THE APPEARANCE OF FIBONACCI AND LUCAS NUMBERS IN THE SIMULATION OF ELECTRICAL POWER LINES SUPPLIED BY TWO SIDES

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1 THE APPEARANCE OF FIBONACCI AND LUCAS NUMBERS IN THE SIMULATION OF ELECTRICAL POWER LINES SUPPLIED BY TWO SIDES Giuseppe Ferri Dipartimeto di Ihgegeria Elettrica-FacoM di Igegeria, Uiversita di L'Aquila Localita Moteluco di Roio, 674 Poggio di Roio, L'Aquila, Italia (Submitted October 995) INTRODUCTION I the aalysis of some physical structures, the possibility of modelig them with a electrical circuit is particularly importat because it allows the determiatio of the characteristic behavior by meas of a simple circuital aalysis. Moreover, it is also iterestig to have a differet method of measuremet evaluatio, comparable with the "direct" oe, which sometimes either is ot simple or requires the use of computer programs which o some occasios do ot go ito covergece. Fially, it ca make a cotributio to the mathematical iterest i testig of etwork software algorithms for solvig liear equatio systems. I this article, a symmetrical ladder etwork is used as a model for the simulatio of electrical power lies. Fiboacci ad Lucas umbers come out from the aalysis of the power distributio amog the users. The electrical characteristics of the ladder etwork have also bee determied i a closed form usig a theory previously developed by the author [].. MODELING OF A POWER ELECTRIC LINE Let us cosider a high voltage electric lie, supplied by the two sides, which gives power to users distributed alog the lie, as i Figure. V. V r user user FIGURE. The Electrical Power Lie Supplied by Two Sides A ladder structure (Fig. ) ca be used as a discrete electrical model of the power lie. For the sake of simplicity, we cosider users who have equal cosumptio, represeted by equal vertical impedaces Z, placed at equidistat poits characterized by equal horizotal impedaces 997] 49

2 V. t \ Z I- i h k f Z h[ X.. - Z + c FIGURE, Ladder Network as a Model of the Power Lie. ANALYSIS OF THE LADDER NETWORK I order to aalyze the etwork of Figure, we ca use the superimpositlo of the effects i the etworks of Figures ad 4. The aalysis of these etworks ca be doe startig from the study of the etwork of Figure 5, by addig a "load impedace. u Z Z X * - Z = oul V A^ ) 7 7 Z FIGURE, Ladder Network Supplied by V A Z Z l + 7 O FIGURE 4. Ladder Network Supplied by V B Z I Z Z - I I VC) 7 7 FIGURE 5. Ladder Network with Idetical Cells 5 [MAY

3 I [] a ew fast method for the ladder etwork characterizatio i Figure 5 was preseted; by usig this method, all the electrical parameters of a ladder etwork formed by idetical cells ca be writte directly by meas of both a parameter that characterizes the sigle cell [the "cell factor K(s) = Z x (s) / Z (s)} ad the polyomials i K whose coefficiets are the etries of two umerical triagles, amed DFF [] ad DFFz [4], here reported: K K 6 K 5 K K 4 K l 4 K 6 K DFF Triagle Etry = + K -K DFFz Triagle Etry = + K + l -K The mathematical properties of triagles ad polyomials have bee preseted i []. Let us call b ad B the polyomials whose coefficiets are the etries of DFF ad DFFz triagles, respectively. These polyomials coicide with the polyomials defied by Morga-Voyce ad the ivestigated by Swamy [7] ad Lahr [5] ad [6]. All the electrical characteristics of the etwork represeted i Figure 5 ca be expressed directly i a closed form by meas of these polyomials if all the cells are equal. The etworks draw i Figures ad 4 are very similar to that of Figure 5. The oly differece is i the fact that the last cell of the Figure 5 etwork has a "load" impedace of ifiite value. It is possible to write the electrical expressios for the Figure ad Figure 4 etworks as simply as for the Figure 5 oes ad also i closed form. For the Figure etwork, we have (see [5], p. 75) that the trasfer fuctio is give by V V A while the voltage at the geerical X th ode is give by D«(A) B (K)> with B_ r (K) =. The voltage behavior for the etwork of Figure 4 is symmetrical. For that reaso, we ca write W ^ B T l i ( < X < ^ + ). () By the applicatio of the superimpositio of the effects, we ca write, for the etwork represeted i Figure, the followig expressio for the ode voltages: () () V X (K) = V>(K) + V>'(K) = V A B (K) B B {K) (<x< + l). (4) 997] 5

4 Deotig by I xl ad I x the currets flowig ito the X th cell horizotal ad vertical impedaces, respectively, we ca write similar expressios, usig the followig property of Morga- Voyce polyomials, b x = B x -B x _ l (see [], [5]-[7]): L a f f l x v K-iJK) B (K) +v R B (K) ( I < x < «+ ); (5) J-vO A x _ B _ X (K) B^jK) Z Z B (K) B B (K) ( < x < ). (6) Let us ow cosider the case of odd, for which the middle poit exists for the voltage ad the vertical curret ad is defied for x = m = ( +) /. I this poit, from (4), we ca write I the middle vertical impedace, we also have A, I ml (v A + v B ) Ḇ, B. '(-l)/ I the case of eve, we ca reaso aalogously by cosiderig the middle horizotal curret, whose value is give by i m.=^(-v A + v B ) ^ (9) ^ ~ beig x = m = ( + )/, while expressios (4)-(6) are always valid. We are maily iterested i determiig the power dissipated i the vertical impedaces (because oly these have a physical meaig), which is give by the voltage-curret product: (7) (8) ^x ~~^~ L y B-xJK), y. A B (K) B B x-l( K ) B (K) (l<x<); () P m -v-( v A + v B y B t \-V)l ( odd). (ii) The Fiboacci ad Lucas umbers appear i the case of K = l, which correspods to Zj = Z = R. I this case, B x = F x+ ad h x = F x+. Cosequetly, we have V =V A F (+l-x) v F x - (w+l) («+l) V m = (V A + V B ) ^ = ( V A + V B ) - i - ^«+ Ai+ ( < x < f i + l), (w odd), () () i.,=x xl R _y t V»l +Vp J^zL [ (w+l)! (*+l) (l<x<w'+l), (4) 5 [MAY

5 A x R K (+l) («+l) ( < X < A? ), (5) I -! 4«t, ' ) r '+l (w eve), (6) I m = ^ ( V A + V B ) I i '+l (w odd), (7) from which: i P x ~ i? V A ^ ± ^ + V B ^ - (77+) 'K (+l) ( l < x < «) ; (8) Pm=^(V A +V B )- Hrc+l ( odd). The last two relatios show that the power cosumptio of the users is also a fuctio of the Fiboacci ad Lucas umbers., EXAMPLE Let us cosider the power dissipatio i the vertical impedaces i the case of =, show i Figure 6 below. - 7 j z j 4 (9) V z) 7! s O FIGURE 6 e Example I the geerical case of differet values betwee the horizotal ad vertical impedaces, we have, from (9): that is, P x ~z, v B^jK) y B^Kl B (K) B (K) ( l < x < ), () ^ A 5 ( ) B!*,(*) p = ^ [ v A + v B r = ^ [ V A + V B ] B (K) l V ^ ^ + ^ + Sj + Vj A : I O + 4 K + K + 6K + lok + 4 () B {K) B (K) P = V B (K) A 5,(*) V B (K +4K + ) + V A K + 6K + lok ] 5

6 I the particular case of Z l = Z = R, we have P x " i? V, F g_ x (K) F x (K) F,(K) B F g (K) ( l < x < ), () from which: P =[8V A + V B ] /44R, P =[V A -fv B ] /49R, P =[V B + 8V A ] /44K. () 4. PARTICULAR SUPPLY VALUES I the aalysis of the symmetrical ladder etwork, which models the power electrical lie, we ca cosider some particular cases for the values of V B ad V A. ) If V B = V A >, the etwork is completely symmetrical ad the curret flows as i the directio, for example, idicated i Figure 6, if is odd. Whe is eve, i the middle horizotal impedace, the curret is zero. ) If V A = -V B, ad oly from the mathematical poit of view, oly the case odd is iterestig. I this case, i the middle poit, all the electrical characteristics (voltage, vertical curret ad power) are zero. ) I the case V B = V A 4- AV, where AV ca be positive or egative ad AV «V A? V B, we have a slightly ubalaced situatio ad, as a cosequece, there is a small differece i the electrical parameter values. This is a real case ad the computatio ca be of practical importace: if oe of the supplies does ot have eough power (owig to a lack of power), the other oe ca provide it. We ca write: ad so that x=" which, i the case of Z x -Z - V = V A B»-* +B *-i +AV^E= L ; AV = A V - ^ (l<x<) A A A, V B A -x + B x-l + AV^i B B AP x = AV x AI x =^-AV z R, is equal to AI V, = 4 - A V ^ L x l x- B m A B m (l<x<) (l<x<), (4) (5) (6) AR =4AV r x R ad, i the middle poit, for odd, is equal to AP m = ^ A V [ x [ «+ J l H+. (7) (8) 54 [MAY

7 This meas that the power variatio is strogly depedet o the umber of cells (i.e., the umber of the users) upo whom the lie is modeled ad is also a fuctio of Fiboacci ad Lucas umbers. For example, if =, for a variatio of %, we have that while, for a variatio of %, we have that where, i the case of cells, we have, for AV = %, ad,forav = %, R-AP m =.4//W-Q (9) R-AP m =.4mW-O, () R-AP m = 5W-Q, () R-AP m = 5/iW-Q. () CONCLUSION A symmetrical ladder etwork with a high umber of cells ca be cosidered as a good model for the ivestigatio of the behavior of a electrical power lie. I the particular case of equal impedaces, the electrical characteristics ca be writte as a fuctio of Fiboacci ad Lucas umbers. REFERENCES. M. Faccio, G. Ferri, & A. D'Amico. "A New Fast Method for Ladder Network Characterizatio,"' IEEE Tras, o Circuits ad Systems 8. (99): M. Faccio, G. Ferri, & A. D'Amico.."The DFF ad DFFz ad Their Mathematical Properties. " I Fiboacci Numbers ad Their Applicatios 5:99-6. Dordrecht: Kluwer, 99.. G. Ferri, M. Faccio, & A. D'Amico. "A New Numerical Triagle Showig Liks with Fiboacci Numbers." The Fiboacci Quarterly 9.4 (99): G. Ferri, M. Faccio, & A. D'Amico. "Fiboacci Numbers ad Ladder Network Impedace." The Fiboacci Quarterly. (99):6~ J. Lahr. "Theorie elektrischer Leituge uter Awedug ud Ermeiterug der Fiboacci- Fuktio." Diss. ETH 6958, Zurich, J. Lahr. "Fiboacci ad Lucas Numbers ad the Morga-Voyce Polyomials i Ladder Networks ad i Electric Lie Theory." I Fiboacci Numbers ad Their Applicatios. Dordrecht: Kluwer, M. N. S. Swamy & B. B. Bhattacharyya. "A Study of Recurret Ladders Usig the Polyomials Defied by Morga-Voyce." IEEE Tras, o Circuit Theory 4 (967):6-64. AMS Classificatio Numbers: B9, 94C5 997] 55