Optimal Inductor Current in Boost DC/DC Converters Regulating the Input Voltage applied to Low-Power Photovoltaic Modules

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1 1 Opimal Inducor Curren in Boos DC/DC Converers Regulaing he Inpu Volage applied o Low-Power Phoovolaic Modules Ferran Reverer and Manel Gasulla, Member, IEEE Absrac In energy-harvesing applicaions, inducor-based swiching dc/dc converers are usually employed o regulae he operaing volage of he energy ransducer and o ransfer he harvesed energy o a sorage uni. In such a conex, his paper analyses he opimal inducor curren of he converer ha leads o maximum power efficiency. This is evaluaed assuming a lowpower phoovolaic (PV) module conneced o a boos dc/dc converer operaing in burs mode so as o reduce he swiching losses. The heoreical analysis and he experimenal resuls repored herein prove ha his opimal inducor curren does no depend on he power generaed by he PV module provided ha he conrol circui is powered from he oupu, bu i does on he oupu volage level of he sorage uni. Experimenal ess wih a commercial boos dc/dc converer show ha he use of his opimal inducor curren provides up o 10% increase in efficiency. Index Terms Boos converer, burs mode, DC/DC converer, efficiency, energy harveser, phoovolaic module. I. INTRODUCTION LTHOUGH swiching dc/dc converers are generally Aemployed o regulae heir oupu volage, hey can also be used o regulae heir inpu volage, which is of ineres in energy harvesers ha power, for insance, he nodes of a wireless sensor nework in smar ciies and buildings. In he firs case, he dc/dc converer is placed beween he energy source (e.g. a baery) and he elecronic circuiry (e.g. sensors, amplifiers, microconrollers and/or ransceivers) wih wo objecives: (i) o power he elecronics wih a sable supply volage, and (ii) o ransfer he energy from he baery o he elecronics in an efficien way. In he second case, he dc/dc converer is placed beween an energy ransducer (e.g. a PV module) and a sorage uni (e.g. a rechargeable baery) wih again wo goals: (i) o mainain he operaing volage of he energy ransducer around is maximum power poin (MPP) [1], and (ii) o ransfer he energy from he ransducer o he sorage uni efficienly. For dc/dc converers regulaing heir oupu volage, he This work was suppored by he Spanish Minisry of Science under projec TEC F. Reverer and M. Gasulla are wih he e-cat Research Group, Deparmen of Elecronic Engineering, Universia Poliècnica de Caalunya (UPC) BarcelonaTech, C/ Eseve Terradas 7, 080 Caselldefels, Spain ( ferran.reverer@upc.edu; manel.gasulla@upc.edu). conrol sraegy applied o he swiching ransisors is seleced according o he oupu power demanded by he elecronic circuiry. Under ligh-load condiions (i.e. for load currens of a few ma), which is quie usual in sensor nodes, he wellknown pulse-widh modulaion (PWM) is no recommended because he fixed swiching frequency causes significan swiching losses and, hence, reduces he efficiency [2]. Such efficiency can be increased by dynamically adjusing he gae driving volage [3], he size of he swiching ransisors [4,5], and he number of acive phases in muliphase dc/dc converers [6]. Anoher way o improve he efficiency is he use of a hybrid conrol whereby he converer operaes in PWM a heavy loads, bu i swiches o a variable-frequency mode a ligh loads so as o reduce he swiching losses. A firs example of ha is he pulse-frequency modulaion (PFM) where he swiching frequency is scaled down wih he load curren. Consan [7] or adapive [8] on-ime, and consan peak inducor curren [9] are wo common conrol echniques based on PFM. A second example is he burs mode (BM) where he ransisors are cyclically swiched on and off a a fixed frequency (he same as in PWM) during an acive period, bu hey are permanenly in off-sae during an inacive period, which becomes longer as he load curren decreases [10]. During he acive period, i is advisable o ransfer he energy from he baery o he elecronics a an opimal value of inducor curren ha can offer an efficiency increase of 10% [11]. For dc/dc converers regulaing heir inpu volage in energy harvesers, he selecion of he conrol sraegy does no depend on he oupu power, bu on he inpu power provided by he energy ransducer. Converers operaing in PWM have been proposed for medium- and high-power PV modules [12, 13], bu oher modulaions are more appropriae for low inpu power levels (e.g. for subwa PV modules) in order o reduce he swiching losses, as also happens when regulaing he oupu volage. For insance: (i) a PFM conrol wih a swiching frequency ha is scaled down wih he PV curren [14], and (ii) a BM conrol wih an inacive period ha increases as he PV curren decreases [15, 16, 17]. The power processing circuis in [15, 16, 17] employed a commercial dc/dc converer (LT1303 [18], MAX1675 [19], and MAX1795 [20]) ha adjused he inducor curren around 1 A, 0.5 A, and 0.25 A during he acive period, respecively. However, he value of ha curren was no seleced in erms of efficiency

2 2 maximizaion, as proposed in [11] for he regulaion of he oupu volage. This paper focuses on dc/dc converers operaing in BM and regulaing heir inpu volage. A he inpu, we assume a lowpower energy ransducer modelled by a DC curren source, such as a PV module [12], ha mus operae around is MPP; he maximum power poin racking (MPPT) mehod ha deermines he MPP volage is ou of he scope of his work and can be found elsewhere [1, 12-17, 21]. On he oher hand, a he oupu, we assume rechargeable baeries ha are charged hrough he dc/dc converer. Following he mehod suggesed in [11], which was applied o dc/dc converers regulaing heir oupu volage, his paper aims o improve he power efficiency of he converer by selecing an opimal value of he inducor curren employed o ransfer he energy from he ransducer o he baeries. Such a case requires a novel sudy of he dc/dc converer because he independen inpu/oupu variables are no he same as in [11]. Whereas in [11] hese variables were he oupu power (volage and curren) demanded by he load and he inpu volage provided by he baery, here hese are he inpu power (volage and curren) generaed by he energy ransducer and he oupu volage provided by he baery. Furhermore, in comparison wih [11], his paper also conribues wih he following. Firs, wo scenarios are considered and compared: conrol circui powered from eiher he inpu or he oupu. Second, he concep of opimal inducor curren is experimenally proved for dc/dc converers having differen levels of fixed, conducion and swiching losses. And hird, experimenal resuls of efficiency when a low-power PV module operaing a is MPP is conneced o a dc/dc converer operaing a is opimal inducor curren are repored and discussed. II. OPERATING PRINCIPLE A power processing circui for a PV module based on a synchronous boos dc/dc converer is shown in Fig. 1. The converer relies on an inducor (L) and wo power MOSFET ransisors (MN and MP). The corresponding gae conrol signals (v c1 and v c2 ) are generaed by a conrol circui wih wo loops [13]: (i) a volage loop ha moniors he inpu volage (v in ) using a comparaor wih a hyseresis of ±V hys and wih a reference volage (V in ) equal o he MPP volage deermined by a MPPT conroller [1], and (ii) a curren loop ha moniors he inducor curren (i L ) by eiher a shun resisance in series wih L or he volage drop across MN or MP. A he inpu of he converer, he PV module provides a DC curren (I in ) and has a high-value inpu capacior (C in ) in parallel ha emporarily sores he energy. On he oher hand, he oupu of he converer is conneced o a rechargeable baery in parallel wih an oupu capacior (C ou ) ha filers ou he highfrequency componens of he oupu curren. Assuming no losses, he average oupu curren injeced o he baery is V in I in /V ou, where V ou is he DC volage level of he baery. The inpu volage (v in ) in Fig. 1 is regulaed around he desired DC volage (V in ) by operaing in BM. This operaing principle involves wo sages (inacive and acive) ha las inacive and acive, respecively, and an overall duy cycle Fig. 1. Power processing circui for a PV module based on a synchronous boos dc/dc converer. v in v cmp v c1 v c1 i L inacive T T zoom on T s acive off acive Burs of pulses 2 V hys V in I L I L0 Fig. 2. Waveforms of ineres from he circui in Fig. 1 operaing in BM- CCM. D T = acive /T T, where T T = inacive + acive, as shown in Figs. 2(a) and 2(b). In he inacive sage, he converer is deacivaed (i.e. MN and MP are off) and I in charges C in, hus increasing v in. When v in = V in + V hys, he comparaor oupu (v cmp ) changes o a high logic level and brings he converer o he acive sage. Then, he energy accumulaed in C in is ransferred o he oupu and v in decreases. When v in = V in V hys, v cmp oggles o a low logic level, he converer is deacivaed and he process sars again. This operaing principle based on iniially soring he energy in C in is very appropriae for low-power PV modules since (i) he converer remains inacive mos of he ime, which reduces he power losses, and (ii) C in provides an operaing volage equal o he MPP volage, which ensures a good impedance maching wih he equivalen impedance of he PV module regardless of he operaing condiions of he converer in acive mode. Power processing circuis wihou a high-value C in where he converer is always acivaed and he impedance maching is carried ou by adjusing he duy cycle of he swiching ransisors [22] are more appropriae for medium- and high-power PV modules. In order o ransfer he energy from he inpu o he oupu during he acive sage, a burs of on/off pulses under PWM conrol is applied o he gae of he ransisors, as shown in Fig. 2(c) and wih more deails in Fig. 2(d) for v c1 ; v c2 is he same as v c1 bu wih some dead ime beween hem o preven cross conducion of he ransisors. As represened in Fig. 2(d), (a) (b) (c) (d) (e)

3 3 TABLE I POWER LOSS COMPONENTS OF THE CIRCUIT IN FIG. 3 WHEN THE CONTROL CIRCUIT IS POWERED FROM THE OUTPUT Power losses Acive mode Inacive mode Fixed Vou I Q,a Vou I Q,i Fig. 3. Equivalen circui model for he analysis of power losses in he circui shown in Fig. 1. v c1 has an on-ime ( on ), an off-ime ( off ), a swiching period T s (= on + off ), a swiching frequency f s (= 1/T s ) and a duy cycle D (= on /T s ). During on (MN on, MP off), he energy previously accumulaed in C in is sored in L and i L increases, whereas during off (MN off, MP on), he energy accumulaed in L is ransferred o he oupu and i L decreases. A currenprogrammed mode conrol in coninuous conducion mode (CCM) is assumed so ha i L has an average of I L0 and a ripple of I L, as shown in Fig. 2(e). In such an operaing mode, we have D = 1 ηv in /V ou, η being he efficiency. The opimal value of I L0 o carry ou such energy ransfer a maximum efficiency is analyzed nex considering he main power losses. III. THEORETICAL ANALYSIS The power efficiency of he circui in Fig. 1 is heoreically analyzed using he same equivalen circui model proposed in [11] and represened in Fig. 3. This circui includes he parasiic resisance (R L, R Ci, R Co, R N, and R P ) of he main componens (L, C in, C ou, MN, and MP, respecively), and he parasiic capaciance (C A, C G1, and C G2 ) of he main nodes (node A, gae of MN and MP, respecively); R S is a shun resisance employed in some dc/dc converers o sense i L. Moreover, he conrol circui has a curren consumpion of I Q,a in acive mode and I Q,i in inacive mode, where I Q,i << I Q,a. The opimal inducor curren is heoreically found in wo differen scenarios ha ake ino accoun he rade-off beween conducion losses and gae-driving losses a differen gaedriving volages [3]. Firs, we assume ha he conrol circui is powered from he oupu, as shown in Fig. 3. This involves a high gae-driving volage (i.e. V ou ) ha decreases he onresisances of MN and MP and, hence, he conducion losses. Second, we consider ha he conrol circui is powered from he inpu. In such a case, he gae-driving volage is lower (i.e. V in ) and, herefore, losses relaed o he charge-discharge process of C G1 and C G2 are also lower. A. Conrol circui powered from he oupu Table I summarizes he power losses (fixed, conducion, and swiching losses [11]) presen in Fig. 3 in boh acive and inacive modes when he conrol circui is powered from he oupu. In acive mode, he equivalen parasiic resisance is R = RCi + RS + RL + RN D+ ( RP + RCo )(1 D). This is assuming ha i L is mosly provided by C in since I L0 >> I in, and ha he curren hrough MP is much higher han he average oupu curren injeced o he baery. In inacive mode, he equivalen parasiic resisance is R = R + R ( V / V ) 2, eq,i Ci Co in ou Conducion Swiching f ( 2 s CeqVou Vou c ) which assumes ha he curren exraced from C ou is V in I in /V ou. The capaciances C G1, C G2, and C A are lumped in one equivalen capaciance, C eq = C G1 + C G2 + C A, because hey have he same charging volage (i.e. V ou ) when he conrol circui is powered from he oupu. As for he swiching losses caused by he volage-curren overlap in MN, c is he average of he urn-on and urn-off ransiion imes. The overall power losses in acive and inacive modes (P L,acive and P L,inacive, respecively) can be calculaed by adding he hree componens indicaed in Table I. Then, he average power losses over a whole period (i.e. T T in Fig. 2) can be expressed and approximaed (assuming P L,acive >> P L,inacive ) as ( 1 ) PL = PL,acive DT + PL,inacive DT = P P D + P P D + P L,acive L,inacive T L,inacive L,acive T L,inacive Since he charge accumulaed in C in in inacive mode is equal o ha exraced from C in in acive mode, we have I = I I and hen D T can also be wrien as in inacive L0 in acive R Iin I L0. Using his relaion in (1), he efficiency can be esimaed as ou Q,i ou c s eq,i in Iin 2 (a) 2 Req,iI in + 0 a The RMS value of i L is approximaed o I L0 since I L < I L0 [23]. (1) 2 Pin PL 1 Vou IQ,a + CeqVou fs η = = 1 R + Pin Vin (2) V I + V f + + R I where P in is he inpu power defined as V in I in. From (2), he higher V in and/or he lower V ou, he higher he efficiency, as in [11]. The effecs of I in depend on which of he las wo erms inside he brackes predominaes. If we assume capaciors wih a low equivalen series resisance (ESR), he las erm in (2) is negligible and hen η should increase wih increasing I in. The efficiency prediced by (2) srongly depends on he seleced value of I L0. According o he firs erm inside he brackes corresponding o conducion losses in acive mode, η decreases wih increasing I L0 a high values of I L0. However, according o he second erm corresponding o fixed losses and swiching losses due o C eq in acive mode, η increases wih increasing I L0 a low values of I L0. Therefore, here is a

4 4 maximum of efficiency a a cerain value of I L0 ha can be calculaed from (2) by equaing η o zero, hus resuling in I L0,op 2 Vou IQ,a + CeqVou fs = (3) R which is independen of boh I in and V in and, hence, of he power generaed by he energy ransducer, bu i increases wih increasing V ou. Replacing (3) in (2) yields he maximum efficiency TABLE II POWER LOSS COMPONENTS OF THE CIRCUIT IN FIG. 3 WHEN THE CONTROL CIRCUIT IS POWERED FROM THE INPUT Power losses Acive mode Inacive mode Fixed (a) VinI Q,a VinI Q,i Conducion 2 (b) 2 Req,iI in Swiching f ( 2 2 s CV G in CV A ou Vou c ) R a I is assumed he same quiescen curren considered in Table I. b The RMS value of i L is approximaed o I L0 since I L < I L0 [23]. 1 2 ηmax = 1 2 R ( Vou IQ,a + CeqVou fs ) V in (4) V I + V f + + R I ou Q,i ou c s eq,i in Iin B. Conrol circui powered from he inpu Table II summarizes he power losses in boh acive and inacive modes when he conrol circui is powered from he inpu. In comparison wih Table I, we have hree main changes: (i) fixed losses are lower since hey depend on V in insead of V ou ; (ii) conducion losses in acive mode are caused by a higher parasiic resisance, R > R, because he on-resisances of MN and MP are higher; and (iii) swiching losses due o he charge-discharge process of he gae capaciances, C G = C G1 + C G2, are lower since he gae volage swing is lower. Following now he same procedure explained in Secion III.A, we can find a new expression for he efficiency, he opimal value of I L0 and he maximum efficiency defined in (5), (6), and (7), respecively. η VI + f CV + CV = 1 R + Vin + Vouc fs + Req,i I in IQ,i Iin in Q,a s G in A ou 2 2 (5) VI in Q,a + fs CV G in + CV A ou,op = (6) R ηmax = 1 2 R VI in Q,a + fs ( CV G in + CV A ou ) Vin (7) + Vouc fs + Req,i I in IQ,i Iin In comparison wih (2), he efficiency resuling from (5) is expeced o be higher a low values of I L0 due o lower fixed and gae-driving losses, bu lower a high values of I L0 due o higher conducion losses, as shown in Fig. 4. Comparing (3) and (6), we also realize ha I L0,op < I L0,op. Furhermore and unlike wha happens in (3), now I L0,op depends on V in and, Fig. 4. Efficiency versus I L0 when he conrol circui is powered from eiher he oupu (in coninuous line) or he inpu (in dashed line). hence, on he operaing poin of he energy ransducer. This means, for a PV module, ha I L0,op should be uned a each irradiance and emperaure level so as o achieve he maximum efficiency of he power processing circui Neglecing power losses due o I Q,i, which are expeced o be he lowes, (4) and (7) can be compared hrough he erms inside he square roo defined in (8) and (9), respecively. 2 Term 1 = R V I + C V f (8) ou Q,a eq ou s Term 2 = R VI + f CV + CV 2 2 in Q,a s G in A ou In order o compare hese erms, we propose o express, in a firs approximaion, he parasiic resisances as R RA + k/ Vou and R RA + k/ Vin, where R A is a resisive componen independen of he gae-driving volage due, for insance, o L, C in, and C ou, whereas k is a consan ha depends, among ohers, on he dimensions of MN and MP. Using hese, he difference beween (8) and (9) can be expressed as 2 2 = RI V V + R fc V V A Q,a ou in A s G ou in V fk 1 CV CV ou s A ou G in Vin (9) (10) If he value of R A is significan, hen he wo firs erms in (10) dominae, becomes posiive and, consequenly, η η. max > max

5 5 Fig. 5. Applicaion circui based on he TPS61252 employed o prove he concep of opimal inducor curren; he numbers given in brackes are he pin numbers of he TPS However, if he value of R A is low enough hanks o he use of capaciors and inducors wih a very low ESR, he las erm in (10) dominaes. If we also have C A V ou > C G V in, hen becomes negaive and, herefore, η max > η max, as represened in Fig. 4. In summary, if he componens around he dc/dc converer are seleced wih a low enough parasiic resisance, i seems preferable o power he conrol circui hrough he oupu so as o achieve a higher efficiency when operaing a he opimal inducor curren. Furhermore, in hose condiions, he opimal value of I L0 does no change wih he power generaed by he energy ransducer, which faciliaes he conrol. TABLE III OPERATING CONDITIONS, INSTRUMENTATION, AND COMPONENTS EMPLOYED TO TEST THE CIRCUIT SHOWN IN FIG. 5 Variable or Value componen I in V in V ou L C in C ou 5.5, 11, and 22 ma (a) provided by Agilen B21 2.5, 2.75, and 3 V (b) provided by Agilen E3631A 4, 5, and 6 V (c) provided by Agilen E3631A (d) 2.2 µh, low-esr 1 mf, analum, low-esr 2 1 mf, analum, low-esr a Emulaing he MPP curren a 25%, 50%, and 100% of he irradiance a STC, respecively. b Emulaing he change of he MPP volage due o changes of boh irradiance and emperaure [25] c Emulaing he differen saes of charge of four cylindrical NiMH secondary baeries in series. d Wih a resisor in parallel o operae in he fourh quadran [26]. IV. MATERIALS AND METHOD A commercial boos dc/dc converer, TPS61252 from Texas Insrumens [24], has been employed o experimenally prove he concep of opimal inducor curren. This converer has a conrol circui powered from he oupu and is I L0 is adjusable from 100 o 1500 ma by an exernal resisor (R LIM ). The inducor curren is measured during he off-ime hrough he volage drop across MP, and a valley curren-mode conrol is applied ha cleverly adjuss he valley curren limi o achieve he desired average inducor curren. In order o have he BM-CCM operaion shown in Fig. 2, an exernal ulralowpower comparaor, LTC1440 from Linear Technology, wih V hys = 50 mv was placed before he feedback (FB) inpu of he converer, as shown in Fig. 5. Using his circui, when v in becomes higher han he desired volage, he comparaor oupu changes o a low level, which brings he converer o acive mode and, hen, i L is regulaed around I L0. Oherwise, when v in becomes lower han he desired volage, he comparaor oupu oggles o a high level and he converer eners ino inacive mode. The circui in Fig. 5 was esed using he operaing condiions, insrumenaion, and componens indicaed in Table III. The values of I in and V in were seleced using as a reference a commercial ulra-hin low-power PV module, SP3-37 from PowerFilm, ha will be under es in Secion VI. A sandard es condiions (STC) involving a solar irradiance of 1000 W/m 2, his PV module has a ypical MPP curren/volage/power of 22 ma/3 V/66 mw, which is adequae o power, for insance, a microconroller-based (a) (b) Fig. 6. From he circui in Fig. 5, experimenal waveforms of (a) he inpu volage (channel 1 in AC coupling) and he comparaor oupu (channel 2) for several acive and inacive periods, and (b) he volage a he swiching node A wihin one acive period. auonomous sensor [27]. Noe ha he maximum MPP curren generaed by he PV module (22 ma) is clearly lower han he minimum value of I L0 ha can be regulaed (100 ma), so he approximaion indicaed in Secion III.A is valid. The inpu power was calculaed as V in I in, whereas he average oupu power (P ou ) was measured by a power analyzer, Yokogawa WT310, wih a sampling frequency of 100 ksa/s and an updae rae of 5 s. Wih he aim of generalizing he concep of opimal inducor curren o oher dc/dc converers wih differen power losses, we also added some exernal componens around he TPS61252, as shown in Fig. 5, so as o raise is fixed, conducion and swiching losses. Fixed losses were increased by connecing a resisor (R i ) beween V ou and he comparaor

6 6 Iin=5.5 ma Iin=11 ma Iin=22 ma 305 ma Vin=2.5 V Vin=2.75 V Vin=3.0 V 305 ma Vou=4 V 200 ma Vou=5 V Vou=6 V 305 ma 370 ma Fig. 7. Experimenal efficiency versus I L0 for differen values of (a) I in, (b) V in, and (c) V ou. Iin=5.5 ma Iin=11 ma Iin=22 ma 275 ma Vin=2.5 V Vin=2.75 V Vin=3.0 V 275 ma Vou=4 V 200 ma Vou=5 V Vou=6 V 275 ma 350 ma Fig. 8. Efficiency calculaed from (2) versus I L0 for differen values of (a) I in, (b) V in, and (c) V ou. oupu, hus generaing an exra curren consumpion of I e (= V ou /R i ) in acive mode. Conducion losses were increased by placing a resisor (R e ) a he oupu of he dc/dc converer, whereas swiching losses were increased by connecing a capacior (C e ) a he swiching node A. All hese ess were conduced a V in = 3.0 V, I in = 22 ma, and V ou = 5.0 V. V. EXPERIMENTAL RESULTS AND DISCUSSION Before evaluaing he efficiency of he circui in Fig. 5, we esed is operaing principle by monioring he volage waveform a he main nodes, as shown in Fig. 6 for V in = 3.0 V, I in = 22 ma, V ou = 5.0 V, and I L0 305 ma. Fig. 6(a) shows he inpu volage and he comparaor oupu for several acive and inacive periods; he laer is he complemenary of ha represened in Fig. 2(b) because his signal is hen invered by he on-chip error amplifier. According o he comparaor oupu, we had D T = 6%, which fairly agrees wih ha prediced by I in /I L0. On he oher hand, Fig. 6(b) shows he volage a he swiching node A wihin one acive period; his signal is also he complemenary of ha represened in Fig. 2(d) since i is invered hrough MN. In Fig. 6(b) we measured f s = 3.5 MHz, insead of he nominal value of 3.25 MHz, and D = 44%, which agrees wih ha calculaed by 1 ηvin Vou assuming η = % (repored laer in Fig. 7). Furhermore, D was very sable during he acive period, which means ha he inducor curren was well regulaed around I L0. Fig. 7 shows he experimenal resuls of efficiency versus I L0 for differen values of (a) I in, (b) V in, and (c) V ou, using I in = 22 ma, V in = 3.0 V, and V ou = 5.0 V as defaul values. The higher he value of boh I in and V in, he higher he efficiency, alhough he effecs of he laer were clearly major. However, he higher V ou, he lower he efficiency. Such effecs of I in, V in, and V ou on he efficiency agree wih (2). Moreover, I L0,op was independen of boh I in [Fig. 7(a)] and V in [Fig. 7(b)], bu i increased (from 200 o 370 ma) wih increasing V ou [Fig. 7(c)], which was already prediced by (3). Wih respec o he case wih minimum efficiency ha was found a he maximum value of I L0, he efficiency increased by 7%, 8%, and 10% in Figs. 7a, 7b, and 7c, respecively, when I L0,op was applied. In order o quaniaively evaluae he model proposed in Secion III, he efficiency was also calculaed from (2) and represened in Fig. 8 for he same operaing condiions discussed before. Noe ha C eq and c were unknown and were exraced by fiing (2) o a se of experimenal resuls, and ha R N and R P were assumed o be dependen on he gae-

7 Iin=5.5 ma Iin=11 ma Vin=2.5 V Vin=2.75 V Vou=4 V Vou=5 V 0.6 Iin=22 ma 0.6 Vin=3.0 V 0.6 Vou=6 V Fig. 9. Difference beween he prediced (Fig. 8) and he experimenal (Fig. 7) values of efficiency. Ie=0 Ie=1mA Ie=2mA 80 Re=0 Re=0.1 Re=0.24 Ce=0 Ce=150pF Ce=330pF Fig. 10. Experimenal efficiency versus I L0 for differen values of (a) I e, (b) R e, and (c) C e. Ie=0 Ie=1 ma Ie=2 ma 80 Re=0 Re=0.1 Re=0.24 Ce=0 Ce=150pF Ce=330pF Fig. 11. Efficiency calculaed from (2) versus I L0 for differen values of (a) I e, (b) R e, and (c) C e. driving volage of he ransisors (i.e. V ou ) in Fig. 8(c) [3,4]. The difference ( η) beween he prediced (Fig. 8) and he experimenal (Fig. 7) values of efficiency is shown in Fig. 9. In mos of he cases under es, η is smaller han 0.5 %, which is small enough o consider he proposed model valid o esimae he efficiency of dc/dc converers regulaing he inpu volage in BM-CCM. Such a small discrepancy can be ascribed o limiaions of: (i) he model, which disregards fixed losses due o he leakage curren of ransisors and capaciors, and swiching losses due o he body diode of MP and o he inducor core; and (ii) he measuremens, especially hose performed by he power analyzer a low power levels. The experimenal resuls of efficiency a oher values of

8 8 power losses emulaed by he exernal componens are represened in Fig. 10. When he dc/dc converer was subjeced o exra fixed losses hrough I e (of 1 and 2 ma) or exra swiching losses hrough C e (of 150 and 330 pf), he efficiency decreased bu especially a low levels of I L0, as shown in Figs. 10(a) and 10(c), respecively. This is because, a low levels of I L0, he converer remains longer in acive mode and, herefore, he effecs of boh fixed and swiching losses are higher. On he oher hand, when he converer suffered from exra conducion losses hrough R e (of 0.1 and 0.24 Ω), he efficiency also decreased bu mainly a high levels of I L0 [see Fig. 10(b)] where conducion losses are more significan. Fig. 10 also shows ha I L0,op increases wih increasing I e and C e, bu decreases wih increasing R e, as prediced by (3). In he wors case esed [i.e. Fig. 10(b) wih R e = 0.24 Ω], he efficiency increased by 13% when I L0,op was applied. For he same esing condiions, we also calculaed he efficiency hrough (2) and he resuls (see Fig. 11) compleely agreed wih he experimenal daa shown in Fig. 10. In a pracical implemenaion, aking ino accoun ha he exac value of some variables involved in (3) or (6) can be unknown, he opimal value of I L0 can be auomaically deermined hrough a conrol algorihm, such as he perurb and observe mehod [17], carried ou by a microconroller. The basic idea would be o slighly perurb he value of I L0 and hen observe how he oupu power changes, assuming he inpu power consan during he conrol cycle. If he oupu power increases, he perurbaion should be kep in he same direcion; oherwise, i should be reversed. For he TPS61252 under es, he value of I L0 could be perurbed using a digial poeniomeer insead of R LIM in Fig. 5. On he oher hand, he oupu power could be observed by sensing: (a) he average oupu curren via a shun resisor and an amplifying low-pass filer [15,28], or (b) he incremen of volage across a small oupu capacior conneced in parallel wih he main sorage device ha would be disconneced for a shor and known ime inerval [29]. VI. APPLICATION TO A LOW-POWER PV MODULE The concep of opimal inducor curren has been furher proved using a commercial low-power PV module, SP3-37 from PowerFilm. This was firs characerized under irradiance-conrolled laboraory condiions o achieve he power-volage (P-V) curve as follows. The PV module was subjeced o hree irradiance levels (idenified as I25, I50, and I100) hrough a LED array, BXRA-C1202 from Bridgelux, powered a differen DC currens and placed a 3 cm [17]. The levels I25, I50, and I100 approximaely correspond o an irradiance of 250, 500, and 1000 W/m 2, respecively, in erms of power generaed by he PV module a he MPP. A each irradiance level, he curren generaed by he PV module was measured a differen applied volages (from 0 V o 4 V in seps of 100 mv) using a source-measuremen uni, Agilen B21. The experimenal resuls of such a characerizaion are represened in Fig. 12(a) showing he MPP a each irradiance P (mw) V (V) I25 I50 I ma (a) V MPP = 3.2V I MPP = 21.2mA V MPP = 3.1V I MPP = 11.8mA V MPP = 3.0V I MPP = 6.0mA (b) Fig. 12. (a) Experimenal P-V curves of he PV module under es for differen irradiance levels. (b) Experimenal efficiency of he circui versus I L0 for differen irradiance levels when he PV module operaes a he V MPP indicaed in Fig. 12(a). level. As expeced, he curren (I MPP ) and he power a he MPP were quie proporional o he irradiance level, whereas he volage (V MPP ) slighly increased wih increasing he irradiance level. Afer characerizing he PV module, his was conneced o he power processing circui shown in Fig. 5 insead of he ideal inpu curren source. Using he same mehodology explained in Secion IV, he efficiency of he circui was measured a differen values of I L0 and for he hree irradiance levels indicaed before. A each irradiance level, V in in Fig. 5 was se o he V MPP value indicaed in Fig. 12(a) so as o exrac he maximum power from he PV module. The experimenal resuls of efficiency are shown in Fig. 12(b) for V ou = 5.0 V. Noe ha he efficiency increased wih increasing he irradiance level. This is because he higher he irradiance, he higher he value of boh V MPP and I MPP (and, hence, V in and I in in Fig. 5) and, herefore, he higher he efficiency, as shown before individually in Figs. 7(a) and 7(b). The resuling value of I L0,op, which was around 305 ma, was he same for he hree irradiance levels. Accordingly, as previously suggesed in Secion III.A, he value of I L0,op seems o be independen of he power generaed by he energy ransducer, hus faciliaing he conrol sraegy. I25 I50 I100

9 9 VII. CONCLUSION This work has gone a sep furher in he field of power processing circuis based on swiching dc/dc converers by proposing an opimal inducor curren o carry ou he energy ransfer from a low-power energy ransducer o a sorage uni. If he conrol circui is powered from he oupu, his opimal inducor curren is independen of boh he inpu volage and he inpu curren. Consequenly, his opimal curren does no depend on he power generaed by he energy ransducer, which has been experimenally proved using a commercial low-power PV module subjeced o differen irradiance levels. However, such a curren depends on he oupu volage, i.e. he volage level of he oupu baeries. Experimenal ess wih a commercial boos dc/dc converer have shown ha he use of his opimal inducor curren provides up o 10% increase in efficiency. Therefore, his is a simple bu effecive way o improve he auonomy of sensor nodes powered by a low-power PV module. REFERENCES [1] H. Kim, S. Kim, C. K. Kwon, Y. J. Min, C. Kim, and S. W. 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