Circuit Types. () i( t) ( )


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1 Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All volages and currens in a dc circui are consan. Capaciors ac like open circuis and inducors ac like shor circuis. Example: Deermine he curren, i, and volage, v( ), for his circui. Soluion: This is a dc circui so he capacior acs like an open circui. The capacior volage, v volage across ha open circui. The inducor acs like a shor circui. The inducor curren, is he curren in ha shor circui. Here s he circui afer replacing he capacior by an open circui and replacing he inducor by a shor circui:, is he i( ), Ohm s law gives and 30 i = = 2.5 A 8+ i( ) ( ) v = = 2.5 = 10 V 1
2 AC Circuis Idenifying feaures: o Sinusoidal inpus: he volages of independen volage sources and currens of independen curren sources are all sinusoidal a he same frequency. o The circui does no conain any swiches. All volages and currens in a dc circui are sinusoidal a he frequency of he sources. Capaciors do no ac like open circuis and inducors do no ac like shor circuis. Example: Deermine he curren, i, and volage, v( ), for his circui. Soluion: This is an ac circui so he capacior does no ac like an open circui and he inducor does no ac like a shor circui. Apply KCL a he op node of he Ω resisor o ge Apply KVL o he lef mesh o ge i ( ) v d = v d d 8i + i + v 30cos2 = 0 d We re no ready o solve hese equaions. We re considering his example now as a conras o he previous example. Capaciors and inducors don always ac like open and shor circuis. Laer we ll be able o show ha v.932cos( ) V = and i( ) = ( ) 2.757cos 2 36 A For now, noice ha even checking his soluion by subsiuing v( ) and Kirchhoff s laws equaions will require quie a bi of effor. i ino he 2
3 Swiched Circuis Idenifying feaure: The circui conains a swich. Open swiches ac like open circuis and closed swiches ac like shor circuis. Example: Deermine he volage, v, for his circui. Soluion: Before ime = 0, he swich is open. An open swich acs like an open circui. An open circui in parallel wih a resisor is equivalen o he resisor. (An open circui is equivalen o an infinie resisance and R = R.) Replacing he open swich by an open circui gives: This is a dc circui, so he capacior acs like an open circui: Using volage division: = 30 = 10 V when < ( ) v Le and le ) v (0 +) v(0 denoe he value of he volage immediaely before he swich closes a ime = 0 denoe he value of he capacior volage immediaely afer he swich closes. In 3
4 he absence of infinie currens, which are physically impossible, he capacior volage is coninuous, so v 0+ = v 0 = 10 V ( ) ( ) We will see ha afer he swich closes he capacior volage is given by 2 e 2 v = 15 5 e V when 0 The erm 5 ges smaller and smaller as ime goes on. Evenually i will be negligible compared o he oher erm. This observaion is imporan enough ha we have vocabulary ha helps us alk abou i 15 is he seady sae (par of he) response 2 e 5 is he ransien par of he response 2 e 15 5 is he complee response A dc circui is said o be a seady sae when all of is currens and volages are consan. An ac circui is said o be a seady sae when all of is currens and volages are sinusoidal a he frequency of he sources. Wih his new vocabulary, we can characerize dc and ac circuis more precisely: DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui is a seady sae. All volages and currens in a dc circui are consan. Capaciors ac like open circuis and inducors ac like shor circuis. AC Circuis Idenifying feaures: o Sinusoidal inpus: he volages of independen volage sources and currens of independen curren sources are all sinusoidal a he same frequency. o The circui is a seady sae. All volages and currens in a dc circui are sinusoidal a he frequency of he sources. Capaciors do no ac like open circuis and inducors do no ac like shor circuis.
5 Using his new vocabulary, he circui is a seady sae before he swich closes a ime = 0. Closing he swich disurbs circui. The circui is no a seady sae immediaely afer he swich closes. Evenually he disurbance dies ou and he circui reurns o a seady sae, bu probably a differen seady sae han before he swich closed. Afer he swich closes, he circui is no a seady sae so he capacior does no ac like an open circui. Le s reurn o he example. Afer ime = 0, he swich is closed. A closed swich acs like a shor circui. A shor circui in parallel wih a resisor is equivalen o he a shor circui. (A shor circui is equivalen o an zero resisance and 0 R = 0.) Replacing he closed swich by a shor circui gives: Apply KCL a he op node of he capacior o ge 30 v v d d = v v + 2v = 30 d d Solving his differenial equaion (using he iniial condiion v ( 0+ ) = 10 V) gives In summary, we have 2 v = 15 5 e V when e V 0 = 2 v 5
6 One more observaion, evenually he ransien par of he response dies ou and he circui reaches seady sae. I is, once again, a dc circui. The capacior acs like an open circui, so we have Consequenly, v = 15 V when As a final summary, here are he circuis ha we considered, ogeher wih he informaion ha we obained from each circui: ( ) = 10 V when < 0 v ( ) v( ) v 0+ = 0 = 10 V d v d v + 2 = 30 2 ( ) v = 15 5 e V when 0 ( ) 15 V when v = 6
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