Analysis and Modeling of Buck Converter in Discontinuous-Output-Inductor-Current Mode Operation *

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1 Energy and Power Engneerng, 3, 5, do:.436/ee.3.54b63 Publshed Onlne July 3 (htt:// Analyss and Modelng of Buck Converter n Dscontnuous-Outut-Inductor-Current Mode Oeraton * Janbo Yang, Weng Zhang,Fars Al-Naem,Xaong Chen, Materals and Engneerng Research Insttute (MERI), Sheffeld Hallam Unversty (SHU), Sheffeld, UK Lab of Green Power & Energy System (GPES), North Chna Unversty of Technology (NCUT), Bejng, Chna Emal: [email protected] Receved Setember, ABSTRACT The Buck converter wth LC nut flter oeratng n dscontnuous outut current mode has a hgh ower factor wth a constant duty cycle. A Buck converter n ths oeraton mode can reduce the reverse recovery loss of the freewheelng dode thus ncrease the effcency. The oeraton, ower factor analyss and modelng of the converter are studed n ths aer. Exermental results are resented to verfy the theoretcal redctons. Keywords: Buck Converter; Power Factor Correcton; Modelng. Introducton AC/DC converter has been studed as a hgh-ower-factor rectfer. The most oular ower stage crcut of the AC/DC converter s a Boost converter [,]. The Boost converter has an nut nductor whch naturally makes the nut current contnuous and less of harmoncs. However, the dsadvantage s the outut has to be hgher than the eak nut voltage. Unlke the Boost AC/DC converters, the Buck AC/DC converters have ste-down characterstcs as the outut voltage s lower than the eak nut voltage. The buck converter wth LC nut flter oeratng n dscontnuous outut nductor current mode has resstve and constant nut medance wth a constant duty cycle D and a constant swtchng cycle T s. Thus, the average nut current of the converter follows the nut voltage wth a constant duty rato control. The low-frequency behavor modelng of the Buck converter wth LC nut flter oeratng n dscontnuous outut nductor mode s resented n ths aer. Based on the model, the characterstc of the converter and the condtons for ower factor correcton s studed. The exermental verfcaton s also gven. The buck converter wth LC nut flter s shown n Fgure. The dscontnuous outut nductor current mode means the current through the nductor L s zero durng art of one swtchng cycle. The converter can stll oerate n dscontnuous caactor voltage mode whch means the voltage across C s zero durng art of a swtchng cycle. These two oeraton modes can be consdered as dual of each other [3]. However, the voltage stress of the swtch and dode wll be very hgh when the converter oerates n dscontnuous caactor voltage mode [4]... Oeraton Prncles The oeraton rncles of the converter and the characterstc waveforms are resented n Fgure. There are three hases for the Buck converter wth LC nut flter oeratng n dscontnuous nductor current mode. Phase : ~DTs. swtch S s turned on. D s reversebased. When the oeraton s n steady state, the average voltage across L, over one swtchng cycle, s zero. The average voltage across C over one swtchng cycle s. Power Stage.. Crcut Confguraton * Project suorted by Natural Scence foundaton of Chna (N. 5774). The Imortaton and Develoment of Hgh-Calber Talents Project of Bejng Muncal Insttutons (No.IDHT35) Fgure. Buck AC/DC converter wth LC nut flter. Coyrght 3 ScRes.

2 J. B. YANG ET AL. 85 (a) ~DTs (b) DTs~DTs (c) DTs~Ts The current through C s I I. When I I, c s ostve, C s chargng. When I I, C s dschargng. Phase : DTs~DTs. S s closed D s forward-based. L s dschargng and the current through L falls to zero at DTs. C s chargng. The eak current through L can also be obtaned as: I D DT s () L Phase 3: DTs~Ts; the swtch S s stll turned off. Current through L falls to zero. Outut s suorted by C ndvdually. Combnaton of () and (), the followng equaton can be obtaned. D D (3) V D If the swtchng erod Ts s much smaller than the nut cycle T, the nut voltage can be consdered constant over one swtchng cycle [3]. Then, the nut voltage V n (), (3) can be drectly relaced by Vsn wt durng half of lne cycle T. ( Vsn wt ) I DTs (4) L D D (5) V sn wt D V can be defned as η. then from equaton (5), the relaton between η and D s sn wt (6) D Therefore, the converter wll oerate n dscontnuous nductor current mode throughout half lne cycle T when D. (d) Swchng waveforms Fgure. Dscontnuous nductor current oeraton and waveforms. equal to the nut voltage. C oerates n contnuous mode that uc s small durng one swtchng cycle. Thus, the average voltage across C over one swtchng cycle can be consdered as the nstantaneous voltage across C. Therefore, L s chargng under constant voltage (V V o ). The eak current through L s: ( V ) I DTs () L 3. Power Analyss As shown n Fgure, the nut current I s the sum of the swtch current s and current through C. The c average current through C s zero durng one swtchng cycle n the steady state. Thus, the average nut current equals to the average swtchng current over one swtchng cycle. Accordng to () and (4), the average nut current, over one swtchng cycle, can be obtaned as, ( Vsn wt ) Iavg DTs (7) L where Iavg means the average nut current over one swtchng cycle. As the swtchng cycle T s s much smaller than the nut cycle T, the nut current can be consdered constant over one swtchng cycle. Buck Coyrght 3 ScRes.

3 85 J. B. YANG ET AL. converter oerates only when the nut voltage s hgher than the outut voltage. Therefore, (7) s vald only when Vsn wt. When the nut rectfed voltage equals to the outut voltage, t arcsn arcsn (8) w V w α s defned as the rato between the outut voltage and the eak of the rectfed nut voltage. Thus, over half nut lne cycle, oeraton s ossble only for T t( t, t ), as shown n Fgure 3. The average nut ower wth constant duty rato and constant swtchng cycle T s s rovded n (9). As referred to (5), the maxmum duty cycle D s η for mantanng the converter oerate n dscontnuous nductor current mode. Substtuton of D for η n (9), the followng relaton can be obtaned n (). P V wt I dt T t sn, avg T t V D Ts = ( arcsn ) 4L s P V T 4L (9) ( arcsn ) () The relaton between actve nut ower and converson rato η wth maxmum duty rato s lotted n Fgure 4. As shown, wth D = η, the converter oerates n the boundary of the dscontnuous nductor current mode and the maxmum actve nut ower wll be obtaned when η s near.58. The root mean square value of the nut current can be obtaned from (7) as I rms T ( Vsn wtv ) T t o t L D Ts dt () 6 VDT s = ( )( arcsn ) L the ower factor s obtaned from () and () as, arcsn P PF. V rms I rms 6 ( )( arcsn ) () where P.F s the ower factor. It s clear that the ower factor s affected only by converson rato η when the duty cycle D s constant. The relaton s lotted n Fgure 5. Fgure 3. Oeraton waveforms durng half nut cycle ower factor & converson rato Fgure 4. Actve Inut Power vs Converson Rato η (from equaton ()). actve nut ower & actve nut ower Fgure 5. Power Factor vs Converson Rato η (from equaton ()). It s shown that the ower factor s reversely roortonal to the converson rato η. When.6 the ower factor wll be hgher than.9. The results are exact the same as [5]. When η s very small, V V sn wt. Then the nut current wll be o I avg Vsn wt DTs (3) L t t Coyrght 3 ScRes.

4 J. B. YANG ET AL. 853 The equvalent nut mendence s resstve, L R (4) DT s The ower factor turns out to be unty. Therefore, wth constant duty cycle D and constant swtchng cycle Ts, the lower outut voltage, the hgher ower factor. where, () 4. Modelng The low-frequency model s obtaned by consderng the relatve voltage or current wll not change over one swtchng cycle snce the swtchng cycle s much hgher than the nut lne frequency. Therefore, the average value, over one swtchng cycle, s consdered as an nstant value durng one swtchng cycle [4,6]. By assumng that the voltage across C s contnuous and constant over a swtchng cycle, the average swtch current s and average current through L over one swtchng cycle can be obtaned from Fgure. V Is DDT s (5) L V I ( D D) DT s (6) () s V() s s outut-nut transfer functon and L () s D() s s the outut-control transfer functon. The As the average voltage across L s zero over one transfer functon between outut voltage and duty cycle swtchng cycle. gven by () has three oles and two zeros. Ths can be aroxmate to have a sngle ole whch s exactly the V V D D (7) same as the tradtonal Boost converters [6,7]. V As shown n Fgure, the followng equatons can be 5. Results obtaned. 5.. Smulatons Is() s ( V() s V()) s V()* s sc (8) Smulatons were carred out to verfy the dscontnuous sl nductor current oeraton of the crcut. The comonents used are: L = 5u. L = u, C = n. C = u, D I() s V()( s sc ) (9) R =., R = 3. The nut voltage s VAC and the outut voltage s 36 V. The results are llustrated as follows () s V () s () (Fgures 6-9), whch show that nut current wll follow Dervaton and Lalace transform of (5) ~ (7) and the nut voltage automatcally when Buck converter substtuton of the results nto (8) ~ (9), the followng wth an nut LC flter oerates n dscontnuous nductor can be obtaned. current wth a constant the duty rato. VP7/ I9* Tme (s) Fgure 6. Inut voltage and nut current (Vac;.5A). Coyrght 3 ScRes.

5 854 J. B. YANG ET AL. 8 V Tme (s) Fgure 7. Outut voltage: 36v (rle: 5v),.5 V Tme (s) Fgure 8. Constant duty cycle (.5). I(L5) Tme (s) Fgure 9. Outut nductor current (L n Fgure : DCM). 5.. Exerments An exerment crcut was also bult and the arameters and the comonents used s the same as the smulatons. The control sgnal was generated usng an UC3854. The results are as follows. The exermental results are n accordance wth the smulatons. The nut ower s 5w and the outut ower s w. Thus the effcency of the system s 8%. The ower factor s.98. As shown n Fgures - 3, the duty cycle D s a smle constant value to gan a hgh ower factor. 6. Conclusons The oeraton and characterstcs of the Buck converter wth LC nut flter oeratng n dscontnuous nductor current mode has been studed. Coyrght 3 ScRes.

6 J. B. YANG ET AL. 855 Fgure. Inut ltage: vac Inut Current:.54mA. Fgure. Outut voltage: 36VDC (rle: 5v). Fgure. Outut nductor current (L n Fgure ) DCM. Coyrght 3 ScRes.

7 856 J. B. YANG ET AL. Fgure 3. Constant duty cycle. It s found that the converter can gan a hgh ower factor when the duty rato mantan constant. A low frequency model of the converter has been develoed. A w exermental crcut has been bult to confrm the theoretcal analyss. REFERENCES [] R. Hammano and R. Nedorff, Imrovng Inut Power Factor - A New Actve Control Smlfes the Task, n Proceedngs of the 9th Internatonal PClM Conference, 989. [] R. Keller and G. Baker, Unty Power Factor Off Lne Swtchng Power Sules, n IEEE INTELEC Record. 984, [3] V. Grgore and J. Kyyra, Hgh Power Factor Rectfer Based on Buck Converter Oeratng n Dscontnuous Caactor ltage Mode, n Proc. IEEE Al. Power Electron. Conf. Exo., Mar. 999, [4] Y. S. Lee, Modelng, Analyss, and Alcaton of Buck Converters n Dscontnuous-Inut-ltage Mode Oeraton, IEEE Trans. Ind. Electron, l., No., 99. [5] H. Endo, T. Yamashm and T. Sugura, A Hgh-Power-Factor Buck Converter, Proc. of the IEEE Power Electroncs Seculsts Conference, R. W. Erckson, Fundamentals of Power Electroncs, M, second edton, [6] PHILIP C. TODD, UC3854 Controlled Power Factor Correcton Crcut Desgn, UNITRODE alcaton note, U34. Coyrght 3 ScRes.