Averaged Modeling of Non-ideal Boost Converter Operating in DCM

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1 Internatnal Jurnal f Cmputer and lectrcal ngneerng l.3 N. February Averaged Mdelng f Nn-deal Bt Cnverter Operatng n DCM Guang-jun Xe Senr Member IACSI Xuan Zha Ha-bn Fang and Hu-fang Xu Abtract he averaged mdel f nn-deal Bt cnverter peratng n dcntnuu mde (DCM crcut a tuded baed n the tchng element mdel the equvalent crcut methd and the energy cnervatn la n hch the duty rat cntrant a ntrduced t mprve the mdel. he pen-lp tranfer functn a derved t buld the mall gnal mdel by addng a vltage-cntrlled feedback lp the cled-lp tranfer functn a al derved. he mulatn reult hed the crrectne f the mdel at lat. Index erm Bt cnverter Nn-deal Cled-lp cntrl I. INRODUCION A general per tchng regulatn ytem nclude per crcut (v. tchng cnverter and cntrl crcut. Bth part nteract and rk tgether. here are many knd f cntrl mde n per tchng regulatn ytem uch a vltage-mde cntrl current-mde cntrl current-lag-lp-mde cntrl charge-mde cntrl and n. ltage cntrl crcut mple n tructure and t ant-nterference ablty trng t utable fr the tuatn hch dn t requre hgh cntrl perfrmance f the crcut[-3]. here an agnable quetn n tch cnverter mdelng that the devatn beteen the tradtnal deal mdel and actual crcut[4-6]. herefre t very neceary t tudy cnverter tch mdelng methd cnderng the nn-deal tuatn. In th paper take Bt cnverter fr example e tudy the average crcut mdel f nn-deal bac cnverter peratng n dcntnuu cnductn mde (DCM and ue the duty rat cntrant t mprve the average mdel and derve tranfer functn then derve the tranfer functn f cled-lp cntrl mdel cmbnng th vltage cntrl feedback lp. Bth ere mulated under the MAAB tl t verfy the crrectne f th mdel. Manucrpt receved May. h rk a upprted by the Intercllegate Key Prject f Nature Scence f Anhu Prvnce. Gang-jun Xe th Schl f lectrnc Scence and Appled Phyc Hefe Unverty f echnlgy Hefe 39 Anhu Chna (crrepndng authr e-mal: gjxe8@hfut.edu.cn. Xuan Zha th Schl f lectrnc Scence and Appled Phyc Hefe Unverty f echnlgy Hefe 39 Anhu Chna. Ha-bn Fang th Schl f lectrnc Scence and Appled Phyc Hefe Unverty f echnlgy Hefe 39 Anhu Chna. Hu-fang Xu th Schl f lectrnc Scence and Appled Phyc Hefe Unverty f echnlgy Hefe 39 Anhu Chna. 79 II. ARG SIGNA ARAG MOD OF NON-IDA BOOS CONRR IN DCM A. arge Sgnal Averaged Mdelng f Nn-deal Bt Cnverter n DCM When e cnder the paratc parameter f the Bt cnverter e can ee per tch MOSF a that deal tch and n retance R n are n ere ee dde a the ere f deal tch frard vltage drp F and n retance R F. R R c are repectvely equvalent ere retance (SR f flter nductance and flter capactr. Accrdng t the energy cnervatn prncple the paratc parameter f the t tch nductance are equvalent t the nductance branch. And then ue the methd f average mdel f tchng element ung current-cntrlled current urce ntead f actve tchng element S ung vltage-cntrlled vltage urce ntead f pave tchng element D. S e can get the large gnal average mdel hn n Fgure. Fg.. arge gnal average mdel f nn-deal Bt cnverter n DCM B. Duty Rat Cntrant he duty cycle frmula hch baed n vltage-ecnd f nductance vltage can nt reflect accurately the frequency charactertc f the crcut and t ha lmtatn. h paper ue the ne duty rat cntrant f Bt cnverter hch a derved n lterature []. d d dv n ( he varable a analyed t drect cmpnent and AC mall-gnal cmpnent. hen the duty rat cntrant ( I ˆ d ˆ d ( D d dv ( D ˆ d( (

2 Internatnal Jurnal f Cmputer and lectrcal ngneerng l.3 N. February d dv ( D dˆ ( d d d vap d d vap D dˆ ap ( I ˆ ( p p ( D d ( (3 ( ( ( ap (4 C. AC Small-Sgnal Mdel If e uppe that the mall-gnal cmpnent much le than the drect cmpnent make the drect cmpnent er and neglect the ecndary mall-gnal prduct term. hen the (3 and (4 can be predgeted t the ( and (6. S e can get the mall-gnal lnear equvalent mdel f nn-deal Bt cnverter n dcntnuu cnductn mdel hn n fg.. ( d d v D dˆ ( d d a a dˆ dv ap ( dd vp ( D dˆ ( ( I ˆ ( D dˆ ( [( I R ( R ˆ ] b v b d b3 b4 v (6 Fg.. mall-gnal mdel f Bt cnverter n DCM Frm fg. e can calculate the nput-t-utput tranfer functn Gv( and the cntrl-t-utput tranfer functn Gvd( are ( Gv( ( dˆ( ( b a( R b3 3 C ( R b ( R C RC b4 RRC ( C tt 4 ( ( tt 3 tt 4 ( Q (7 G vd ( ( dˆ (ˆ v ( b a( R b3 ( R b3( RCC RC b4 RRC ( C tt 4 ( ( 3 tt 3 tt 4 ( Q (8 he parameter f the actual Bt cnverter a fllng: nput-vltage utput-vltage ad current I.A R Ω 9.μH R 3.6mΩ C μf RC mω. he mdel number f the MOSF hch e ued n th paper SK69. Rn mω fr Schttky dde RF mω tchng frequency f kh PWM utput the Peak vltage. hen there parameter are put nt the frmula a fllng R 4 DR n DR F ( R 3( D D D D D D ap I D IO D I ( D D D and D R the ttal equvalent retance f the three paratc retance hch ere calculated t branch f nductance. hen e can get D.3 R.69mΩ.84 I.43A. Put the fur parameter nt frmula(7and (8 then e can get frmula (9 and (. ( ( G ( v (9 ( ( G ( vd ( 8

3 Internatnal Jurnal f Cmputer and lectrcal ngneerng l.3 N. February III. MODING OF BOOS CONRR WIH OAG FDBACK-OOP CONRO Nn-deal Bt cnverter th vltage-mde cntrl hn n fg.3. Cntrl crcut cnttuted by the cntrller the PWM cmparatr the clck crcut and the trgger. he crcut take a ample f the utput vltage and then the ample gnal a the feedback gnal put nt the nput t make the nput vltage table. Fg.3. Sytem tructure f Bt cnverter th feedback-lp A. Sample Crcut and the Pule Wdth Mdulatr ( he tranfer functn f the ample crcut H and H( v vref ref H( ( he pule dth mdulatr a vltage cmparatr actually. v c the utput gnal f errr amplfer and Cmpenatn netrk.. It cmpared th clck gnal then the cmparatr putut a pule th the duty cycle d that cntrl t man tch f the cnverter. he tranfer functn f the PWM cmparatr d ˆ ( Fm( c( ( B. PID Cmpenatr he mt mprtant cmpnent f the degn cmpenatr n cntrl crcut. ranfer functn f the man crcut Gvd( f there n cmpenatr the tranfer Gc( put t nt frmula (3 the pen-lp functn tranfer functn f the ytem. hen e can get the pen-lp tranfer functn (4 f the ytem thut cmpenatr. ( H( Gc( Gvd( (3 8 ( Gvd ( m ( And ( dB. We can calculate ut the crver frequency c 383 rad/. c f c. H and the phae margn 9. It π can be een that the crver frequency t l repne peed f the ytem t l and the phae margn t large. hee are dadvantaged fr ele dynamc charactertc. S the cmpenatn netrk neceary. he PID cmpenatr nt nly ha the advantage that can mprve the teady perfrmance f ytem but al ha the mre uperrty t mprve ytem dynamc perfrmance. S e che the PID cmpenatn netrk hch hn n fg.4. Fg.4. PID cmpenatr n th paper Frm the fg.4 e can get the tranfer functn f the PID cmpenatr hch frmula (. hen the ytem Open-lp tranfer functn after cmpenated frmula (6. ( ( Gc( G c ( ( p p. ( c ( G (( GcG m vd p c p Q 3 (6 he next tep t calculate parameter f the Cmpenatn netrk. If e make p rad / t cancel the er pnt hch a caued by utput capactance (SR then the

4 Internatnal Jurnal f Cmputer and lectrcal ngneerng l.3 N. February pen-lp tranfer functn frmula (7 n. ( ( ( GG c vd 3 c( G c( ( ( [ ( ] p Q (7 If e make crver frequency fc f pen-lp tranfer functn after cmpenatn t ne ffth f the tchng f frequency that fc f khz and 4 ϕ m 4 uppe that phae margn then er-pnt frequency and ple frequency repectvely are nϕ m n 4 f fc 4.4kH nϕ m n4 fp fc nϕ m n4 4.4kH nϕ m n4 fc f kh and make. Frm c( c ( c c( c db db e can get Gc( jc 33dB 44.8 hen db ( f f ( f f c p c p Gc Gc ( jc ( fc f ( f fc ( ( G ( 6 c ( ( 6 GG c vd c dB ( ( ( ( c ( [ ] (8 (9 ( Frm p rad / RC 68 rad / RC C C3 p 6 rad / RCC 3 We can get R Ω C nf R 3.3kΩ C 4.nF R3.8kΩ and C3.9nF repectvely. And they meet the fllng frmula. G c RC 8.6 R3( C C3 I. SIMUAION AND XPRIMNA RSUS he mall-gnal f pen-lp and cle-lp Bt cnverter mulated by ung MAAB t verfy the accuracy f mdelng hch prped n th paper. he Bde f pen-lp tranfer functn and nput-t-utput tranfer functn are hn n fg. (a (b repectvely. Fg.6 h the tep repne avefrm f cntrl-t-utput n pen-lp and cle-lp crcut repectvely. Frm the fgure; e kn that crver frequency and phae margn f pen-lp tranfer functn n the ytem are mre reanable after cmpenatn. And ha decreaed teady-tate errr n l frequency decreaed ampltude f nput-t-utput tranfer functn and utput mpedance n l and medum frequency trengthened the dturbance f nhbtry ablty and mprved the tranent repne peed and tablty. Magntude (db Phae (deg (a Magntude (db Phae (deg Bde Dagram Frequency (rad/ec he pen-lp tranfer functn(he brken lne and real lne tand fr tuatn f befre and after cmpenatn repectvely. Bde Dagram Frequency (rad/ec (b he nput-t-utput tranfer functn(he brken lne and real lne tand fr tuatn f pen-lp and cle-lp crcut repectvely Fg.. he mall-gnal tranfer functn f Bt cnverter 8

5 Internatnal Jurnal f Cmputer and lectrcal ngneerng l.3 N. February Step Repne Ampltude me (ec x -3 Fg.6. he tep repne avefrm f cntrl-t-utput n pen-lp and cle-lp crcut.(he brken lne and real lne tand fr tuatn f befre and after cmpenatn repectvely. CONCUSION Baed n the average mdel f tchng element methd an average f the equvalent crcut methd and the dea f energy cnervatn la e take Bt cnverter fr example t tudy the average mdel f the bac nn-deal cnverter hch added vltage cntrl feedback lp n DCM crcut. And e mprve the mdel by ung the duty rat cntrant; derve the pen-lp tranfer functn and cled-lp tranfer functn f the mdel. Bth t be verfed the crrectne f the mdel buldng under the MAAB mulatn tl. In th paper e have tuded the mdelng f vltage-cntrl Bt cnverter cndered paratc parameter n DCM. he mdel ueful and ntutntc and t phycal gnfcance clear. RFRNCS [] Sun J Mtchell D M Greuel M F. Averaged mdelng f PWM cnverter peratng n dcntnuu cnductn mde[j]. I ran. Per lectrn 6(4: [] Carkk D Kamercuk M K. nergy-cnervatn apprach t mdelng PWM dc-dc cnverter[j]. I ran Aerp lectrn Syt 993 9(3:9 63. [3] Davud A Jatkevch J Chapman P. Averaged mdellng f tched-nductr cell cnderng cnductn le n dcntnuu mde[j]. I lectr. Per Appl 7 (3: [4] Zhang Wepng. Mdel and Cntrl f Stchng Cnverter [M]. Bejng: Chna Per Pre. [] Wang Zhengln Wang Kaheng Chen Guheng. MAAB/Smulnk and Cntrl Sytem Smulatn[M]. Bejng: lectrnc Indutry Pre. [6] Sun J Mtchell D M Greuel M F. Averaged mdelng f PWM cnverter peratng n dcntnuu cnductn mde[j]. I ran. Per lectrn 6(4: Guang-jun Xe receved the B.S. degree and the M.S. degree frm Hefe Unverty f echnlgy n 99 and 99 repectvely. He receved the Ph.D. degree frm Unverty f Scence and echnlgy f Chna n. He n a Prfer f Hefe Unverty f echnlgy and h reearch nteret nclude IC degn cmputatnal ntellgence. 83

, where. This is a highpass filter. The frequency response is the same as that for P.P.14.1 RC. Thus, the sketches of H and φ are shown below.

, where. This is a highpass filter. The frequency response is the same as that for P.P.14.1 RC. Thus, the sketches of H and φ are shown below. hapter 4, Slutn. H ( H(, where H π H ( φ H ( tan - ( Th a hghpa lter. The requency repne the ame a that r P.P.4. except that. Thu, the ketche H and φ are hwn belw. H.77 / φ 9 45 / hapter 4, Slutn. H(,