Piezoelectric Transformers in Power Electronics

Transcription

1 Ben-Guion Univesity of the Negev Depatent of Electical and Copute Engineeing Piezoelectic Tansfoes in Powe Electonics Thesis subitted in patial fulfillent of the equieents fo the degee DOCTOR OF PHILOSOPHY by Svetlana Bonstein Subitted to the Senate of Ben-Guion Univesity of the Negev Octobe 005 תשס "ו תשרי Bee-Sheva

2 ABSTRACT The inceasing utilization of piezoelectic tansfoes (PT) in powe electonics equies fundaental analysis and design of PT powe convetes. In addition, the possibility of ovecoing the essential liitations of PT powe ability by a cobined opeation of PTs needs to be exained. These studies could povide wide oppotunities fo PT exploitation in powe electonic cicuits. The ain objective of this eseach is to develop geneic odels of PT powe convetes. We exaine the output and input netwoks of the PT convete by utilizing a PT equivalent cicuit. Obtaining PT equivalent cicuit paaetes fo physical easueents is a pobleatic issue. Theefoe, the fist concen of the pesent wok is to ty to ipove the ealie published ethods of PT paaetes deteination in ode to ake the suitable fo evey kind of PT and to the conditions unde which the given PT opeates. The study poposes and veifies two novel ethods fo extacting PT equivalent paaetes fo easueents. A copaative analysis of the diffeent types of ectifies connected to the PT output is caied out in ode to obtain geneic odels of PT AC/DC powe convetes. The ethodology applied in this study follows the appoach in which the PT output ectifie netwok is epesented by an equivalent cicuit that loads the PT as the coesponding ectifie. The unique natue of a PT allows innovative cicuit design that ensues soft switching without atching inductos. This design substantially educes the costs of electonic coponents in the cicuit. Theefoe anothe ai of the eseach is to develop a geneic appoach fo achieving soft switching with no additional eactive eleents at the input. To coplete the wok, the geneal equieents fo the cobined opeation of seveal PTs ae investigated. Keywods: piezoelectic tansfoe, equivalent cicuits, paaete estiation, geneic appoach, ectifies, soft switching, powe supplies.

3 CONTENTS Chapte. Intoduction. Backgound - A. Piezoelectic eleents - B. Opeational pinciples -3 C. Physical stuctue of PTs -3. Models fo piezoelectic tansfoe powe convetes -5 A. Electic equivalent cicuit -5 B. Geneal opeation chaacteistics of PT -6 C. Input netwoks -7 D. Output netwoks -7 E. Powe liitations -7.3 Motivation -9.4 Objective of the eseach and ethods of appoach -0.5 Dissetation outline and ajo esults -.6 Publications - Chapte. Estiating the paaetes of the PT equivalent cicuit. Shot-cicuit test -. Measueent pocedue -.3 Measueent of adial vibation ode PT saple -4.4 Tansfe function test -6.5 Extaction of odel paaetes -8.6 Measueent of a Rosen-type PT saple -.7 Conclusions -7 Chapte 3. A piezoelectic tansfoe AC/DC convetes' output netwoks I 3. Analysis and design of a piezoelectic tansfoe AC/DC convete in a low voltage application 3-3. Pinciple of opeation of a half-way two diodes ectifie The unified analysis of ovelapping and non-ovelapping odes 3-4

4 A. Lossless PT (R =0) 3-4 B. Real PT (R >0) 3-7 C. Diodes losses 3-8 D. Voltage ange and powe chaacteistics of a eal convete Expeiental esults Conclusions 3-6 Design Exaples 3-7 Chapte 4. A piezoelectic tansfoe AC/DC convetes' output netwoks II 4. A copaison of piezoelectic tansfoe AC/DC convetes with cuent-double and voltage-double ectifies 4-4. Copaative cicuits and copaison pinciples 4- A. Voltage-double ectifie 4-5 B. Cuent-double ectifie 4-7 C. Chaacteistics of the copaed ectifies Geneal equations of a PT loaded by an output ectifie Main copaison esults Siulation and Expeiental Results Conclusions 4-9 Chapte 5. Design consideations fo achieving ZVS in a half-bidge invete fo diving a piezoelectic tansfoe with a no seies inducto 5. Analysis and Design of ZVS PT Powe Invete 5- A. Analysis of ZVS condition fo half-bidge invete 5- B. The ain assuptions 5-4 C. Noalized odel fo a soft switching PT invete 5-5 D. Design guidelines Siulation and expeiental esults Conclusions 5-3 Chapte 6. Cobined opeation of Piezoelectic Tansfoes 6. Paaetes of a single PT 6-6. Possible connection of PTs fo cobined opeation (identical PTs) Paallel-paallel connection of non-identical PTs 6-5

5 A. The influence of non-equality of C o 6-5 B. The Influence of non-equality of R 6-7 C. The Influence of non-equality of 6- D. The Influence of non-equality of C and L while is kept constant Siulations and Expeients Conclusions 6-6 Conclusions and Futue Wok Refeences

6 ASKNOWLEDGEMENTS I would like to thank fist and foeost y Ph.D. adviso Pof. Sa Ben-Yaakov, fo his invaluable guidance and continued suppot duing the couse of this eseach wok. I a also exteely gateful to Pof. Gegoy Ivensky fo his help and encouageent. Without his assistance, I would have neve accoplished this wok. Special thanks go to all the PEL students and staff fo thei help duing y gaduate caee. Last, I would like to expess y pofound gatitude to y faily fo thei constant suppot and encouageent in y educational pusuits.

7 CHAPTER INTRODUCTION Powe electonic cicuits have taditionally been based on agnetic technology, and until ecently, have not been pat of the tide of iniatuization and integation advances fo which signal-pocessing integated cicuits have benefited. The tansfoes and inductos in conventional convetes ae usually bulky copaed to othe eleents. In an effot to iniatuize powe coponents a piezoelectic, athe than a agnetic tansfoe could be eployed in the powe supplies. Utilization of piezoelectic tansfoes in powe electonics becae possible owing to new piezoelectic ateials that have ecently been developed. These ateials exhibit ipoved piezoelectic ceaic chaacteistics. In the past few decades piezoelectic tansfoes used widely in any applications, such as DC/DC convetes, adaptes, and electonic ballasts fo fluoescent laps... BACKGROUND A. Piezoelectic Eleents Piezoelectic ceaics ae chaacteized as sat ateials and have been widely used in the aea of actuatos and tansduces. Both piezoelectic actuatos and tansduces ae ade of piezoelectic ceaic eleents. A piezoelectic eleent is coposed of two electode plates and a piezoelectic ceaic ateial. It can be side o end-plated and ay be built as single o ulti-laye stuctue (Fig..). Multi-laye constuction inceases entie capacitance of the ceaic eleent and deceases the volue of each laye and thus echanical losses. Geneally speaking, piezoelectic eleents can opeate in eithe a longitudinal o tansvese ode with a coesponding esonant fequency. In the longitudinal ode, the diection of the echanical stess, T, is paallel to the electic o polaization diection, P, as shown on Fig... In the tansvese ode the diection of the echanical stess, T, is pependicula to the electic o polaization diection, P, as shown in Fig..3. A piezoelectic eleent can opeate as eithe an actuato o a tansduce. -

8 Piezoelectic ceaic ateial Side-plated ba Piezoelectic ceaic ateial End-plated ba (a) (b) Fig.. Piezoelectic eleent. This eleent is coposed of two electode plates (side (a) o end plated (b)), ade out of a piezoelectic ceaic ateial. The plate can be single (a, b) o ulti-layeed (c). V P T Fig.. Longitudinal ode piezoelectic eleent. In this type of eleent, the diection of the opeating stess, T, is paallel to the polaization diection, P, with a coesponding esonant fequency. V P T Fig..3 Tansvese ode piezoelectic eleent. In this type of eleent, the diection of the opeating stess, T, is pependicula to the polaization diection, P, with a coesponding esonant fequency. -

9 B. Opeational Pinciples The opeation pinciple of a piezoelectic tansfoe (PT) is that actuatos and tansduces function togethe so that enegy can be tansfoed fo an electical fo to an electical fo via echanical vibation. In othe wods, in a piezoelectic tansfoe the convese and diect piezoelectic effects (in the actuato and tansduce, coespondingly) ae used to acoustically tansfo powe fo one voltage and cuent level to anothe though a vibating stuctue. The convese piezoelectic effect, in which an applied electic field poduces a esulting stain in a body, is used to convet an oscillating electic field to the piay pat of a stuctue into a vibation ode of the entie ba. When diven at esonance, alost standing wave distibutions of lage aplitudes of stess and stain esult. The esonantly aplified stain in the piay half of the ceaic ba is conveted to a voltage acoss the output teinals by a diect piezoelectic effect. Depending upon the geoetical and ateial paaetes, voltage aplifications of vaious agnitudes can be obtained, with associated step-ups/downs in cuent levels. C. Physical Stuctue of PTs On the whole, piezoelectic tansfoes can be classified into diffeent categoies based on thei vibation odes o opeating fequencies. They ae coonly classified into thee ain goups: high voltage o Rosen-type PT [-7], low voltage o thickness vibation ode PT [8-3], and adial vibation ode PT [4-6] (Figs..4,.5, and.6 espectively). The Rosen-type PT is ade by a cobination of a tansvese ode piezoelectic actuato and a longitudinal ode piezoelectic tansduce. Vibation occus along its length diection and its esonant fequency is athe low. Because of the inheent high voltage gain associated with the Rosen-type PTs, they ae coonly efeed to as high voltage PTs. To achieve even highe voltage gain, the actuato, o tansduce, o both of the Rosen-type PT ay be ade ulti-layeed [, 3, 5-6]. The ost suitable application fo a Rosen-type PT is a dive fo high voltage laps, such as cold cathode fluoescent laps used as a backlight souces fo flat panel displays of notebook coputes [4-5, 7-8]. The thickness vibation ode PT is a cobination of a longitudinal ode actuato and a longitudinal ode tansduce and both can be constucted as a single o a ulti-layeed stuctue. This type of PT is also known as a low voltage PT because of its inheent low -3

10 voltage gain. Since the vibation is occus along its height the esonant fequency is high. Its ain applications include DC/DC convetes and adaptes [9-]. The adial vibation ode PT is a cobination of piezoelectic actuato and tansduce (single o ulti-laye) both of which opeate in the tansvese ode. This type of PT can be used in diffeent applications such as DC/DC convetes, adaptes, and electonic ballasts [3-5]. V in P T P T V out Fig..4. Rosen-type piezoelectic tansfoe. Poling diection Output Input V out P T V in P T O 4 O 3 O O I I (a) (b) Fig..5. Thickness vibation piezoelectic tansfoe: (a) single-laye, (b) exaple of a ulti-laye stuctue. T V out P P V in Fig..6. Radial ode piezoelectic tansfoe. -4

11 .. MODELS FOR PIEZOELECTRIC TRANSFORMER POWER CONVERTERS A. Electic Equivalent Cicuit It is obvious that to analyze and design a PT-based convete, a PT odel has to be ceated in the fo of a siple electic cicuit. Geneally, the behavio of PT can be epesented by a tansission line with distibuted paaetes, which in tun would be tansfeed into a laped paaetes cicuit [5-6, 6, 4, 6]. Since a PT is a non-linea eleent and its paaetes ae stongly dependent on the opeational conditions such as the electic field, echanical stess, tepeatue, depolaization, aging etc., ceating a convenient electical equivalent cicuit that will epesent the PT behavio, is not a siple atte. Such a cicuit could estiate the equivalent PT paaetes only unde the vey naow ange of fequencies and electical loads. Seveal studied descibe the non-linea behavio of piezoelectic ateials [7-9], but the odels ae quite coplicated. The PT is epesented as an electo-echanical syste that is descibed by a high ode syste of diffeential equations. This appoach is essentially theoetical and does not attept to obtain the siple electical equivalent cicuit. Because of the diffeent vibation odes and echanical stuctues, the vaious PT categoies have diffeent echanical and electical chaacteistics. Howeve, fo a specific fequency bandwidth aound the coesponding echanical esonant fequency and a naow load ange, a single-banch equivalent cicuit odel can be developed (as shown in Fig..7), which can chaacteize diffeent types of PTs. In this odel the cicuit is identical to a seiespaallel esonant netwok, which has been widely applied to a esonant convete, invete o electonic ballast cicuits. The paaetes L, C, and R epesent the echanical behavio of the PT; C in and C o ae the input and output dielectic capacitances; and n is the gain that epesents electical-echanical and echanical-electical enegy tansfoations. In geneal, these tansfoations can be epesented by eithe of two equivalent cicuits as shown on Fig..8, both of which avoid DC coupling shoting as occus in PT. Howeve, the Type netwok puts an inductance L in seies with a dependent cuent souce, as well as a capacitance C o in paallel to a dependent voltage souce, which ay lead to convegence -5

12 pobles in PSPICE siulations. Since this study uses PSPICE siulations fo suppoting analytical odels of PT convetes, the Type equivalent netwok was applied. I L C R V in C in V o n + - I n C o V o Fig..7 Equivalent cicuit of PT. C -L -R ae the equivalent electical paaetes that epesent the echanical esonant cicuit of PT, C in and C o ae the input and the output capacitances, n is the gain. i i i v i V V n n ni nv V V Type Type Fig..8 Models fo ideal tansfoe. Diffeent echanical stuctues associated with diffeent types of PT esult in diffeent equivalent cicuit paaetes. Vaious studies [5, 4, 6] have deonstated how to obtain this equivalent cicuit and how to estiate its paaetes fo the echanical diensions and ceaic ateial constants. Howeve the non lineaity of the paaetes of piezoelectic ceaic ateials confines these esults to a vey naow voltage-tepeatue-load ange. In addition, thee ae soe uncetainties egading the estiation of electodes esistance and disposition. All these factos indicate that fo estiating PT behavio in the electic cicuit, its equivalent paaetes need to be obtained expeientally. -6

13 The poble of obtaining PT equivalent paaetes fo physical easueents has been widely discussed in the liteatue, but still eains an open issue. The ajoity of the poposed ethods ae based on the shot-cicuit test, which ceates conditions that hadly eseble eal opeational conditions. Soe woks [30-3] use the tansfe function test, but the pocess of extacting the equivalent paaetes fo expeiental esults with this test is fa too coplicated. One of the objectives of the pesent investigation is to develop a siple pocess to find equivalent PT paaetes fo voltage gain and phase easued unde pactical opeational conditions. B. Geneal Opeation Chaacteistics of PT A geneic PT odel was developed in [3]. The odel consists of a syste of closed fo appoxiate equations that descibe the ain PT chaacteistics (efficiency, voltage atio, output powe) as a function of load and all PT paaetes. This geneic odel poved to be vey useful in engineeing design. The pesent study uses this appoach fo futhe developent of geneic odels of PT powe convetes and invetes. C. Input Netwoks A geneal appoach to designing input and output atching netwoks is pesented in [4, 3]. The ain equieent fo input netwok is that thee ae inial losses in the switches. This can be achieved by soft switching. Most of the applications use additional inductive o capacitive-inductive netwoks at the input in ode to achieve zeo-voltage switching conditions [9,, 33-34]. Howeve, a disadvantage when using this appoach is that PT specific popeties ae not fully utilized. The unique natue of PT povides the oppotunity fo innovative cicuit design that allows soft switching without copensating inductos. This eans that PTs can be eployed without any additional agnetic devices. Recent publications [3-4] have shown the effectivity of PT powe convetes built with no additional inductance in the input netwok. The issue of a geneic appoach to achieve soft switching with no additional eactive eleents at the input is still an open poble and is one of the focuses of the pesent study. -7

14 D. Output Netwoks Low output voltage PT convetes ae usually built on a thickness o adial vibation ode PTs with two diode ectifie netwok. The convetes use eithe a half-way o a cuent double ectifie with an inductive-capacitive filte (Figs..9,.0). Convetes fo high output voltage typically use a Rosen-type PT and ectifie with capacitive filte as shown on Figs.. and.. Diffeent kinds of ectifies wee widely discussed in the liteatue [4-5, 33, 35-36], but thee wee no publications exist on the geneal opeational chaacteistics of a PT AC/DC convete. The advantages and disadvantages of each type of output netwoks and thei chaacteistics when opeating in a PT convete is anothe issue we addess. E. Powe Liitations PTs ae especially suitable fo low powe applications, since its powe ability is liited by echanical losses. Fo highe powe tansfo, in this eseach we suggest using the cobined opeation of seveal PTs. Its possibilities and geneal equieents ae exained in this wok. D Lf PT Vin L o D Cf RL Fig..9 Low voltage PT convete with half-way two diodes ectifie. L V in PT D D R L Cf L Fig..0 Low voltage PT convete with cuent double ectifie. -8

15 D D 3 PT V D 4 in D C f R L Fig.. High voltage PT convete with full way ectifie and capacitive filte. D V in PT D C f R L Fig.. High voltage PT convete with voltage double ectifie..3. MOTIVATION The iniatuization of powe supplies has been an ipotant issue in the powe electonics industy duing the last decade. The tansfoes and inductos used in conventional convetes ae usually bulky copaed to othe eleents. Theefoe low-pofile tansfoes [38] wee integated to educe the height and size of convetes. PTs have seveal inheent advantages ove low-pofile tansfoes, such as a vey low pofile, high powe density, no winding (which eans suitability fo autoated anufactuing), high insulation popeties, and potentially low cost. Nueous studies have concentated on the poble of how to substitute an electoagnetic tansfoe with a PT in powe supplies. Because of the adically diffeent natue of PT (esonant tank with high quality facto and essential capacitances at the input and the output) diect eplaceent of an electoagnetic tansfoe by PT in the electic cicuit is ipossible. As such, DC-DC convetes odels ae not appopiate fo PT powe supplies. -9

16 Pevailing odels of PT convetes usually elate to a concete PT and hence pesent only patial solutions to the poble. The inceasing populaity of PTs in powe electonics equies fundaental analysis and design of PT powe convetes, based on geneic chaacteistics of the device. In addition, the essential liitation of PT powe ability could be ovecoe by the cobined opeation of PTs. By ealizing these conditions PT technology can be widely exploited in powe electonic cicuits..4. OBJECTIVE OF THE RESEARCH AND METHODS OF APPROACH The objective of the pesent eseach is to develop a geneic odel of PT powe convetes and to ovecoe the poble of PT powe tansfe liitations by cobined opeation of a nube of PTs. We adhee to geneic PT odel and utilize the geneic PT convete odel fo analysis and design of a PT convete with the desied chaacteistics soft switching, axiu efficiency, and adequate voltage gain. This appoach otivated the following plan: ) To coplete the ethods of PT paaetes deteination that will suit to evey kind of PT and to the specific opeational conditions of a given PT. ) To develop geneic analytical odels of the pevailing ectifies. A copehensive copaison of the diffeent ectifies in PT powe convetes is to be caied out esulting in a ecoendation of the pefeed applications aeas fo each ectifie in tes of output cuent and voltage, powe handling capability, and load esistance. The esults of this study will be useful fo optial design of PT output netwoks o fo the optial design of PT fo convetes with desied chaacteistics. 3) To analyze the inheent soft switching capability of PTs. A geneic analytical odel should estiate the boundaies of soft switching in a powe invete/convete built aound a given PT and should povide a PT design fo soft-switching with no additional eleents at the input. 4) To study the ethods fo cobining identical and non-identical PTs, and to delineate the equieents fo cobined opeation of PTs. In paticula, we identify and quantify -0

17 the effect of non-identities in the PT paaetes on cuent shaing and oveall efficiency. This will pevent oveheating of one o oe of the cobined PTs..5. DISSERTATION OUTLINE AND MAJOR RESULTS This dissetation is divided into six chaptes, conclusions, and efeences. On the following we offe a bief desciption of each chapte. Chapte biefly eviews the backgound and application of piezoelectic tansfoes. Then, we set fo the objectives and eseach plan. Chapte is devoted to obtaining equivalent cicuit paaetes fo the physical easueents. Chaptes 3-4 discuss the behavio of a PT powe convete with diffeent kinds of output ectifies as a paticulaly odels of PT convetes opeating at eithe echanical esonant fequency o fequency of axiu output powe. Chapte 3 studies steady-state pocesses in a powe PT convete with a halfway two diodes output ectifie un at a echanical esonant fequency. It is shown that such a powe convete can opeate in two odes: with o without ovelap of the conduction intevals of the diodes. The analytical equations and chaacteistics deived in this study wee found to be in good ageeent with siulation and expeiental esults. The study highlights the effect of load and paasitic esistances on output voltage and the losses fo both odes. Expessions obtained in the study can be used in engineeing design. Chapte 4 pesents a copaative analysis of PT-based powe convetes that apply cuent double and voltage double types of ectifies and opeate at the fequency of axiu output voltage. The copaison ethodology follows an appoach developed ealie by which the output capacito of PT and the ectifie sections ae epesented by an equivalent cicuit. The equivalent cicuit is linea and is coposed of a paallel RC netwok that loads the PT as the coesponding ectifie does. We applied geneic analysis to copae two types of ectifie, yielding geneic odels of PT AC/DC powe convetes. Chapte 5 pesents a geneal pocedue fo aintaining soft switching in an inductoless half-bidge PT invete. Soft switching capability of the PT was exained and detailed guidelines ae given fo the load and fequency boundaies, voltage tansfe function and -

18 output voltage to aintain opeation unde ZVS conditions. The analysis takes into account the axiu powe dissipation of the PT that is used to contol the peissible powe tansfe though the device. The analytical esults fo the half-bidge inductoless PT invete ae veified by siulation and expeients fo a adial vibation ode PT. In Chapte 6 the behavio of paallel-connected PTs was analyzed by applying a geneic PT odel that epesents a cobined netwok. The influence of a deviation in each of the PT paaetes was exained with espect to total output powe and losses. It was found that the equieents fo identifying the equivalent cicuit paaetes R and C o ae not stong. The ost citical paaete fo equal loss in the PTs is the echanical esonant fequency of PTs, which has to be equal in the paallel units, even though the values of the paaetes C and L can diffe. Since esonant LCC-type invetes ae pactically identical to PT equivalent electical cicuit, the esults of this study ae elevant to cobined connected LCC invetes. The analytical odel veified by SPICE siulation and expeients. The expeients wee done on paallel-connected PTs (T-, Face Co., VA, USA) and LCC invetes. We coplete the study with a suay of the ain esults and soe suggestions fo futue eseach..6. PUBLICATIONS The thesis is based on ateial pesented in papes published in efeeed jounals and in poceedings of intenational confeences, efeence nubes [39-4, 44, 46]. -

19 CHAPTER ESTIMATING THE PARAMETERS OF THE PT EQUIVALENT CIRCUIT.. SHORT-CIRCUIT TEST A single-banch equivalent cicuit odel (shown in Fig..7) can chaacteize all types of PTs within a specific fequency bandwidth aound the coesponding echanical esonant fequency. The echanical diensions and ateial paaetes of PT deteine the paaetes of this cicuit [5-6, 6, 4, 6]. Howeve, equivalent paaetes should be obtained expeientally. The poble of obtaining PT equivalent paaetes fo physical easueents has been widely discussed in the liteatue, but still eains an open issue. Consideing the fact that a PT is a non-linea eleent, a linea netwok can descibe its behavio only ove sall fequency and cuent/voltage (load) anges. Measueent pocedues fo paaete extaction should theefoe be caied out aound the opeational fequency of the device and at noinal cuent levels when the paticula device is not that sensitive to the voltage level. The poposed ethod [39] extends the ethods pesented ealie [5, 4, 3]. When the input o output teinals of a PT ae shoted, the electical equivalent cicuit is identical to the Y-paaete cicuit odel shown in Fig... This Y-paaete cicuit has been used to easue and chaacteize PT equivalent cicuit paaetes. L Y in C in C R Fig.. Y-paaete equivalent cicuit of PT with shot cicuited output teinals. -

20 .. MEASUREMENT PROCEDURE Measueents have to be caied out fo the input and output sides of the PT while the copleentay teinals ae shoted. The equivalent cicuit paaetes of an expeiental PT ae obtained as follows: ) The total input and output capacitances C Tin and C To ae easued at low fequencies. These capacitances include in addition to C in and C o the seies capacitance C and all paasitic capacitances in paallel to the seies netwok C -L -R. ) The input adittance Y in (agnitude and phase) can be obtained by the setup shown in Fig.. applying the HP Netwok Analyze. The input voltage of the PT should be chosen so as to keep the input cuent within the opeational ange of the device. The fequency is swept aound the echanical esonant fequency. The easued data V A /V R is ecalculated to find the actual agnitude of adittance of the PT using the following expession: Y in 50 + R ( 50 + R ) i ef i R = (.) 50 VA 50 + R R ef V V V A R 3) The esonant f and anti-esonant f a fequencies, coesponding to the zeo iaginay pat of the adittance, ae found fo both input and output teinals (in/o), espectively. 4) Input and output capacitances ae estiated fo the easued values of C T, f and f a [5], as follows: C f (in / o) ( in / o) C T(in / o) f a(in / o) = (.) This equation is valid when the quality facto Q of the PT is sufficiently high (Q >300), and if the paasitic capacitances paallel to the R -C -L and C in netwok ae negligibly sall. 5) The seies cicuit paaetes C -L -R (in/o) ae calculated fo the easued agnitude and the phase of the PT adittance by the ethod pesented in [3] fo the piezoelectic esonant blade eleents. The ethod is based on two adittance values Y in (f i ), easued slightly befoe the esonant fequency. We calculate -

21 R C eqi pi = Re = C eqi { Y ( f )} in C T i ; ; C C eqi si I Y = = { ( f )} i [ + ( icp R ) ] i eqi C pi ( C R ) i in pi i eqi (.3) Applying f and f in (.3), we obtain paaetes C -L -R : L Cs C s Cs R eq = (.4) C C s s i i ; C = ; R = ( ) + C L + ( icp R eq ) i si i i 6) The tansfe atio n is extacted fo the calculated values of the seies inductances obtained fo the input and output easueents as follows: L (out) n = (.5) L (in) Aplifie A Netwok Analyze output input 50Ω 50Ω RF A R V out,ap. I PT R ef Y in R i PT Fig.. Expeiental setup. -3

22 .3. MEASUREMENT OF THE RADIAL VIBRATION MODE PT SAMPLE The expeiental setup used a HP4395A Netwok Analyze. The expeiental saple was a adial vibation ode PT, Face Co., VA, USA. The input s voltage was appoxiately V in =3V to keep the input cuent within opeational ange (50A). The easued data V A /V R (Fig..3) was ecalculated to find the actual agnitude of adittance by using the expeiental R i =Ω and R ef =466Ω. The fequency was swept fo 00 to 50 khz. Following the pocedue descibed above, the equivalent paaetes of the expeiental PT wee found to be as follows: C in =.7nF, C o =.33nF, L =0.5H, C =7.5µF, R =Ω, n=.08. Fig..3 shows the easued adittance (agnitude and phase) and the econstucted adittance obtained by PSPICE siulation of the PT equivalent cicuit applying the estiated odel paaetes. The slight diffeence between the expeiental and econstucted cuves in the egion of anti-esonant fequency is pobably due to inaccuacies in estiating the input capacitance C in. An eo in the value of this capacitance has a lage effect on f a than on f because of the high ipedance of the netwok in the anti-esonant egion. The value of C in can be coected by iteation but this is not essential because the noinal opeational egion of the PT is nea f whee the atching between the easued and econstucted adittance is excellent. -4

23 Y in,db ϕ, o x 0 5 f, Hz Fig..3 Input adittance (uppe agnitude, lowe phase) of the expeiental PT. Solid line easued, dotted line econstucted by siulation of PT equivalent cicuit based on the calculated equivalent cicuit paaetes. -5

24 .4. TRANSFER FUNCTION TEST In geneal, PT pefoance is dependent on the opeating conditions (voltage/cuent) and load. When diving at a high level of vibation in a esonant ode with a stong electic field and high echanical stess, paaetes of piezoelectic ateials becoe significantly nonlinea. Seveal attepts have been ade to ceate a non-linea PT odel o to coect the conventional equivalent PT cicuit of PT [7-9], but no significant ipoveent was achieved. The ajoity of ealie ethods fo obtaining PT equivalent paaetes ae based on the shot-cicuit test, which iposes conditions that ae fa fo eal opeational conditions. A few woks [30-3] used the tansfe function test, but the pocess of extaction of the equivalent paaetes fo the expeiental esults, was too coplicated. Futheoe, fo high esistance loads and high output voltages (which ae essential fo Rosen-type PTs) the shot cicuit test could not estiate the paaetes coectly because zeo load esistance and zeo output voltage could not eet noinal conditions. In addition, extaction of PT paaetes using only two points fo the entie easueent is not efficient. Although the equivalent cicuit obtained fo the two easued points of input ipedance of shoted PT adequately chaacteizes the aea aound the chosen points, it ight incoectly estiate othe aeas. To illustate this shotcoing, we estiated the equivalent paaetes of the Rosen-type PT (ELS-60) by algoith pesented in section.. They wee found to be: C =33pF, L =78H, R =77.3Ω, n=5.06, C in =700pF, C o =pf. Fig..4, which shows the easued and calculated input ipedance of the shoted PT, indicates that the aea aound the echanical esonant fequency is estiated accuately, but in geneal thee is poo ageeent between the theoetical and easued esults. PT paaetes ae dependent on nueous conditions that cannot be taken into account in one siple analytical odel. Theefoe the only way to accuately estiate PT behavio in a netwok is to extact the equivalent paaetes fo an expeient caied out unde noinal opeational conditions (noinal voltage and load anges). This can be accoplished by easuing the output to input voltage gain and phase of a loaded PT. -6

25 The objective of this section is to ceate a siple algoith fo obtaining the equivalent PT paaetes fo the easueents of the voltage atio (aplitude and phase) of the loaded PT, and to veify the dependence of these PT paaetes on the load esistance. Fig..4 Measued (cicles) and calculated (solid line) input ipedance of a PT with shoted output teinals. The PT equivalent paaetes fo the calculated cuve wee extacted fo the shot cicuit test. -7

26 .5. EXTRACTION OF MODEL PARAMETERS Measueents ae caied out fo the input and output sides as the copleentay teinals ae loaded by the noinal load esistances (R L at the output and R L /n at the input teinals). The voltage atio (agnitude and phase) can be obtained by the setup shown in Fig..5. The fequency is swept aound the echanical esonant fequency. The agnitude of voltage atio V out /V in is ecalculated fo the easued data V A /V R as follows: V V out in ef ( j) VA V R R ef (i) = 0 (.6) 50 + R whee i=, j= fo the fowad voltage atio and i=, j= fo the backwad voltage atio. The fowad voltage tansfe atio k o of a PT with a voltage souce connected at the input pot and a load connected at the output pot (Fig..6, a) is Vo k = = (.7) o Vin nr nco nl n n nlco j + nr Co R f C R f CR f whee is the angula opeating fequency =πf, and R f =R ef +50Ω is the fowad load esistance. The eal and the iaginay pats of the inveted voltage atio k f =/k o ae Re I k ( k ) f ( ) f nr = n + R nl = R f f nc + C + nr o C o nl C o n CR f (.8) The backwad voltage atio k i of a PT with a voltage souce connected at the output pot and a load connected at the input pot (Fig..6, b) is V o k = = (.9) i Vin nr ncin nl n nlcin j + nr Cin n R C R CR -8

27 whee R =R ef +50Ω is the backwad load esistance. The eal and iaginay pats of the inveted atio k =/k i ae given espectively by the following equations: Re I k ( k ) ( ) = nr + n R nl = R nc + C + nr in C in nl C n CR in (.0) Aplifie A V out,ap. Netwok Analyze output input 50Ω 50Ω RF A R R ef V in PT V out R ef Fig..5. Expeiental setup fo PT equivalent paaete easueents. The aplifie is connected to the input/output sides while the copleentay teinals ae loaded by the noinal load esistances R ef and R ef. -9

28 I L C R V V C o + I in n - C in n R V o f o (a) I L C R V o R C in V o n + - I n C o V in (b) Fig..6. The equivalent cicuit of the loaded PT: (a) when the voltage souce is connected at the input pot and load the esistance is connected at the output pot, (b) when the voltage souce is connected at the output pot and the load esistance is connected at the input pot. Applying Eq. (.8) and (.0) we obtain expessions fo tangents of the phases of the fowad and backwad voltage atios as follows: nl n nr Co I ( ) ( k ) + f R f CR f tan ϕ f = = (.) Re( k f ) nr nco n + + nlco R C f nl n nr Cin I ( ) ( k ) + R CR tan ϕ = = (.) Re( k ) nr ncin + + nlcin n R C -0

29 These expessions can be tansfoed into ( ) ( LC + R CoCR f ) 3 ( C R + R C + C R ) LCoCR f tan ϕ f = (.3) f o ( L C + R C C R ) f in tan( ϕ ) = (.4) 3 C R + R C + CinR LCinCR n Applying a = L CoCRf a = L C + R CoCR f a 3 = C Rf + R C + CoRf (.5) and b = L CinCR b = L C + R CinCR b 3 = CR + R C + CinR (.6) n we siplify (.3-.4) to a b 3 3 tan tan ( ϕf ) + a a 3tan( ϕf ) ( ϕ ) + b b tan( ϕ ) 3 = = (.7) Solving (.7) by the least ean squae ethod using the easued phase of the tansfe function and applying the attained a-b values to (.5-.6) we obtain six equations that allow us to ake an initial estiation of six PT paaetes. Howeve, because of the ational behavio of the phase tangent the estiated aplitude is popotional to the actual aplitude of the voltage atio with a constant coefficient k. A constant k can be obtained fo the easued voltage atio k o(exp) and the estiated voltage atio k o(est) as k=k o(exp) /k o(est). Applying (.)-(.), taking into account that phase angle is estiated coectly and assuing that C o and C in do not depend on the load esistance, we can see that paaetes R and L ae popotional to k, C is popotional to /k and n is popotional to k. Hence, the actual PT equivalent paaetes ae ecalculated fo the initial estiation as follows: R =k R (ini), L =k L (ini), C =C (ini) /k, n=kn (ini). -

30 .6. MEASUREMENT OF A ROSEN-TYPE PT SAMPLE The expeiental setup applied a HP4395A Netwok Analyze. The expeiental saple was a Rosen-Type PT, ELS-60. The objective of this test was to find the load dependence of the PT paaetes. The expeient was caied out unde the Following conditions: ) The s voltage applied to the input pot was about V in s =4V. The s voltage applied to the output pot was adjusted so as to obtain 4V s at the input at esonance fequency. ) The fequency was swept fo 6 to 75 khz. 3) The load esistances R ef connected at the output pot while easuing the output/input voltage atio vaied fo 0kΩ to 500kΩ. 4) Coespondingly, the load esistances connected at the input pot while easuing the input/output voltage atio vaied fo 400 to 0kΩ. 5) The easued data V A /V R (Fig..4) was ecalculated to find the actual agnitude of the voltage atio. 6) Paaetes wee calculated fo the phase easueents and then coected following the pocedue descibed above. To confi the obtained esults, the easued and calculated voltage atios and phase wee copaed. The calculated voltage atio and phase values wee obtained by applying (.7) and (.9). Fig..7 shows easued and calculated aplitude and phase of voltage atio befoe and afte coection of the PT paaetes (R L =56.9kΩ). One can see that afte coection, thee is good ageeent between the calculated and easued voltage atios. Fo veification of the poposed ethod an independent easueent of the loaded PT input ipedance was caied out. Fig..8 depicts the easued ipedance and the ipedance calculated fo the paaetes obtained fo R L =56.9kΩ. Thee is good ageeent between the expeient and the calculation. Fig..9 shows the PT paaetes n, R, C, L, and the echanical esonant fequency as a function of the load esistance R L. As can be seen, paaetes L and C ae slightly dependent on the load while the echanical esonant fequency is noticeably dependent on it. The paaete n is soewhat independent fo the load esistance wheeas R is stongly dependent on it. -

31 Fig..7. Measued and econstucted voltage atio and phase as a function of fequency. Solid lines ae the calculated and cicles ae the expeiental esults. Lowe solid line in the voltage atio plot (paaetes with subscipt ini) is obtained befoe paaete coection, uppe solid line is obtained afte paaete updating, -3

32 Fig..8. Measued and calculated input ipedance (a aplitude, b phase) as a function of fequency. -4

33 (a) (b) -5

34 (c) (d) -6

35 (e) Fig..9. PT equivalent paaetes n, R, C, L (a)-(d), and the echanical esonant fequency of PT (e) as a function of the load esistance..7. CONCLUSIONS Two ways fo obtaining PT equivalent paaetes fo physical easueents: the shot cicuit test and the tansfe function tests ae poposed in this section. Each test was veified by siulation and expeient. It was shown that the shot cicuit test, which is based on utilizing the chaacteistic points of the input ipedance, although easie to ipleent, is ecoended fo those PTs that ae not sensitive to voltage levels. On the othe hand, the tansfe function test, which is based on the all the infoation obtained fo the easueents of the voltage tansfe atio (aplitude and phase), is oe accuate and suits evey type of PT. We also deonstated the dependence of PT equivalent paaetes on the load esistance. -7

36 CHAPTER 3 A PIEZOELECTRIC TRANSFORMER DC/DC CONVERTERS' OUTPUT NETWORKS I With the advance developent of PT technology, this tansfoe ay tun into a viable altenative to electoagnetic tansfoes in vaious applications such as DC/DC convetes. The issue of optial PT convete design becoes, theefoe, an ipotant issue that needs to be addessed. 3.. ANALYSIS AND DESIGN OF A PIEZOELECTRIC TRANSFORMER DC/DC CONVERTER IN A LOW VOLTAGE APPLICATION The typical output ectifie netwoks that can be applied to PT convetes, as discussed in Chapte (Figs..-.4) ae ) The cuent double ectifie; ) The half-way ectifie with two diodes and an LC-filte; 3) The full-way fou diodes ectifie with the capacitive filte; 4) The voltage double ectifie with the capacitive filte. PT powe convetes coonly opeate in one of two odes: at a echanical esonant fequency o at a fequency of axiu output voltage (o powe, which is the sae). When we use a half-way, two diode ectifie with an LC filte at the output (Fig..) we have to ensue that thee is a diect cuent path because a DC cuent cannot flow though the output teinals of the PT. To deal with this poble we connected an additional inductance L o paallel to the PT output. This inductance has to be chosen so that the paallel banch L o C o and the seies banch L C have the sae esonant fequencies, so as to iniize the ciculating enegy in the eactive eleents of the cicuit. Fo this eason, the convete should be opeating at a echanical esonant fequency [4-5, 0]. 3-

37 To date, only one opeational ode of this convete has been copehensively studied, naely the ode with no ovelapping intevals of the diodes cuents. Howeve, when caying out expeients on this cicuit, one sees that the output voltage is dependent on the load, in contast to what was expected accoding to the siple theoetical odel. When the load esistance deceases the output voltage becoes stongly dependent on the load esistance. In this chapte the behavio of a low voltage PT convete with a half-bidge two-diode ectifie ove the full load ange is investigated. We develop a closed-fo equations odel that ebaces all opeating odes of the convete. 3.. PRINCIPLE OF OPERATION OF A HALF-WAY TWO DIODES RECTIFIER The conventional equivalent cicuit of the PT was used to analyze the low voltage DC/DC convete topology (Fig. 3.). This cicuit is coposed of L, C, R, C, C, in O a v Co i voltage souce and a cuent souce (n is the tansfoe tun atio) [3]. A halfway n n ectifie Rect and an inducto L o ae connected at the output of the PT. The ectifie includes two diodes ( D, D ) and an L f, Cf filte. R L is the load esistance. The function of L o in this cicuit is to povide a path fo the DC cuent and also to cancel the ciculating enegy caused by C o [5]. The esonant fequency of the paallel cicuit L o, C o has to be equal to the seies esonant fequency of the PT. i L C R PT D L f Rect I o ico ilo V in C in v Co n + - i n Co Lo v Co D C f RL v out Fig. 3. The PT convete loaded by a halfway two diodes ectifie. 3-

38 i n ϑ i n ϑ v CO i LO i CO λ V CO p γ γ γ ϑ ϑ v CO i LO λ = π ϑ ϑ ϑ i CO ϑ i D i D id i D 0 ϑ ϑ ϑ ϑ 3 4 (a) ϑ i D i D ϑ 0 ϑ = ϑ3 id i D (b) ϑ 4 ϑ Fig. 3. Rectifie s cuent and voltage wavefos in the ovelapping (a) and nonovelapping (b) odes. The siulated cuent and voltage wavefos of the ectifie opeating in ovelapping and non-ovelapping odes ae shown in Fig. 3. a, b ( ϑ = t is the noalized tie and is the opeating angle fequency). The PT cuent i is assued to be sinusoidal in both odes because of the high Q of pactical PTs. The capacito C o voltage is pactically sinusoidal in the non-ovelapping ode but has a discontinuous behavio (intevals of zeo voltage) in the ovelapping ode. In this chapte we focus on the ovelapping ode. Duing the inteval ϑ ϑ = λ the voltage v Co acoss the capacito C o is positive, theefoe the diode D is ON and the diode D is OFF. At ϑ, the polaity of the voltage v Co changes and theefoe D tuns on. If just befoe ϑ, the instant cuent of D (i D ) is highe than the absolute value of the capacito C o cuent (i Co ) the ectifie will opeate unde ovelapping conditions. In this case the diodes D and D conduct siultaneously duing the inteval ϑ ϑ3, and the cuent i D inceases while the cuent i D deceases (due to changing i ). Duing this ovelapping inteval, the cuent i Co =0, because the diodes have shoted the capacito. At ϑ 3, i D =0 and D tuns off. Duing ϑ 3ϑ4, only D is ON and v Co <0. Inductance 3-3

39 L o chages by the input supply and it s cuent inceases. At ϑ 4, v Co changes polaity and an ovelapping inteval occus again, but now i D inceases while i D deceases, and so on UNIFIED ANALYSIS OF OVERLAPPING AND NON-OVERLAPPING MODES The opeation fequency of the convete,, is equal to the echanical esonant fequency of a PT and to the esonant fequency of the paallel netwok C o L o = = (3.) L C L C o o The analysis is caied out unde the following assuptions: ) The diodes and eactive eleents ae ideal. ) The output filte is ideal ( L =, C = ). f f 3) The cuent i has a sinusoidal wavefo. 4) The ipulses of v Co have a sinusoidal shape and the duation of each ipulse is appoxiately λ. whee A. Lossless PT (R =0) Accoding to the last assuption, the voltage v Co is π v Co ( ϑ) = VC cos ϑ O p (3.) λ V C O p is the peak of the voltage acoss the capacito C o. The peak of the fist haonic of this voltage is λ cos 4 = V COp λ (3.3) π λ VCO ()p 3-4

40 The aveage output voltage V out is equal to the DC coponent of the positive value of v Co and is expessed as V out λ = VCOp (3.4) π Cobining (3.4) in (3.3) we obtain VC O ()p λ cos = 4Vout (3.5) λ π Since the opeating fequency is equal to the esonant fequency of the seies cicuit C, L VC O ()p = Vin.p (3.6) n The tansfe atio of the output voltage of the ectifie V out to the peak of the input voltage V in.p fo an ideal PT is ko V = nv λ π = λ 4 cos ideal out (3.7) in.p In the non-ovelapping ode ko = li(ko ) = π NOM ideal ideal (3.8) λ π At the instant ϑ the diode D caies a cuent that was flowing via capacito C o. This cuent that stats as a step can be found fo (3.) I ch dv Co = C o = C ovcop (3.9) dϑ ϑ π λ o applying (3.4) I ch 3 π = V outco (3.0) λ 3-5

41 The output cuent I o at ϑ ay be witten as I = sin( γ ) i L (3.) O n I O Ich + + I i whee sin γ is the cuent at the instant ϑ, I is the peak of the input cuent, γ - see n n Fig. 3. (a), and i L is the inducto L O o cuent. The inducto L o is shoted by the diodes D and D duing the whole inteval ϑ ϑ3. That is why its voltage is pactically zeo and its cuent i L O has a constant value up to the instant ϑ 3 when D tuns off. Taking into account this condition, we use ϑ 3 to deive i. At ϑ 3 C = i D = 0 and theefoe L O i O i L O i n I n ( γ ) = = sin (3.) whee γ - see Fig. 3. (a). Consequently, the cuent balance at ϑ (3.) ay be witten as 3 π I γ γ γ IO = V C + sin cos out o (3.3) λ n whee γ = γ + γ is the ovelapping angle, γ = π λ. Taking into account that (3.3) can be siplified to γ λ γ γ sin = cos and assuing that cos equation 3 Vout π I λ I O = = V C + cos out o (3.4) R λ n L Fo the enegy balance it follows that V in.p I out V = (3.5) R L Solving (3.7), (3.4) and (3.5) togethe, we obtain the equation fo duation of the ipulses of the voltage v Co in the ovelapping ode: 3-6

42 4 5 λ = π C or L (3.6) In the non-ovelapping ode λ = π. Duation of the capacitance C o voltage ipulses λ as a function of the load facto C o R L based on (3.6) is depicted in Fig The ectifie opeates in ovelapping ode when C R < π o L. λ 3 π OM NOM C R O L π Fig. 3.3 Duation λ as a function of the noalized load facto C o R L. OM - ovelapping ode, NOM- non-ovelapping ode. B. Real PT (R >0). We use the following additional assuptions: ) The voltage acoss the esonant cicuit L, C is zeo. ) The voltage acoss R and the fist haonic of the input voltage of the ectifie ae in phase with the input voltage of the PT. Unde these assuptions V in.p VCO ()p I R = (3.7) n 3-7

43 o, accoding to (3.5) V I R 4V λ cos out in.p = (3.8) n λ π It can be shown that equation (3.4) deived above fo an ideal PT, is also coect fo a lossy PT. Taking λ cos fo (3.4) and inseting it to (3.8) we obtain V in.p I 3 Vout π Vout CO I R R L λ = (3.9) λ π Since the left side of this equation is the output powe, (3.9) ay be ewitten as V out R L = V out R L V out λ π C o π λ 3 (3.0) Equation (3.0) can now be used to obtain the conduction angle λ, whose expession is identical to that of (3.6). We find hee that even in the lossy case, λ is independent of the equivalent loss esistance of the PT -R. C. Diode Losses In low output voltage ectifies the voltage dops of the diodes have be taken into account fo designing the convete. The powe losses P D in each diode ae calculated fo the following equation [43]: P = V I + R I (3.) D F D(ave) F D(s) whee: I D(ave) and I D(s) ae the aveage and the s diode cuents, and R F and V F ae the diode paaetes (R F is the equivalent fowad esistance and V F is the fowad voltage). 3-8

44 Accoding to Kichoff s law, the diodes cuents i D and i D duing the ovelapping intevals can be descibed by the following equations: i i D D i = i L + (3.) O n i = I O i L (3.3) O n Duing the non-ovelapping intevals, the cuent of one of the diodes is I o and the cuent of the othe diode is zeo. The aveage diode cuent I D(ave) is appoxiately equal to half of the output DC cuent (I D(ave) =0.5I o ). The s diode cuent I D(s) was deteined using the assuption that duing the ovelapping intevals the diode cuents ae changing linealy. Applying (3.0), (3.4) and (3.0), following expession fo I D(s) is obtained: I D(s) I O = ϕ( λ) (3.4) whee λ π π 3 ( ) = + ( π λ) ϕ λ λ π + λ (3.5) Hence, the geneal equation of the powe losses in each diode P D (3.) can be ewitten in the following fo: P D = I ( ( )) OVF + I OR F ϕ λ (3.6) In non-ovelapping ode this equation is siplified to P I OR F = I OVF (3.7) NOM D + Assuing that the diode s powe dissipations is the ajo powe loss, the ectifie efficiency is expessed as P out η ect = (3.8) Pout + PD 3-9

45 whee P = V I (3.9) out out O is the output powe. Applying (3.6) we tansfo (3.8) into η ect = (3.30) VF R F ( ϕ( λ) ) + V out + R L The value of R F in oden diodes is usually sall and in ost pactical cases it is uch lowe than the load esistance of the ectifie R L. This consideation and the fact that the function ϕ(λ) fo evey λ suggest that the te R F (ϕ(λ)) /R L in the last equation can be neglected. Hence, η ect V (3.3) F + V out As this equation shows, the ectifie efficiency η ect deceases as the output voltage V out gets lowe. This is especially noticeable when V out becoes close to the diodes' fowad voltage V F. D. Voltage Range and Powe Chaacteistics of a Real Convete Applying the powe balance condition we can eplace the loaded output ectifie by an equivalent esistance R eq connected to the output of the PT (Fig. 3.4) [8, 36, 39, 4]. The powe balance condition including the ectifie losses can be expessed as V C O () p R eq V V out out = + VFI O = (3.3) R L R Lηect fo which R eq VC ()p R Lη O ect = V (3.33) out 3-0

46 We found that the voltage atio of the last equation can be calculated by an expession that is siila to (3.5), but which takes into account the ectifie s efficiency * λ V cos C O ()p 4 = V η * (3.34) out ect λ π The angle λ * is defined by an expession siila to (3.6) 5 * π C or L 4 λ = (3.35) η ect In the non-ovelapping ode, expessions (3.33) and (3.34) convege to R NOM eq π = R L (3.36) η ect Applying (3.3)-(3.34), we obtain the output to input voltage atio of a convete with losses in the ectifie and the PT as follows: * λ 0.5η ect π ko = * λ n R + cos R eq (3.37) i L C R PT V in C in vc O n + - i n i CO Co i LO Lo Req Fig. 3.4 A siplified equivalent cicuit of the convete. 3-