MODEING AND SIMUATION OF SWITHING MODE D/D ONVETES Željko JAKOPOVIĆ Feth KOONIĆ Ntko SOIĆ Fulty of eletril engineering nd omuting Unsk 3, H10000 ZAGEB, OATIA Abstrt Modeling nd simultion of swithing mode D/D onverters n be done on severl wys, deending on the desired gol of modeling, omlexity of onverter model, vilbility of softwre tools, simultion time onsumtion et. There re three bsi rohes to the roblem. iruit oriented modeling (stti modeling) enbles simultion of onverter's behviour by mens of stndrd or dedited eletril iruit simultors. Dynmi modeling of ower onverters enbles nlysis of onverter's dynmi behviour for design nd imlementtion of onverter ontrol. Dynmi modeling n be divided into iruit verging roh nd sttese roh. Eh roh hs its own dvntges nd drwbks. The im of the er is to omre iruit oriented modeling nd iruit verging dynmi modeling of swithing mode D/D onverters on severl bses (model omlexity, ury of the results, simultion time et.) 1. INTODUTION Oertion riniles of swithing mode D/D onverters re bsilly very simle. Just 2 or 3 (in disontinuous urrent mode, DM) hses of oertion n be identified, resulting with rorite iruit toologies. In eh oertion hse, onverter iruit is usully TI system (liner time invrint) nd there should be no roblem with simultion of suh iruit with eletril iruit simultor. The roblem lies in the ft tht oertion hses re hnged very quikly t high oerting frequenies. To hndle the roblem of numeril onvergene, iruit simultor should hve good numeril integrtion lgorithm, but very smll integrtion stes re required to obtin stisftory ury. This leds to very long simultion times. On the other hnd, iruit oriented modeling nd simultion of swithing mode D/D onverters is very nturl for eletril engineers. The ide of vergediruit modeling is trying to retin the ossibility of simultion the onverter behviour with eletril iruit simultor, but enbling the dynmil nlysis of onverter. The resulting simultion times re smller omred with stti modeling roh. In vergediruit modeling roh TI system roerty of KV nd K vlidity for instntneous s well s verged vribles is used. So ll TI omonents fter verging re remining the sme vlues. Power swith (BJT, IGBT, et.) is obviously not the TI omonent. For swithing mode D/D onverters, ower swith toology n usully be modeled s nonil swithing ell [1]. Under smll rile ssumtion nd slow vrition ssumtion for system vribles of nonil swithing ell it n be reled with verged swithing ell onsisting of TI elements nd deendent voltge nd urrent soures. Averged swithing ell n be roximted with idel trnsformer. All nmed elements n esily be used in eletril iruit simultor. Trnsformtion roedure for ower swith is grhilly desribed on Fig.1 (,b,,d). Sttese modeling roh rovides omlete solution to the tsk of nlyzing nd ontrolling the dynmis of ower onverter iruits. Sttese models re imortnt in nlyzing, simulting nd ontrolling both stedy
i d v q(t) i di dv ) b) ) d) Fig. 1. Trnsformtion of onverter's ower swithes ) tive nd ssive swith in D/D onverter (BJT nd diode) b) nonil swithing ell with swithing funtion q(t) ) roximte verged swithing ell for ontinuous ondution d) roximte verged swithing ell, using idel trnsformer stte behviour of ower onverter iruits nd erturbtions wy from it. But unfortuntely, for simultion of sttese models so lled dynmi system simultors (e.g. MATAB, MATIXx) should be used insted of eletril iruit simultors. Anlysis of the sttese modeling roh is beyond the soe of this er. As the bse for omrison for iruit oriented nd vergediruit modeling roh the bukboost (u/down) D/D onverter is used, Fig.2. Bsi dt re: V in = 12 V, = 250 μh, = 220 μf, = 2 Ω, f = 50 khz. Duty rtio for ower swith is D, 0<D<1. For simultion, dedited ower eletroni iruit simultor SIMPOE is used on PENTIUM 166 MHz P. v in i d v model) n signifintly slow down the simultion, not ontributing to the ury of the observed bsi system vribles, suh s outut voltge V O nd indutor urrent I. How to hoose n integrtion ste h in simultion? If deresing of integrtion ste does not hnge signifintly the results, then the lst hosen integrtion ste is dequte. As oerting frequeny of onverter is 50 khz, e.g. eriod is T = 20 μs, mximum integrtion ste h MAX should be t lest 10 times smller thn eriod T, nd minimum integrtion ste h MIN should be 10 times smller thn h MAX Integrtion ste signifintly ontributes to the time of simultion s well s to the ury of the results. With suh onditions, first 5 ms of bukboost onverter strtu were simulted. equired simultion time ws 28 s. For the omrison uroses, the resonses on the ste hnge of duty rtio D, resistor vlue (outut urrent) hnge nd inut voltge V IN hnge were simulted. hnges were of suh n extend tht lrge signl behviour should be suosed. Fig. 2. Bukboost D/D onverter iruit 2. IUIT OIENTED MODEING APPOAH Bukboost D/D onverter n esily be modeled by mens of stndrd or dedited eletril iruit simultor, e.g. SIMPOE. There re severl ossible omlexity levels of ower swith modeling in SIMPOE. The simlest model of ower swith is idel swith with ON = 0 nd OFF =, nd this model is used in this simultion. The use of more omlex ower swith models (e.g. VI stti hrteristis or SPIE 3. AVEAGED IUIT MODEING APPOAH As mentioned in the introdution, for vergediruit modeling ower swithes of onverter (trnsistor nd diode) re trnsformed ording to Fig. 1., resulting with verged iruit for n bukboost onverter in ontinuous ondution mode (M), Fig.3. omlexity of verged iruit model is similr to the omlexity of iruit oriented nd n esily be relized in stndrd or dedited iruit simultor. Swithing ell is reled with idel trnsformer with d : d rtio (d = 1 d, 0 < d < 1). Pssive omonents (,,) retined their vlues s they re TI elements. It
should be noted tht x _ mens verge vlue of vrible x over swithing eriod T. v in d' : d Fig. 3. Averged iruit for bukboost onverter (M) Simultion of verged iruit model is not so integrtion ste sensitive s stti simultion. For the sme integrtion ste intervl s in stti simultion (200 ns < h < 2 μs), simultion time required for the strtu of onverter ws 8 s, signifintly smller thn 28 s in stti simultion. Even more, rtilly the sme results were obtined using 500ns < h < 5 μs, resulting with just 3 s of simultion time. For the omrison uroses, the resonses on the ste hnge of duty rtio D, resistor vlue (outut urrent) hnge nd inut voltge V IN hnge were simulted s in the se of stti simultion. Averged iruit model of D/D onverter is vlid only for onstnt duty rtio D, beuse the reltionshi between inut nd outut voltge is bsilly nonliner V IN D D = =. (1) V 1 D D O 4. OMPAISON OF SIMUATION ESUTS iruit oriented nd verged iruit models of bukboost swithing mode ower onverter from Fig.2. were simulted. Arorite simultions re nmed stti (for iruit oriented model) nd dynmi (for verged iruit model). A 20 ms oerting yle ws simulted onsisting of onverter strtu nd lition of ste hnge (D, or V IN ) fter 10 ms. In the first se, the ste hnge of duty rtio from D = 0.5 to D = 0.6 ws nlyzed. esults of simultion (Fig.4.) re lerly showing tht simle verged iruit model of bukboost onverter is not rorite for the nlysis of the ontrol vrible d(t) influene. Assumtion is tht the results of stti simultion (iruit oriented) re referene. In the seond se, the ste hnge of the inut voltge V IN ws nlyzed. There is lso n exellent greement between the results of stti nd dynmi simultion (Fig.5.), wht n be seen on the enlrged detil. In the lst se, ste hnge of the lod (in rtio 1:2) ws nlyzed. There is n exellent greement between the results of stti nd dynmi simultions (Fig.6.), wht n be seen on the enlrged detil. Time of simultion required for stti simultion ws 2'33'' nd for dynmi simultion only 30''. Nturlly, time of simultion deends on defined integrtion ste nd this vlues should only serve s n indition. 5. INEAIZATION OF AVEAGED IUIT MODE After nlysis of simultion results it n be onluded tht verged iruit model of swithing D/D onverter is not suitble for the nlysis of duty rtio d(t) vrition influene on onverter behviour. Beuse of nonliner nture of reltionshi between V IN nd V O, lineriztion roedure should be rovided to extend the lition field of verged iruit model even in the field of ontroller design. ineriztion roedure is desribed in detils in literture [1,2,3], so only bsi fts re resented. ineriztion is mde for smllsignl onditions, round nominl, stedystte oerting oint. Nominl vlues re denoted by uerse letters nd smll devitions from the nominl vlues re denoted by the ~ suersrit. So we n write for duty rtio d(t) = D d ~ (t) (2) d'(t) = D' d ~ (t) (3) The sme nlogy n be lied for other vribles. ineriztion is bsed on the ft tht terms tht involve squres or roduts of the smll erturbtions n be negleted, resulting with exressions used in building linerized verged iruit model d(t) i DI Di~ I d ~ (t) (4) d(t) v DV Dv~ V d ~ (t ) (5) After lineriztion roedure, bukboost D/D onverter from Fig.2 n be reled by linerized verged iruit model in ontinuous urrent mode (M), Fig.7.
Fig. 4. omrison of ste resonses on duty rtio hnge (D = 0.5 0.6) Full oerting yle t left nd enlrged detil t right. S stti simultion; D dynmi simultion Fig. 5. omrison of ste resonses on the inut voltge hnge (V IN = 12 V 18 V ) Full oerting yle t left nd enlrged detil t right S stti simultion; D dynmi simultion Fig. 6. omrison of ste resonses on the lod hnge ( = 2 Ω 1 Ω ) Full oerting yle t left nd enlrged detil t right S stti simultion; D dynmi simultion
v in I d D D' : D (V in V o ) d D Fig. 7. inerized verged iruit model of bukboost onverter (M) Simultion results of ste resonses for smll nd lrge duty rtio vritions re omred with rorite results obtined on the bsis of iruit oriented nd verged iruit models. esults re lerly showing tht linerized verged iruit model of D/D swithing onverter n give stisftory results even for 5% rnge of duty rtio vrition (e.g. D = 0.5 to 0.525), enbling the use of suh model in the ontroller design roess. I 6. ONUSION Eh modeling nd simultion roh for swithing mode D/D onverters hs its dvntges nd drwbks. It is shown tht tking into ount limittions of verged iruit model nd linerized verged iruit model of D/D swithing ower onverter, fst nd urte simultion results n be obtined. It is ossible to nlyze different sets of onverter behviour. Averged iruit modeling roh to D/D onverters modeling is eseilly desirble in edution, tehing of ower eletronis, beuse it gives fst nd urte results. EFEENES [1] Ksskin, Shleht, Verghese; "Priniles of Power eletronis", AddisonWesley, 1992. [2] V.Vorerin; "Simlified Anlysis of PWM onverters Using Model of PWM Swith, Prt I: ontinuous ondution Mode", IEEE Trns. on AES, Vol.26, No.3, My 1990.,.490496 [3] V.Vorerin; "Simlified Anlysis of PWM onverters Using Model of PWM Swith, Prt II: Disontinuous ondution Mode", IEEE Trns. on AES, Vol.26, No.3, My 1990.,.497505 [4] SIMPOE, eferene Mnul, SIME GmbH, 1997. I Fig. 8. omrison of ste resonses for iruit oriented model (S), verged iruit model (D) nd linerized verged iruit model (). eltive hnge of D is 5%