Software Makes Transfer Functions More Manageable

Transcription

1 Softwre Mkes Trnsfer Funtions More Mngeble By Ekrem Cengeli, Senior System Engineer, Mirosemi, Grden Grove, Clif. Combining PWM swith models with liner iruit-nlysis softwre nd mth softwre retes fully omuter-bsed roh to obtining nd nlyzing the trnsfer funtions of d-d onverters. Smll-signl nlysis of swithing d-d onverters determines their seifi trnsfer funtions, for exmle, for stbility nlysis or design of roer inut filter. Trnsfer funtions of interest for swithing d-d onverters re inut imedne, outut imedne, duty yle (or ontrol voltge) to outut voltge, nd inut voltge to outut voltge. Two iees of literture bout omlishing smll-signl nlysis of swithing d-d onverters suggests two fundmentl methods: 1. The stte-se verging method, by Dr. Middlebrook et l, emloys lgebri mniultion of set of stte-se equtions to derive verge model equtions of swithing onverter toology. Linerizing the verged model t d oerting oint derives the desired trnsfer funtion. With this method, the omlete iruit must be verged nd linerized every time the toology is hnged. 2. The method of PWM swith model, by Dr. Vorerin, verges nd linerizes the d-d onverter s swith ir, whih is its only nonliner ortion, nd invrintly uses this model in different toologies. The min dvntges of PWM swith roh re: l The PWM swith model is lredy verged nd linerized, so no verging nd lineriztion is neessry, whih differs from the stte-se verging method. l As iruit-oriented roh, it is very onvenient for use with numeril or symboli iruit-nlysis softwre. l The PWM swith model s roerties re the sme for given ondution nd ontrol mode, so they n be used invrintly from one toology to nother. Beuse of these dvntges, rtiulrly tht it is iruit oriented, the PWM swith model is best suited for obtining symboli nd numeri smll-signl equtions of swithing d-d onverters. The smll-signl model of swithing onverter with Power Eletronis Tehnology June

2 PWM swith model yields liner iruit, rl1 nd finding the trnsfer funtion of interest requires nlyzing this liner iruit. PSPICE hs built-in PWM swith models for numeril nlysis of swithing d-d onverters. The models re simle V in enough to remodel on ny other numeril eletril simultion tools, even if PSPICE is not vilble. However, obtining iruit equtions of suh liner iruit (symboli nlysis) is done by mnully nlyzing the iruit. Deending on the level of omlexity with the equivlent liner iruit, obtining the nlytil trnsfer funtion of interest my turn out to be very diffiult tsk. This n our if there re lrge number of terms in the resultnt trnsfer funtion when using stndrd liner iruit-nlysis tehniques (nodl or loo). Alterntively, Middlebrook suggested the extr element theorem (EET) to nlyze liner iruits in more effiient mnner. This method redues the number of mthemtil mniultions nd simlifies obtining the trnsfer funtion of interest. Vorerin s book, Fst Anlytil Tehniques for Eletril nd Eletroni Ciruits, is good referene for the EET nd rovides vrious lition exmles. Although the EET method is muh more rtil nd useful omred to nodl or loo nlysis methods, it still requires mnul mniultion of iruit equtions nd beomes imrtil when the number of iruit elements exeeds ertin number. For exmle, let s sy designer is interested in the smllsignl PWM swith-equivlent iruit model of buk onverter to obtin duty-yle to outut-voltge trnsfer funtion with voltge-mode ontrol in ontinuous ondution mode (CCM). By inluding the indutor s d resistne nd the itor s ES, there re eight iruit elements in the equivlent iruit, nd the resultnt trnsfer funtion ontins only bout 25 terms of iruit rmeters. EET esily nlyzes this iruit to find the trnsfer funtion of interest. On the other hnd, the omlexity inreses when determining the smll-signl PWM swith-equivlent iruit model of, for exmle, the nonisolted, single-ended rimry indutor onverter (SEPIC) to obtin its duty-yle to outut-voltge trnsfer funtion with voltge-mode ontrol in disontinuous ondution mode (DCM). Now, by inluding the indutors d resistnes nd the itors ES, there re 14 iruit elements, nd the resultnt trnsfer funtion ontins more thn 700 terms of iruit rmeters when exnded nd olleted with s. Obviously, it is mjor tsk to obtin the trnsfer funtion of the SEPIC onverter toology mnully even with the EET method. Besides, it is good ossibility to indvertently introdue errors during this mnul nlysis roess. Needless to sy, it lso requires lerning nd digesting the EET method. Besides, the omlexity level of the equivlent iruit nlysis is muh higher with the urrent-mode ontrol if ll the rsiti elements of the retive omonents nd subhrmoni osilltion model rmeters re inluded (the finl trnsfer funtion ontins thousnds of rmeters with the SEPIC onverter toology). Anlyzing suh iruit even with the EET method is n extremely long (or imossible) tsk unless you omit some rsiti elements to simlify the iruit. There is omuter-bsed method to nlyze suh liner iruits nd obtin nlytil exressions for the trnsfer funtions of interest. The method works even for trnsfer funtions ontining thousnds of rmeters in it. It is fully omuter bsed, so designer n obtin nd nlyze the resultnt trnsfer funtions muh fster nd more relibly thn mnul nlysis methods. Tools needed for suh nlysis re: l Liner iruit-nlysis softwre tht n erform symboli iruit nlysis l Mth softwre to ost-roess the trnsfer funtion eqution generted by the iruit-nlysis rogrm l An exeutble rogrm to formt the trnsfer funtion eqution generted by the iruit-nlysis rogrm in the form required by the mth softwre. The detils of this method re exlined nd demonstrted next. Softwre Progrms Severl liner iruit-nlysis rogrms n erform symboli iruit nlysis. Among these re TINA, by Design Wre In.; Anlog Insydes, by ITWM (Germny), whih works on Mthemti by Wolfrm eserh s n dd-on rogrm; nd SAPWIN, written by the eletril engineering dertment t the University of Florene (Itly). The first two rogrms re suorted by omnies, so designer hs to urhse them to use. However, SAPWIN is vilble s free downlod on the University of Florene s eletril engineering dertment s website. Although ny of these rogrms n be used for the urose of symboli iruit nlysis, SAPWIN will be used here beuse it is free nd suffiient for the uroses of smll-signl nlysis of swithing d-d onverters. After the designer obtins the trnsfer funtion through 15 Power Eletronis Tehnology June 2008 L1 C1 esr1 esr Fig. 1. SEPIC toology. The swith ir is shown with terminls, nd. r C I out V out

3 smll-signl nlysis V I µ I I i v g i i V µ V V k i d v g o g f v k o d I i () (b) Fig. 2. DC () nd smll-signl (b) PWM swith models with voltge-mode ontrol in DCM. I in I V in µ I µ V Fig. 3. DC equivlent iruit of the SEPIC in Fig. 1 with voltgemode ontrol in DCM. I I I out V out the symboli iruit-nlysis rogrm, it must be mniulted by mth softwre rogrm to obtin its ttributes (oles, zeros, d gins, et.) symbolilly nd numerilly. There re severl mth softwre rogrms, nd lmost ll of them hve symboli mth funtions built into them, suh s Mtlb, Mthd, Mthemti, Mle nd MuPAD. The funtionlities, bilities nd ese of use of these mth rogrms in symboli mode differ nd some hve dvntges over others. Mthd (version 13.1) will be used here simly beuse it seems to be the most ommon mth softwre used by d-d onverter design engineers. SAPWIN solves liner iruits nd dislys the result (trnsfer funtion eqution) in seifi formt. The trnsfer funtion generted by SAPWIN must be modified to trnsfer it into Mthd for mniultion. This modifition n be done mnully by oying the SAPWIN trnsfer funtion into text editor nd then modifying it. However, mnul modifition wouldn t be rtil nd deendble (for exmle, it would be rone to introduing errors) if the number of rmeters in the trnsfer funtion exeeded ertin level (for exmle, if 100 ws the limit). Therefore, n exeutble ode is needed tht formts the SAPWIN iruit eqution into form tht n be trnsferred to Mthd eqution. Suh ode signifintly imroves the effetiveness of the method, redues the nlysis time nd elimintes the ossibility of introduing n error into the eqution. PWM Swith Model Vorerin derived the PWM swith models for voltgemode ontrol in CCM nd DCM nd for urrent-mode ontrol in CCM. PWM swith models won t be exlined here, rther n exmle will be given tht shows how to use the PWM swith models in symboli nlysis of swithing d-d onverters. The method will be demonstrted on smll-signl nlysis of SEPIC with voltge-mode ontrol in DCM. Wht follows is brief exlntion of PWM swith models of voltge-mode ontrol in DCM. Fig. 1 shows the nonisolted SEPIC onverter toology. With the PWM swith models, the swith irs re modeled s three-terminl element, so the diode nd swith hve to be onneted t one end to exress the toology with the PWM swith model. Therefore, the diode in Fig. 1 is drwn on the bottom rther thn on the to. Terminl is the ommon terminl to whih both swithes re onneted, terminl is the tive terminl to whih the ontrol swith is onneted nd terminl is the ssive terminl where the diode is onneted. For given ondution nd ontrol mode, the PWM swith model hs three-terminl d nd smll-signl models. The d model is used to solve for rmeters of d oerting oint, whih re then substituted into the smllsignl model rmeters. Fig. 2 shows the PWM swith d nd smll-signl models with voltge-mode ontrol in DCM. The uerse terminl urrents nd voltges in Fig. 2 reresent d quntities, nd lower-se terminl urrents nd voltges in Fig. 2b reresent the erturbed quntities (or smll-signl quntities). The d quntities re solved by substituting the d model into the SEPIC toology in Fig. 1 nd oening itors C nd C1 nd shorting indutors L1 nd. Power Eletronis Tehnology June

4 smll-signl nlysis Then by using SAPWIN, substitute the smll-signl model into the SEPIC toology to obtin the trnsfer funtion of interest. Besides terminl urrents nd voltges, Figs. 2 nd 2b lso ontin model rmeters, whih re exressed by the d terminl urrents nd voltges s: 2 D V µ=, L1 (Eq. 1) 2 f I S L1 where f S is the swithing frequeny nd D is the stedystte duty yle of the onverter. g i I = (Eq. 2) V I ki = 2 D (Eq. 3) I k o = 2 (Eq. 4) D I go = (Eq. 5) V g f I = 2. (Eq. 6) V Smll-Signl Anlysis of SEPIC Converter Now see how the symboli smll-signl nlysis of the SEPIC in Fig. 1 with voltge-mode ontrol in DCM n be done by using the PWM swith model in Fig. 2, SAP- WIN nd Mthd. First, substitute the three-terminl d model in Fig. 2 into Fig. 1, oen the itors nd short the indutors to solve for the d quntities. The resultnt equivlent iruit is shown in Fig. 3. Note tht omitting the d resistne of the indutor hs negligible effet on the ury of the d quntities on rtil exmles, nd inluding them in the d model signifintly omlites the smll-signl trnsfer funtion with no notieble benefit. Therefore, neglet indutor d resistnes in the d models in this exmle. Solve for V, µ, I, V, I nd D using the equivlent iruit in Fig. 3 to exress the model rmeters g i, k i, g o, k o nd g f in Eqs. 2, 3, 4, 5 nd 6. These rmeters n be solved from Fig. 3 by insetion: V = V out (Eq. 7) µ= V out Vin V I = Iout = out (Eq. 8) (Eq. 9) V = V in (Eq. 10) Vout I= µ I= µ. (Eq. 11) Using Eqs. 1, 7 nd 11: D = 2 L1 L1 2 µ ( ) fs. (Eq. 12) For suerior solutions in industril eletronis

5 smll-signl nlysis L1 v i g i If Eqs. 7 to 11 re substituted into Eqs. 2 to 6, this results in: (Eq. 13) gi = µ2 2µ Vout ki = D 2 Vout k o = D rl1 k i d k o d i C1 i g f v esr1 g o v (Eq. 14) (Eq. 15) i out esr Fig. 4. Smll-signl equivlent iruit of the SEPIC in Fig. 1 with the PWM swith model with voltge-mode ontrol in DCM. r C v out g = 1 o (Eq. 16) g = 2µ. (Eq. 17) f Now tht the d quntities hve been solved, substitute the smll-signl model in Fig. 2b into Fig. 1 nd short the d inut-voltge soure to obtin the trnsfer funtion from d to v out. The resultnt equivlent iruit is shown in Fig. 4. Fig. 5 shows the equivlent iruit in Fig. 4 s drwn using the SAPWIN shemti ture tool. The little irulr element t node vout in Fig. 5 defines the node whose voltge is to be solved by SAPWIN. Therefore, in Fig. 5 SAPWIN solves for node vout in terms of iruit rmeters. By dividing the SAPWIN eqution of vout by d nd substituting Eqs. 13 to 17 into it, the trnsfer funtion of v out (s)/d(s) n be derived. Note tht some iruit rmeters in Fig. 5 re in squre brkets. SAPWIN is limited in nming omonents in the iruit. For exmle, ll resistor nmes hve to strt with itl. A designer who wnts to nme resistor rl, for exmle, would need to hve itl t the strt, so the resistor would tully hve to be nmed rl. Seeing the resistor nme s rl in the finl eqution 3.3 kw to 750 kw DC Power Sulies M P E M AGNA-POWE ELECTONICS HIGH-POWE AC TO DC POWE SUPPLIES Vd, Ad, high ower density, erth grounded user interfe POGAMMABLE Advned omuter interfes: S232, Ethernet, GPIB, nd USB interfes FEATUES AND SPECS THAT LEAD THE INDUSTY Wter ooling otion, 50/60/400 Hz inut, high effiieny, nd more... Power Eletronis Tehnology June

6 would be inonvenient for the user. Therefore, the ode written to simlify the SAPWIN eqution is done in suh wy tht it lso simlifies ny rmeter nmes where squre brkets re used, by tking only the hrters within the brkets nd removing the hrters in the rest of the rmeter nme. The user n nme the iruit rmeters with ny hrters on the keybord by utting them within squre brkets. One the ode simlifies the SAPWIN eqution, the iruit rmeter nmes will be shown in the eqution with the SEPIC in Fig 4. hrters in squre brkets. The nlytil eqution of node vout in Fig. 5 is lulted by SAPWIN, s shown in Fig. 6 (this figure n be viewed lrger in the online version of this rtile). When the SAPWIN eqution in Fig. 6 is simlified by the exeutble ode, it looks like the eqution in Fig. 7 (whih lso n be viewed lrger in the online version of this rtile). As seen in Fig. 7, the ode removes the hrters outside the brkets in the referene designtors of the iruit elements, dds multilition sign (*) between ny djent rmeters if there is no mth oertor between them, les ll the terms on the sme line, utomtilly detets smll-signl nlysis Fig. 5. Sreen ture of SAPWIN shemti of the smll-signl equivlent iruit of the the division sign tht sertes the numertor from the denomintor, nd dislys the eqution in string formt. These stes n be done mnully modifying the eqution in Fig. 6 in text editor. However, it would be umbersome nd rone to introduing n error into the finl eqution for long eqution s in Fig. 6. The eqution in Fig. 7 is in string formt tht n be onverted into Mthd eqution. However, when this eqution is oied into Mthd, the rogrm will not tret it s n eqution but s string, insted. So, Mthd srit is needed tht onverts n eqution in string formt into Mthd eqution. Suh Mthd srit s written by Tke look t our brod offering of qulity ower modules nd fi nd out why Bel is now the referred soure for d-d onverters. Finlly, you n get ost effetive roduts in industry stndrd form ftors without srifi ing erformne. To lern more bout how Bel n hel you ower your next system, visit us t High Effiieny Power Modules Isolted Converters- Single & Dul Outut 1/16, 1/8, 1/4, 1/2 Briks u to 120A Bus Converters- 4:1, 5:1, 6:1 Fixed tios 1/16, 1/8, 1/4 Briks u to 500W VMs- Solutions for most Miroroessors U to 150A Outut; Goldfi nger nd TH Non-Isolted POL Modules- Boost, Buk nd Inverting 1A to 150A Outut; Vertil Mount or SMT BELFUSE 19 Power Eletronis Tehnology June 2008

7 smll-signl nlysis Fig. 6. SAPWIN eqution for the voltge of node vout in Fig. 5. Dr. Vlery F. Ohkov, rofessor of the Mosow Power Engineering Institute (Tehnil University), ussi is vilble on the Mthd Collbortory offiil online forum. Now, to simlify the eqution of vout in Fig. 7 by the ode, trnsfer it into Mthd, divide it by d nd substitute the d quntities into it from Eq. 13 to Eq. 17. The eqution of vout in Figs. 6 or 7 is rther long nd not useful in its resent form. But it tells the orders of the numertor nd the denomintor, so its known how mny oles nd zeros there re. Both the numertor nd the denomintor in Fig. 6 re fourth order, so there re four zeros nd four oles. One signifint benefit of the EET method is tht the resultnt eqution is exressed in terms of series rllel ombintion of the iruit omonents, whih is very onvenient nd useful in determining the resultnt eqution with ertin reltions between the iruit rmeters. The eqution obtined by SAPWIN doesn t look s elegnt s the eqution obtined through the EET method. However, it n be rrnged in muh better nd useful form thn how it looks in Fig. 7 by using the symboli keywords of Mthd, inluding ftor, ollet, simlify, substitute, ssume nd oeffs. The first symboli keyword to ly to the eqution in Fig. 7 should be ftor. By lying the keyword ftor to trnsfer funtion, the zeros nd oles of the trnsfer funtion tht n be ftored out re found. Therefore, if there re ny zeros or oles tht ftor out in the trnsfer funtion, the roximte exressions don t hve to be used for them. Power Eletronis Tehnology June

8 Next, nlytil formuls re needed of the roximte oles nd zeros of the trnsfer funtion, whih is exlined in Chter of Fundmentls of Power Eletronis (by.w. Erikson nd D. Mksimovi) with the ondition tht the oles nd zeros re well-serted. Of interest re the lotion of the oles nd zeros if they re rel nd the resonnt frequeny nd qulity ftor if they re omlex. To find these quntities nlytilly, the oeffiients of the numertor nd denomintor of the trnsfer funtion re ssigned to rrys using oeffs. One these quntities re nlytilly exressed in Mthd, they n further be simlified by using simlify, they n be olleted with ertin iruit rmeters sequentilly using ollet, nd they n be exressed with ertin onditions of iruit rmeters using ssume nd substitute. Using these keywords will signifintly mke the nlytil formuls look better nd more useful if not s good s low-entroy exressions obtined through EET. By lying the keyword ftor to the trnsfer funtion in Fig. 7, the ES zero set by the outut itor C nd esr ftors out s (1s 3 C 3 esr). The oeffiients of the remining third-order olynomil of the numertor nd the fourth-order olynomil of the denomintor re ssigned to two rrys to find roximted nlytil exressions for the zeros nd oles. Smll-Signl Anlysis Fig. 7. The eqution of vout in Fig. 6 s simlified by the exeutble ode. At this oint, one needs to deide where to le the oles nd zeros of the ower trnsfer funtion on the s- lne. Then, the oles nd zeros n be roximted to nlytil exressions rovided they re well-serted. Alterntively, if the ower stge of the SEPIC hs lredy been designed nd therefore the vlues of the ower trin omonents hve been defined the nlytil exressions of the oles nd zeros n be obtined with the vlues in hnd rovided tht the vlues yield oles nd zeros wellserted from eh other. The Power Eletronis Tehnology website, eletronis.om, shows numeril exmle of this roess emloyed for SEPIC. The online version of this rtile lso inludes useful referenes. PETeh th ssn of ltronis omonnts.systms.utomotiv.mbddd.wirlss.mironno-systms Power Eletronis Tehnology June 2008