Dynamic Analysis of Frequency-Controlled Electronic Ballasts

Transcription

1 3 4 6 Dynamic Analyi o Frequencyontrolled Electronic Ballat Yan Yin, egan Zane, obert Erickon olorado Power Electronic enter Unierity o olorado at Boulder Boulder, O 8394 AbtractThi aer reent analytical tool aimed at imroing and imliying the deeloment o requencycontrolled dimming electronic ballat. A modiied haor tranormation i rooed that conert a requencymodulated ignal into an equialent timearying haor. The rooed tranormation i alied to deelo a comlete mallignal haor model o the reonant ballat, which exlicitly model the eect o the requency modulation on the eneloe o the outut. A Sicecomatible imlementation o the model i reented that acilitate A analyi o the ballat in addition to eneloe tranient imulation, and i eriied through comarion o exerimental and imulation reult. A cloedorm olution o the controltooutut current traner unction or the ballatreitor ytem i reented, along with key oberation o the ole location and lowrequency gain that acilitate imle and intuitie comenator deign. Finally a deign examle or the eedback controller i gien to eriy the theoretical analyi. Key Word haor tranormation; requency modulation; reonant ballat; mallignal haor model I. INTODUTION With the growing demand or more intelligent and unieral lighting roduct, there exit a igniicant need or imroed analytical tool or the deeloment o lighting electronic ballat. Thi aer reent tool or analyzing the dynamic behaior o requencycontrolled dimming luorecent ballat, which aid the deign o eedback controller and gie an intuitie undertanding o the eect o comonent and load change on the loo tability. In many dimming ballat deign, a imle lowrequency (F eedback loo i emloyed to tabilize the lam current and imroe the cret actor. In ractice, ocillation are requently obered at low ower leel, requiring additional deign iteration and more comlex eedback control. To redict and comenate or thee henomena, dynamic analyi o the controltooutut characteritic i needed. A hown in Fig., the control inut can be bu oltage ariation (amlitude modulation, AM or requency ariation (requency modulation, FM, and the deired outut i the reulting eneloe ariation in the lam current. Thee iue were inetigated and modeled uing a timearying haor tranormation in [] or the ecial cae o oeration at reonance with AM, and in [,3] a Sice model wa deeloed Thi work i onored by eneral Electric o. lobal eearch, through the olorado Power Electronic enter and i counded by the Deartment o Energy' National Energy Technology aboratory under ooeratie Agreement DEF6NT4. F V John laer eneral Electrical omany lobal eearch enter One eearch ircle Nikayuna, NY 39 V Fig. Highrequency ballat drien by AM/FM inut uing the haor tranormation to enable eneloe tranient imulation o AM and hae modulation (PM. Thi aer ocue on the dynamic analyi o the eneloe behaior in the requencycontrolled ballat ytem. In Section II, a modiied haor tranormation that conert an FM ignal into an equialent haor i irt rooed, and then i alied to derie the comlete mallignal haor model o the requencycontrolled dimming ballat, in which the eect o the requency modulation on the eneloe o the outut current/oltage are exlicitly modeled. In Section III a Sicecomatible mallignal model i deeloed or Sice A analyi, and i eriied by the meaurement. Section IV gie a cloedorm olution o the requencytooutut current traner unction o the mallignal ballat model with the reitie load, along with key oberation o the ole location and F gain that acilitate imle and intuitie olution o the traner unction. The eect o the negatie incremental imedance o the luorecent lam [] on the dynamic o the tudied ballat ytem are alo dicued. Section V reent a eedback controller deign examle. The concluion are ummarized in Section VI. II. MODIFIED PHASO TANSFOMATION AND ITS APPIATION TO THE FEQUENYONTOED BAAST The object in erorming the haor tranormation i to remoe the witching requency comonent o the inuoidal waeorm and analyze the dynamic o the ignal eneloe. For the tyical halbridge ballat, the inuoidal aroximation can be alied a long a the conerter oerate near

2 reonance with high Qactor [4]. The tranormation aroach wa irt rooed in [], where a inuoidal ignal can be rereented by a comlex haor: jω x( e[ x( e t ], ( where x ( i the timearying haor correonding to x (. Thi tranormation aroach i well uited to mallignal modeling o amlitude modulation, where the witching requency ω i contant. Small ariation in the amlitude o x ( are then modeled by mall ariation in x (. Hence a mallignal timeinariant haor model can be deried. Howeer, requency modulation caue x ( to contain a rotating comonent. The ariation in x ( are no longer mall, and mallignal analyi cannot be alied. To accommodate requency modulation, we need to conider a time arying requency ω (t. Baed on the deinition o requency modulation (auming the carrier i inuoidal xfm ( Aco( ω ( d, ( we rooe a modiied haor tranormation o the orm j ( x( e[ x( e ω ]. (3 Thi rooed new aroach i ueul or mallignal modeling when x ( contain both amlitude and requency modulation. Frequency modulation no longer lead to a rotating comonent in x (. Small ariation in the amlitude o x ( lead to mall ariation in x (, which atiie the mallignal aumtion. By alying (3 to the baic circuit element i equation, we can get the largeignal haor model or thee circuit element a hown in Fig.. A an examle, we aly (3 to the dierential equation o the inductor to demontrate how to get the haor model. The i equation o the inductor i di( (. (4 The haor tranormation o current and oltage are: j ( e[ ( ω ( i t i t e ]. ( j ( ( e[ ( ω t t t e ] Uon alication o ( to (4, the haor equation or the inductor i obtained: di( jω ( i( (. (6 Equation (6 can be conerted to a circuit model a an inductor in erie with an imaginary reitor, a hown in Fig.. Through a imilar aroach, the largeignal haor model or an FM oltage ource, reitor and caacitor can alo be obtained. Thee model are imilar to thoe in [], excet that here the imaginary reitor can be time arying. To derie the mallignal haor model, we need to erturb and linearize the largeignal haor model. With the aumtion that: V, i I i, and ω Ω ω, then the mallignal haor model or the inductor i obtained a: d i jω i jω I. (7 Similarly, the mallignal haor model or other circuit element can be deried. They are alo hown in Fig.. The haor model o any reonant toology can then be ound by imly relacing the baic element according to Fig.. Fig. 3 how the mallignal haor model or a tyical reonant ballat with the lam initially modeled a a reitor. It i notable that in the mallignal haor model hown in Fig. 3, the eect o requency modulation on the reonant tank i exlicitly modeled a deendent ource that are aociated with each reactie element (oltage ource with inductor and current ource with caacitor. Thee deendent ource alter the outut haor, reulting the amlitude modulation o the timedomain ignal. Hence the FM inut, ater aing through the reactie network, i conerted to an AM/FM outut. Once the haor o any ignal i obtained by oling the haor model, the original ignal eneloe can be recoered ( co( ω ( d ( j ω ( t j ω ( t i j Ω jω jω I c jv c ω Fig. argeignal and mallignal haor model or baic circuit element i jω jω j ω I c jvc jvc ω ω c jω Fig. 3 Smallignal haor model or ballat i out

3 by inding the magnitude o the haor [,3]: x en ( x( x ( x (, (8 t where x ( and x( are the real and imaginary art o x (, reectiely. By erturbing and linearizing (8, we can ind the mall ignal ariation o the eneloe a: where X, x en X x X x X x X x, (9 X X X X and X are teadytate alue o the real art, imaginary art and magnitude o the haor, x and x are the mall ignal comonent o the real and imaginary art o the haor. III. SPIEOMPATIBE MODE AND EXPEIMENTA VEIFIATION Owing to the exitence o imaginary reitor, the model in Fig. 3 cannot be directly realized in a Sicecomatible imulator. Howeer, by ue o the ame aroach reented in [], thi comlex circuit can be lit into real and imaginary art, a hown in Fig. 4, and imlemented in a Sice imulator or tranient and A analyi. Fig. comare the tranient imulation reult or an FM inut ( i et to a inuoidal ignal with the original timedomain waeorm (only the oitie ortion i hown here. It can be een that the haor model gie exactly the eneloe o the original timedomain waeorm without witching leel imulation. ω The model in Fig. 4 can alo be imlemented in Sice or A analyi to ind any traner unction correonding to inut oltage ( or requency ( ω or. Then the imulation reult can be comared with the meaurement. Fig. 6 how the exerimental etu or the meaurement o the traner unction o requencytooutut current. The oltagecontrolled ocillator (VO i ued to generate the requency modulation. The witching requency i then controlled by the VO inut oltage with F kvo( Vb. ( The inut ilter o the network analyzer ilter all the highrequency comonent in and only the comonent with en the modulating requency i meaured. Hence, the meaured traner unction i roortional to the traner unction rom requency to outut current eneloe _ en/. i out Fig. 7 comare PSice imulation and exerimental reult o the requencytooutut current traner unction with F khz, 3Ω and k VO khz/volt. It can be een that the imulation reult i in good agreement with the meaurement. i Ω i Ω Ω i c ω I ω V ω I c i ω Vc c Ω c c c c Ω ω V c c Ω ω V Fig. 4 Sicecomatible mallignal haor model or ballat Fig. Tranient analyi gien by the haor model (to and the original ballat Drier F VO D _ Bia V b Network Analyzer en en c c i out Fig. 6 Exerimental etu or controltooutut current traner unction meaurement (39uH, 4.3nF, 3.8nF

4 Fig. 7 Simulation and meaurement reult o requencytooutut current traner unction with reitie load 3Ω real art o ballat model _ In the dicuion aboe, the imulation and exerimental reult are ound baed on the reitie load in order to eriy the theoretical haor model or the ballat. To model the dynamic eneloe behaior o the ballatluorecent lam ytem, the lam model hould be incororated into the ballat model uing the aroach imilar to [3], a hown in Fig. 8. The lam model hown in Fig. 8 i imilar to the one in [3] excet the olynomial cure itting i relaced by the exonential unction. The imulation and exerimental reult are comared in Fig. 9, where a E Biax TM 6W comact luorecent lam (F i ued a the load with oerating requency F khz and P lam 4W. Again, there i good agreement between the imulation and meaurement reult. IV. E ANAYSIS The Sice model o Section III acilitate A imulation o dimming ballat, but till lack intuitie aid ueul or the deeloment o ballat ytem. In thi ection, we highlight a ew key oberation rom the theoretical analyi that gie a more intuitie eel or the ytem oeration. By combining the reactie element with their correonding imaginary reitor in Fig. 3, we can ind that in the i eq i E rm rm i i i Imaginary art E E o ballat model _ i krm k ke eq eq rm Fig. 8 Sice model combining the ballat model with the luorecent lam model Fig. 9 Simulation and meaurement reult o requencytolam current traner unction with a E Biax TM 6W F (P lam4w Sdomain the reulting mallignal model toology i imilar to that o the original reonant tank excet that jω relace. The imilarity in the toology mean that the mallignal model in Fig. 3 ha the ame denominator a the reonant tank excet that the ariable i relaced by jω. Uon oling the model in Fig. 3, the traner unction rom requency to outut current haor can be maniulated into the ollowing cloedorm: h i out N( jω N( jω D( jω ( D( jω D( jω D( jω j (. ( h _ re ( h _ im where D( rereent the characteritic equation o the original tank, and N( i a econd order olynomial. Then, according to (9, the traner unction rom requency to outut current eneloe can be exreed a: i ( out _ en I out I out ( I out h _ re( I out h _ im (. ( where I and out I are the teadytate real and imaginary out art o the outut current haor, reectiely. Although the reulting cloedorm olution i an intimidating ixthorder traner unction, when ut in the orm o ( and (, it can be een that the ole o the ytem are imly the ole o the original tank hited u and down by jω. The ole diagram or a tyical ballat i hown in Fig., where * rereent the ole o the reonant tank, and Δ rereent the ole o the mallignal model. A the ballat i generally oerated near and aboe the reonant requency, thi create a air o comlex ole at low requency ( ω in Fig., which dominate the oenloo reone oer the bandwih o a

5 Im * ω * ω * jω e jω (a Traner unction o controltooutut current Fig. Pole o the reonant tank and the mallignal haor model I out (b Incremental imedance o the luorecent lam F Fig. F gain o the mallignal haor model or ballat tyical eedback controller. From Fig., ω can determined a ω ω, (3 Ω Ωω /(4Q where ω and Q are the reonant requency and quality actor o the reonant tank. Aroximately, i the ballat oerate with high Qactor, ω can be etimated a Ω ω. Thi aroximation i in agreement with the argument in [6]. Other highrequency (HF ole, in general, can be ignored rom the controller deign oint o iew. The F gain o the traner unction can alo be determined rom the cloedorm olution, although it may be een imler to determine the F gain directly rom the teadytate olution o the ballat by inding the loe o the cure o outut current magnitude. witching requency ealuated at the quiecent oerating requency, a hown in Fig. or the ballat, Fig. Simulation and meaurement reult or ballat with E Biax TM 6W F at low dimming leel (3% o the ull ower I out F. (4 While the aboe analyi i baed on the reitie load, or more accurate reult, the dynamic o the luorecent lam hould be taken into account to obtain the comlete dynamic o the ballat ytem. A een by comaring Fig. 7 & 9, the dynamic o the lam hae negligible eect on the ytem dynamic at high ower leel, which i a key aumtion in mot nondimming ballat deign. Howeer, a een in Fig. (a, the actual traner unction with a lam load can deiate igniicantly rom the reitie load aumtion at low dimming leel. For the ballat ytem being meaured, it can be een that an additional F ole/zero air come into the traner unction with the luorecent lam load, which alter the F gain and hae hit and comlicate the controller requirement. We beliee that the increaed eect o the lam on the ytem at low dimming leel can be intuitiely undertood by noting the igniicantly reduced outut imedance o the ballat and increaed eectie imedance o the lam. It hould be noted that the deiation between

6 meaurement and imulation i more ronounced at low dimming leel (comare Fig. 9 & (a due to nonidealitie o the lam model, which can be een by comaring the imulated and meaured incremental imedance o the luorecent lam in Fig. (b. To obtain more accurate imulation reult, a higher order lam model and more comlicated cure itting could be ued. In order to account or the dynamic o the lam analytically, a mallignal haor model o the lam would need to be deeloed to relace the reitie load in Fig. 3, which could then be oled and maniulated into the orm imilar to ( and (. While thi may be included in our uture work, at thi oint it aear that uch an exercie will reult in a traner unction that can only be oled numerically, roiding little inight into the ytem oeration. We are currently uruing alternatie analyi aroache or extenion that may acilitate generalized analytical reult or the comlete lamballat ytem. Although we do not yet hae comlete analytical reult, we can make a ew key oberation rom the aboe analyi regarding the eect o the lam on the dynamic o thi ballat toology (Fig. 6 at low dimming leel: The F ole and zero in the traner unction nearly match the zero and ole in the incremental imedance o the luorecent lam, reectiely. The lam mainly aect the ytem dynamic in the F range. For HF erturbation, the lam behae eentially like a ure reitor. Baed on the aboe reult, we can deine a uitable eedback controller deign rocedure or the dimming ballat a ollow: Determine the F gain and the requency o the double ole o the requencytooutut current traner unction rom the haor ballat model (with equialent reitie load. Select the crooer requency well below the requency o the double ole. 3 Determine the magnitude and hae eect o the lam on the ytem at low dimming leel rom the imle meaurement o incremental imedance o the luorecent lam. 4 Deign the comenator to achiee high F gain, deired crooer requency and hae margin. V. FEEDBAK ONTOE DESIN EXAMPE In thi ection, we reent a deign examle o the eedback controller or the ballat hown in Fig. 3 with E Biax TM 6W F according to the dynamic analyi and dicuion in the reiou ection. A tyical comenator i the imle integrator a π c ( ( IF4 K VO khz/v D V IF4 IF4 39u with Hz to achiee high F gain and deired crooer requency. Thi controller work well at high ower leel, but ome ocillation may occur at low dimming leel a hown in Fig. 4, where the lam ower i teed rom 3% to % o ull ower. By ollowing the rocedure in the reiou ection, we can eaily deign a new comenator to obtain better erormance: From the haor model, we know the F gain o the requencytooutut current traner unction (with the gain o the VO and current enor i below db and the requency o the double ole i aboe khz. Select the crooer requency around khz or ull ower oeration. 3 The zero and ole o the incremental imedance o the lam are about l khz and khz, reectiely, and the gain dierence between F and HF i about ~db. 4 From the inormation gien by te to 3, we can chooe the comenator a the ollowing orm 3.8n b 4.3n en E 6W F _ Vre c ( Fig. 3 loedloo ytem o ballat with E Biax TM 6W F uer trace: current reerence mv/di lower trace: lam current ma/di time: 4u/di Fig. 4 Ste reone o the lam current at low dimming leel with the integrator

7 π π z c ( π with Hz, 3386Hz, z 73Hz. (6 Fig. how the te reone o lam current (rom 8mA to 3mA in eak alue or thi deign at low dimming leel (rom 3% to % o the ull ower. It can be een that the ytem i table a redicted. VI. ONUSION Studie o the dynamic behaior o the eneloe ignal are imortant or the deeloment o the lighting electronic ballat. For dimming ballat, requency control i a igniicant cae. In thi aer, analytical tool are reented that are ueul or dynamic analyi o requencycontrolled dimming electronic ballat and deeloment o eedback comenation network. A mallignal haor model i reented or A analyi o the eect o the requency modulation on the lam current eneloe baed on a modiied haor tranormation by conerting an FM ignal into an equialent timearying haor. The good agreement between the imulation reult and meaurement how that thi model can accurately redict the eneloe dynamic. To aid deeloment, key oberation are reented that gie intuitie analytical reult on the F gain and ole location o the controltooutut current traner unction or the ballatreitor ytem. The eect o the incremental imedance o the luorecent lam on the ballat ytem dynamic at low dimming leel are alo dicued. Baed thee analytic and exerimental reult, a eedback controller deign rocedure or the tudied ballat ytem i rooed and a deign examle i gien. The exerimental reult eriie that thi controller work ery well, een at ery low dimming leel. uer trace: current reerence mv/di lower trace: lam current ma/di time: 4u/di Fig. Ste reone o the lam current at low dimming leel with the PID comenator EFEENES [] E. Deng,. Negatie Incremental Imedance o Fluorecent am, Ph.D. Thei, aliornia Intitute o Technology, Paadena, 99. [] S. BenYaako, S. lozman, and. abinoici, Eneloe Simulation by SPIEomatible Model o inear Electric ircuit Drien by Modulated Signal, IEEE Tran. on Ind. Al., Vol. 37, No., March/Aril., 733. [3] S. lozman and S. BenYaako, Dynamic Interaction Analyi o HF Ballat and Fluorecent am Baed on Eneloe Simulation, IEEE Tran. on Ind. Al., Vol. 37, No., Setember/October., [4]. Erickon, D. Makimoic, Fundamental o Power Electronic, econd edition, Maachuett: Kluwer Academic Publiher,. []. T. im and. H. ho, Phaor Tranormation and it Alication to the D/A Analye o Frequency Phaeontrolled Serie eonant onerter (S, IEEE Tran. on Power Electronic, Vol., No. Aril 99,. [6] V. Vorerian, Aroximate SmallSignal Analyi o the Serie and the Parallel eonant onerter, IEEE Tran. on Power Electronic, 4.