MODELING OF CROSS-REGULATION IN MULTIPLE-OUTPUT FLYBACK CONVERTERS

Transcription

1 MODELING OF CROSS-REGULATION IN MULTIPLE-OUTPUT FLYBACK CONVERTERS Dragan Maksimovićand Rober Erickson Colorado Power Elecronics Cener Deparmen of Elecrical and Compuer Engineering Universiy of Colorado, Boulder, CO hp://ece- pwrelec Phone: (303) , Fax: (303) , Absrac In his paper, complex operaion of muliple-oupu flyback converers is explained in erms of he exended canilever magneics model. Analyical resuls are derived ha can accuraely predic seady-sae cross-regulaion properies of a converer wih any number of oupus and wih arbirarily complex magneics configuraion. Two snubber configuraions are considered: a passive volage-clamp snubber and an acive-clamp snubber. Predicions of he model are verified by experimens. The analyical models are he basis for a discussion of magneics design guidelines ha can resul in improved cross-regulaion in muliple-oupu flyback converers. Inroducion The muliple-oupu flyback converer such as he example shown in Fig. is one of he mos frequenly used configuraions in lowcos, relaively low power applicaions. If all windings are perfecly coupled, and if conducion losses are small, he oupu volages in coninuous conducion mode are simply proporional o he respecive urns raios and closed-loop regulaion of one oupu resuls in perfec regulaion of all oupus. In pracice, however, perfec coupling is impossible o achieve, he observed operaion is much more complex, and poor cross-regulaion resuls are ofen obained. Poor cross-regulaion can only in par be aribued o conducion losses. If an oupu is heavily loaded, is oupu drops due o increased volage drops across conducing devices and winding resisances. If his oupu is closed-loop regulaed, he duy raio increases o compensae for he load-induced volage drop, and all oher unregulaed oupu volages increase accordingly. Conducion losses and heir effecs on cross-regulaion in a muliple-oupu converer can be modeled using sandard averaging echniques. I has been recognized ha ransformer leakage inducances play he key role in operaion of a muliple-oupu flyback converer [, 2, 3]. Previous analyses were limied o a wo-oupu case, or were based on simplifying assumpions abou he ransformer model. Transformer leakage inducances are usually considered second-order effecs, and any analysis ha would aemp o ake heir effecs ino accoun would yield inracable resuls, especially if he ransformer had more han wo secondary windings. However, secondary curren waveforms and he seady-sae soluion in he converer are in fac very srongly dependen on he ransformer leakages. Because of he complex converer operaion, and lack of general and complee mag- R s 4kΩ V g 30V _ C s 00µF Q s IRF640 D s i v v Q q IRF640 _ W W 2 v 2 MUR80 _ C 2 330µF MUR80 i W 3 3 v 3 C 3 330µF MUR80 _ W 4 v 4 C 4 330µF Figure : Experimenal 3-oupu flyback converer wih passive or acive-clamp snubber. neics model suiable for cross-regulaion analysis, very few resuls are now available o aid he designer. The purpose of his paper is o give complee qualiaive and quaniaive explanaion of he muliple-oupu flyback converer operaion, and o derive seady-sae cross-regulaion models valid for any number of oupus and arbirarily complex magneics. Our analysis is based on he exended canilever magneics model [4], which has earlier been applied o modeling cross-regulaion in forward-ype converers wih coupled inducors [5]. In Secio, he exended canilever model is briefly described using a flyback ransformer example for he experimenal converer in Fig.. General analyical resuls ha give predicions of seady-sae oupu volage variaions as funcions of load currens in a converer wih arbirary number of oupus are derived in Secio. The resuls are obained for a converer wih a passive volage-clamp snubber, and for a converer wih an acive-clamp snubber. Experimenal ver- R o2 R o3 R o4 V 2 _ V 3 V 4

2 ificaion of he cross-regulaion models is presened in Secio. In Secion 5 we discuss model implicaions and design guidelines. 2 Exended canilever magneics model: experimenal example The exended canilever model is a general circuodel for muliplewinding magneics [4]. Fig. 2 shows he winding geomery and he corresponding exended canilever model for he ransformer used in he experimenal 3-oupu, 00kHz flyback converer of Fig.. The model parameers are a magneizing inducance L, effecive leakage inducances l ij beween he windings, and effecive urns raios n j. The model can be used for arbirarily complex magneics wih any number of windings. Mos imporanly, he leakage inducances ha are essenial for analyses and design of muliple-oupu converers are direcly exposed in he model, and all model parameers can be direcly measured, as described in [4]. Leakage inducance values in he model can be relaed o he winding geomery. Winding W 4 occupies a small porion of he bobbin, and is farhes away from he primary winding W. As a resul, he effecive leakage inducance is relaively large. Winding W 2 is closes o he primary and he effecive leakage inducance is he smalles of he hree primary-o-secondary leakage inducances. Noice ha a leakage inducance in he model can have a negaive value. In Fig. 2, l 34 is negaive. This can be relaed o he winding arrangemen where he W 4 curren and he curren induced in he W 2 winding resul in he opposie polariy of he induced W 3 curren [4]. In he experimenal converer, he inpu is V g = 30V, and he nominal oupus are: V 2 = V 3 = 2V, V 4 = 3.3V. The 3.3V (winding W 4) oupu is aken o be he main oupu ha would be regulaed by a feedback loop, while V 2 (winding W 2)andV 3(winding (W 3) are auxiliary oupus. 3 Modeling of cross-regulaion in muliple-oupu flyback converers Muliple-oupu flyback converers have complex operaion and characerisics ha srongly dependen on he ransformer leakage inducances. Secondary currens can have posiive or negaive slopes, and muliple disconinuous conducion modes may occur depending on operaing condiions and relaive values of leakage inducances. Changes in he load curren a one of he oupus srongly affec all oupu volages. This secion has wo main objecives: () o qualiaively explain complex operaion of a muliple-oupu flyback converer using equivalen circuodels based on he exended canilever magneics model; (2) o derive analyical resuls ha can be used o predic how load variaions a one oupu affec volage variaions a any oher oupu. To derive general resuls valid for any number of oupus, i is convenien o use marix/vecor noaion. For a hree-oupu example, V 2 V 2/ I 2 I 2 V = V 3, V = V 3/, I= I 3, I = I 3, () V 4 V 4/ I 4 I 4 are he vecors of oupu volages and currens, referred o he secondary or he primary side, respecively. The reference polariies for he currens and volages are as shown in Figs. and 2. W 3 : 5T #22AWG n2 :0.42 i 4.5 µh l v 2 3 v L 220 l µh W 2 µh µh W 0.4 : l 34 :0.42 i 3 35 µh v 4 W 4 EC4-3C80 core (a) 4 µh (b) air gap air gap 3 µh W 4 : 5T #6AWG W 2 : 5T #8AWG layer W : 36T #8AWG 2 layers Figure 2: Winding geomery (a) and he exended canilever model (b) of he ransformer used in he flyback converer of Fig.. The seady-sae cross-regulaion models we derive in his secion will show how changes in he DC load currens I produce variaions in he DC oupu volages V: V = R I, (2) where R is a marix of Thevenin equivalen oupu resisances for a given DC operaing poin. In he analysis and discussion of crossregulaion properies, i is ofen useful o express he model (2) in erms of he volages and currens referred o he primary side: V = (N RN ) I = R I (3) where N is a diagonal marix wih he effecive urns raios,, ec., on he main diagonal. In general, i is clear ha smaller enries in R resul in beer cross-regulaion. Because of he leakage inducances, however, he oupu resisances are nonzero even if all conducion losses are negleced. One should also noe ha perfec closed-loop cross-regulaion can be achieved even wih non-zero oupu resisances. For example, suppose ha wo rows of R have equal erms, so ha arbirary load variaions resul in he same volage variaions (referred o he primary) on boh oupus corresponding o hese rows. Then, if one of he wo oupus is closed-loop regulaed, he oher oupu will be ighly regulaed as well. The models derived in his secion will show how R and R can be found in general, in erms of he ransformer exended canilever model. Referring o Fig., we consider muliple-oupu flyback converers wih wo commonly used snubber configuraions: () a passive volage-clamp snubber where a resisor R s loads he snubber capacior C s, and he auxiliary device Q s is removed, and (2) an acive-clamp W 3 v 3

3 L l 24 V 2 / - V 2 2 l 34 i 3 i 3 n3 V 4 / - V 3 V 3 / Q off, Q s /D s on (a) Commuaion inerval 2 V x n4 - V 4 L l 24 l 34 V 2 / i 3 Q on Q off D s on c 2 Diode conducion inerval (-D)T Q off D s off V 4 / V 3 / Q off, Q s /D s off (b) Diode conducion inerval Figure 3: Circuodel during pars of he swiching cycle when he main swich Q is off: (a) he commuaion inerval and (b) he diode conducion inerval. snubber where he snubber resisor R s is removed and he auxiliary device Q s is used o enable bidirecional curren flow o and from he snubber capacior C s. Seady-sae soluion and cross-regulaion model parameers in hese wo cases can be significanly differen. 3. Muliple-oupu flyback converer wih passive volageclamp snubber The converer operaion is explained wih reference o he circui models shown in Fig. 3 and he idealized waveforms shown in Fig. 4. A se of experimenal waveforms observed wih he passive volageclamp snubber in he experimenal converer of Fig. is shown in Fig. 5. When he ransisor Q urns off, a commuaion inerval sars when he snubber diode D s urns on, conducing he magneizing curren and clamping he primary volage o v =. The secondaryside diodes also begin conducing. The circuodel during he commuaion inerval is shown in Fig. 3(a). A secondary curren increases a he rae deermined by he snubber volage, he oupu volages, and he effecive leakage inducances. For example, for he winding W 2 curren, we have: d d = Vs V2/n2 V3/n3 V2/n2 V4/n4 V2/n2 l 24, (4) and similar expressions can be wrien for he oher winding currens. Given ha he refleced oupu volages are nearly equal, V 2/ V 3/ V 4/, he rae of change is dominaed by he firs erm, d d Vs V2/n2. (5) Figure 4: Idealized secondary curren waveforms in he converer wih a passive volage-clamp snubber. A/div i 3 A/div A/div v q 50V/div Figure 5: Experimenal waveforms in he flyback converer wih he passive volage-clamp snubber a he operaing poin: D = 0.52, V 2 = 3.5V, V 3 = 4.2V, V 4 = 3.4V, R o2 = 25Ω, R o3 = 50Ω, R o4 = Ω. Therefore, during he commuaion inerval c, he secondary currens increase a he raes deermined by he effecive leakage inducances,,and, as shown in Fig. 4. These raes are no direcly relaed o he individual load currens. A he end of he commuaion inerval c, he sum of he refleced secondary currens becomes equal o he magneizing curren, i 3 =, (6) and he snubber diode D s urns off. The values of he individual secondary winding currens a his ime are deermined by he relaive values of,, and, and again are no direcly relaed o he

4 individual load currens, n k i k = Lo l k, (7) where =, (8) and k =, 2, 3. In he magneics example of Fig. 2, winding W4 is poorly coupled o he primary and is relaively large. As a resul, he curren a he end of he commuaion is relaively small. Since winding W 2 and W 3 are beer coupled o he primary, he currens, i 3 a he end of he commuaion inerval are relaively large, as illusraed by he experimenal waveforms in Fig. 5. During he remainder of he swiching period, which is called here he diode conducion inerval, he secondary currens increase or decrease a he raes ha depend on he differences beween he refleced oupu volages, and he volage V x across he magneizing inducance. The circuodel during he diode conducion inerval is shown in Fig. 3(b). If an oupu is heavily loaded and he curren a he end of he commuaion inerval is relaively small, he slope of he corresponding winding curren will be posiive, which means ha he corresponding oupu volage is reduced. In Fig. 5, his can be observed in he curren of he V 4 oupu. Since he currens, i 3 a he end of he commuaion inerval are relaively large and since hese oupus are no heavily loaded, he corresponding winding currens decrease during he diode-conducion inerval, which indicaes ha he refleced oupu volages V 2/, V 3/ mus be greaer han V x. From his qualiaive discussion, i follows immediaely ha because of he ransformer leakage inducances, each oupu exhibis a nonzero oupu resisance, even in coninuous conducion mode, and even if all losses are negleced. During he diode conducion inerval, he raes of change of he refleced secondary currens are di d = B(V uv x), (9) where B = u T = [ ],and L o2 l 24 L o3 l 34 l 24 l 34 L o4, (0) L o2 = l 24, () L o3 = l 34, (2) L o4 = l 24 l 34, (3) are he Thevenin equivalen oupu inducances of he secondary windings, referred o he primary side. The secondary winding currens may have posiive or negaive slopes. A winding curren wih negaive slope may drop o zero before he end of he diode conducion inerval (for example, see curren i 3 in Fig. 5), which resuls in a disconinuous conducion mode and even larger variaions in he oupu volages. In general, depending on he load condiions and values of he magneics model parameers, an oupu may operae in coninuous or disconinuous mode, which adds o he complexiy of he converer operaion. We firs derive a seady-sae cross-regulaion model assuming ha all oupus operae in he coninuous conducion mode. Assuming ha he commuaion inerval c is a small fracion of he swiching period, averaging of a secondary winding curren over a swiching period gives a relaion ha includes he iniial value of he curren i k a he sar of he diode conducion inerval, which is given by (7), he rae of change of he curren, which is given by (9), and he average oupu curren I k, I k ( D)i k ( D) 2 di k 2f s d. (4) The magneizing curren a he end of he commuaion inerval is relaed o he load currens: Vx( D) I m 2L f s Vx( D) (n2i2 n3i3 n4i4) D 2L f s = D (ut I ) Vx( D). 2L f s (5) Combining (7), (9), (4), and (5) yields he seady-sae soluion for he refleced DC oupu volages V in erms of he refleced load currens I and he duy raio D: B (V uv x)= where B is given by (0), and B 2 = b 2 = 2f s ( D) 2B2I b 2 L V x, (6), (7). (8) Volage V x across he magneizing inducance L during he diode conducion inerval can be found from he vol-second balance on L, V x = DVg Vscfs D cf s DVg Loimfs, (9) D assuming ha he commuaion inerval c is shor compared o ( D)T, and ha he volage across he effecive primary-osecondary leakage inducances during he commuaion inerval is approximaely equal o he snubber volage. From (5) and (9), and using << L, i follows ha D V x V g D fslo ( D) 2 (ut I ) (20) The seady-sae soluion (6), (20) is a general resul ha shows how he dc oupu volages depend on he dc load currens in a mulipleoupu flyback converer wih any number of oupus and wih arbirarily complex magneics. For a given DC operaing poin, he marix of Thevenin equivalen oupu resisances follows from (3), (6) and (20): ( ) R = 2fs B ( D) 2 B 2 Lo L 2 (B b o 2 u)u T (2) L

5 Assuming ha effecive leakage inducances are of he same order of magniude, and ha << L, he expression for R can be simplified: ( ) R = 2fs B ( D) 2 B 2 Lo 2 uut (22) Here, B (given by (0)), B 2 (given by (7)), and (given by (8)), all depend only on he exended canilever magneics model parameers. The oupu resisances are direcly proporional o he swiching frequency f s, which is no surprising since he oupu resisances are relaed o he impedances of he ransformer leakage inducances. Also, one may noe ha he oupu resisances depend on he seadysae operaing poin hrough he duy raio D. When he converer is operaed open loop a consan duy raio D, he oupu resisances are consan, and he model predics ha oupu volage variaions are linear funcions of load variaions. In he analysis above, we assumed ha all oupus operae in he coninuous conducion mode (CCM). I is of ineres o find a condiion for CCM operaion because even larger variaions in oupu volages can be expeced if an oupu moves from CCM o disconinuous conducion mode (DCM). For a secondary winding, he condiion for operaion in CCM is ha he peak curren ripple is smaller han he average value during he diode conducion inerval. In vecor form, I D D di 2f s d > 0 (23) The relaion beween he curren slopes and he average currens can be deduced from (9) and (6), di d = 2fs L ( D) 2 B2I b o 2 V x. (24) L Eqs. (23) and (24) yield he CCM condiion: I > ( D)2 V x 2f sl u. (25) Noe ha he CCM condiion for each oupu depends no only on he oupu s load curren bu also on he load currens a he oher oupus. Increasing he load curren on one oupu evenually drives he oher oupus ino disconinuous conducion mode. This is because increasing he load curren on one oupu increases he magneizing curren (as shown by (5)), which in urn increases he currens a he end of he commuaion inerval on all oupus (as shown by (7)). For an oupu wih consan load, he larger iniial value a he sar of he diode conducion inerval implies ha he final value a he end of he diode conducion inerval mus decrease. If he winding curren drops o zero before he end of he diode conducion inerval, his oupu eners disconinuous conducion mode. The resul (25) quanifies his behavior. 3.2 Muliple-oupu flyback converer wih acive-clamp snubber If an acive-clamp snubber is used, he volage v is clamped o during he enire inerval when he main ransisor Q is off. The commuaion inerval and he diode conducion inerval merge ino i 3 Q on 2I 3 -D 2I 2 -D Q off, Q s /D s on (-D)T 2I 4 -D Figure 6: Idealized secondary curren waveforms in he converer wih an acive-clamp snubber. one inerval of lengh ( D)T, wih he equivalen circuodel shown in Fig. 3(a). Idealized secondary curren waveforms are shown in Fig. 6. As an example, Fig. 7 shows experimenal waveforms for he same operaing condiions as in Fig. 5, bu wih he acive-clamp snubber. Noe ha he secondary curren waveshapes are significanly differen han in he passive-snubber case. Wih he acive-clamp snubber, all oupus can operae only in coninuous conducion mode, which simplifies he analysis. The secondary currens sar from zero and increase a he raes ha depend on leakage inducances and differences beween refleced oupu volages: di d = B(V u). (26) Winding currens a he end of he diode conducion inerval can be easily relaed o he DC load currens, as shown in Fig. 6, and o he curren slopes from (26). This gives a general seady-sae soluion: B (V 2f s u)= (27) ( D) 2I where = D Vg (28) D follows immediaely from he vol-second balance on L. The oupu resisance marix R is obained direcly from (27): R = 2fs ( D) 2 B. (29) I can be compared o he resul (22) obained for he passive snubber case. Because of he absence of he commuaion inerval and disconinuous modes, operaion and he seady-sae model of a muliple oupu flyback converer wih acive-clamp snubber are simpler. 4 Experimenal verificaion of he cross-regulaion models In his secion, various predicions of he cross-regulaion models derived in Secio are compared wih experimenal resuls obained

6 A/div i 3 A/div A/div v q 50V/div Figure 7: Experimenal waveforms in he flyback converer wih he acive-clamp snubber a he operaing poin: D = 0.52, V 2 = 3.5V, V 3 = 4.2V, V 4 = 3.4V, R o2 = 25Ω, R o3 = 50Ω, R o4 = Ω V 2 [V] V 3 [V] I 4 [A] Figure 8: Prediced (solid lines) and experimenal (doed lines) cross-regulaion V 2(I 4) and V 3(I 4) for he case when he auxiliary oupus operae a I 2 = I 3 = 0.4A, and he main oupu load varies beween I 4 = 0.2A and I 4 = 2A. The converer is operaed open-loop a f s = 00kHz, D = on he converer in Fig. wih he ransformer configuraion and model shown in Fig. 2. We firs consider he case when he converer is operaed wih he passive volage-clamp snubber. Fig. 8 compares he resuls prediced by he model wih experimenal resuls for he case when he auxiliary oupus V 2, V 3 are loaded a I 2 = I 3 = 0.4A and he main oupu load varies beween I 4 = 0.2A and I 4 = 2A. The converer is operaed open-loop a consan duy raio D = Throughou his load range on he V 4 oupu, all oupus operae in he coninuous conducion mode. One may observe ha oupu volage variaions V 2, V 3 are linear funcions of he load I 4, as prediced by he model. The model also correcly predics ha he oupu V 2 decreases and he oupu V 3 increases wih increasing load on he V 4 oupu. Alhough our cross-regulaion model included only magneics parameers while conducion and oher losses were negleced, he prediced resuls correlae very well wih he experimens. A he operaing poin D = 0.52, I 2 = I 3 = 0.4A, I 4 = A, he oupu resisance marix (referred o he primary side) is found from (22): R = Ω (30) and referred o he secondaries, R = NR N = Ω (3) The measured oupu resisance marix is: R experimen = Ω (32) One can observe very good correlaion beween prediced and measured off-diagonal erms. This indicaes ha he coupling erms are indeed deermined mainly by he ransformer leakage inducances. The diagonal erms, which are he Thevenin equivalen resisances of he individual oupus, are affeced by he winding and diode conducion losses, which were no included in he model. Hence, he measured erms on he main diagonal are higher han he values obained from he model. In he experimenal example, he CCM condiion (25) resuls in: I 2 I 3 I 4 > 0.7A 0.7A 0.7A. (33) The V 2 oupu has he smalles erm (0.73) on he main diagonal, and is mos likely o operae in DCM because he winding W 2 is bes coupled o he primary, i.e., because he leakage in he magneics model of Fig. 2 is smaller han or. In general, increasing load on one oupu while keeping he oher wo loads consan causes his oupu o move from DCM o CCM, and he oher wo oupus o move from CCM o DCM. For example, for I 2 = 0.6A, I 4 = A, and D = 0.52, he model predicion is ha all oupus operae in CCM for 0.9A <I 3 <0.32A. In he experimenal prooype he V 3 oupu enered DCM when I 3 dropped below 0.2A, while he V 2 oupu enered DCM once I 3 exceeded 0.45A. Nex, we consider he experimenal converer of Fig. wih he acive-clamp snubber. A he same operaing poin as in he passive snubber case, D = 0.52, I 2 = I 3 = 0.4A, I 4 = A, he oupu resisance marix (referred o he primary side) is found from (29): R = Ω (34) and referred o he secondaries, R = NR N = Ω (35) Some of he oupu resisance values are significanly differen compared o he passive-snubber case. This example shows ha even

7 changes in he snubber configuraion or snubber parameers can have significan effecs on he cross-regulaion performance of he flyback converer. The measured oupu resisance marix is: R experimen = Ω (36) Conclusions are similar as in he passive snubber case: off-diagonal erms in he model correlae well wih he experimens, while he erms on he main diagonal are higher because of he conducion losses which were no included in he model V 3 [V] V 2 [V] I 4 [A] Design consideraions In Secio, we have shown ha complex behavior of a mulipleoupu flyback converer can be explained in erms of he exended canilever magneics model. The general seady-sae soluions (6), (22) and (27), (29) allow one o deermine he seady-sae crossregulaion performance for any given magneics design, and a given converer configuraion wih arbirary number of oupus. Furhermore, using he derived analyical resuls, various magneics design approaches can be evaluaed and compared. Several general design guidelines o improve cross-regulaion can be deduced: Oupu resisances are direcly proporional o leakage inducances (, l 24, l 34) beween he secondaries. Therefore, igher coupling beween he secondaries yields improved cross-regulaion. Minimizaion of leakage inducances beween he secondaries and he primary is no essenial for good cross-regulaion. In fac, larger,, inducances increase he CCM load ranges, which may improve cross-regulaion. Relaive values of he effecive leakage inducances beween he primary and he secondaries (,, ), are imporan for good cross-regulaion and he bes windings arrangemen depends on he specified load ranges a he oupus. The winding of he oupu wih he wides load range should have he bes coupling o he primary, i.e. i should have he smalles value of he effecive leakage inducance. Winding arrangemens ha resul in maching rows of he oupu resisance marix R referred o he primary lead o good closed-loop cross-regulaion even if leakage inducances are relaively large. If wo oupus have he same erms in he corresponding rows of he R marix, closed-loop regulaion of one of he oupus yields excellen regulaion of he oher oupu. The las poin in he design guidelines above can be illusraed using he experimenal example of Figs. and 2. Couplings among he secondary windings are such ha all erms in he firs and he hird row of R have he same sign, as shown in (30) or (34). As a resul, he oupus V 2 and V 4 change in he same direcion for any change of load currens. In conras, he signs of he erms corresponding o V 3 and V 4 are opposie. If, for example, he load I 4 increases, he oupus V 4 and V 2 decrease, while he oupu V 3 increases, as shown in Fig. 8. Suppose now ha he oupu V 4 is closed-loop regulaed so ha load-induced variaions in V 4 are compensaed by variaions Figure 9: Cross-regulaion V 2(I 4) and V 3(I 4) for he case when he auxiliary oupus operae a I 2 = I 3 = 0.4A, and he main oupu load varies beween I 4 = 0.2A and I 4 = 2A. The converer is operaed closed-loop a f s = 00kHz, V 4 = 3.3V. in duy raio D. From our model we can expec improved crossregulaion on he V 2 oupu compared o he V 4 oupu. Fig. 9 shows he measured closed-loop cross-regulaion V 2(I 4), V 3(I 4), forhe converer wih passive volage-clamp snubber. As expeced, he oal volage variaions on he V 2 oupu is smaller han on he V 3 oupu. The winding arrangemens ha resul in opposie-sign erms in he resisance marix clearly give poor cross-regulaion resuls. A beer mach beween he oupu resisance erms for he V 2 and V 4 oupus could be obained by improving he coupling beween he primary and he W 4 oupu, which would resul in improved cross-regulaion on he V 2 oupu. 6 Conclusions Operaion and characerisics of muliple-oupu flyback converers depend srongly on he ransformer leakage inducances. In paricular, seady-sae oupu volages and cross-regulaion properies are deermined by how well individual ransformer windings are coupled. Taking ino accoun he effecs of leakage inducances has been considered inracable in all bu he simples cases limied o wo secondary windings or based on oher simplifying assumpions abou he ransformer model. In his paper, complex operaion of muliple-oupu flyback converers is explained in erms of he exended canilever magneics model. This general circuodel allows easy qualiaive and quaniaive explanaion of he observed secondary curren waveforms, as well as a general seady-sae analysis in erms of he exended canilever model parameers. Two snubber configuraions are considered: a passive volage-clamp snubber and an acive-clamp snubber. In boh cases, analyical models are derived ha can accuraely predic seady-sae cross-regulaion properies of a converer wih any number of oupus and wih arbirarily complex magneics configuraion. Predicions of he models are verified by experimens. The analyical models are he basis for a discussion of magneics design guidelines ha can resul in improved crossregulaion in muliple-oupu flyback converers. References [] T. Wilson, Jr,. Cross regulaion in an energy-sorage DC-o-DC converer wih wo regulaed oupus, IEEE PESC, 997 Record, pp

8 [2] K. Liu, Effecs of leakage inducances on he cross regulaion in a disconinuous conducion mode flyback converer, Proc. High Frequency Power Conversion Conference, 989, pp [3] J. Marrero, Improving cross regulaion of muliple oupu flyback converers, PCIM 996. [4] R. W. Erickson, D. Maksimović, A muliple-winding magneics model having direcly measurable parameers, IEEE PESC 998. [5] D. Maksimović, R. W. Erickson, C. Griesbach, Modeling of crossregulaion in converers conaining coupled inducors, IEEE APEC 998, pp [6] Uceda & Co. COULD NOT FIND THEM IN PESC 98