electrical circuits electronic circuit form taken network circuit components different specific values ratings components regarded concerned physical layout components circuit positions circuit concerned connections exist components numerous physical circuit strictly speaking replacing component entirely different type contexts loosely described different instance low pass filter results high pass filter described high pass low pass network identical correct term classes object network type component specified absolute value prototype network electronic network related mathematical particular networks contain terminal devices circuit viewed application theory network analysis circuit point view network theory network branches edges theory standard theory extended deal active components multi terminal devices integrated circuits analysis infinite networks circuit circuit article follow usual conventions electronics lines represent filled small circles represent junctions open small circles represent connection outside world cases represented practical circuit use specific symbols etc concerned type component network symbol general instead theory section article gives alternative method representing networks names names relate appearance drawn circuits drawn variety ways consequently variety names instance circuits shown figure look different identical example demonstrates common convention naming letter alphabet resemblance greek alphabet letters way example delta series parallel network branches possible series parallel simplest variations way circuit presented network branches possible note parallel series representation delta discussed later series parallel continue constructed greater greater numbers branches number unique obtained branches total number unique obtained branches important linear network analysis simplest possible terminal networks transform available linear circuits transform important networks analysed terms series parallel combinations example network figure consisting network connected parallel network desired network networks successive applications rules combination series parallel possible case transform needed addition series parallel rules called star star refer general case branches connected simple filter shown figure commonly filter designs section identical potential section identical section identical viewed short section ladder longer sections normally described ladder kinds circuits commonly analysed characterised terms port network bridge bridge important uses linear non linear applications including bridge bridge phase ways bridge rendered circuit rendering figure traditional depiction bridge circuit second rendering clearly shows bridge derived series parallel combinations rendering commonly known obvious equivalent seen left moved right right normal network bridge port network input output ports consisting pair opposite box figure seen identical bridge case filter input output ports pair adjacent loading indication component output port bridge included bridge shown figure twin derived bridge way explained network article derivative discussed article twin practical applications desirable input output share common ground terminal instance input output connections connecting input output terminal normal bridge reason twin bridge balance measurement applications twin wave generator lower figure shows twin emphasise connection bridge infinite ladder extended limit filter designs variations ladder discussed electronic filter composite image filter articles balanced form ladder viewed arbitrary order anti forms sense anti ladder anti ladder finds application voltage circuits particular generator wave version generator uses double anti ladder infinite formed multiple sections simple bridge sections infinite chains sections occur theoretical analysis artificial simulation transmission lines rarely practical circuit implementation components circuits containing components greatly increase number possible conversely number different circuits represented cases circuit easily specific components identified complex circuits description proceed specification transfer function ports network components theory theory branch mathematics dealing network analysis extensively represent network analysed network captures certain aspects network aspects related words useful representation network network networks includes derived laws history theory network analysis linear passive networks moment laws formulated abstract representation network loop analysis circuits approach later circuits replacing provided dual analysis analysis responsible matrix equal sum tree products introduced idea representing matrix founding field applied analysis responsible introduction spanning tree aid choosing compatible set network comprehensive network apply electrical circuits began engineer friendly article limited survey series parallel combinations called chains rank provided charts small number work grew earlier survey foster collaborating port telephone produced distinct long time electrical circuit theory remained concerned linear passive networks recent developments devices circuits required new tools deal enormous increases circuit complexity led use theory improve efficiency computer calculation circuit networks commonly classified kind electrical elements making circuit element kinds specifically drawn unique symbol networks element kind networks consisting elements likewise networks element kind circuits simple element kind networks circuit simplest element kind network ladder network commonly low pass filters elements example element kind network conversely concerned geometric relationship elements network kind elements heart representation network network elements represented edges edge drawn line terminating dots small circles edges elements circuit analysis edges called branches dots called represent network terms discussing networks figure shows representation circuit figure network analysis usually addition directed capture direction current flow voltage labelled capture branches instance consisting square branches branches branches uniquely labelled directed branch connects designated source target typically indicated arrow drawn branch basic properties edge connected said incident captured matrix format matrix called matrix fact matrix alternative mathematical representation need kind drawing matrix rows correspond matrix columns correspond branches elements matrix zero branch direction directed indicated sign element equivalent transformed include operations translation rotation reflection bending stretching branches crossing branches equivalent said field electrical networks additional transforms considered result equivalent produce interchange series connected branches dual interchange parallel connected branches achieved need special rule second concerned divided separate parts sets branches incident set separate parts considered equivalent parts joined combining single likewise split separate parts splitting considered equivalent trees links tree connected directly indirectly branches forming closed loops closed loops currents tree network analysis interested spanning trees trees connect present network article spanning tree meant tree stated given network contain number different trees branches removed order form tree called links branches remaining tree called twigs number branches tree formula important relationship circuit analysis formula number branches number links removed form tree tie sets cut sets goal circuit analysis determine branch currents network network independent branch related branch currents transfer function elements composed complete solution network terms branch currents branch branch currents independent minimum number branch currents required complete solution consequence fact tree links removed currents tree remaining branches tree zero current independent link currents branch currents chosen set independent set associated links tree choose branches terms branch complete solution network obtained branch consequence fact short branches tree results voltage zero link independent tree branch common analysis approach solve loop currents branch currents branch currents found terms loop currents set loop currents chosen guarantee set independent loop currents associated certain set loops set loops consists loops formed replacing single link given tree circuit analysed replacing single link tree forms exactly unique loop number loop currents defined equal term loop context usual meaning loop theory set branches forming given loop called tie set set network formed loop currents sum tie set branch currents possible choose set independent loop currents reference trees tie sets sufficient necessary condition choosing set independent loops ensure chosen loop includes branch previously included loops chosen particularly straightforward choice analysis loops chosen analysis applied possible map plane sphere branches crossing called ability map plane sphere equivalent conditions finite mapped plane map small region sphere conversely mapped sphere stretched space inside occupies nearly sphere entire occupies small region sphere case map plane approach choosing network analogous dual loop current method voltage associated pairs primary branch found terms method particular tree chosen order ensure independent dual tie set cut set tie set formed allowing links open circuit cut set formed allowing tree branches short circuit cut set consists tree branch short links short tree branches cut set produces cuts parts minimum set branches needed set network formed pair sum cut set branch dual special case analysis analysis rank separate parts branches defined formula represents number degrees freedom set network equal number rank defined formula rank plays role analysis plays analysis gives number voltage required rank dual concepts related formula solving network set independent chosen state network expressed terms result set independent linear need solved simultaneously order find values network set expressed matrix format leads characteristic matrix network form matrix formed loop analysis basis matrix formed analysis basis solved number known ways method systematic elimination method involves use known rule provides direct expression unknown variable terms useful provides compact expression solution trivial networks greater calculation effort required method working manually dual relationship branches pairs relationship branches loops dual found entirely graphical method dual given tree set branches branches tree form tree dual set current loop associated tie sets original tree identical set voltage pair associated cut sets dual following table lists dual concepts related circuit theory dual tree called maze consists spaces connected links way tree consists connected tree branches formed requires tie set dual cut set dual condition met sphere branches crossing note tie set required tie portions dual cut set required cut portions finite network map sphere require fold tie set passes hole fail tie parts consequently dual cut parts contain required cut set consequently formed networks containing mutual corresponding element equivalent circuits developed dual formed mutual directly elimination operations set network meaning aid happening elimination voltage set network corresponds elimination connected corresponds known transform transform extended greater numbers connected known star transform transform transform corresponds elimination current corresponds elimination elimination current branches common arbitrary number general result transform general star map sphere contains star multiple dual exist required represent elimination mutual conventional representation circuits means explicitly representing mutual occurs components result disconnected separate convenience analysis multiple parts combined single single makes difference theoretical behaviour circuit analysis carried valid practical difference circuit implemented way destroy isolation parts example primary secondary functions voltage ratio longer isolation recent techniques theory able deal active components problematic conventional theory new techniques able deal mutual active components basic approaches available dealing mutual active components introduced signal flow signal flow weighted directed circuits containing mutual active networks weight directed edge represents gain possessed general signal flow unlike regular directed described correspond physical arrangement components second approach extend classical method includes mutual active components methods proposed achieving constructed representing currents circuit representing passive components identical branches trees active components method relies identifying spanning trees common alternative method extending classical approach requires proposed method based rooted tree way extending classical theory active components use electronic components represented naturally connection points normal branch connect modern integrated circuits connections problem overcome instead regular conventional representation components represented edges connects components represented connect arbitrary number connect graphical representation box compared edge line representations lines box connected directed carry labels determined label conventional directed thought labelled source target usually indicated arrow general complex required characterised regular containing terminal components exactly non zero entries row matrix non zero entries row representation number non zero entries row rank corresponding branch highest branch rank rank matrix non classical network analysis develops set network network current loop analysis voltage analysis set network found necessarily minimum necessary form set independent difference number loop analysis analysis cases minimum number possible requirement relaxed mix current voltage allowed result absolute minimum number required describe behaviour network given maximum distance spanning forests network network synthesis theory applied network synthesis classical network synthesis realises required network number canonical forms examples canonical forms driving point canonical ladder network foster canonical form positive real functions methods hand start given canonical form form result mathematical representation canonical forms require mutual major aim methods network synthesis eliminate need mutual come driving point mutual minimal loops theory powerful network synthesis elements network represented real numbers element kind networks networks states switching networks infinite networks earliest network infinite studied ladder network represent transmission lines developed final form certainly early studies infinite networks limited periodic structures elements repeated late 20th century tools infinite networks arbitrary available infinite networks largely theoretical interest infinite networks constrained real world restrictions properties instance laws fail cases infinite defined driving point depends termination infinity property theoretical infinite networks general dissipate infinite power constraints placed addition usual network laws laws real world applications transmission line example class practical problems modelled elements distributed element model examples launching waves continuous medium field problems measurement resistance points substrate networks extend idea infinite networks infinite network branch connected leading network new network infinite constructed pairs finite path networks infinite networks called networks
Topology (electrical circuits)