0//06 lon0t438a.pptx ESSON 0: THE APAE TANSFOM ET 438a Automatic ontrol Sytm Tchnology arning Objctiv Aftr thi prntation you will b abl to: Explain how th aplac tranform rlat to th tranint and inuoidal rpon of a ytm. onvrt tim function into th aplac domain. U aplac tranform to convrt diffrntial quation into algbraic quation. Tak th Invr aplac tranform and find th tim rpon of a ytm. U Initial and Final Valu Thorm to find th tadytat rpon of a ytm. lon0t438a.pptx
0//06 Th aplac Tranform 3 aplac tranform convrt tim domain problm into function of a complx variabl,, that i rlatd to th frquncy rpon of th ytm aplac Tranform Tim domain lon0t438a.pptx omplx frquncy 4 omplx Frquncy and Th aplac Tranform a. omplx frquncy combin tranint rpon with inuoidal tady-tat rpon to gt total rpon of ytm to input j omplx Frquncy b. a.) Highr frquncy b.) Fatr tim contant = xponntial dcay/incra contant that i rlatd to tim contant of ytm tranint rpon. = / = in circuit analyi t t Exponntia lly incraing function Exponntia lly dcraing function lon0t438a.pptx ovr tim ovr tim
0//06 5 Sinuoidal pon From omplx Frquncy Th radian frquncy j = jpf am frquncy ud in phaor analyi aplac rlatd to in rpon through Eulr' Idntity. Eulr' rlat complx xponntial to in and coin tim function jt jt co( t) jin( t) co( t) jin( t) Adding and ubtracting th abov rlationhip giv th xponntial form of in and coin lon0t438a.pptx Exponntial Form of Sinuoid 6 Add th idntiti Subtract th idntiti jt jt co( t) jin( t) co( t) jin( t) + jt jt co( t) jin( t) co( t) jin( t) - jt jt co( t) jt jt jin( t) jt jt co( t) jt j jt in( t) Exponntial form of oin Exponntial form of Sin Sinc t ( j)t t jt aplac can giv complt rpon: dc tranint and tady-tat inuoidal lon0t438a.pptx 3
0//06 Baic aplac Tranform Pair 7 Tim Domain Function aplac Domain Function t (t) Impul (t) u Unit Stp at at in(t) co(t) inar ramp (lop ) a a Not: tim function multiplid by contant giv aplac function multiplid by contant Exampl: 5u (t) 3in( 4t) 5 4 3 4 6 lon0t438a.pptx aplac Tranform Exampl 8 Match th following tim function to corrct aplac domain function uing th tranform pair. a.) 0 t 3 5 3 b.) c.) d.) t 5t at t aplac tabl 3. txtbook ( ) 3 5 4 0 ( a) 4 5.) 3co(t) lon0t438a.pptx 4
0//06 aplac Thorm 9 aplac of an unknown function (t) F () f apitaliz unknown function nam plac t with aplac Oprator Symbol Exampl (t) I () i (t) V () v inarity of tranform - can multiply by contant f (t) F () and f (t) F () If a f (t) bf (t) a F () b F () Thn lon0t438a.pptx 0 aplac Tranform of alculu Oprator aplac Tranform turn drivativ into multiplication by If Thn d dt f (t) F () f(t) F () f(0) Subtract any non-zro initial condition For highr ordr drivativ 0 initial condition rduc formula to d dt f(t) ( F () f(0)) d dt f (0) d dt f(t) F () lon0t438a.pptx 5
0//06 aplac Tranform of alculu Oprator aplac turn intgration into diviion by If f (t) F () Thn f (t) dt F () Exampl from circuit analyi: apacitor voltag v(t) v(t) i(t) dt V () I lon0t438a.pptx i(t) dt () I () Mor Exampl From ircuit Analyi Find th aplac rlationhip for inductor voltag d v(t) i(t) dt d v(t) i(t) dt V () I () aplac rlationhip for ritor voltag v (t) i (t) v (t) i (t) V () I () lon0t438a.pptx 6
0//06 aplac Tranform and Impdanc 3 mmbr phaor analyi i only valid for inuoidal tady-tat. Turn ac analyi into an analyi imilar to th dc. (Ohm' law) itanc Inductiv actanc apacitiv actanc X X j j j pf j 90 - j 90 j Sinc aplac variabl rprnt frquncy, it' poibl to rplac j with and with j. If i rplacd with j, analyi rvrt to phaor W can find th frquncy rpon of a dynamic ytm by convrting diffrntial quation into aplac domain and rplacing with j. Swping frquncy produc Bod plot of ytm. lon0t438a.pptx 4 aplac Tranform and Impdanc aplac Impdanc (Ohm aw) Impdanc (Phaor) Inductor apacitor itor V () I () V () I () V () I () Inductor apacitor itor V (j) j I (j) V (j) j I (j) V (j) I (j) lon0t438a.pptx 7
0//06 5 aplac prntation of OP AMP ircuit Find I/O rlationhip of intgrator uing aplac rlationhip I in (t) I f (t) U OP AMP thory and olv. No I ntr invrting nod and V + =V - =0 du to ground connction. U K at invrting nod K Subtitut into K quation lon0t438a.pptx 6 aplac prntation of OP AMP ircuit Intgrat both id of abov quation to gt V o (t). Intgration i invr of diffrntiation Tak aplac of Equation lon0t438a.pptx an u gnralizd gain formula of invrting OP AMP and aplac Impdanc 8
0//06 7 aplac prntation of OP AMP ircuit Exampl 0-: Find th input/output rlationhip for th circuit hown blow. Gnralizd gain formula U aplac impdanc rlationhip to find gain For inductor lon0t438a.pptx 8 Exampl 0- Solution () Subtitut into gnralizd gain formula And intgrator action Diviion by man Intgration in tim lon0t438a.pptx 9
0//06 9 ET 438a Automatic ontrol Sytm Tchnology End on 0: Th aplac Tranform lon0t438a.pptx 0