Bjop ol. o. Decemer 008 Byero Journl of Pure nd Applied Science, ():70 75 Received: Octoer, 008 Accepted: Decemer, 008 SIMULATIO OF TRASIET EQUILIBRIUM DECAY USIG AALOGUE CIRCUIT *Adullhi,.., Ango U.S. nd Btu M. A Applied Science Deprtment, College of Science & Technology, dun Polytechnic, P.M.B.0, dun, Sokoto Energy Reerch Centre, Umnu Dnfodio Univerity, Sokoto *Correpondence Author: Emil: nirmn@yhoo.com ABSTRACT An nlogue circuit which could e ued for imulting Trnient Equilirium decy w contructed. The Trnient Equilirium decy conidered w. Th P decy.d.8min erie. An integrting time contnt of milliecond w choen o to minimize integrting error nd mximum voltge level of 0 w choen for the deign in order to void turtion of the Opertionl Amplifier ued. The differentil eqution decriing the decy erie to e mechnized on the nlogue circuit w converted to voltge eqution through mplitude cling. Scle fctor, =.x0, = 0., = 6.5x0, =.95x0 were clculted from rnge of prolem vrile. To peed up the decy erie i.e. to cue mchine time to e mller thn the prolem, time cling w employed with cle fctor, given y 7.x0 9. When the contruction w teted, it imulte the equilirium decy with percentge error of ±%. ey word: nlogue, imultion, trnient, opertionl mplifier, mplitude cling ITRODUCTIO An nlog computer cn e ued to olve vriou type of prolem. It olve them in n nlogou wy. Mot generl purpoe nlog computer ue n ctive electricl circuit the nlogou ytem ecue it h no moving prt, high peed of opertion, good ccurcy nd high degree of vertility. Active electricl network coniting of reitor, cpcitor, nd op mp connected together re cple of imulting ny liner ytem ince the forwrd voltge trnfer chrcteritic of thee network re nlogou to the ic liner mthemticl opertion encountered in the ytem mthemticl model. The norml procedure for imulting ytem trt with determining the mthemticl model decriing the phyicl quntitie of interet. An nlog lock digrm i mde to relte the equence of mthemticl opertion nd to id A B in cling the vrile. From the nlog lock digrm the electricl component re connected together (ptched).the computer i operted nd the computer vrile oerved on recorder or ocillocope (Pz, 00). The preent work i n ttempt to deign imple electronic nlogue circuit to imulte n rtificil rdioctive decy. The deign will ue 7 opertionl mplifier connected externlly to reitor, cpcitor nd potentiometer to olve et of coupled differentil eqution decriing rdioctive decy from nuclide A with decy contnt to form nuclide B which in turn decy with decy contnt decy i repreented : C to form tle nuclide C. Such () The growth nd decy of the numer of tom of the differentil nuclide preent i repreented y the ction of coupled eqution (ujic, 006) d d The mthemticl model of n nlog computer progrmmed to imulte pecific phyicl ytem i identicl to the mthemticl model of the ytem. Differentil nd /or lgeric eqution in order to e () mechnized on the nlog computer mut firt e converted to voltge eqution, cle fctor or volt per phyicl unit rtio mut e choen for ll the dependent vrile (Rlton nd Meek, 976). 7
Bjop ol. o. Decemer 008 The voltge trnfer chrcteritic of the electricl network re nlogou to the deired mthemticl opertion. The input nd output voltge (computer vrile) re nlogou to the correponding mthemticl vrile (prolem vrile) of the prolem. Becue of limittion of the computer or it ocited input/output equipment, it i uully necery to chnge the cle of the computer vrile, thu forcing the vlue of computer vrile to differ from the correponding prolem vrile vlue. It i importnt to undertnd tht n nlog computer olution i imply voltge wve form whoe time dependency i the me tht of the deired vrile (Pz, 00). The prolem will e cled in mplitude well in time. Amplitude cling i done to void ny ritrry choice of the cle fctor ecue thi my led to the olution eing outide the voltge rnge of opertion or lterntively eing o mll to e lot mong the inherent noie. Time cling enure tht the prolem cn e olved in horter or longer time the ce my e (Dwe, 980). The ccurcy nd tility of the nlogue circuit i eentilly et y the chrcteritic of the externl component connected to the opertionl mplifier uch reitor, cpcitor nd potentiometer. The tolernce of the externl component ued in cloe loop et limit on cloed loop ccurcy (Jcoowitz nd Bford, 97). Methodology The following rdioctive decy illutrting rtificil trnient equilirium decy w conidered: Th P decy erie..d.8min The deign eek to otin (t) nd (t) in the form integrtion technique (Pz, 00). In order to limit of equivlent voltge level which could e meured the integrting error which incree with time, n on the propoed nlogue circuit. For prcticl integrting time contnt of milliecond w reon, integrtion opertion i eier to implement crefully choen for ech integrtor, uch tht the thn the differentition opertion. The reon lie in feed ck cpcitor nd input reitor tke vlue of the fct tht computer ignl re rel voltge nd,. µf nd k repectively. To void turtion of therefore, re corrupted y noie to ome extent. the 7 opertionl mplifier ued mximum Since integrtion h tendency to verge out the voltge level of 0 volt w choen for thi deign. effect of noie (while differentition will ccentute it), more precie olution cn e otined uing Amplitude cle fctor re defined follow: Scle fctor = d mchine unit M.U mximum vlue of vrile mx mx (), (Oroge, 998): Thu, for the preent prolem, mx () A mx d mx mx (5), mx (6) mx Generlly, time cling for the derivtive of x i given y []: n n n d x dx n n (7) Where t = time t which phenomenon ctully occur nd T = time required for the phenomenon to occur on the computer, i the cling fctor. Comining mplitude nd time cling, the computer ptched progrmme from eqution i otined : d d (8) 7
Bjop ol. o. Decemer 008 Term in rcket re computer term nd thoe in qure rcket re potentiometer coefficient nd d Integrtor () mplifier gin. The correponding ptch progrmme i hown in Figure. Inverter Amplifier Integrtor A Potentiometer Figure : Computer Ptch Progrmme decriing the imultion of eqution 8 B Amplifier From fig, the function d i pplied t the input of the firt integrtor nd inverted in ign o tht it output voltge i, i the gin of the integrtor mrked. The voltge i finlly inverted in ign to through n inverter, whoe input reitnce equl it feed ck d reitnce. The output repreent the olution of the differentil eqution i.e. firt decy. The function. The function d nd re pplied to the input network of umming integrtor. The input reitor nd the integrtor cpcitor re choen o tht i multiplied y the contnt cle fctor nd y their um i.e. nd therefore, the defining reltion i given y i integrted nd i inverted in ign y the econd integrtor o tht it output voltge i + B. The output B repreent the olution of the econd decy. potentiometer coefficient, while nd re the gin of the firt nd econd mplifier repectively. Deign conidertion Uing n initil numer of nuclide 00, the cle vrile nd other prmeter for the computer 0 mx ptched progrmme were otined follow: 99.97 7.8x0 t 05..99x0 9.788x0 6 mx t rel time.00675x 0 i 7
Bjop ol. o. Decemer 008 T computer t ime X0 T t 7.x0 d d 57 5. 75 mx mx 9 mx.x0 d mx mx 6.5x0 d x, mx 0. mx, mx.95x0 mx RESULT AD DISCUSSIO Uing n integrting time contnt of milliecond, the vlue of the time cling fctor,, w otined 7.x0 9.Thi vlue otined which i le thn unity men tht the phenomen will occur fter in the computer thn it doe in nture(dwe 980).The potentiometer coefficient otined were pproximtely 0. with gin of 0 for ech integrtor. Thi vlue will enure tht the cle fctor would e limited within the voltge rnge nd the olution would not e outide the voltge rnge of opertion. Scled quntitie in the form of were compred with correponding decy contnt rtio nd good greement w otined within percentge error of ±.% on the verge for the trnient equilirium decy (tle ). Thi gree with the trnient equilirium pproximtion i.e. ince (ujic, 006). Thi deign tet w primrily done to ee if the circuit would imulte the Trnient decy with firly reonle ccurcy if it i eventully contructed. Tle : Teting rtificil Trnient Equilirium Decy. 5.0.5.5 0.65 0. 5.08.509.7 0.67 0..006.006 0.0098.00.00.000000 When the circuit w contructed nd the voltge w meured on the ocillocope, the vlue of the output otined were 0. for the prent nuclide nd 9.7 for the dughter nuclide. Thi give good greement to within percentge error of± %. Fig how the circuit digrm for teting the nlogue circuit. Power upply unit Amp 7 +v CRO in out 0 olt 0 olt v Figure : Circuit Digrm for teting the nlogue computer 7
Bjop ol. o. Decemer 008 COCLUSIO Although the deign eek to otin ignl output of 0 which i nlogou to the phyicl ytem eing imulted, the vlue of 0. nd 9.7 otined for the prent nd dughter nuclide repectively, men tht the contructed circuit imulted the differentil REFERECES Dwe, D.F. (980). Anlogue computing technique. Wirele World, July. pp 55 59 Jcoowitz, S nd Bford, A. (97). Electronic computer mde imple. Howrd nd Wyndhm Ltd Oroge, C.O. (998). Control ytem engineering. Univerity Pre Plc. Pp 095 Pz, R. (00). Anlog computing technique, chpter. Aville t http:// eqution decriing the Trnient equilirium decy with firly reonle ccurcy. Hence, undertnding the equilirium for given decy erie through imultion will help cientit to etimte the mount of rdition tht will e preent t vriou tge of the decy. www.ece.nmu.edu/nmn. Pp 55. Retrieved th Ferury, 007 Rlton, A nd Meek, L.C. (976). Encyclopedi of Computer Science. Litton Pulihing Compny. Pp 5986 ujic, J. (006). Rdioctive decy, Lecture. Aville t http:// www.nucmed conultnt.com/tutoril/lecture note. Pp.Retrieved th Ferury, 007 75