foster foster important fields electrical network analysis synthesis states passive terminal port network strictly increases frequency easily seen individually increase frequency basis proof passive networks generally constructed proof presented principle published earlier foster colleagues extended encompassing concept consequence foster poles alternate frequency foster property develop canonical forms realising networks foster work important starting point development network synthesis possible construct networks active components generate equivalent negative negative example circuit explanation imaginary complex electrical possess opposite sign frequency dependent specification network passive implies energy sources passive network network consequently consist entirely purely imaginary number zero real foster applies equally network imaginary passive port increases frequency result easily proved formula formula formula imaginary unit given formula formula increasing frequency decreasing consequently increasing proved increasing case network theory principle procedure applies equally reflecting principle electric networks convenient circumstances use concept mean mathematics carried units desired specific example foster stated general form foster form imaginary passive port strictly increases frequency foster general particular applies distributed element networks foster formulated terms applicable microwave frequencies lower frequencies examples following examples illustrate number simple circuits given formula formula formula angular frequency formula inspection seen increasing frequency given formula formula formula increasing frequency function identical function vice versa general result dual function foster follow foster series circuit series circuit sum formula low frequencies dominated large negative increases zero magnitude smaller passes zero point equal frequency continues increase progressively dominant parallel circuit parallel circuit dual series circuit function form function series circuit formula function formula low frequencies dominated small positive increases pole anti frequency equal opposite cancel past pole large negative increasing zero dominated poles consequence foster poles passive function alternate frequency increases passing pole function negative obliged pass zero reaching pole increasing poles function completely determine frequency characteristics foster network foster networks identical poles equivalent circuits sense functions identical scaling factor difference elements multiplied scaling factor shape functions identical consequence foster phase increase frequency consequently plot foster function smith chart travel chart clockwise direction increasing frequency port passive consisting elements distributed element circuit represented rational function formula formula formula real positive formula transform variable replaced formula dealing steady state signals follows fact elements simple rational functions combination rational functions results rational function referred driving point place network external circuit connected drives signal paper foster describes rational function realised ways foster form consists number series connected parallel circuits foster second form driving point consists number parallel connected series circuits driving point means unique foster advantage poles directly associated particular circuit known ladder filter design networks foster network passive active network containing power source obey foster called networks particular circuits containing positive feedback declines frequency example possible create negative negative circuits circuits function phase like positive negative slope frequency interest accomplish tasks foster network example usual passive foster matching networks match antenna transmission line frequencies limits antenna non foster network match antenna continuous band frequencies allow creation compact wide violating limit practical non foster networks active area research history developed ongoing investigations improved filters telephone applications work commercially important large sums money saved increasing number telephone conversations carried line published campbell proof great use immediately filter design appears prominently proof landmark paper summarised state art filter design time foster published paper following year included canonical forms germany importance foster work foundation network synthesis innovations extension foster work element kind networks discovering interested finding necessary sufficient condition rational port network function condition known positive real function reverse problem networks equivalent function important problems network theory filter design foster networks networks
Foster's reactance theorem