ J2_5. ∗ mag_H_5jwsoPlus2wm ; Cu2Conj_mag = V_s5∗ J2_5. ∗ mag_H_5jwsoPlus2wmConj ; %5minus2 [mag_H_5jwsoMinus2wm phase_H_5jwsoMinus2wm ] = bode (H_5jwsoMinus2wm , logspace (1 ,6 ,200) ) ; mag_H_5jwsoMinus2wm = squeeze (mag_H_5jwsoMinus2wm) ; phase_H_5jwsoMinus2wm = squeeze (phase_H_5jwsoMinus2wm) ; [ mag_H_5jwsoMinus2wmConj phase_H_5jwsoMinus2wmConj ] = bode (H_5jwsoMinus2wmConj , logspace (1 ,6 ,200) ) ; mag_H_5jwsoMinus2wmConj = squeeze (mag_H_5jwsoMinus2wmConj) ; phase_H_5jwsoMinus2wmConj = squeeze (phase_H_5jwsoMinus2wmConj) ; Cl2_mag = −V_s5∗ J2_5. ∗ mag_H_5jwsoMinus2wm; Cl2Conj_mag = −V_s5∗ J2_5. ∗ mag_H_5jwsoMinus2wmConj ; %fifthHarmonic , sideband3 alpha3_5 = 3; J3_5 =b e s s e l j ( alpha3_5 , mag_beta) ; %5 plus3 [ mag_H_5jwsoPlus3wm phase_H_5jwsoPlus3wm ] = bode (H_5jwsoPlus3wm , logspace (1 ,6 ,200) ) ; mag_H_5jwsoPlus3wm = squeeze (mag_H_5jwsoPlus3wm) ; phase_H_5jwsoPlus3wm = squeeze (phase_H_5jwsoPlus3wm) ; [ mag_H_5jwsoPlus3wmConj phase_H_5jwsoPlus3wmConj ] = bode (H_5jwsoPlus3wmConj , logspace (1 ,6 ,200) ) ; mag_H_5jwsoPlus3wmConj = squeeze (mag_H_5jwsoPlus3wmConj) ; phase_H_5jwsoPlus3wmConj = squeeze (phase_H_5jwsoPlus3wmConj) ; Cu3_mag = V_s5∗ J3_5. ∗ mag_H_5jwsoPlus3wm ; Cu3Conj_mag = V_s5∗ J3_5. ∗ mag_H_5jwsoPlus3wmConj ; %5minus3 [mag_H_5jwsoMinus3wm phase_H_5jwsoMinus3wm ] = bode (H_5jwsoMinus3wm , logspace (1 ,6 ,200) ) ; mag_H_5jwsoMinus3wm = squeeze (mag_H_5jwsoMinus3wm) ; phase_H_5jwsoMinus3wm = squeeze (phase_H_5jwsoMinus3wm) ; [ mag_H_5jwsoMinus3wmConj phase_H_5jwsoMinus3wmConj ] = bode (H_5jwsoMinus3wmConj , logspace (1 ,6 ,200) ) ; 72  450 451 452 453 454 455 456 457 458 459 460 461  mag_H_5jwsoMinus3wmConj = squeeze (mag_H_5jwsoMinus3wmConj) ; phase_H_5jwsoMinus3wmConj = squeeze (phase_H_5jwsoMinus3wmConj) ; Cl3_mag = −V_s5∗ J3_5. ∗ mag_H_5jwsoMinus3wm; Cl3Conj_mag = −V_s5∗ J3_5. ∗ mag_H_5jwsoMinus3wmConj ; c1 = (Co_mag. ∗ ClConj_mag)+(Cu_mag. ∗ CoConj_mag)+(Cl_mag. ∗ Cl2Conj_mag)+(Cu2_mag. ∗ CuConj_mag) ; c2 = (Co_mag. ∗ Cl2Conj_mag)+(Cu_mag. ∗ ClConj_mag)+(Cl_mag. ∗ Cl3Conj_mag)+(Cu2_mag. ∗ CoConj_mag)+(Cu3_mag. ∗ CuConj_mag) ; c3 = (Co_mag. ∗ Cl3Conj_mag)+(Cu_mag. ∗ Cl2Conj_mag)+(Cu2_mag. ∗ ClConj_mag)+(Cu3_mag. ∗ CoConj_mag) ; fifthHarmonicOutput = ( sqrt ((1/3) ∗ ( c1.^2+c2.^2+c3 .^2) ) ) ./(Co_mag) ; fifthHarmonicOutput = ((1/ delta_w ) ∗ fifthHarmonicOutput ) ;  462 463 464 465 466 467 468 469 470 471 472 473  finalOutput = ( fundamentalOutput+thirdHarmonicOutput+fifthHarmonicOutput ) ∗ 2 ∗ pi ∗ Tratio . ∗ ( mag_ESRzero) .^1. ∗ (mag_RHPzero) .^2. ∗ (mag_ZOHpole) .^3; % plots plota = semilogx (wout/(2 ∗ pi ) ,20 ∗ log10 ( fundamentalOutput ) , 'b ' ) ; hold on , plotb = semilogx (wout/(2 ∗ pi ) ,20 ∗ log10 ( thirdHarmonicOutput ) , 'g ' ) ; hold on , plotc = semilogx (wout/(2 ∗ pi ) ,20 ∗ log10 ( fifthHarmonicOutput ) , ' r ' ) ; hold on , hold on , plotd= semilogx (wout/(2 ∗ pi ) ,20 ∗ log10 ( finalOutput ) , 'k ' ) hold on , bode ( [ 0 ] , [ 1 ] , [ 2 ∗ pi ∗ 600 ,2 ∗ pi ∗ 1e4 ] ) hold on legend ( [ plota , plotb , plotc , plotd ] )  73  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.24.1-0073435/manifest

Comment

Related Items