The Generalized Switched Accounting or Conduction. Inductor Model (GSIM): Losses. Sam Ben-Yaakov. Isaac Zafrany 1

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1 The enerlized Swihed Aouning or Conduion s Zfrny 1 1 Tehnil Suppor Avn! Corporion 11 lgley Hpld S. P. O. Bo Herzeli SRAE Tel: ; F: ; Emil: s_zfrny@vniorp.om; Wesie: hp:// nduor Model (SM): osses Sm Ben-Ykov *Power Eleronis orory eprmen of Eleril nd Compuer Engineering Ben-urion Universiy of he Negev P. O. Bo 653, Beer-Shev SRAE Tel: ; F: ; Emil: sy@ee.gu..il; Wesie: hp:// Asr - The mehod of verge modeling nd simulion of PWM onverers ws eended o inlude onduion losses. The mehod overs losses due o he induor's resisne nd due o he volge drops ross he swih nd he diode. The sme model is pplile o oh CCM nd CM operions, nd he swihing from one o he oher is uomi. The lrge-signl model is SPCE ompile nd s suh n e used o run C (sedy se), TRAN (lrge signl, ime domin) nd AC (smll signl, frequeny domin) nlyses.. NTROUCTON As previously shown [1-6], lssil PWM onverers shre ommon, opology independen swihing module: swihed induor (Fig. 1). Alhough normlly relized y swih nd seering diode, he swihing ion wih idel diode n e desried y oggle swih whih lernely onnes one end of he induor o wo erminls nd (Figs. 1, ). A eneri Swihed nduor Model (SM) whih inludes hree dependen urren soures (Fig. 1) ws used o desrie he inerion of he swihed induor module wih he inerfe iruiry. The dependen soures of he (SM) represen he verge erminl urrens of he swihed induor ssemly [1-6]. The SM models desried erlier did no urely ke ino oun onduion losses. These ffe no only he effiieny u n lso hnge he lrge nd smll signl rnsfer funions. The ojeive of his sudy ws o epnd he SM mehodology o inlude onduion losses due o he induor's resisne, he volge drops ross he swih nd diode.. THE BASC SM WTH NO OSSES The SM opology-independen equivlen irui is developed y onsidering he verge volge ross he induor V. Eminion of he swihed induor ssemly (Figs. 1,) revels h V is funion of he volge ross he erminls V (,), V (,) nd he ime h he swih is in he ON (T ON ) nd OFF (T OFF ) posiions: V(,) + V(,) E V = = Ts = V(,) on + V(,) off (1) where T s is he period of he swihing yle, on =T on /T s, off =T off /T s. The verge volge impressed on he induor will develop he orresponding verge urren. One he urren is ville, he dependen urren soures of he SM model (Fig. 1) n e redily defined y onsidering he wy y whih he urren is spli eween he erminls. Sine he urren of erminl is idenil o he induor urren ( v ) he dependen urren soure n e defined s: Corresponding uhor

2 S V(,) V(,) () idel ' E Fig.1. The verge simulion mehod. Swihed induor in PWM opologies. Swihed induor ssemly. () The eneri Swihed nduor Model (SM) [1-6]. v (2) The urren is hen divided eween erminls nd () (Figs. 1,) ording o he frion of ime he induor is onneed o hem, nmely: v on on + off (3) v off on + (4) off where in se of oninuous onduion mode (CCM) (Fig. 2): ( on + off ) = 1 (5) nd herefore: off = 1-on (6) n he se of disoninuous onduion mode (CM): off < 1-on. From geomeril onsiderion [2-4] he epression for off will e: 2v f s off = on V(,) on (7) For he se of CM he dependen urren soures nd (eqs. 3,4) re normlized y ( on + off ) in order o oin he orre verge urren during he onduion inervls of he induor: T on nd T off [2-4]. The SM model n e mde o uomilly follow CCM-CM hnges y seleing off ording o he rule: 2 = vf s off min (1 on ), on V(, ) (8) on where f s = 1/T s - swihing frequeny n SPCE implemenion of he SM, nlog ehviorl dependen soures re used o emule E,, nd. Furher, sine SPCE reognizes ime dependen vriles whih re eiher volge or urrens, on nd off re oded ino volge. The minimum funion (8) is reognized s vlid epression y mos modern irui simulors. Consequenly, he simulion model will uomilly follow CCM o CM hnge y seleing he orre off for eh se. Anoher imporn feure of he SM is h i n e used s-is o run C (sedy se), TRAN (lrge signl, ime domin) nd AC (smll signl, frequeny domin) nlyses. The ler is due o he f h he SM is formuled s SPCE ompile lrgesignl model h is uomilly linerized y he simulor efore running he AC nlysis. No furher derivions re needed eyond he formulion of he lrge signl model.. THE SM WTH OSSES The irui of Fig. 2 represens he prolem on hnd. shows h he si swihed induor ssemly inludes ddiionl elemens: he resisne of he induor, he volge drop of he rnsisor nd he volge drop ross he diode. These effes n e ken ino oun y dding he ddiionl volge drops o he originl por's volges (', ', ' in Fig 2). However, re should e ken o oun for he f h he volge drops re, in generl,

3 ' R ind p ' ' v + - Vgson ' V' V(,) V' + - ' E' ' + - ' E' E' V(,) + - ' V' Fig. 2. Proposed modeling seps. Swihed induor in PWM opologies wih onduion losses. enerlized model of swihed induor ssemly. no funion of he verge urren h is pssing hrough given por, u rher o he verge urren during he ime h he induor is onneed o given por. For emple, he model should ke ino oun he f h when he induor is onneed o he rnsisor (Fig. 2), he relevn volge drop is funion of he verge volge drop ross he rnsisor during he T on ime. This fine deil is espeilly imporn in CM h my e hrerized y high urren during T on u hving low verge urren (when ompued over T s ). follows hen h ddiionl dependen soures: E, E ' nd E ' (Fig 2) h oun for he prsii volge drops need o e formuled in erm of he dded volge seen y he induor when onneed o given por. Under hese ondiions, he verge volge ross he induor (V ) will e: V = {V(', ' ) V(' ) V(' )}on + + {V(', ' ) V(' ) V(', )} (9) off The mehod used o derive he epressions for E, E ' nd E ' is eplined wih referene o Fig. 3. Here we disinguish eween he verge induor urren ( v ), whih is verged over T on, nd, he verge urren during he onduing inervl T on +T off (Fig. 3). From si geomeril onsiderions we find: v s = on + off n he prive se of CCM: = v (10) (11) v v v T s T s Fig. 3. Wveforms of erminl urrens of he swihed induor ssemly: under CCM, under CM.

4 Bse on he ove we n now define dependen volge soures h will emule he prsii volge drops: E ' V' = s * Rind E ' V' = V(Swih _ Model)@ E ' V' = V(iode _ Model)@ s s (12) (13) (14) where V(Swih_Model)@ nd V(iode_Model)@ re he ul volge drops ross he rnsisor nd diode respeively when he urren is pssing hrough hem. These volge soures oun for he rel volge drops h re used y he ul urren flowing hrough he prsii elemens: induor's resisne, rnsisor nd diode. By his mehod we n epnd he generlized model (Fig. 1) o inlude he effe of onduion losses. The pproh kes ino oun he physil hrerisis of he elemens y pplying he SPCE models of he ul swih (MOSFET, BT e.) nd diode nd y luling he volge drops for eh insne of he simulion. n he sen of SPCE model, one n use liner models o ke ino oun he resisnes nd volge offse (if ny) of he elemens. should e noed h eep for he dded volge soures E, E ' nd E ', he model is idenil o he originl SM (Fig. 1). V. SPCE MPEMENTATON The proposed generlized model n e used o develop inpu files for generl purpose eleroni irui simulor whih inludes lgeri ehviorl models. To simule given opology, he model hs o e pled in he orresponding orienion. The proposed model is demonsred y onsidering Boos onverer wih pril prsii losses R ind, nd S (Fig. 4). The power sge is redued o SPCE-ompile irui (Fig. 4) y pplying he generlized model (Fig. 2). The dependen volge soures E, E nd E duplie he volge drops V Rind, V SWon nd V, respeively. These volge drops re genered on 'he fly' during he simulion y pssing he urren (luled eh insne) vi he models of he diode nd swih. Consequenly, he volge drops represen he ul volge h he induor will see s i onnes o he orresponding elemens. The definiion of he dependen s R ind V in ' V + in - V Rind E' R ind R onv E S E ' v s C o R esr ' E ' ' C o R esr V ou Vou V SWon V gson + - s R o R o V Fig. 4. Simulion irui. Benhmrk irui used o demonsre he proposed simulion pproh. SPCE ompile verge model for enhmrk irui. See e for definiion of ehviorl dependen soures. soures re s given ove. R onv is smll resisor pled o preven shor C (when he simulor shors he induor during he evluion of he is poin). Missing in Fig. 4 is he volge junions h represen on nd off [5,6]. on is he primry drive h n e genered y n independen volge soure (for open loop simulion) or funion of he oupu volge of he error mplifier for lose loop simulion [2-4]. Simulion n e rried ou for volge mode onrol or urren mode onrol (verge or pek) y pplying he proper uy Cyle eneror (C) [5, 6]. V. SMUATON RESUTS The proposed generlized model (Fig. 4) ws verified gins ime domin (yle-yyle) simulion of he omplee irui. The prmeers of he onverer were s follows: V in =10V, on =0.25, R ind =0.08Ω, =75µH, pril diode MUR405 (from Ord Version 9 lirry), C o =220µF, R esr =0.07Ω nd swihing frequeny f s =100KHz. n hese runs, he rnsisor model ws repled y resisor

5 0.8 off V ou Vin off Rlod [O hm ] 1.6 V ou Vin Fig.5. C rnsfer funion nd off of lossless onverer (solid line) nd lossy onverer (dshed line) verified gins yle-y-yle simulion resuls (rosses). (R SWon =1Ω) rher hn SPCE model s shown in Fig. 4. This demonsres he versiliy of he model h n eiher pply full SPCE models or pieewise liner models. The solid line in Fig. 5 shows he C rnsfer funion V ou /V in nd off s funion of he vlue of he lod resisne R lod - under oninuous nd disoninuous opering ondiions of he idel onverer. For he onverer wih onduion losses, very good greemen eween he proposed generlized model (Fig. 5, dshed line) nd he yle-y-yle simulion resuls (depied y X) is oserved over he whole lod rnge. The open-loop smll-signl rnsfer funions eween duy yle nd oupu volge V ou /V d (f) (Fig. 6) were oined y running wo ses of simulions: for CM nd CCM. The resuling Bode plos of V ou /V d (f) nd phse under deep CM ondiion (R lod =200Ω,) re shown in Fig. 6, nd under CCM ondiion (R lod =10Ω) - in Fig. 6. The solid lines in Fig. 6 represen he rnsfer funions of he lossless onverer nd he dshed lines re for he onverer wih prsii losses. Crosses nd irles mrk he yle-y-yle simulion resuls, whih were oined y Sr-Hspie (Avn! Corp.). Eellen greemen is oserved eween he simulion resuls oined y he proposed generlized model (lines) nd he yle-y-yle simulion resuls over mos of he frequeny rnge. The disrepny high frequeny refles he inury of he verge modeling mehod when he frequeny of he moduled signl pprohes he swihing frequeny Vou Vd (f) M gniude [ B ] P hse [ eg.] H z 10H z 100H z 1.0KH z 10K H z 100K H z 100 V ou Vd (f) M gniu d e [ B ] P hse [ eg.] Frequeny oo o H z 10H z 100H z 1.0KH z 10KH z 100KH z Frequeny o o Fig. 6. Smll signl oupu-volge o duyyle rnsfer funion of Boos onverer (Fig. 4) under CM nd CCM. Solid lines: idel onverer. shed lines: lossy onverer. isree poins (rosses nd irles): resuls of yle-y-yle simulion. o V. CONCUSONS The modified SM mehodology presened here n e used o esime onduion losses, lrge signl rnsfer funion nd he rel smll signl rnsfer funions h ke ino oun he dmping effe of he losses. The mehod provides n esime of he power dissipion of he induor, swih nd diode used y onduion losses. By king he rio of oupu o inpu power he effiieny (ssoied wih onduion losses) n e lso esily oined. These ould help o opimize he design of swih mode sysems.

6 REFERENCES [1] S. Ben-Ykov, "SPCE simulion of PWM C-C onverer sysems: Volge feedk, oninuous induor onduion mode," EE Eleroni eers, vol. 25, pp , [2] Y. Amrn, F. Huliehel nd S. Ben-Ykov, "A unified SPCE ompile verge model of PWM onverers," EEE Trns. on Power Eleronis, vol. 6, pp , [3]. Kimhi nd S. Ben-Ykov, "A SPCE model of urren mode PWM onverers opering under oninuous induor urren ondiions," EEE Trns. on Power Eleronis, vol. 6, pp , [4] S. Ben-Ykov nd Z. on, "Unified SPCE ompile model of urren feedk in swih mode onverers," EE Eleroni eers, vol. 28, pp , [5] S. Ben-Ykov nd. Adr, "Averge models s ools for sudying he dynmis of swih mode C-C onverers," EEE Trns. PESC Re., pp , Tipei, [6] S. Ben-Ykov, "Averge simulion of PWM onverers y dire implemenion of ehviorl relionships," n. J. Eleronis, vol. 77, pp , 1994.