Dynamical Profile of Switched-Mode Converter Fact or Fiction?

Transcription

1 Julkasu 687 Publcaton 687 Mkko Hankanem Dynamcal Profle of Swtched-Mode Converter Fact or Fcton? Tampere 27

2 Tampereen teknllnen ylopsto. Julkasu 687 Tampere Unversty of Technology. Publcaton 687 Mkko Hankanem Dynamcal Profle of Swtched-Mode Converter Fact or Fcton? Thess for the degree of Doctor of Technology to be presented wth due permsson for publc examnaton and crtcsm n Rakennustalo Buldng, Audtorum RG22, at Tampere Unversty of Technology, on the 3th of November 27, at 12 noon. Tampereen teknllnen ylopsto - Tampere Unversty of Technology Tampere 27

3 ISBN (prnted) ISBN (PDF) ISSN

4 Abstract Ths thess proposes a dynamcal profle for a swtched-mode DC-DC converter. The developed concept and defnton of the dynamcal profle s ndependent of the topology, conducton mode and control prncple, as long as the regulated quantty remans the same. The profle conssts of certan transfer functons that descrbe the dynamcal propertes of a sngle converter. The bass of the dynamcal profle for the voltage-output converter s the modfed g-parameter set and for the current-output converter the modfed y-parameter set, respectvely. In addton, two specal admttance parameters that are mportant n the nteracton analyss are also ntroduced. These parameters, formng the dynamcal profle, manly defne how a swtched-mode converter would behave as a part of an nterconnected system and how t would affect the other subsystems. Consstent formalsms for evaluatng the stablty and performance of a converter mposed by the load and supply nteractons are provded. It s shown that the nteractons are manly reflected va the open-loop parameters. The dynamcal profle can be derved n two dstnct ways; analytcal modelng methods can be used or the transfer functons that characterze the profle can be measured. The exstence of the dynamcal profle, for the voltage-output converters, s demonstrated by developng the profle for a buck converter wth dfferent control prncples. Operatons n dscontnuous and contnuous conducton modes are also dscussed. It s notced that the control method and operaton mode strongly affect the dynamcal propertes. It s verfed both analytcally and expermentally that these propertes can be easly deduced by studyng the parameters of the profle. The dynamcal profle for the current-output converters s also proposed. The profle can be derved by usng conventonal modelng methods or from the correspondng voltage-output-converter profle by applyng dualty. It s dscovered that the dynamcs of a current-output converter are totally dfferent than n the correspondng voltage-output converter. The prevalng assumpton has been that the current-output converter has a pecular characterstc of ncreased gan crossover frequency, when usng a low mpedance load. Ths phenomenon s addressed to be due to a wrong control desgn and the use of a resstve load as the ntal load. The prevalng method seems to be to use the resstve load n modelng and analyzng swtched-mode converters. However, the true nomnal load for the voltageoutput converter s a constant current snk and for the current-output converter a pure voltage source. Illustratve examples are provded, whch explctly show the adverse effect of the resstve load hdng the real dynamcal profle. As a concluson, the ntroduced concept of the dynamcal profle provdes valuable tool and framework n analyzng and ensurng the performance and stablty of a swtched-mode converter, or any electrcal devce, as a part of a larger system. Its use can sgnfcantly save the desgn and prototype testng tmes and pecular phenomena can usually be avoded. Several examples and aspects presented n the thess explctly prove that the unque dynamcal profle of any gven converter s a fact not a fcton.

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6 Preface Ths work was carred out at the Insttute of Power Electroncs at Tampere Unversty of Technology (TUT) durng the years The research was funded by TUT, Fnnsh Fundng Agency for Technology and Innovaton (TEKES), Efore Oyj, Salcomp Oyj and Patra Oyj. Ther contrbutons are greatly apprecated. Fnancal supports n the form of personal grants from Noka Foundaton, Foundaton of Technology, Eml Aaltonen Foundaton and Ulla Tuomnen Foundaton are also greatly acknowledged. I want to express my deepest grattude to Professor Teuvo Sunto for supervsng the thess and provdng nterestng research topcs. It has been a pleasure to work under hs gudance and many frutful conversatons wth hm on the topc (and also off the topc) have nspred me. Matt Karppanen, M.Sc., deserves specal thanks for the help n the lab and answerng numerous questons. Professor Mummad Veerachary and Dr. Vesa Tuomanen revewed the thess and ther constructve comments and recommendatons that mprove the qualty of the text n ths thess are greatly acknowledged. I would also lke to thank Antt Hankanem, c.phl., for fndng and correctng the grammatcal errors n the manuscrpt of ths thess. I wsh to thank my parents and ssters for ther support and love, not only durng my studes, but throughout my whole lfe. A year ago, when I was startng to wrte ths thess, I faced an unpredctable event n my own famly. Wthout the help of my parents, ssters, other relatves, frends and colleagues t would not have been possble to complete ths thess n such a dscplned manner as I have now done. Thanks for beng there for me! Above all, my beloved daughter Amla deserves very specal thanks. Playng wth you at home and the sunshne n your eyes always make me feel happy, even f the skes are gray! Tampere, October 27 Mkko Hankanem v

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8 Contents Abstract... Preface...v Contents... v st of Publcatons... x Author s Contrbuton...x st of Notatons, Symbols and Abbrevatons... x 1 Introducton Background and Motvaton Voltage-Output Converter Current-Output Converter A Revew of Exstng Methods to Analyze the Performance and Stablty and Model Swtched-Mode Converters Structure of the Thess Summary of Scentfc Contrbutons Dynamcal Profle Defnton of the Dynamcal Profle Two-Port Representaton oad and Supply Interacton Formalsms Internal and Input-Output Stablty Dscusson Dynamcal Revew Modelng of Swtched-Mode Converter Steady-State Operaton State-Space Averagng Effect of Control Prncple Interacton Analyss oad Interactons Supply Interactons Double Interactons Effect of oad Resstance Expermental Evdence Mxed-Data Method Mxed-Data Control Desgn Mxed-Data Nomnal Model Measured Internal oop-gan oad Interactons Supply Interactons...87 v

9 4.5 Dscusson Current-Output Converters Dervaton of Dynamcal Profle State-Space Averagng Applyng Two-Port Representaton oad Interactons Supply Interactons Dynamcal Issues and Expermental Evdence Conclusons Summary of Papers Fnal Conclusons and Man Contrbutons Future Topcs References Appendces Appendx A Appendx B Appendx C Appendx D Publcatons v

10 st of Publcatons The thess s based on the followng publcatons, whch are referred to as P1, P2, P3, P4, P5, P6, P7, P8, P9, P1 and P11 n the text. [P1] [P2] [P3] [P4] [P5] [P6] [P7] [P8] [P9] T. Sunto, M. Hankanem, Unfed small-sgnal model for PCM control n CCM: untermnated modelng approach, HAIT Journal of Scence and Engneerng, vol. 2, ssues 3-4, pp , 25. T. Sunto, M. Hankanem, M. Karppanen, Analysng dynamcs of regulated converters, IEE Proc. Electrc Power Applcatons, vol. 153, ssue 6, pp November 26. M. Hankanem, M. Karppanen, T. Sunto, oad mposed nstablty and performance degradaton n a regulated converter, IEE Proc. Electrc Power Applcatons, vol. 133, ssue 6, pp November 26. M. Hankanem, M. Sppola, T. Sunto, oad-mpedance based nteractons n regulated converters, n Proc. IEEE Internatonal Telecommuncatons Energy Conference, Berln, Germany, 25, pp M. Hankanem, M. Sppola, T. Sunto, Characterzaton of regulated converters to ensure stablty and performance, n Proc. IEEE Internatonal Telecommuncatons Energy Conference, Berln, Germany, 25, pp M. Hankanem, T. Sunto, M. Karppanen, oad and supply nteractons n VMC-buck converter operatng n CCM and DCM, n Proc. IEEE Power Electroncs Specalsts Conference, Jeju, Korea, 26, pp M. Hankanem, M. Karppanen, T. Sunto, Converter senstvty to load mposed nstablty and performance degradaton, n Proc. IEEE Power Electroncs Specalsts Conference, Jeju, Korea, 26, pp M. Hankanem, M. Karppanen, T. Sunto, EMI-flter nteractons n a buck converter, n Proc. Internatonal Power Electroncs and Moton Control Conference, Portoroz, Slovena, 26, pp M. Hankanem, M. Sppola, T. Sunto, Analyss of the load nteractons n constant-current-controlled buck converter, n Proc. IEEE Internatonal Telecommuncatons Energy Conference, Provdence, USA, 26, pp x

11 [P1] [P11] M. Hankanem, T. Sunto, Small-sgnal models for constant-current regulated converters, n Proc. Annual Conference of the IEEE Industral Electroncs Socety, Pars, France, 26, M. Hankanem, M. Karppanen, T. Sunto, A. Altowat, K. Zenger, Source-reflected load nteracton n a regulated converter, n Proc. Annual Conference of the IEEE Industral Electroncs Socety, Pars, France, 26, Author s Contrbuton The author planned and carred out the expermental tests, and was responsble for fndng the nternal profle n [P1]. The modellng of the peak-current-mode controlled converter n [P1] s done by the frst author. In [P2], the author partcpated n the theoretcal analyss and was responsble for performng the expermental evdence together wth the co-authors. Publcatons [P3] - [P11] were manly contrbuted by the author of ths thess. The author performed the theoretcal analyss as well carred out the measurements. Professor Teuvo Sunto, the supervsor of the thess, gave valuable and constructve comments regardng publcatons [P3] - [P11]. M.Sc. Matt Karppanen helped wth the experments and measurements. x

12 st of Notatons, Symbols and Abbrevatons NOTATIONS ˆx dx dt y x x x x arg x Small-sgnal component of x Tme dervate of x Partal dervate of y wth respect to x Absolute value,.e. magntude of x Tme averaged value of x Angle of x Argument of x x Vector x x1... xn Degree SYMBOS A System matrx B Input matrx C Capactor C Output matrx C Control varable ĉ Perturbed control varable D Input-output matrx D Dode D Averaged duty-rato D Averaged complementary duty-rato (.e. 1 D ) d Instantaneous duty-rato d Complement of the nstantaneous duty-rato (.e. 1 d ) ˆd e o F m Perturbed duty-rato Voltage-source load n current-output converter Duty-rato gan x

13 f f c Frequency vector Crossover frequency f res Resonant frequency G a G cc G c G co G o G se Gs () H I I n I Control gan Controller transfer functon Control-to-nput transfer functon Control-to-output transfer functon Forward transfer functon Sensor gan Transfer functon Output-current-feedback gan Average current Averaged nput current Averaged nductor current Io MAX Maxmum output current I o n o j Averaged output current Instantaneous current Instantaneous nput current Instantaneous nductor current Instantaneous output current Imagnary unt j o j s () Output current snk Inductor Transfer functon of the subsystem Subsystem (matrx) oop gan CO oop gan of the current-output converter VO oop gan of the voltage-output converter M c M( D ) Q r r C Compensaton ramp Converson rato Swtch Magntude of a complex number Equvalent seres resstance of capactor C x

14 r d r E Dynamc resstance of dode D Equvalent resstance r ds( on) Dynamc on-tme resstance of the swtch Q R R eq R s R s1 R s2 S j S s T o T s off 1 Resstor Equvalent load resstance Output current sensng resstor Equvalent nductor current sensng resstor Equvalent output current sensng resstor Transfer functon of the subsystem S Subsystem (matrx) aplace varable Output-to-lne current transfer functon ength of swtchng perod t, t Swtch off-tme off t off 2 Swtch off-tme when nductor current s zero (n DCM) t on Swtch on-tme u u c Instantaneous voltage Output (voltage) of voltage controller u cco, Output (voltage) of voltage controller n current-output converter u C u n u u o u r U U n U U o U E U m Instantaneous (output) capactor voltage Instantaneous nput voltage Instantaneous nductor voltage Instantaneous output voltage Instantaneous reference voltage Average voltage Averaged nput voltage Averaged nductor voltage Averaged output voltage Equvalent voltage PWM-sawtooth waveform ampltude,.e. the PWM-gan Uo MAX Maxmum output voltage x

15 U ref Reference voltage Y n Input admttance Yn oc Open-crcut nput admttance Yn sc Short-crcut nput admttance Yn zz, Ideal nput admttance Intermedate parameters Z, Z, Z Flter output mpedances f 1 f 2 f 3 Z, Z, Z, Z oad mpedances Z o Z S Output mpedance Source mpedance Infnty Dampng factor n Undamped natural frequency (rad/s) Phase n radans SUBSCRIPTS n Integer number off Off-tme -c Closed-loop -dcm Refers to the VMC-DCM converter -o Open-loop -ocf Refers to the PCMC-OCF converter -pcmc Refers to the PCMC converter SUPERSCRIPTS oad-affected transfer functon S Supply-affected transfer functon - Refers to the current-output converter -v Refers to the voltage-output converter ABBREVIATIONS AC AC-DC CC CCM CO Alternatve current AC-to-DC rectfer Constant current Contnuous conducton mode Current-output xv

16 CST Control system toolbox DC Drect current DC-DC DC-to-DC converter DCM Dscontnuous conducton mode DPA Dstrbuted power archtecture DPS Dstrbuted power system EET Extra element theorem EMI Electromagnetc nterference FRA Frequency response analyzer GM Gan margn IEEE Insttute of Electrcal and Electroncs Engneers IVFF Input-voltage feedforward TI near tme-nvarant NRO Negatve resstor oscllaton OCF Output-current feedforward PCMC Peak-current-mode control PI Proportonal-ntegral control PID Proportonal-ntegral-dervatve control PM Phase margn PO Pont of load PWM Pulse wdth modulaton RHP Rght half plane S1 Subsystem 1 S2 Subsystem 2 TUT Tampere Unversty of Technology SSA State-space averagng VMC Voltage-mode control VO Voltage-output xv

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18 1 Introducton Ths chapter provdes the bass for the thess. Essental background nformaton and fundamental ssues behnd the topc of the thess are presented. The basc operatng prncples of swtched-mode converters are also dscussed. An extensve lterature revew of the prevous work related to the topc s presented, pontng out several prevalng ambgutes. Fnally, the man contrbutons and short summares of the followng chapters are presented. 1.1 Background and Motvaton Electronc power processng and converson have nterested researchers and engneers snce the 17 th century. Nowadays, varous electrcal apparatuses are used and the consumpton of electrcal energy s ncreasng year by year. However, the electrcal energy has to be produced somewhere and somehow, and at the same tme the amount of produced energy has to be consumed. Between the ponts of producton and consumpton dfferent knds of electronc power processng and converson methods are needed. Generators, transmsson lnes, transformers, AC-DC rectfers and DC-DC converters are the core components n the electrc power dstrbuton. Ths thess concentrates on the swtched-mode DC-DC converters, whch are usually close (n physcal and conceptual sense) to the end user or applcaton. Swtched-mode converters have replaced the lnear regulators n the modern DC-DC converson. The hstory of swtched-mode converters dates back to the md-6s, when the actve power swtches started to replace the mechancal swtches and relays [1]. Whle the lnear regulators are qute smple and have low effcency, swtched-mode converters have a nonlnear nature due to the swtchng acton, and hence, they are more complcated to analyze and model. On the other hand, they are usually smaller, more effcent and wegh less [2]. The scentfc research on swtched-mode converters was started n the md-7s by Dr. Mddlebrook [3]-[5]. Hs semnal work 1

19 Chapter 1 Introducton n modelng and analyzng the dynamcs and nput-flter nteractons are stll qute relevant and mportant, although scentfc research has been carred out for more than 3 years. Although the evoluton of desgn and modelng of swtched-mode converters, and power electroncs n general, have been rapd snce the 7s there are stll some open questons and msunderstandngs watng for an answer and a correcton. IBA PO-converters 1-12V Supply system Isolated bus converter u n 48V 8-16V Fg Dstrbuted power system. In a modern electronc devce (e.g. telecom power supply) varous DC-voltage and DC-current levels are usually requred. To power these devces dstrbuted power systems/archtectures (DPSs/DPAs) are wdely employed [6]-[9]. An ntermedate bus archtecture (IBA), shown n Fg. 1.1 nsde the dashed lne, has become the most used DPA n new applcatons [1]. The IBA conssts of an solated bus converter, whch produces an ntermedate bus voltage (8 16 V) and several pont-of-load (PO) converters. Usually, an EMI flter has to be placed before every power stage and a storage battery may be connected to the system after the front-end rectfer n order to provde energy to the load system durng the power outages. It s obvous, that the system shown n Fg. 1.1 s complcated both from a dynamcal and desgn vewpont. In order to desgn a stable system wth adequate performance margns the functonng of each buldng block n the system has to be known. Each DC-DC converter n a DPS, or n any applcaton, s always a part of an nterconnected system. Ths actually means that the source and/or load system may 2

20 Chapter 1 Introducton sgnfcantly affect the stablty of an ndvdual converter, and hence, the stablty of the entre system. Therefore, an mportant and nterestng queston arses: How to perform the nteracton analyss to ensure stablty and adequate performance of the converter and the whole system? The canoncal dynamcal profle and nteracton formalsm presented n ths thess wll answer ths queston and provde a powerful faclty to analyze the performance and stablty of DC-DC converters. The terms performance, stablty and also the crossover frequency or the bandwdth contnuously appear n ths thess. Therefore, t s necessary to defne the meanng of these terms n the scope of the thess n order to avod confuson. The terms performance and stablty can be addressed to both the tme and frequency domans. The tme doman performance s typcally studed by means of a step response (.e. a transent response n swtched-mode converters), where a step change s ntroduced nto the reference sgnal and the output of the system s montored. Typcal characterstcs of the step response are the rse tme, over shoot and settlng tme. However, the classcal step response analyss ncorporates the dsturbance sgnal (.e. the step) nto the reference sgnal, whch s typcally constant and even physcally unavalable n the modern converters. The transent response analyss of the swtchedmode converters s typcally done by ntroducng the step change e.g. nto the load current or nput voltage and the output voltage s montored. From a dynamcal vewpont, ths approach does not gve the same performance characterstcs as the classcal step response method. The tme doman performance s not dscussed n ths thess, but the frequency doman performance s often consdered. The frequency doman performance relates to the loop gan s ( ) of the converter. The performance of the converter s judged by means of the gan margn (GM) and phase margn (PM). In the bode plot, the GM can be expressed as the vertcal dstance of the loop gan magntude from the unty gan (.e. db) at the frequency, where the phase s -18. Consequently, the PM s defned as the phase of the loop gan at the unty gan frequency added wth 18. To guarantee adequate performance the GM and PM are typcally requred to be at least 6 db and 45, respectvely. The nstablty occurs f the GM < db or the PM <. The bandwdth of the system and the (gan) crossover frequency f c are sometmes confusngly defned n power electroncs. Accordng to the system theory, the bandwdth s related to the senstvty or complementary senstvty functon provdng two dfferent defntons of the bandwdth. The gan crossover frequency f c s naturally the frequency, where the gan of the system s unty (.e. db). In ths thess, the gan crossover frequency f c s used, when comparng the propertes of dfferent loop gans and transfer functons n order to avod confuson.[11] 3

21 Chapter 1 Introducton An extensve revew of the exstng methods to analyze and model the dynamcs of a swtched-mode converter wll be gven n Secton 1.3. However, t s worth mentonng already at ths pont that most of the exstng modelng and analyzng methods do not reveal the true nternal dynamcs of a sngle converter. Ths s manly due to the wrong ntal modelng wth a resstve load, whch may hde the nternal dynamcs of the converter. The nternal dynamcs for e.g. a voltage-output converter can be derved by usng a constant-voltage source at the supply sde and a constantcurrent snk as a load. Illustratve examples of the effect of a wrong ntal load wll be gven later. New converter topologes and control methods are contnuously developed and publshed n academa and ndustry, but the focus seems to be only on certan advantages of these new topologes. The dynamcal ssues and senstvtes for nteractons are not usually dscussed. So, usually the performance of these new topologes or control methods as a part of an nterconnected system, lke one shown n Fg. 1.1, s unknown. It has been found out durng the research that the true nature of swtched-mode converters relates to the frequency doman. By studyng certan frequency responses (.e. transfer functons) t s easy to conclude the possble senstvtes for the load and/or supply nteractons as wll be shown e.g. n Chapters 3 and 4. However, only tme doman smulatons and measurements are usually presented e.g. n converter manufacturers datasheets [12]-[15]. If frequency responses are presented lke n [12] only the magntudes are shown but not the phase plots, whch are just as mportant as the magntudes. Agan, the performance of these commercal converters as a part of an nterconnected system, lke one shown n Fg. 1.1, remans unknown. The concept of the dynamcal profle of the swtched-mode power supply s ntroduced n ths thess. It provdes a straghtforward method and a physcal nsght nto the converter nternal or the nomnal dynamcs. By analyzng the dynamcal profle, the dynamcal propertes (.e. load and supply senstvtes and nsenstvtes and control-loop stablty) can be concluded by analyzng certan transfer functons n the frequency doman. Consequently, the stablty and performance of a converter wth known and analyzed dynamcal profle as a part of the nterconnected system can be easly derved. Chapters 2 and 3 wll dscuss more n detal the concept of dynamcal profle and how to derve t. 4

22 Chapter 1 Introducton 1.2 Voltage-Output Converter The basc confguraton of a voltage-output (VO) converter, llustrated n Fg. 1.2, conssts of a power stage, load and supply systems and a control crcutry. The power stage contans nductor(s), capactor(s), swtches and ther parastes. If electrcal solaton s needed, a transformer may be ncluded n the power stage. Dependng on the power-stage-crcut structure (.e. topology), the output voltage U o s ether smaller or larger than the nput voltage U n. The output voltage s adjusted by means of the control crcut by typcally applyng a pulse wdth modulaton (PWM) to control the power stage swtch on-tme t on. In Fg. 1.2, the swtch on-tme t on s ndependent varable formng a control scheme called a voltage-mode control (VMC) or drect duty-rato control. Other control schemes can be formed by takng a feedback or feedforward e.g. from the nput voltage, nductor current or load current. The control crcutry ncludes an error amplfer or a controller and a PWMcomparator, whch controls the swtch on-tme t on. The nomnal load for the voltageoutput converter conssts of a current snk j o. The mpedance Z n Fg. 1.2 connected parallel to the constant-current snk represents the non-deal load system. Respectvely, the mpedance Z S at the supply sde represents the effects of the nondeal source. Supply system u n + - n Z S Power stage + u C - o Z oad system + uo - j o t on _ + Um u c controller U ref Control crcut Fg Basc confguraton of voltage-output converter. 5

23 Chapter 1 Introducton The basc steady-state analyss of swtched-mode converters relates to the prncples called an nductor volt-second balance and capactor amp-second balance [18]. Accordng to the nductor volt-second balance the net change of the nductor voltage over one swtchng perod T s s zero. The capactor amp-second balance states that the net change of the capactor current over one swtchng perod T s s zero. The steady-state and small-sgnal modelng and analyss are dscussed more n detal n Chapter Current-Output Converter Instead of controllng the output voltage U o, the output current I o s regulated n current-output (CO) converters. Current-output converters are typcally used n applcatons, where an overload protecton s needed. The need for current lmtng arses e.g. n DPS system f a storage battery s connected to the system and charged/dscharged, and hence, large currents are drawn. Fg. 1.3 shows the typcal output-voltage/current characterstcs of a converter, where the constant voltage and current controls are needed. When the output current s lower than the maxmum set level, the converter s n the voltage-output mode and regulates only the output voltage. Consequently, when the maxmum allowed output current s reached the converter enters nto the current-output mode and starts regulatng the output current. u o Uo - MAX VO CO Io - MAX o Fg Typcal output-voltage/current characterstcs of a converter n applcatons where constant voltage and current control s needed. The constant-current lmtng can be accomplshed ether by usng only an output current as the feedback sgnal or both the output voltage and current n cascade, where the nner loop s the voltage loop and the outer loop s the current loop [16]. The overload protecton can also be accomplshed by usng a modfed constant 6

24 Chapter 1 Introducton power lmtng, whch s typcally mplemented n such a way that the reference of the current loop s gradually ncreased along the decrease n the output voltage untl the maxmum defned output current s reached after whch the lmtng scheme follows the constant-current scheme [16], and [17]. The study of the cascade operaton and the modfed-constant-power lmtng are left out of ths thess and only the mode, where the output current s used as the feedback sgnal, s consdered. The basc confguraton of the current output converter s shown n Fg The only dfference compared to the voltage-output converters s the load system, whch conssts of an deal voltage source e o n seres wth the load mpedance Z. However, the nternal dynamcal profle and system nteractons are totally dfferent n the current-output converters. The correspondng dynamcal profle, dynamcs and nteractons are dscussed n detal n Chapter 5. Supply system u n n Power stage oad system Z + S Z u C + u eo o - - o - R s t on _ Um + u cco, controller R so U ref Control crcut Fg Basc confguraton of current-output converter. 1.4 A Revew of Exstng Methods to Analyze the Performance and Stablty and Model Swtched-Mode Converters As t was stated earler, the scentfc research on the topc started n the 7s. The basc topologes (.e. buck, boost and buck-boost) were analyzed and modeled for the frst tme n [19]. Averaged and lnearzed general power-stage models were also ntroduced. The semnal paper [4] publshed by Mddlebrook and uk proposed a 7

25 Chapter 1 Introducton state-space averagng (SSA) modelng technque, whch produces both contnuoustme steady state and dynamc lnearzed models. The basc dea behnd the SSA s to average the swtch on- and off-tme state-space equatons over one swtchng perod. Crcut averagng and hybrd modelng were consdered as alternatve modelng methods that gve the same canoncal crcut model as the SSA method. The SSA has become popular snce ts ntroducton bascally due to ts smple and clear methodology. It s commonly known that the SSA gves accurate open-loop models up to half the swtchng frequency, when the converter operates n contnuousconducton-mode (CCM) under drect-duty-rato control or VMC [2]. The canoncal crcut model ntroduced n [4] was argued for beng a useful tool for analyzng smallsgnal dynamcs of swtched-mode converters regardless of the topology. However, t contans a resstve load as well as parastc loss elements n the duty rato dependent generators hdng the true nternal dynamcs. In spte of the practcalty and smplcty of the SSA, several other modelng approaches have also been developed. In crcut averagng, the voltage and current waveforms are averaged nstead of averagng the state equatons as n SSA [19], and [21]. Average models for the PWM-swtch were ntroduced n [22] and [23]. nearzaton of the averaged crcut and PWM-swtch yelds approprate small-sgnal models of the converter. It s clear that the rpple nformaton s lost n averagng. However, the averaged models are also usable n tme doman smulatons and transent analyses. Swtched crcuts can be equally used, f the rpple nformaton s needed [21]. Because the swtchng acton s actually dscrete, a sampled-data modelng has been proposed. The bass of the sampled data modelng s presented n [21]. The modelng s based on the contnuous tme state-space model and the standard matrx exponental expresson for lnear tme-nvarant (TI) systems. Accordng to [21], [24] and [25], the prevalng method s to derve the dscrete-tme model from the contnuous-tme state and swtchng equatons. In [26], the dscretedoman model s derved from the correspondng model n the aplace-doman by usng z-transformaton. The sampled-data modelng s derved n [27]-[29] by usng a dscrete-tme state-space model. The sampled-data modelng typcally nvolves tedous calculatons and therefore t s not wdely adopted. However, the sampleddata models mght become useful, when dgtal controllers replace the analog controllers. It should be clear that the true nternal or nomnal dynamcs can be derved from the power stage model and from the control crcut model. However, t seems that the defnton of the nomnal power stage or model s not clear among the scentsts and engneers. There are numerous examples of modelng and analyzng swtched-mode 8

26 Chapter 1 Introducton DC-DC converters wth a resstve load (these are only example papers, not the complete lst: [4], [18] and [3]-[35]). The actual load s very seldom a pure resstor but should be treated as an external system not ncluded n the nomnal model. The semnal paper [4] actually uses a resstve load when ntroducng the SSA method and the canoncal equvalent crcut. It s obvous that ncludng the load resstor n the socalled canoncal model mght lose the nformaton of the nomnal dynamcs. Even the fundamental power electroncs text books such as [36] and [18] use the resstve load n ther analyses and provde ncorrect nformaton for the reader. So, what type of load should be used to get the nomnal dynamcs? A voltage-output DC-DC converter s known to have current source nput and voltage source output ports [P5], so the natural nomnal load connected to the voltage source output port s obvously a current-snk. Consequently, the nomnal load for a current-output converter wth a current source output port s a pure voltage-source. In spte of the prevalng technque to use the resstve load, a few attempts to defne the nomnal or general load have been presented. A general load mpedance s treated as an alternatve for the resstve load n [37]. Frst t s stated that the load can be seen as a current source, but later the load s replaced wth the general load mpedance. The dea of the general load (mpedance) s actually correct, but the authors seem to lack the understandng of the true nomnal dynamcal behavor of swtched-mode converters. It s explctly stated n [38] that the nomnal load refers to the use of ether a resstve or dc current snk load. It s true that the nternal output mpedance can be measured ether by usng a resstve or current snk load, but when measurng or analyzng e.g. the loop gan ths does not apply. It seems that the authors of [38] are confused wth the termnology of the nomnal dynamcs or nomnal load and provde vague nformaton. An approach known as an untermnated modelng was ntroduced n [39] treatng a converter as a stand-alone module wthout consderng the load mpedance, but usng a current snk load. The untermnated modelng method has been appled n [4]-[42] for studyng the load nteractons. In [43], the method was successfully used for analyzng the nput flter nteractons. The untermnated model was derved n [44] by frst constructng the models wth the load resstor R and then lettng R. However, the most convenent way of gettng the untermnated model s to use the constant-current-snk load as an ntal load system as t was done e.g. n [P4] and [P5]. As a sgnfcant contrbuton of ths thess, t was found that the dervaton of the nomnal dynamcs, and hence, the dynamcal profle by usng the correct ntal load s a startng pont for understandng the behavor of swtched-mode converters under varous condtons. It s well known that the load and supply nteractons can sgnfcantly affect the converter dynamcs. Typcally, the converter s equpped wth an nput EMI flter, 9

27 Chapter 1 Introducton whch may deterorate the performance of the converter. The nput flter nteractons were frst studed by Mddlebrook n the semnal paper [3]. A converter wth an nput flter was modeled and desgn crtera for nput flter were developed. The dervaton of the nput-flter-affected transfer functons were based on the method known later as an extra element theorem (EET) [18], and [45]-[47]. The EET provdes a tool to analyze the change of transfer functons, when mpedance s added to the network. However, the EET nvolves tedous calculatons, and therefore, may not be sutable for practcal usage. The load and nput flter nteractons can be easly concluded from a two-port lnear crcut representaton of the converter wth load and supply (e.g. flter mpedance) mpedances [48] and [43]. The two-port modelng technque based on g-parameters [49] s revewed and dscussed more n detal n Chapter 2. The nput-flter nteractons have been under extensve research snce the Mddlebrook s paper. It has been notced that dfferent topologes and control methods have dfferent senstvtes for nstablty or performance degradaton due to the nput flter or supply mpedance [33], and [5]-[56]. Obvously, the converter dynamcs are also affected by the load. The load nteractons have also been studed n varous papers such as n [38]-[42], [44], [57], and [58]. The load nteracton formalsm s smpler to understand than the correspondng supply sde formalsm. In Chapter 2, t wll be shown that the performance of a converter may be deterorated f the load mpedance and the open-loop output mpedance of the converter overlap. Although, the supply and load-sde-nteracton formalsms are dfferent, the stablty of the converter wth load and/or supply system can be concluded from the mpedance rato known as a mnor-loop gan [3], [39] and [41]. If studyng the supply nteractons, the mnor-loop gan s defned as the rato of the supply mpedance (e.g. nput flter output mpedance) and the closed-loop nput mpedance of the converter. The correspondng mnor-loop gan at the load sde s the rato between the closed-loop output mpedance of the converter and the load mpedance. In order to guarantee the stablty, the mnor loop gan must satsfy the Nyqust stablty crteron [3], and [59]. Varous forbdden regons n the complex half plane, out of whch the mnor loop gan should stay, have been presented n the lterature [41], and [6]-[63]. It s clamed that these forbdden regons provde certan phase (PM) and gan margns (GM) for the nterconnected system. However, the PM and GM of the mnor loop very seldom concde wth the correspondng margns n the load or supply-affected loop gan of the converter [P3] and [P7]. Ths means that the performance of the converter may be drastcally deterorated even f the correspondng margns of the mnor loop gan are adequate. The mnor-loop-gan analyss s only sutable for ensurng the stablty but the performance and true margns should always be checked from the true major loop gan of the converter. 1

28 Chapter 1 Introducton A typcal method to analyze the converter performance n the tme doman s a transent response analyss. There are numerous papers clamng that the hgher the crossover frequency f c of the loop gan s the faster s the transent response [64]- [68]. Accordng to the classcal control theory ths s true, because the reference s step-changed. However, n swtched-mode converters the reference s usually kept constant but the load current s changed ntroducng a transent nto the output voltage. In [P2], [69] and [7], t was demonstrated that even f the peak-currentmode controlled (PCMC) converter and PCMC converter wth an output current feedforward (OCF) have the same loop gan, the PCMC-OCF converter has a consderably faster transent response. The reason for ths s the smaller open-loop output mpedance of the PCMC-OCF converter. Therefore, the transent response actually relates to the open-loop output mpedance of the converter and ts behavor. Ths was also notced n [71], where a larger closed-loop output mpedance of the PCMC converter at lower frequences compared to the VMC converter yelded also a longer settlng tme. The transent response as a functon of tme usng nverse aplace transformaton of the closed-loop output mpedance, when a certan load step change occurs, was computed n [72]. The transent response of a parallel RC-crcut was consdered n [67] and [68] to mmc the transent response of a swtched-mode DC-DC converter. The results of the above analyses seem to be, however, a bt unrelable because of the smplfcatons made n the analyses. The true relaton between the frequency and tme doman stll seems to be fuzzy and needs further research n order to put the relaton n a correct mathematcal from. Actually, the challenge s the complex structure (.e. the numerator and denomnator are hgh-order polynomes n the aplace varable s ) of the closed-loop output mpedance makng the computaton of the nverse aplace transformaton a dffcult and tedous task. Nevertheless, the relaton can be mplctly studed as t was dscussed above (.e. larger closed-loop output mpedance at lower frequences longer settlng tme and small mpedance fast response). In addton, the amount of peakng n the closedloop output mpedance at the phase or gan crossover frequency dctates the PM and GM of the converter loop gan [72]: The peakng s related to senstvty functon (.e. 1/ 1 s ( ) ), whch s a elementary part of the equaton of the closed-loop output mpedance by defnton [11]. The peakng n the senstvty functon due to a low PM or GM would naturally be observable n the transent response. The frst basc courses on power electroncs at unverstes are typcally based on fundamental text books such as [18], [36] and [73]. They all present the fundamentals of swtched-mode power converson, but n [18] the study s done more n detal. Common to all these text books s that the modelng and even the basc operatng 11

29 Chapter 1 Introducton prncples are dscussed wth the resstve load. The text book [18] s maybe the most often used ntroductory level book on swtched-mode converters, but t loses the pont of presentng the true canoncal model (both steady-state and small-sgnal) and dynamcal ssues by ncorporatng the resstve load nto the models. The contents of these fundamental text books may explan the reason, why the prevalng technque stll strctly reles on the use of the resstve load n the analyses both among academa and ndustry. Chapter 5 of ths thess s solely dedcated to the current-output converters, whch are typcally used n battery-powered applcatons. The modelng and analyss of the current output converter n [74]-[76] are based on the use of resstve load, although the real load typcally conssts of a back-up battery wth low nternal mpedance [77]. The pecular behavor of the ncreasng crossover frequency n the loop gan wth battery-type load observed e.g. n [74] and [79] was shown to be due to the use of wrong ntal load (.e. resstve) n [P9]. 1.5 Structure of the Thess The thess contans sx chapters. The man contrbutons and short summares of the followng chapters are: Chapter 2: Dynamcal Profle The basc concept of the dynamcal profle s presented. The chapter revews the papers [P2], [P3], [P5], [P7], and [P11]. It s found that the g-parameter set characterzes effectvely the dynamcal propertes of a swtched-mode converter. In addton, the nteracton formalsm ntroduces two specal admttance parameters. The two-port representaton and the load and supply nteracton formalsms are presented. Both open- and closed-loop operatons are consdered. It s shown that the open-loop parameters reflect the nteractons, and therefore, ther behavor s the man nterest. The chapter concentrates only on the voltage-output converters, but the man dea behnd the dynamcal profle apples also to the current-output converters, whch are studed n Chapter 5. It s found out that the true nternal dynamcs can be derved by usng a pure voltage source at the nput and a constant current snk at the output actng as a load. The prevalng technque to use a passve resstor as the ntal load s argued for beng a conservatve and erroneous approach. Fnally, an nternal and nput-output-stablty formalsm s presented yeldng to an mpedance rato (.e. mnor-loop gan) between the nterfaces of two subsystems from whch the stablty can be verfed. 12

30 Chapter 1 Introducton Chapter 3: Dynamcal Revew Ths chapter starts wth ntroducng the basc prncples n modelng a swtched-mode converter. Frst, the steady-state analyss s derved by applyng the volt- and amperesecond balances and then the SSA method s used to compute the dynamcal profle of a VMC-buck converter operatng n CCM. The effect of a control prncple on the converter dynamcs s dscussed by usng VMC-CCM, VMC-DCM, PCMC and PCMC-OCF as examples. It s found out that the dynamcal propertes can be sometmes deduced drectly from the analytcal model, but t s preferred to analyze also the frequency responses. The open-loop profles of the three control modes are analyzed and based on the nternal dynamcal profles, the load and supply nteractons are effectvely predcted and analyzed. It s demonstrated that the load nteractons are reflected va the open-loop output mpedance. Even though the loop gans of the PCMC and PCMC-OCF converters are the same, the transent responses are found out to be dfferent (.e. PCMC-OCF provdes superor response compared to PCMC). Ths dfference can be addressed to the small output mpedance of the PCMC-OCF converter. The load and supply nteractons are studed by usng Ccrcuts as the load and supply mpedances. The study of ths chapter s manly related to papers [P1]-[P8] and [P11]. Chapter 4: Expermental Evdence The basc procedures of analyzng the performance and stablty that was ntroduced n Chapter 3 are verfed n practcal stuatons. A buck converter wth VMC and PCMC control modes are studed. The VMC converter s desgned to operate n CCM and DCM. A frequency response analyzer s used to measure the g-parameter set. Some of the measured parameters are compared to the analytcal model and t s observed that they match well wth each other. The non-dealtes makng the measurements and predctons slghtly to dffer are also dscussed. In cases, where the njecton sgnal goes through the modulator, t s observed that the phase starts to lag more than predcted due the modulator crcut and the snusodal njecton sgnal. A mxed-sgnal method, whch uses both analytcal and expermental data, s also ntroduced. Ths method can be used e.g. n control desgn by measurng the transfer functon between the control sgnal and output voltage at the open loop and then desgnng the controller based on the measurement. The method s also useful n defnng the nternal profle n some cases. Furthermore, t can be appled to calculate varous non-dealtes so that they can be taken nto account and ncluded n the dynamcal profle. The study of ths chapter s manly related to papers [P1]-[P8] and [P11]. 13

31 Chapter 1 Introducton Chapter 5: Current-Output Converters The dynamcal ssues of the current-output converters are dscussed. The chapter s based on papers [P9] and [P1]. It s shown that the dynamcal profle can be derved from the correspondng voltage-output converter by applyng dualty. A SSA modelng example s also ntroduced for computng the dynamcal profle. It s dscovered that the bass of the profle s the modfed y-parameter set. It s demonstrated that the most convenent way of presentng the dynamcal profle s to use the correspondng parameters of the voltage-output converter, because they are usually well known and avalable. The pecular phenomenon of the ncreasng crossover frequency, when havng a low mpedance load e.g. a battery-back, s shown to be due to the wrong ntal modelng wth a resstve load. Therefore, the nomnal load for the current-output converters s an deal voltage source (.e. low mpedance load). Expermental evdence s provded to llustrate the effect of the ncreasng crossover frequency on the loop gan, when changng from a resstve load to a low mpedance load. The practcal load gvng the desred result s proposed to be a parallel connecton of a resstor and a capactor, where the capactor provdes the dynamcal short crcut and recovers the nternal behavor at hgher frequences. Chapter 6: Conclusons Ths chapter concludes the thess. A short summary of each paper s gven. The fnal conclusons of the work behnd the thess are put together. The scentfc contrbutons of the thess are also gven and dscussed n detal. Fnally, potental future research topcs that have arsen durng the work are presented. 1.6 Summary of Scentfc Contrbutons The man scentfc contrbutons of the thess can be lsted and summarzed brefly as follows: The concept of dynamcal profle s proposed and ts exstence s proven. It s shown that every converter has ts unque dynamcal profle whch characterzes ts dynamcal features. Open-loop parameters are shown to be the man faclty to study the nteractons and senstvtes to them. The nomnal loads of the voltage- and current-output converters, nvokng the nomnal dynamcs, are stated to be a constant-current snk and a pure voltage source, respectvely. 14

32 Chapter 1 Introducton The dynamcal profle of the current-output converter s presented for the frst tme. It fully explans the observed pecular behavor. The thess wll defntvely show that the exstence of the dynamcal profle s a fact not a fcton. 15

33 2 Dynamcal Profle The concept of dynamcal profle s ntroduced and ts capablty of revealng the nternal dynamcs of a swtched-mode DC-DC converter s shown. The two-port representaton s revewed and argued for beng the most useful method for dervng the dynamcal profle, yeldng the true nternal and canoncal model of a swtchedmode converter. The load and supply nteracton and nternal stablty formalsms of an nterconnected system are also presented n a consstent way. The focus of the chapter s manly on the voltage-output converters. The dynamcal profle of currentoutput converters s dscussed n Chapter Defnton of the Dynamcal Profle A dynamcal profle defnes the dynamcal propertes of a swtched-mode DC-DC converter. Due to the observed frequency-doman nature, the proposed profle conssts of varous open and closed-loop transfer functons. These transfer functons are defned as the relaton between certan voltage(s) and/or current(s) characterzng the dynamcs of a gven converter. It s mportant to note that the nterest s partcularly n the nternal dynamcs, whch means that the nterfaces between the converter and nterconnected subsystems should be explctly defned. The defnton of the nternal dynamcal profle ncludes the followng: For a gven topology, the nternal dynamcal profle conssts only of the correspondng power stage components and the necessary control crcutry. Addtonal components, such as nput capactors and extra output flters should be consdered as external subsystems. The dynamcal profle can be derved by usng an deal voltage source at the nput and a constant-current snk load at the output n the case of voltage- 16

34 Chapter 2 Dynamcal Profle output converters and an deal voltage source both at the nput and output n the case of current-output converters. At closed loop, the addtonal components or subsystems nsde the feedback loop may adversely change the nternal dynamcal profle and ther effect should be consdered. Bascally, there are two ways of extractng the nternal dynamcal profle; an analytcal model of the converter can be derved or the correspondng transfer functons can be measured by usng a frequency response analyzer (FRA). Typcally, a practcal converter contans some non-dealtes, whch cannot be modeled accurately. Ths mples that by measurng the transfer functons the real dynamcal profle could be obtaned. However, sometmes t may be mpossble to measure every transfer functon or the nternal dynamcs drectly (.e. a resstve load has to be used). In these cases the measurement and analytcal data can be used together to compute the nternal dynamcal profle. Ths technque s known as a mxed-data method and t wll be ntroduced n Chapter 4. Obvously, f the model and measurements are known to be n a good agreement, t s reasonable and convenent to use the model to study the dynamcal profle analytcally and guarantee the functonng, stablty and performance of the converter before constructng the entre prototype or startng the mass producton. Measurements can be used to verfy the analytcal results. The rest of ths chapter wll concentrate on showng the capablty of the nternal dynamcal profle. It can be proven that every converter topology, conducton mode and control method produces dfferent dynamcal profles. The control method and conducton mode may sgnfcantly change the dynamcal propertes, although the chosen topology can retan some common dynamcal features. However, f the dynamcal profles under dfferent control methods or conducton modes and the source and load subsystems are known, the most sutable converter can be easly chosen for the applcaton as well as the stablty and performance of the nterconnected system can be guaranteed Two-Port Representaton Almost any electrcal system can be seen as a black box consstng of nput and output ports. Two-port networks are typcally used to represent such black boxes [49] and [8]. Two-port models have also been appled to analyze DC-DC converters 17

35 Chapter 2 Dynamcal Profle [43], [44], [48], [51], [52], [81] and [82], but the dea of descrbng the true nternal dynamcal profle has not been explctly presented earler. A two-port model for a voltage-output converter s shown n Fg. 2.1 nsde the dashed lne. The nput port of the model s a Norton equvalent crcut and the output port s a Thevenn equvalent crcut mplyng that the two-port model consttutes of g-parameters [49]. The use of g-parameters s well justfed, because ther exstence s always guaranteed [49]. The nput port of the model n Fg. 2.1 corresponds to ˆ Y uˆ T ˆ G cˆ (2.1) n no n oo o c at open loop (.e. YN and ˆ T ˆ G cˆ ) and the output port to Yn o N oo o c uˆ G uˆ Z ˆ G cˆ (2.2) o oo n oo o co at open loop (.e. ZT Zo o and uˆt Goouˆn Gcocˆ). The mnus-sgn before the term Z ˆ o oo s due to the drecton of the output current. It s worth notng that here open loop refers to a stuaton where the outer feedback loop s dsconnected (.e. the output voltage feedback path s dsconnected n the case of voltage-output converters). Inner feedback or feed-forward loops are connected n the case of the open loop. The hat over the varables represents the small-sgnal component of the varable. The general control varable s denoted by ĉ. ˆn Z T ˆo + uˆn + _ YN ˆN + _ uˆt uˆo ˆo _ ĉ Fg Two-port model of voltage-output converter. The equatons for nput n (2.1) and output n (2.2) can be equally presented n a matrx form yeldng 18

36 Chapter 2 Dynamcal Profle uˆ n ˆ n Yno To o Gc ˆ o uˆ G o oo Zoo G co cˆ (2.3) ˆn Zo - o ˆo + uˆn + _ Yn - o T ˆ Gcˆ o- o o c + _ G ˆ coc uˆo + _ G ˆ o- oun _ ˆo ĉ Fg Two-port model of open loop voltage-output converter wth g-parameters. It should be noted that the general form of the g-parameter set typcally conssts only of four parameters (see e.g. [49]), but here the general control varable s also ncluded n the parameter set. The set n (2.3) s represented as the correspondng two-port crcut n Fg. 2.2 and the correspondng open-loop parameters n (2.1), (2.2) and (2.3) are denoted as follows: Yn o = nput admttance,.e. ˆ uˆ n To o = reverse or output-to-nput current transfer functon,.e. ˆ ˆ G c = control-to-nput current transfer functon,.e. ˆ n cˆ n Go o = forward, nput-to-output, lne-to-output transfer functon or u audosusceptblty,.e. ˆ o uˆ n u Zo o = output mpedance,.e. ˆ o ˆ G co = control-to-output transfer functon,.e. ˆ cˆ o u o n o 19

37 Chapter 2 Dynamcal Profle An equvalent way of representng (2.3) and the model n Fg. 2.2 s to use controlblock dagrams. The block dagrams for the open-loop voltage-output converter are shown n Fg. 2.3 descrbng the output and nput dynamcs. The block dagrams are useful, when dervng the closed-loop dynamcal profles as wll be shown next. ˆo Zo - o ˆo To - o uˆn Go - o uˆo uˆn Yn - o ˆn G co G c ĉ ĉ a) b) Fg Control-block dagrams for voltage-output converter at open loop: a) output dynamcs and b) nput dynamcs. The closed-loop dynamcal model of the voltage-output converter can also be presented as a two-port network as shown n Fg The equvalent g-parameter set at closed loop can be expressed by uˆ n ˆ n Ync To c Gc c ˆ o uˆ G o oc Zoc G coc u ˆ r (2.4) ˆn Zo - c ˆo + uˆn + _ Yn - c T ˆ G ˆ o- c o c- cur + _ G co- c uˆ r uˆo + _ G ˆ o- cun _ ˆo Fg Two-port model of closed-loop voltage-output converter wth g-parameters. 2

38 Chapter 2 Dynamcal Profle The general descrpton for the converter loop gan VO s typcally gven by G G G G (2.5) VO se cc a co where Gse s the output-voltage sensor gan, G a s the control gan (note: G cc s the controller transfer functon and G a s the gan between u ˆco and ĉ as t s shown n Fg For the VMC converters t s the PWM generator gan, but for nstance for the PCMC converters t s 1/ R s, where R s s the current sensng resstor. Because Ga mght be dfferent for certan converters (.e. control modes), t s named as control gan ). The loop gan can be used to study the stablty and performance of the converter. It should be noted that the controller desgn and also the loop gan are strongly affected by the behavor of G co. ˆo Zo - o ˆo To - o uˆn Go - o uˆo uˆn Yn - o ˆn Open-loop uˆco ĉ G co G a G se Open-loop uˆco ĉ G c G a Gse uˆo Closed-loop G cc uˆr Closed-loop G cc uˆr a) b) Fg Control-block dagrams for voltage-output converter at closed loop: a) output dynamcs and b) nput dynamcs. The control-block dagrams are effcent for computng the dynamcal closed-loop profle consstng of the open-loop parameters. Accordng to Fg. 2.5 a) and [18], the closed-loop output voltage u ˆo can be gven by Goo Zoo ˆ ˆ ˆ GccGaGco u u uˆ 1G G G G 1G G G G 1G G G G o n o r se cc a co se cc a co se cc a co (2.6) 21

39 Chapter 2 Dynamcal Profle Usng the defnton for the loop gan n (2.5) yelds Goo Zoo ˆ 1 VO uˆ uˆ uˆ 1 1 G 1 o n o r VO VO se VO (2.7) Accordng to Fg. 2.5 b), the closed-loop nput current ˆn can be gven by ˆ Y uˆ T ˆ G G G G uˆ G G G G uˆ (2.8) n no n oo o se cc a c o cc a c c r Substtutng u ˆo n (2.8) wth (2.6) and applyng the defnton for the loop gan n (2.5) yelds G ˆ o ogc VO Zo ogc VO ˆ ˆ n Yno un To o o Gco 1VO Gco 1VO Gc VO uˆ r GG 1 se co VO (2.9) The closed-loop parameters are shown separately n (2.1) - (2.15), where the subscrpt c refers to the closed loop. As the voltage reference u r (.e. uˆr ) s typcally constant n sngle-loop converters (whch are consdered n ths thess) the closed-loop parameter set reduces to consst only of (2.1) - (2.13). Y nc GooGc VO Yno G 1 co VO (2.1) T oc ZooGc VO Too G 1 co VO (2.11) Go o G oc 1 VO (2.12) Z Z oc 1 o o VO (2.13) Gc VO G c c GG 1 se co VO (2.14) 22

40 Chapter 2 Dynamcal Profle G 1 VO G 1 co c se VO (2.15) It s obvous that the closed-loop parameters consst manly of the correspondng open-loop parameters. Therefore, the controller desgn cannot much reduce any senstvty for dfferent loads or supply systems. Consequently, the partcular nterest s n the open-loop parameters oad and Supply Interacton Formalsms The mportance of the dynamcal profle s revealed, when analyzng the effects of the load and source subsystems. Every practcal converter s a part of an nterconnected system; the dynamcs of a stand-alone converter may be affected by external mpedances e.g. EMI flters, cablng nductances, addtonal capactors and closedloop nput and output mpedances of the other converters n the system. The openloop g-parameter set provdes the bass for analyzng the nteractons. However, the supply nteracton formalsm ntroduces two specal admttance parameters that have to be consdered also n the analyss. The two-port model of the nterconnected system s shown n Fg. 2.6, where the load and supply systems are modeled as external mpedances (.e. the load as Z and the supply as Z S ). The load nteractons can be found by computng ˆo at the presence of the mpedancetype load Z from Fg. 2.6 by applyng basc crcut theory, yeldng G uˆ Z ˆj G cˆ Z Z ˆ o o n o co o oo (2.16) The load-affected open-loop transfer functons can be found by replacng ˆo n (2.4) by means of (2.16) gvng a load-affected transfer functon matrx as GooToo ZToo GcoToo Yno Gc ˆn ˆ Z Z u oo Z Zoo Z Z oo n G ˆj oo Zoo Gco o uˆ o Zoo Zoo Z oo cˆ Z Z Z (2.17) 23

41 Chapter 2 Dynamcal Profle The two-port model of the nterconnected system n closed-loop s smlar to the open-loop system shown n Fg Only the subscrpt o, whch refers to open loop s changed to c, referrng to closed loop. Therefore, the correspondng load-affected set at closed loop assumng that the voltage reference u r (.e. uˆr ) s constant can be gven by Go ctoc ZTo c Ync ˆ ZZo c ZZ oc uˆ n n Go c Zo c uˆ ˆj o o Zo c Z oc 1 1 Z Z (2.18) ˆn ˆo uˆns + _ Z s + uˆn _ Yn - o T ˆ o- o o Gcˆ c + _ + _ G ˆ coc G Zo - o uˆ o- o n + uˆo _ Z ˆo j ĉ Fg Two-port model of voltage-output converter wth load and supply subsystems. It s apparent that the load nteractons on the output dynamcs are drectly reflected va the open-loop output mpedance. Accordng to (2.17) and (2.18), the nternal output dynamcs would stay ntact f the open-loop output mpedance s small. The load nteractons on the nput dynamcs are not as straghtforward. The output mpedance clearly has an effect on the nternal transfer functons, but the open-loop forward (.e. Go o ) and reverse (.e. T o o ) transfer functons have to be consdered too. Accordng to (2.17), a small open-loop output mpedance would actually result only n ntact reverse transfer functons, because ZT Z Z oo oo Z o o T oo (2.19) 24

42 Chapter 2 Dynamcal Profle The nput admttance stays ntact f (.e. Y no G Z oo oo T Z oo Y Goo and/or Too no ). Go o or To o s close to zero Consequently, a zero T o o would make G c nsenstve to load nteractons (.e. G c G T Z co oo T o o Zoo G c ). The load-affected loop gan VO s a combnaton of the load affected control-tooutput transfer functon n (2.17) and the nternal loop gan n (2.5), yeldng VO VO Z 1 Z oo (2.2) Accordng to (2.2), the load nteractons n the loop gan are reflected va the nomnal open-loop output mpedance Zo o. The supply or source nteractons on the converter dynamcs can be found by computng u ˆn at the presence of the mpedance-type source yelds Z S from Fg. 2.6, whch uˆ n uˆ ˆ ˆ ns ZsTooo ZsGcc 1 ZY s no (2.21) The source-affected open-loop transfer functons can be found by replacng u ˆn n (2.4) by means of (2.21) gvng a source-affected transfer functon matrx as Yno To o Gc uˆ ns ˆ 1ZY s n o 1 s n o 1 n ZY ZY s no ˆ o uˆ Go o 1 ZsYn sc 1 Z o sy n Zoo Gco cˆ 1 ZY s no 1 ZY s no 1 ZY s no (2.22) Because the two-port model of the nterconnected system n closed-loop s smlar to the open-loop system shown n Fg. 2.6 (only the subscrpt o, whch refers to open loop s changed to c, referrng to closed loop), the correspondng source-affected set 25

43 Chapter 2 Dynamcal Profle at closed loop, assumng that the voltage reference u r (.e. uˆr ) s constant can be gven by Yn c To c ˆ 1ZY s n c 1 s n c ˆ n ZY uns uˆ G 1 ˆ o o c ZsY nsc o Z oc 1 ZY s nc 1 ZY s nc (2.23) The specal admttances Yn and Yn sc are defned n (2.24) and (2.25), respectvely. They are the same at open and closed loop. Y n s the nput admttance n a specal condton, where both u ˆo and ˆo are zero. Equaton (2.24) can be obtaned by lettng u ˆo and ˆo zero n (2.1) and (2.2), and then solvng ĉ from (2.2) and replacng t n (2.1) n order to compute ˆ n / uˆ n Yn. Yn s also known as an deal or nfntebandwdth nput admttance because of the closed-loop nput admttance defned n (2.1). Therefore, t s symbolcally the same for a converter wth a certan topology regardless of the conducton and control modes as well as load [P5], [43] and [52]. For a buck converter t also s physcally the same (.e. I / U [P5]). In other converters the physcal correspondence mght be lost, because certan parameters can be dependent on the value of the actual crcut elements [52]. Y nsc s known as the short-crcut nput admttance and beng dependent on the operaton and control modes [P2] and [43]. In a short-crcuted converter uˆo s zero. Therefore, Yn sc can be computed by frst lettng u ˆo to zero n (2.2), then solvng (2.2) for ˆo and replacng t n (2.1) n order to compute ˆ n / uˆ n Yn sc. n n Y n G G oo c Yn o (2.24) Gco Y nsc G T oo jo Yno (2.25) Zoo Accordng to (2.22) and (2.23), the supply nteractons on the nput dynamcs are reflected va the open-loop nput admttance (.e. Yn o ). Ths apples also to the openloop forward transfer functon (.e. Go o ). However, on the open- and closed-loop output mpedances the supply nteractons are reflected va the open-loop nput 26

44 Chapter 2 Dynamcal Profle admttance (.e. Yn o ) and the short crcut admttance Yn sc. On the control-to-output transfer functon (.e. G co ) the supply nteractons are reflected va the open-loop nput admttance and the deal nput admttance Yn. The source affected loop gan can be presented as 1 ZY 1 ZY S s n VO s no VO (2.26) Obvously, the source nteractons n the loop gan are also reflected va the nomnal open-loop nput admttance Y n o and the deal nput admttance Yn. In order to have ntact output mpedance and loop gan, t s obvous (see (2.24) and (2.25)) that Go o must be zero. Ths would make Y n and Y n sc to be the same as Yn o and Y n c even f Yn o or Yn c exhbts a resonant behavor. It s mportant to note that the load may also change the nput-port parameters and hence the supply nteractons by changng the nput mpedances [P11]. It s also evdent that the source system may have effect on the load nteractons by changng e.g. the output mpedances n (2.22) (va Yn o and Yn sc ) and, hence n (2.17). It should also be noted that t s not always necessary to analyze all the transfer functons of the profle. The four most meanngful and mportant transfer functons to be analyzed are Y n o, Zo o, Go o and G co. These four parameters are the key elements reflectng the load and source nteractons as can be concluded e.g. from (2.17) and (2.22). If a complete understandng and characterzaton of a converter s needed, all the transfer functons n the dynamcal profle are worth analyzng. 2.2 Internal and Input-Output Stablty An nterconnected system consstng of two subsystems S and s shown n Fg The nput, output and ntermedate varables are denoted as ( u ˆ ˆn 1, o 2), ( ˆ n1, u ˆo2) and ( zz),, respectvely. Accordng to Fg. 2.7 the ntermedate parameters ( zz), also represent two physcal parameters, whch are the same by defnton (.e. ˆ ˆ o1 n2 z and uˆ 1 uˆ 2 z ). The nternal and nput-output stabltes [11] can be analyzed by o n 27

45 Chapter 2 Dynamcal Profle developng the mappngs from the nput varables to the ntermedate and output varables, respectvely. [P2], [P8], [84] and [85] ˆn1 + u ˆn1 _ S ( z) ˆo1 ˆn2 ˆo 2 uˆo1 + ( z ) _ uˆn2 + u ˆo2 _ Fg Interconnected (cascaded) system. The cascaded system llustrated n Fg. 2.7 can be represented n a matrx form as ˆ ˆ n1 S11 S12 un1 z S21 S 22 z z z uˆ ˆ o o2 (2.27) The subsystems S and can be both at open or closed loop. For the stablty to exst, all the transfer functons n the mappng have to be stable [P2] and [11]. The mappng from the nput to ntermedate varables can be gven by S z 1S ˆ S22 11 un1 z S ˆ 21 S22 12 o2 1 S S22 11 (2.28) and the mappng from the nput to output varables can be expressed by S S S S ˆ 1S S22 ˆ n 11 un1 uˆ S ˆ o S o2 22 1S2211 1S22 11 (2.29) Accordng to system theory, f no rght half plane (RHP) pole-zero cancellatons occur n the transfer functon S 22 11, and, f one of the four transfer functons n (2.28) and (2.29) s stable, the others are also stable [11] and [85]. Ths means that the 28

46 Chapter 2 Dynamcal Profle stablty depends on the stablty of 1/(1 S2211), correspondng to apply the Nyqust stablty crteron [59] to S2211. Accordng to (2.27), the transfer functons S 22 and 11 are the correspondng output mpedance and nput admttance of the subsystems S and (.e. S 22 and 1 11 n n Z os Y Z ). The stablty analyss concretzes to an mpedance rato Z / Z os n, whch s also known as the mnor-loop gan [3]. It s obvous that nstablty occurs f ZoS Zn. Ths relaton can be used to derve safe load and source profles for a gven converter to guarantee stablty. For nstance, f the subsystem S s the converter and the subsystem s a load system, the correspondng mnor-loop gan s Zo c/ Z and the stablty s assured f Z Z oc and 18. Consequently, f the subsystem S s the supply system (e.g. an Z Zo c EMI flter) and the subsystem s the converter, the correspondng mnor-loop gan s ZS / Zn c and the stablty s assured f ZS Zn c and 18 ZS Zn. It s c worth remndng that the presented nternal and nput-output stablty methods are sutable only for ensurng the stablty as dscussed n Secton 1.4. The true GM and PM should be checked from the correspondng load (2.2) or source (2.26) affected loop gans. Actually, the margns assocated to the mnor-loop gan should be ncreased at the vcnty of the converter crossover frequency n order to avod performance degradaton compared to the correspondng margns of the voltage-loop gan as dscussed n [P3] and [P7]. However, f Go o of the converter s zero or very small, the loop gan would stay ntact even f the converter s unstable. Therefore, t s well justfed to use the nternal and nput-output stablty method, and sometmes t mght also be the only method to verfy the stablty of the converter [P1] and [86]. 2.3 Dscusson The ntenton of ths chapter was to ntroduce the dynamcal profle of a swtchedmode DC-DC converter. It was proposed that the profle conssts of certan transfer functons/parameters at open loop (.e. (2.3)) or closed loop (.e. (2.4)). It was also observed that the supply nteracton formalsm ntroduces two specal admttances (.e. Yn and Y n sc ), whch should be ncluded n the dynamcal profle. Actually, wthout knowng the load and supply nteracton formalsms the profle s useless. The nteracton formalsms evoke the usefulness of the profle by showng explctly the open-loop parameters, whch reflect the load and supply nteractons. It should be 29

47 Chapter 2 Dynamcal Profle emphaszed that the nteractons are always reflected va the open-loop parameters, even n the closed-loop nteractons as can be seen after replacng the correspondng defnton n (2.18) and (2.23) wth (2.1) - (2.15). However, t was proposed e.g. n [38], [4], [42], [5] and [55] that the nteractons and performance of the converter loop gan can be studed va the closed-loop output and nput mpedances. The prevalng method to nclude the closed-loop mpedances n the equatons for loadand source affected loop gans can be argued for beng false, because t loses the crucal nformaton of the open-loop mpedances. The closed-loop mpedances only act as a stablty boundary, as wll be shown later. Zn - c Yn - sc, Y n- o, Y n- Zo - c u n + - Subsystem Subsystem 2 u o1 uoc, u o Converter Z j out Zo 1 - c Z o- o, VO Zn2 - c Fg Interconnected system showng the parameters needed for load and supply nteracton analyses. Fg. 2.8 shows an nterconnected system wth three subsystems connected n seres [P5]. The subsystem n the mddle s assumed to be a DC-DC converter and the Subsystems 1 (S1) and 2 (S2) can be composed of e.g. other converters, flters, cablngs, etc. The drecton of the arrows pontng out of or nto the block Converter defnes the parameter needed from other systems n order to study the nteractons. For nstance, S1 needs the Zn c of the converter, because t represents the load mpedance Z for S1. Consequently, Zo 1 c represents the supply mpedance Z S for the converter. The curved lnes nsde the Converter-block defne the nternal parameters requred for the nteracton analyss. Accordng to Fg. 2.8, t s obvous that f a complete understandng of the converter dynamcal propertes s desred, the whole parameter set of open-loop transfer functons (2.3) has to be derved. Ths would also enable the dervaton of the two specal admttances,.e. Y n and Y n sc. Sometmes t s not feasble to e.g. measure all the open-loop g-parameters. However, at least Zo o, Yn o, Go o and G co should 3

48 Chapter 2 Dynamcal Profle be measurable or they should be derved n order to analyze the nteractons. Knowng the open-loop output mpedance ( ) enables the analyss of the load nteractons. Zo o Dervaton of open-loop nput admttance ( Y n o ) partally facltates the analyss of the supply nteractons, but the behavor of the forward transfer functon ( G ) would show the attenuaton propertes of the converter for the nput-sde nteracton (.e. small Go o mples a good attenuaton and large Go o a magnfyng effect for the supply nteractons. The behavor of G co provdes nformaton of the control loop propertes,.e. what type of controller should be used. Nevertheless, the analyss should always be performed at least at the operatng pont whch s the most crtcal to the converter dynamcs. o o The dscusson of the dynamcal profle n ths chapter has been carred out at a very abstract level. Therefore, the explct evdence that verfes the exstence of the profle needs to be provded. The next two chapters wll concentrate on that. However, the abstract dscusson actually means that we have not consdered the topology, control method or conducton mode n the profle. As a consequence, the ntroduced dynamcal profle s generally applcable or n other words, canoncal. As the term canoncal means somethng that s generally or unversally applcable, the prevous canoncal models ntroduced e.g. by Mddlebrook n [4] and [83] lack for beng truly canoncal or general, because they are proposed to be vald only n a contnuous conducton mode (CCM) or dscontnuous conducton mode (DCM) under drectduty-rato or voltage-mode control and they use a resstve load. The canoncal model ntroduced n ths chapter s vald only for the voltage-output converters (.e. converters, where the output voltage s regulated). The canoncal model of the current-output converter s ntroduced n Chapter 5. 31

49 3 Dynamcal Revew Ths chapter starts wth an ntroducton to the basc modelng of a swtched-mode converter. A modelng example of a voltage-output buck converter s gven. Frst, the steady state operaton s brefly analyzed and then the small-sgnal g-parameter set s derved by applyng the popular SSA method. Dynamcal revew s done for VMC- CCM, VMC-CCM, PCMC and PCMC-OCF converters and t s explctly shown how the control method or operaton mode can have a sgnfcant effect on the dynamcal profle, and hence, on the performance and stablty. In Secton 3.3, the capablty of the dynamcal profle s revealed by showng a few analytcal examples of the nteracton analyss based on the nteracton formalsms derved n Chapter 2. Secton 3.4 dscusses the effect of usng a resstve load as the ntal load n the modelng and nteracton analyss. 3.1 Modelng of Swtched-Mode Converter As t was dscussed n Secton 1.3, the modelng can be done n varous ways. However, the basc steady-state modelng s qute straghtforward, ncludng the applcaton of the nductor volt-second and capactor amp-second balances. The most convenent way of computng the small-sgnal model s to apply the SSA method. The classcal SSA method ntroduced n [4] faled to produce accurate small-sgnal models for converters operatng n DCM. The method presented n [44] ntroduced a consstent formalsm for modelng the small-sgnal behavor of a converter regardless of the operaton mode. Generally, n the scope of ths thess, the ntenton of the modelng s to develop the analytcal dynamcal profle of the converter and, therefore, the chosen modelng method s rrelevant as long as the model s accurate. 32

50 Chapter 3 Dynamcal Revew Steady-State Operaton Under steady-state analyss, the small AC rpple supermposed to the DC components s assumed to be zero. The steady-state analyss gves mportant nformaton of the converter operaton, but the dynamcal propertes are not revealed. The voltages and currents of the converter can be computed by applyng the nductor volt-second and capactor amp-second balances. The net change of the nductor current can be expressed as T s 1 ( T ) () u dt (3.1) s Under steady-state condtons the net change of the averaged nductor current over one swtchng perod T s s zero,.e. the ntal value () and the fnal value ( T ) are the same. Hence, accordng to the nductor volt-second balance: s T s udt (3.2) Smlarly, the net change of the capactor voltage over one swtchng perod T s can be expressed as T s 1 u ( T ) u () dt (3.3) C s C C C Under steady-state condtons the net change of the averaged capactor voltage over one swtchng perod T s s zero,.e. the ntal value u C () and the fnal value u ( T ) are the same. Hence, accordng to the capactor amp-second balance: C s uc T s dt (3.4) C A buck converter wth the relevant parastc elements from the modelng perspectve s llustrated n Fg, 3.1. The equvalent swtch on- and off-tme sub-crcuts are shown n Fgs. 3.2 and 3.3, respectvely. 33

51 Chapter 3 Dynamcal Revew n Q r ds ( on ) r + un + _ uo t on - U + D C u C r d + - u + r c - - o Fg Buck converter wth parastc elements. n r ds ( on ) + - u r + r c un + _ C + uo o C u C - - Fg On-tme crcut structure of buck converter. n r + un r d + - u + _ uo - C U + D C u C + r c - - o Fg Off-tme crcut structure of buck converter. Accordng to Fg. 3.2, the on-tme nductor voltage and capactor current can be expressed as u u u ( r r ), on n o ds( on) Con, o (3.5) 34

52 Chapter 3 Dynamcal Revew Accordng to Fg. 3.3, the off-tme nductor voltage and capactor current can be expressed as u u U ( r r ) off, o D ds( on) Coff, o (3.6) Averagng the on- and off-tme equatons (3.5) and (3.6) over one swtchng perod T s by multplyng (3.5) wth the duty-rato D and (3.6) wth ts complement D (.e. 1 D ) and then summng them together yelds U DU U DU ( Dr Dr r ) I n o D ds( on) d I I I C o (3.7) Note that n the above equaton, the captal letters denote varables, whch contan only the DC-component (.e. the large-sgnal component wthout the rpple nformaton). Applyng the nductor volt-second and capactor amp-second balances to (3.7) yelds DU U DU ( Dr Dr r ) I I n o D ds( on) d I o (3.8) From the above equaton t s easy to fnd the converson rato M( D) U / U by lettng the parastes to zero. Ths procedure gves M( D) D for the buck converter. The same procedure presented n ths secton can be appled to any converter operatng n the CCM. In the DCM, a slghtly dfferent procedure must be appled as t s demonstrated e.g. n [44]. o n State-Space Averagng The varable-structure nature of the swtched-mode converters produces non-lnearty, whch may make the analyss and modelng dffcult. However, the correspondng systems dependng on the status of the semconductor swtches are typcally lnear. The SSA method has been wdely adopted to capture the dynamcs of the swtchedmode converters by averagng the converter behavor over one swtchng perod and then lnearzng t at the desred operaton pont to obtan a lnear model [18]. 35

53 Chapter 3 Dynamcal Revew The general expresson for the state space s dx() t Ax() t Bu() t dt y() t Cx() t Du() t (3.9) where x ( t) s the state vector, u ( t) the nput vector and y ( t) the output vector. For the voltage-output converter the state, nput and output varables are typcally chosen as un n x( t), ( t) o, ( t) u u y C u (3.1) o c The buck converter operatng n the CCM and shown n Fg. 3.1 s used as a modelng example. The control method s the tradtonal VMC. The constructon of the statespace representaton starts wth solvng the equatons for u,, C n and u o durng both sub-cycles. Ths yelds for the on-tme from Fg 3.2: u u ( r r ) u n ds( on) o C o n u u r o C C C (3.11) and for the off tme from Fg. 3.3: u ( r r ) u U d o D C o n u u r o C C C (3.12) The state space formalsm requres the dervates of the current (.e. ) and voltage (.e. u C ) of the storage elements. Recallng that rearrangng (3.11) and (3.12) yelds for the on-tme: u d dt du C and dt C, C 36

54 Chapter 3 Dynamcal Revew d ( rc r rds( on) ) uc un rc dt duc o dt C C n u r u r o C C C o o (3.13) and for the off tme: d ( r rd rc) uc rc U o dt duc o dt C C n u r u r o C C C o D (3.14) The averaged state-space equatons can be computed by multplyng the on-tme equatons wth d and the off-tme equatons wth d, addng them together and applyng d d 1 f applcable. The descrbed procedure above yelds: d ( r rc drds( on) dr d ) uc dun rco du dt duc o dt C C d n u r u r o C C C o D (3.15) The obtaned averaged state space s nonlnear due to products of two varables. The state space can be lnearzed by computng the Jacoban matrx of the functon y f(, t x), where x contans the state and nput varables, respectvely. The Jacoban matrx nvolves the partal dervatves of the functon y f( t, x) n respect to all the other varables. The lnearzed functon yˆ f( t, xˆ) can be computed from 37

55 Chapter 3 Dynamcal Revew y y yˆ xˆ1... xˆ x x 1 n xˆ 1. f f yˆ (, t X)... (, t X). x x n 1 n. xˆ n (3.16) Where X contans the steady-state values of the correspondng value at the gven operatng pont. The lnearzed small-sgnal state space obtaned from applyng the partal dervaton s dˆ ( ) ˆ ˆ rc re uc D rc ˆ ˆ U E ˆ un o d dt duˆ ˆ ˆ C o dt C C ˆ Dˆ I dˆ n uˆ r ˆ uˆ r ˆ o C C C o (3.17) where r r r Dr D E ds( on) d U U U ( r r ) I E n D d ds( on) (3.18) The hat over the varables represents the perturbed value around the correspondng steady-state value (.e. the small-sgnal component). The steady-state operatng pont can be obtaned from (3.15) by lettng the dervates equal to zero and replacng the varables wth ther steady-state values, yeldng ( r r ) I U DU r I U D I I n I o DI C E C n C o E U r I U r I o C C C o (3.19) The dervaton of the steady-state duty rato D from (3.19), when excludng the parastes, yelds the same as (3.8). 38

56 Chapter 3 Dynamcal Revew The output voltage n the lnearzed small-sgnal state space (.e. (3.17)) s convenent duˆ C to replace wth uˆ ˆ o uc rcc, because the nterest s n frequency doman dt representaton, whch requres the use of aplace transformaton. The resultng fnal state-space equatons are dˆ ( ) ˆ ˆ rc re uc D rc ˆ ˆ U E ˆ un o d dt duˆ ˆ ˆ C o dt C C ˆ Dˆ I dˆ n duˆ uˆ ˆ o uc rcc dt C (3.2) Equaton (3.2) can be equally represented n the typcal state-space formalsm as dxˆ( t) Axˆ() t Buˆ() t dt yˆ() t Cxˆ() t Duˆ() t (3.21) where uˆ n ˆ ˆ ˆ, ˆ ˆ n x u, ˆ o, uˆ y (3.22) ˆ C uo dˆ The matrces A, B, C and D are defned as follows rc re 1 D rc U E A, B, 1 1 C C D I C, D duˆ C 1 rc C dt (3.23) 39

57 Chapter 3 Dynamcal Revew The derved tme-doman representaton can be converted to frequency doman by applyng aplace transformaton: sx(s) AX(s) BU(s) Y(s) CX(s) DU(s) (3.24) Applyng basc matrx algebra the system can be solved, yeldng ( s ) 1 X(s) I A BU(s) 1 Y(s) ( C( si A) B D) U(s) (3.25) Accordng to (3.25), solvng ( s ) varables to the state varables and solvng 1 I A B gves the transfer functons from the nput 1 C( si A) B D yelds the transfer functons from the nput varables to the output varables. The nput-to-output descrpton s actually the g-parameter representaton of the converter dynamcs and, hence, forms the bass of the dynamcal profle automatcally. The g-parameter set of the VMC buck converter, after solvng 1 C( sia) BD s 2 Ds D(1 src C) C D(1 src C ) ( re s)(1 src C ) Yn o T oo C C G 2 re rc 1 oo Z oo s s C (3.26) sdu E UE(1 srcc) Gc C I G 2 re rc 1 co s s C (3.27) The converter dynamcs can be analytcally studed at the desred operatng pont by means of the g-parameter set presented n (3.26) - (3.27) by usng computatonal software such as Matlab, Mable and Mathematca. The above presented modelng example was based on the voltage-output VMC buck converter, but the basc modelng dea s the same for the other topologes as well. Other control 4

58 Chapter 3 Dynamcal Revew methods (e.g. PCMC) may requre a slghtly dfferent modelng approach, but as long as the dynamcal profle, and hence, the model s correct t s reasonable to say that the dynamcal analyss wll produce accurate results. 3.2 Effect of Control Prncple Choosng the converter topology for a certan applcaton s typcally a trval case. Bascally, the power supply desgner only needs to know n whch range the nput and output voltages are and then choose the topology. For nstance, the system shown n Fg. 1.1 clearly mples the need of buck converters or ts dervates. Consequently, the low voltages produced e.g. by the fuel cells and solar panels have to be ncreased ndcatng the need of boost converters or ts dervates. However, choosng the control method s not trval from the dynamcal vewpont. The control method can sgnfcantly affect the converter performance and stablty. Typcally, dfferent control prncples have both pros and cons, whch have to be taken nto account, when desgnng the converter. Studyng the parameters of the dynamcal profle at the desred operatng pont wll reveal the dynamcal propertes and ts senstvtes to the load and supply nteractons. Consequently, f the load and supply systems are known, the nteracton analyss reveals the sutablty of the chosen control method for the applcaton. As an example, a buck converter wth three dfferent control methods, VMC, PCMC and PCMC wth OCF are studed. In the case of VMC, both CCM and DCM are dscussed. un + _ Z S fs = 1kHz r ds ( on ) 4m U D.3V r d 55m r.6m r C 33m C F o + u o 1V - Z j o VMC PCMC PCMC-OCF PWM R s1 Control system u o R s2 o u R R s2 1V Fg Voltage-output buck converter wth VMC, PCMC and PCMC-OCF control (CCM: = 15 μh, DCM: = 5 μh). 41

59 Chapter 3 Dynamcal Revew The buck converter used n the analyss s shown n Fg The VMC forms the basc control prncple takng the output voltage ( u o ) as a feedback sgnal and comparng the error voltage of the controller to the sawtooth ramp, makng the dutyrato (d) as an ndependent varable. In PCMC, the sawtooth ramp s replaced wth the sgnal derved from the nductor current, whch s typcally sensed va a current transformer and then converted nto a voltage sgnal by means of a resstor R s1. In PCMC-OCF the output current o s feedforwarded (.e. added to the control sgnal) by usng a current sensng resstor R s2 [7] and [87]. The VMC control system s qute smple, ncludng only an error amplfer producng the control sgnal for the PWM generaton. In PCMC, the up-slope of the nductor current and the control sgnal s compared. The basc PCMC s prone to operate n subharmonc mode at the duty ratos exceedng.5 and, therefore, an artfcal compensaton ramp M c s typcally summed to the control sgnal to extend the dutyrato range [P1]. The PCMC has become a popular control method manly due to ts hgh nput-nose attenuaton and a feature to lmt the swtch current pulse-by-pulse [P1]. However, the PCMC buck converter has large open-loop output mpedance, whch makes the converter prone to load nteractons. To overcome ths dsadvantage, the output current can be taken as a feedforward sgnal and added to the control sgnal formng the PCMC-OCF. Theoretcally, ths can make converter nsenstve to both the supply and load nteractons [87]. The open-loop parameters are convenent to analyze frst. The open-loop here means that the feedback from the regulated sgnal s dsconnected. In the voltage-output converters, ths sgnal s the output voltage and, consequently, n the current-output converters the regulated sgnal s the output current. In other words, the control sgnal s kept constant at a certan value that produces a duty rato, whch, n turn sets the output as desred at the steady-state condtons. The orgnal SSA technque [4] results n reduced order models for converters operatng n DCM. The full-order model and g-parameter set for the VMC-DCM converter can be obtaned by applyng a generalzed modelng method descrbed n [44]. Before startng the modelng, t should be notced that the tme-averaged nductor current s a contnuous functon of tme wthn a swtchng cycle regardless of the operaton mode, as depcted n Fg. 3.5 a) and b). 42

60 m 1 - m 2 - m m 2 1 t t t t on off 1 t t t on off1 off 2 T s T s a) b) Fg Inductor current waveforms. a) CCM b) DCM Accordng to Fg. 3.5 a), the average nductor current durng the on-tme and on off-tme may be expressed as off on off 1 t on t ton t on t off 1 off 1 t off 1 (3.28) For developng the averaged state space, the dervatves of the tme-averaged state varables (.e. nductor current and capactor voltage) have to be defned. Accordng to Fg. 3.5 a) or b), the dervatve of the averaged nductor current s d t t m dt T T m on off (3.29) s s where m 1 and m 2 are the correspondng up and down slopes of the nductor current, respectvely. It was notced e.g. n [44] that the parastc elements do not have consderable effect on the converter dynamcs n DCM. Therefore, only the ESR of the output capactor s consdered hereafter. For a buck converter m 1 and m 2 can be computed from Fgs 3.2 and 3.3, yeldng 43

61 Chapter 3 Dynamcal Revew m m 1 2 u u n o u o (3.3) Ths thess concentrates only on the fxed-frequency operaton modes. Therefore, the cycle tme T s s constant and the dynamcs assocated wth the on tme t on and off tmes t off 1 and toff 2 may be equally captured by usng the duty rato d ton / Ts and ts complements d / 1 t 1 T and d 2 t 2 / T. Typcally, the complement of the duty off s off rato s denoted by d (CCM), but here subscrpts 1 and 2 are used n order to avod confuson between CCM and DCM. It should be noted that n CCM d d1 1, but n DCM d d1 1. Accordng to these assumptons, (3.29) can be represented as s d dt dm d m (3.31) The dervatve of the tme-averaged capactor voltage can be computed e.g. from Fg. 3.2, yeldng d uc Q C o (3.32) dt T C C s The tme-averaged nput current n for a buck converter equals the on-tme nductor current. Therefore, the frst equaton n (3.28) apples. In fxed- on frequency operaton modes the nput current n s n d d d 1 (3.33) Accordng to the tradtonal SSA method (see Secton 3.1), the tme-averaged output voltage u o can be presented as d u u u r C dt C o C C (3.34) 44

62 Chapter 3 Dynamcal Revew Equatons (3.31)-(3.33) form the general averaged state space representaton for the buck converter. The small-sgnal model for VMC-CCM converter wthout the parastc elements (except the output capactor ESR) can be computed by followng the same procedure presented n Secton 3.1 (startng from (3.15)). When consderng the general state-space equatons, the only unknown varable s the length of the offtme1 ( t off 1 ). The dynamcs assocated to t off 1 can be recovered by computng ts relaton to accordng to the waveforms of Fg. 3.5 b), yeldng T s 1 1 () t dt mt t t (3.35) 1 on on off 1 Ts 2T s whch under fxed-frequency operaton equals to 1 md 1 d d1ts (3.36) 2 Solvng the above equaton for d1 yelds d 1 2 d (3.37) dt m s 1 In order to fnd the fxed-frequency averaged models n DCM, d 1 has to be replaced wth (3.37) n the general averaged state-space equatons (3.31)-(3.33). Therefore, the non-lnear averaged state-space equatons for a VMC-DCM buck converter can be wrtten as d d un 2 uc dt dt u u d uc o dt C C 2 dts un uc n 2 d u uo uc rcc dt s n C C (3.38) The steady-state operatng pont can be solved from (3.38) by settng the dervatves to zero. Ths yelds 45

63 Chapter 3 Dynamcal Revew I I U n o I o MI U c K D M 1 M o (3.39) where M Uo / Un and the dmensonless value K 2 / ReqTs. The equvalent load resstor R s defned as R U / I. These notatons are defned n order to smplfy eq the fnal equatons. eq o o The lnearzed small-sgnal state space wthout the losses can be computed from (3.38) by applyng (3.16). Applyng the defntons for steady-state n (3.39) and the notatons M and K, the lnearzed small-sgnal state space can be wrtten as dˆ R eq K ˆ 1 K ˆ u dt 1 M (1 M ) 1M M(2 M) K 2U n u ˆ ˆ n d (1 M) 1M duˆ ˆ ˆ C o dt C C ˆ 2U 1 uˆ uˆ dˆ 2 2 M M o M n C n Req (1 M) Req (1 M) Req K duˆ uˆ ˆ o uc rcc dt C C (3.4) Equaton (3.4) can be equally represented n the state-space formalsm shown n (3.21). The matrces A, B, C and D for the VMC-DCM converter are defned as follows Req K 1 K 1 M (1 M) 1M A 1 C M(2 M) K 2U (1 M) 1M B 1 C n (3.41) 46

64 Chapter 3 Dynamcal Revew 2 2 M M 2U o 1 M Req (1 M) Req (1 M) Req K C, D duˆ (3.42) C 1 rc C dt Solvng 1 C( sia) BD yelds the transfer functons from the nput varables to the output varables and gves the g-parameter representaton of the converter dynamcs as t was dscussed n Secton 3.1. The g-parameter set of the VMC-DCM buck converter [88], after solvng 1 C( sia) BD s Y G R nodcm oodcm T oodcm Z oodcm M eq 3 2 M M K M Req K s CR 1M CR 1M 1 eq M K s Req 1 src C M 2 M K 1 M 1srC C C 1M 1M C R 2 eq K 1 K 1 s s 1 M C 1 M 1 M eq 2 M 1 M (3.43) 2 2UnM CReq 1 M 2Un 1 src C 2U o 1 M Gcdcm C Req K G codcm R 2 eq K 1 K 1 s s 1 M C 1 M 1 M (3.44) The PCMC s a drect extenson of VMC. Ths means that under fxed-frequency operaton mode and n CCM the basc averaged and small-sgnal state space equatons (.e. (3.15) and (3.17), respectvely) are the same, but the duty rato n PCMC s not anymore ndependent, but dynamcally dependent on the nput and 47

65 Chapter 3 Dynamcal Revew output voltage, control current co as well as other crcut elements [P1]. The dynamcal dependence s commonly known as duty-rato constrants [18], and can be expressed as dˆ F ( ˆ q ˆ quˆ q uˆ ) (3.45) m co c n o o where F m s the duty-rato gan, q c s the nductor-current-feedback gan, q s the nput-voltage-feedforward gan and q o s the output-voltage-feedback gan. The duty rato n PCMC converter s establshed, when the on-tme nductor current reaches the compensated control current co as shown n Fg co - M c m 1 - m 2 t dt s ' dt s T s Fg Duty-rato generaton n PCMC based on the nductor current up-slope. The state varable s the average nductor current equaton determnng the duty-rato can found from Fg. 3.6, yeldng. Therefore, the comparator co M dt (3.46) c s where s the dynamc dstance between the peak nductor current and the average nductor current as shown n Fg. 3.6 and man task s to fnd expresson for M c s the compensaton ramp [P1]. The. In CCM, the tme averaged nductor current s always n the mddle of the rpple band. The tme-varyng averaged nductor current may also be expressed as a frst-order functon of tme wthn the swtchng cycle T s. Therefore, can be expressed as 48

66 Chapter 3 Dynamcal Revew dd ' Ts dm1d' m2t m1 m2 (3.47) 2 can be found by computng the dfference between the nductor current up-slope and the averaged nductor current n detal. As a consequence, n (3.47) at s can be expressed as t dt. Ths s presented n [P1] mdt s 1 s dt d () t T dd u U r r () t T 2 n D d ds( on) s dd m m T s (3.48) In (3.47) and (3.48) d ' denotes the duty-rato complement n fxed-frequency and CCM operaton modes,.e. d' 1 d. m 1 and m 2 correspond to the up- and downslopes of the nductor current ncludng the effect of the parastc elements. After replacng n (3.46) wth (3.48) and lnearzng the result by applyng (3.16), the duty-rato constrants can be expressed as ˆ 1 ˆ ˆ DD ' Ts d co uˆ n D' DUE 2 TsMc 2 (3.49) where the correspondng gans F m, q c, buck converter as follows: q and q o accordng to (3.45) are defned for a F q m c o T 1 s M DD ' Ts q 2 q c 1 D D U 2 E (3.5) 49

67 Chapter 3 Dynamcal Revew The lnearzed small-sgnal state space of the PCMC converter can be computed by replacng the small-sgnal duty-rato ˆd n (3.2) wth ts defnton shown n (3.49). Ths procedure yelds dˆ ( ) ˆ ˆ re rc FmUE uc DFmUEq ˆ u dt rc ˆ FmUE ˆ o co duˆ ˆ ˆ C o dt C C ˆ ( DF I ) ˆ F qi uˆ F I ˆ n m o m o n m o co duˆ uˆ ˆ o uc rcc dt C n (3.51) Equaton (3.51) can be equally represented n the state-space formalsm shown n (3.21). The matrces A, B, C and D for the PCMC (CCM) converter are defned as follows ( re rc FmUE) 1 D FmUEq rc FmUE A, B, 1 1 C C D FmIo FIq m o FI m o C, D duˆ C 1 rc C dt (3.52) The g-parameter set of the PCMC buck converter, after solvng the transfer functons 1 from the nput varables to the output varables (.e. C( sia) BD) s Y G T nopcmc oopcmc Z oopcmc oopcmc DF qu DF I s DF I 1sr C m E m o m o C D F qu 1sr C r F U s1sr C m E C E m E C C s 2 C C re FmUE rc 1 s C 5

68 Chapter 3 Dynamcal Revew FmqI o co pcmc 2 E m E C s s FU m E D FI m o s FU m E1 src C Gc pcmc C FI m o G r F U r 1 C (3.53) (3.54) The compensaton ramp M c s typcally chosen n such a way that a good nputoutput attenuaton s accomplshed. Ths can be acheved by havng Go o (.e. DF qu ) yeldng m E M c DU E DUn 2 2 (3.55) PCMC PCMC ˆo Zo - o ˆo To - o uˆn Go - o uˆo uˆn Yn - o ˆn G co G c ˆco ˆco R s2 G a R s2 G a H () s H () s uˆco uˆco a) b) Fg Block dagrams of the dynamcs of PCMC-OCF converter. a) output dynamcs. b) nput dynamcs. The PCMC-OCF s a drect extenson of the PCMC. The g-parameter set for the PCMC-OCF converter can be obtaned by constructng the correspondng block 51

69 Chapter 3 Dynamcal Revew dagrams of the output and nput dynamcs, whch are shown n Fg. 3.7 [7] and [87]. The g-parameter set can then be computed from the block dagrams, yeldng Y Y noocf nopcmc T T R H G G ooocf opcmc s2 a cpcmc G G G G cocf a cpcmc G ooocf oopcmc Z Z R H G G G ooocf oopcmc s2 a copcmc G G coocf a co (3.56) It s obvous that the parameter set of the PCMC-OCF converter s manly constructed from the PCMC parameters. The control gan n G a s equal to 1/ R s1, where R s1 s the equvalent nductor current sensng resstor. Accordng to (3.56), the outputcurrent feedforward changes only Zo o ocf and T oo ocf leavng the other parameters vrtually ntact [P4], [7] and [87]. Therefore, the hgh nput-nose attenuaton of the PCMC converter s mantaned also n the PCMC-OCF converter. In order to obtan the load nvarance the open-loop output mpedance Zo o ocf should be zero makng the loop gan stay ntact, when the load Z s connected (see (2.2)). Ths can be acheved by lettng Zo o ocf to zero n (3.56) and solvng the output-current-feedback gan H, yeldng H 1 Zoopcmc (3.57) R G G s2 a co pcmc However, the practcal mplementaton of such a gan would be dffcult due to the dependence of the nomnal transfer functons (.e. Zo o pcmcand Gco pcmc ) and, hence, on the operaton pont of the PCMC converter [7]. For that reason, a unty-gan feedforward scheme (.e. H 1) can be used to obtan small open-loop output mpedance [87]. As a consequence, Zo o ocf can be expressed as R Z Z R G G Z G (3.58) s2 ooocf oopcmc s2 a copcmc oopcmc copcmc Rs 1 Substtutng Zo o pcmc and G co pcmc n (3.58) wth ther expressons n (3.53) and (3.54) yelds 52

70 Chapter 3 Dynamcal Revew Z ooocf R s2 re 1 FmUE s 1 srcc R s1 C 2 re FmUE rc 1 s s C (3.59) Generally, the rato of the equvalent current sensng resstors (.e. Rs 2 / R s1) should equal to 1 (the current sensng resstors R s1 and R s2 n Fg. 3.4 are equal to 75 m). Ths would make the numerator n (3.59) to resemble the correspondng numerator of the open-loop output mpedance of the VMC-CCM converter shown n (3.26), 1 R / R F U. Consequently, the open-loop output mpedance because s2 s1 m E Zo o ocf of the PCMC-OCF converter would resemble the open-loop output mpedance of the VMC-CCM converter at lower frequences wthout the resonant behavor. If the rato s dfferent the mpedance would also act dfferently, as t s dscussed n [88]. The open-loop reverse transfer functon T oo ocf of the PCMC-OCF converter can be computed by substtutng T oo pcmc and G c pcmc n (3.58) wth ther expressons n (3.53) and (3.54) yeldng T ooocf R s2 UED D R s2 D R s2 re rc 1 1 R Rs 1 Io FmIO Rs 1 FmIO Rs 1 R C 2 re FmUE rc 1 s s C s2 2 FI m o s s s1 (3.6) The unty rato of the equvalent current sensng resstors (.e. Rs 2 / R s1) do not ntroduce smlar behavor n T oo ocf than n Zo o ocf. Havng Rs 2 Rs 1 would make To o ocf to resemble the correspondng T o o of the VMC-CCM converter at lower frequences, but at hgher frequences the magntude of To o ocf s ncreased due to the addtonal zeros n (3.6) compared to (3.26). The ncreased magntude can boost the load nteractons to the supply sde and back to the load sde accordng to the load-affected nput admttance (2.17) and supply-affected output mpedance (2.22) & 53

71 Chapter 3 Dynamcal Revew (2.25). Ths can be avoded by compensatng the PCMC-OCF converter to have G. o o Matlab wth control system toolbox (CST) can be used to study the dynamcs of the converters presented above. The voltage-output converter shown n Fg. 3.4 s used to evaluate the dynamcs. The g-parameters at desred operatng pont can be found by developng a specfc program (m-fle), whch contans the component values and symbolc representatons of the parameters. The equatons for the load and supply nteractons can also be ncluded n the program. An example lstng of the m-fle code for VMC-CCM converter s shown n Appendx A. Bode-plots and other functons can be used by applyng the CST commands. Smulnk s not used n ths thess for the tme doman analyss, although t can be a valuable tool, when smulatng e.g. the transent responses. The reason for not usng Smulnk or any other smulaton tool s partcularly the nterest n the frequency doman behavor, when consderng the scope of the thess. Internal G co U o = 1 V 4 VMC CCM 5 V Magntude (db) 2 2 5V 2 V PCMC & PCMC OCF 2 V VMC DCM Phase (deg) PCMC & PCMC OCF VMC CCM VMC DCM Frequency (Hz) Fg Internal control-to-output transfer functons of VMC-CCM (sold lne = 5 V, dotted lne = 2 V), VMC-DCM (crcles = 5 V, plus-sgns = 2 V) and PCMC & PCMC- OCF (dashed lne = 5 V, dash-dot lne = 2 V). The nput voltage of the converter llustrated n Fg. 3.4 s assumed to be n the range of 2 V 5 V n the analyses done n ths chapter. The control-to-output transfer functons of the three dfferent control methods are shown n Fg. 3.8 at a low and hgh lne. It s apparent that the magntude varaton as a functon of the nput voltage s dfferent n VMC (both CCM and DCM) and PCMC (PCMC-OCF). Reason for 54

72 Chapter 3 Dynamcal Revew ths can be concluded from the symbolcal representaton of G co. The numerator of the VMC-CCM converter G co n (3.27) shows a strong dependency only on the nput voltage ( U ). Ths mples that the nput voltage varaton wll cause manly a n constant magntude varaton and the poles and zeros are only slghtly moved. Accordng to (3.44), the same dependency on the nput voltage ( U ) s also evdent n the VMC-DCM converter. In the PCMC (and hence, n PCMC-OCF), the nput voltage varaton affects the product of the duty-rato gan F m and n U E, whch, accordng to (3.54) s both n the numerator and denomnator of G co of the PCMC converter. Ths affects the locaton of the poles n the denomnator and produces totally dfferent behavor than observed n the VMC. However, the effects of the nput voltage varaton are observable only at the low and hgh frequences, whch n turn makes the controller desgn more straghtforward, because the nput voltage varaton s not needed to be consdered n the desgn as s the case n the VMC. It s also evdent that wthout compensaton (.e. M c ) the DC-gan (.e. s ) wll be nfnte n the PCMC at the mode lmt D.5 [P1]. Accordng to Fg. 3.8, only the VMC-CCM converter exhbts resonant behavor. The reason for ths can be deduced from the denomnator of the parameter set n (3.26) and (3.27). The second order transfer functon can be expressed as Gs () s 2 n 2 2 s2 n n (3.61) where n s the undamped natural frequency and the dampng factor [59]. The converter wll have resonant behavor f the dampng factor s between and 1 (.e. underdamped case). The coeffcent of the frst-order term n the VMC-CCM denomnator s actually rather small compared to 1/ C (.e. 2 n ) resultng complex conjugate roots and, hence, small. In the case of VMC-DCM, PCMC and PCMC- OCF, the roots are well separated and real. Ths result n 1 and, consequently, the converters are well damped (.e. overdamped case) and no resonant behavor exsts. The ntenton of these short examples was to show that t s possble to draw certan concluson from the converter performance n frequency doman from the symbolcal transfer functon equatons. Ths method apples not only to the G co but also to the other parameters n the dynamcal profle. 55

73 Chapter 3 Dynamcal Revew As t was dscussed earler the controller desgn and the resultng loop gan are based on the behavor of G co. Bascally, the desgn s trval; the poles and zeros of the controller transfer functon are placed n such a way that the desred crossover frequency, phase and gan margns are acheved [89], [18] and [73]. In practce, however, the desgn procedure may nvolve a few teraton steps. The most often used controllers are a proportonal-ntegral (PI) (.e. Type-2) and proportonal-ntegraldervatve (PID) (.e. Type-3). The PI compensator can theoretcally boost the phase up to 9, but sometmes ths s not suffcent and the PID compensator, whch can have phase boost of 18, must be used. For nstance, the phase of G co (see Fg. 3.8) n the VMC-CCM converter tends towards -18 near the resonant frequency, mplyng the need of suffcent phase boost from the compensator, and hence, the PID compensator s typcally used n the VMC-CCM converters to assure stablty and adequate margns. Consequently, n the VMC-DCM and PCMC (PCMC-OCF) converters the phase of G co vares only between and 9 and the PI compensator can be used. oop gans for the VMC-CCM, VMC-DCM, PCMC and PCMC-OCF are plotted n Fg The controllers were desgned n such a way that mnmum of 5 phase margn s acheved and the crossover frequency f c was set near 1 khz. The prevalng understandng s that transent response n tme doman strctly relates to the loop-gan crossover frequency f c [64] and [65]. However, t was demonstrated e.g. n [P2] that the converter open-loop output mpedance s the key factor reflectng the load changes. Actually, the open-loop mpedance sets the lmt for performance changes and the closed-loop mpedance stands for the boundary for the reducton of the crossover frequency f c [P3] and [P7]. The measured transent responses for the three control prncples (.e. VMC-CCM, PCMC and PCMC-OCF) are shown n Fg. 3.1 and the dfferences are obvous. ookng at the open- and closed-loop output mpedances n Fgs and 3.12 reveal the reason; the mpedance of the PCMC- OCF converter s the smallest at f c causng only a small voltage dp. Consequently, the mpedances n the VMC-CCM and PCMC converters are close to equal at f c makng the voltage dp also equal. The longer set-up tme n the PCMC transent response can be addressed to the larger low-frequency mpedance. The open-loop output mpedance of the VMC-DCM converter s also qute large at the lower frequences and, n general, ts behavor s smlar to the PCMC converter. The dfferent phase behavor of the output mpedances mght also reveal senstvtes to certan load. As t was observed n [P3] the VMC-CCM converters are typcally 56

74 Chapter 3 Dynamcal Revew senstve to capactve loads up to the resonant frequency f res, because the phase s. The VMC-DCM and PCMC converters have ths senstvty only at low frequences. Magntude (db) Phase (deg) Internal VO U n = 5 V U o = 1 V 1 PCMC & 8 PCMC OCF 6 VMC CCM VMC DCM VMC DCM VMC CCM 15 PCMC & PCMC OCF Frequency (Hz) Fg Internal loop gans of VMC-CCM (sold lne), VMC-DCM (dash-dot lne) and PCMC & PCMC-OCF (dashed lne) converters at hgh lne 5 V Transent responses to load change from.2 A to 2.5 A (25 ma/μs) VMC CCM ΔU o 169 mv x 1 3 Voltage (V) PCMC ΔU o 178 mv x PCMC OCF 1. ΔU o 43 mv Tme (s) x 1 3 Fg Measured transent responses of VMC-CCM, PCMC and PCMC-OCF converters. The load s changed from.2 A to 2.5 A at the rate of 25 ma/μs. 57

75 Chapter 3 Dynamcal Revew Phase (deg) Magntude (dbω) Internal open loop output mpedance. U n = 5 V 4 PCMC 2 VMC DCM VMC CCM 2 PCMC OCF VMC CCM 5 PCMC OCF VMC DCM 5 PCMC Frequency (Hz) Fg Internal open-loop output mpedances of VMC-CCM (sold lne), VMC-DCM (dash-dot lne), PCMC (dashed lne) and PCMC-OCF (dotted lne) converters at hgh lne 5 V. Magntude (dbω) PCMC Internal closed loop output mpedance. U n = 5 V VMC DCM PCMC OCF 1 VMC CCM Phase (deg) PCMC VMC CCM VMC DCM PCMC OCF Frequency (Hz) Fg Internal closed-loop output mpedances of VMC-CCM (sold lne), VMC-DCM (dash-dot lne) PCMC (dashed lne) and PCMC-OCF (dotted lne) converters at hgh lne 5 V. 58

76 Chapter 3 Dynamcal Revew Magntude (dbω 1 ) PCMC & PCMC OCF VMC DCM Internal open loop nput admttance. U n = 5 V VMC CCM Phase (deg) 1 VMC DCM VMC CCM PCMC & PCMC OCF Frequency (Hz) Fg Internal open-loop nput admttances of VMC-CCM (sold lne), VMC-DCM (dash-dot lne) and PCMC & PCMC-OCF (dashed lne) converters at hgh lne 5 V. Accordng to (2.22), the open-loop nput admttance Y n o s the key parameter reflectng the supply (source) nteractons. However, n Z o o and G co, the short crcut nput admttance Y n sc and deal nput admttance Y n has to be taken nto account as well when studyng the nteractons and senstvtes. The common factor n Y n sc and Y n s the forward transfer functon G o o, and, as was dscussed n Chapter 2 the smaller t s the smaller are the reflected nteractons. The nternal openloop Yn o s shown n Fg for the VMC-CCM, VMC-DCM and PCMC & PCMC-OCF converters. The resonant behavor of the VMC-CCM converter s obvous, makng the converter senstve to supply nteractons near the converter output flter resonant frequency f res, because Yn o and Y n c are not equal. Just by lookng at Yn o one could say that the VMC-DCM and PCMC & PCMC-OCF converters are senstve to supply nteracton because of the large admttance elsewhere than at the vcnty of f res, f compared to the correspondng open-loop nput admttance of the VMC-CCM converter. However, f studyng e.g. the supplyaffected Gco we must study also Go o and/or Yn. Go o of the three dfferent control methods s shown n Fg It s clear that the PCMC and PCMC-OCF converters have substantally smaller Go o than the VMC converters. Accordng to (2.22) the supply nteractons would be mnmzed f Y n o Y n. Ths s acheved f 59

77 Chapter 3 Dynamcal Revew Go o. Yn for buck converter can be expressed as I / U DI / U I / U yeldng about -4 db for converter wth n E E n n Un 5 V, Uo 1 V, I Io 2.5A and D.2. ookng agan at Y n o of the PCMC and PCMC-OCF converters shown n Fg t s evdent that the magntude stays almost constant n the whole frequency range and s about -4 db, yeldng Yno Yn. The same behavor can be observed also n the VMC-DCM converter. However, the condton Y n o Y n s obtaned even f Go o s larger than n the VMC-CCM converter (elsewhere than at the vcnty of the resonant frequency). It was explaned n [88] that the reason for ths s the smaller rghtmost part of Y n (.e. Go ogc / Gco ). The supply nteractons wll be treated n the next subsecton more n detal. The closed-loop forward transfer functon G o c of the VMC-CCM converter s also shown n Fg It s evdent that G o c s much smaller than the correspondng open-loop transfer functon up to the loop gan crossover frequency (.e. 1 khz). Ths s due to the large loop gan, and hence, large denomnator n the defnton of G G /1 ) hdng the G (.e. o c oc oo VO nformaton n G o o. Therefore, the closed-loop forward transfer functon ( Go c ) cannot be used n the analyss of the nput-output attenuaton propertes. Ths emphaszes the use of the open-loop parameter n the dynamcal revew. Internal forward transfer functon. U n = 5 V Magntude (db) 5 1 VMC CCM PCMC & PCMC OCF VMC CCM at closed loop VMC DCM Phase (deg) 1 PCMC & PCMC OCF VMC CCM VMC DCM VMC CCM at closed loop Frequency (Hz) Fg Internal open-loop forward transfer functons of VMC-CCM (sold lne), VMC- DCM (dash-dot lne) and PCMC & PCMC-OCF (dashed lne) converters at hgh lne 5 V. Closed-loop forward transfer functon of VMC-CCM s marked wth dotted lne. 6

78 Chapter 3 Dynamcal Revew Magntude (dbω 1 ) Phase (deg) Internal closed loop nput admttance. U n = 5 V VMC DCM VMC DCM PCMC & PCMC OCF PCMC & PCMC OCF VMC CCM VMC CCM Frequency (Hz) Fg Internal closed-loop nput admttances of VMC-CCM (sold lne), VMC-DCM (dash-dot lne) and PCMC & PCMC-OCF (dashed lne) converters at hgh lne 5 V. It was dscussed n Chapter 2 that the closed-loop nput mpedance sets the lmt for the stablty and crossover frequency reducton. However, accordng to the formalsm of the dynamcal profle, the relaton of the nput current and voltage s presented as admttance and can be easly transformed to mpedance by takng the nverse of the admttance. The closed-loop nput admttance Y n c of the VMC-CCM, VMC-DCM and PCMC & PCMC-OCF converters s shown n Fg The admttances seem to be equal almost throughout the whole frequency range. It s mportant to notce that even f the senstvtes for supply nteractons mght be totally dfferent, the stablty boundares can be the same or very near to equal as shown n Fg Interacton Analyss The nternal dynamcal profle has propertes to defne the nomnal dynamcs of a voltage-output swtched-mode converter havng an deal voltage source at the supply sde and an deal current snk at the load sde. However, a sngle converter s always a part of a larger system, whch typcally alters the nomnal dynamcs. The nteracton formalsm presented n Chapter 2 can be effectvely used when predctng the nteractons. The load and supply sde nteractons were successfully and extensvely studed n [P3], [P4], [P6]-[P8] and [P11]. Therefore, the ntenton of ths secton s to gve a short revew of how to perform the nteracton analyss and then summarze the man results. The converter shown n Fg. 3.4 s equpped wth non-deal load and 61

79 Chapter 3 Dynamcal Revew supply mpedances, Z and Z S, respectvely. The load and supply nteractons are studed separately, meanng that the nternal dynamcal profle s used n both cases oad Interactons The load nteractons are studed by means of a seres connected, resonant-type Ccrcut, the mpedance Z of whch s calculated based on the component values shown n Table 3.1. The nductor s assumed to be deal and the component values are chosen so that the desred resonant frequency f res, and desred load effects are acheved. oad Table 3.1. Component values of the C-load. C r CC, f res, Z1 1.8 mh 2.35 mf 3 m 1 Hz Z2 324 μh 47 μf 1 m 4 Hz Z3 216 μh 47 μf 1 m 5 Hz 4 Internal open loop output mpedances and the load mpedances. U n = 5 V Magntude (dbω) 2 PCMC VMC DCM VMC CCM 2 PCMC OCF Phase (deg) VMC DCM VMC CCM PCMC PCMC OCF Frequency (Hz) Fg Internal open-loop output mpedances of VMC-CCM (sold lne), VMC-DCM (dash-dot lne) and PCMC & PCMC-OCF (dashed lne) converters at hgh lne 5 V and load mpedances Z 1 (crcles), Z (stars) and Z (plus-sgn)

80 Chapter 3 Dynamcal Revew 5 Internal closed loop output mpedances and the load mpedances. U n = 5 V Magntude (dbω) PCMC VMC DCM VMC CCM 5 PCMC OCF Phase (deg) VMC CCM PCMC OCF VMC DCM PCMC Frequency (Hz) Fg Internal closed-loop output mpedances of VMC-CCM (sold lne), VMC-DCM (dash-dot lne) and PCMC & PCMC-OCF (dashed lne) converters at hgh lne 5 V and load mpedances Z (crcles), Z (stars) and Z (plus-sgn) Accordng to the load-nteracton formalsm ntroduced n Chapter 2, the load wll change the nomnal dynamcs f the denomnator 1 Z / Z 1. Obvously, the load nteractons would be mnmzed f Z Zo o, but generally, Z Zo o can be set as a boundary for the performance degradatons. Consequently, the closed-loop output mpedance was found to have relaton to the converter stablty and crossover frequency reducton. The nternal open-loop and closed-loop output mpedances of the three converters are re-plotted n Fgs and 3.17, respectvely, together wth the three dfferent load mpedances. oo et s consder the VMC-CCM converter frst. Accordng to Fgs 3.16 and 3.17, the load mpedance Z 1 and the output mpedances do not overlap. However, the magntudes of Z 1 and Zo oare qute close to each other predctng a possble mnor performance degradaton. The load-affected loop gans of the VMC-CCM converter are shown n Fg It s clear that the load Z 1 has only a small effect on the loop gan as the above analyss predcted. The load Z 2 has the smallest ESR, mplyng small mpedance at the resonant frequency. Accordng to Fg. 3.16, the mpedance overlappng s obvous. The closed-loop and load mpedances nearly overlap as can be seen from Fg The loop gan should now be sgnfcantly affected and the crossover-frequency reducton should almost occur at the load resonant frequency (.e. 4 Hz). These mplcatons can be verfed from Fg The mpedance curve 63

81 Chapter 3 Dynamcal Revew of the load Z 3 clearly mples affected loop gan, but no crossover frequency reducton as t can also be seen from Fg Magntude (db) Phase (deg) oad affected loop gans of VMC CCM converter at U n = 5 V Frequency (Hz) Fg oad-affected loop gans of VMC-CCM converter (dashed lne = Z, 1 dotted lne = Z and sold lne Z 2 3 ). Magntude (db) Phase (deg) oad affected loop gan of VMC DCM converter at U n = 5 V Frequency (Hz) Fg oad-affected loop gans of VMC-DCM converter (dashed lne = Z, 1 dotted lne = Z and sold lne Z ). 2 3 The load-affected loop gans of the VMC-DCM converter are shown n Fg Followng the same procedure as n VMC-CCM converter for analyzng the load 64

82 Chapter 3 Dynamcal Revew nteractons, the effects of the loads are obvous. However, a few nterestng thngs should be mentoned. The load Z 2 ntroduces mpedance overlappng also n the closed-loop mpedance, mplyng loop gan crossover reducton. Accordng to Fg. 3.19, the crossover frequency s, ndeed, reduced near 4 Hz. The ncreasng phase behavor of the load-affected loop gan near the load resonant frequency actually ncreases the phase margn e.g. n the case of the load Z 2. The opposte behavor can be observed n the VMC-CCM converter due to the phase behavor of the open-loop output mpedance, whch makes the VMC-CCM converter senstve to capactve loads at lower frequences. The load-affected loop gans of the PCMC converter are shown n Fg Due to the smlar behavor of the PCMC and VMC-DCM converter output mpedances the load-affected loop gans have also analogous behavor to the VMC-DCM. Magntude (db) oad affected loop gans of PCMC converter at U n = 5 V Phase (deg) Frequency (Hz) Fg oad-affected loop gans of PCMC converter (dashed lne = Z, 1 dotted lne = Z and sold lne Z ). 2 3 It s evdent from Fgs and 3.17 that the PCMC-OCF converter has the lowest mpedances at open and closed loop. It s also clear that no mpedance overlappng takes place. Ths mples that the loop gan would stay almost ntact as t can be seen from Fg

83 Chapter 3 Dynamcal Revew Magntude (db) Phase (deg) oad affected loop gans of PCMC OCF converter at U n = 5 V Frequency (Hz) Fg oad-affected loop gans of PCMC-OCF converter (dashed lne = Z, 1 dotted lne = Z and sold lne Z ) Supply Interactons The source mpedance Z S conssts of a resonant-type EMI-flter C-crcut. Three dfferent flters (.e. Z f 1, Z f 2 and Z f 3 ) are used accordng to Table 3.2. The desgned flters are not optmzed for EMI suppresson. The component values are chosen n such a way that the supply nteractons can be effcently demonstrated. Accordng to the defnton of the dynamcal profle and nteracton formalsm the reflectng parameters of the supply nteractons are orgnally defned as admttances. However, t s most convenent to take the nverse of the admttances and use the correspondng mpedances n the nteracton analyss. Table 3.2. Component values of the C-flter used to study the supply nteractons. Flter f C f rf r Cf f res, S Z f 1 11 μh 5 μf 1 m 1 m 68 Hz Z f 2 2 μh 13 μf 5 m 1 m 99 Hz Z f 3 2 μh 1 μf 2 m 1 m 11 khz 66

84 Chapter 3 Dynamcal Revew It was mentoned earler that the supply nteracton analyss s not as straghtforward as the load nteracton analyss. It was found out that the supply affects the loop gan performance through a combned effect of the open-loop nput admttance/mpedance and the deal nput admttance/mpedances. It s obvous (see (2.22)) that f these two admttances/mpedances are equal the supply system wll not change the nomnal dynamcs. The deal nput admttance/mpedance for a buck converter can be computed from (2.24) yeldng a smple and general expresson Y n DI DI I U n n Zn (3.62) UE Un Un In In the example buck converter used n the thess Yn ( Zn ) at hgh lne s exactly (when ncludng the parastes) db (39.4 db). Magntude (dbω) Internal open loop nput mpedances and supply mpedances. U n = 5 V 6 VMC DCM VMC CCM 4 2 PCMC & PCMC OCF Phase (deg) 2 1 PCMC & PCMC OCF VMC DCM VMC CCM Frequency (Hz) Fg Internal open-loop nput mpedances of VMC-CCM (sold lne), VMC-DCM (dash-dot lne) and PCMC & PCMC-OCF (dashed lne) converters at hgh lne 5 V and supply mpedances Z (crcles), Z (stars) and Z (plus-sgn). f 1 f 2 f 3 The supply mpedances, denoted as Z f 1, Z f 2 and Z f 3, are plotted n Fg together wth the open-loop nput mpedances of VMC-CCM, VMC-DCM, PCMC and PCMC-OCF (note that the open- and closed-loop nput mpedances/admttances are the same for the PCMC and PCMC-OCF converters). It s clear that the flter mpedance Z f 1 and the open-loop mpedances of the converters overlap. The peak value (~ 4 db) of the flter mpedance Z f 1 at the resonant frequency also nteracts 67

85 Chapter 3 Dynamcal Revew wth deal nput mpedance. Flter Z f 2 nteracts only wth the VMC-CCM converter open-loop nput mpedance and the flter Z f 3 has hardly any nteracton on the openloop mpedances beng, however, very close to the VMC-DCM and PCMC & PCMC-OCF mpedances and Zn. The flter mpedances are shown n Fg together wth the closed-loop nput mpedances. It s obvous that the mpedance overlappng takes place when consderng the flter Z f 1. The mpedance of the flter Z f 2 stays well below the closed-loop mpedances and the flter mpedance Z f 3 has hardly any nteracton on the mpedances of the VMC-CCM and PCMC & PCMC-OCF converters. In the case of the VMC-DCM converter the mpedance overlappng due to the Z f 3 s obvous. The stablty of the converters wth the dfferent flters could be checked from Fg. 3.23, but let us frst consder the stablty va the loop gans. Magntude (dbω) Phase (deg) 6 4 Internal closed loop nput mpedances and supply mpedances. U n = 5 V 2 VMC DCM PCMC & PCMC OCF PCMC & PCMC OCF VMC CCM 1 VMC CCM VMC DCM Frequency (Hz) Fg Internal closed-loop nput mpedances of VMC-CCM (sold lne), VMC-DCM (dash-dot lne) and PCMC & PCMC-OCF (dashed lne) converters at hgh lne 5 V and supply mpedances Z (crcles), Z (stars) and Z (plus-sgn). f 1 f 2 f 3 68

86 Chapter 3 Dynamcal Revew Supply affected loop gans of VMC CCM converter. U n = 5 V Magntude (db) Phase (deg) Frequency (Hz) Fg Supply-affected loop gans of VMC-CCM converter Z (crcles), Z (stars) and f 1 f 2 Z (plus-sgn). f 3 Supply affected loop gans of VMC DCM converter, U n = 5 V Magntude (db) Phase (deg) Frequency (Hz) Fg Supply-affected loop gans of VMC-DCM converter Z (crcles), Z (stars) and f 1 f 2 Z (plus-sgn). f 3 69

87 Chapter 3 Dynamcal Revew Supply affected loop gans of PCMC and PCMC OCF converters. U n = 5 V Magntude (db) Phase (deg) Frequency (Hz) Fg Supply-affected loop gans of PCMC & PCMC-OCF converters. The loop gans of the supply-affected converters are plotted n Fg for the VMC-CCM, n Fg for the VMC-DCM and n Fg for the PCMC & PCMC-OCF converter, respectvely. Accordng to the loop gans n Fg and Fg. 3.25, the VMC-CCM and VMC-DCM converters are unstable wth flter Z f 1. Ths s because of the postve gan margn as the phase starts to decrease towards 18 near the flter resonant frequency. The nstablty could also be observed from Fg. 3.23, relatng to the mnor loop gan Z Z f 1 nc nc Z f 1. (3.63) Zf1 Znc Z The phase of the flter Z f 1 s 9 at the lower frequences (.e. from 1 Hz to the flter resonant frequency) and the phase of the closed-loop mpedance of the VMC- CCM and VMC-DCM converters s near -18. Accordng to (3.63), the resultng phase of the mnor loop gan s a bt lower than 27, where the mpedance overlappng takes place. Ths volates the stablty boundares derved n Chapter 2 and ndcates an unstable converter. The supply-affected loop gans wth flter Z f 2 clearly have no nstablty problems. The loop gan s only slghtly changed near the flter resonant frequency. Ths could be also verfed from Fgs and The VMC-CCM converter wth flter Z f 3 also seems to be stable. The resonant frequency of the flter Z f 3 s actually slghtly beyond the converter crossover frequency, 7

88 Chapter 3 Dynamcal Revew ndcatng that the performance mght not be changed at all. However, t s evdent that f the nput voltage s ncreased the crossover frequency would be lmted because of the dp n the magntude. Accordng to Fg. 3.23, the output mpedance of the VMC-DCM converter and the flter mpedance Z f 3 overlap. It can be seen from Fg that the crossover frequency s slghtly reduced, but the converter s stll stable, although PM s reduced a bt. The supply-affected loop gans of the PCMC (PCMC-OCF) converter are shown n Fg It s obvous that they are not changed at all by the flter. Accordng to the Bode-plots, the converters seem to be stable. However, the mnor-loop study reveals that the converter s unstable wth flter Z f 1. The reason s the same as n the case of the VMC converters, but now the phase of the nput mpedance s -18. Explanaton for the unchanged loop gans s that n the PCMC (PCMC-OCF) the Go o, makng the open-loop nput admttance (or mpedance) Y n o and the deal nput admttance (or mpedance) Y n equal. Therefore, accordng to the defnton of the supply-affected loop gan n (2.26), the loop gan wll not change. Ths mples that the stablty should be verfed from the mnor-loop gan to accurately verfy the stablty. Accordng to the supply nteracton formalsm n (2.22), the supply mpedance changes the nomnal output mpedance va Yn o and the short crcut nput admttance ( Y n sc ). Based on the defnton n (2.25), Y n sc for the VMC-CCM, VMC-DCM, PCMC and PCMC-OCF converters can be computed from the correspondng g- parameters (.e. (3.26), (3.43), (3.53) and (3.56) ) yeldng Zs Y nsc 2 D r s E (3.64) Y M o nscdcm U U n o I (3.65) Y DF qu DF I F qi m E m o nscpcmc m o re FmUE s (3.66) 71

89 Chapter 3 Dynamcal Revew Y nscocf R r sf U R R F I q r R s D R R qu DR D R 2 m o E s1 s1 s2 E s1 s1 s1 E m E s1 s2 (3.67) Short crcut nput admttance. U n = 5 V Magntude (dbω 1 ) VMC DCM PCMC&PCMC OCF VMC CCM Phase (deg) 2 1 PCMC&PCMC OCF VMC CCM VMC DCM Frequency (Hz) Fg Short-crcut nput admttances of VMC-CCM (sold lne), VMC-DCM (dash-dot lne), PCMC (dashed lne) and PCMC-OCF (dotted lne) converters at hgh lne 5 V. It s evdent from (3.64)-(3.67) that Y n sc s dependent on the control and also on the conducton mode. Yn sc for the dfferent converters s plotted n Fg Accordng to the supply nteracton formalsm the output mpedance would stay ntact f Yno Ynsc. From Fg we can see that Y n o Y n sc s obtaned n VMC-DCM, PCMC and PCMC-OCF converters. Ths s due to the small rghtmost part of Y nsc (.e. Go oto o / Z ), whch s obtaned as a result of small o o Go o n the case of PCMC and PCMC-OCF converters. In VMC-DCM, Yno Ynsc s acheved due to the small product of GooTo o [P6] and [88]. The VMC-CCM converter has a tendency to change the nomnal output mpedance, because Yno Ynsc Double Interactons It has already been mentoned but not explctly shown that the load may change the senstvty for the supply nteractons by changng the nput admttance. Consequently, the supply mpedance may change the output mpedance of the converter and senstvty for the load nteractons [P11]. In the prevous analyses, we 72

90 Chapter 3 Dynamcal Revew have assumed that the parameters (.e. Z o o, Yn o, Yn sc and Y n ) that reflect the nteractons are nternal. However, t was found out that only Y n sc and Y n are ndependent of the load or supply. Ths mples that the partcular nterest s n the Zo o and Y n o. So far, we have assumed that the load of the converter remans nomnal, when the supply nteractons are dscussed. Consequently, the supply system s assumed to be nomnal n the case of the load nteracton analyss. However, nstead of usng the nomnal parameters (.e. Zo oand Yn o ) n the analyss the supply affected open-loop output mpedance (.e. Z S o o) or the load affected nput admttance (.e. Yn o ) should be used f the supply or load system s not deal. The mpedance of the flter Z f 1 was used as the supply mpedance Z S and the supply S affected open-loop output mpedance Zo o was computed accordng to (2.22) and s plotted n Fg It s obvous that the supply-affected mpedance has resonant behavor at the resonant frequency (.e. 68 Hz) of flter mpedance Z f 1 makng the converter more prone to load nteractons just before the resonant frequency, because of the larger mpedance compared to the nomnal output mpedance (.e. dashed lne). Magntude (dbω) 1 1 Internal and supply affected open loop output mpedances S Z o o Z o o Phase (deg) 1 5 S Z o o 5 S Z o o Frequency (Hz) Fg Internal (dashed lne) and supply-affected (sold lne) open-loop output mpedances of the VMC-CCM converter. The mpedance of the load Z 3 was used as the load mpedance Z and the load affected open-loop nput admttance Yo o was computed accordng to (2.17) and s 73

91 Chapter 3 Dynamcal Revew plotted n Fg The load Z 3 ncreases the admttance at the lower frequences makng the converter more senstve to the supply nteractons. The resonant behavor s also slghtly dfferent than n the correspondng nomnal parameter, whch may ntroduce performance degradaton f resonant-type supply mpedance s present. Magntude (dbω 1 ) Internal and load affected open loop nput admttances Y n o Y n o Phase (deg) 5 5 Y n o Y n o Frequency (Hz) Fg Internal (dashed lne) and load-affected sold lne) open-loop nput admttances of the VMC-CCM converter. 3.4 Effect of oad Resstance It was dscussed n Chapter 1 that the prevalng method n modelng swtched-mode converters s to use a resstve load as the ntal load. However, n practcal applcatons the load s very seldom a sngle resstor. It may be easy to model the load as a sngle resstor, but as a consequence, the nternal dynamcs can be hdden. Ths could lead to wrong deductons of the converter senstvtes, to dfferent nteractons and pecular phenomena may occur. To facltate the understandng of the effect of the resstve load, a few examples wll be provded. In [P1] both nternal (.e. G & Z wth a current-snk load) and load affected ( G R & Z R, where superscrpt co o o R stands for the resstve load) transfer functons were ntroduced for the PCMC R R converter. It was found out that the maxmum gans of G co and Zo oare equal to the load resstor R. However, the maxmum gans of the nternal G co and Zo oare equal to FU m E, whch s typcally much hgher than the load resstance R. Evdence of ths wll be provded n Chapter 4, when the mxed-data method s ntroduced. As a consequence, the use of the resstve load damps the maxmum (low frequency) gans co o o 74

92 Chapter 3 Dynamcal Revew of the transfer-functons. Therefore, the nomnal dynamcs are hdden, and, f the nteracton analyss s based on the load affected parameters the true performance cannot be verfed, because the dampng makes the converter more nsenstve to the nteractons. The smlar dampng effect can also be observed n the VMC-DCM converter. In VMC-CCM converter, the denomnator of load affected parameters equals to 2 ( Rrc RrE rcr E) C R re s s C( R r ) C( R r ) c c (3.68) Accordng to (3.26) and (3.61), the dampng factor of the VMC-CCM converter wth the nomnal current-snk load equals re rc C (3.69) 2 C Consequently, accordng to (3.26) and (3.68), the dampng factor R of the load affected VMC-CCM converter equals R ( Rrc RrE rcr E) C C( R r ) ( rc re) C c R R re 2 C 2 C( R r ) c (3.7) If we compare the dampng factors and R, t s obvous that because of the term / R n the numerator of R, the resultng dampng factor R s always larger than. Consequently, the load-affected parameters are damped more than the correspondng nternal parameters. Ths effect s llustrated n Fg. 3.3, where the open-loop output mpedance s plotted both at constant-current snk load (sold lne) and resstve load (dashed lne). If the real load of the converter has a resonant behavor wth the resonant frequency near the converter resonant frequency the use of the load-affected output mpedance n the nteracton analyss would, msleadngly, show smaller senstvty to the performance degradaton due to the dampng effect. Obvously, ths dampng effect s present n all the g-parameters of the VMC-CCM converter. 75

93 Chapter 3 Dynamcal Revew Magntude (dbω) Effect of the resstve load on the open loop output mpedance nt. R Phase (deg) R nt Frequency (Hz) Fg Internal (sold lne) and load-affected (dashed lne) open-loop output mpedances of the VMC-CCM converter. Internal and load affected control to output transfer functons of self oscllatng flyback converter 4 U = 2 V 2 G n co Magntude (db) 2 G R co Frequency (Hz) Phase (deg) 45 G R co G co Frequency (Hz) Fg Internal (sold lne) and load-affected (dashed lne) control-to-output transfer functons of self-oscllatng flyback converter. The control-to-output transfer functons ( G & G R ) of a certan self-oscllatng flyback converter both at nomnal and resstve load are plotted n Fg [9]. The magntude varaton s smlar to what was observed earler e.g. n the case of the PCMC converter, but the phase behavor has nterestng characterstcs. Accordng to Fg. 3.31, the phase of the load-affected G R co s close to at lower frequences makng a PI-controller sutable (.e. a suffcent phase boost can be obtaned and the co co 76

94 Chapter 3 Dynamcal Revew phase of the loop gan would start at -9 ). However, the phase of the nomnal starts at -9 mplyng that the converter would be condtonally stable f the PIcontroller s used (.e. the phase of the loop gan would start at -18 ). Gco As t was dscussed earler, the nput-output attenuaton propertes can be studed by means of the forward transfer functon G o o. A certan 4 th -order PCMC buck converter (see [91]) was mplemented and both nternal and load-affected forward transfer functons were measured and are plotted n Fg It s obvous that the R load-affected Go o shows better attenuaton propertes at the lower frequences than the correspondng nternal Go o. Although the nternal Go o s also rather small, the even smaller load-affected G R o o hdes the correct nformaton of the converter dynamcal propertes. Internal and load affected forward transfer functons of 4 th order PCMC buck converter Magntude (db) 2 4 G o o G R o o Frequency (Hz) 9 Phase (deg) 9 G o o G R o o Frequency (Hz) Fg Internal (dashed lne) and load-affected (sold lne) open-loop forward transfer functons of 4 th -order PCMC buck converter. Although the above presented examples cover only a narrow ntroducton to the effect of the resstve load, t should be clear that the nternal parameters should be used n order to systematcally analyze the converter dynamcal behavor. Generally, the resstve load reduces the possble senstvtes for load and supply nteractons, whch, n turn, may cause performance degradaton or even nstablty f the nternal profle s recovered at a current-snk-type load. In the current-output converters, the use of the resstve load as the ntal load may cause even more pecular characterstcs as t wll be shown n Chapter 5. 77

95 4 Expermental Evdence The analytcal dervaton of the dynamcal profle of a swtched-mode converter was dscussed n the prevous chapters. However, sometmes t may be mpossble, dffcult or unnecessary to analytcally derve the g-parameter set, and hence, the dynamcal profle. In these stuatons, the g-parameters can be measured n a frequency doman by usng a frequency response or network analyzer. Consequently, the measured transfer functons can gve mportant nformaton of the model accuracy f the analytcal model s derved and compared wth the measurements. In some cases, t may be physcally mpossble to measure all the transfer functons of the g- parameter set, but f the model s known to be accurate, the lackng measurements can be substtuted wth the correspondng parameters, derved from the analytcal model. Ths s one form of a mxed-data method, whch combnes both the measurements and analytcal data. The method s useful also n the control desgn, nteracton analyss and n computng the two specal admttances of the dynamcal profle as wll be shown n Secton 4.1. The measured and analytcally derved control-to-output transfer functons are compared n Secton 4.2. The comparson of the measurements and analytcal models n the same fgures are omtted after the Secton 4.2, because the key dea of ths chapter s to show that the procedures ntroduced n the prevous chapters are vald also n practce. The match between the measurements and analytcal model s evdent when comparng the correspondng frequency responses n Chapter 3 and 4. The load and supply nteractons are dscussed n Sectons 4.3 and 4.4 n a smlar way to Chapter 3 but now the load and supply mpedances are composed of real components. The studed control methods are VMC-CCM, VMC- DCM and PCMC. The PCMC-OCF converter s left out from the study. Fnally, nterestng practcal ssues are dscussed n Secton

96 Chapter 4 Expermental Evdence 4.1 Mxed-Data Method The frequency doman analyss of a contnuous tme system s usually done n a aplace-doman. Software packages such as Matlab and Mathematca are useful tools, when plottng and analyzng varous frequency responses. Typcally, a Bodeplot s plotted from an s-functon by usng a specfed software functon. However, when applyng the mxed-data method, the magntude and phase at certan frequences have to be extracted from the s-functon n order to provde three vectors (.e. magntude, phase and frequency). Ths s easy to do wthn the software. The frequency response analyzers usually allow exportng the measurement data nto e.g. Matlab. The data are typcally n three vectors; frequency, phase and magntude. If the frequency vectors are set equal n the software and measured data, varous computatons can be performed. The phase (radans) and magntude r (real number), extracted from the s-functon, form a complex number z n polar form as j z r(cos jsn ) re (4.1) For operatons lke multplcaton and dvson t s easer to use the polar form shown j rghtmost n (4.1) (.e. re ). Sometmes summng or subtracton operatons are requred, and therefore the Cartesan form (.e. r(cos jsn ) ) would make the calculatons easer. However, the software packages perform the dfferent operatons automatcally and only one form s actually requred n the software envronment. After the necessary calculatons are completed, the new magntude r n and phase n of the resultng complex number z can be calculated from r ( rcos ) ( rsn ) n n 2 2 sn 1 arg z tan cos (4.2) Note that the equatons (4.1) and (4.2) represent the phase and magntude only at a certan frequency f. Ths s done only for the sake of smplfcaton of the dea behnd the method. The true formaton s actually a vector z composng of complex numbers z at the frequences n the vector f. Fnally, the resultng frequency response s easy to plot by usng the derved phase and magntude vectors and the correspondng frequency vector f. 79

97 Chapter 4 Expermental Evdence Applyng the presented method, the nteracton analyss can be done beforehand based e.g. on the measured power stage transfer functons and theoretcal load and/or supply system or vce versa. Also, the analog control desgn can be done accurately based on the measured control-to-output transfer functon and theoretcal controller crcut model. The resultng mxed-data loop gan corresponds very accurately to the measured loop gan, as t wll be shown n the next subsecton. Other applcatons for the presented method nclude e.g. recoverng the nomnal dynamcs from load/source affected measurements and ncludng the effect of an error-amplfer bandwdth Mxed-Data Control Desgn Accordng to (2.5), the control-to-output transfer functon G co plays a sgnfcant role n the loop gan equaton. In practce, the measurement from the control sgnal to the output voltage typcally ncludes also the effect of the modulator (.e. G a ), the sensor gan (.e. unknown part. G se ) and non-dealtes, leavng the controller transfer functon G cc the only Magntude (db) Phase (deg) oop gan. U n = 5 V Frequency (Hz) Fg Comparson of the measured (dashed lne) and mxed-data (sold lne) loop gans. Consder the expermental VMC-CCM buck DC-DC converter studed n ths chapter. The measured G co, shown n Fg. 4.2 at the hgh lne (.e. Un 5 V ), ncludes the effects of the modulator and sensor gans. The measurement data were exported nto Matlab and a specfed program (.e. m-fle, see Appendx B) was created to perform the control desgn by usng the mxed-data. The desred phase 8

98 Chapter 4 Expermental Evdence margn (PM) was at least 5 deg and the gan margn (GM) more than 6 db. The desred crossover frequency f c was near 1 khz. An analog Type-3 controller was used and the component values of the control crcut were chosen from the correspondng E-seres [59] and [89]. The mxed-data loop gan s shown n Fg. 4.1 together wth the measured loop gan matchng well wth each other. The advantage of ths method s that the measured G co contans the non-dealtes n the crcut, whch the model perhaps cannot predct. Consequently, the non-dealtes n the control crcut are usually neglgble, so the mxed-data control desgn provdes a very accurate loop gan predcton, avodng the tme and money consumng redesgn of the controller. Accordng to Fg. 4.1, the desred margns are well met. The msmatch n the phase at the lower frequences s due to the reduced dynamcs of the measurement equpment Mxed-Data Nomnal Model When modelng the swtched-mode DC-DC converters, the nomnal/nternal model should be computed by usng a current-snk load [P5]. However, measurng the nomnal transfer functons at open loop can sometmes be mpossble wth a constantcurrent load. Ths apples to e.g. the PCMC converter due to ts constant-current nature at open loop. Therefore, the measurements have to be carred out by usng a resstve load and the nomnal dynamcs are, of course, hdden. The load-nteracton formalsm was explaned n detal n Chapter 2 gvng the load-affected control-tooutput transfer functon G co as G co Gco Z 1 Z oo (4.3) Solvng the nomnal G co from above yelds G co G co Z 1 Z oo (4.4) Due to the measurement confguraton, the nomnal open-loop output mpedance s possble to measure drectly wth the load the PCMC buck converter s shown n Fg The load Z. The measured load affected Zo o G co of Z was a pure 4- resstor. 81

99 Chapter 4 Expermental Evdence Accordng to Fg. 4.3, the measured and predcted G co match well wth each other. The predcted and measured nomnal Z o o are shown n Fgs and 4.4, respectvely, mplyng a good match. A Matlab program (see appendx B) was created to calculate the nomnal G co from (4.4) by usng the mxed-data (.e. measured G co and Zo o and theoretcal 4- resstor). The resultng nomnal G co s shown n Fg. 4.3 together wth the predcton from the computed nomnal model. Accordng to Fg. 4.3, t s obvous that the nomnal dynamcs can be accurately recovered from the measurement data by usng the ntroduced mxed-data method. Ths method was successfully appled n [P1] to compute the nomnal model. 4.2 Measured Internal oop-gan The buck converter shown n Fg. 3.4 was mplemented n practce. A photograph and schematcs of the expermental converter are shown n Appendx D. The control modes were the PCMC and VMC. The converter was ntally bult to operate n the CCM, but n the case of the VMC the value of the nductor was lowered to obtan also the DCM. The dynamcal profle was measured by usng Venable Instruments Model 312 frequency response analyzer wth an mpedance measurement kt. Appendx D shows an llustratve photograph of the practcal measurement test set-up. An electronc load was used n a constant-current mode when measurng the nomnal/nternal parameters. However, a passve resstor had to be used n the case of an open-loop PCMC converter due to ts current-output nature. The measured and predcted control-to-output transfer functons of the VMC-CCM and VMC-DCM converters are shown n Fg. 4.2 at the hgh lne (.e. U 5 V ). It s obvous that the measured and predcted curves match well up to 1 khz and beyond that the measured phase starts to decrease. The reason for ths msmatch s most lkely due to the combned effect of the phase lag of the modulator crcut and the snusodal njecton sgnal. The observed phase lag deserves a more detaled analyss and t s addressed to be one of the future research topcs. The measured and predcted control-to-output transfer functon of the PCMC converter s shown n Fg The nternal open-loop behavor of the PCMC converter s mpossble to measure wth a constant-current type load due to the constant-current nature of the converter at open loop. Therefore a resstve load was used and the nternal G co was computed by applyng the mxed-data method, whch was ntroduced n Secton 4.1. Accordng to Fg. 4.3, there s a sgnfcant dfference between the nternal and load-affected G co, especally at lower frequences. Ths phenomenon was dscussed n the prevous n 82

100 Chapter 4 Expermental Evdence chapter. It was dscovered n [P1] that most of the prevous models of the PCMC converters are naccurate because of the use of the resstve load as the ntal load hdng the nternal dynamcs. Ths s, agan, a good example of the mportance of defnng the true nternal model, not the load-affected one. Magntude (db) Control to output transfer functon. U n = 5 V 6 DCM CCM Phase (deg) DCM CCM Frequency (Hz) Fg Measured and predcted control-to-output transfer functons of VMC converter. The DCM and CCM curves are ndcated wth arrows. Dashed lne represents the measurement and sold lne the predcton. Magntude (db) Control to output transfer functons of PCMC converter. U n = 5 V R nternal R Phase (deg) 5 1 nternal Frequency (Hz) Fg Measured and predcted control-to-output transfer functons of PCMC converter wth resstve and nomnal loads. Dashed lne represents the measurement and sold lne the predcton 83

101 Chapter 4 Expermental Evdence The phase behavor of the control-to-output transfer functons reveal the type of the controller that should be used, as t was already dscussed n Chapter 3. For nstance, the phase of the VMC-CCM converter approaches -18 after the output flter resonant frequency ndcatng a need of a Type-3 (.e. a PID) controller to provde a suffcent phase boost. In the VMC-DCM and PCMC the suffcent phase boost can be acheved by usng a Type-2 (.e. a PI) controller. The measured loop gans of the converters are shown n Fg The control loops were desgned to have at least 5 deg of a phase margn and a crossover frequency near 1 khz. Accordng to Fg. 4.4, these crtera are met. It s obvous that there are some non-dealtes n the measurement; the saturaton of the magntude at lower frequences (.e. f < 1 Hz) s due to the reduced dynamcs of the analyzer and the phase lag observed n G co s drectly reflected nto the loop gan. It s mportant to recognze the factors that can have effect on the measurement n order to be sure of the valdty of the obtaned frequency responses. 6 Internal loop gans. U n = 5 V Magntude (db) 4 2 Phase (deg) Frequency (Hz) Fg Measured nternal loop gans of VMC-CCM (sold lne), VMC-DCM (dotted lne) and PCMC (dashed lne) converters. 4.3 oad Interactons The load nteractons are studed by means of a resonant type load Z (.e. a C-flter wth resonant frequency fres 5 Hz and element values as follows: = 5 H, C = 2 F, r, =.2, and r C, = 45 m), whch s connected n parallel to the output-current snk (.e. the nomnal load) of the converter. The 84

102 Chapter 4 Expermental Evdence measured nput mpedance of the load and the nternal open-loop output mpedances of the converters are shown n Fg It s obvous that the there s a good match between the analytcal (see Fg. 3.11) and measured curves of the VMC-CCM, VMC- DCM and PCMC converters. The output mpedance of the VMC-DCM converter has a smlar behavor to the PCMC converter makng the converter prone to performance degradaton at lower frequences due to the large output mpedance [P6]. Accordng to the load nteracton formalsm, the resonant-type load n Fg. 4.5 would mostly affect the magntudes of the VMC-DCM and PCMC converters. However, the postve phase behavor of the open-loop output mpedance of the VMC-CCM converter at lower frequences shows senstvty to a capactve load and ntroduces a phase lag nto the load affected loop gan. If the crossover frequency s decreased, the VMC-CCM converter may become unstable. Magntude (dbω) Measured C load mpedance and open loop output mpedances Frequency (Hz) 1 PCMC VMC DCM VMC CCM 5 Hz Phase (deg) VMC CCM PCMC VMC DCM Frequency (Hz) Fg Measured nternal open-loop output mpedances of VMC-CCM (sold lne) VMC- DCM (dotted lne) and PCMC (dashed lne) and C-load mpedance (dash-dot lne). It was shown n Chapter 3 that the crossover frequency of the converter loop gan wll be reduced f the nomnal closed-loop output mpedance and the load mpedance overlap. Accordng to Fg. 4.6, ths would not happen. Agan, the measured and analytcally derved (see Fg. 3.12) output mpedances are n good agreement. However, the mpedances are so small at lower frequences that the resoluton of the current probe saturates the magntude at some pont (near -6 db) makng the measurement unrelable at the frequences lower than 1 Hz. The nternal closedloop output mpedance of the VMC-DCM converter exhbts smlar behavor to the PCMC converter. Accordng to Fg. 4.6, the PCMC converter has the largest closed- 85

103 Chapter 4 Expermental Evdence loop output mpedance at the low frequences makng t also prone to the crossoverfrequency reducton. Magntude (dbω) Phase (deg) 2 Measured C load mpedance and closed loop output mpedances 4 VMC DCM VMC CCM Frequency (Hz) PCMC 5 Hz VMC DCM PCMC VMC CCM Frequency (Hz) Fg Measured nternal closed-loop output mpedances of VMC-CCM (sold lne), VMC-DCM (dotted lne) and PCMC (dashed lne) and C-load mpedance (dash-dot lne). 6 oop gans wth C load. f res = 5 Hz Magntude (db) 4 2 Phase (deg) Frequency (Hz) Fg Measured load-affected loop gans of VMC-CCM (sold lne), VMC-DCM (dotted lne) and PCMC (dashed lne). The load-affected loop gans at the hgh lne (.e. U 5 V ) are shown n Fg It s apparent that the magntude of the PCMC converter loop gan s most affected f n 86

104 Chapter 4 Expermental Evdence compared to the nternal loop gan n Fg Ths s due to the largest output mpedances n Fgs. 4.5 and 4.6. However, the output mpedances and loop gan of the PCMC buck converter are not senstve to nput voltage varaton as t was dscussed n Chapter 3. Ths means that the mpedances and loop gans at the low and hgh lnes are equal. The nput voltage varaton s evdent e.g. n the VMC converters, and generally, the varaton should always be taken nto account. In the VMC-CCM and VMC-DCM converters studed here, the low lne (.e. U 2V ) mpedances are a bt larger than at the hgh lne, but wll not cause any severe nteractons or stablty problems. As t was predcted above, the capactve load would lead the phase of the VMC-DCM and PCMC converters and lag the phase of the VMC-CCM converter. Ths can also be verfed from Fg n 4.4 Supply Interactons The source mpedance Z S used n the followng supply nteracton analyss composes of an output mpedance of a sngle-secton C-EMI flter, whch was desgned to meet the EMC requrements n the case of VMC-CCM-converter and havng a resonant frequency f res S 5 Hz and element values as follows: f = 5 H, C f = 2 F, r f =.2, and r Cf = 45 m. The measured flter output mpedance and the nternal nput mpedances of the three converters are shown n Fg. 4.8 at the low lne (.e. U 2 V ). n 4 Internal open loop nput mpedances and EMI flter mpedance Z S Magntude (dbω) 2 Z S Frequency (Hz) 2 Phase (deg) 1 1 Z S Frequency (Hz) Fg Measured nternal open-loop nput mpedances of VMC-CCM (sold lne), VMC- DCM (dotted lne) and PCMC converters (dashed lne) and EMI-flter mpedance (dash-dot lne). 87

105 Chapter 4 Expermental Evdence Obvously, resonant behavor n the open-loop nput mpedance of the VMC-CCM converter makes t prone to performance degradaton. Resonant behavor s not present n the PCMC and VMC-DCM converters as t was also dscussed n Chapter 3. It was notced n Chapter 2 that the supply nteractons are reflected solely va the 1 nput mpedance n four parameters (.e. Y, n o To o, G c and Go o ). However, n the supply-affected output mpedance and control-to-output transfer functon the nteractons are also reflected va the short-crcut nput admttance/mpedance and the deal nput admttance/mpedance. In order to study the supply nteractons on the output mpedance and the control-to-output transfer functon, and hence, on the loop gan these two specal admttances/mpedances should be able to be measured. It should be clear that ths cannot be done drectly, because of the need of a short crcuted and nfnte-bandwdth-controlled converter. However, t s possble to compute these specal parameters by measurng the parameters that defne these admttances/mpedances and then smply applyng (2.24) or (2.25) [P5], [P6] and [P8]. Magntude (dbω) 4 2 Internal closed loop nput mpedances and the EMI flter mpedance Z S Z s Frequency (Hz) 1 Phase (deg) 1 Z s Frequency (Hz) Fg Measured nternal closed-loop nput mpedances of VMC-CCM (sold lne), VMC- DCM (dotted lne) and PCMC converters (dashed lne) and EMI-flter mpedance Z (dash-dot lne). S The closed-loop nput mpedance 1 Y n c and the flter mpedance are shown n Fg. 4.9 mplyng stable operaton, because the mpedance overlappng does not exst. The nput mpedances of the three converters are the same up to about 1 khz and beyond that the VMC-DCM converter mpedance has deterorated behavor n the mpedance. 88

106 Chapter 4 Expermental Evdence It was dscussed n Chapter 2 that f the forward transfer functon Go o s small the nteractons nto the loop gan and output mpedance would be mnmzed, because Yno Ync Yn Ynsc. It s mportant to note that even f Y n o n a certan converter has resonant behavor, t s not prone to performance degradaton f Go o s zero or close to zero. Naturally, the resonant behavor of the converter can make t more prone to nstablty. Go o of the three control modes are shown n Fg It s apparent that the PCMC converter has capablty to reduce the supply nteractons more than the VMC converters. The compensaton ramp M c s not deal n the practcal mplementaton, and therefore, Go o of the PCMC converter s larger than predcted n Fg The measured closed-loop forward transfer functon Go c of the VMC-CCM converter s also shown n Fg. 4.1 n order to show that the nput-output attenuaton propertes are hdden n the closed-loop parameter as t was dscussed n Chapters 2 & 3. Magntude (db) VMC CCM at closed loop Internal open loop forward transfer functon. U n = 2 V PCMC VMC CCM VMC DCM Phase (deg) Frequency (Hz) 1 1 PCMC VMC CCM VMC DCM VMC CCM at closed loop Frequency (Hz) Fg Measured open-loop forward transfer functon of VMC-CCM (sold lne), VMC- DCM (dotted lne) and PCMC converters (dashed lne). Measured closed-loop forward transfer functon s marked wth dash-dot lne. Y ( Z ) of the buck converter used n ths thess s at the low lne about db n n (23.5 db). The computed Y n s shown n Fg 4.11, expermentally confrmng that. Accordng to Fgs. 4.8 and 4.9, only the mpedances of the PCMC converter are close to 23.5 db almost throughout the whole frequency range. Ths mples that the loop gan should stay ntact or has only a mnor change. Consequently, the VMC-CCM 89

107 Chapter 4 Expermental Evdence converter should have qute a large change n the loop gan. The supply affected loop gans are shown n Fg It s clear that the mplcatons stated above are correct. Ideal nput admttance. U n = 2 V Magntude (dbω 1 ) Phase (deg) Frequency (Hz) Fg Computed deal nput admttances of VMC-CCM (sold lne), VMC-DCM (dotted lne) and PCMC converters (dashed lne). The predcton s marked wth small crcles. Magntude (db) 4 2 oop gans wth EMI flter VMC CCM VMC DCM PCMC Phase (deg) VMC CCM 5 1 PCMC VMC DCM Frequency (Hz) Fg Measured supply-affected loop gans of VMC-CCM (sold lne), VMC-DCM (dotted lne) and PCMC converters (dashed lne). The transfer functons that defne the short-crcut admttance Y n sc were measured and Yn sc was computed n a smlar manner as Y n. Yn sc for the VMC-CCM, VMC-DCM and PCMC converters are shown n Fg It s evdent that the 9

108 Chapter 4 Expermental Evdence predcton and measurement-based computatons match well wth each other. The assumptons that Y n would be specfc for certan topology, but ndependent on the load, control- and operaton modes and that Yn sc would be dependent on the controland operaton mode are expermentally verfed n Fgs and Magntude (dbω 1 ) Short crcut nput admttance. U n = 2 V VMC DCM PCMC VMC CCM Phase (deg) 2 1 PCMC VMC DCM 1 VMC CCM Frequency (Hz) Fg Computed short crcut nput admttances of VMC-CCM, VMC-DCM and PCMC converters. Dashed lne represents the measurement and sold lne the predcton. 4.5 Dscusson There are several mportant ssues relatng to the practcal dervaton of the dynamcal profle, and therefore, they are worth dscussng. When defnng the concept of the dynamcal profle n Chapter 2, t was stated that the supply system should be a pure voltage source and the load should be composed of a constant-current snk n the voltage-output converters and of a pure voltage source n the current-output converters, respectvely. However, n practcal cases the supplyng and loadng devces are never deal. Consder, for nstance, the supply of the converter, whch profle s under dervaton. The supply s typcally another power supply, producng a constant steady state voltage. From the dynamcal vewpont the supply converter has an output mpedance, whch accordng to the nteracton formalsm, may have an effect on the nternal dynamcs. Consequently, the same apples also to the load sde; the electronc loads that are typcally used to mplement the constant-current load (and also e.g. constant-resstance, -voltage and -power) mght have pecular dynamcal characterstcs, whch may affect the nternal dynamcs of the converter beng under study [P9]. To be sure of the possble mpacts of the non-deal supply and 91

109 Chapter 4 Expermental Evdence load systems ther output and nput mpedances should be measured. In order to avod or at least mnmze the effects of the supply and load, large capactors can be placed to the nput and load sde to dynamcally solate the converter that s under study. As t has been dscussed, a resstve load has to be used n some cases and the nternal dynamcal profle can be then computed by usng the mxed-data method. In Secton 4.1, the mxed data method was only appled to the dervaton of the control-to-output transfer functon. However, f the dynamcal profle s measured wth the resstve load the effect of the resstor has to be removed from every parameter. If not dong so, the nherent dynamcal propertes cannot be obtaned and studed. It s also evdent that the long connector cables mght ntroduce extra resstance (and also nductance at hgher frequences), whch s seen as a resstor (or nductance) even f the non-deal supply and load systems are used. However, ths effect s typcally neglgble but may be sgnfcant n converters wth a hgh swtchng frequency. The functonng of the frequency response analyzer should also be understood. The general operatng prncple s that a snusodal sgnal s njected nto a loop or sgnal that s of nterest and then the phase and magntude of the snusodal sgnal s measured at the desred pont and compared to the njected reference sgnal n order to compute the magntude and phase of the loop gan or transfer functon. Bascally, the procedure s smple. The analyzer s typcally equpped wth software, whch eases the use. However, there are a few mportant ssues that should be ponted out. The ampltude of the njected snusodal sgnal has to reman snusodal throughout the whole frequency range. Otherwse, the measurements are not relable, because of the dstorted reference sgnal. An osclloscope can be used to verfy the qualty of the sgnal. It s also mportant to defne the physcal locatons of the loops and be sure that a correct loop or transfer functon s measured. In the modern converters, the mpedances can be very small, makng the resoluton of the voltage and current measurement the lmtng factor. Therefore, the magntude of the measured response can saturate to a certan level, whch does not correspond to the true nternal profle. When measurng the loop gan of the converter, an njecton transformer has to be used and, as t was shown n ths chapter, the dynamcs of the njecton transformer reduces the frequency range that can be accurately measured. Ths affect should also be understood. 92

110 5 Current-Output Converters The dynamcal ssues of the current-output converters are dscussed n ths chapter. The voltage-output converters regulate the output voltage and, consequently, n the current-output converters the output current s regulated. A typcal applcaton, where the current-output converter s needed, s a DPA system havng a storage battery connected n parallel to provde back-up power durng the power outages. The need for regulatng or lmtng the current may rse as the battery s charged. Generally, the converter n these applcatons s n the voltage-output mode, but when the current lmt s actvated, the converter wll enter nto the current-output mode as llustrated n Fg The dynamcal profle of the current-output converters can be easly derved from the correspondng voltage-output converter profle by applyng dualty or e.g. the basc SSA modelng method can be equally used. However, the dynamcal profle of the current-output converter dffers sgnfcantly from the voltage-output converter. Ths chapter provdes the true dynamcal profle of the current-output converter and analyzes certan dynamcal ssues and revses the prevalng nadequate knowledge. 5.1 Dervaton of Dynamcal Profle The dynamcal profle can be derved n two dstnct ways. The SSA (or any other modelng method) can be used n a smlar manner to what was presented n Chapter 2. The profle can also be derved from the voltage-output converter parameters by applyng dualty and changng the Thevenn s equvalent output port to the Norton s port. The latter s preferred, because the dynamcal profle of the voltage-output converters s rather easy to obtan and ts behavor s typcally well known makng the dynamcal analyss of the current-output converter straghtforward. The prevous attempts to analyze the dynamcs of the current-output converters were based on the use of a resstve load [74], [75] and [76]. However, the real nternal or nomnal load for the current-output converters s a pure voltage-source. The use of the resstve 93

111 Chapter 5 Current-Output Converters load only leads to wrong deductons of the converter dynamcs as t was dscussed n [P9] and [P1]. A current-output buck converter s llustrated n Fg The power stage and source system of the converter are naturally the same as n the correspondng voltage-output converter, but the load s now a seres connecton of the pure voltage source e o and the load mpedance Z. The output current has to be sensed and typcally a voltage across a small resstor ( R n Fg. 5.1) s measured to form the equvalent current s measurement sgnal. The control system and PWM blocks are smlar to the correspondng blocks of the voltage-output converter. The appled control prncples are a constant-current (CC) VMC and a CC-PCMC. The converter operates n CCM n both control modes. u n + _ Z S fs = 1kHz r ds ( on ) 4m U D.3V r d 55m r.6m r C 33m C F + u o - R s o = 2.5A Z + _ eo R s1 CC-VMC CC-PCMC PWM Control system R so u R 2.5V Fg Current-output buck converter wth CC-VMC and CC-PCMC prncples State-Space Averagng The basc procedures of the SSA ntroduced n Chapter 3 can also be appled to the current-output converters. The state varables reman the same n the current-output converters, but n the nput varables the output current s replaced wth the output voltage, and consequently, the output voltage n the output varables s replaced wth the output current. The constructon of the state-space s omtted here, because of ts trvalty. After lnearzaton of the averaged state space equatons, the fnal lnearzed state-space equatons for the CC-VMC converter can be presented as 94

112 Chapter 5 Current-Output Converters dˆ re ˆ D 1 U E ˆ ˆ ˆ un uo d dt duˆ C 1 1 uˆ ˆ C uo dt rc rc ˆ Dˆ I dˆ c n ˆ ˆ 1 1 uˆ uˆ r r o C o C C c (5.1) Equaton (5.1) can be equally represented n the typcal state-space formalsm by dxˆ( t) Axˆ() t Buˆ() t dt yˆ() t Cxˆ() t Duˆ() t (5.2) where uˆ n ˆ ˆ n xˆ, uˆ uˆ, ˆ o uˆ y ˆ (5.3) C o dˆ The frequency doman representaton and, hence, the dynamcal profle of the current-output converter can be solved from the state-space representaton by applyng the aplace transformaton n a smlar way to what was shown n Chapter 3. In the case of voltage-output converters, we used modfed g-parameters, but now the representaton consttutes of the modfed y-parameters [49]. The general form of the y-parameter set of the current-output converters s uˆ n ˆ n Yno To o G c uˆ o ˆ o Goo Yo o G co cˆ (5.4) where the general control varable s denoted by ĉ. For the CC-VMC converter shown n Fg. 5.1 the y-parameter set s 2 D D DU E 2 re rc 1 C( s s ) D C U E Y 1 src C I no To o G c G ( E ) oo Yo o Gco r s (5.5) 95

113 Chapter 5 Current-Output Converters Accordng to (5.5), the CC-VMC converter exhbts nternally frst-order behavor except the open-loop output admttance Yo o[p1]. It s clear that the y-parameter set and also the dynamcal profle of the CC-VMC current-output converter n (5.5) are totally dfferent than the correspondng parameters of the voltage-output converter n (3.26) and (3.27). The correspondng y-parameter set for the CC-PCMC s shown n (5.6)-(5.7). It s evdent that the same frst-order behavor, observed n the CC-VMC converter, apples also to the CC-PCMC converter (except the open-loop output admttance Yo o pcmc ). It turns out that the good nput-output attenuaton propertes of the correspondng voltage-output PCMC converter can also be acheved n the CC- PCMC converter, because the term D FmqU E can be made to zero as t was dscussed n Chapter 3. Comparng the denomnators of the CC-VMC and CC-PCMC converters show that due to the term FU m E the pole s located at hgher frequences n the CC-PCMC converter. Accordng to (5.7), the low frequency magntude of FU Gco pcmc s close to unty, because m e E frequency magntude of G co s qute large. r. In the CC-VMC converter, the low Y G T nopcmc oopcmc oopcmc Yo opcmc DF qu DF I DF I m E m o m o (5.6) 2 re FmUE rc 1 DFmqU E s s C FqI m o re FmUE s FU m E D FmIo G c pcmc FU m E FI m o Gco pcmc re FmUE s (5.7) Applyng Two-Port Representaton It s possble to derve the dynamcal profle of the current-output converter from the correspondng voltage-output converter model by applyng dualty. Ths can be done smply by changng the Thevenn s output port model to the equvalent Norton s crcut and replacng the voltage-output converter current-snk load wth a pure voltage source e ˆo. The resultng two-port model s llustrated n Fg. 5.2 nsde the 96

114 Chapter 5 Current-Output Converters dashed lne. The modfed y-parameter set constructed from the modfed g- parameters of the voltage-output converter s shown n (5.8). The use of the voltageoutput converter parameters n the current-output converter model s well justfed, because they are usually avalable and easy to obtan. Ths makes the dynamcal revew easer. For nstance, the open-loop nput admttance of the current-output converter s the short crcut admttance Y n sc, whch s obvous because of the short crcuted converter. It s also clear from (5.8) that the open-loop output mpedance s the same as n the correspondng voltage-output converter. ˆn Z s + GT co o- o ( Gc + )ˆ c Z o- o ˆo + Z uˆns + _ uˆn _ Yn - sc - T Z uˆ o- o o o- o G cˆ co Zo - o G Z uˆ o- o n o- o Zo - o uˆo _ + _ eˆo ĉ Fg Two-port model of a current-output converter wth load and supply subsystems. The transfer functons correspond to the model of voltage-output converter. Too GcoToo Ynsc Gc uˆ n ˆ Z n o o Z oo uˆ o ˆ Go o 1 G o co cˆ Zoo Zoo Z oo (5.8) The most convenent way of dervng the closed-loop profle for the current-output converters s to use the control-block dagrams. The control-block dagrams for the output and nput dynamcs are shown n Fg. 5.3 a) and b), respectvely. The G se s the sensor gan (.e. typcally the current sensng resstor R s ), G cc s the controller transfer functon and G s the PWM-modulator gan (.e. typcally 1/ U ). a Computng the output current ˆo, and hence, the output dynamcs, from Fg. 5.3 a) yelds m 97

115 Chapter 5 Current-Output Converters ˆ Goo Yo o GccGaGco uˆ uˆ uˆ 1G G G G 1G G G G 1G G G G o n o r se cc a co se cc a co se cc a co (5.9) Applyng the defnton G G G G (5.1) CO se cc a co for the current-output converter loop gan, (5.9) can be represented as ˆ Goo Yo o 1 CO uˆ uˆ uˆ 1 1 G 1 o n o r CO CO se CO (5.11) Computng the nput current ˆn and, hence, the nput dynamcs, from Fg. 5.3 b) yelds ˆ Y uˆ T uˆ G G G G ˆ G G G uˆ (5.12) n no n oo o se cc a co o cc a co r After substtutng ˆo n (5.12) wth ts defnton n (5.11) the nput dynamcs can be represented as ˆ Go ogc CO Yo ogc CO n Yno uˆn To o uˆ o Gco 1CO Gco 1CO G GG CO 1 c se co CO uˆ r (5.13) The closed-loop parameter set of the current-output converters s presented n (5.14). When the parameters are constructed from the output and nput dynamcs (.e. (5.11) & (5.13)) the parameters of the closed-loop current-output converter can be computed to be as defned n (5.15) - (5.2). Typcally, the reference u ˆ r s kept constant and Gc c and Gco c can be neglected. uˆ n ˆ n Ync To c G cc uˆ o ˆ o Goc Yo c G coc cˆ (5.14) 98

116 Chapter 5 Current-Output Converters Y G G Y G 1 o o c CO nc n o co CO (5.15) T Y G T G 1 o o c CO oc o o co CO (5.16) G G 1 c CO r GG se co CO (5.17) Go o G oc 1 CO (5.18) Y Y oc 1 o o CO (5.19) G ro 1 G se CO 1 CO (5.2) uˆo Yo - o uˆo To - o uˆn Go - o ˆo uˆn Yn - o ˆn Open-loop uˆco ĉ G co G a G se Open-loop uˆco ĉ G c G a Gse ˆo Closed-loop G cc uˆ r Closed-loop G cc uˆ r a) b) Fg Control-block dagrams for current-output converter at open and closed loop: a) output dynamcs and b) nput dynamcs. 99

117 Chapter 5 Current-Output Converters oad Interactons The current-output-converter load nteractons at the presence of the load mpedance Z can be found from Fg. 5.2 by solvng u ˆo : uˆ o Z G uˆ G cˆ eˆ 1 ZY o o n co o oo (5.21) and then replacng t n (5.8). Ths procedure yelds the load affected parameters as ZT o ogo o To o ZGT co oo Yno Gc 1ZY oo 1ZY oo 1ZY oo uˆ n ˆ n G 1 ˆ o o G co e o ˆ o Z Z Z cˆ Yoo Yoo Yoo (5.22) From above we can conclude that f Y then G G, T oo oo o o co co G oo G and T. It s nterestng that due to the dualty Z n the voltage-output converter produces the same relaton. o o oo Expressng the load affected parameters as a functon of the correspondng voltageoutput converter parameters yelds TooGoo Too GcoTo o Yno Gc uˆ n ˆ Z n Zoo Z Zoo Z Z oo eˆ o ˆ Go o 1 G o co cˆ Z Zoo Z Zoo Z Z oo (5.23) Typcally, the current-output converters are used n applcatons where the load s a low mpedance battery. Therefore and accordng to (5.23), the nternal/nomnal behavor wll be recovered (.e. small or zero Z ). Ths may set challenges to the control desgn as wll be shown n Secton Supply Interactons The current-output converter supply nteractons at the presence of the supply/source mpedance Z S can be found from Fg. 5.2 by solvng u ˆn : 1

118 Chapter 5 Current-Output Converters uˆ n uˆns ZS To ouˆo Gccˆ 1 ZY S no (5.24) and then replacng t n (5.8). Ths procedure yelds the supply-affected parameters as Yn o To o G c ˆ ˆ uns 1ZY S n o 1 S n o 1 n ZY ZY S no uˆ o ˆ o Go o 1 ZSYn oc 1 ZSY n Y ˆ oo G co c 1ZY S no 1ZY S no 1ZY S no (5.25) Where the open-crcut nput admttance Yn oc s defned as G T Y Y Y (5.26) oo oo noc no no Yo o The study of the dynamcal profle of the voltage-output converter ntroduced a shortcrcut nput admttance Y n sc, but agan, due to the dualty the specal admttance parameter s actually the open-crcut admttance Y. It s nterestng that n oc Yn oc corresponds to the nternal open-loop nput admttance Y n o of the voltage-output converter. Consequently, the open-loop nput admttance Yn o of the current-output converter corresponds to the short-crcut nput admttance Yn sc of the voltage-output converter. The deal nput admttance Yn can be shown to reman the same n both converter types. If a good nput-output attenuaton exsts.e. G then t can be concluded that f G Y Y Y Y then also Y S oo Y and Y oo no oo no nc noc n Y. no o o G S co G, co Expressng the supply affected parameters as a functon of the correspondng voltageoutput converter parameters yelds Too GcoTo o Gc Yn o Zo o Z oo ˆ uns ˆ 1 S n o 1 S n o 1 n ZY ZY ZY S no uˆ o ˆ G o oo cˆ Zoo 1ZSYno 1 1ZSYn Gco 1 ZY S no 1 ZY S no Zoo 1 ZY S no Z oo (5.27) 11

119 Chapter 5 Current-Output Converters It s apparent from above, that the supply nteractons are reflected mostly va Yn o, whch was found out to correspond to the short-crcut nput admttance Y n sc of the current-output converters. However, Y n sc was found out to be load ndependent n Chapter 4. Accordng to (5.23), the load can change n (5.27) nstead of usng Yn o s not justfed. Yn o. Therefore, the use of Yn sc 5.2 Dynamcal Issues and Expermental Evdence Bascally, the dynamcal analyss of the current-output converters can be carred out n the smlar way as t was presented earler for the voltage-output converters. However, the dynamcal behavor of the current-output converters dffers sgnfcantly from the correspondng voltage-output converters, and therefore, t deserves a lttle more detaled analyss. Also, the pecular phenomenon of the ncreased crossover frequency, when applyng battery as a load and observed both n academa [74] and ndustry, can be explaned. Control to output transfer functons of the VMC current output converter. U n = 5 V Magntude (db) G (R = 4 Ω) co G (R = 5 mω) co 1.9 khz G co (nom) 24.3 khz 13.4 khz Phase (deg) G co (R = 5 mω) G co (R = 4 Ω) G co (nom) Frequency (Hz) Fg Control-to-output transfer functons of CC-VMC converter (sold lne = nomnal, dashed lne = R as a load and dash-dot lne = 4- load). s 12

120 Chapter 5 Current-Output Converters Magntude (db) Control to output transfer functons of the PCMC current output converter. U n = 5 V G (R = 4 Ω) co G (R = 5 mω) co G co (nom) Phase (deg) G (R = 4 Ω) co G (R = 5 mω) co Frequency (Hz) G co (nom) Fg Control-to-output transfer functons of CC-PCMC converter (sold lne = nomnal, dashed lne = R as a load and dash-dot lne = 4- load). s The control-to-output transfer functons of the CC-VMC and CC-PCMC buck converters are shown n Fgs. 5.4 and 5.5, respectvely. It should be noted that the modulator gan G 1/ 3 s ncluded n the model, because t has a sgnfcant effect a on the control desgn. As t has been dscussed throughout the thess, t s common to nclude the load resstor nto the nomnal/nternal model. However, t has been shown that the true nomnal load should be defned and used. For the voltage-output converters the nomnal load was a current snk, but n the case of the current-output converters the nomnal load composes of a voltage source. G co wth both a resstve load (.e. dash-dot lne) and a pure voltage source load (.e. sold lne) are plotted n Fgs. 5.4 and 5.5. The dfference between the crossover frequences of the CC-VMC converter s obvous and the magntude of the resstve G co of the CC-PCMC converter decreases almost two decades before the nomnal G co. A battery-type load can usually be seen as havng a rather low mpedance [77]. Accordng to (5.23), a low mpedance load would recover the nomnal behavor and, consequently, f the ntal control desgn s done wth the resstve load t s clear that the crossover frequency wll be very hgh wth the battery load and could lead to nstablty. In current-output converters, the output current has to be sensed. Typcally ths s done by measurng the voltage across a small current sensng resstor (.e. R s ). The R s actually behaves as a small resstve load and therefore t should be ncluded n G co when startng to desgn the controller. G co wth R 5 m are shown n Fgs. 5.4 and 5.5. It s s 13

121 Chapter 5 Current-Output Converters obvous that even though the resstor s rather small t has an effect on G co n both converters and ths should be taken nto account, when desgnng the controller [78]. Another nterestng feature n the CC-VMC converter s that the phase of the nomnal G co decreases only to near -1 makng the PI-controller (Type-2) sutable. The use of a resstve load would exclude ths, but n practcal applcatons the resstve load may not be used. The deductons (.e. smaller low-frequency magntude and the locaton of the pole n the denomnator n hgher frequences n the CC-PCMC converter) made n Secton 5.5.2, from the symbolcal parameters are explctly observable n Fgs. 5.4 and 5.5. The control loop was desgned based on the analytcal G co shown n Fgs. 5.4 and 5.5. The desred margns were the same as n the correspondng voltage-output converters (.e. PM > 5, GM > 6 db and the crossover frequency at the hgh lne near 1 khz). Because the current sensng resstor R s decreases the magntude near 1 khz, ts effect on the magntude was taken nto account, when desgnng the controller. To make the measurement of the nternal behavor possble a pure voltage source load should be avalable. However, the constant-voltage mode of the electronc load that was used n the laboratory was found to be unrelable from the dynamcal vewpont, as t was dscussed n [P9]. The nternal behavor was recovered by usng a RC-crcut ( R 4 and C 4 F) as the load, makng the large capactor act lke a dynamcal short-crcut at the hgher frequences. The measured and predcted loop gans are shown n Fg It s clear that they are n good agreement. Accordng to the fgure, the CC-VMC converter has frst-order behavor wth no resonant peakng. The load resstor damps the low-frequency loop gan of the CC-VMC converter, makng the measurement devatng from the predcton. The desred margns and crossover frequency are, however, clearly met. The fast decreasng phase at hgher frequences s due to the same reason as n the correspondng voltage-output converter (.e. the modulator crcut and snusodal njecton sgnal). The loop gans of the current-output converters were also measured wth a resstve load and are shown n Fg. 5.7 together wth the predctons. Agan, there s a good agreement between the measurements and predctons. Now, the resonant nature of the CC-VMC converter s exposed nto effect due to the load. The decreased crossover frequences are evdent. It should be clear that f the control desgn s mplemented n such a way that the crossover frequency of the loop gan wth the resstve load s set to 1 khz the crossover frequency of the nternal loop gan can be ncreased up to 1 khz, whch s not desrable. 14

122 Chapter 5 Current-Output Converters 6 Current output converter loop gans wth CC VMC and CC PCMC, U n = 5 V Magntude (db) 4 2 CC PCMC CC VMC Phase (deg) Frequency (Hz) Fg Measured and predcted loop gans of current-output converter (sold lne = CC- VMC, dashed lne = CC-PCMC, crcles = CC-VMC measurement and stars = CC-PCMC measurement. Magntude (db) 5 CC VMC and CC PCMC loop gans wth 4 Ω load, U n = 5 V CC VMC CC PCMC Phase (deg) Frequency (Hz) Fg Measured and predcted loop gans of current-output converter (sold lne = CC- VMC, dashed lne = CC-PCMC, crcles = CC-VMC measurement and stars = CC-PCMC measurement. The load s 4- resstor. The measured and predcted nternal closed-loop output mpedances are shown n Fg The resonant behavor of the CC-VMC output mpedance s obvous ndcatng also that the nternal loop gan does not have t as t was dscussed n [P1]. 15

123 Chapter 5 Current-Output Converters Magntude (dbω) CO converter closed loop nternal output mpedances, U n = 5 V CC VMC CC PCMC Phase (deg) Frequency (Hz) Fg Measured and predcted closed-loop output mpedances of current-output converter (sold lne = CC-VMC, dashed lne = CC-PCMC, crcles = CC-VMC measurement and stars = CC-PCMC measurement. 16

124 6 Conclusons 6.1 Summary of Papers [P1] The man contrbuton of the paper s the accurate small-sgnal modelng of the PCMC converters. However, n the scope of ths thess the most mportant contrbuton s the dervaton of the open-loop dynamcal profle from practcal measurements. Due to the current-source nature of the PCMC at open-loop, a resstve load had to be used n the measurements. It was notced that the resstve load effectvely hd the nomnal behavor, whch could lead to wrong deductons of the model accuracy. To recover the nternal dynamcs n the control-to-output transfer functon a mxed-data method was developed. As a result, t was dscovered that the analytcal model and the profle derved by usng the mxed-data method concded. [P2] The characterzaton of regulated converters was addressed to enable the assessment of the stablty, performance, supply and load nteractons as well as the transent responses. A canoncal model of a converter was proposed and appled to create a set of parameters that were able to fully descrbe the dynamcs assocated wth a converter as such as well as under external nteractons. System theory was used to develop methods to study the nternal stablty of cascaded subsystems such as an EMI flter, load and other converters. A framework that allowed the evaluaton of dfferent converter topologes under dfferent operaton and control modes was establshed. [P3] The stablty and performance of a regulated converter was analyzed based on ts closed-loop output mpedance. System theory was used to obtan a set of transfer 17

125 Chapter 6 Conclusons functons that defned the nternal stablty of an nterconnected system consstng of the source and load converters. The nternal stablty was descrbed n terms of the rato of the output mpedance of the source converter and the nput mpedance of the load converter known as the mnor-loop gan. Thus, the closed-loop output mpedance of a source converter could be used to defne the safe operatng areas that avoded nstabltes mposed by the load mpedance. It was shown that the margns assocated wth the mnor-loop gan (.e. the gan and phase margns) dd not generally match wth the margns of the output-voltage loop gan. The relatonshp was especally weak at the frequences close to and beyond the crossover frequency of the loop gan. Ths means that the margns specfed to the mnor-loop gan should be gradually ncreased as the voltage-loop-gan crossover frequency s approached n order to avod performance degradaton (.e. changes n margns and crossover frequency) n the supply converter. Expermental evdence was provded based on a buck converter under VMC and PCMC. [P4] The effect of the load mpedance on the dynamcs and performance of a regulated converter was nvestgated. Theoretcal formulaton was derved utlzng two-port representaton. It was defntvely shown that the load nteractons were reflected nto the converter dynamcs va the nternal open-loop output mpedance. At the frequences, where the loop gan was much hgher than unty, the nternal closed-loop output mpedances acted as a boundary for the control-bandwdth reducton. The loop gan was always affected, whenever the nternal open-loop output mpedance was equal or greater than the load mpedance. It was found out that the converters were senstve especally to the capactve and resonant-type loads. The senstvty was dependent on the control mode, and could not be much reduced by means of the basc controller desgn. [P5] The characterzaton of regulated converters was nvestgated n order to establsh a set of dynamcal parameters defnng the nteractons arsng n the nterconnected systems such as DPS systems. The commercally avalable converters are usually vaguely specfed n respect to those nteractons. Provded nformaton do not suffce for predctng the stablty and performance. It was notced that there were certan double reflectons, whch were not prevously recognzed but could ncrease the load senstvty f not properly consdered. The defned parameter set could also be used to desgn the converters to be more nsenstve to dfferent nteractons. 18

126 Chapter 6 Conclusons [P6] The paper nvestgated the dynamcal dfferences of the load and supply nteractons n the drect-duty-rato or VMC buck converters n the contnuous and dscontnuous conducton modes. The dynamcal parameters.e. the phase margn and control bandwdth of the CCM and DCM converters were desgned to be the same. It was shown that the senstvty for the nteractons could be concluded from the measured frequency responses. The nvestgatons showed that a buck converter operatng n the CCM was more senstve to capactve loads than a converter n the DCM, whle the DCM converter was more prone to nstablty caused by the load. The supply nteractons caused e.g. by an EMI flter were shown to be smaller n the CCM converter except near the converter-output-flter resonant frequency, where the forward-voltage transfer functon had an amplfyng effect on the supply nteractons. [P7] The paper studed the stablty and performance analyss of a regulated converter based on the mpedance rato known commonly as the mnor-loop gan. System theory was used to prove that the mnor-loop gan could be addressed to the nternal stablty of an nterconnected system n a scentfcally sound manner. As a consequence, the output mpedance of the source system could be used to specfy the stablty boundary of the source converter n respect to the load mpedance. The margns assgned to the mnor-loop gan dd not, however, concde wth the margns assocated to the loop gan of the converter. Therefore, the converter performance can be deterorated unless the mnor-loop gan margns are suffcent for avodng that. Expermental evdence was provded to support the deas presented n the paper. [P8] The paper nvestgated the effects of an EMI flter on the dynamcs of a buck converter. It was shown theoretcally that the EMI flter could ncrease sgnfcantly the load senstvty of the VMC converter, but the PCMC converter was qute nsenstve to the EMI flter nteractons. Expermental valdatons were carred out usng a buck converter wth three dfferent control modes - VMC, PCMC and PCMC- OCF. The nvestgatons showed that the phenomenon causng the nstablty under PCMC was the negatve-resstor oscllaton (NRO) phenomenon, and confrmed also the excess EMI-flter senstvty of the VMC converter. [P9] The paper nvestgated the load nteractons n a constant-current-controlled buck converter. The nomnal transfer functons and the load nteracton formalsm were 19

127 Chapter 6 Conclusons derved from the voltage-output-converter model by applyng dualty. It was observed that the constant-current control made the converter very senstve to load nteractons mplyng that a proper controller desgn s a necessty. The hgh crossover frequency n the nomnal control-to-output transfer functon may lead to performance degradaton f the ntal control desgn s done wth a resstve load, because the typcal load for the constant-current-controlled converters s a low mpedance storage-battery recoverng the nomnal feature. It was shown that electronc loads, typcally used n the prototype testng and verfcaton, mght have unexpected characterstcs. Ths could lead to wrong deductons of the converter stablty and performance. [P1] The paper nvestgated the dynamcal propertes of the current-output converter, and the set of transfer functons representng the dynamcs was developed based both on the state-space averagng method and on the two-port model of the correspondng voltage-output converter. Accordng to the nvestgatons, the observed tendency to ncreasng crossover frequency could be addressed to the nternal dynamcs of the converter, whch the storage battery nvoked nto effect. Ths meant that the orgn of the problem was an erroneous control desgn. [P11] The mechansm and characterzng parameters causng the source reflected nteractons were nvestgated n ths paper both theoretcally and expermentally usng a buck converter under a VMC, PCMC and nput voltage-feedforward (IVFF) control. The reflected nteractons would be elmnated f the forward-voltage transfer functon could be made zero, but n practce such a condton could not be fully acheved. The nvestgatons showed that the VMC converter was very senstve to the source-reflected nteractons. 6.2 Fnal Conclusons and Man Contrbutons The concept of a dynamcal profle of a swtched-mode converter was ntroduced n the thess. It was composed of sx g-parameters at open and/or closed loop and two specal admttances fully characterzng the dynamcal propertes of a sngle swtched-mode converter. The prevous works presented e.g. n [44], [82], [48] and [43] have set the startng pont for the thess. However, the true nternal dynamcal profle has not been clearly defned earler. It was found that the nternal or nomnal dynamcs could be obtaned for the voltage-output converters by usng a pure voltage 11

128 Chapter 6 Conclusons source at the supply sde and a constant-current snk as a load. The two-port representaton and the supply and load nteracton formalsms were revewed. It was clamed that the stablty and performance of a sngle converter as a part of a larger system could be studed by means of the nternal dynamcal profle and nteracton formalsms. Many exstng converter topologes and control prncples have dfferent dynamcal propertes, and hence, dynamcal profles. By studyng the dynamcal profles t s easy to evaluate the senstvtes for dfferent nteractons and the choosng of a proper topology and control mode becomes smple. It was found out that the open-loop parameters predomnantly reflected the nteractons. The open-loop output and nput mpedances were shown to manly reflect the nteractons e.g. onto the loop gan. However, the two specal admttances had to also be studed n the supply nteracton analyss. It was also dscovered that the load or supply mpedance could cause certan double nteractons that change the nternal parameters, and hence, the dynamcal propertes. The closed-loop parameters could not be effectvely used n the nteracton analyss, because the nformaton of the dynamcal propertes was typcally hdden. On the other hand, the closed-loop nput and output mpedances could be used to derve safe load and supply profles n order to guarantee the stablty. Ths was based on the nput-to-output and nternal stablty formalsm that resulted n an mpedance rato of the closed-loop nput or output mpedance and the nput or output mpedance of the nterconnected subsystem. Ths mpedance rato, whch s also known as the mnor-loop gan, s sometmes the only way to check the stablty of the converter (.e. the loop gan can be unchanged even f the converter s unstable). As examples, the dynamcal profles of the VMC-CCM, VMC-DCM, PCMC and PCMC-OCF buck converters were analyzed analytcally and the expermental evdence was provded by obtanng the profles for the VMC-CCM, VMC-DCM and PCMC buck converters. Even f the thess concentrated on the buck converters the proposed methods are naturally applcable to other topologes and control modes as well. The dervaton of the dynamcal profle can be carred out analytcally or t can be measured. The analytcal profle derved by modelng the small-sgnal behavor of the swtched-mode converter. The selected modelng method s rrelevant as long as the model accuracy can be verfed. It s also mportant that the model reveals the true nternal dynamcs, not the dynamcs e.g. wth a resstve load. The dynamcal profle can be measured by usng a frequency response analyzer. Agan, t s mportant to be sure that the nternal dynamcs are measured. However, sometmes t s not possble to use the constant-current-snk load (.e. PCMC at open loop) and a resstve load must be used. A mxed-data method was developed to solve ths problem. The 111

129 Chapter 6 Conclusons method combned the analytcal and expermental data. Specfc software could be used to compute the nternal profle. The mxed-data method was also useful e.g. n the controller desgn, studyng the non-dealtes and recoverng the nternal profle from the supply/load-affected profle. It was obvous that f the nternal dynamcal profle was derved both analytcally and expermentally and they were n a good agreement, the accuracy of the profle could be maxmzed. On the other hand, t s not always possble or reasonable to compute the converter model due to ts complexty. Therefore, the only way to obtan the dynamcal profle s to use the expermental measurements. It s noteworthy that t s not always necessary or even physcally possble to measure all the parameters n the dynamcal profle. Typcally, the open-loop nput and output mpedances, open-loop forward transfer functon, control-to-output transfer functon and the loop gan provde enough nformaton to analyze the performance and stablty of the converter. The prevalng method to use a resstve load n modelng and analyzng swtchedmode converters was argued for beng ncorrect. Several examples were shown to verfy that. Typcally, the resstve load damps the magntude and resonances, f present, makng the converter to look more nsenstve to the nteractons than the nternal dynamcal profle would be. The resstve load can also change the phase of the transfer functon. Ths may cause severe problems f e.g. the control-to-output transfer functon changes the phase 9 between resstve and nomnal load. The basc deas of the dynamcal profle were frst appled to the voltage-output converters. However, the current-output converters, where the output current s regulated, are used n numerous applcatons. The dynamcal ssues of the currentoutput converters have been prevously analyzed only n a few papers [74], [75] and [76], but they have not provded the true nternal dynamcal profle due to the use of the resstve load. In ths research, the true nternal or nomnal profle of the currentoutput converter was derved by applyng conventonal modelng methods and from the two-port representaton of the correspondng voltage-output converter by applyng dualty. The parameter set was composed of modfed y-parameters and two specal admttance parameters. The nomnal load for the current-output converters was found out to be a pure voltage source. The load and supply nteracton formalsms were derved and t was found that a low mpedance load recovers the nomnal features. The dynamcal revew of the CC-VMC and CC-PCMC converters was carred out. The prevously observed pecular phenomenon of the ncreasng crossover frequency n the loop gan, when havng the low mpedance load (.e. typcally a back-up 112

130 Chapter 6 Conclusons battery), was dscovered to be the result of a wrong ntal modelng wth the resstve load. Both analytcal and expermental analyses were provded to confrm ths. The man contrbutons of the thess can be summarzed as follows: The concept of dynamcal profle was proposed and shown to be an effcent tool and framework for analyzng stablty, performance and external nteractons of a swtched-mode converter ndependent of the topology or control mode. A buck-converter under three control modes was used as a source of evdence supportng the exstence of the dynamcal profle. It was shown that the open-loop parameters fully characterze the defned dynamcal profle. The closed-loop output and nput mpedances can be used to stablty assessment under external nteractons. The use of a resstve load as the nomnal load was shown to be a wrong approach. The nomnal loads of the voltage- and current-output converters were stated to be a constant-current snk and a pure voltage source, respectvely. The dynamcal profle of the current-output converter was presented for the frst tme. The profle was obtaned from the two-port representaton of the voltage-output converter by applyng dualty. The a pror nformaton from the voltage-output converter makes ths approach extremely useful. It was notced that the most sutable nomnal practcal load for the currentoutput converter was a RC-crcut, where the large capactor behaved as a dynamcal short-crcut at the hgher frequences exposng the nternal dynamcs. The ttle of ths thess ncludes an open queston. Many examples and aspects presented n the thess provded the answer and explctly proved that the unque dynamcal profle of any gven converter s a fact not a fcton. 113

131 Chapter 6 Conclusons 6.3 Future Topcs At the moment the concept of the dynamcal profle s manly appled only n the academa. The natural nterest should be, however, to make practcal the use of the dynamcal profle and to promote ts superorty as a valuable tool n analyzng and ensurng performance and stablty. The lack of understandng and knowledge of the dynamcal ssues among the engneers are the man obstacles n makng the use of the dynamcal profle as a common tool n the ndustry. Tranng courses and semnars should be organzed and targeted on the ndustry. Ths thess analyzed only the buck converter under the three dfferent control modes. However, there are numerous topologes and control prncples whch are used n varous applcatons wthout completely understandng ther dynamcal behavor. Therefore, at least a boost and buck/boost converter and ther (solated) dervates should be analyzed. Consequently, the dynamcal profles should be derved to every new topology and/or control prncple that s ntroduced. The cascaded control was not consdered n the thess. The proposed methods provde a faclty to analyze the dynamcs of a converter, where the voltage and current loops are cascadng. Also, the relaton between the frequency and tme domans assocated wth transent response should be studed. The use of dgtal control has begun to replace ts analog counterparts. The dgtal control can provde e.g. adaptve control, but the effects of the dgtalzaton of the control block on the converter dynamcs have not been extensvely analyzed. The analog and correspondng dgtal dynamcal profle should be compared and analyzed n order to understand the effects of dgtal control. A potental research topc s also the effect of the exctaton sgnal on the transfer functons, when makng the measurements. A snusodal sgnal s typcally used as the exctaton sgnal, but t s observed to cause phase lag n hgher frequences f the sgnal goes through the modulator. Ths effect should be put n a mathematcal form and dfferent exctaton sgnals should be used to compare the phase lags and other propertes n order to fnd the best type of sgnal to be njected nto a loop n order to get accurate frequency responses. 114

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140 Appendces Appendx A %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Parameters needed for the dynamcal revew of VMC-CCM % % buck converter % % Copyrght Mkko Hankanem % % Tampere Unversty of Technology % % Insttute of Power Electroncs % % 27 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s=tf('s'); % Specfyng the aplace varable "s" % Inductor and capactor values and ther ESRs C=316e-6; r_c=33.3e-3; =18e-6; r_=6e-6; % Swtch on-tme resstance r_ds r_ds=.4; % Dode on-tme resstance r_d and max forward voltage Ud r_d=.55; Ud=.3; Un=5; % Input voltage 2V-5V Uout=1; Iout=2.5; % Nomnal load R=Uout/Iout; % Value of the load resstor needed to obtan the nomnal load current Ts=1/(1); % Swtchng perod D=(Uout+(r_+r_d)*Iout+Ud)/(Un+Ud+(r_d-r_ds)*Iout); % Duty-rato I=Iout; %Controller component values and transfer functon Rb=3.3e3; R1=7.87e3; R2=3; R3=1.15e3; C_1=47e-9; C_2=47e-12; C_3=11.6e-9; K=1/(R2*(C_1+C_2)); Gcc=(K*((1+s*R1*C_1)*(1+s*(R2+R3)*C_3)))/(s*(1+s*R3*C_3)*(1+s*((R1*C_1*C_2)/(C_1+C_2)) )); Vm=1; % Modulator gan, n practcal mplementaton Vm=3 % Denomnator of the transfer functons.e. g-parameters: den=(s^2+((s*(r_+r_c+(r_ds*d)+(r_d*(1-d))))/)+(1//c)); % Nomnal/nternal open-loop g-parameters: Gco_vmc_ccm_nom=(((((Un+Ud+(r_d-r_ds)*I)*(1+s*r_c*C))//C)/den)/Vm); Gc_vmc_ccm_nom=((((((D*(Un+Ud+(r_d-r_ds)*I))*s)/)/den)+I)/Vm); Zoo_vmc_ccm_nom=((((r_+r_ds*D+r_d*(1-D)+s*)*(1+s*r_c*C))//C)/den); Too_vmc_ccm_nom=(((D*(1+s*r_c*C))//C)/den); Goo_vmc_ccm_nom=Too_vmc_ccm_nom; Yno_vmc_ccm_nom=((((D^2)*s)/)/den); % Nomnal/nternal closed-loop parameters: oop_nom=gcc*gco_vmc_ccm_nom; YnC_vmc_ccm_nom=Yno_vmc_ccm_nom- (oop_nom/(1+oop_nom))*((goo_vmc_ccm_nom*gc_vmc_ccm_nom)/gco_vmc_ccm_nom); ToC_vmc_ccm_nom=Too_vmc_ccm_nom- (oop_nom/(1+oop_nom))*((zoo_vmc_ccm_nom*gc_vmc_ccm_nom)/gco_vmc_ccm_nom); GoC_vmc_ccm_nom=Goo_vmc_ccm_nom/(1+oop_nom); ZoC_vmc_ccm_nom=Zoo_vmc_ccm_nom/(1+oop_nom); 123

141 Appendces % Specal admttance parameters: Ynf=(Yno_vmc_ccm_nom-((Goo_vmc_ccm_nom*Gc_vmc_ccm_nom)/Gco_vmc_ccm_nom)); Ynf_real=-D*I/Un; Ynsc=Yno_vmc_ccm_nom+((Goo_vmc_ccm_nom*Too_vmc_ccm_nom)/Zoo_vmc_ccm_nom); Ynsc_real=(D^2)/(r_+D*r_ds+(1-D)*r_d+s*); % oad crcut parameters: _=18e-6; C_=2.35e-3; r_c_c=1e-3; r_8k=8e-3; % oad transfer functons Z_1=(s*_+((1/(s*C_))+3*r_C_C)); Z_4=(s*3*_+((1/(s*.2*C_))+.1*r_C_C)); Z_5=(s*2*_+((1/(s*.2*C_))+r_C_C)); Z=1; % Set the desred load mpedance here! % Resonance at 1 Hz % Resonance at 4 Hz % Resonance at 5 Hz % Supply-flter parameters 1=11e-6; 2=2e-6; 3=2e-6; r_f=.1; C1=5e-6; C2=13e-6; C3=1e-6; r_cf=.1; % Flter transfer functons % Resonance at 68 Hz Z_f68=((r_f+s*1)*(1+s*r_Cf*C1))/(s^2*1*C1+s*(r_Cf+r_f)*C1+1); % Resonance at 99 Hz Z_f99=((5*r_f+s*2)*(1+s*1*r_Cf*C2))/(s^2*2*C2+s*(1*r_Cf+5*r_f)*C2+1); % Resonance at 11 khz Z_f11k=((2*r_f+s*3)*(1+s*1*r_Cf*C3))/(s^2*3*C3+s*(1*r_Cf+2*r_f)*C3+1); Zs=1; % Set the desred supply mpedance here! % oad nteractons %%%%%%%%%%%%%%%%%%%% % Open-loop: Yno_vmc_ccm_=Yno_vmc_ccm_nom+ ((Goo_vmc_ccm_nom*Too_vmc_ccm_nom)/(Z+Zoo_vmc_ccm_nom)); Too_vmc_ccm_=(Z*Too_vmc_ccm_nom)/(Z+Zoo_vmc_ccm_nom); Goo_vmc_ccm_=Goo_vmc_ccm_nom/(1+(Zoo_vmc_ccm_nom/Z)); Zoo_vmc_ccm_=Zoo_vmc_ccm_nom/(1+(Zoo_vmc_ccm_nom/Z)); Gc_vmc_ccm_=Gc_vmc_ccm_nom+(Gco_vmc_ccm_nom*Too_vmc_ccm_nom)/(Z+Zoo_vmc_ccm_nom); Gco_vmc_ccm_=Gco_vmc_ccm_nom/(1+(Zoo_vmc_ccm_nom/Z)); % Closed-oop: oop_=oop_nom/(1+(zoo_vmc_ccm_nom/z)); YnC_vmc_ccm_=Yno_vmc_ccm_- (oop_/(1+oop_))*((goo_vmc_ccm_*gc_vmc_ccm_)/gco_vmc_ccm_); ToC_vmc_ccm_=Too_vmc_ccm_- (oop_/(1+oop_))*((zoo_vmc_ccm_*gc_vmc_ccm_)/gco_vmc_ccm_); GoC_vmc_ccm_=Goo_vmc_ccm_/(1+oop_); ZoC_vmc_ccm_=Zoo_vmc_ccm_/(1+oop_); % Supply nteractons %%%%%%%%%%%%%%%%%%%% % Open-loop: Yno_vmc_ccm_Snom=Yno_vmc_ccm_nom/(1+Zs*Yno_vmc_ccm_nom); Too_vmc_ccm_Snom=Too_vmc_ccm_nom/(1+Zs*Yno_vmc_ccm_nom); Goo_vmc_ccm_Snom=Goo_vmc_ccm_nom/(1+Zs*Yno_vmc_ccm_nom); Zoo_vmc_ccm_Snom=((1+Zs*Ynsc)/(1+Zs*Yno_vmc_ccm_nom))*Zoo_vmc_ccm_nom; Gc_vmc_ccm_Snom=Gc_vmc_ccm_nom/(1+Zs*Yno_vmc_ccm_nom); Gco_vmc_ccm_Snom=((1+Zs*Ynf)/(1+Zs*Yno_vmc_ccm_nom))*Gco_vmc_ccm_nom; % Closed-oop: oop_snom=((1+zs*ynf)/(1+zs*yno_vmc_ccm_nom))*oop_nom; YnC_vmc_ccm_Snom=Yno_vmc_ccm_Snom- (oop_snom/(1+oop_snom))*((goo_vmc_ccm_snom*gc_vmc_ccm_snom)/gco_vmc_ccm_snom); ToC_vmc_ccm_Snom=Too_vmc_ccm_Snom- (oop_snom/(1+oop_snom))*((zoo_vmc_ccm_snom*gc_vmc_ccm_snom)/gco_vmc_ccm_snom); 124

142 Appendces GoC_vmc_ccm_Snom=Goo_vmc_ccm_Snom/(1+oop_Snom); ZoC_vmc_ccm_Snom=Zoo_vmc_ccm_Snom/(1+oop_Snom); % Double nteractons %%%%%%%%%%%%%%%%%%%%%%% % Open-loop: Yno_vmc_ccm_S=Yno_vmc_ccm_/(1+Zs*Yno_vmc_ccm_); Too_vmc_ccm_S=Too_vmc_ccm_/(1+Zs*Yno_vmc_ccm_); Goo_vmc_ccm_S=Goo_vmc_ccm_/(1+Zs*Yno_vmc_ccm_); Zoo_vmc_ccm_S=((1+Zs*Ynsc)/(1+Zs*Yno_vmc_ccm_))*Zoo_vmc_ccm_; Gc_vmc_ccm_S=Gc_vmc_ccm_/(1+Zs*Yno_vmc_ccm_); Gco_vmc_ccm_S=((1+Zs*Ynf)/(1+Zs*Yno_vmc_ccm_))*Gco_vmc_ccm_; % Closed-loop: oop_s=((1+zs*ynf)/(1+zs*yno_vmc_ccm_))*oop_; YnC_vmc_ccm_S=Yno_vmc_ccm_S- (oop_s/(1+oop_s))*((goo_vmc_ccm_s*gc_vmc_ccm_s)/gco_vmc_ccm_s); ToC_vmc_ccm_S=Too_vmc_ccm_S- (oop_s/(1+oop_s))*((zoo_vmc_ccm_s*gc_vmc_ccm_s)/gco_vmc_ccm_s); GoC_vmc_ccm_S=Goo_vmc_ccm_S/(1+oop_S); ZoC_vmc_ccm_S=Zoo_vmc_ccm_S/(1+oop_S); 125

143 Appendces Appendx B %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Mxed-Data Controller Desgn % % Copyrght Mkko Hankanem % % Tampere Unversty of Technology % % Insttute of Power Electroncs % % 27 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Measured control-to-output transfer functon data are converted to % a complex number gan_1=1.^((gan)./2); phase_f=(phase/18)*p; Gco_compx=gan_1.*(cos(phase_f)+j*sn(phase_f)); % Analytcally desgned controller transfer functon Gcc s % converted to a complex number w=2.*p.*f; [mag ang]=bode(gcc, w); mag1=mag(:,:); ang1=ang(:,:); ang1_f=(ang1/18)*p; f1=w/2/p; Gcc_compx=mag1.*(cos(ang1_f)+j*sn(ang1_f)); % Calculatng the loop gan oop=gcc_compx.*gco_compx; % Extractng the magntude n decbels and phase n degrees from the % complex number oop oop_g=abs(oop); oop_gdb=2*log1(oop_g); oop_ang_f=atan2(mag(oop),real(oop)); oop_ang=18*oop_ang_f/p; 126

144 Appendces Appendx C %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Nomnal Gco Calculaton from Measured Data % % Based on Mxed-Data Method % % Copyrght Mkko Hankanem % % Tampere Unversty of Technology % % Insttute of Power Electroncs % % 27 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Resstve load R=4; % Measured control-to-output transfer functon (wth resstve load) % data are converted to a complex number gan_1=1.^((gan)./2); phase_f=(phase/18)*p; GcoR_compx=gan_1.*(cos(phase_f)+j*sn(phase_f)); % Measured open-loop output mpedance (nomnal) % data are converted to a complex number gan_2=1.^(gan2./2); phase_f2=(phase2/18)*p; Zoo_compx=gan_2.*(cos(phase_f2)+j*sn(phase_f2)); % Computng the nomnal/nternal Gco basng on the load nteracton % formalsm Gco=GcoR_compx.*(1+(Zoo_compx./R)); % Extractng the magntude n decbels and phase n degrees from the % complex number oop Gco_g=abs(Gco); Gco_gdB=2*log1(Gco_g); Gco_ang_f=atan2(mag(Gco),real(Gco)); Gco_ang=18*Gco_ang_f/p; 127

145 Appendces Appendx D Auxlary power supples Osclloscope Current probes Computer nterface Electronc load Buck converter Injecton transformers Frequency response analyzer and lnear amplfer Fg. D.1. aboratory test setup. Power stage VMC-CCM & VMC-DCM PCMC & PCMC-OCF VMC & PCMC of current-output converter Fg. D.2. Power stage and control crcuts of the expermental buck converter. 128