RC (ResistorCapacitor) Circuits. AP Physics C


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1 (ResisorCapacior Circuis AP Physics C
2 Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED ( 0 and he curren hrough he resisor is zero. A swich (in red hen closes he circui by moving upwards. The uesion is: Wha happens o he curren and volage across he resisor and capacior as he capacior begins o charge as a funcion of ime? Which pah do you hink i akes? V C Time(s
3 Volage Across he Resisor  Iniially V Resisor ε If we assume he baery has NO inernal resisance, he volage across he resisor will be he EMF. (sec Afer a very long ime, V cap ε, as a resul he poenial difference beween hese wo poins will be ZERO. Therefore, here will be NO volage drop across he resisor afer he capacior charges. Noe: This is while he capacior is CHARGING.
4 Curren Across he Resisor  Iniially I max ε/r (sec Since he volage drop across he resisor decreases as he capacior charges, he curren across he resisor will reach ZERO afer a very long ime. Noe: This is while he capacior is CHARGING.
5 Volage Across he Capacior  Iniially V cap ε (sec As he capacior charges i evenually reaches he same volage as he baery or he EMF in his case afer a very long ime. This increase DOES NOT happen linearly. Noe: This is while he capacior is CHARGING.
6 Curren Across he Capacior  Iniially I max ε/r (sec Since he capacior is in SERIES wih he resisor he curren will decrease as he poenial difference beween i and he baery approaches zero. I is he poenial difference which drives he value for he curren. Noe: This is while he capacior is CHARGING.
7 Time Domain Behavior The graphs we have jus seen show us ha his process depends on he ime. Le s look hen a he UNITS of boh he resisance and capaciance. Uni for Resisance Ω Vols/Amps Uni for Capaciance Farad Coulombs/Vols Vols Coulombs R xc x Amps Vols Coulomb 1 Amp Sec Coulombs R xc Coulombs Seconds Coulombs Amps SECONDS!
8 The Time Consan I is clear, ha for a GIVEN value of "C, for any value of R i effecs he ime rae a which he capacior charges or discharges. Thus he PRODUCT of R and C produce wha is called he CIUIT Capaciive TIME CONSTANT. We use he Greek leer, Tau, for his ime consan. The uesion is: Wha exacly is he ime consan?
9 The Time Consan The ime consan is he ime ha i akes for he capacior o reach 63% of he EMF value during charging.
10 Charging Behavior Is here a funcion ha will allow us o calculae he volage a any given ime? ε Le s begin by using KVL V cap ε (sec We now have a firs order differenial euaion.
11 Charging funcion ε How do we solve his when we have 2 changing variables? To ge rid of he differenial we mus inegrae. To make i easier we mus ge our wo changing variables on differen sides of he euaion and inegrae each side respecively. Rearranging algebraically. Geing he common denominaor Separaing he numeraor from he denominaor, Cross muliplying. Since boh changing variables are on opposie side we can now inegrae.
12 Charging funcion 0 d Cε Cε ln( Cε C ε e Cε Cε Cε e 1 0 d ( Cε Cε e Cε (1 e ε However if we divide our funcion by a CONSTANT, in his case C, we ge our volage funcion. As i urns ou we have derived a funcion ha defines he CHARGE as a funcion of ime. ( C ε (1 e C C V ( ε (1 e
13 Le s es our funcion V V V ( ε (1 e (1 ε (1 e (1 ε (1 e ε ε 0.95ε 0.98ε 0.86ε 0.63ε Transien Sae Seady Sae V (2 0.86ε V V (3 (4 0.95ε 0.98ε Applying each ime consan produces he charging curve we see. For pracical purposes he capacior is considered fully charged afer 45 ime consans( seady sae. Before ha ime, i is in a ransien sae.
14 Charging Funcions ( Cε (1 e V ( ε (1 e V ( R ε (1 e R I ( Io (1 e o Likewise, he volage funcion can be divided by anoher consan, in his case, R, o derive he curren charging funcion. Now we have 3 funcions ha allows us o calculae he Charge, Volage, or Curren a any given ime while he capacior is charging.
15 Capacior Discharge Resisor s Volage Suppose now he swich moves downwards owards he oher erminal. This prevens he original EMF source o be a par of he circui. ε V Resisor A 0, he resisor ges maximum volage bu as he capacior canno keep is charge, he volage drop decreases. (sec
16 Capacior Discharge Resisor s Curren Similar o is charging graph, he curren hrough he resisor mus decrease as he volage drop decreases due o he loss of charge on he capacior. Iε/R I Resisor (sec
17 Capacior Discharge Capacior's Volage The discharging graph for he capacior is he same as ha of he resisor. There WILL be a ime delay due o he TIME CONSTANT of he circui. In his case, he ime consan is reached when he volage of he capacior is 37% of he EMF.
18 Capacior Discharge Capacior s Curren Similar o is charging graph, he curren hrough he capacior mus decrease as he volage drop decreases due o he loss of charge on he capacior. Iε/R I cap (sec
19 Discharging Funcions 0 IR V cap d R d C d R d C o 1 d d d 0 d Once again we sar wih KVL, however, he reason we sar wih ZERO is because he SOUE is now gone from he circui.
20 Discharging Funcions o 1 d 1 0 d ln( o o e ( Dividing by "C" hen "R" o e V ( I( ε e I o o e We now can calculae he charge, curren, or volage for any ime during he capaciors discharge.
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