A DETAILED ANALYSIS OF THE IMPERFECTIONS OF PULSEWIDTH MODULATED WAVEFORMS ON THE OUTPUT STAGE OF A CLASS D AUDIO AMPLIFIER

Transcription

1 A DETAILED ANALYSIS OF THE IMPERFECTIONS OF PULSEWIDTH MODULATED WAVEFORMS ON THE OUTPUT STAGE OF A CLASS D AUDIO AMPLIFIER Francois Koeslag Disseraion presened in parial fulfilmen of he requiremens for he degree of Docor of Philosophy in Engineering a he Universiy of Sellenbosch Supervisor: Prof. H. du T. Mouon Co-supervisor: Dr. H.J. Beukes December 8

2 DECLARATION By submiing his disseraion elecronically, I declare ha he enirey of he work conained herein is my own, original work, ha I am he owner of he copyrigh hereof (unless o he exen explicily oherwise saed) and ha I have no previously in is enirey or in par submied i for obaining any qualificaion. - i -

3 SUMMARY Alhough he Class D opology offers several advanages, is use in audio amplificaion has previously been limied by he lack of compeiiveness in fideliy compared o is linear counerpars. During he pas decade, echnological advances in semiconducor echnology have awakened new ineres since compeiive levels of disorion could now be achieved. The oupu sage of such an amplifier is he primary limiing facor in is performance. In his disseraion, four non-ideal effecs exising in his sage are idenified and mahemaically analysed. The analyical analysis makes use of a well-esablished mahemaical model, based on he double Fourier series mehod, o model he imperfecions inroduced ino a naurally sampled pulsewidh modulaed waveform. The analysis is complemened by simulaion using a sraegy based on Newon s numerical mehod. The heory is verified by a comparison beween he analyical-, simulaed- and experimenal resuls. - ii -

4 OPSOMMING Die klas D opologie bied verskeie voordele, maar die oepassing daarvan in oudio verserkers was beperk o op hede as gevolg van onvergelykbare vlakke van disorsie in vergelyking me analoog verserkers. Tegnologiese vooruigang in halfgeleier egnologie oor die laase dekade he o nuwe belangselling gelei in die oepassing van die klas D opologie in oudio verserkers, siende da kompeerende vlakke van disorsie nou haalbaar was. Die uireesadium van hierdie verserkers is die beperkende fakor in disorsie. Hierdie proefskrif idenifiseer en analiseer vier nie-ideale effeke wiskundig. Daar word gebruik gemaak van n wel bekende meode, gebaseer op die dubbele Fourier reeksuibreiding, om die nie-idealieie in n nauurlik gemonserde pulswyde gemoduleerde golfvorm e modelleer. Die analise word aangevul deur simulasies gebaseer op n sraegie wa gebruik maak van Newon se numeriese meode. Die eorie word geverifieer deur n vergelyking ussen die analiiese-, simulasie- en eksperimenele resulae. - iii -

5 ACKNOWLEDGEMENTS I would like o hank Prof. H. du Toi Mouon and Dr. H.J. Beukes who boh iniiaed his projec and organized he necessary funding. Thank you for he echnical suppor and paience hroughou he projec. The Naional Research Foundaion (NRF) for heir financial suppor. My parens Ronald and Ilse for he suppor and he valuable opporuniy hey gave me. Finally, I would like o hank God for graning me he academic abiliy o complee his disseraion. - iv -

6 TABLE OF CONTENTS 1 Lieraure Inroducion 1 1. Basic Concep and Developmen Sysem Imperfecions Circui Definiions, Scope and General Assumpions Exising Lieraure and Conribuions Analyical Deerminaion of he Specrum of NPWM Dead Time Non-Zero Turn-On and Turn-Off Delays Non-Zero Turn-On and Turn-Off Swiching Transiions Parasiics and Reverse Recovery Disseraion Ouline.. 13 A Fundamenal Analysis of PWM.15.1 Inroducion Fundamenal Conceps of PWM The Analyical Specrum of PWM Specral Plos and General Discussion Simulaion Sraegy The Newon-Raphson Numerical Mehod Cross-Poin Calculaion Using Newon-Raphson s Mehod Simulaion Resuls.46 General Discussion Summary Incorporaion of PTEs in he Double Fourier Series Mehod Inroducion Realisic Inducor Curren Model Incorporaion of Time Delays Modulaed Edge v -

7 3.3. Unmodulaed Edge Incorporaion of a Sinusoidal Curren Polariy Dependency Polariy Dependency Sampled on he Trailing Edge Polariy Dependency Sampled on he Leading Edge Incorporaion of a Purely Sinusoidal Non-Linear Inducor Curren Magniude Dependency in he 3-D Uni Area The Unmodulaed Leading Edge The Modulaed Trailing Edge Incorporaion of Secion 3. wihin he 3-D Uni Area Summary Swiching Device Characerisics Inroducion Power MOSFET Srucure Power MOSFET Operaion Characerisic Curves Power MOSFET Dynamic Model Power MOSFET Swiching Waveforms Summary The Effec of Dead Time Inroducion Analysis of Dead Time A Disincly Posiive and Negaive Inducor Curren (Scenario ) Neiher Disincly Posiive nor Negaive Inducor Curren (Scenario ) Analyical Model Simulaion Sraegy Analyical and Simulaion Resuls Harmonic Composiion of TENPWM wih Dead Time General Relaion o Circui Parameers Summary. 9 6 The Effec of he MOSFET Turn-On and Turn-Off Delays 93 - vi -

8 6.1 Inroducion Analysis of he Turn-On and Turn-Off Delays Error Descripion Simulaion Sraegy Simulaion Resuls Baseband Harmonics Sideband Harmonics Summary.1 7 The Effec of Non-Zero, Non-Linear Swiching Transiions Inroducion Analysis of Non-Zero Rise and Fall Swiching Transiions Soluions o he Expressions of he Swiching Curves Consan Gae-o-Drain Capaciance Dynamic Gae-o-Drain Capaciance Error Descripion Simulaion Sraegy Simulaion Resuls Baseband Harmonics Sideband Harmonics Summary The Effec of Parasiics and Reverse Recovery Inroducion Analysis of he Parasiics and Reverse Recovery Analysis of a Half-Bridge Topology Analysis for a Full-Bridge Topology General Observaions and Commens Simulaion Sraegy Simulaion Resuls Baseband Harmonics Sideband Harmonics Summary vii -

9 9 Combinaion Model and Experimenal Verificaion Inroducion Overview of he Individual Non-Lineariies The Dead Time The Turn-On and Turn-Off Delays The Non-Zero, Non-Linear Swiching Transiions The Parasiics and Reverse Recovery The Inclusion of Noise Combinaion Models Inducor Curren Scenario Inducor Curren Scenario Experimenal Verificaion Summary Conclusions and Fuure Work Inroducion A Fundamenal Analysis of PWM Incorporaion of PTEs in he Double Fourier Series Mehod Swiching Device Characerisics The Effec of Dead Time The Effec of he MOSFET Turn-On and Turn-Off Delays The Effec of Non-Zero, Non-Linear Swiching Transiions The Effec of Parasiics and Reverse Recovery Combinaion Model and Experimenal Verificaion Fuure Work viii -

10 SUMMARY OF PAPERS PRESENTED Conference Papers F. Koeslag, H. du T. Mouon, H.J. Beukes and P. Midya, A Deailed Analysis of he Effec of Dead Time on Harmonic Disorion in a Class D Audio Amplifier, Africon 7, Windhoek, Namibia, 6-8 Ocober 7. F. Koeslag, H. du T. Mouon and H.J. Beukes, The Isolaed Effec of Finie Non-Linear Swiching Transiions on Harmonic Disorion in a Class D Audio Amplifier, 17 h Souh African Universiies Power Engineering Conference (SAUPEC), Durban, Souh Africa, January 8. F. Koeslag, H. du Toi Mouon, H.J. Beukes, An Invesigaion ino he Separae and Combined Effec of Pulse Timing Errors on Harmonic Disorion in a Class D Audio Amplifier, 39 h Annual IEEE Power Elecronics Specialiss Conference (PESC), Rhodes, Greece, June 8. Journal Papers F. Koeslag, H. du Toi Mouon, H.J. Beukes, Analyical Calculaion of he Oupu Harmonics in a Power Elecronic Inverer wih Curren Dependen Pulse Timing Errors, Submied o Journal. - ix -

11 LIST OF FIGURES Figure 1.1 Basic circui parameers and definiions...4 Figure.1 (a) Leading edge, (b) railing edge and (c) double edge modulaion [17] Figure. Generaion of (a) LENPWM, (b) TENPWM and (c) DENPWM Figure.3 (a) LEUPWM, (b) TEUPWM, (c) symmerical DEUPWM and (d) asymmerical DEUPWM Figure.4 Definiion of he 3-D area inroduced by W.R. Benne [11] Figure.5 Appropriae scaling of he 3-D uni area of Figure Figure.6 (a) Sawooh carrier waveform and (b) modulaing waveform for LENPWM... Figure.7 3-D uni area for LENPWM... 3 Figure.8 (a) Sawooh carrier waveform and (b) modulaing waveform for TENPWM... 6 Figure.9 3-D uni area for TENPWM... 7 Figure.1 (a) Triangular carrier waveform and (b) modulaing waveform for DENPWM... 3 Figure.11 3-D uni area for DENPWM... 3 Figure.1 Analyical volage specrum of (a) LENPWM (or TENPWM), and (b) DENPWM Figure.13 Geomeric represenaion of he Newon-Raphson numerical mehod Figure.14 Generaion of TENPWM Figure.15 Generaion of DENPWM... 4 Figure.16 (a) Time domain and (b) magniude specrum represenaion of a recangular pulse Figure.17 The sinc funcion Figure.18 Time shifing of he recangular pulse of Figure.16 (a) Figure.19 (a) Analyical and (b) simulaed specrum of TENPWM wih c / = Figure 3.1 Volage across and curren hrough he inducor Figure 3. Definiion of he inducor curren for (a) Scenario and (a) Scenario... 5 Figure 3.3 Definiion of d inroduced on he modulaed edge Figure 3.4 Definiion of he consan ime delay mapping d ' wihin he 3-D uni area x -

12 Figure 3.5 -D represenaion of he error inroduced for he modulaed edge Figure D uni area for TENPWM wih delay d Figure 3.7 Definiion of d inroduced on he unmodulaed edge Figure 3.8 -D represenaion of he error inroduced for he unmodulaed edge Figure 3.9 Proposed arbirary inducor curren polariy dependency sampled on he ideal railing edge Figure 3.1 Definiion of he various curren zones wihin he 3-D uni area Figure 3.11 Combinaion of Figure 3.9 and Figure 3.1 illusraing he sampling process... 6 Figure D uni area for TENPWM wih curren polariy dependency Figure 3.13 Proposed arbirary inducor curren polariy dependency sampled on he ideal leading edge... 6 Figure 3.14 Definiion of he various curren zones wihin he 3-D uni area Figure 3.15 Combinaion of Figure 3.13 and Figure 3.14 illusraing he sampling process Figure D uni area for TENPWM wih curren polariy dependency Figure 3.17 Simulaed TENPWM specra for sampling on he (a) railing and (b) leading edge Figure D uni area for TENPWM wih curren magniude dependency Figure D uni area for TENPWM wih curren magniude dependency Figure 3. Definiion of he various curren zones for i L(upper_env) wihin he 3-D uni area... 7 Figure 3.1 Definiion of he various curren zones for i L(lower_env) wihin he 3-D uni area Figure 4.1 Verical cross-secional view of a power MOSFET [4] Figure 4. (a) Oupu and (b) ransfer characerisic curves [4] Figure 4.3 Verical cross-secional view of a power MOSFET wih parasiic capaciances [4] Figure 4.4 C GD as a funcion of v DS [4] Figure 4.5 Circui model when power MOSFET is in he (a) acive and (b) ohmic region [4] Figure 4.6 Swiching characerisics in a single phase leg for a posiive inducor curren [3] xi -

13 Figure 4.7 Swiching characerisics in a single phase leg for a negaive inducor curren [3]... 8 Figure 5.1 Commuaion sequence in a single phase leg for (a) i LA > and (b) i LA <... 8 Figure 5. Swiching waveforms for a posiive inducor curren. (a) Low side gae-osource volage. (b) High side gae-o-source volage. (c) Inducor curren. (d) Swiched oupu volage Figure 5.3 Swiching waveforms for a negaive inducor curren. (a) High side gae-osource volage. (b) Low side gae-o-source volage. (c) Inducor curren. (d) Swiched oupu volage Figure 5.4 Change in curren polariy (a) ouside inerval d and (b) wihin inerval d [35] Figure D uni area for TENPWM wih dead ime Figure 5.6 Generaion of TENPWM wih dead ime for (a) i L > and (b) i L < Figure 5.7 (a) Analyical (m=) and (b) simulaed baseband harmonics for TENPWM Figure 5.8 Analyical specrum for (a) m=1 and 383n 364 and (b) combinaion wih Figure 5.7 (a) Figure 5.9 Simulaed specrum showing he firs wo carrier harmonics and is respecive sidebands... 9 Figure 5.1 Simulaed THD as a funcion of (a) d and (b) M Figure 5.11 Simulaed THD as a funcion of L fil for (a) d =15ns and (b) d =5ns Figure 6.1 Swiching waveforms for a posiive inducor curren. (a) High side gae-osource volage. (b) Low side gae-o-source volage. (c) Inducor curren. (d) Swiched oupu volage Figure 6. Swiching waveforms for a negaive inducor curren. (a) Low side gae-osource volage. (b) High side gae-o-source volage. (c) Inducor curren. (d) Swiched oupu volage Figure 6.3 d(vr) for (a) Scenario and (b) Scenario Figure 6.4 d(vf) for for (a) Scenario and (b) Scenario Figure 6.5 Generaion of TENPWM wih urn-on and urn-off delays xii -

14 Figure 6.6 Simulaed TENPWM baseband harmonics for (a) Scenario and (b) Scenario... 1 Figure 6.7 Simulaed TENPWM specra for (a) Scenario and (b) Scenario Figure 7.1 Swiching waveforms for a posiive inducor curren. (a) High side gae-osource volage. (b) Low side gae-o-source volage. (c) Inducor curren. (d) Swiched oupu volage Figure 7. Swiching waveforms for a negaive inducor curren. (a) Low side gae-osource volage. (b) High side gae-o-source volage. (c) Inducor curren. (d) Swiched oupu volage Figure 7.3 (a) Approximae and acual curve of C GD as a funcion of v DS on a log scale. (b) Approximae curve of C GD as a funcion of v DS on a linear scale for par IRFI419H-117P Figure 7.4 Rise ime for (a) Scenario and (b) Scenario Figure 7.5 Fall ime for (a) Scenario and (b) Scenario Figure 7.6 Difference in fall ime for (a) Scenario and (b) Scenario Figure 7.7 Generaion of TENPWM wih non-zero rise and fall imes Figure 7.8 Simulaed TENPWM baseband harmonics for (a) Scenario and (b) Scenario Figure 7.9 Simulaed TENPWM specra for (a) Scenario and (b) Scenario Figure 8.1 Equivalen circui model of he urn-off process [31] Figure 8. Measured swiching oupu volage ransiions from (a) on o off and (b) off o on Figure 8.3 Analyically mached waveforms of (a) Figure 8. (a), and (b) Figure 8. (b)... 1 Figure 8.4 Measured volage envelopes and inducor curren for (a) M=.1 and (c) M=.. Measured peak overvolage a he cres of he overvolage envelope for (b) M=.1 and (d) M= Figure 8.5 Analyically reconsruced volage envelopes of Figure 8.4 for (a) M=.1 and (b) M= Figure 8.6 Measured volage envelopes and inducor curren for (a) M=.5 and (c) M=.8. Measured peak overvolage a he cres of he overvolage envelope for (b) M=.5 and (d) M= xiii -

15 Figure 8.7 (a) Measured volage envelopes for M=.8. (b) Measured peak overvolage a he cres of he overvolage envelope wih V d =V Figure 8.8 Inrinsic power diode curren swiching characerisic during urn-off [4] Figure 8.9 Analyically reconsruced volage envelopes of Figure 8.6 for (a) M=.5 and (b) M= Figure 8.1 Measured swiching oupu volage ransiions from (a) on o off and (b) off o on Figure 8.11 Analyically mached waveforms of (a) Figure 8.1 (a), and (b) Figure 8.1 (b) Figure 8.1 Measured volage envelopes for (a) M=. and (b) M= Figure 8.13 Measured volage envelopes a (a) M=. and (b) M=.8 for V d =V Figure 8.14 Measured volage envelopes for M=.8 for (a) V GS =11.5V and (b) V GS =1V Figure 8.15 Generaion of TENPWM wih v DS Figure 8.16 Simulaed baseband harmonics wih v DS for (a) Scenario and (b) Scenario Figure 8.17 Simulaed sideband harmonics wih v DS for (a) Scenario and (b) Scenario Figure 9.1 Simulaed THD vs. M wih non-zero d for (a) Scenario and (b) Scenario..139 Figure 9. Simulaed THD vs. M wih non-zero d(vr) and d(vf) for (a) Scenario and (b) Scenario Figure 9.3 (a) Rising and (b) falling swiching ransiion curves for L fil =. 14 Figure 9.4 (a) Rising and (b) falling swiching ransiion curves for Scenario Figure 9.5 Swiching curves during (a) vr for i L <, (b) vr for i L >, (c) vf for i L > and (d) vf for i L < Figure 9.6 Noise vs. M for (a) Scenario and (b) Scenario Figure 9.7 Combinaion 3-D uni area for TENPWM for Scenario Figure 9.8 Combinaion simulaion model for Scenario Figure 9.9 (a) Analyical (m=) and (b) simulaed baseband harmonics for TENPWM for M= Figure 9.1 Rising egde ransiion a (a) i L(lower_env) = and (b) i L(lower_env) = xiv -

16 Figure 9.11 Variaion in (a) vr and (b) vf over a complee cycle of he modulaing waveform..149 Figure 9.1 Combinaion 3-D uni area for TENPWM for Scenario...15 Figure 9.13 Combinaion simulaion model for Scenario, (a) i L(lower_env) > and (b) i L(upper_env) < Figure 9.14 (a) Analyical (m=) and (b) simulaed baseband harmonics for TENPWM for M= Figure 9.15 THD+N vs. M deermined by (a) measuremen and (b) simulaion Figure 9.16 Various measured and simulaed specra Figure 9.17 Duy cycle error for (a) M=. and (b) M= Figure 9.18 THD+N vs. M deermined by (a) measuremen and (b) simulaion for various L fil xv -

17 LIST OF TABLES Table.1 Comparison of analyical and simulaion resuls for TENPWM 46 Table. Comparison of analyical and simulaion resuls for DENPWM Table.3 Analyical and simulaed harmonic magniudes of he specrum shown in Figure...48 Table 3.1 Definiion of he basic variables used hroughou his Chaper..53 Table 3. Comparison of analyical and simulaion resuls for TENPWM wih a consan ime delay inroduced on he modulaed edge..56 Table 3.3 Comparison of analyical and simulaion resuls for TENPWM wih inducor curren polariy condiion sampled on he railing edge..61 Table 3.4 Comparison of analyical and simulaion resuls for TENPWM wih inducor curren polariy condiion sampled on he leading edge. 65 Table 3.5 Comparison of analyical and simulaion resuls for TENPWM wih inducor curren magniude dependency sampled on he railing edge. 68 Table 3.6 Comparison of analyical and simulaion resuls for TENPWM wih inducor curren magniude dependency sampled on he Leading edge Table 4.1 Descripion of he acion on he various ime insans of Figure Table 5.1 Analyical and simulaed magniude of he baseband harmonics for TENPWM Table 6.1 Simulaed THD for TENPWM for various R G..1 Table 6. Simulaed THD for TENPWM for various V GS Table 7.1 Simulaed THD for linear and non-linear swiching ransiions for TENPWM for various R G Table 7. Simulaed THD for linear and non-linear swiching ransiions for TENPWM for various V GS Table 8.1 Various variables for expressing he upper envelope of he overvolage for Scenario Table 8. Various variables for expressing he Lower envelope of he undervolage for Scenario. 15 Table 8.3 Various variables for expressing he upper envelope of he overvolage for Scenario xvi -

18 Table 8.4 Table 9.1 Table 9. Table 9.3 Various variables for expressing he lower envelope of he undervolage for Scenario Definiion of he basic variables used hroughou his Chaper 138 Pracical Device Descripion and Measuremen Seup.153 Definiion of he scaling variables for various L fil xvii -

19 LIST OF ACRONYMS -D Two Dimenional 3-D Three Dimenional DC Direc Curren LEPWM Double Edge Pulsewidh Modulaion (Uniform or Naural) DEUPWM Double Edge Uniformly Sampled Pulsewidh Modulaion FPGA Field Programmable Grid Array HF High Frequency LEPWM Leading Edge Pulsewidh Modulaion (Uniform or Naural) LEUPWM Leading Edge Uniformly Sampled Pulsewidh Modulaion MOSFET Meal Oxide Semiconducor Field Effec Transisor MOSFETs Meal Oxide Semiconducor Field Effec Transisors NPWM Naurally Sampled Pulsewidh Modulaion PAE Pulse Ampliude Error PAEs Pulse Ampliude Errors PCM Pulse Code Modulaion PDM Pulse Duraion Modulaion PNPWM Pseudo Naurally Sampled Pulsewidh Modulaion PTE Pulse Timing Error PTEs Pulse Timing Errors PWM Pulsewidh Modulaion TEPWM Trailing Edge Pulsewidh Modulaion (Uniform or Naural) TENPWM Trailing Edge Nauarally Sampled Pulsewidh Modulaion TEUPWM Trailing Edge Uniformly Sampled Pulsewidh Modulaion UPWM Uniformly Sampled Pulsewidh Modulaion THD Toal Harmonic Disorion - xviii -

20 GLOSSARY BV DSS C fil C GD C GS C iss D D A1 D B1 D A D B f f c g m i A i B i L i L I L i D I D i o I o L fil M r DS(on) R G R load T A1 T B1 T A breakdown volage demodulaion filer capacior gae-o-drain capaciance gae-o-source capaciance inpu capaciance duy cycle diode of high side swich of phase leg A diode of high side swich of phase leg B diode of low side swich of phase leg A diode of low side swich of phase leg B reference frequency carrier/swiching frequency ransconducance inducor curren flowing in phase leg A inducor curren flowing in phase leg B inducor curren ripple componen inducor curren inducor curren (scalar) drain curren drain curren (scalar) oupu load curren oupu load curren (scalar) demodulaion filer inducor modulaion index on-sae resisance gae resisance load resisance high side swich of phase leg A high side swich of phase leg B low side swich of phase leg A - xix -

21 T B d d(vr) d(vf) vf vr T c V d v DS v o V DF v GS V GS V GS(h) c low side swich of phase leg B dead ime urn-on delay (rising volage) urn-off delay (falling volage) volage fall ime volage rise ime swiching period DC bus volage drain-o-source volage oupu load volage diode forward volage gae-o-source volage gae-o-source volage (scalar) gae-o-source hreshold volage angular modulaing frequency (rad/s) angular swiching frequency (rad/s) - xx -

22 1 LITERATURE 1.1 INTRODUCTION The Class D mode of operaion was originally inroduced in 1959 by Baxandall for he poenial applicaion in oscillaor circuis [1]. Since hen i has found several widespread applicaions in power elecronics. More recenly, audio amplifiers implemening his opology have emerged on a large scale. Alhough he Class D opology offers several advanages, is use has previously been limied by he lack of compeiiveness in fideliy compared o is linear counerpars. Unil recenly, his drawback has been he resul of limiaions in semiconducor echnology []. Due o echnological advances in his field during he pas decade, however, new ineres has been awakened in he applicaion of his opology in audio amplificaion, since compeiive levels of disorion could now be achieved []. Two primary moivaions currenly drive he research in his field of which efficiency can be regarded as he firs and mos imporan [3]. This increased efficiency over convenional analogue amplifiers has he effec of decreasing supply requiremens. Moreover, he lower power loss is also decreasing or even eliminaing he use of heasinks. The resulan higher levels of efficiency ranslae ino smaller, lower cos designs. The second moivaion is ha audio is increasingly derived from digial sources. This is an advanage since he oupu sage can be driven direcly from a digial signal afer pulse code modulaion (PCM) o pulsewidh modulaion (PWM) conversion, creaing a purely digial audio amplifier wihou he need for any digial-o-analogue conversion [4]. 1. BASIC CONCEPT AND DEVELOPMENT Class D sages operae in swiched mode, which means ha he power ransisors in he oupu sage are eiher fully on or fully off. Since only wo possible saes exis, such an oupu sage relies on he amplificaion of some binary inermediae signal

23 CHAPTER 1 INTRODUCTION The audio signal hus needs o be modulaed in and ou of his inermediae signal. PWM is currenly he mos popular form of his binary signal used in Class D audio amplifiers [5]. Early research in his field was based on his mehod in combinaion wih Class D power sages o accomplish he amplificaion [6], [7]. Several modulaion mehods currenly exis for creaing PWM, of which he earlies and mos basic is called naural sampling. This mehod uilizes he normal form of analogue signals direcly. The analogue audio inpu is compared o a reference waveform (modulaed) whose frequency is muliples higher han ha of he audio bandwidh in order o represen he inpu signal accuraely. The resuling swiching oupu is hen fed o he power sage, which performs he necessary amplificaion. The amplified audio signal is hen recovered (demodulaed) once he oupu waveform has been passed hrough a low-pass filer. Alernaively, when digial signals are available, he convenional mehod is firs o conver o analogue and hen o proceed wih he above menioned modulaion process. Since audio is increasingly derived from digial sources, he logical nex sep was o generae he PWM direcly from digial code. Early publicaions on his subjec proposed sysem archiecures running a high modulaor speeds of ens of GHz [6]. Such sysems were clearly impracical due o he limiaions posed by he power sage. The digial couner frequency was brough down o several ens of MHz by he inroducion of noise shaping echniques, bu performance was sill limied due o disorions inroduced by he modulaion process [5]. This problem was addressed by he inroducion of a echnique called he enhanced concep of power digial-o-analogue conversion [8]. This mehod relies on preprocessing of he inpu signal in order o compensae for he disorion inroduced by he modulaion process. Resuls from simulaions on his subjec were published as early as 199 [9]. The nex major advance in his field was he implemenaion of sigma-dela modulaion. However, early research again idenified he problem of high pulse frequencies which were degrading audio performance during amplificaion in he power sage [8]. This problem was addressed by reducing he pulse frequency wih a echnique called bi flipping. A complee design and implemenaion of his archiecure can be found in [1]. - -

24 CHAPTER 1 INTRODUCTION 1.3 SYSTEM IMPERFECTIONS I is well known ha PWM inroduces harmonic disorion. Early research based on he double Fourier series revealed his fac [11]. The analysis in [11] confirms ha he oupu specrum capures he inpu specrum, bu signal dependan harmonics proporional o ordinary Bessel funcions are creaed a muliples of he swiching frequency, accompanied by heir respecive sidebands. These sidebands appear in he audio band if he carrier frequency is no high enough, which resuls in disorion. However, by selecing a high carrier o fundamenal raio hese harmonics can be minimized o negligible values. A swiching frequency of more han en imes he modulaing frequency resuls in harmonic levels below 144 db [1]. From he previous secion i is eviden ha early research paid special aenion o he digial implemenaion of he conversion process o PWM. Several schemes were inroduced o minimize and overcome he limiaions posed by he uniformly sampled PWM (UPWM) modulaion process. Such an example is he cross-poin deecor found in he enhanced concep of power digial-o-analogue conversion. The original analogue waveform is represened by he uniform samples of he digial signal. However, from he specral analysis i is eviden ha uniform sampling inroduces disorion wihin he baseband [1], [13]. The linearizer improves (i.e. reduces) his disorion by approximaing naural sampling using a echnique called pseudo naurally sampled PWM (PNPWM). This is done by esimaing he crossing poin of he modulaing and reference waveform hrough inerpolaion of addiional daa poins using numerical mehods. An analogue PWM process is also prone o disorion. This can eiher be a resul of noise appearing in he modulaing signal or of non-ideal effecs associaed wih he reference waveform. Carrier non-lineariy leads o iming errors wihin he sampling process. Previous work has shown ha minor deviaions in carrier lineariy influence disorion significanly [14]. I is eviden ha naural sampling provides a basis for achieving low disorion. This suggess ha he power sage is he primary limiing facor in he performance of Class D audio amplifiers [1]. Several imperfecions in he oupu sage conribue o disorion. These non-ideal effecs can be caegorized ino wo main groups [15], i.e. pulse-iming errors (PTEs) and pulse-ampliude errors (PAEs). The former group is a resul of hree sources of which he firs is disorion due o power supply imperfecions. The oupu volage of a Class D audio - 3 -

25 CHAPTER 1 INTRODUCTION amplifier is direcly proporional o he supply volage. Any volage flucuaions caused as a resul of curren drawn by he amplifier inroduce an error in he oupu [16]. The second pulse ampliude error (PAE) resuls from he non-linear swich impedance. Finally, he ringing effec caused by he resonan acion beween he parasiic inducance and capaciance also leads o ampliude errors. PTEs exis as a resul of he non-ideal swiching characerisics of he power devices. These errors can be sub-divided ino wo groups, i.e. errors occurring as a primary or secondary consequence of he non-ideal swiching behaviour. Typical PTEs resuling from he primary consequence, assuming a power meal oxide semiconducor field effec ransisor (MOSFET) as swiching device, include he non-zero urn-on and urn-off delays, as well as he non-zero, non-linear urn-on and urn-off swiching ransiions. The well-known pulse iming error (PTE) resuling from dead ime can be classified as a secondary consequence of he swiching behaviour. The scope of his disseraion as well as general assumpions are considered in he following secion. 1.4 CIRCUIT DEFINITIONS, SCOPE AND GENERAL ASSUMPTIONS In his disseraion an invesigaion is launched ino he parameers affecing oal harmonic disorion (THD) in he oupu sages of Class D swiching audio amplifiers. A general sysem represenaion, discussed below, is shown in Figure 1.1. V d TA1 T A A ila D A1 DA L fil C fil R load L fil D B1 DB ilb B TB1 T B Figure 1.1: Basic circui parameers and definiions

26 CHAPTER 1 INTRODUCTION The focus of his invesigaion is limied o a discree, open loop sysem. The waveform generaed by he digial source is assumed o be a resul of a perfec, single-sided, wo-level PNPWM process, effecively reproducing NPWM. The oupu sage opology is assumed o be eiher a half-bridge or full-bridge configuraion wih he curren flow in each phase leg (denoed A and B) defined in Figure 1.1. The gae drivers are assumed o be ideal, i.e. hey swich in zero ime wih no propagaion delay exising beween he gaing signals supplied o he upper and lower meal oxide semiconducor field effec ransisors (MOSFETs) of each phase. The MOSFETs are considered perfecly mached. The imperfecions associaed wih he power supply and he filer are negleced. The basic definiions of he circui parameers defined in Figure 1.1 remain unchanged for he res of his disseraion. 1.5 EXISTING LITERATURE AND CONTRIBUTIONS A deailed sudy of exising lieraure falling wihin he scope of each aspec invesigaed in his disseraion will now be considered. This includes a general overview on each subjec involved herein, afer which a more focussed review of highly relevan repors is presened. Each sub-secion concludes wih he lis of conribuions made by his disseraion ANALYTICAL DETERMINATION OF THE SPECTRUM OF NPWM Pulsewidh modulaion is a non-linear process which resuls in a non-periodic pulse rain. This means ha one-dimenional Fourier analysis canno be applied. This complicaes he analyical deerminaion of he specrum significanly. W.R. Benne [11] and H.S. Black [17] inroduced a mehod for deermining he modulaion producs analyically by represening he pulse rain as a hree-dimensional (3-D) uni area. The analysis was originally proposed for use in communicaion sysems, and expanded o power converer sysems by S.R. Bowes [18] and B. Bird [19]. A furher sudy by D.G. Holmes [] involved he derivaion of an expression for uniform and naural sampling where he reference waveform is sinusoidal, requiring only one Bessel funcion muliplicaion for each harmonic. An alernaive mehod for deermining he specrum of PWM analyically has been inroduced by Z. Song and D.V. Sarwae [1]

27 CHAPTER 1 INTRODUCTION The specrum of PWM can also be found by applying he fas Fourier ransform (FFT) o a simulaed ime-varying swiched waveform (such as PSpice). This approach has boh advanages and disadvanages. One major advanage is he reducion in mahemaical effor compared o analyical compuaion. The downside is ha he ime resoluion of he simulaion has o be very high in order o produce accurae crossing poins beween he modulaing and carrier waveforms. This in urn requires significan compuing power, which is very ime-consuming. In conras, he analyical soluions exacly idenify he frequency componens creaed by he modulaion process. Moreover, he harmonic composiion of he waveform is also shown, i.e. he individual conribuions of he fundamenal low frequency componen, baseband harmonics, carrier harmonics as well as sideband harmonics o he specrum. This informaion canno be supplied via simulaion. Relevan Lieraure The analyical analysis presened in [] employs W.R. Benne s [11] mehod o esablish he harmonic composiion of PWM in he presence of a non-zero dead ime. The analysis shows how he 3-D uni area can be modified o accommodae his delay. A publicaion, Analyical Calculaion of he Oupu Harmonics in a Power Elecronic Inverer wih Curren Dependen Pulse Timing Errors, by he auhor [3] addresses he limiaions posed by he curren model in []. Limiaions posed by exising Lieraure The invesigaion in [] effecively demonsraes he modificaions necessary o incorporae consan ime delays wihin he 3-D uni area. However, he proposed model canno be applied direcly o he analysis in his disseraion since he inducor curren model in [] is very limied. Furhermore, as will be discussed laer in his secion, he majoriy of PAEs and PTEs are dependen on he curren magniude which resuls in varying delays. The model in [] only includes a consan ime delay. As menioned, W.R. Benne s [11] mehod has wo advanages over he simulaion of a ime-varying waveform, i.e. he exac magniudes of he harmonic componens can be deermined rapidly from he coefficiens, and secondly, he harmonic composiion of he specrum can be deermined. Wih he inclusion of non-ideal - 6 -

28 CHAPTER 1 INTRODUCTION effecs he analyical inegraion becomes edious, which is apparen from he soluions in []. As will be shown in Chaper 3, no closed form soluion describing he Fourier coefficiens can be obained wih he inclusion of a more realisic inducor curren model wihin he 3-D uni area, and should be solved numerically. The firs advanage of using his mehod is hus slighly more complex in he presence of non-ideal effecs han for he ideal case. Moreover, as shown in [], dead ime dependen modulaion producs are creaed, which mean ha he sideband harmonics exending wihin he audible band migh no decay as rapidly as for he ideal case. Since he THD is calculaed from he harmonic magniudes up o a cerain frequency (ypically khz for audio), each harmonic componen should be added individually wihin his band. This resricion holds for all analyical mehods of calculaing he specrum of PWM. This limiaion is overcome by simulaion. However, as menioned, a very accurae crossing poin beween he naural inersecion of he reference and carrier waveforms is required. This is especially rue in audio applicaions where he non-lineariies are very suble. To conclude, boh he analyical calculaion and he simulaion are useful. Whereas he analyical soluion gains insigh ino he harmonic composiion, he simulaion (if fas and accurae) is more useful in pracice. Conribuions in his Disseraion The analyical analysis for he incorporaion of he consan ime delay repored in [] is generalised, afer which i is exended o include non-linear curren dependen delays wih a more realisic inducor curren model. A fas, accurae simulaion mehod is inroduced, which allows for rapid calculaion of he specrum of PWM wih he inclusion of he non-lineariies DEAD TIME Dead ime, ofen referred o in he lieraure as blanking ime [4], can be regarded as he mos dominan source of disorion in inverers wih swiching frequencies greaer han 15kHz [5]. Since pracical swiching devices have non-zero urn-on and urn-off imes, an immediae ransiion in a phase leg resuls in he flow of a cross-conducion curren beween he volage rails. In order o avoid his shoo-hrough condiion, a urn-on delay is inroduced - 7 -

29 CHAPTER 1 INTRODUCTION a each on o off ransiion of he swiching devices o preven simulaneous conducion. This delay is referred o as dead ime. Dead ime has been an acive opic of research wihin power elecronics for many years. Various analyical approaches for he modelling of is effec for differen sampling mehods have been published o dae, noably [] and [6]. Lieraure on is effec wihin swiching audio amplificaion is also well esablished [15], [5]. Anoher publicaion [7] inroduces a mehod in which he dead ime is effecively reduced o zero. Relevan Lieraure The analysis in [5] models he dead ime wihin he ime domain by varying he inpu duy cycle and measuring he corresponding oupu duy cycle. The Fourier ransform of he duy cycle error is deermined nex; from his he harmonic disorion can be calculaed. The dead band referred o in [5] is a consequence of a change in curren polariy during he dead ime. The average duy cycle remains more or less consan in his region, which leaves he oupu volage floaing. The consan variable k (expressed as a percenage of he duy cycle above 5%) describes he level a which he dead band exiss. As a saring poin, he iniial heory repored in [15] calculaes he Fourier ransform of he square wave resuling from he average error inroduced over a single swiching cycle, i.e. for a purely sinusoidal inducor curren. The analysis is hen exended in which an inducor curren model wih a non-zero ripple componen, expressed as a scaled raio of he peak oupu and ripple curren, is considered. The iniial expression obained for a purely sinusoidal inducor curren is adaped o accommodae he addiional consrain. The analyical analysis presened in [] uilises he double Fourier series o calculae he harmonic componens of naurally sampled PWM (NPWM) wih dead ime. The analysis is performed for an inducor curren which is eiher purely sinusoidal, or i has a ripple componen ha saisfies he consrain of only changing polariy once over one half-cycle of he modulaing waveform. Limiaions posed by exising Lieraure The firs complicaion regarding he direc applicaion of he analysis in [5] wihin an open loop sysem is ha, in order o achieve accepable levels of disorion, pracical values of - 8 -

30 CHAPTER 1 INTRODUCTION dead ime need o be orders of magniude smaller. This means ha he inducor curren rarely changes polariy during he dead ime, effecively eliminaing he dead band. The consrain is me by seing ime = 1 in [5]. Secondly, k is direcly correlaed o he inducor curren ripple. This suggess ha, for any given circui, a measuremen firs needs o be performed in order o esablish he value of k before he disorion can be calculaed. If i is assumed ha = 1, he analysis in [15] corresponds o ha proposed in [5] wih he indirec relaion o he filer inducor expressed in erms of peak curren. The shorcoming of boh he above menioned models is ha he analysis is performed in he ime domain, i.e. by varying he swiching frequency while keeping he remaining parameers consan, he disorion will no necessarily remain unaffeced. This dependence of he cross modulaion producs on dead ime was noed in []. The analyical mehod considered in [] overcomes he limiaion posed by [15] and [5]. However, he analysis is effecively limied o a purely sinusoidal inducor curren. Alhough he effec of dead ime is well esablished, he limiaions posed wihin curren models sugges ha here is sill no complee model for predicing he isolaed effec of dead ime on disorion wihin open loop applicaions. Conribuions made by his Disseraion An analyical model is inroduced, in which a realisic inducor curren model is incorporaed. For a given dead ime, he harmonic composiion of he specrum can be deermined direcly from a given se of circui parameers. The analyical model is accompanied by an equivalen simulaion model NON-ZERO TURN-ON AND TURN-OFF DELAYS The urn-on and urn-off delays exis as a resul of he ime required for he charge or discharge of he MOSFET s inpu capaciance. I is well known ha he analyical expressions describing hese delays are dependen on he curren polariy and he curren magniude [4], [8]

31 CHAPTER 1 INTRODUCTION Relevan Lieraure The fundamenal analysis presened in [15] firs noed ha disorion arises from hese delays (referred o as delay disorion). Moreover, i suggesed ha he disorion exiss as a resul of wo conribuions, of which he firs is due o he differenial delay resuling from he inheren polariy dependency. The second conribuion resuls from he non-linear curren modulaion. The proposed soluion was o minimize he exernal gae resisance and o opimize he applied gae volage such ha hese delays cancel each oher ou. The curren modulaion was considered negligible compared o oher error sources afer which he analysis concluded ha delay disorion is generally no a limiing facor in swiching oupu sages. The ime domain analysis of he effec of he urn-on and urn-off delays presened in [9], respecively referred o as finie speed urn-on and finie speed urn-off, considers he individual impac of each delay on he average volage during he dead ime. The analysis describes he scenario in which he above menioned delays offer o minimize he average error volage wihin he dead ime. Limiaions posed by exising Lieraure Alhough he disorion mechanism was idenified in [15], here was no deailed analysis illusraing is exac effec. This shorcoming was addressed in [9] o a carain degree. However, he analysis focused on he ineracion beween he iming errors raher han on quanifying he individual effec of he urn-on and urn-off delays. Alhough he end goal wihin a sysem s design remains low overall disorion, insigh is gained ino he disorion mechanisms by considering he individual effecs. Furhermore, he analysis in [9] was performed in he ime domain, which means ha he sideband swiching harmonics resuling from he modulaion process were unknown. This, however, is a concern since he differenial delay noed in [15] suggess ha an effec similar o dead ime exiss, which has been shown o influence he modulaion producs []

32 CHAPTER 1 INTRODUCTION Conribuions made by his Disseraion The isolaed effec of he urn-on and urn-off delays on THD is esablished. An simulaion model is inroduced in which he inheren curren polariy and non-linear curren magniude dependencies are modelled NON-ZERO TURN-ON AND TURN-OFF SWITCHING TRANSITIONS The non-zero inrinsic gae-o-drain capaciance wihin he power MOSFET srucure leads o non-zero swiching ransiions. Since his capaciance is a non-linear funcion of he drain-o-source volage, a non-linear swiching curve is inroduced. Early work in [4], [8] has shown is dependence on boh curren polariy and magniude. Relevan Lieraure The non-linear swiching characerisic was noed in [15]. However, for purposes of simpliciy, a linear ransiion wih equal rise and fall swiching imes were assumed. A brief analysis followed, which illusraed a moderae influence. The analysis concluded ha, in pracice, he effec of he swiching ransiions conribues o noise and disorion, bu is less dominan han oher error sources. The analysis in [9] conains an invesigaion deermining he effec of he swiching node capaciance on he rising and falling edge ransiions during he dead ime. This is achieved by esablishing he average error volage a he swiching node resuling from a consan capaciance, i.e. a linear ransiion. Various swiching scenarios are presened during he period of dead ime from which a ime domain represenaion of he error volage as a funcion of he duy cycle can be esablished. Limiaions posed by exising Lieraure Like he mehods menioned in Secions 1.5. and 1.5.3, he analysis in [15] and [9] was performed in he ime domain. The error a he swiching node in [9] was found from he average error resuling from he charge or discharge of he swiching node capaciance during

33 CHAPTER 1 INTRODUCTION he dead ime. This only has an effec on he rising edge for a negaive inducor curren, and he falling edge for a posiive inducor curren. The remaining edges were assumed o swich in zero ime beween he various volage levels. Moreover, he swiching node capaciance s effec becomes less dominan a low curren. This, in urn, means ha he MOSFET s swiching charerisic dominaes in his sae. Conribuions made by his Disseraion A closed form soluion describing he MOSFET s swiching curve in he presence of a non-linear gae-o-drain capaciance is derived, from which a simple approximaion o he swiching curve for boh edges can be esablished. Disorion analysis of he non-linear swiching ransiion compared o a linear swihing ransiion is performed via simulaion PARASITICS AND REVERSE RECOVERY I is well known ha he sray parasiic elemens exising wihin pracical power MOSFETs lead o unwaned volage ransiens when swiched a high speeds. The curren lieraure conains several deailed invesigaions on he sources giving rise o his effec. Analyical expressions have been derived in which he swiching behaviour of he MOSFET is modelled in he presence of boh he common source and swiching loop inducance, addressing rade-offs beween overshoo, swiching speed and energy loss [3]. Anoher publicaion conains analyical soluions for overshoo in he presence of PCB sray inducances [31]. The analysis of reverse recovery in lieraure [3], [33] has mosly been limied o he influence on efficiency and swiching device raings. Disorion analysis resuling from is effec has only been menioned briefly in previous work [15], [3], [33]. Relevan Lieraure In [3] i was menioned ha he ringing effec mainly alers he high frequency (HF) specrum; i was hus concluded ha he exac influence is no easily generalized due o is srong dependence on pracical implemenaion

34 CHAPTER 1 INTRODUCTION The effec of he reverse recovery on he swiching waveform was considered in [3]. The analysis was limied o he impac on he swiching waveforms, and i was concluded ha i would only marginally affec sysem performance. The power loss analysis in [33] included he effec of he parasiic componens in he analysis. Reverse recovery was included in [33] as par of a power loss analysis, noing ha his effec only occurs during forced commuaion, i.e. during he dead ime. The disorion analysis included in [33] saed ha he effec was no easily quanified heoreically, and i was hus modelled as a curren dependen delay prior o he swiching ransiion. Limiaions posed by exising Lieraure No analysis was included on eiher effec in [3]. The above menioned repors [3], [33] on boh subjecs were mosly limied o power loss raher han disorion. The inclusion of is effec ino he model menioned in [33] was in erms of a PTE. To knowledge, he effec of reverse recovery as a PAE on disorion has ye o be esablished. Conribuions made by his Disseraion The effec of reverse recovery on disorion is deermined by means of a simulaion model. Is effec is modelled as an addiional consrain wihin he analysis of he parasiics. 1.6 DISSERTATION OUTLINE This secion conains a broad ouline of he srucure and research mehodology used in his disseraion. Firsly, a review of he double Fourier series analyical soluion for ideal NPWM is considered. This review is necessary since he inegral limis are modified in laer chapers o ake accoun of PTEs. An accurae simulaion sraegy is nex developed for NPWM which allows for rapid calculaion of he specra. A general analysis for he incorporaion of PTEs wihin he double Fourier series mehod of analysis is inroduced. The findings achieved are used o exend he analyical NPWM soluion as much as possible o accoun for PTEs. The ime based simulaion sraegy is applied o validae boh analysis approaches. When he analyical soluion becomes oo complex, he simulaion sraegy can

35 CHAPTER 1 INTRODUCTION be applied wih confidence because of he mach achieved. Finally, he analyical and simulaion resuls are compared o experimenal resuls o verify he validiy of he research

36 Tc A FUNDAMENTAL ANALYSIS OF PWM.1 INTRODUCTION Communicaion sysems require a message signal o shif ino anoher frequency range o make i suiable for ransmission over a communicaion channel. Power elecronics also uilize such a frequency shif o conrol he swiching device(s) of a converer in order o realize a arge reference volage or curren. This frequency shif is ermed modulaion and can be defined as he process by which some characerisic of a carrier is varied in accordance wih a modulaing wave [36]. The inverse process, corresponding o a shif back ino he original frequency range, is known as demodulaion. PWM, also referred o in ex as pulse duraion modulaion (PDM) or pulse lengh modulaion [17], is a very well esablished modulaion sraegy for conrolling he oupu of power elecronic converers. I can be described as he modulaion of a pulse carrier in which he value of each insananeous sample of a coninuously varying modulaing wave is caused o produce a pulse of proporional duraion [17]. This chaper focuses on he fundamenal conceps of PWM and serves as a foundaion o he following chapers. Firsly, some well-known conceps involving he various mehods of modulaion are reviewed, afer which he double Fourier series mehod of analysis, originally inroduced by W.R. Benne [11], is considered. A novel simulaion sraegy, which allows for accurae and rapid calculaion of he specrum, is hen inroduced.. FUNDAMENTAL CONCEPTS OF PWM The primary crierion of all modulaion schemes is o creae an inermediae signal ha has he same fundamenal vol-second average as he reference waveform a any insan in ime [13]. PWM hus requires he calculaion of he exac duraion of each pulse, which is necessary o preserve he original modulaing waveform

37 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM The pulse widh is generaed by a simple comparison beween he reference waveform and a high frequency carrier (sawooh or riangular) waveform. The sampling process used o deermine he pulse duraion can be eiher naural or uniform, wih hree possible mehods of modulaing he pulse widh. Eiher he leading, railing or boh edges of he modulaed waveform can be varied o produce he desired pulse widh as illusraed in Figure.1. The grey lines represen he modulaed edges. (a) (b) (c) Figure.1: (a) Leading edge, (b) railing edge and (c) double edge modulaion [17]. NPWM, someimes referred o in lieraure as analog PWM [4], is he earlies and mos simple PWM sraegy [13]. I is generaed whenever he sample insan occurs a he naural inersecion of he modulaing and carrier waveform. Figure. illusraes NPWM for leading edge (LENPWM), railing edge (TENPWM) and double edge (DENPWM) modulaion. (a) (b) (c) Figure.: Generaion of (a) LENPWM, (b) TENPWM and (c) DENPWM

38 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM UPWM is achieved whenever swiching occurs a he inersecion of a regular or uniformly sampled reference waveform and he carrier waveform. Figure.3 shows UPWM for leading edge (LEUPWM), railing edge (TEUPWM) and double edge (DEUPWM) modulaion. For LEUPWM and TEUPWM, illusraed in Figure.3 (a) and (b) respecively, sampling of he reference waveform respecively akes place a he verical rise (leading) or fall (railing) following he sawooh ramp. The crosspoin is hen deermined by direcly comparing he ampliude of he sampled reference wih he carrier waveform. (a) (b) (c) (d) Figure.3: (a) LEUPWM, (b) TEUPWM, (c) symmerical DEUPWM and (d) asymmerical DEUPWM. For DEUPWM, sampling can be symmerical or asymmerical. Symmerical UPWM resuls when he sampled reference is aken a eiher he posiive or negaive peak of he riangular waveform wih is ampliude held consan over he carrier period. This concep is illusraed in Figure.3 (c) and (d). The exen o which a pulse can be modulaed is also known as he modulaion index. This variable, denoed by M, is usually referred o as eiher a fracion wih uniy as is maximum value, or as a percenage. Noe ha, in he following secions, when referring o LEPWM, TEPWM or DEPWM, i is applicable o boh naural and uniform sampling. As menioned in Chaper 1, single-sided NPWM is assumed. The jusificaion of his assumpion wihin Class D applicaions (employing digial PWM) is ha he PNPWM process approximaes NPWM using numerical calculaion of he inersecion beween he reference and carrier waveforms. Thus, approximaing DENPWM requires wice he amoun of inersecions, which in urn increases he amoun of logic cells required

39 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM.3 THE ANALYTICAL SPECTRUM OF PWM The specrum of a signal provides an alernaive viewpoin as a funcion of frequency ha is ofen more meaningful and revealing han he original funcion of ime. PWM is a nonlinear process, which resuls in disorion of he modulaing signal. Deermining he specrum of PWM is hus very helpful, since i creaes a beer undersanding of he non-lineariies involved. However, hese non-lineariies also complicae he analyical analysis significanly. The remainder of his secion conains a summary of he double Fourier series mehod of analysis presened in [17] and [13]. This-well esablished analyical mehod was originally inroduced by W.R. Benne [11] for purposes of communicaion sysems [17], [13]. S. Bowes and B. Bird [19] expanded his o power converer sysems [13]. The fundamenal concep of his heory is explained for ideal wo-level PWM. As a saring poin, he analysis assumes he exisence of wo independenly periodic ime funcions given by: x= ω (.1) c y = ω (.) These wo funcions of ime represen a high frequency carrier wave and low frequency modulaing waveform respecively. The pulse rain creaed by he comparison of hese wo funcions is generally non-periodic [17]. This poses a problem for Fourier analysis. W.R. Benne [11] addressed his problem by represening he pulse rain by a 3-D area. B A C S D Figure.4: Definiion of he 3-D area inroduced by W.R. Benne [11]

40 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM The configuraion defined in Figure.4 corresponds o a TENPWM signal and was arbirarily chosen for purposes of illusraion of he concep which will now be explained. The area defined conains idenical walls wih fla ops a he same heigh. These walls are parallel o each oher and all perpendicular o he surface S which hey res upon. Nex, assume ha he walls are scaled ino square cells in such a way ha one wall exiss for every unis in he x-direcion, and ha one complee cycle of he waveform defining he righ hand side of each wall exiss for every unis in he y-direcion. This makes i possible o represen he heigh of he cells by a double Fourier series wih x and y as inpu argumens, denoed by F(x,y). Figure.5 shows an exracion of Figure.4 wih he appropriae scaling. y Fx,y ( ) π π x Figure.5: Appropriae scaling of he 3-D uni area of Figure.4. The Fourier series can now be developed. Consider wo planes ha are boh perpendicular o plane S, denoed by A and B in Figure.4. Boh planes are parallel o he x- axis. Wih B fixed, he projecion of he inersecion of plane A ono plane B produces a series of recangular pulses in he x-direcion all wih equal duraion, shown a he op of Figure.4. By moving plane A o a new poin of inersecion on he y-axis while sill keeping i parallel o he x-axis, anoher projecion of equal pulses is creaed. I can hus be concluded ha he inersecion he laer plane a any arbirary poin on he y-axis (denoed y 1 ) will always produce a periodic funcion in he x-direcion. This makes i possible o describe hese pulses wih a simple Fourier series: 1 F( x, y ) = a 1 ( y1) + a ( y1) cos( mx) b ( y1) sin m m ( mx) + (.3) m=

41 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM Wih coefficiens: π 1 a ( y ) = F( x, y ) cos ( mx) dx, m=,1,,..., m π (.4) 1 1 π 1 b ( y ) = F( x, y ) sin ( mx) dx, m= 1,,3,..., m π (.5) 1 1 These coefficiens depend on a specific poin of inersecion wih he y-axis. Since hey are also periodic wih respec o y, i is possible o represen hem wih anoher Fourier series for all possible values of y. This is given by: 1 a ( y) = c + cos ( ) sin m m c ny d mn mn ( ny) + (.6) n= 1 1 b ( y) = e + cos ( ) sin m m e ny f mn mn ( ny) + (.7) n= 1 Wih he coefficiens defined as: π 1 c = a cos mn m ( ny) dy π (.8) π 1 d = a sin mn m ( ny) dy π (.9) π 1 e = b cos mn m ( ny) dy π (.1) π 1 f = b sin mn m ( ny) dy π (.11) By subsiuing he Fourier expansion for he coefficiens in Eq. (.6) and (.7) ino he original series and expanding for he coefficiens in Eqs. (.8) o (.11), he double Fourier series can be deermined by rigonomeric manipulaion of he erms as: 1 F( x, y) = A + cos A ( ny) B sin n n ( ny) + (.1) n= 1 m= 1 ( ) sin ( ) + A cos mx + B mx m m - -

42 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM ± m= 1 n=± 1 ( ) sin ( ) + A cos mx+ ny + B mx+ ny mn mn Where: π π 1 A = F ( x, y) cos mn ( mx + ny) dx dy π (.13) π π 1 B = F ( x, y) sin mn ( mx + ny) dx dy π (.14) The complex form is given by: π π 1 j( mx+ ny) C = A + jb = F ( x, y mn mn mn ) e dx dy π (.15) The Fourier series of Eq. (.1) can be relaed o ime by subsiuing for Eqs. (.1) and (.). Also, for each momen of ime insered ino Eqs. (.1) and (.), a specific poin is defined wihin he area. The combinaion of hese equaions for equal ime corresponds o a sraigh line wih slope / c. Again, consider wo planes, denoed by C and D in Figure.4. Boh hese planes are perpendicular o plane S. Plane C includes he origin while plane D is fixed a a poin parallel o he x-axis. The projecion of he inersecion of plane C wih he walls ono plane D will produce a series of pulses of varying duraion. This projecion is shown a he boom of Figure.4. Since he Fourier series represens he heigh a any poin wihin he defined area, i mus also define he heigh along he sraigh line corresponding o he ime funcions of Eqs. (.1) and (.). This makes i possible o represen a series of pulses wih varying duraion by means of a double Fourier series. The significance of each erm in Eq. (.1) will now be discussed. The carrier index variable and baseband index variable are defined as m and n respecively. The ineger values of hese variables define he absolue frequency of he harmonic componens by he relaion m c +n. The firs erm in Eq. (.1) exiss a a frequency where boh m and n are equal o. This corresponds o he DC offse of he modulaed wave. The frequency componens of he second and hird erm represen special groups of harmonics. For he case where m= (second erm), he harmonics are defined by n alone. This corresponds o he baseband harmonics creaed by he modulaing wave. Noe ha he desired fundamenal oupu is defined when n=1, wih all oher ineger values of n unwaned. In a similar manner, for he case where n= - 1 -

43 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM (hird erm), he frequency componens are defined by m alone. This corresponds o he harmonics creaed by he carrier wave and exiss a muliples of he swiching frequency. The fourh and final erm is formed by a combinaion of all possible harmonic pairs formed by he sum and difference of he carrier and modulaing wave wih he excepion of he special case where n=. These combinaions are generally referred o as sideband harmonics [13]. The analysis and soluions which now follow are summarized from [17] and [13]. The Specrum of LENPWM The sawooh carrier and modulaing waveform which will be used o derive an expression for LENPWM are shown in Figure.6. The carrier waveform of (a) is defined by: x f ( x) = + 1 for he region x< π (.16) π Wih he modulaing waveform of (b) given by: ( ) ( ) f y = M cos y for < M < 1 (.17) f( x) f ( y) +1 x f ( x ) = x y f ( y ) = Mcos( y) 1 1 (a) (b) Figure.6: (a) Sawooh carrier waveform and (b) modulaing waveform for LENPWM. The oupu volage equals V d (half-bridge) whenever he reference waveform is greaer han he carrier waveform, and i equals zero whenever he carrier waveform is greaer han - -

44 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM he reference waveform. As a resul of hese wo condiions f (x,y) can ake on wo values over he region x. Saed mahemaically: x f ( x, y) = Vd when M cos( y) > + 1, or x> πm cos( y) + π (.18) π x f ( x, y) = when M cos( y) < + 1, or x< πm cos( y) + π (.19) π Figure.7 illusraes he 3-D uni area defined by Eqs. (.18) and (.19). y V d x= πmcos( y)+ π x Figure.7: 3-D uni area for LENPWM. The complex Fourier coefficien of Eq. (.15) can now be evaluaed by insering he limis defined in Figure.7. The expression yields: π πm cos( y) + π V π d jmx jmx jny Cmn = e dx + e dx e dy π πmcos( y) + π (.) - 3 -

45 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM Eq. (.) can now be evaluaed for differen values of he index variables m and n. For m=n=, Eq. (.) can be simplified o: π πmcos( y) + π V π d A + jb = dx dx dy π + πmcos( y) + π π VM d = cos( y) dy+ V π d = V d (.1) Rewriing Eq. (.1) ino is real and complex pars yield: A = V and B = (.) d The DC componen can now be found by subsiuing Eq. (.) ino Eq. (.1): A V d = (.3) For m=, n> he inner inegral of Eq. (.) gives: π πm cos( y) + π π Vd jny An + jbn = dx dx e dy π + πmcos( y) + π VM d V = + π π π π jny d jny cos( y) e dy e dy (.4) Subsiuing for cos(y)=(e jy +e jy )/ Eq. (.4) leads o: π π VM d jn ( + 1) y VM d jn ( 1) y = e dy+ e dy 4π 4π (.5) The firs erm in Eq. (.5) always inegraes o zero for all values of n>. The second erm inegraes o a non-zero value only when n=1. Evaluaing he laer for his condiion, Eq. (.5) gives: VM d A1 + jb1 = (.6) - 4 -

46 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM Eq. (.6) represens he magniude of he fundamenal low frequency harmonic componen. Noe ha his is he ideal desired oupu since he modulaing waveform of Eq. (.17) is preserved wih no addiional baseband harmonics resuling from he modulaion process (Eq. (.6) equals zero for all n>1). For m>, n=, he inegral of Eq. (.) leads o: π πmcos( y) + π V π d jmx jmx Am + jbm = e dx e dx dy π + πm cos( y) + π V d π = 1 e jπ m jmπm cos( y) + jmπ dy π Vd Vd jmπ jmπ M cos( y) = j + j e e dy πm π m (.7) The inegral in Eq. (.7) corresponds o a Bessel funcion of he firs kind. Subsiuing for he relaion in Eq. (A1.11) wih =M and =y and noing ha: ( π ) ( π ) ( ) jπ m e = cos m + jsin m = 1 m (.8) Eq. (.7) can be simplified o: V d ( ) ( π ) = j 1 1 m J Mm π m (.9) Eq. (.9) represens he carrier harmonics. For m>, n, Eq. (.) can be evaluaed: π πmcos( y) + π V π d jmx jmx jny Amn + jbmn = e dx e dx e dy π + πmcos( y) + π V d π jm( πm cos( y) + π) jny = 1 e e dy jπ m V d π jmπ jmπ M cos( y) jny = j e e e dy π m (.3) The inegral in Eq. (.3) again corresponds o a Bessel funcion of he firs kind. Subsiuing for he relaion in Eq. (A1.7) resuls in: π V j n d m = j ( 1) e Jn ( π Mm) (.31) π m - 5 -

47 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM Eq. (.31) defines he sideband harmonics. The complee soluion o LENPWM can be found by subsiuing Eqs. (.1), (.6), (.9) and (.31) ino he double Fourier series of Eq. (.1) o give: Vd VdM flenpwm () = cos( ω) (.3) Vd π m= sin m m ( ) J ( πmm) ( m[ ωc] ) ± Vd 1 m π + ( 1) Jn( πmm) sinm[ ωc] + n[ ω] n π m= 1n=± 1m The Specrum of TENPWM Figure.8 (a) and (b) respecively represen he sawooh carrier waveform and modulaing waveform ha will be used o consruc he 3-D uni area for TENPWM. f ( x) f ( y) +1 x f ( x ) = 1 +1 x y f ( y ) = Mcos( y) 1 1 (a) (b) Figure.8: (a) Sawooh carrier waveform and (b) modulaing waveform for TENPWM. The carrier waveform of (a) is defined by: x f ( x) = 1 for he region x< π (.33) π - 6 -

48 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM While he modulaing waveform of (b) is given by: ( ) ( ) f y = M cos y for < M < 1 (.34) Using he same swiching consrains described in he previous secion, he following wo mahemaical condiions can be consruced: x f ( x, y) = Vd when M cos( y) > 1, or x< πm cos( y) + π (.35) π x f ( x, y) = when M cos( y) < 1, or x> πm cos ( y) + π (.36) π y x= πmcos( y)+ π V d x Figure.9: 3-D uni area for TENPWM. The uni area can now be consruced using he consrains defined in Eqs. (.35) and (.36). This is illusraed in Figure

49 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM Rewriing Eq. (.15) by insering he limis defined in Figure.9, he expression yields: π πmcos( y) + π V π d jmx jmx jny Cmn = e dx + e dx e dy π πmcos( y) + π (.37) Eq. (.37) can now be evaluaed for differen values of he index variables m and n o deermine he magniude of he harmonic componens creaed. For m=n=, Eq. (.37) can be simplified o: π πmcos( y) + π V π d A + jb = dx dx dy π + πmcos( y) + π π VM d Vd = cos( y) dy π + V = d (.38) The resul obained in Eq. (.38) is exacly he same as ha of he previous secion. The DC componen can hus be wrien as: A V d = (.39) For m=, n>, he inner inegral of Eq. (.37) gives: π πm cos( y) + π π Vd jny An + jbn = dx dx e dy π + πmcos( y) + π VM d V = + π π π π jny d jny cos( y) e dy e dy (.4) The firs erm of Eq. (.4) is he exac negaive of he firs erm of Eq. (.4). The soluion can hus be obained direcly by adaping Eq. (.6). This gives: A VM d + jb = (.41) 1 1 Similar o Eq. (.6), Eq. (.41) defines he magniude of he fundamenal low frequency harmonic componen

50 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM For m>, n=, Eq. (.37) can be wrien as: π πmcos( y) + π V π d jmx jmx Am + jbm = e dx e dx dy π + πmcos( y) + π V d π jmπm cos( y) + jmπ = e 1dy jπ m π = Vd Vd jmπ jmπ M cos( y) j j e e dy πm π m (.4) Once again he inegral in Eq. (.4) corresponds o a Bessel funcion of he firs kind. The soluion is given by: V d ( ) ( π ) = j 1 1 m J Mm π m (.43) Eq. (.43) defines he carrier harmonics. For m>, n, Eq. (.37) inegraes o: π πmcos( y) + π V π d jmx jmx jny Amn + jbmn = e dx e dx e dy π + πmcos( y) + π V d π jmπm cos( y) + jmπ jny = e 1e dy jπ m V d π jm( πm cos( y) + π) jny = j e e dy π m (.44) Again, he inegral in Eq. (.44) represens a Bessel funcion of he firs kind. Subsiuing for he relaion in Eq. (A1.7), Eq. (.44) can be wrien as: π V j n d m = j ( 1) e Jn ( π Mm) (.45) π m Eq. (.45) represens he sideband harmonics siuaed around muliples of he carrier harmonics

51 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM By nex subsiuing he resuls achieved in Eqs. (.38), (.41), (.43) and (.45) ino he double Fourier series of Eq. (.1) and relaing he resul o ime using Eqs. (.1) and (.), he complee soluion o TENPWM can be expressed as: Vd VdM ftenpwm () = + cos( ω) (.46) Vd sin π m m= 1 m ( ) J ( πmm) ( m[ ωc] ) ± Vd 1 m π ( 1) Jn( πmm) sinm[ ωc] + n[ ω] n π m= 1n=± 1m The Specrum of DENPWM Figure.1 illusraes he carrier waveform and reference waveform ha will be used o derive he derivaion of he expression for DENPWM. f ( x) f ( y) +1 x f ( x ) = 1 x f ( x ) = x y f( y ) = Mcos( y) 1 1 (a) (b) Figure.1: (a) Triangular carrier waveform and (b) modulaing waveform for DENPWM. The riangular carrier wave shown in Figure.1 (a) is defined over wo regions: x f ( x) = 1 for he region x< π (.47) π - 3 -

52 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM x f ( x) = + 3 for he region π x< π (.48) π The modulaing wave illusraed in Figure.1 (b) is defined by: ( ) ( ) f y = M cos y for < M < 1 (.49)Once again he oupu volage swiches sae under he same condiions as described for LENPWM and TENPWM. For he region defined by x<: x π π f ( x, y) = Vd when M cos( y) > 1, or x< M cos( y) + (.5) π x π π f ( x, y) = when M cos( y) < 1, or x> M cos( y) + (.51) π Also, for he region bounded by x<: x π 3π f ( x, y) = Vd when M cos( y) > + 3, or x> M cos( y) + (.5) π x π 3π f ( x, y) = when M cos( y) < + 3, or x< M cos( y) + (.53) π Eqs. (.5) o (.53) define he limis of he uni area shown in Figure.11. Insering he limis defined in Figure.11 ino he complex Fourier coefficien defined by Eq. (.15), he expression yields: π π π 3π M cos( y) + Mcos( y) + π π Vd jmx jmx jmx jny Cmn = e dx e dx e dx e dy π + + π π π 3π Mcos( y) + Mcos( y) + (.54) Eq. (.54) can now be evaluaed for he differen values of he index variables m and n. For m=n=, he inegral of Eq. (.54) simplified o: π π π 3π Mcos( y) + Mcos( y) + π π Vd A + jb = dx dx dx dy π + + π π π 3π Mcos( y) + Mcos( y) + π VM d = cos( y) dy+ V π d = V d (.55)

53 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM y π π x= Mcos( y) + π x= Mcos( y) + 3 π V d V d 3 x Figure.11: 3-D uni area for DENPWM. The resul achieved in Eq. (.55) is exacly he same as ha of he previous wo secions. The DC componen can hus once again be wrien as: A V d = (.56) For m=, n>, evaluaion of Eq. (.54) leads o: π π π 3π Mcos( y) + Mcos( y) + π π Vd jny An + jbn = dx dx dx e dy π + + π π π 3π Mcos( y) + Mcos( y) + VM d V = + π π π π jny d jny cos( y) e dy e dy (.57) The inegral of Eq. (.57) corresponds o ha of Eq. (.4)

54 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM Evaluaing Eq. (.57) using he same echnique discussed in he previous secion gives: A VM d + jb = (.58) 1 1 The laer soluion represens he fundamenal low frequency harmonic componen. For m>, n=, Eq. (.54) can be wrien as: π π π 3π Mcos( y) + Mcos( y) + π π Vd jmx jmx jmx Am + jbm = e dx e dx e dx dy π + + π π π 3π Mcos( y) + Mcos( y) + π π π π cos( ) cos( ) 3 π V j mm y j m j mm y j m d = e e e e dy jπ m V V = j e e dy+ j e e dy π π π π 3π cos( ) π π j m j mm y j m j mm cos( y) d sw m π m (.59) Boh inegrals in Eq. (.59) correspond o Bessel funcions of he firs kind. By subsiuing again for he relaion in Eq. (A1.11) resuls in: 3π π V j m j m d π = j J Mme e π m (.6) Rewriing he exponenial funcions in brackes as he sum of sine and cosine funcions, Eq. (.6) can be simplified o: V π π J Mm sin m π m d = (.61) Eq. (.61) represens he carrier harmonics. For m>, n, Eq. (.54) inegraes o: π π π 3π Mcos( y) + Mcos( y) + π π Vd jmx jmx jmx jny Amn + jbmn = e dx e dx e dx e dy π + + π π π 3π Mcos( y) + Mcos( y) + π π π π cos( ) cos( ) 3 π V j mm y j m j mm y j m d jny = e e e e e dy jπ m V V jny = j e e e dy + j e e e dy π π π π π π π j m j mm cos( y) j m j mmcos( y) d jny d m π m (.6)

55 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM By subsiuing for he Bessel funcion relaion given in Eq. (A1.7), Eq. (.6) can be wrien as: 3π π π π V j m n j m n d π + = j Jn Mm e e π m (.63) Eq. (.63) in urn can be simplified o: V d π π π = J n Mm sin m + n π m (.64) The laer soluion represens he sideband harmonics. Subsiuing for Eqs. (.55), (.58), (.59) and (.64) he double Fourier series describing he uni area of Figure.11 can be found: Vd VdM fdenpwm () = + cos( ω) (.65) V 1 π π J Mm m cos m d + sin π m= 1 m ( [ ωc ]) ± V d 1 π π π + Jn Mmsin m+ ncos m + n π m= 1n=± 1m ( [ ωc ] [ ω ]).4 SPECTRAL PLOTS AND GENERAL DISCUSSION The specra for he analyical mehod described in he previous secion will now be briefly discussed. The resuls presened in his secion are once again summarized from [17] and [13]. The Fourier coefficiens defined in Eq. (.15) represen he magniudes of he harmonic componens. The specrum can hus easily be found hrough evaluaion of he index variables m and n for he differen values described in Secion.3. Noe ha he parameers used in his secion correspond o M=.85, V d =1V and a carrier o fundamenal raio c / =384. The volage specrum for LENPWM is shown in Figure.1 (a). From his illusraion he fundamenal low frequency harmonic componen can be seen o be presen a order 1. The firs wo carrier harmonics exising a ineger muliples of he swiching frequency, ogeher wih heir respecive sidebands are also shown in his specral plo

56 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM 4 Carrier 1 wih Sidebands Carrier wih Sidebands 4 Carrier 1 wih Sidebands Carrier wih Sidebands Fundamenal Componen Fundamenal Componen - - Magniude [dbv] Magniude [dbv] Harmonic Number [-] Harmonic Number [-] (a) Figure.1: Analyical volage specrum of (a) LENPWM (or TENPWM), and (b) DENPWM. (b) Upon examinaion of Figure.1 i is eviden ha LENPWM is achieved when reversing he ime scale of TENPWM and vice versa [17]. Saed mahemaically: () ( ), and () ( ) f = f f = f (.66) LENPWM TENPWM TENPWM LENPWM As a resul TENPWM produces exacly he same magniude specrum as LENPWM. Figure.1 (b) represens he harmonic volage specrum for DENPWM. The fundamenal low frequency harmonic componen represened by Eq. (.58) is shown ogeher wih he carrier and sideband harmonics a muliples one and wo of he swiching frequency. Noe ha, upon inspecion of Eq. (.61), i is eviden ha he sine funcion equals zero for all even values of he index variable m. Similarly, Eq. (.64) equals zero when eiher boh m and n are even or when boh of hem are odd. This has he ne effec of eliminaing all even sideband harmonics around even muliples of he carrier frequency as well as cancelling all odd sideband harmonics around odd muliples of he carrier frequency..5 SIMULATION STRATEGY The analyical soluions obained in Secion.3 idenify exacly he frequency componens creaed by he modulaion process. These harmonic componens can also be found by applying he FFT o a simulaed ime-varying swiched waveform (such as PSpice) [13]. This approach has boh advanages and disadvanages. One major advanage is he reducion in mahemaical effor compared o analyical compuaion. The downside is ha he

57 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM ime resoluion of he simulaion has o be very high in order o produce accurae crossing poins beween he reference and carrier waveforms. This in urn requires significan compuing power, which is very ime-consuming. The use of his simulaion sraegy is eliminaed for applicaion in his disseraion since he errors in pulse widh inroduced by a lack of precision could exceed he suble non-ideal effecs ha some of he less dominan sources of disorion migh have. The laer requiremen calls for a very precise, ime effecive simulaion mehod o insure accurae modelling of he non-ideal effecs. In his secion a novel simulaion sraegy based on accurae cross-poin calculaion using Newon Raphson s numerical mehod is inroduced o mee he simulaion requiremen..5.1 THE NEWTON-RAPHSON NUMERICAL METHOD In general, he soluion o he roos of a non-linear equaion canno be found analyically and he equaion is solved by approximae mehods [37]. These esimaes are usually based on he concep of ieraion which, as a saring poin, requires an iniial guess. From his iniial value an ieraive procedure generaes a se of approximaions, which are assumed o presumably converge o he desired roo. The Newon-Raphson mehod, also referred o in lieraure as Newon s mehod [38], uilizes a sraigh-line approximaion o obain he roos of a non-linear equaion. Figure.13 shows an arbirarily chosen non-linear funcion f () for which a single roo exiss. The fundamenal concep of Newon s mehod is o replace his non-linear funcion wih a linear approximaion in he viciniy of he roo and hen o solve for he laer. This linear funcion is chosen o be he angen line o f (). Assume ha represens he iniial guess o he roo of f (). The inersecion of he angen line o f ( ) wih he x-axis produces a beer esimae o he roo han ha of he iniial guess. A convergen se of approximaions can hus be formed by calculaing he angen a each new poin of inersecion produced by he previous ieraion

58 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM y f () f( ) 1 Figure.13: Geomeric represenaion of he Newon-Raphson numerical mehod. Through geomeric analysis of Figure.13 an expression for he approximaion o he roo can now be derived. The slope of he line angen o f ( ) can be found by calculaing he raio of variaion in boh he and y direcion (y/). By noing ha he derivaive of f ( ), denoed by f '( ). also represens he gradien he following can be saed: f ( ) ( ) 1 = f ' ( ) (.67) Solving Eq. (.67) for 1 leads o: ( ) '( ) f = 1 f (.68) The general case of Eq. (.68) can be wrien as: ( n ) '( ) f = (.69) n + 1 n f n Eq. (.69) represens he soluion o Newon-Raphson s numerical mehod. Noe ha he number of ieraions depends on he required accuracy. By defining an error crierion n+1 n o which each ieraion can be compared, he process can erminae as soon as he error becomes smaller han he required accuracy

59 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM.5. CROSS-POINT CALCULATION USING NEWTON-RAPHSON S METHOD This secion will show how Newon-Raphson s numerical mehod can be uilized o deermine he specra of TENPWM and DENPWM. TENPWM The single pulse illusraed in Figure.14 corresponds o he p h pulse of a TENPWM waveform, denoed by p TENPWM. The low frequency modulaing wave is represened by m TENPWM () while he carrier waveform is denoed by c TENPWM (). The oupu swiches o he posiive rail whenever he modulaing wave is greaer han he carrier waveform wih he inverse siuaion corresponding o a ransiion o he negaive rail. Since he leading edge of he swiched oupu volage remains fixed for TENPWM he widh of he pulse is solely deermined by he railing edge. The crossing poin of he laer wo waveforms is given by: () () () () m = c, or alernaively m c = (.7) TENPWM TENPWM TENPWM TENPWM ntenpwm ctenpwm () mtenpwm () W p TENPWM V d p TENPWM ( p TENPWM -1T ) c T c p TENPWM T c Figure.14: Generaion of TENPWM

60 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM The modulaing wave of Eq. (.7) is defined by: () ( ω ) m = M cos wih < M < 1 (.71) TENPWM The carrier waveform, which produces he p h pulse of he swiched oupu volage, can be wrien direcly from Figure. as: c () = ( p 1) for ( p 1 TENPWM TENPWM TENPWM ) T < < p T (.7) c TENPWM c T c The nonlinear funcion f TENPWM () can now be defined in such a way ha: () () () f = m c (.73) TENPWM TENPWM TENPWM Subsiuing for Eqs. (.71) and (.7), Eq. (.73) can be simplified o: = M cos( ω ) + ( p 1) for ( p 1 TENPWM TENPWM ) T < < p T (.74) c TENPWM c T c The roos of Eq. (.74) can be deermined by using he general soluion o he Newon Raphson numerical mehod described in Eq. (.69). The approximaion o he crossing poin for TENPWM for he same limis bounding Eq. (.74) is given by: ntenpwm + 1 M cos p 1 TENPWM TENPWM Tc = ntenpwm ω M sin ( ω n ) TENPWM T ( ω ) + n n ( TENPWM ) c (.75) Wih he origin aken as reference, Eq. (.75) represens he inersecion of he modulaing wave wih he p h pulse of he carrier waveform. The widh of he corresponding swiched oupu volage can hus be found hrough geomeric analysis of Figure.14 as: ( ) W = p T (.76) 1 ptenpwm ntenpwm + 1 TENPWM c According o he definiion of he carrier waveform given by Eq. (.7), he poin of inersecion of he p h pulse can occur anywhere wihin he inerval bounded by (p TENPWM 1)T c <<p TENPWM T c. Wih he iniial guess (n=1) chosen as one half of he laer inerval quick convergence o he roo will be assured

61 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM Thus: 1 = p T 1TENPWM TENPWM c (.77) DENPWM Consider he p h pulse of he DENPWM waveform shown in Figure.15. The swiching condiions are exacly he same as for TENPWM. The carrier waveform is denoed by c DENPWM (), while he reference wave is represened by m DENPWM (). For his mehod of modulaion he pulse widh is deermined by he poins of inersecion of boh he leading and railing edge of he modulaed waveform. The crossing poin can be deermined by: () () () () m = c, or alernaively m c = (.78) DENPWM DENPWM DENPWM DENPWM n DENPWM +1 n DENPWM mdenpwm () cdenpwm () W p DENPWM V d ( p -1T ) DENPWM c p DENPWM T c p DENPWM T c Figure.15: Generaion of DENPWM. Wih he reference waveform of Eq. (.78) defined by: () ( ω ) m = M cos wih < M < 1 (.79) DENPWM - 4 -

62 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM Since he carrier consis of wo line segmens, i mus be defined over wo regions as: c DENPWM () 4 4( p 1) 1 DENPWM T + c = 4 + ( 4p 1 DENPWM ) Tc (.8) For he respecive regions: Tc ( p 1) T < < ( p 1 DENPWM c DENPWM ) = Tc ( p 1 DENPWM ) < < p T DENPWM c (.81) Defining he non-linear funcion f DENPWM () gives: () () () f = m c (.8) DENPWM DENPWM DENPWM By subsiuing for he modulaing and carrier waveforms respecively defined in Eqs. (.79) and (.8), Eq. (.8) can be defined over he wo inervals as: 4 M cos( ω ) + 4( p 1) 1 DENPWM T + c = 4 M cos( ω ) + ( 4 p 1 DENPWM ) Tc (.83) For he respecive regions: Tc ( p 1) T < < ( p 1 DENPWM c DENPWM ) = Tc ( p 1 DENPWM ) < < p T DENPWM c (.84) An approximaion o he roos of Eqs. (.83) can be found by applying Newon s mehod in Eq. (.69)

63 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM Thus, he esimae of he inersecion of he modulaing and carrier waveforms for DENPWM over he region (p DENPWM 1)T c <<(p DENPWM 1)T c / can be defined as: 4 M cos( ω ) + 4 ( p 1) 1 ndenpwm 1 ndenpwm 1 DENPWM T + c = 4 ω M sin ( ω n ) DENPWM 1 T ndenpwm 1+ 1 ndenpwm 1 c (.85) For he inerval bounded by (p DENPWM 1)T c /<<p DENPWM T c he inersecion is given by: 4 M cos 4 p 1 DENPWM DENPWM Tc = 4 ω M sin ( ω n ) DENPWM T ndenpwm + 1 n DENPWM ( ω ) n n ( DENPWM ) c (.86) From Figure.15 i is eviden ha he widh of he p h pulse can be found by calculaing he difference beween Eqs. (.85) and (.86), which leads o he expression: W = (.87) p DENPWM ndenpwm + 1 ndenpwm 1+ 1 Using a similar argumen as for TENPWM, he iniial guess (n=1) of Eqs. (.85) and (.86) can be esablished as: Tc = 4 1 ( p 1) + 1 DENPWM 1 DENPWM 4 (.88) Tc = 1 ( 4p 1) (.89) DENPWM DENPWM 4 Specra of TENPWM and DENPWM Consider he single recangular pulse, denoed by g i () where <i<+, wih ampliude V d and widh W p illusraed in Figure.16 (a), generaed as a resul of a NPWM process. In order o simplify he mahemaical represenaion and analysis of his funcion, g i () was chosen o be siuaed around he origin wih a DC offse equal o V d /

64 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM g () i G ( f ) i V d VW d p W p W + p 3 W p W p 1 W p 1 + W p + W p 3 + W p f (a) (b) Figure.16: (a) Time domain and (b) magniude specrum represenaion of a recangular pulse. The recangular funcion of Figure.16 (a) can be described mahemaically by: g i () V for he region d = elsewhere W W p < < p (.9) The Fourier Transform of g i (), denoed by G i ( f ), can be compued as: W p jπ f G ( f i ) = V e d d W p ( π fwp ) sin = VW d p π fwp (.91) To simplify he resul, he funcion in brackes can be furher reduced by defining he sinc funcion ofen found in communicaions heory [4]. I is defined as: sinc ( λ ) sin ( πλ ) = (.9) πλ Where represens he independen variable. As shown in Figure.17, he funcion decays as increases, oscillaing hrough posiive and negaive values wih is zero crossings exising a ineger values of

65 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM sinc( λ) λ Figure.17: The sinc funcion. I has a maximum value of uniy a =. By noing ha =fw p Eq. (.91) reduces o: ( ) sinc( ) G f = VW fw (.93) i d p p Figure.16 (b) shows he magniude specrum G i ( f ) of he recangular pulse g i (). Consider he pulse of Figure.18 (a). Suppose ha his pulse is shifed by a consan ime in such a way ha is leading edge coincides wih ha of Figure.18. For he p h pulse of a TENPWM waveform his ime shif corresponds o: Wp = ( p 1) T + TENPWM (.94) TENPWM c In a similar manner he DENPWM of Figure.15 can also be relaed o he pulse of Figure.16 (a). This is given by: Wp DENPWM = + (.95) ndenpwm + 1 Figure.18 illusraes he ime shifing of he recangular pulse of Figure.16 (a) by a consan amoun. Depending on he mehod of modulaion, he widh W p can eiher correspond o Eq. (.76) or (.87)

66 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM g () i g () i V d V d W p W + p p + W (a) (b) Figure.18: Time shifing of he recangular pulse of Figure.16 (a). The effec of he ime shif on he specrum can be deermined by calculaing he Fourier Transform of he pulse illusraed in Figure.18 (b) as: + W p jπ f Gi( f ) = V e d d jπ f ( ) = VWsinc fw e (.96) d p p From Eq. (.96) i is eviden ha, if he funcion g i () is shifed by a posiive consan ime, i is he equivalen of muliplying is Fourier Transform by a facor exp( f ) [4]. The muliplicaion has he effec of only alering he phase of G i ( f ) by an amoun corresponding o f while he magniude remains unaffeced. From his discussion i is eviden ha each individual pulse of a PWM waveform can be represened by a recangular wave shifed by some consan ime. The complee pulse rain, denoed by g(), can be found as: g = g (.97) () () i= i Where g i () is defined in Eq. (.96). Applying he Fourier Transform, he resul yields: I g() =I gi () (.98) i=

67 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM From Lebesgue s Convergence Theorem 15 [41] i can be shown ha: I g = Ig (.99) () () i i=.5.3 SIMULATION RESULTS Table.1 and Table. respecively compare he analyical resuls o he simulaions for TENPWM and DENPWM using he proposed sraegy. The condiions used correspond o hose of Secion.4. The fundamenal componen and firs carrier wih is respecive sideband harmonis are shown in boh ables. TABLE.1 COMPARISON OF ANALYTICAL AND SIMULATION RESULTS FOR TENPWM. Harmonic Number [-] Analyical Magniude [V] Simulaed Magniude [V] The resuls of he simulaions of boh TENPWM and DENPWM are consisen wih he analyical soluions, which confirms he validiy of he proposed model. Noe ha he simulaions were performed wih en ieraions of Newon s mehod (accurae o 8 db). As already menioned in Secion.4, all even sideband harmonics around even muliples of he carrier frequency as well as all odd sidebands around odd muliples of he carrier frequency are eliminaed for ideal DENPWM

68 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM TABLE. COMPARISON OF ANALYTICAL AND SIMULATION RESULTS FOR DENPWM. Harmonic Number [-] Analyical Magniude [V] Simulaed Magniude [V] GENERAL DISCUSSION Consider he analyical and simulaed specra of TENPWM respecively shown in Figure.19 (a) and (b) for c / =8. The remaining parameers correspond o hose of he previous secion. The fundamenal low frequency harmonic componen and he firs wo carrier harmonics (m=1 and m=) ogeher wih heir respecive sidebands are shown. Since he index variables of he analyical soluions of Secion.4 define a specific harmonic according o he relaion m c +n i is possible o plo he individual conribuion of m=1 and m= wih is respecive sidebands. This is shown in Figure.19 (a). 4 Carrier 1 wih Sidebands Carrier wih Sidebands 4 Carrier 1 wih Sidebands Carrier wih Sidebands Fundamenal Componen Fundamenal Componen - - Magniude [dbv] Magniude [dbv] Harmonics of m=1 Harmonics of m= Harmonic Number [-] (a) Harmonic Number [-] (b) Figure.19: (a) Analyical and (b) simulaed specrum of TENPWM wih c / =

69 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM The specrum shown in Figure.19 (b) differs significanly from ha shown in (a). The reason for his is ha he various overlapping harmonics of he analyical soluion have been ploed individually, and no summed. The harmonic magniudes of he specra of Figure.19 (a) and (b) are compared in Table.3. Noe ha he analyical soluion in (a) will produce he same resul as in (b) if he various overlapping harmonics are summed. The value of c / used was merely for purposes of illusraion of he concep and is no realisic for implemenaion in pracical sysems. Moreover, he phasor summaion of he harmonic componens is no of grea concern when considering he ideal case a high carrier o fundamenal raios due o is negligible ampliude caused by he rapid decay of he sidebands. However, as will be shown in Chapers 5 o 9, when considering non-ideal effecs hese sidebands do no decay as quickly. TABLE.3 ANALYTICAL AND SIMULATED HARMONIC MAGNITUDES OF THE SPECTRUM SHOWN IN FIGURE.19. Harmonic Number [-] Analyical Magniude [V] Simulaed Magniude [V] The analyical analysis provides a ool wih which he sole conribuion of each harmonic componen is esablished, more specifically, he individual conribuion of he baseband and sideband harmonics wihin he audible band..7 SUMMARY Fundamenal conceps of PWM were considered, afer which he double Fourier series mehod of analysis for calculaing he specrum of PWM analyically was reviewed. A simulaion sraegy, based on Newon s numerical mehod, was inroduced which allows for

70 CHAPTER A FUNDAMENTAL ANALYSIS OF PWM rapid and accurae calculaion of he specrum of PWM. The resuls of he proposed simulaion sraegy correlaed o hose of he analyical mehod, verifying is validiy

71 vf 3 v sw INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD 3.1 INTRODUCTION A mehod, inroduced in he paper Analyical Calculaion of he Oupu Harmonics in a Power Elecronic Inverer wih Curren Dependen Pulse Timing Errors [34] by he auhor, for he incorporaion of curren dependen PTEs wihin he double Fourier series mehod of analysis is considered in his chaper. The incorporaion of dead ime wihin W.R. Benne s [11] mehod has been repored in [], which effecively includes a curren polariy dependen delay wihin he ideal analysis. In his chaper he analysis in [] is generalised and exended. This serves as a general analyical ool in which he harmonic componens of a NPWM waveform can be calculaed in he presence of curren dependen PTEs. The analysis sars off wih a brief overview of a known approximaion o he inducor ripple curren in Secion 3.. The analysis presened in Secions 3.3 o 3.6 covers he incorporaion of he PTEs wihin he 3-D uni area. A general analysis of ha presened in [] for he inclusion of a consan delay which is independen of curren is given in Secion 3.3. A purely sinusoidal curren dependency is included in Secions 3.4 and 3.5, respecively covering a polariy dependency and non-linear magniude dependency. In addiion o he analysis in [] he former secion disinguishes beween sampling on he leading or railing edge of he NPWM waveform. Secion 3.6 shows how he inducor curren model of Secion 3. can be incorporaed ino he 3-D uni area. 3. REALISTIC INDUCTOR CURRENT MODEL Since he expressions describing he various PTEs and PAEs are curren dependen, i is imperaive ha a proper and accurae inducor curren model firs be esablished before proceeding wih he consrucion of he analyical and simulaion models in Chapers 5 o

72 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD Consider he inducor volage v L(ideal) and curren i L(ideal) waveforms for an ideal swiching ransiion illusraed in Figure 3.1. This illusraion represens a half-bridge opology wih V d being he applied bus volage. v Lideal ( ) V d o v Vd v o V d Δi L i Lideal ( ) DT c ( 1-D) T c T c Figure 3.1: Volage across and curren hrough he inducor. By nex assuming ha v o remains consan over one cycle of he swiching period an expression describing Δi L can be consruced from Figure 3.1 as: DT 1 c Vd 1 Vd L o o c L fil L fil Δ i = v d = v DT (3.1) Where v o is assumed o follow he reference volage exacly. Thus: v o VM d = cos( π f) (3.) By nex subsiuing Eq. (3.) ino Eq. (3.1) and noing ha D=((1+Mcos(f )/), he expression in Eq. (3.1) is simplified o: Δ Vd i = L ( D D ) L f (3.3) fil c The resul achieved was also noed in [33]. From Eq. (3.3) i can be seen ha he maximum deviaion in Δi L occurs a D=5% while he minimum is esablished a eiher he

73 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD maximum or minimum value of D. The remainder of his secion idenifies wo possible scenarios in which he inducor curren can manifes iself. Figure 3. defines he upper and lower envelope of he inducor curren which is respecively represened by i L(upper_env) and i L(lower_env). Curren i o is assumed o be purely sinusoidal as well as in phase wih he oupu volage. Variables i L(upper_env) and i L(lower_env) can respecively be defined as: ΔiL il( upper _ env) = Io cos( π f) + (3.4) ΔiL il( lower _ env) = Io cos( π f) (3.5) Where Δi L is defined in Eq. (3.3) and: I o VM d = (3.6) R load From Figure 3. (a) and (b) wo scenarios of he inducor curren can be defined. For (a) M was chosen such ha i L(upper_env) is always disincly posiive and i L(lower_env) is always disincly negaive. This condiion governs inducor curren Scenario Upper Envelope 3 Upper Envelope Curren [A] -.5 Curren [A] Lower Envelope Time [ms] (a) - -3 Lower Envelope Time [ms] (b) Figure 3.: Definiion of he inducor curren for (a) Scenario and (a) Scenario. Scenario occurs when boh i L(upper_env) and i L(lower_env) cross hrough zero a some poin in ime over one complee cycle of he modulaing waveform. This is shown in Figure 3. (b). Noe ha during his sae i L(upper_env) and i L(lower_env) are sill defined by Eqs. (3.4) and (3.5)

74 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD TABLE 3.1 DEFINITION OF THE BASIC VARIABLES USED THROUGHOUT THIS CHAPTER. Variable V d R L L fil f f c Value 1 V uh 1 khz 384 khz Unless saed oherwise, he basic variables used wihin he res of his chaper correspond o hose defined in Table 3.1. When referring o resuls wihin Scenario or wihin Scenario, he governing condiion described holds. 3.3 INCORPORATION OF TIME DELAYS Suppose ha he duy cycle of an ideal TENPWM waveform needs o be alered in order o produce a consan posiive DC offse V DC(offse) wih respec o he ideal case. A possible soluion for achieving his requiremen is o adap he swiching sraegy in such a way ha he modulaed (railing) edge of he swiched oupu volage s urn-on is delayed by some consan ime d. Similarly, sill for a TENPWM waveform, assume ha he same negaive consan DC offse V DC(offse) relaive o he ideal case is required. This can be achieved by delaying he unmodulaed (leading) edge by d. The relaion beween V DC(offse) and d for he wo scenarios can be found by averaging he delay over a complee swiching cycle which resuls in: V DC ( offse) + V f = V f d d c d d c for he railing edge for he leading edge (3.7) The following wo sub-secions show how hese requiremens can be incorporaed ino he 3-D uni area

75 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD MODULATED EDGE Figure 3.3 illusraes an exrac of pulses of a TENPWM waveform wih he modulaed edge delayed by d. The ideal modulaed edge is represened by he dashed line. d d d V d T c Figure 3.3: Definiion of d inroduced on he modulaed edge. The incorporaion of ime delays wihin he double Fourier series mehod of analysis necessiaes a sligh modificaion wihin he uni area. Since he pulse rain is defined along he inersecion of he sraigh line y=( / c )x wih he various walls wihin he 3-D uni area, he corresponding mapping of d (in radians) d ' needs o be defined along he his line. Figure 3.4 shows a visual represenaion of his concep. Φ ' d ω y= x ω c Figure 3.4: Definiion of he consan ime delay mapping d ' wihin he 3-D uni area. Consider Figure 3.5 which shows a wo-dimensional (-D) represenaion of Figure 3.4. The funcion defining he modulaed edge hus needs o be shifed in boh he x and y

76 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD direcions in such a way ha poin A is now locaed a A' wih he disance from A o A' equaling d '. However, since he pulse rain is represened by he projecion of he inersecion of he funcion defined along he sraigh line y=( / c )x ono a plane parallel o he x-axis he acual ime delay inroduced is d insead of d '. This resuls in an error err being inroduced. Ideal Modulaed Edge Φ ' d ω y= x ω c A' y offse A Delayed Modulaed Edge Φ err Φ d Figure 3.5: -D represenaion of he error inroduced for he modulaed edge. The correc delay can be included ino he uni area by shifing he ideal modulaed edge by he corresponding radians d =f c d (since he mapping of T c = wihin he uni area) in he posiive x-direcion, and adjusing he y-offse (denoed y offse ) in such a way ha poin A coincides wih A', i.e. y offse =( / c ) d. If i is assumed ha he ideal modulaed edge is described by x=mcos(y)+, he delayed edge is given by: ( ) x= πm y y + +Φ (3.8) cos offse π d This is illusraed in Figure 3.6. By nex insering he limis defined in Figure 3.6 he complex Fourier coefficien of Eq. (.15) can be wrien as: π M cos( y y offse ) d V π + π+φ d jmx jny Cmn = e dxe dy π (3.9)

77 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD y V d x= πmcos( y y offse ) + π + Φ d x= πmcos( y) + π Φ d x Figure 3.6: 3-D uni area for TENPWM wih delay d. Table 3. compares he analyical and simulaion resuls for TENPWM wih he addiion of he delay. Noe ha d =1ns and M=.8. I is eviden ha he harmonic magniudes of he analyical and simulaion resuls correlae, which verifies he validiy of he argumen. TABLE 3. COMPARISON OF ANALYTICAL AND SIMULATION RESULTS FOR TENPWM WITH A CONSTANT TIME DELAY INTRODUCED ON THE MODULATED EDGE. Harmonic Number Analyical Magniude [V] Simulaed Magniude [V]

78 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD 3.3. UNMODULATED EDGE Figure 3.7 shows an exrac of he pulses of a TENPWM waveform wih he leading edge delayed by d. The ideal modulaed edge is once again represened by he dashed line. d d d V d T c Figure 3.7: Definiion of d inroduced on he unmodulaed edge. Consider Figure 3.8 which represens a -D skech of he error inroduced on he unmodulaed edge of a TENPWM waveform. In a similar manner as described in Secion 3.3.1, he unmodulaed edge needs o be shifed in he x direcion by an amoun d. However, since he leading edge is unmodulaed, i.e. independen of y, he projecion of he inersecion of he funcion defined along he sraigh line y=( / c )x wih poin A' produces he same projeced disance d ono a plane parallel o he x-axis as he projecion of he inersecion of a funcion defined along a line parallel o he x-axis and poin B. Ideal Unmodulaed Edge Φ d A' ω y= x ω c y offse A Delayed Unmodulaed Edge B Φ err Φ d Figure 3.8: -D represenaion of he error inroduced for he unmodulaed edge

79 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD I is hus eviden ha, for an unmodulaed edge (no y-dependency), no correcion is necessary in he y direcion. The corresponding delay wihin he uni area is hus simply defined by d =f c d. 3.4 INCORPORATION OF A SINUSOIDAL CURRENT POLARITY DEPENDENCY Consider a TENPWM waveform which is governed by some inducor curren polariy dependency. Suppose ha, during he negaive half-cycle of a purely sinusoidal inducor curren, he railing edge is delayed by an arbirarily chosen ime (denoed d ). This secion shows how his condiion can be incorporaed ino he 3-D uni area. Since he curren polariy can be sampled on eiher edge of he TENPWM waveform he analysis needs o be divided ino wo pars POLARITY DEPENDENCY SAMPLED ON THE TRAILING EDGE In his secion i is assumed ha he curren polariy is sampled on he ime insan of swiching of he ideal modulaed railing edge. This sampling process is illusraed in Figure 3.9 for a carrier o fundamenal raio chosen as c / =5. Noe ha he ideal modulaed edge is represened by he dashed line. d d d T c V d i < L i L i > i > L Figure 3.9: Proposed arbirary inducor curren polariy dependency sampled on he ideal railing edge. L

80 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD This condiion can nex be inegraed ino he 3-D uni area. Firsly, he various areas wihin he uni area, during which he inducor curren is posiive and negaive, need o be idenified. Since he polariy is assumed o be sampled on he swiching insan of he ideal modulaed edge he curren polariy can be defined over he regions as shown in Figure 3.1. y Posiive Curren Negaive Curren 3 Pivo Pivo x= πmcos( y) + π x Figure 3.1: Definiion of he various curren zones wihin he 3-D uni area. The slope wih pivos exising a he polariy ransiion sampled on he railing edge illusraed (y=/ and y=3/) is necessary in order o keep he sampled curren polariy consan over he region <x. Figure 3.11 represens a combinaion of Figure 3.9 and Figure 3.1 wih he required mapping. The relaion wihin he 3-D uni area using he proposed polariy sampling process is clearly visible from his illusraion. Consider he region wihin Figure 3.11 for which he inducor curren is negaive. In his area he railing edge is delayed by d in accordance wih he original requiremen. A corresponding mapping equalling d =f c d hus needs o be inroduced in accordance wih he analysis described in Secion In he region where he inducor curren is posiive, boh he leading and railing edges correspond o hose of ideal TENPWM

81 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD d d d π V d i < L i L i > i > L Figure 3.11: Combinaion of Figure 3.9 and Figure 3.1 illusraing he sampling process. L The 3-D uni area describing hese condiions is shown in Figure 3.1. Noe ha he linear secions exising wihin inervals y 1 <y y and y 3 <y y 4 are defined a a slope / c o ensure a single crossing poin a he inersecion of line y=( / c )x wih he walls. The complex Fourier coefficien of Eq. (.15) can nex be wrien by insering he limis defined in Figure 3.1 such ha: ωc y ( ) ( 1) 1 πm cos y + π y y y + π ω V d jmx jny jmx jny Cmn = e dx e dy e dx e dy + + π y1 (3.1) ωc y πm cos( y y ) ( ) ( 3 ) 3 offse π d π πm cos y π y y y + π 4 ω jmx jny jmx jny jmx jny + e dxe dy+ e dxe dy+ e dxe dy y y3 y3 + +Φ + Where y 1 =/, y =/+( / c ) d, y 3 =3/ and y 4 =3/+( / c ) d

82 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD y V d y 4 y 3 ωc x= ( y- y 3 )+ π ω x= πmcos( y- y offse ) + π + Φ d y y 1 ωc x= ( y- y 1 )+ π ω x= πmcos( y) + π Φ d x Figure 3.1: 3-D uni area for TENPWM wih curren polariy dependency. Table 3.3 shows a comparison beween he analyical and simulaion resuls for TENPWM including he curren polariy condiions. The parameers used correspond o d =1ns and M=.8. The analyical and simulaion resuls correlae. TABLE 3.3 COMPARISON OF ANALYTICAL AND SIMULATION RESULTS FOR TENPWM WITH INDUCTOR CURRENT POLARITY CONDITION SAMPLED ON THE TRAILING EDGE. Harmonic Number Analyical Magniude [V] Simulaed Magniude [V]

83 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD 3.4. POLARITY DEPENDENCY SAMPLED ON THE LEADING EDGE In his secion i is assumed ha he curren polariy is sampled on he ime insan of swiching of he ideal unmodulaed leading edge. This is shown in Figure 3.13 for a carrier o fundamenal raio c / =5. The ideal modulaed edge is represened by he dashed line. d d T c V d i < L i L i > i > L Figure 3.13: Proposed arbirary inducor curren polariy dependency sampled on he ideal leading edge. L Similar o mehodology followed in Secion 3.4.1, he firs sep in inegraing he condiion described in Figure 3.13 ino he 3-D uni area is o define he various regions for which he inducor curren is posiive and negaive. Since he curren polariy is assumed o be sampled on he leading edge he various regions need o be defined a a slope / c as shown in Figure 3.14, wih he pivo poin as indicaed. This ensures ha he polariy sample is held consan over a single swiching period <x

84 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD y 3 Pivo Posiive Curren Negaive Curren ω Slope = ω c Pivo ω Slope = ω c x= πmcos( y) + π x Figure 3.14: Definiion of he various curren zones wihin he 3-D uni area. A combinaion of Figure 3.13 and Figure 3.14 is shown in Figure From his illusraion i can be seen how he polariy is sampled and held over a single swiching period. From Figure 3.13 i is eviden ha he railing edge need o be delayed by d if he curren polariy is negaive on he swiching insan of he ideal unmodulaed leading edge. Once again a corresponding mapping of d = d f c needs o be inroduced wihin he region in Figure 3.14 for which he inducor curren is negaive

85 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD d d π V d i < L i L i > i > L Figure 3.15: Combinaion of Figure 3.13 and Figure 3.14 illusraing he sampling process. L The 3-D uni area represening his condiion described is shown in Figure No closed form soluion for variables y 1 4 can be achieved and hus needs o be solved numerically. Variable y 1 can be calculaed from x=mcos(y)+ and y=( / c )x+/. Also, y 3 can be deermined from x=mcos(y)+ and y=( / c )x+3/. The remaining variables y and y 4 are found by respecively adding ( / c ) d o y 1 and y 3. The limis of Figure 3.16 correspond o hose defined in Eq. (3.1) wih only y 1 4 being differen

86 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD y y 4 y 3 ωc 3π x= y- ω x= πmcos( y-y offse ) + π + Φ d y y 1 ω π x= y- ω c x= πmcos( y) + π Φ d x Figure 3.16: 3-D uni area for TENPWM wih curren polariy dependency. Table 3.4 shows a comparison beween he analyical and simulaion resuls for TENPWM including he curren polariy condiions. The parameers used correspond o d =1ns and M=.8. I can be seen ha similar resuls are produced. TABLE 3.4 COMPARISON OF ANALYTICAL AND SIMULATION RESULTS FOR TENPWM WITH INDUCTOR CURRENT POLARITY CONDITION SAMPLED ON THE LEADING EDGE. Harmonic Number Analyical Magniude [V] Simulaed Magniude [V]

87 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD 4 Carrier 1 wih Sidebands Fundamenal Componen Carrier wih Sidebands 4 Carrier 1 wih Sidebands Fundamenal Componen Carrier wih Sidebands - - Magniude [db] Baseband Harmonics Magniude [db] Baseband Harmonics Frequency [khz] (a) Frequency [khz] (b) Figure 3.17: Simulaed TENPWM specra for sampling on he (a) railing and (b) leading edge. Figure 3.17 (a) shows he simulaed specra using he sampling process of Secion while (b) shows he specra wih sampling on he leading edge described in his secion. The difference in clearly visible. 3.5 INCORPORATION OF A PURELY SINUSOIDAL NON-LINEAR INDUCTOR CURRENT MAGNITUDE DEPENDENCY IN THE 3-D UNIT AREA Consider a TENPWM waveform of which eiher he leading or railing edge is governed by some non-linear purely sinusoidal inducor curren magniude dependency. The curren dependen delay d inroduced is assumed o be governed by he arbirarily chosen relaion: ( ω ) = k I cos (3.11) d scaling Where k scaling is a consan scaling variable. In his secion i is shown how his condiion can be incorporaed ino he 3-D uni area by applying he mehodology inroduced in Secions 3.3 and 3.4. Noe ha he curren magniude is assumed o be sampled on he same edge on which he error is being inroduced THE UNMODULATED LEADING EDGE In his secion he unmodulaed leading edge is delayed in accordance wih he relaion defined in Eq. (3.11). Since he curren magniude is sampled on he leading edge, he curren

88 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD polariy zones defined in Figure 3.14 hold. Nex, he equivalen mapping of Eq. (3.11) wihin he uni area can be esablished as d =f c k scaling and y=. As menioned in Secion 3.3., he ideal leading edge of a TENPWM waveform is independen of y. By adding Eq. (3.11) o he leading edge a y dependency is inroduced. This means ha he adjusmen (y offse ) discussed in Secion 3.3 mus be included. The expression describing he curren dependen leading edge is hus given by: Φ =Φ I y y (3.1) cos ( ) d scaling offse Where y offse =( / c ) d. This allows for he 3-D uni area o be consruced, which is illusraed in Figure y x=φ d x= πmcos( y) + π V d x Figure 3.18: 3-D uni area for TENPWM wih curren magniude dependency

89 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD By insering he limis defined in Figure 3.18 ino Eq. (.15) yields: π πm cos( y yoffse ) + π d Φd V jmx jny Cmn = e dxe dy π (3.13) Table 3.5 shows a comparison beween he analyical and simulaion resuls for TENPWM including he curren polariy condiions. The parameers used correspond o d =1ns, M=.8. The analyical and simulaion resuls correlae well. TABLE 3.5 COMPARISON OF ANALYTICAL AND SIMULATION RESULTS FOR TENPWM WITH INDUCTOR CURRENT MAGNITUDE DEPENDENCY SAMPLED ON THE TRAILING EDGE. Harmonic Number Analyical Magniude [V] Simulaed Magniude [V] THE MODULATED TRAILING EDGE In his secion he modulaed railing edge is considered. The curren magniude is sampled on he ideal railing edge which means ha he polariy zones defined in Figure 3.1 apply. The equivalen mapping of Eq. (3.1) wihin he uni area is he same as in Secion The 3-D uni area represening hese condiions is shown in Figure The complex Fourier coefficien of Eq. (.15) can be wrien as: π πm cos( y yoffse ) + π d Φd V jmx jny Cmn = e dxe dy π (3.14) Where d is defined in Eq. (3.1). A comparison beween he analyical an simulaion resuls are shown in Table

90 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD y V d x= πmcos( y y offse ) + π + Φ d x= πmcos( y) + π x Figure 3.19: 3-D uni area for TENPWM wih curren magniude dependency. The condiions correspond o hose of Secion From Table 3.6 i can be seen ha he analyical and simulaion resuls correlae well. TABLE 3.6 COMPARISON OF ANALYTICAL AND SIMULATION RESULTS FOR TENPWM WITH INDUCTOR CURRENT MAGNITUDE DEPENDENCY SAMPLED ON THE LEADING EDGE. Harmonic Number Analyical Magniude [V] Simulaed Magniude [V]

91 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD 3.6 INCORPORATION OF SECTION 3. WITHIN THE 3-D UNIT AREA The previous secion assumed a purely sinusoidal inducor curren. In his secion i will be shown how he curren model of Secion 3. can be inegraed ino he uni area o provide a ool where he ripple curren can be parsed as a parameer o model more realisic polariy dependencies. The objecive is achieved via simple manipulaion of Eq. (3.3) in which D needs o be described as a coninuous funcion wih he required mapping of T c = wihin he uni area. Since y=f and D=((1+Mcos(f ))/) Eq. (3.3) can be rewrien as: V Δ = 1 cos ( ) (3.15) d il M y 8Lfil fc Since i L(upper_env) and i L(lower_env) are respecively defined on he railing and leading edges he implemenaion has o be done wih Secion 3.4 in mind. Figure 3. defines he various curren polariy zones for i L(upper_env) wihin he 3-D uni area according o he analysis of Secion y Posiive Curren Negaive Curren y 3 y i L ( upper_env ) x Figure 3.: Definiion of he various curren zones for i L(upper_env) wihin he 3-D uni area

92 CHAPTER 3 INCORPORATION OF PTES IN THE DOUBLE FOURIER SERIES METHOD In a similar manner Figure 3.1 defines he various curren polariy zones for i L(lower_env) wihin he 3-D uni area according o he analysis of Secion Noe ha no closed form soluion can be achieved for variables y 1 4 and ha i mus hus be solved numerically. y y 4 Posiive Curren Negaive Curren i L ( lower_env ) y 1 x Figure 3.1: Definiion of he various curren zones for i L(lower_env) wihin he 3-D uni area. A general power elecronic ool has been inroduced in his chaper. However, wihin swiching audio applicaions, he analysis can be simplified by noing ha y offse if f c >>f. This assumpion will be applied for he res of his disseraion. 3.7 SUMMARY A general analyical ool has been developed which allows for he calculaion of he harmonic composiion of naurally sampled PWM in he presence of curren dependen pulse iming errors. The analysis was verified by simulaion. The analysis firsly considered a purely sinusoidal inducor curren afer which a more realisic curren model wih a non-zero ripple was included. The proposed analyical models and simulaion resuls correlaed

93 4 C GD C GS SWITCHING DEVICE CHARACTERISTICS 4.1 INTRODUCTION The power MOSFET has been available since he early 198 s [4]. Even hough he power BJT was developed during he 195 s [8] and has coninued o evolve ever since, fundamenal drawbacks exis in is operaing characerisics. The developmen of he power MOSFET was primarily driven by he limiaions posed by he power BJT. The BJT is a curren conrolled device. This means ha a coninuous curren mus be applied o is base in order o keep he device in he on sae. Even larger reverse base currens are required o ensure rapid urn-off. The base drive circuiry required hus ends o be complex and expensive. The required gae drive characerisics compromise boh efficiency and lineariy. The power BJT s safe operaing area is limied by is inabiliy o handle high volages and currens simulaneously. Anoher limiaion is he difficulies of using he devices in parallel. This can be ascribed o he decrease in forward volage drop for increasing emperaure, which in urn leads o curren diversion o a single device. In conras, he MOSFET is a volage conrolled device. Is insulaed gae srucure need only be supplied wih curren in order o charge and discharge he inpu capaciance during swiching ransiions. This means ha no seady sae curren is required in he on or off sae. The gae drive requiremens are hus grealy simplified as a resul of he high inpu impedance. Furhermore, since curren conducion occurs only hrough ranspor of majoriy carriers, no delays are inroduced as a resul of he recombinaion process associaed wih he injecion of minoriy carriers. This leads o faser swiching imes han he power BJT. Since he power MOSFET is no subjec o second breakdown, he safe operaing area is also much greaer han ha of he power BJT. Lasly, hese devices can easily be mouned in parallel because he forward volage drop increases wih increasing emperaure. This resuls in even curren disribuion amongs paralleled devices

94 CHAPTER 4 SWITCHING DEVICE CHARACTERISTICS From he brief discussion above i is eviden ha he power MOSFET represens a closer approximaion o an ideal swich han he power BJT. I hus finds is applicaion in audio swiching amplifiers in which fideliy is he primary design parameer. This chaper describes he fundamenal operaion of he power MOSFET on a semiconducor level. The inclusion of his overview is essenial since he non-ideal effecs invesigaed in Chapers 5 o 8 are eiher a primary or secondary consequence of is operaion. The firs four secions of his chaper (4. o 4.5) conain a fundamenal overview of he basic srucure, operaion and dynamic characerisics, which are summarized from [4]. Secion 4.6 describes he operaion wihin a single phase leg, in which all non-ideal effecs considered in his disseraion are included. Noe ha he above menioned analysis is adaped and summarized from [3]. 4. POWER MOSFET STRUCTURE Figure 4.1 illusraes a verical diffused MOSFET (VDMOS). This device is fabricaed using a n + pn - n + srucure and is ermed an enhancemen mode n-channel MOSFET. This verical srucure consiss of alernaing n-ype and p-ype doped semiconducor layers wih he n + op layer labelled he source and he n + boom ermed he drain. The p-ype region is known as he body wih he n - layer called he drif region. Source Gae Field Oxide Gae Oxide n + p + + n n p n + Parasiic BJT n - Inegral Diode n + Drain Figure 4.1: Verical cross-secional view of a power MOSFET [4]

95 CHAPTER 4 SWITCHING DEVICE CHARACTERISTICS From Figure 4.1 i should be noed ha a parasiic BJT exiss wihin he n + pn - n + srucure wih he p body region represening is base. This BJT mus be kep in he off sae a all imes, since forward biasing would lead o unwaned conducion, which in urn can resul in device breakdown. The source meallizaion connecs he p body region o ha of he n + op layer, creaing a shor circui beween he base and he emier of he parasiic BJT. This shor leads o an inegral diode being conneced beween he drain and source regions. 4.3 POWER MOSFET OPERATION From Figure 4.1 i seems ha curren flow beween he gae and source erminals is impossible since one of he pn juncions will always be reverse biased. Moreover, he gae meallizaion is isolaed by a layer of silicon dioxide (SiO ), which rules ou he possibiliy of minoriy carrier injecion. In order o have curren flow beween he drain and source erminals, i is imperaive o esablish a conducive pah beween he n + source and n - drif regions. This pah is creaed by applying a posiive volage o he gae erminal (wih respec o he source). The posiive charge exising on he gae elecrode requires an equal bu inverse negaive charge on he silicon beneah he SiO layer. The elecric field esablished in his way aracs elecrons o he p base region jus beneah he SiO while repelling majoriy carriers (holes). This creaes a depleion region. An increased gae volage aracs more elecrons o he surface while repelling more holes. This resuls in an increased depleion region wih he holes pushed ino he p base area beyond he depleion boundary. These addiional holes are neuralized by elecrons from he n + source region. An equilibrium is esablished once he densiy of he free elecrons beneah he gae oxide is equal o he densiy of free holes beyond he depleion region. This channel of free elecrons beneah he SiO is known as he inversion layer. The value of applied gae volage a which his inversion layer is formed is known as he gae-o-source hreshold volage. In his sae he free elecrons jus below he gae oxide are highly conducive and can be regarded as a n-ype semiconducor. The applicaion of a drain volage will resul in curren flow hrough he channel region. The power MOSFET is swiched ino he off sae by urning he gae volage back o zero. This is done by creaing a shor circui beween he gae and source erminals. Since he elecrons are no longer araced by he induced elecric field, he channel beween he n + source and n - drif region is broken

96 CHAPTER 4 SWITCHING DEVICE CHARACTERISTICS 4.4 CHARACTERISTIC CURVES Figure 4. (a) shows a plo of i D as a funcion of v DS for various values of he V GS for a n- channel device. The power MOSFET is cu off whenever v GS is smaller han V GS(h). In his off sae he pn-juncion is reverse biased and he device is able o block any volage smaller han BV DSS. As shown in Figure 4. (a), when operaing in he acive region he drain curren is only dependen on v DS. The power MOSFET eners he ohmic region whenever v GS V GS(h) >v DS >. i D i D Ohmic Acive V GS4 V GS3 V GS V GS1 BVDSS Cuoff (a) v DS V GS ( h ) (b) v GS Figure 4.: (a) Oupu and (b) ransfer characerisic curves [4]. This condiion is achieved by driving he device wih a relaively large V GS. In his sae v DS is small, and i is deermined by he device s on-sae resisance and operaing curren. Figure 4. (b) illusraes i D as a funcion of v GS. 4.5 POWER MOSFET DYNAMIC MODEL The swiching speed of he power MOSFET is limied o he charge and discharge of he inpu gae capaciance due o is insulaed gae srucure. Figure 4.3 shows a cross-secional view of he power MOSFET srucure, illusraing he locaion of he parasiic capaciances. These capaciances can be divided ino wo primary groups of which he firs is labelled C GS and he second of which is labelled C GD. The former capaciance can again be subdivided ino hree separae componens. The firs wo componens of C GS arise as a resul of he overlap of

97 CHAPTER 4 SWITCHING DEVICE CHARACTERISTICS he gae elecrode wih he n + source and p base regions, respecively denoed as C n+ and C p. The hird capaciance exiss due o he overlap of he gae elecrode wih he source meallizaion, labelled C m. Noe ha he laer is deermined by he ype of insulaor used beween he gae elecrode and he source meallizaion as well as is hickness. Source Gae C m Cn+ Cp n + p n + C GD Depleion Layer Boundary n - n + Drain Figure 4.3: Verical cross-secional view of a power MOSFET wih parasiic capaciances [4]. The second capaciance exiss as a resul of he overlap of he gae elecrode wih he n - drif region. As menioned, i is labelled C GD and consiss of wo componens in series. The firs componen arises from he capaciance creaed by he gae elecrode and oxide layer, which is once again deermined by he hickness and ype of insulaing maerial used. The second componen is a resul of he capaciance inroduced by he depleion layer. Wih zero volage applied o he gae elecrode, he depleion layer beneah he gae oxide will be a a minimum. By applying a posiive gae volage, he depleion layer will sar forming, increasing for higher values of applied volage. This in urn leads o a decrease in he capaciance of he depleion layer because i is widened. This process coninues unil he surface inversion layer is formed and he depleion layer is a is maximum value. In his sae he depleion layer capaciance exhibis is lowes value. I is hus eviden ha hese capaciances are no consan and change wih volage due o he parial conribuion from he depleion layer. Figure 4.4 shows a plo of C GD as a funcion of v DS

98 CHAPTER 4 SWITCHING DEVICE CHARACTERISTICS C GD v DS Figure 4.4: C GD as a funcion of v DS [4]. Figure 4.5 shows he circui diagrams used o model he parasiic capaciances, wih (a) represening he MOSFET when i is eiher cuoff or in he acive region. During he inerval when v GS <V GS(h) he MOSFET is cu off and i D =. Once he hreshold volage is reached he device eners he acive region and he curren increases according o he curren-volage ransfer characerisic of Figure 4. (b) wih he MOSFET s ransconducance g m defining he slope. In his sae he drain curren is hus defined by g m (v GS <V GS(h) ). Drain Drain Gae C GD C GS C GD i D = f ( V GS ) Gae r DS ( on ) C GS Source Source (a) (b) Figure 4.5: Circui model when power MOSFET is in he (a) acive and (b) ohmic region [4]. The MOSFET eners he ohmic region once v GS V GS(h) >v DS >. In his sae he inversion layer has a nearly uniform hickness and effecively shors he drain and source erminals. This means ha he circui model of Figure 4.5 (a) is no longer valid. Figure 4.5 (b) shows he equivalen circui in he ohmic region. Noe he inclusion of a non-zero r DS(on) beween he drain and source erminals. This is a resul of he resisances associaed wih he differen regions of he semiconducor maerial

99 CHAPTER 4 SWITCHING DEVICE CHARACTERISTICS 4.6 POWER MOSFET SWITCHING WAVEFORMS This sub-secion is adaped and summarized from [3] and serves as a brief overview of he MOSFET swiching characerisics. Figure 4.6 illusraes he swiching waveforms wihin one leg of a single phase inverer for a posiive i L. V d T A1 T A i L V d T A1 T A i L V d T A1 T A i L V d T A1 TA i L (a) V GS ( h ) (b) V GS d d (c) I d d ( vf ) vf d( vr) vr (d) V DF ir L ds( on) V d ir L ds( on) (e) Figure 4.6: Swiching characerisics in a single phase leg for a posiive inducor curren [3]. The solid bold waveforms of Figure 4.6 (a) and (b) respecively represen he pracical gaing signals of he high side and low side MOSFETs, whereas he dashed lines represen he

100 CHAPTER 4 SWITCHING DEVICE CHARACTERISTICS ideal waveforms. The low side MOSFET s i D is shown in Figure 4.6 (c) wih v DS represened by (d). The ideal swiching oupu measured over T A is shown in Figure 4.6 (e). Wih he circui diagrams represening he devices in play during he specified inervals a deailed descripion of each inerval can now be consruced. This is given in Table 4.1. TABLE 4.1 DESCRIPTION OF THE ACTION ON THE VARIOUS TIME INSTANTS OF FIGURE 4.6. Time Descripion 1 The gaing signal of T A1 swiches from high o low and he discharge of C GS resuls in d(off). v GS becomes emporarily clamped o a fixed value needed o mainain I L according o he ransfer characerisic of Figure 4. (b). Since v GS remains consan during his inerval, all of he inpu curren supplied by V GS flows hrough C GD. This causes he v DS o drop rapidly as i ransverses hrough he acive oward he ohmic region of Figure 4. (a). 3 D A sars is commuaion sequence once v DS of T A goes low enough, which in urn causes v GS o fall rapidly. V DF of D A causes v DS of T A o fall o a value V DF. 4 v GS =V GS(h) and T A1 is cu off. i L now flows compleely hrough D A d is over and he gaing signal of T A ransiions from low o high. Since he incepion of his inerval boh v DS and v GD equals zero, which means ha T A never eners he acive region. This resuls in rapid swiching. The only requiremen is he shif of charge from he D A o T A. Once he complee curren is ransferred o T A volage v DS falls o i L r DS(on). The gaing signal of T A ransiions from high o low. v GS =V GS(h) and T A is cuoff. Due o he curren polariy he curren will ransfer direcly o D A again allowing for a rapid ransiion.v DF causes v DS o fall o V DF. Afer d he gaing signal of T A1 changes from low o high. v GS =V GS(h) and hus T A1 eners he acive region. The combinaion of inervals 8 and 9 leads o d(on). v GS becomes emporarily clamped o a fixed value needed o mainain I L according o he ransfer characerisic of Figure 4. (b). Since v GS remains consan during his inerval all of he inpu curren supplied by V GS flows hrough C GD. This causes he v DS o rise rapidly as i ransverses hrough he acive o he ohmic region. 11 T A eners he final inerval once v DS has reached is on-sae value a i L r DS(on) below V d. Figure 4.7 shows he swiching waveforms wihin one leg of a single phase inverer for a negaive i L. The differen swiching inervals can be deermined in a similar manner as for he posiive inducor curren. As a resul, i is no necessary o discuss i in deail. The non-ideal swiching characerisics described in Figure 4.6 and Figure 4.7 serve as a basis for he analysis which follows in he remainder of his disseraion

101 CHAPTER 4 SWITCHING DEVICE CHARACTERISTICS V d T A1 T A i L V d T A1 T A i L V d T A1 T A i L V d T A1 TA i L (a) V GS ( h ) V GS (b) V GS V GS ( h ) d d (c) I d d ( vf ) vf V DF d( vr) vr (d) ir L ds( on) V d ir L ds( on) (e) Figure 4.7: Swiching characerisics in a single phase leg for a negaive inducor curren [3]. 4.7 SUMMARY The firs par of his chaper (Secions 4. o 4.5) summarized he basic srucure, operaion and dynamic characerisics of he power MOSFET, which have been summarized from [4]. The final par (Secion 4.6) was adaped from [3] and considered he non-ideal swiching behaviour wihin a single phase leg. This chaper hus served as a foundaion in which he primary and secondary consequences of is operaion on he swiching waveform wihin a single phase leg were considered

102 d 5 v sw THE EFFECT OF DEAD TIME 5.1 INTRODUCTION This chaper conains a deailed analysis of he effec of dead ime on harmonic disorion in open loop class D audio amplifiers, adaped from he papers in [3] and [46] by he auhor. A fundamenal analysis is considered in Secion 5.. Employing he findings in Chaper 3, an analyical model describing TENPWM wih dead ime is proposed in Secion 5.3. A simulaion model based on he sraegy inroduced in Chaper.5 is considered in Secion 5.4, afer which a deailed discussion of he analyical and simulaion resuls follows in Secion 5.5. This chaper concludes wih a summary. 5. ANALYSIS OF DEAD TIME The analysis will be divided ino wo secions. Secion 5..1 conains a deailed analysis for he region during which boh i L(upper_env) and i L(lower_env) are posiive or boh of hem are negaive (inducor curren Scenario in Secion 3.). In Secion 5.. he remaining region where i L(upper_env) is posiive and i L(lower_env) is negaive is invesigaed (inducor curren Scenario in Secion 3.) A DISTINCTLY POSITIVE AND NEGATIVE INDUCTOR CURRENT (SCENARIO ) The effec of dead ime on he swiched oupu volage can bes be described using one leg of he single phase inverer shown in Figure 5.1. Through simple inspecion of he his illusraion i is eviden ha he oupu volage depends on he polariy of he oupu curren when boh swiches (T A1 and T A ) are off. Addiional saes exis when he swiches ransiion from off o on and vice versa

103 CHAPTER 5 THE EFFECT OF DEAD TIME Consider Figure 5.1 (a), which represens he region during which i LA is disincly posiive in he presence of a non-zero dead ime. Referring o he illusraion in (a), consider he ransiion of T A1 from on (curren direcion ) o off and T A from off o on. During d diode D A conducs (curren direcion ) while diode D A1 blocks he curren flow o he posiive rail. This resuls in he same oupu volage as for an immediae ransiion of T A from off o on, which in urn corresponds o he correc oupu volage. Nex, wih he oupu curren sill posiive, consider he change of T A from on (curren direcion ) o off and T A1 from off o on. Again, D A conducs (curren direcion ) while D A1 blocks he curren flow o he posiive rail. This condiion leads o a loss of volage, since he oupu remains clamped o he negaive rail during d. T A1 D A1 T A1 D A1 V d V d i LA i LA T A D A T A D A (a) (b) Figure 5.1: Commuaion sequence in a single phase leg for (a) i LA > and (b) i LA <. The remaining wo condiions exis when he oupu curren is disincly negaive. Refer o Figure 5.1 (b). For a ransiion of T A from on (curren direcion ) o off and T A1 from off o on wih he addiion of d, D A1 conducs (curren direcion ) while D A is reverse biased. This resuls in he correc volage a he oupu erminals, since he same resul is achieved for a ransiion wih zero dead ime. The final condiion is given by a ransiion of T A1 from on (curren direcion ) o off and T A from off o on for a negaive oupu curren. The conducion sequence of D A1 and D A during d (curren direcion ) is exacly he same as for he laer case. Since he volage is clamped o he posiive rail during d insead of a ransiion o he negaive rail, a volage gain is achieved. Figure 5. illusraes he swiching waveforms influencing d for a posiive inducor curren. Since only d is considered, he remaining waveforms are considered as ideal. During he ransiion of v GS(high_side) from on o off he volage is assumed o swich insanly from V GS o zero

104 CHAPTER 5 THE EFFECT OF DEAD TIME v GS ( high_side ) (a) V GS d( vf ) v d ( vr ) GS ( low_side ) (b) V GS d d i L ( approx ) (c) vf i Lideal ( ) vr (d) v sw Vd Figure 5.: Swiching waveforms for a posiive inducor curren. (a) Low side gae-o-source volage. (b) High side gae-o-source volage. (c) Inducor curren. (d) Swiched oupu volage. This leads o a value of d(vf) =. Using similar condiions v GS(high_side) resuls in d(vr) = during a change of v GS(high_side) from off o on. This also resuls in insananeous swiching ransiions, i.e. vr = vf =. The inducor curren polariy is sampled a he ime insan of swiching on he ideal oupu volage ransiion, which occurs on eiher i L(upper_env) or i L(lower_env). This resuls in an approximaion i L(approx) wihin region d as shown in Figure 5. (c). The volage loss associaed wih i L > is shown in in Figure 5. (d). Figure 5.3 illusraes he swiching waveforms affecing d for a negaive inducor curren. The swiching waveforms are idealized in exacly he same way as for Figure 5.. The volage gain associaed wih i L < is shown in in Figure 5.3 (d)

105 CHAPTER 5 THE EFFECT OF DEAD TIME v GS ( low_side ) (a) V GS d( vf ) d ( vr ) v GS ( high_side ) (b) V GS d d (c) i L ( approx ) i Lideal ( ) vf vr (d) v sw Vd Figure 5.3: Swiching waveforms for a negaive inducor curren. (a) High side gae-o-source volage. (b) Low side gae-o-source volage. (c) Inducor curren. (d) Swiched oupu volage. 5.. NEITHER DISTINCTLY POSITIVE NOR NEGATIVE INDUCTOR CURRENT (SCENARIO ) Two possible sub-scenarios exis in he region where he inducor curren is neiher disincly posiive nor negaive. Firsly, consider Figure 5.4 (a) which illusraes a change in polariy of i L occurring wihin he region LE+ d <<TE or TE+ d <<LE. From he analysis in he previous secion i is eviden ha he commuaion sequence resuls in reproducing he ideal swiching waveform, i.e. he dead ime has no effec. The remaining sub-scenario exiss when i L changes polariy during d. This is shown in Figure 5.4 (b) for a pracical sysem, wih he magena and urquoise waveform respecively represening he swiched oupu volage and inducor curren

106 CHAPTER 5 THE EFFECT OF DEAD TIME T c i L LE TE LE d d d (a) (b) Figure 5.4: Change in curren polariy (a) ouside inerval d and (b) wihin inerval d [35]. The change in swiching sae of he oupu volage is a resul of reverse recovery. However, his disseraion assumes open loop sysems d <<T c and, as a resul, he laer scenario will be ignored in he analysis. 5.3 ANALYTICAL MODEL In his secion a model is inroduced for TENPWM which allows for he harmonic componens creaed by he dead ime o be calculaed analyically. Since inducor curren Scenario reproduces he ideal case under he assumpion described in he previous secion, i is only necessary o consider Scenario. Implemening he findings in Chaper 3 he uni area for TENPWM wih dead ime can be consruced. The leading edge wih polariy sampled on i L(lower_env) needs o be delayed by he corresponding mapping d =f c d wihin he region where boh i L(upper_env) and i L(lower_env) are posiive. In he region where boh i L(upper_env) and i L(lower_env) are negaive, he railing edge needs o be delayed by d. The 3-D uni area represening hese condiions is shown in Figure 5.5. Noe ha variables y 1, y, y 3 and y 4 are solved numerically, as menioned in Chaper

107 CHAPTER 5 THE EFFECT OF DEAD TIME y x=φ d y 4 x= πmcos( y) + π i L ( lower_env ) y 3 V d x= πmcos( y) + π + Φ d y i L ( upper_env ) y 1 Φ Φ d d x Figure 5.5: 3-D uni area for TENPWM wih dead ime. By nex insering he limis defined in Figure 5.5, he complex Fourier coefficien of Eq. (.15) can be wrien as: y1 π+ πm cos( y) y π+ πm cos( y ) jmx ( + ny) jmx ( + ny) Vd Cmn = e dx dy e dx dy π + Φ d y 1 π + π cos( ) +Φ d y π + π jmx ( ny) y3 M y 4 M cos y + jmx ( + ny) + e dx+ e dxdy y y 3 ( ) π π+ πm cos( y) jmx ( + ny) e dxdy dy (5.1) + y4 Φd The complee soluion defining he uni area for TENPWM wih dead ime is achieved by subsiuing Eq. (5.1) ino Eq. (.1). Noe ha Eq. (5.1) is solved numerically

108 CHAPTER 5 THE EFFECT OF DEAD TIME 5.4 SIMULATION STRATEGY The simulaion echnique inroduced in Secion.5 will now be adaped o deermine he specrum of TENPWM wih dead ime. The remainder of his sub-secion describes he aleraions necessary o he ideal simulaion sraegy of Secion.5 o represen he pulses wih dead ime. The volage loss and gain inroduced in he TENPWM waveform wih dead ime is respecively shown in Figure 5.6 (a) and (b). Noe ha he parameers used in Figure 5.6 correspond o hose of he ideal case originally inroduced in Figure.14. n TENPWM +1 n TENPWM ctenpwm () m TENPWM () 1 1 ctenpwm () mtenpwm () W p TENPWM W p TENPWM V d d V d V d V d d ( p -1T ) TENPWM c p TENPWM T c ( p -1T ) TENPWM c p TENPWM T c p TENPWM T c p TENPWM T c (a) (b) Figure 5.6: Generaion of TENPWM wih dead ime for (a) i L > and (b) i L <. Through evaluaion of Figure 5.6 (a) and (b) an expression for he widh of he p h pulse can be found. Since he widh is dependen on he curren polariy, i mus be defined over wo regions. This leads o he equaliy: W ptenpwm, d Wp when TENPWM d il > = Wp + when TENPWM d il < (5.)

109 CHAPTER 5 THE EFFECT OF DEAD TIME Noe ha he ime shif for he ideal case of Eq. (.94) is also dependen on he pulse widh. Thus, redefining he ime shif in he presence of blanking ime yields: W ( p ) T i = W ( p ) T + + i < ptenpwm, d d TENPWM 1 c + when L > ptenpwm, d d TENPWM 1 c when L (5.3) 5.5 ANALYTICAL AND SIMULATION RESULTS The analyical soluions derived in Secion 5.3 and he simulaion model of Secion 5.4 will now be compared and discussed for TENPWM. The parameers used correspond o hose defined in Chaper 1.1 unless saed oherwise. The parameers used in his secion correspond o V d =3V M=.8, L fil =1.4μH and c / =384 unless saed oherwise HARMONIC COMPOSITION OF TENPWM WITH DEAD TIME In his secion he effec of d wihin he audible band is considered. The plo shown in Figure 5.7 (a) was consruced using Eq. (5.1) in which only he baseband harmonics were considered, i.e. m=. The simulaion sraegy of Secion 5.4 was employed o consruc he specrum of Figure 5.7 (b). Noe ha d =15ns in boh plos. 4 4 Fundamenal Componen Fundamenal Componen Magniude [dbv] - -4 Magniude [dbv] Frequency [khz] Frequency [khz] (a) (b) Figure 5.7: (a) Analyical (m=) and (b) simulaed baseband harmonics for TENPWM

110 CHAPTER 5 THE EFFECT OF DEAD TIME From (a) i can be seen ha he analyical specrum only conains baseband harmonics a odd orders of he modulaing waveform. The simulaion in (b), however, conains addiional even order harmonics. This suggess ha he sideband swiching harmonics appear as disorion in he audible band. 4 4 Fundamenal Componen Analyical Specrum, m= Analyical Specrum, m=1 Magniude [dbv] - -4 Magniude [dbv] Frequency [khz] (a) Frequency [khz] (b) Figure 5.8: Analyical specrum for (a) m=1 and 383n 364 and (b) combinaion wih Figure 5.7 (a). The conribuion of he cross modulaion producs of he firs carrier (m=1 and 383<n< 364) is shown in Figure 5.8 (a). A combinaion of Figure 5.7 (a) and Figure 5.8 (a) is illusraes in Figure 5.8 (b). The laer plo was consruced by simply ploing Figure 5.7 (a) on op of Figure 5.8 (a), hus no allowing for phasor summaion of he various harmonics. TABLE 5.1 ANALYTICAL AND SIMULATED MAGNITUDE OF THE BASEBAND HARMONICS FOR TENPWM. Harmonic Number [-] Analyical Magniude [dbv] Simulaed Magniude [dbv]

111 CHAPTER 5 THE EFFECT OF DEAD TIME The analyical specrum of Figure 5.8 (b) correlaes very well wih he simulaion of Figure 5.7 (b). Noe ha, heoreically, he complee analyical specrum wihin he specified range can be obained by phasor summaion of all he modulaion producs wihin he baseband. Table 5.1 compares he magniudes of he analyical and simulaion resuls of he odd order harmonics of Figure 5.7, which correlae very well. 4 Carrier 1 wih Sidebands Fundamenal Componen Carrier wih Sidebands Magniude [dbv] Baseband Harmonics Frequency [khz] Figure 5.9: Simulaed specrum showing he firs wo carrier harmonics and is respecive sidebands. Figure 5.9 is a specral plo consruced using he simulaion sraegy of Secion 5.4. The firs wo carrier harmonics wih is respecive sidebands are shown. From his illusraion i is eviden ha he sideband swiching harmonics do no decay a such a rapid rae in he presence of d as occurs in he ideal case discussed in Chaper. This confirms he findings GENERAL RELATION TO CIRCUIT PARAMETERS Figure 5.1 (a) plos he simulaed percenage THD as a funcion of d (which is represened as a percenage of he swiching period) for M=.8. The increase in disorion for increasing d is a well known resul. Figure 5.1 (b) illusraes he relaion beween he THD and M. Since he dead ime only has an effec wihin inducor curren Scenario, he lower limi of M equals.5 o avoid Scenario being enering. From his plo i is apparen ha disorion resuling from he dead ime canno be lowered by lowering M, also noed in [5]

112 CHAPTER 5 THE EFFECT OF DEAD TIME THD [%] Dead Time [%] (a) THD [%] Modulaion Index (b) Figure 5.1: Simulaed THD as a funcion of (a) d and (b) M. The percenage THD as a funcion of L fil for d =15ns is illusraed in Figure 5.11 (a). Consider he iniial decrease in disorion beween L fil =H and L fil =7H. In his region he various harmonics wihin he baseband add in such a way ha he THD decreases for increasing values of L fil. Above L fil =7H hey add in such a way o increase he overall disorion THD [%] THD [%] Inducance [uh] (a) Inducance [uh] (b) Figure 5.11: Simulaed THD as a funcion of L fil for (a) d =15ns and (b) d =5ns. A similar plo o Figure 5.11 (a) is shown in Figure 5.11 (b) for d =5ns. The same relaion holds. Neglecing all oher effecs, his suggess ha for a given dead ime, an opimum value of L fil can be deermined

113 CHAPTER 5 THE EFFECT OF DEAD TIME 5.6 SUMMARY The analysis in his chaper considered he isolaed effec of dead ime on harmonic disorion. An overview was given of his well-known effec, afer which an analyical model based on he double Fourier series mehod of analysis was inroduced for TENPWM. A simulaion model was nex consruced. The proposed analyical and simulaion models allow for L fil as well as f c o be parsed as parameers. The analyical and simulaion resuls achieved showed very good correlaion. I was shown ha he dead ime produces even, as well as odd order harmonics wihin he audible band. Moreover, i was esablished ha he former harmonics are a resul of he modulaion producs, which in urn dependend on he swiching frequency. I was also shown ha, for a specific dead ime, he THD can be opimized by varying L fil

114 d( vf ) 6 v sw THE EFFECT OF THE MOSFET TURN-ON AND TURN-OFF DELAYS 6.1 INTRODUCTION I will be shown in his chaper ha he MOSFET urn-on and urn-off delays lead o disorion wihin he baseband. A simulaion model is presened, which is followed by a deailed discussion of he resuls achieved for TENPWM. This chaper conains a more deailed analysis of ha presened in [46] by he auhor. 6. ANALYSIS OF THE TURN-ON AND TURN-OFF DELAYS Figure 6.1 illusraes various swiching waveforms for a posiive inducor curren. Since only he urn-on and urn-off delays are considered, he remainder of he swiching waveform is idealized in an aemp o isolae he non-ideal effec under invesigaion. Two separae values of dead ime, denoed d(low_side) and d(high_side) are assumed where he respecive values represen he dead ime inroduced for he low side and high side MOSFET. The waveform is idealized if d(low_side) = d(vf) and d(high_side) =. This is necessary o avoid simulaneous conducion of he swiching devices. The swiching ransiion of v GS(low_side) is also considered o be ideal. If all ampliude errors are negleced, for a posiive inducor curren, he swiched oupu volage of Figure 6.1 (d) wihin he urn-on and urn-off delay is independen of v GS(low_side). This waveform has however been included in Figure 6.1 as (b) o clarify he above menioned iming requiremens.laer in his secion i will be shown ha he inverse siuaion holds for a negaive inducor curren, i.e. he swiched oupu volage wihin he urn-on and urn-off delay is independen of v GS(high_side). Since he curren polariy deermines wheher v GS(low_side) or v GS(high_side) swiches he oupu volage, he urn-on and urn-off delays are labelled in accordance wih he swiched oupu volage raher han relaing hem o v GS(low_side) or v GS(high_side). The designaions used are d(vf) and d(vr) (volage fall and volage rise)

115 CHAPTER 6 THE EFFECT OF THE MOSFET TURN-ON AND TURN-OFF DELAYS v GS ( high_side ) (a) V GS ( h ) V GS d( vf ) v d ( vr ) GS ( low_side) (b) V GS d( low_side) d ( high_side ) i L ( approx ) (c) i Lideal ( ) v sw 1 (d) V d Figure 6.1: Swiching waveforms for a posiive inducor curren. (a) High side gae-o-source volage. (b) Low side gae-o-source volage. (c) Inducor curren. (d) Swiched oupu volage. Consider v GS(high_side) shown in Figure 6.1 (a). An insananeous ransiion in gae drive from v GS =V GS o v GS = and vice versa is assumed, wih V GS well above V GS(h) according o he analysis in [4]. This ideal characerisic is represened by he dashed line in Figure 6.1 (a). The solid line represens he acual par of v GS which leads o d(vf) and d(vr). Firsly, consider d(vf). Once v GS ransiions from V GS o zero, he acual gae-o-source volage decreases exponenially as a resul of he discharge of C iss hrough R G. This is given by he relaion [8]: GS GS RGCiss v = V e (6.1) The gae-o-source volage decreases up o he poin where he drain curren equals he inducor curren. Similar o he analysis in Chaper 5, i L is approximaed by a consan value wihin he inerval d(vf). This value is calculaed on he ideal swiching ransiion of he oupu

116 CHAPTER 6 THE EFFECT OF THE MOSFET TURN-ON AND TURN-OFF DELAYS volage, which occurs on i L(upper_env) and is represened by i L(approx) in Figure 6.1 (c). The relaion beween v GS and I L can be found from he ransfer characerisic of he MOSFET as: v GS I L = VGS ( h) + (6.) g fs From Eqs. (6.1) and (6.) d(vf) can be calculaed as: V GS d( vf ) = RGCissln for IL> IL VGS ( h) + g fs (6.3) Using a similar approach, an expression for d(vr) can be found for a posiive inducor curren. Again, an immediae ransiion in he applied gae drive volage from zero o V GS resuls in he charge of C iss hrough R G by he exponenial relaion: vgs = V GS e RGCiss 1 (6.4) During he iniial rising period (defined as 1 ) he MOSFET is in he cu-off region since v GS <V GS(h). This inerval ends once v GS =V GS(h) which yields: V GS 1 = RGCissln VGS V GS ( h) (6.5) A he laer insan in ime he drain curren will sar o flow in he MOSFET and v GS will coninue o increase in relaion wih Eq. (6.4). The drain-o-source volage will increase in accordance wih he ransfer characerisic of he MOSFET up o he poin where I D =I L. Subsiuing for Eq. (6.) he equaliy in (6.5) can be wrien as: V GS d( vr) = RGCissln for IL> IL VGS VGS ( h) g fs (6.6) Using minor aleraions o hose of he previous analysis, similar expressions defining d(vf) and d(vr) can now be consruced for a negaive inducor curren. Figure 6. illusraes he swiching waveforms

117 CHAPTER 6 THE EFFECT OF THE MOSFET TURN-ON AND TURN-OFF DELAYS v GS ( low_side ) (a) V GS V GS ( h ) d( vf ) d ( vr ) v GS ( high_side ) (b) V GS d ( low_side ) d ( high_side ) (c) i L ( approx ) i Lideal ( ) v sw (d) V d Figure 6.: Swiching waveforms for a negaive inducor curren. (a) Low side gae-o-source volage. (b) High side gae-o-source volage. (c) Inducor curren. (d) Swiched oupu volage. In order o idealize he swiching waveforms he dead ime requiremens change o d(low_side) = and d(high_side) = d(vr). The swiching waveform of v GS(high_side) is assumed o be ideal. As already menioned, for a negaive inducor curren he oupu volage wihin he urn-on and urn-off delays is independen of v GS(high_side). The relaion defining d(vf) for a negaive inducor curren is: V GS d( vf ) = RGCissln for IL< IL VGS VGS ( h) g fs (6.7)

118 CHAPTER 6 THE EFFECT OF THE MOSFET TURN-ON AND TURN-OFF DELAYS While d(vr) can be defined as: V GS d( vr) = RGCissln for IL< IL VGS ( h) + g fs (6.8) 6.3 ERROR DESCRIPTION The soluions obained in Secion 6. are analysed in his secion. The parameers used for he remainder of his chaper correspond o V d =1V M=.85, R G =1, V GS =1V, V GS(h) =4.9V, g fs =11, C iss =81pF, L=1.7μH and c / =384 unless saed oherwise. The plos illusraed in Figure 6.3 (a) and (b) respecively represen d(vr) for Scenario (R L =1) and Scenario (R L =8.). Volage Rise Delay [s] 5.89 x Time [ms] (a) Volage Rise Delay [s] 6.1 x Time [ms] (b) Figure 6.3: d(vr) for (a) Scenario and (b) Scenario. From Figure 6.3 (a) i is eviden ha Scenario resuls in a coninuous curve. This can be ascribed o he fac ha i L(lower_env) remains negaive over he complee swiching cycle, which in urn resuls in d(vr) being only defined by Eq. (6.8). For Scenario he curren changes polariy, which means ha d(vr) is eiher defined by Eq. (6.6) or Eq. (6.8), leading o he abrup change in d(vr) illusraed in Figure 6.3 (b)

119 CHAPTER 6 THE EFFECT OF THE MOSFET TURN-ON AND TURN-OFF DELAYS 5.78 x x Volage Fall Delay [s] Time [ms] Volage Fall Delay [s] Time [ms] (a) (b) Figure 6.4: d(vf) for for (a) Scenario and (b) Scenario. Figure 6.4 shows a plo of d(vf) for he various scenarios of he inducor curren. Similar o Figure 6.3 he curve in Figure 6.4 (a) is only defined by Eq. (6.3) since i L(upper_env) remains posiive over he complee swiching cycle. For Scenario shown in Figure 6.4 (b) he soluion is eiher described by Eq. (6.3) or Eq. (6.7). The analyical soluions of Eqs. (6.3), (6.6), (6.7) and (6.8) can nex be used o consruc he simulaion model. 6.4 SIMULATION STRATEGY Wih he soluions for d(vf) and d(vr) defined in Secion 6. i is now possible o incorporae he delays ino he simulaion sraegy inroduced in Secion.5. The pulse in Figure 6.5 represens he p h pulse of a TENPWM waveform wih he addiion of a non-zero urn-on and urn-off delay. The remaining parameers correspond o hose used in Figure.14. From Figure 6.5 i can be seen ha he pulse widh changes o: W = W + (6.9) ptenpwm ( vr _ vf ) ptenpwm d( vr) d( vf ) Since he ime shif of Eq. (.94) is also dependen on he pulse widh, i needs o be adaped. Subsiuing for Eq. (6.9) he ime shif of Eq. (.94) can be wrien as: Wp TENPWM d( vr) + d( vf ) = ( ptenpwm 1) Tc + (6.1)

120 CHAPTER 6 THE EFFECT OF THE MOSFET TURN-ON AND TURN-OFF DELAYS ntenpwm ctenpwm () mtenpwm () W p TENPWM V d V d d ( vr ) d ( vf ) ( p TENPWM -1T ) c p TENPWM T c p TENPWM T c Figure 6.5: Generaion of TENPWM wih urn-on and urn-off delays. 6.5 SIMULATION RESULTS In his secion he simulaion model of Secion 6.4 will be uilized o deermine he effec of he non-zero d(vr) and d(vf) on disorion. The resuls achieved will be discussed for boh scenarios of he inducor curren. Furhermore, noe ha he parameers used correspond o hose defined in Secion 6.3 unless saed oherwise BASEBAND HARMONICS Firsly, he effec wihin he baseband is deermined in his secion. Figure 6.6 (a) shows a specral plo of he baseband harmonics for inducor curren Scenario. From Figure 6.6 i can be seen ha Scenario produces a much higher level of disorion han ha of Scenario. The reason for he disorion in Figure 6.6 (b) being so much higher han ha of (a) can be explained when referring o Figure 6.3 and Figure 6.4. The abrup change in d(vr) and d(vf) associaed wih Scenario (Figure 6.3 (b) and Figure 6.4 (b)) resuls in he more prominen baseband harmonics shown in Figure 6.6 (b) han hose in (a)

121 CHAPTER 6 THE EFFECT OF THE MOSFET TURN-ON AND TURN-OFF DELAYS 4 4 THD=.114% Fundamenal Componen Fundamenal Componen - - Magniude [dbv] Baseband Harmonics Magniude [db] Baseband Harmonics THD=.59% Harmonic Number [-] (a) Harmonic Number [-] (b) Figure 6.6: Simulaed TENPWM baseband harmonics for (a) Scenario and (b) Scenario. Table 6.1 compares he THD of Scenario and for various values of R G. I can be seen ha he disorion increases for increasing values of R G. This is a resul from he fac ha d(vr) and d(vf) increase for increasing values of R G, which is eviden upon simple inspecion of Eqs. (6.3), (6.6), (6.7) and (6.8). TABLE 6.1 SIMULATED THD FOR TENPWM FOR VARIOUS R G. R G [] Simulaed THD, Scenario 1 [%] Simulaed THD, Scenario [%] Table 6. shows a comparison beween Scenario and for various values of V GS. Since Eqs. (6.3) and (6.8) are independen of V GS for Scenario he THD remains consan. For Scenario he disorion increases for increasing values of V GS. The following secion focuses on he conribuion of he sideband swiching harmonics wihin he baseband

122 CHAPTER 6 THE EFFECT OF THE MOSFET TURN-ON AND TURN-OFF DELAYS TABLE 6. SIMULATED THD FOR TENPWM FOR VARIOUS V GS. V GS [V] Simulaed THD, Scenario 1 [%] Simulaed THD, Scenario [%] SIDEBAND HARMONICS In his secion he analysis focuses on he sideband harmonics. Figure 6.7 compares he specra for Scenario and. From he laer illusraion i can be seen ha he sideband harmonics decay a a rapid rae for Scenario. 4 Carrier 1 wih Sidebands Carrier wih Sidebands 4 Carrier 1 wih Sidebands Carrier wih Sidebands Fundamenal Componen Fundamenal Componen - - Magniude [dbv] Baseband Harmonics Magniude [dbv] Baseband Harmonics THD=.59% Harmonic Number [-] (a) -14 THD=.114% Harmonic Number [-] (b) Figure 6.7: Simulaed TENPWM specra for (a) Scenario and (b) Scenario. For Scenario he sidebands do no roll off ha quickly and hus conribue o disorion wihin he baseband, yielding an effec similar o ha of dead ime

123 CHAPTER 6 THE EFFECT OF THE MOSFET TURN-ON AND TURN-OFF DELAYS 6.6 SUMMARY The isolaed effec of he MOSFET urn-on and urn-off delay on harmonic disorion was considered in his chaper. Well-known analyical expressions describing hese delays [4], [8] were implemened ino a simulaion model for TENPWM. The simulaion resuls obained for TENPWM showed ha inducor curren Scenario produces much higher levels of disorion han Scenario due o he abrup change in he delays upon a change in curren polariy. Furhermore, i was shown ha he THD increases for increasing values of he delays. Lasly, he analysis showed ha he sideband swiching harmonics only increase disorion under Scenario

124 vf 7 v sw THE EFFECT OF NON-ZERO NON-LINEAR SWITCHING TRANSITIONS 7.1 INTRODUCTION This chaper conains he analysis presened in [47] by he auhor, in which he isolaed effec of he non-zero swiching ransiions is invesigaed. The analysis is based on wellknown analyical expressions describing he swiching behaviour [4], [8]. A mehod for he modelling of he non-linear swiching curve based on a specific MOSFET case sudy is hen inroduced and compared o ha of a linear ransiion o deermine is significance. A simulaion model is also inroduced, which is followed by a deailed discussion of he resuls achieved. The analysis shows ha he disorion is highly dependen on curren polariy raher han modulaion and ha i increases for longer swiching imes. Furhermore, he proposed non-linear swiching model resuls in significanly higher levels of disorion han hose of he linear ransiions. 7. ANALYSIS OF NON-ZERO RISE AND FALL SWITCHING TRANSITIONS The swiching waveforms ha influence inervals vf and vr are shown in Figure 7.1 for a posiive inducor curren. The analysis in his chaper is only concerned wih he effec of he non-zero rise and fall ransiions, and hus only he porions of he swiching waveforms falling wihin hese inervals are considered o be non-ideal. This requires condiions in which he remainder of he waveform is idealized

125 CHAPTER 7 THE EFFECT OF NON-ZERO NON-LINEAR SWITCHING TRANSITIONS Firsly, during he iniial ransiion of v GS(high_side) from on o off he volage is assumed o swich in zero ime from V GS o v GS(IL ). This leads o a value of d(on) =. Using similar condiions v GS(high_side) resuls in d(off) = during a change of v GS(high_side) from off o on. In order o avoid simulaneous conducion of he swiching devices, he dead ime requiremen has o be adaped in such a way ha d(low_side) = vf and d(high_side) = as shown in Figure 7.1. v GS ( low_side ) V GS (a) V GS ( h ) v GS ( IL ) d( on) d ( off ) v GS ( high_side ) (b) V GS d ( low_side ) d ( high_side ) i Lapprox ( ) (c) i L ( ideal ) vf vr (d) v sw Vd Figure 7.1: Swiching waveforms for a posiive inducor curren. (a) High side gae-o-source volage. (b) Low side gae-o-source volage. (c) Inducor curren. (d) Swiched oupu volage. As already menioned in Chaper 6, for a posiive inducor curren wih all ampliude errors negleced, he oupu volage is independen of v GS(low_side). However, i has once again been included in Figure 7.1 (b) o clarify he iming requiremens of he dead ime. Consider vf. Upon compleion of inerval d(on) he swiched oupu volage sars o fall. As in he analysis of Chaper 6, i L is again approximaed by i L(approx) wihin region vf. This value is

126 CHAPTER 7 THE EFFECT OF NON-ZERO NON-LINEAR SWITCHING TRANSITIONS calculaed on he ideal oupu swiching ransiion from off o on which occurs on i L(upper_env). This is shown in Figure 7.1 (c). According o he MOSFET ransfer characerisic v GS will also remain consan a a value: v I L GS ( IL ) = VGS ( h) + (7.1) g fs Since v GS(IL ) is consan, he full gae curren will charge C GD. The expression describing I G can hus be wrien as: I G G IL ( VGS ( h) + g ) fs vgs ( IL ) = = (7.) R R The volage v DG across C GD is given by: G dv d DG I C G = (7.3) GD Since v GS is consan, he rae of change of v DS equals he rae of change of he v DG. Saed mahemaically: dv d DS dv d DG = (7.4) Subsiuing for Eq. (7.3) he equaliy in Eq. (7.4) leads o he expression: dv d DS IL ( VGS ( h) + g ) fs = for IL > (7.5) R C G GD An expression describing v DS during vr can now be found in a similar manner. Since he drain curren remains consan a a value calculaed on i L(lower_env) he gae volage is sill given by Eq. (7.1), which in urn resuls in he complee I G flowing hrough C GD. The gae curren is given by: I G IL VGS V V ( ) GS V GS I GS ( h) g L fs = = (7.6) R R G G

127 CHAPTER 7 THE EFFECT OF NON-ZERO NON-LINEAR SWITCHING TRANSITIONS Noe ha v GS(IL ) is defined in Eq. (7.1). The volage v DG across C GD is given by Eq. (7.3) and since he rae of change of v DS equals he rae of change of v DG he expression for v DG during inerval vr can be consruced from Eqs. (7.3) and (7.6) as: dv d DS IL VGS VGS ( h) g fs = for IL > (7.7) R C G GD v GS ( high_side ) V GS (a) V GS ( h ) v GS ( IL ) d( on) v d ( off ) GS ( low_side ) (b) V GS d( low_side) d ( high_side ) (c) i L ( approx ) vf i L ( ideal ) vr (d) v sw V d Figure 7.: Swiching waveforms for a negaive inducor curren. (a) Low side gae-o-source volage. (b) High side gae-o-source volage. (c) Inducor curren. (d) Swiched oupu volage. Figure 7. illusraes he swiching waveforms affecing inervals vf and vr. For a negaive inducor curren he dead ime changes o d(low_side) = and d(high_side) = vr in order o isolae he swiching ransiions from he remaining non-ideal effecs. Noe ha v GS(high_side) is assumed o be ideal and ha i is once again included o sae he dead ime requiremens

128 CHAPTER 7 THE EFFECT OF NON-ZERO NON-LINEAR SWITCHING TRANSITIONS Following a similar approach o ha of he previous analysis for a posiive inducor curren, an expression describing v DS during vf can be consruced as: dv d DS IL ( VGS VGS ( h) g ) fs = for IL < (7.8) R C G GD In a similar manner, an expression for v DS for inerval vr can be obained as: dv d DS IL VGS ( h) + g fs = for IL < (7.9) R C G GD The expressions describing he swiched oupu volage during inervals vr and vf obained will be solved in he following secion. 7.3 SOLUTIONS TO THE EXPRESSIONS OF THE SWITCHING CURVES The soluions o he expressions of he swiching curves derived in he previous secion will now be deermined. Iniially, a soluion for a consan C GD is obained, afer which a more realisic soluion for a dynamic C GD is considered. As will be shown in Secion 7.3.1, a consan C GD resuls in a linear swiching curve, whereas he proposed dynamic approximaion of C GD leads o a non-linear swiching characerisic. The reasoning behind he inclusion of boh models is o show how he proposed model differs from ha inroduced in previous work [9], as well as o deermine wheher he non-linear swiching characerisic has a negligible effec on disorion compared o he linear swiching curve. Noe ha he soluions obained in his secion are applicable o boh half-bridge and full-bridge opologies CONSTANT GATE-TO-DRAIN CAPACITANCE Since C GD is dependen on v DS i is necessary o esablish an approximaion o achieve he objecive of a linear swiching characerisic in his secion. By defining C GD1 a v DS = and C GD a v DS =V d he bes consan approximaion is achieved by: C C C GD1 GD GD = (7.1)

129 CHAPTER 7 THE EFFECT OF NON-ZERO NON-LINEAR SWITCHING TRANSITIONS Consider a posiive inducor curren. The soluion o v DS during vf can be found hrough inegraion of Eq. (7.5) as: IL VGS ( h) + g fs vds = Vd for IL > RC G GD (7.11) By noing ha = vf if v DS is zero, he soluion o vf can be found as: vf VRC d G GD = for IL > IL V + GS ( h) g fs (7.1) The soluion o v DS during vr can be found hrough inegraion of Eq. (7.7), which yields: IL VGS VGS ( h) g fs vds = for IL > RC G GD (7.13) Since v DS =V d when = vr he solulion o vr can be expressed as: vr = VRC for IL V > d G GD IL GS VGS ( h) g fs (7.14) Using a similar approach he soluions o v DS during vr and vf can now also be deermined for a negaive inducor curren. The soluion o v DS during vf can be found hrough inegraion of Eq. (7.8) as: IL VGS VGS ( h) g fs vds = Vd for IL < RC G GD (7.15) Since v DS equals zero when = vf he soluion o vf can be wrien as: vf = VRC for IL V < d G GD IL GS VGS ( h) g fs (7.16) The soluion o v DS during vr can be found hrough inegraion of Eq. (7.9), which leads o: IL VGS ( h) + g fs vds = for IL < RC G GD (7.17)

130 CHAPTER 7 THE EFFECT OF NON-ZERO NON-LINEAR SWITCHING TRANSITIONS By nex noing ha v DS =V d when = vr he solulion can be expressed as: vr VRC d G GD = for I IL L < V + GS ( h) g fs (7.18) The following secion conains a more complex model of C GD. I is shown ha he prosposed model leads o a non-linear soluion o v DS during he swiching inervals DYNAMIC GATE-TO-DRAIN CAPACITANCE Since he C GD vs. v DS characerisic is MOSFET specific, a case sudy approach will be used in his secion o obain he soluion o he swiching curves for a varying C GD. Consider Figure 7.3. The bold curve in (a) represens he acual C GD vs. v DS characerisic for par IRFI419H-117P obained from he device s daashee [48]. x Approximaion Acual Curve Gae-o-Drain Capaciance [F] 1-11 Gae-o-Drain Capaciance [F] Drain-o-Source Volage [V] (a) Drain-o-Source Volage [V] (b) Figure 7.3: (a) Approximae and acual curve of C GD as a funcion of v DS on a log scale. (b) Approximae curve of C GD as a funcion of v DS on a linear scale for par IRFI419H-117P. From Figure 7.3 (a) i can be seen ha a good approximaion o he acual curve is achieved via a power law of he form: kc GD GD CGD DS C = m v (7.19) The approximaion is shown in (a) wih parameers corresponding o m = 15 pf and k =.75. Figure 7.3 (b) represens he plo of (a) on a linear scale. The relaion in Eq. C GD C GD

131 CHAPTER 7 THE EFFECT OF NON-ZERO NON-LINEAR SWITCHING TRANSITIONS (7.19) can nex be used o obain he soluions o he expressions derived in Secion 7.. Firsly, consider a posiive inducor curren. The soluion o v DS during vf can be found hrough inegraion of Eq. (7.5) as: ( kc + 1)( VGS( h) g fs + IL) 1 k CGD + 1 GD vds = Vd for IL > Rgm G fs C GD (7.) Since v DS = when = vf he soluion o vf can be wrien as: vf ( kc GD + 1) Rgm G fs C V GD d = for IL > k + 1 V g + I ( C )( GS( h) fs L) GD (7.1) Eq. (7.) can nex be wrien in erms of vf by rewriing Eq. (7.1) and subsiuing he resul ino Eq. (7.). Since k C GD =.75 Eq. (7.) simplifies o: V v = V I > (7.) d 4 DS d for 4 L vf From Eq. (7.) i is eviden ha he swiching curve can be approximaed by a polynomial of order four. Using resul of his, he remaining soluions can easily be obained. For a posiive inducor curren, v DS during vr is given by: Vd v ( ) 4 DS = Vd for 4 vr IL > (7.3) Where vr can be wrien as: vr vr 1 4 G fs C Vd 4Rgm = for I > GD ( VGS g fs VGS ( h) g fs IL ) L (7.4) Consider a negaive inducor curren. The soluion o v DS wihin inerval vf can be wrien as: Vd v ( ) 4 DS = for 4 vf IL < (7.5) vf

132 CHAPTER 7 THE EFFECT OF NON-ZERO NON-LINEAR SWITCHING TRANSITIONS Wih vf defined by: vf 4Rgm G fs C Vd = for I < GD ( VGS g fs VGS ( h) g fs IL ) 1 4 L (7.6) Finally, he soluion o v DS during inerval vr is given by: V vds = I < (7.7) d 4 for 4 L vr While vr in Eq. (7.7) can be wrien as: vr 1 4 G fs C Vd 4Rgm = for I < GD ( VGS ( h) g fs + IL ) L (7.8) 7.4 ERROR DESCRIPTION In his secion he soluions of he swiching curves obained in Secion 7.3 are analysed. Noe ha he parameers used for he remainder of his chaper correspond o V d =1V M=.85, R G =1, V GS =1V, V GS(h) =4.9V, g fs =11, L=1.7μH and c / =384 unless saed oherwise. Figure 7.4 (a) and (b) respecively show a plo of vr for Scenario (R L =1) and Scenario (R L =8.) using he soluions obained in Secion x x Rise Time [s] Rise Time [s] Time [ms] (a) Time [ms] (b) Figure 7.4: Rise ime for (a) Scenario and (b) Scenario

133 CHAPTER 7 THE EFFECT OF NON-ZERO NON-LINEAR SWITCHING TRANSITIONS From Figure 7.4 (a) i can be seen ha Scenario resuls in a coninuous curve. This can be ascribed o he fac ha i L(lower_env) remains negaive over he complee swiching cycle which in urn resuls in vr being only defined by Eq. (7.8). For Scenario he curren changes polariy, which means ha vr is eiher defined by Eq. (7.4) or Eq. (7.8), leading o he abrup change in vr shown in Figure 7.4 (b)..178 x 1-9 x Fall Time [s] Fall Time [s] Time [ms] Time [ms] (a) (b) Figure 7.5: Fall ime for (a) Scenario and (b) Scenario. Figure 7.5 shows a plo of vf for he various scenarios of he inducor curren. Similar o Figure 7.4 he curve in (a) is only defined by Eq. (7.1) since i L(upper_env) remains posiive over he complee swiching cycle. For Scenario shown in Figure 7.5 (b) he soluion is eiher described by Eq. (7.1) or Eq. (7.6) x 1-1 x Difference in Fall Time [s] Difference in Fall Time [s] Time [ms] (a) Time [ms] (b) Figure 7.6: Difference in fall ime for (a) Scenario and (b) Scenario

134 CHAPTER 7 THE EFFECT OF NON-ZERO NON-LINEAR SWITCHING TRANSITIONS Figure 7.6 illusraes a plo of he difference in vf for he soluions obained in Secion 7.3. and Secion This serves as a reference for discussing of he resuls, which follows in Secion SIMULATION STRATEGY The simulaion sraegy of Secion.5 can now be adaped o accommodae for vr and vf. Since a comparison beween he linear and non-linear swiching characerisic will be included, he simulaion model needs o be se up for boh ses of soluions obained in Secion 7.3. Noe ha models for boh TENPWM and DENPWM are presened. TENPWM The p h pulse of a TENPWM waveform in he presence of a non-zero urn-on and urn-off swiching ransiion is shown in Figure 7.7. The solid curves represen he non-linear ransiions for a posiive inducor curren during vr and vf, whereas he dashed curves represen swiching for a negaive inducor curren. The linear ransiions are shown as grey lines wihin he laer inervals. ntenpwm +1 ctenpwm () mtenpwm () W p TENPWM vr vf V sw ( ptenpwm ) -1 T c p TENPWM T c p TENPWM T c Figure 7.7: Generaion of TENPWM wih non-zero rise and fall imes

135 CHAPTER 7 THE EFFECT OF NON-ZERO NON-LINEAR SWITCHING TRANSITIONS The Fourier Transform of v DS () during vr including he necessary ime shif is given by: vr jπ f jπ f jπ f I vds () e = vds () e de (7.9) Where: ( ) = p T (7.3) TENPWM 1 c Depending on he curren polariy, v DS () in Eq. (7.9) is defined by Eqs. (7.14) or (7.17) for linear swiching curves and Eq. (7.3) or (7.7) for he non-linear approximaion. The Fourier Transform of v DS () for inerval vf is given by: vf jπ f jπ f jπ f I vds () e = vds () e de (7.31) Wih defined as: ( ) = p T + W (7.3) TENPWM 1 c p TENPWM In a similar manner as for vr v DS () is defined by Eq. (7.11) or (7.15) for a linear swiching curve, whereas Eq. (7.) or (7.5) defines i for a non-linear ransiion. From Figure 7.7 i is eviden ha he pulse widh also needs o be adaped o: W = W (7.33) ptenpwm ( vr _ vf ) ptenpwm vr The ime shif of Eq. (.94) is a funcion of he pulse widh and hus needs o be alered as well. By subsiuing for Eq. (7.33) he ime shif of Eq. (.94) changes o: Wp TENPWM vr = ( ptenpwm 1) Tc + (7.34) 7.6 SIMULATION RESULTS The simulaion model inroduced in he previous secion can nex be implemened. In his secion a deailed analysis of he resuls achieved will be considered for TENPWM. A discussion of he difference beween he linear and non-linear swiching ransiions is also

136 CHAPTER 7 THE EFFECT OF NON-ZERO NON-LINEAR SWITCHING TRANSITIONS included. Once again, hroughou his secion he resuls will be discussed for boh scenarios of he inducor curren. The parameers used correspond o hose defined in Secion 7.4, unless saed oherwise BASEBAND HARMONICS The effec of vr and vf wihin he baseband is considered in his secion. Figure 7.8 is a comparaive specral plo of he baseband harmonics of boh linear and non-linear swiching ransiions, which were ploed using he respecive soluions derived in Secions and From Figure 7.8 i can be seen ha Scenario produces much higher levels of disorion han Scenario. The reason why he disorion in Figure 7.8 (b) is so much more prominen can be explained when referring o Figure 7.4 and Figure 7.5. The abrup change in vr and vf associaed wih Scenario (Figure 7.4 (b) and Figure 7.5 (b)) resuls in more prominen baseband harmonics shown in Figure 7.8 (b) han in (a). 4 4 Fundamenal Componen Fundamenal Componen Magniude [dbv] Baseband Harmonics Magniude [db] Baseband Harmonics Linear Transiions, THD=.5% Non-Linear Transiions, THD=.13% Harmonic Number [-] (a) -14 Linear Transiions, THD=.153% Non-Linear Transiions, THD=.371% Harmonic Number [-] (b) Figure 7.8: Simulaed TENPWM baseband harmonics for (a) Scenario and (b) Scenario. A comparison beween he linear and non-linear swiching ransiions for various values of R G is represened in Table 7.1. I is eviden ha he disorion increases for increasing values of R G. Through simple inspecion of he soluions in Secion 7.3 i can be observed ha vr and vf increase for increasing values of R G, suggesing ha longer swiching ransiions resul in higher levels of disorion

137 CHAPTER 7 THE EFFECT OF NON-ZERO NON-LINEAR SWITCHING TRANSITIONS TABLE 7.1 SIMULATED THD FOR LINEAR AND NON-LINEAR SWITCHING TRANSITIONS FOR TENPWM FOR VARIOUS R G. R G [] Simulaed THD, Linear [%] Scenario 1 Scenario Simulaed THD, Non-Linear [%] Simulaed THD, Linear [%] Simulaed THD, Non-Linear [%] Table 7. shows a similar comparison as Table 7.1, only for various values of V GS. The soluions in Secion 7.3 describing vr (Eqs. (7.18) and (7.8)) and vf (Eqs. (7.1) and (7.1)) for Scenario are independen of V GS, which resuls in a THD ha remains consan. TABLE 7. SIMULATED THD FOR LINEAR AND NON-LINEAR SWITCHING TRANSITIONS FOR TENPWM FOR VARIOUS V GS. V GS [] Simulaed THD, Linear [%] Scenario 1 Scenario Simulaed THD, Non-Linear [%] Simulaed THD, Linear [%] Simulaed THD, Non-Linear [%] For Scenario he disorion decreases for increasing values of V GS for he linear ransiions, while he inverse holds for he non-linear swiching ransiions. In boh Table 7.1 and Table 7. i can be seen ha he proposed non-linear model of Secion 7.3. resuls in higher levels of disorion han he linear model of Secion This can parially be explained when referring o Figure 7.6 which shows he difference in vf beween he linear and non-linear cases. The soluions of he proposed non-linear model yields higher swiching

138 CHAPTER 7 THE EFFECT OF NON-ZERO NON-LINEAR SWITCHING TRANSITIONS imes han ha of he linear model, which in urn resuls in higher levels of disorion, as suggesed in Table 7.1. The following secion focuses on he conribuion of he sideband swiching harmonics wihin he baseband SIDEBAND HARMONICS The following analysis shows under which circumsances he sideband harmonics are mos likely o deeriorae disorion by conribuing o he baseband harmonics. Figure 7.9 compares he specra of he linear and non-linear swiching ransiions for he various scenarios of he inducor curren. From he laer illusraion i can be seen ha he sideband harmonics decay a a rapid rae for Scenario. 4 Carrier 1 wih Sidebands Carrier wih Sidebands 4 Carrier 1 wih Sidebands Carrier wih Sidebands Fundamenal Componen Fundamenal Componen - - Magniude [dbv] Baseband Harmonics Magniude [dbv] Baseband Harmonics Linear Transiions,.5% THD Non-Linear Transiions,.1% THD Harmonic Number [-] (a) -14 Linear Transiions,.15% THD Non-Linear Transiions,.371% THD Harmonic Number [-] (b) Figure 7.9: Simulaed TENPWM specra for (a) Scenario and (b) Scenario. However, for Scenario he sidebands of he firs carrier do no roll off a such a rapid rae and hus conribue o disorion wihin he baseband. This slow decay associaed wih Scenario can be ascribed o abrup change in vr and vf upon ransiion, as described in he previous secion. 7.7 SUMMARY This chaper focussed on he isolaed effec of non-zero linear, as well as non-zero nonlinear swiching ransiions on harmonic disorion. The analysis included a linear model

139 CHAPTER 7 THE EFFECT OF NON-ZERO NON-LINEAR SWITCHING TRANSITIONS based on well-known analyical expressions [4], [8] as well as a newly proposed non-linear model based on an approximaion of he non-linear v DS vs. C GD curve of a specific power MOSFET. A simulaion model was consruced for TENPWM. From he resuls obained for TENPWM i was shown ha inducor curren Scenario yields much higher levels of disorion han Scenario due o he abrup change in ransiion imes upon a change in curren polariy. I was also shown ha he THD increases for increasing swiching imes. For Scenario he sideband harmonics conribue o disorion wihin he baseband. The proposed non-linear model resuled in much higher levels of disorion han he linear model for boh cases of he inducor curren

140 8 v sw THE EFFECT OF PARASITICS AND REVERSE RECOVERY 8.1 INTRODUCTION This chaper focusses on he effecs of he urn-on and urn-off volage ransiens on harmonic disorion. Due o he vas amoun of variables involved, as well as he dependence on pracical implemenaion, i is very difficul o characerise he volage ransien s exac parameers [31], [3]. Thus, he proposed analysis makes use of an exising analyical soluion and focus is shifed owards he parameers ha give rise o disorion wihin he baseband raher han predicing he exac characerisic of he overvolage for a given pracical seup. The condiions under which he reverse recovery increases disorion are hen deermined, afer which a mehod for modelling is effec is esablished. A simulaion models is inroduced for TENPWM afer which a deailed discussion of he resuls achieved are considered. 8. ANALYSIS OF THE PARASITICS AND REVERSE RECOVERY As menioned in he previous secion, he analysis will focus on he mechanism giving rise o disorion wihin an exising pracical seup raher han analyically esimaing he ampliude and frequency of he ringing effec for ha specific circui. This approach calls for some exising analyical soluion describing he overvolage wihin curren lieraure. The analysis in [31] conains a deailed sudy of he MOSFET urn-off overvolage ransien in he presence of various parasiics. A closed form analyical soluion for his volage is obained from equivalen circui models of he power MOSFET and he PCB. A brief summary of he analysis and soluion obained in [31] now follows

141 CHAPTER 8 THE EFFECT OF PARASITICS AND REVERSE RECOVERY Lsray Rloop LD + C oss sink v DS I L L S R D Figure 8.1: Equivalen circui model of he urn-off process [31]. In order o obain an analyical expression he analysis uilizes a relaively simple MOSFET model in which essenial design parameers are included. This model s simpliciy is based on he assumpion ha C oss is fully charged o he supply volage (and hus consan during he ringing phase) before he curren sars o fall. This MOSFET model is shown in he righ hand side recangle of Figure 8.1. Boh he PCB and he componen leads conribue o he oal sray inducance. The former is modelled using a compuer aided design (CAD) ool (InCa) in which a mehod (he Parial Elemen Equivalen Circui) is implemened ha permis modelling of he inerconnecions. The various self-and-muual parial inducances and resisances are respecively represened by a single inducance and a single resisance, all conneced in series. This circui is shown in he lef hand recangle of Figure 8.1. The sray inducance L sray is a combinaion of he oal self-and-muual parial inducances ogeher wih he lead inducances of he upper swiching device while R loop is he oal parial resisance of he PCB racks. From Figure 8.1 he expression describing he MOSFET s exernal overvolage akes he general form [31]: Where: LsrayIL Δ vds = e 1 ζ sink ( ωovζ) sin ( ω ov 1 ζ ) (8.1) Ro Coss 1 ζ =, ωov = and Lo = Lsray + LD + LS (8.) L L C o o oss - 1 -

142 CHAPTER 8 THE EFFECT OF PARASITICS AND REVERSE RECOVERY A single swiching phase leg is considered as a saring poin from where he analysis is exended o a full-bridge opology. I will be shown ha he analyical soluion of Eq. (8.1) is applicable o boh scenarios of he inducor curren as well as boh opologies, hence he reason for i being inroduced as a general soluion a encepion of his secion. The analysis of he half-bridge opology now follows ANALYSIS OF A HALF-BRIDGE TOPOLOGY The half-bridge opology provides a simple saring poin for modelling he effec of he parasiics and reverse recovery. The wo scenarios of he inducor curren inroduced in Chaper 3 once again govern he analysis, resuling in i being divided ino wo sub-secions. Inducor Curren Scenario Figure 8. illusraes a pracical swiched oupu volage wih a duy cycle of 5% measured wihin a single phase leg of a full-bridge inverer. Figure 8. (a) represens a swiching ransiion of he lower swich from on o off while (b) shows a ransiion from off o on. Noe ha D was chosen a 5% since i provides a simple measuremen environmen of he waveforms in which Scenario is reproduced Volage [V] Volage [V] Time [s] x 1-7 (a) Time [s] x 1-6 (b) Figure 8.: Measured swiching oupu volage ransiions from (a) on o off and (b) off o on

143 CHAPTER 8 THE EFFECT OF PARASITICS AND REVERSE RECOVERY The values of he circui parameers used correspond o V d =1V, R G =1, V GS =14V, L fil =1.4μH, C=47nF and f c =384kHz. The waveforms of Figure 8. (a) and (b) can nex be mached using he analyical soluion of Eq. (8.1). Firsly, consider Figure 8. (a). The required damping and frequency of he oscillaion can be deermined ieraively from R o, C oss and L o as 5, 7pF and 4nH respecively. The peak overvolage is measured a 9.3V and exiss a ime peak =.6ns (relaive o incepion of he ringing period). Subsiuing for = peak ino Eq. (8.1) wih v DS =v DS(peak) =9.3V and noing ha he sanding curren produced corresponds o I L =.31A, he only unknown variables are L sray and sink = if. Assuming a value of L sray =3nH, if can be deermined as.85ns. 18 Analyical Soluion Pracical Measuremen 4 3 Analyical Soluion Pracical Measuremen Volage [V] 1 1 Volage [V] Time [s] x 1-8 (a) Time [s] x 1-8 (b) Figure 8.3: Analyically mached waveforms of (a) Figure 8. (a), and (b) Figure 8. (b). Using a similar approach, he measured undervolage of Figure 8. (b) can nex be mached. The parameers used for he damping and frequency of he ringing effec are he same as for Figure 8.3 (a). The peak overvolage is measured a 4.9V and once again exiss a.6ns due o he frequency being he same. Subsiuing for he laer peak undervolage and noing ha he sanding curren again corresponds o I L =.31A, sink = ir can be deermined as 1.6ns. I is hus eviden ha Eq. (8.1) provides an accurae soluion for he modelling of boh he over- and undervolage. In boh Figure 8.3 (a) and (b) he frequency and damping of he oscillaion correlae well. Since I L is known and peak is consan, he laer argumen suggess ha he required waveform can be deermined by simply esablishing he pracical relaion beween v DS(peak) and sink for a given seup

144 CHAPTER 8 THE EFFECT OF PARASITICS AND REVERSE RECOVERY (a) (b) (c) (d) Figure 8.4: Measured volage envelopes and inducor curren for (a) M=.1 and (c) M=.. Measured peak overvolage a he cres of he overvolage envelope for (b) M=.1 and (d) M=.. Figure 8.4 illusraes he measured envelopes of he over- and undervolage as well as he inducor curren for (a) M=.1 and (c) M=. over wo cycles of a 1kHz modulaing waveform. Noe ha he laer values of M were chosen in such a way as o ensure ha he inducor curren remains wihin he boundaries of Scenario. The corresponding peak overvolage ransiens measured on he cres of v DS(upper_env) are respecively shown in (b) and (d). These measuremens are included since he aliasing effec of he oscilloscope resuls in a possible misinerpreaion of he acual measuremen. The solid curves shown in (a) and (c) have been included in order o illusrae he exac measured envelopes. From boh Figure 8.4 (a) and (c) i is apparen ha a direc relaion exiss beween he envelopes of he over- and undervolage and he inducor curren

145 CHAPTER 8 THE EFFECT OF PARASITICS AND REVERSE RECOVERY This relaion will now be esablished. Firsly, consider he overvolage. Upon inspecion of Figure 8.4 a general expression defining v DS(upper_env) can be deermined by offseing i L(lower_env) by some consan value and applying he appropriae scaling, ha is: Δ v = k + k i (8.3) DS ( upper _ env) offse scaling L( lower _ env) The wo unknown consans k offse and k scaling can be solved by measuring he peak overvolage a he cres of v DS(upper_env) and he corresponding I L, which is deermined a he rough of i L(lower_env). Anoher measuremen of he peak overvolage a he rough of v DS(upper_env) and he corresponding I L a he cres of i L(lower_env) makes i possible o solve Eq. (8.3). TABLE 8.1 VARIOUS VARIABLES FOR EXPRESSING THE UPPER ENVELOPE OF THE OVERVOLTAGE FOR SCENARIO. M Measured Peak Volage [V] Curren Magniude [A] Cres Trough Cres Trough k offse k scaling The various variables esablished for M=.1 and M=. are summarized in Table 8.1. Noe ha he magniude of he curren was calculaed using he soluion obained in Eq. (3.5). Nex, consider he undervoage. From Figure 8.4 (a) and (c) i is eviden ha a similar relaion o ha esablished for he overvolage exiss for he undervolage, which means ha: Δ v = k + k i (8.4) DS ( lower _ env) offse scaling L( upper _ env) However, for some par of he inerval displayed in Figure 8.4 (a) and (b) i can be seen ha v DS(lower_env) becomes sauraed (respecive measured peak undervolaage levels of 1.5V and.v for M=.1 and M=.), and hus does no follow i L(upper_env) according o he relaion defined in Eq. (8.4). Assuming ha he same laer linear relaion exiss during sauraion, he cres of v DS(lower_env) can be esimaed. This allows for he calculaion of k offse and k scaling

146 CHAPTER 8 THE EFFECT OF PARASITICS AND REVERSE RECOVERY For he remaining region a simple condiion can be applied o v DS(lower_env) in order o clamp i o he corresponding volage during clipping. Table 8. conains a summary of he various variables. TABLE 8. VARIOUS VARIABLES FOR EXPRESSING THE LOWER ENVELOPE OF THE UNDERVOLTAGE FOR SCENARIO. M Measured Peak Volage [V] Curren Magniude [A] Cres (es.) Trough Cres Trough k offse k scaling Figure 8.5 illusraes he analyical reconsrucion if he measuremens obained in Figure 8.4 are using he parameers obained in Table 8.1 and Table Overvolage Envelope 15 Overvolage Envelope 1 1 Volage [V] 5 Volage [V] Undervolage Envelope -1 Undervolage Envelope Time [s] x 1-3 (a) Time [s] x 1-3 (b) Figure 8.5: Analyically reconsruced volage envelopes of Figure 8.4 for (a) M=.1 and (b) M=.. Since v DS(upper_env) and v DS(lower_env) define v DS(peak) he appropriae scaling needs o be applied in accordance wih Eq. (8.1) in order o derive an expression for v DS. This requires he derivaion of an expression for sink as a funcion of ime. Since v DS(peak) =v DS(upper_env) a = peak an expression defining sink = if can be found from Eq. (8.1): L i = e ( ωovζ peak ) sin ( ω 1 ζ ) sray L ( lower _ env ) if ov peak ΔvDS ( upper _ env) 1 ζ (8.5)

147 CHAPTER 8 THE EFFECT OF PARASITICS AND REVERSE RECOVERY In a similar manner v DS(peak) =v DS(lower_env) which means ha an expression defining sink = ir can be esablished from Eq. (8.1): L i = e ( ωovζ peak ) sin ( ω 1 ζ ) sray L ( upper _ env ) ir ov peak ΔvDS ( lower _ env) 1 ζ (8.6) By nex subsiuing he resuls of Eqs. (8.5) and (8.6) ino Eq. (8.1) an expression for v DS can be deermined for inducor curren Scenario. Inducor Curren Scenario Figure 8.6 illusraes measuremens of he volage envelopes as well as he inducor curren for (a) M=.5 and (c) M=.8. In boh cases Scenario is reproduced a some sage of he swiching inerval. Noe ha he remaining condiions used correspond o hose of Figure 8.4. Figure 8.6 (b) and (d) respecively represen he corresponding overvolage ransisiens of (a) and (c) measured on he cres of v DS(upper_env). Consider boh measuremens of v DS(upper_env) and v DS(lower_env). In he region where he inducor curren falls wihin he limis governing Scenario, he same linear relaion inroduced in he previous secion holds. However, in Scenario i can be observed ha v DS(upper_env) and v DS(lower_env) are governed by some oher consrain. This is eviden by noing ha he peak overvolage measuremens shown in (b) and (d) are boh more or less equal. (a) (b)

148 CHAPTER 8 THE EFFECT OF PARASITICS AND REVERSE RECOVERY (c) (d) Figure 8.6: Measured volage envelopes and inducor curren for (a) M=.5 and (c) M=.8. Measured peak overvolage a he cres of he overvolage envelope for (b) M=.5 and (d) M=.8. Figure 8.7 (a) and (b) show a measuremen similar o ha of Figure 8.6 (c) and (d) wih M=.8 and V d =V. Noe ha he bus volage was increased in order o magnify he effec observed. The addiional curves insered ono v DS(upper_env) will now be explained. (a) (b) Figure 8.7: (a) Measured volage envelopes for M=.8. (b) Measured peak overvolage a he cres of he overvolage envelope wih V d =V. The dashed curves in Figure 8.7 (a) represen an exension of he esimae of he relaion described for Scenario wihin he boundaries of Scenario while he solid curves denoe he acual measured value of v DS(upper_env). I is clear ha a mechanism differen o ha observed for Scenario governs v DS(upper_env) in his region. Noe ha he same phenomenon

149 CHAPTER 8 THE EFFECT OF PARASITICS AND REVERSE RECOVERY is observed for v DS(lower_env). An indicaion as o which non-ideal effec is responsible wihin he laer inerval can be esablished by noing ha i occurs wihin he condiions of he dead ime menioned in Chaper 5, or during forced commuaion as also observed and described by F. Nyboe [33]. For a disincly posiive inducor curren he volage loss occurs on he leading edge (he edge a which he overvolage is produced). Refer o Figure 5.1 (a). A he ime insan jus before T A1 swiches from off o on D A conducs. Once he complee curren flows hrough T A1, a minoriy carrier reverse recovery charge Q rr mus be removed before D A becomes reverse biased. An ineviable shoo hrough curren is produced unil he laer charge is removed. In a similar manner he volage gain occurs on he railing edge (he edge a which he undervolage is produced). Following he same mehology as for a disincly posiive inducor curren, i can be concluded from Figure 5.1 (b) ha he shoo hrough curren resuls from he reverse recovery of D A1. A relaion once again exiss beween v DS(upper_env) and i L(lower_env), as well as beween v DS(lower_env) and i L(upper_env), and i will now be esablished. I F di D d Q rr I rr Figure 8.8: Inrinsic power diode curren swiching characerisic during urn-off [4]. The urn-off swiching characerisics of he inrinsic power diode is shown in Figure 8.8. A well known approximae expression describing I rr is given by [4]: 6 did Irr.8 1 BVDSS IF (8.7) d The swiching imes as well as he waveform of Figure 8.8 are dependen on he properies of he semiconducor as well as he pracical circui in which i finds iself [4]. Moreover, as emphasized in [4], he soluion in Eq. (8.7) is based on various assumpions and hus represens a very rough esimae ha essenially summarizes he rade-offs beween he respecive variables involved. Aemping o analyze he reverse recovery effec

150 CHAPTER 8 THE EFFECT OF PARASITICS AND REVERSE RECOVERY analyically wih he given amoun of unknown facors will hus be edious. One very useful observaion, however, can be made from Eq. (8.7) by noing ha I rr is direcly proporional o he square roo of I F for an assumed consan value of di D /d. This allows for v DS(upper_env) and i L(lower_env) o be inerrelaed once again. Adaping Eq. (8.3) he expression yields: Δ v = k + k i (8.8) DS ( upper _ env) offse scaling L( lower _ env) Variable k offse can be deermined by noing ha i L(lower_env) = upon a ransiion from inducor curren Scenario o Scenario. Thus, by simply measuring he peak overvolage of v DS(upper_env) a he ime insan where i L(lower_env) changes polariy from negaive o posiive, k offse can be found. The remaining unknown variable k scaling can be esablished by measuring he peak overvolage a he cres of v DS(upper_env) and he corresponding value of I L. TABLE 8.3 VARIOUS VARIABLES FOR EXPRESSING THE UPPER ENVELOPE OF THE OVERVOLTAGE FOR SCENARIO. M Measured Peak Volage [V] Curren Magniude [A] Cres i L(lower_env) = Cres i L(lower_env) = k offse k scaling The various variables for M=.5 and M=.8 are summarized in Table 8.3. In a similar way he relaion describing he undervolage can be wrien as: Δ v = k + k i (8.9) DS ( lower _ env) offse scaling L( upper _ env) The unknown variables k offse and k scaling are deermined from measuremen in he same way as for v DS(upper_env). TABLE 8.4 VARIOUS VARIABLES FOR EXPRESSING THE LOWER ENVELOPE OF THE UNDERVOLTAGE FOR SCENARIO. M Measured Peak Volage [V] Curren Magniude [A] Trough i L(lower_env) = Trough i L(lower_env) = k offse k scaling

151 CHAPTER 8 THE EFFECT OF PARASITICS AND REVERSE RECOVERY The various variables obained for he undervolage are summarized in Table 8.4 for M=.5 and M=.8. Before v DS(upper_env) and v DS(lower_env) can be reconsruced i is firs necessary o deermine he values of k offse and k scaling for Scenario. A simple way of deermining k offse is o noe ha i serves as he boundary condiion beween Scenario and Scenario and is hus equal o each oher. The remaining variable k scaling can nex be esablished from a simple measuremen of v DS(upper_env) and v DS(lower_env) wih is corresponding curren during Scenario. Two furher measuremens need o be made o deermine he level a which he over-and-undervolage clip during Scenario. This allows for he analyical reconsrucion of he measured volage envelopes of Figure 8.6 which is shown in Figure Volage [V] 1 5 Undervolage Envelope Overvolage Envelope Volage [V] 1 5 Undervolage Envelope Overvolage Envelope Time [s] x 1-3 (a) Time [s] x 1-3 (b) Figure 8.9: Analyically reconsruced volage envelopes of Figure 8.6 for (a) M=.5 and (b) M=.8. Consider Figure 8.6 (b) and (d). A final remark can be made by observing ha he frequency and waveform of he ringing effec during Scenario correlae well o hose of Scenario shown in Figure 8.4 (b) and (d). This implies ha he analyical soluion of Eq. (8.1) can once again be applied o describe he over- and undervolage waveforms. 8.. ANALYSIS FOR A FULL-BRIDGE TOPOLOGY Figure 8.1 illusraes a swiched oupu volage a D=5% measured wihin a full-bridge inverer for a swiching ransiion of he low side swich from (a) on o off and (b) off o on

152 CHAPTER 8 THE EFFECT OF PARASITICS AND REVERSE RECOVERY Volage [V] 5 Volage [V] Time [s] x Time [s] x 1-6 (a) (b) Figure 8.1: Measured swiching oupu volage ransiions from (a) on o off and (b) off o on. I will now be deermined wheher he soluion of Eq. (8.1) can sill be used o mach he measured waveform wihin his opology. Figure 8.11 (a) and (b) respecively show a closeup of he waveforms illusraed in Figure 8.1 (a) and (b). The values of he circui parameers correspond o hose used in Secion Analyical Soluion Pracical Measuremen - -4 Analyical Soluion Pracical Measuremen Volage [V] 1 1 Volage [V] Time [s] x Time [s] x 1-6 (a) (b) Figure 8.11: Analyically mached waveforms of (a) Figure 8.1 (a), and (b) Figure 8.1 (b). Using he same approach as for a single phase leg, he measured waveform of Figure 8.1 (a) can be mached. The parameers deermining he frequency and damping are he same as for a single phase leg. The peak overvolage is measured a 9.6V and exiss a ime peak =.6ns. Since I L =.6A if can be deermined as 1.66ns if L sray is assumed o ake on a value of 3nH. Nex, consider Figure 8.1 (b). Using he same parameers for he frequency

153 CHAPTER 8 THE EFFECT OF PARASITICS AND REVERSE RECOVERY and damping as for (a), he undervolage can be mached by measuring is peak value, which corresponds o 8.4V. Since he magniude of I L equals.6a, ir can be deermined as 1.89ns. I is hus eviden ha he soluion of Eq. (8.1) sill serves as an accurae approximaion o he over-and undervolage curve wihin he full-bridge opology. (a) (b) Figure 8.1: Measured volage envelopes for (a) M=. and (b) M=.8. Consider he measuremens of v DS(upper_env) and v DS(lower_env) shown in Figure 8.1 for (a) M=. and (b) M=.8. These measuremens respecively reproduce inducor curren Scenario and Scenario. From Figure 8.1 (a) i is eviden ha he relaion beween v DS(upper_env) (or v DS(lower_env) ) and i L(lower_env) (or i L(upper_env) ) esablished wihin a single phase leg in Secion 8..1 also holds wihin he full-bridge opology. Nex, consider Figure 8.1 (b). The relaion beween v DS(lower_env) and i L(upper_env) correlaes wih ha achieved wihin a single phase leg. The sligh kink observed in he curve of v DS(upper_env) is a resul of he swiching ransiions of he various phase legs no execuing on he exac same ime insan. This mismach is also observed in Figure 8.1 (a). 8.3 GENERAL OBSERVATIONS AND COMMENTS The analysis in Secion 8. was performed for a given pracical seup in which he parameers were mosly kep consan. In his secion V GS and V d will be varied in order o esablish wheher he approach used in he above analysis sill holds. Noe ha he remaining parameers correspond o hose of he previous secion. Figure 8.13 (a) and (b) respecively

154 CHAPTER 8 THE EFFECT OF PARASITICS AND REVERSE RECOVERY represen he measured volage envelopes wihin a single leg of a full-bridge inverer for M=. and M=.8 a V d =V. (a) (b) Figure 8.13: Measured volage envelopes a (a) M=. and (b) M=.8 for V d =V. From hese measuremens i is apparen ha he same relaion esablished in Secion 8. holds a higher oupu power levels oo. However, in he region where v DS(upper_env) and v DS(lower_env) were consan for V d =1V (Figure 8.5 and Figure 8.9) i can now be observed ha some relaion o i L(lower_env) and i L(upper_env) once again exis. This relaion can easily be esablished by simply deermining wheher swiching occurs during he condiions of reverse recovery or no, i.e., during a ransiion from on o off under he condiion i L(lower_env) > (or a ransiion from off o on under he condiion i L(upper_env) <) v DS(upper_env) and v DS(lower_env) is respecively described by Eqs. (8.8) and (8.9). For all oher ransiions a direc relaion in accordance wih Eqs. (8.3) and (8.4) can be esablished. Figure 8.14 (a) and (b) respecively represen a measuremen of he volage envelopes wihin a full-bridge opology for V GS =11.5V and V GS =1V, wih M equalling.8 in boh illusraions

155 CHAPTER 8 THE EFFECT OF PARASITICS AND REVERSE RECOVERY (a) (b) Figure 8.14: Measured volage envelopes for M=.8 for (a) V GS =11.5V and (b) V GS =1V. By varying V GS he swiching characerisic of each phase leg is alered, which resuls in he cancellaion of v DS(upper_env) and v DS(lower_env) o a cerain exen. This was also briefly menioned by F. Nyboe [9]. However, he degree o which he parasiics conribue o disorion remains unknown. Furhermore, opimizing V GS o minimize he laer effec alers d(on), d(off), vr and vf, which migh have a more dominan effec on he disorion. 8.4 SIMULATION STRATEGY In his secion i is shown how he soluions of v DS obained in Secion 8. can nex be incorporaed ino he simulaion sraegy of Secion.5. Consider he p h pulse of a TENPWM waveform shown in Figure 8.15, in which v DS is included. The Fourier Transform of he overvolage during inerval W ptenpwm including he necessary ime shif is given by: Where: IΔ DS = +Δ W ptenpwm V DS (8.1) jπ f d j [ ] π f j π f v e v e de ( ) = p T (8.11) TENPWM 1 c Noe ha v DS is defined in Eq. (8.1) wih sink = if which is defined by Eq. (8.5)