EQUIVALENT ELECTRIC CIRCUIT MATLAB SIMULINK OF A BIFACIAL SOLAR CELL IN TRANSIENT STATE: APPLIED MAGNETIC FIELD EFFECT

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1 Journal des Scences EQUIVAENT EECTRIC CIRCUIT MATAB SIMUINK OF A BIFACIA SOAR CE IN TRANSIENT STATE: APPIED MAGNETIC FIED EFFECT S.MBODJI, A.Y DIAO, I. Y. F. I. BARRO, F. ZOUGMORE, G.SISSOKO ([email protected]) aboratore des Semconducteurs et d Energe solare, Département de Physque, Faculté des Scences et Technques, Unversté Chekh Anta Dop, Dakar, SENEGA aboratore des Matéraux et Envronnement, Département de Physque, UFR-SEA, Unversté de Ouagadougou, BURKINA FASO Résumé Dans ce paper, nous présentons une étude du régme transtore obtenu par varaton du pont de fonctonnement d une photople bfacale au slcum de type n + -p-p + soumse à un champ magnétque constant. a photople bfacale est connectée à un transstor Mosfet et est éclarée par une lumère multspectrale constante. e régme dynamque transtore est observé entre deux ponts de fonctonnement en régme statque dépendant de deux résstances varables. Cec permet d avor la réponse transtore en tout pont de la caractérstque I- V de la cellule solare du crcut ouvert jusqu au court crcut. Dans le but de fare des smulatons de ce régme transtore avec Matlab Smulnk, nous avons utlsé un modèle de crcut électrque équvalent de la photople en régme statque et l nfluence du champ magnétque sur le photocourant transtore a été étudée. Nous avons consdéré tros modes d éclarement : éclarement par l émetteur, éclarement par la face arrère et l éclarement smultané par les deux faces. Ces smulatons nous ont perms de proposer un crcut électrque équvalent de la photople en régme dynamque transtore obtenu par varaton du pont de fonctonnement. Abstract The bfacal cell wth an n + -p-p + structure and under constant magnetc feld s placed n a fast-swtchnterrupted crcut and submtted to a constant multspectral llumnaton. The transent decay occurs between two steady states through operatng ponts dependng on two varables resstors; ths allows us to obtan a transent decay at any operatng pont of the I- V curve of the solar cell, from the short crcut to the open one. To perform the smulaton wth Matlab Smulnk, the bfacal cell has been replaced by an equvalent electrcal crcut and the nfluence of the magnetc feld on the transent photocurrent has been studed. Three llumnaton modes have been consdered here: the front llumnaton, the back-sde llumnaton and the smultaneous llumnaton of both front and back-sde of the bfacal solar cell. These smulatons lead to an equvalent crcut of the bfacal cell n transent state assumng that the photocurrent s the dffused. Mots clés Bfacale, Cellule solare; Champ magnétque. Keywords Bfacal, Solar cell; Magnetc feld.. INTRODUCTION Ths work presents a transent study of a bfacal slcon solar cell under magnetc feld and submtted to constant whte lght (Fg. ). To perform the smulaton wth Matlab Smulnk, the bfacal cell n the expermental setup [] has been replaced by an equvalent electrcal crcut and the nfluence of the magnetc feld on the transent photocurrent has been studed. Ths equvalent electrcal crcut ncludes the photocurrent source [], the devce depleton capactance, the nductance, S. MBODJI et al / J. Sc.Vol.6, N (006) 99

2 Journal des Scences the seres resstance Rs, and the shunt resstance Rsh. Rc s the external load. TEORY The excess mnorty carrer concentraton s obtaned by solvng the equaton contnuty (Eq.) wth boundary condtons (Eq. and 3) whch nclude nterface states n the bfacal solar cell. D * D (x, t) (x, t) (x, t) = x t () and τ represent respectvely mnorty carrer dffuson and lfetme under constant magnetc feld [3]. The boundary condtons are: At the p-n nterface ( x = 0 ) D D (x, t) = Sf (0, t) () x x = 0 and at the back sde of the cell ( x = ) (x, t) x x = = Sb (, t) (3) Sf s the juncton recombnaton velocty whch takes nto account the ntrnsc recombnaton velocty Sf 0 and Sb s the back surface recombnaton velocty. + (x) = A 3 = wth: x cosh( ) + B K [ ε exp( b ( x)) + γ exp( x b )] x snh( ) Illumnaton mode ε γ Front sde llumnaton ( = fr) 0 Rear sde llumnaton ( = re) 0 Smultaneous llumnaton ( = s) K s gven by : double K = a D b (4) where a and b are coeffcents deduced from modellng of the generaton rate overall radatons n the solar spectrum [4] The subscrpt ndcates how the solar cell s llumnated: front sde ( = fr), rear sde ( = re) or smultaneous llumnaton ( = s). Constants A and B are determned by mean of boundary condtons (Eq. ) and (Eq. 3) where the mnorty carrer densty s replaced wth (Eq. 4). (Eq. ) and boundary condtons (Eq. and 3 ) consttute a typcal Sturm-ouvlle s system [5] from whch we obtan the mnorty carrers densty n transent state: (x, t) = X n (x)t,n (0)exp( n, t) c,, n (5) X,n and T,n (0) are deduced from (x,t) by normalzaton and Fourer transform. Fgure :Bfacal solar cell under constant magnetc feld The steady state excess mnorty carrer densty s: τ c,,n s the decay tme constant gven by : = τ τ c,,n ω,n (6) where ω,n s the egenvalue of the transcendental equaton: S. MBODJI et al / J. Sc.Vol.6, N (006) 00

3 Journal des Scences tan( D D (Sf + Sb ) ) = D Sf Sb (7) To perform the smulaton wth Matlab Smulnk, the bfacal cell n the orgnal setup [] has been replaced by an equvalent electrcal crcut and the nfluence of the magnetc feld on the transent photocurrent has been studed. Fgure 4: Illumnated I-V curve We present on Fgures 5, 6 and 7 the transent photocurrent decay for dfferent solar cells and dfferent magnetc felds Fgure :Solar cell equvalent Crcut wth R,R and the swtch. Fgure 3 :Smulnk model of the Solar cell equvalent Crcut wth R,R and the swtch Fgure 5: Transent photocurrent decay : front sde llumnaton :B= 0 T, B=0.00 T,B =0.005 T At tme t<0 the solar cell s n parallel wth the resstor R gvng the potental V correspondng to the operatng pont F. At tme t=0, the fast swtch begns turnng on and at tme t>0 s completely turned on. It then connects the resstor R n parallel wth the solar cell and the resstor R; The voltage V drops from V to V correspondng to the new operatng pont F as t can be seen on Fg. 4. Fgure 6 : Transent photocurrent decay : rear sde llumnaton : B= 0 T,B=0.00 T,B =0.005T S. MBODJI et al / J. Sc.Vol.6, N (006) 0

4 Journal des Scences = 4 ( Rc Rs) C () We only nvestgate solutons correspondng to >0 and <0. When >0 the soluton can be expressed by: I (t) = Aexp( b.t) + A.exp( c.t) + K.exp( a.t) () Fgure 7: Transent photocurrent decay : smultaneous llumnaton : B= 0 T, B=0.00 T, B =0.005 T It s shown that the transent photocurrent decreases f the magnetc feld ncreases for each llumnaton. Based on Fg. 8 we can wrte the followng equaton:... I (t) I (t) + (Rc Rs ) I (t) + = (Iph (t) Ish (t)) (8) C C Neglectng I sh (t), Eq. 8 becomes :... I(t) I(t) + (Rc Rs) I(t) + = I C C where J ( t) = q D B,0,0 T,0(0) exp[ ( ω, 0 ph (t) + ) t τ ω (0) The seres resstance Rs and the capactance C depend on the magnetc feld [6-] (9) where R 0 b =. c = + R 0..C, C a = ω, 0 + τ and R 0 = Rc-Rs Constants A and A can be calculated for two ponts : at the maxmum current (di /dt = 0, begnnng of the decay) and when the current I s approxmately constant correspondng to the end of the decay ( greater value of tme, see for example Fg. 6 ) I(t)=Is at t= ts. Constants A, A and K depend on C, Rs, ω,0, ts, τ, and Rc. and τ are fxed at 0,00 and 4,5µs respectvely Based on the relaton V (t)= Rc.I (t), we plotted the photovoltage V versus the tme t for each llumnaton mode. Fgure 8: Electrcal equvalent crcut of the solar cell n transent state Solutons of equaton (9) depend on the sgn of the descrmnant expressed below: Fgure 9 : Transent photovoltage decay : front sde llumnaton (=0.0cm,τ = 4,5µm) S. MBODJI et al / J. Sc.Vol.6, N (006) 0

5 Journal des Scences photovoltage versus the tme t(s) when the solar cell s llumnated respectvely by front sde, back sde and both front and back-sde. Fgure 0: Transent photovoltage decay : rear sde llumnaton (=0.0cm,τ=4,5µm) Fgure : Transent photovoltage decay : front sde llumnaton (=0.0cm,τ = 4,5µm). Fgure : Transent photovoltage decay : smultaneous llumnaton (=0.0cm,τ =4,5µm) When s negatve, the photovoltage s snusodal for each llumnaton (Fg. to 4) and we notce three parts: the frst s where the photovoltage ncreases wth the magnetc feld, n the second the transent photovoltage doesn t follow the ncrease of the magnetc feld and the last part s where the photovoltage s approxmately constant for the three magnetc feld. Fgures 9, 0 and show that under the nfluence of the magnetc feld the photovoltage decay ncreases when the solar cell s llumnated by front sde, rear sde and both front sde and back-sde. In the case, <0 the soluton can be expressed by: I 0 (t) = exp( )[A'.cos( β.t) + A'.sn( β.t)] + K.exp( a.t) (3) where. 0 =., β =.C. Fgure 3 : Transent photovoltage decay: rear sde llumnaton (=0.0cm,τ=4,5µm). A and A are calculated by the same way as A and A. Fg. to 4 present the transent S. MBODJI et al / J. Sc.Vol.6, N (006) 03

6 Journal des Scences back-sde; b and c can be taken as a system of equatons whch system lead to the values of and C for these two llumnaton modes. Wth the value of a the lfetme τ s calculated assumng that the egenvalue of the fundamental mode ω,0 s known []. 3. CONCUSION. Fgure 4: Transent photovoltage decay : smultaneous llumnaton (=0.0cm,τ=4,5µm). Thus, the case >0 s more nterestng than the case <0. The exponental ft of the transent photovoltage when >0 can gve values of b, c and a. Value of Rc s fxed, Rs s neglgble compared to Rc for front sde llumnaton and for smultaneous llumnaton of both front and In ths work, we presented a theoretcal study of a bfacal solar cell under constant magnetc feld and under constant llumnaton. The effect of the magnetc feld has been exhbted on both photocurrent and photovoltage for each llumnaton mode. Based on theoretcal and smulaton results and Fg. 5,6 et 7the electrcal parameters, C, R S and the mnorty carrers lfetme τ are determned va parameters a, b and c for each llumnaton mode. 4. REFERENCES [] F.I. Barro, S. Mbodj, A.. Ndaye, S. Madougou, I. Zerbo, F. Zougmore, G. Sssoko, Proc.9 th European Photovoltac Solar Energy Conference and Exhbton, june 004, Pars, France, Poster AV.57. [] C.M.Colomb,S.A.Stockman,S.Varadarajan and G.E.Stllman,Appl Phys.ett.60,6 january 99 [3] Y. Betser, D Rtter, G. Bahr, S. Coen et J. Sperlng: Appl. Phys. ett, Vol 67.No.3,5 septembre 995,pp [4] S.M.SZE,Physcs of semconductor devces, nd Edton,Wley Interscence New-York (USA), p.45(98) [5] P.K. BASU, S.N. SING : Solar Energy Materals and Solar Cells, 33(994), pp [9]G. Sssoko, B. Deng, A. Corréa, M. Adj, D. Azlnon : Proc.World Renewable Energy Congress (998), pp [0] W.Shockley: Bell syst.techn.j.8,p 435 (949) [] R. S. Muller and T.I. Kamns: «Devce Electroncs for Integrated Crcut», nd Edton Wley, New york (986). [] A.Berhals, A. Metz and R.ezel: Proc.6 th European photovoltac solar conference,-5 May 000,Glasgow,UK,pp4-45 [3] A. Rcaud: «Photoples solares». Presses polytechnques et Unverstares romandes, ausanne, Susse, 997 [4] Physcs of semconductors devces Edton Wley, New York (969).p640 S. MBODJI et al / J. Sc.Vol.6, N (006) 04