RLC Transient Response

Transcription

1 . Introuction RC Transint Rspons Th stunt will analyz an RC circuit. A unit stp input will xcit this circuit, proucing a transint voltag rspons across all circuit lmnts. Ths rsponss will b analyz by thory, simulation an xprimntal rsults. Th primary rspons proprtis of concrn ar initial valu an final voltag valus along with voltag masurmnts at intrmiat stps. Equations that govrn slct componnt transint rsponss will b comput picwis. To construct ths quations th stunt will n to trmin th amping cofficint, natural ampning frquncy, initial capacitor voltag an final capacitor voltag. Knowing ths funamntal RC circuit proprtis, th stunt will b abl to trmin th roots of th circuit s charactristic quation, basic circuit rspons, an corrsponing transint quation. Onc all ths pics ar brought togthr to form a transint rspons quation, th stunt will comput th xpct voltag at prioic tim intrvals. Th circuit will b simulat by th stunt an thn construct, whras oscilloscop masurmnts will b takn at intrvals corrsponing to thos us in thortical calculations.. Backgroun A sris RC circuit may b mol as a scon orr iffrntial quation. Fining th solution to this scon orr quation involvs fining th roots of its charactristic R quation. Th charactristic quation moling a sris RC is s s =. C This quation may b writtn as s αs ω = with th following rprsntations: R Damping Factor = α = Unamp Natural Frquncy = Charactristic Equation Roots = s, = α ± α ω ω = Knowing th abov RC circuit proprtis along with initial an final transint valus for voltag or currnt thn nabls th transint capacitor voltag or transint inuctor currnt to b calculat. This is on using th following quations (whr infinity in this cas will rfr to th valu whn th circuit rachs stay-stat bfor changing again):.. Transint Capacitor Voltag for a Unit Stp Input to a Sris RC Circuit Ovramp: α > ω, v C (t) = v C ( ) A S t A S t V Critically Damp: α = ω, v C (t) = v C ( ) (A A t) -αt Unramp: α < ω, v C (t) = v C ( ) (A cos ω t A sin ω t) -αt V * *Not th iffrnc btwn ω o an ω in th quations. V C ECE Dpartmnt Pag April 9/, 7

2 .. Transint Inuctor Currnt for a Unit Stp Input to a Sris RC Circuit Ovramp: α > ω, i (t) = i ( ) B S t B S t A Critically Damp: α = ω, i (t) = i ( ) (B B t) -αt A Unramp: α < ω, i (t) = i ( ) (B cos ω t B sin ω t) -αt A whr th amp natural frquncy is xprss as. In orr to calculat th initial conitions, th quation rlating th voltag an currnt through an inuctor will also b n. Sinc th two lmnts ar in sris, this can b rwrittn [ VC ()] I () as: = t C 3. Prlab Th prlab is brokn own into a st of stps. For ach stp, you shoul rcor your answr in th appropriat tabl in th Data Entry sction at th n of this ocumnt. Also, obtain your lab instructor s signatur bfor procing to Sction Ovramp voltag transint rspons of capacitor in RC circuit v R Figur : RC Circuit with kω Rsistor A. Suppos th RC circuit in Figur has componnt valus as isplay in th figur. Assum th function gnrator proucs a squar wav with a pak-to-pak amplitu of -5 to 5 volts, an a frquncy of 5 Hz. Comput th amping factor, α, an th unamp natural frquncy, ω. Dtrmin th initial voltag across th capacitor, v C ( ), an th final voltag across th capacitor, v C ( ). Comput th charactristic roots, s,, utilizing α an ω, an stat th typ of amping. B. Th amping factor α shoul b gratr than th unamp natural frquncy ω, thus th circuit is ovramp an th capacitor s voltag transint can b xprss by th following quation with th two unknowns A & A : ECE Dpartmnt Pag April 9/, 7

3 v C (t) = v C ( ) A S t A S t Volts Evaluating at tim, th following rlationship is form: v C () = = v C ( ) A A Display all intrmiat stps laing to a solution for A A. C. Anothr quation is rquir to solv for both unknowns, thus th rivativ of th capacitor voltag transint quation is takn. [ V C ] t = s A S t s A S t Again, valuating at tim, th following rlationship is form: [ V C ( )] t = i () = = s A s A C Display all intrmiat stps laing to a solution for A & A. D. Having solv for th two unknown variabls, writ th full quation for v C (t). Evaluat v C (.5 ms), v C ( ms) an v C ( ms) 3.. Unramp voltag transint rspons of capacitor in RC circuit v R Figur : RC Circuit with Ω Rsistor E. Comput th amping factor, α, an th unamp natural frquncy, ω, for th circuit in Figur. Dtrmin th initial voltag across th capacitor v C ( ) an th final voltag across th capacitor v C ( ). Comput th charactristic roots, s,, utilizing α an ω, an stat th typ of amping. ECE Dpartmnt Pag 3 April 9/, 7

4 F. Th amping factor, α, shoul b lss than th unamp natural frquncy, ω. Thus, th circuit is unramp an th capacitor s transint voltag can b xprss by th following quation with th two unknowns, B & B. v C (t) = v C ( ) (B cos ω t B sin ω t) -αt Volts Evaluating at tim, th following rlationship is form: v C () = = v C ( ) (B cos() B sin()) Volts Display all intrmiat stps laing to a solution for B. G. Anothr quation is rquir to solv for B. Thus, th rivativ of th capacitor voltag transint quation is takn. [ V C ] = (B cos ω t B sin ω t) (-α -αt ) t (-B ω sin ω t B ω cos ω t ) ( -αt ) Again, valuating at tim, th following rlationship is form: [ V C ( )] i ( ) = = = (B cos ω t B sin ω t) (-α -αt ) t C (-B ω sin ω t B ω cos ω t ) ( -αt ) Display all intrmiat stps laing to a solution for B & B. H. Having solv for th two unknown variabls, writ th full quation for v C (t). Evaluat v C (.5 ms), v C ( ms) an v C ( ms) Unramp voltag transint rspons of rsistor in RC circuit Suppos w want to fin th voltag transint rspons of th Ω rsistor in Figur. Th simplst way to solv for th rsistor voltag transint is to fin th transint circuit currnt an multiply by th rsistor s rsistanc. Sinc w ar analyzing a sris circuit, w will fin th transint inuctor currnt an multiply by Ω. I. Comput th amping factor, α, an th unamp natural frquncy, ω. Dtrmin th initial currnt through th rsistor, i R ( ) (= i ( )), an thn convrt this valu to th initial voltag, v R ( ), using Ohm s aw. Similarly, th final voltag across th rsistor in stay-stat, i R ( ) (= i ( )). Comput th charactristic roots, s, utilizing α an ω, an stat th typ of amping., ECE Dpartmnt Pag 4 April 9/, 7

5 J. Th amping factor, α, shoul b lss than th unamp natural frquncy, ω. Thus, th circuit is unramp an th inuctor s transint currnt can b xprss by th following quation with th two unknowns C & C : i = i ( C cosωt C sinωt) Amps Evaluating at tim, th following rlationship is form: i ( ) = = i ( C cos C sin ) Amps Display all intrmiat stps laing to a solution for C. K. Anothr quation is rquir to solv for C. Thus, th rivativ of th inuctor currnt transint quation is takn: [ i ] = ( C t cosω t C sinω t)( α Evaluating at tim, th following rlationship is form: ) ( C ω sinω t C ω cosω t)( [ i ] v ( ) = = = ( C cos C sin )( α ) ( Cω sin Cω cos )( ) t Display all intrmiat stps laing to a solution for C & C.. Having solv for th two unknown variabls, writ th full quation for v R (t). Rmmbr that you n to multiply v R (t) by th Ω. Evaluat v R (.5 ms), v R ( ms), an v R ( ms). 4. Exprimntal Procur During th xprimntal procur, mak sur to fill in th appropriat portions of th Tabls -3 in th Data Entry sction an obtain your lab instructor s signatur upon compltion of th sction (showing scrn shots to him/hr whr appropriat). 4.. Equipmnt Tktronix TDS 3B Digital Phosphor Oscilloscop Agilnt 33A Wavform Gnrator Rsistors, Inuctors, Capacitors as N 4.. Ovramp voltag transint rspons of capacitor in RC circuit Construct th circuit in Figur. St th function gnrator to prouc a squar wav with a pak-to-pak amplitu of -5V to 5V. Slct th prio larg nough for th ) ECE Dpartmnt Pag 5 April 9/, 7

6 circuit to rach stay-stat (.g. f 4-6 Hz). Using th horizontal bars, masur th initial capacitor voltag, v C ( ), th final capacitor voltag, v C ( ), an th capacitor voltag at.5 ms, ms an ms. Also, scrib th typ of amping you obsrv. Inclu a scrnshot of th circuit rspons in your lab writ-up Unramp voltag transint rspons of capacitor in RC circuit Construct th circuit in Figur. Th function gnrator shoul again mol a stp of sufficint prio as in Subsction 4.. Using th horizontal bars, masur th initial capacitor voltag v C ( ), th final capacitor voltag, v C ( ), an th capacitor voltag at.5 ms, ms an ms. Also, scrib th typ of amping you obsrv. Inclu a scrnshot of th circuit rspons in your lab writ-up Unramp voltag transint rspons of rsistor in RC circuit Using th horizontal bars, masur th initial rsistor voltag, v R ( ), th final rsistor voltag, v R ( ), an th rsistor voltag at.5 ms, ms an ms. Also, scrib th typ of amping you obsrv. Inclu a scrnshot of th circuit rspons in your lab writ-up. 5. Simulat Procur Gnrat a computr simulation moling th capacitor voltag transint of th RC circuit for Figurs an. Masur th initial capacitor voltag, v C ( ), th final capacitor voltag, v C ( ), an th capacitor voltag at.5 ms, ms, an ms. Rcor your valus in th appropriat sctions of Tabls an 3. Inclu a scrnshot of th schmatic an a scrnshot of th transint analysis an compar any iffrnc btwn your pr-lab, simulation, an xprimntal rsults in your lab writ-up for all fiv valus in ach of th thr scnarios (incluing th rsistor voltag scnario scrib blow). Gnrat a computr simulation moling th rsistor voltag transint of th RC circuit in Figur. Masur th initial rsistor voltag, v R ( ), th final rsistor voltag, v R ( ), an th rsistor voltag at.5 ms, ms an ms. Rcor your valus in th appropriat sctions of Tabl. 6. Conclusions This conclus th lab. Mak sur to rturn all componnts to thir appropriat bins. Th writ-up shoul inclu all livrabls an xplanations ask for in th procur as wll as th Data Entry an ab Instructor Signatur Pag with all rcor numbrs, quations, tc. ECE Dpartmnt Pag 6 April 9/, 7

7 Prlab Sction Data Entry an ab Instructor Signatur Pag(s) Attach this/ths pag(s) to your writ-up. Tabl : Ovramp RC Circuit Quantity Calculat Valu Masur, Simulat Valu(s) A α N/A A ω N/A A Typ of Damping A S, N/A A v c ( ) A B v c ( ) v c ( ) = st st vc = vc A A B C A A = v c ( )/t = ( t ) vc t st st = s A s A C A = A = D D D D v c (t) v c (.5 ms) v c ( ms) v c ( ms) ECE Dpartmnt Pag 7 April 9/, 7

8 Prlab Sction Tabl : Unramp RC Circuit Quantity Calculat Valu Masur, Simulat Valu(s) E α N/A E ω N/A E Typ of Damping E S, N/A E ω N/A E v c ( ) E F v c ( ) v c ( ) = v = ω sin ω t) c vc ( B cos t B F G B = vc = ( B cos ωt B sin ωt)( α t ω B sinω t ω B cosω t) ( αt ) v c ( )/t = G B = B = H H H H v c (t) v c (.5 ms) v c ( ms) v c ( ms) ECE Dpartmnt Pag 8 April 9/, 7

9 Prlab Sction Tabl 3: Unramp RC Circuit, Rsistor Voltag Quantity Calculat Valu(s) Masur, Simulat Valu(s) I α, ω N/A I Typ of Damping I S, N/A I ω N/A I v R ( ) I J v R ( ) i = ω sin ω t) ir = i ( C cos t C i () = J C = K i ( ) v ( ) = = ( C cos C sin )( α ) t ( ω C C sin ω cos ) K v ( ) = C = C = v R (t) v R (.5 ms) v R ( ms) v R ( ms) Prlab Signatur Exprimntal Signatur ECE Dpartmnt Pag 9 April 9/, 7