Capacitors and inductors


 Frederick Bond
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1 Capaciors and inducors We coninue wih our analysis of linear circuis by inroducing wo new passive and linear elemens: he capacior and he inducor. All he mehods developed so far for he analysis of linear resisive circuis are applicable o circuis ha conain capaciors and inducors. Unlike he resisor which dissipaes energy, ideal capaciors and inducors sore energy raher han dissipaing i. Capacior: In boh digial and analog elecronic circuis a capacior is a fundamenal elemen. I enables he filering of signals and i provides a fundamenal memory elemen. The capacior is an elemen ha sores energy in an elecric field. The circui symbol and associaed elecrical variables for he capacior is shown on Figure. i C v Figure. Circui symbol for capacior The capacior may be modeled as wo conducing plaes separaed by a dielecric as shown on Figure 2. When a volage v is applied across he plaes, a charge q accumulaes on one plae and a charge q on he oher. insulaor plae of area A q and hickness s v E d q s Figure 2. Capacior model 6.07/22.07 Spring 2006, Chanioakis and Cory
2 If he plaes have an area A and are separaed by a disance d, he elecric field generaed across he plaes is and he volage across he capacior plaes is q E = (.) εα qd v= Ed = (.2) ε A The curren flowing ino he capacior is he rae of change of he charge across he dq capacior plaes i =. And hus we have, d dq d εa εa dv dv i = = v = = C (.3) d d d d d d The consan of proporionaliy C is referred o as he capaciance of he capacior. I is a funcion of he geomeric characerisics of he capacior plae separaion (d) and plae area (A) and by he permiiviy (ε) of he dielecric maerial beween he plaes. ε A C = (.4) d Capaciance represens he efficiency of charge sorage and i is measured in unis of Farads (F). The currenvolage relaionship of a capacior is dv i = C (.5) d The presence of ime in he characerisic equaion of he capacior inroduces new and exciing behavior of he circuis ha conain hem. Noe ha for DC (consan in ime) dv signals ( 0 d = ) he capacior acs as an open circui (i=0). Also noe he capacior does no like volage disconinuiies since ha would require ha he curren goes o infiniy which is no physically possible. If we inegrae Equaion (.5) over ime we have 6.07/22.07 Spring 2006, Chanioakis and Cory 2
3 dv id = C d (.6) d v= id C (.7) = id v(0) C 0 The consan of inegraion v(0) represens he volage of he capacior a ime =0. The presence of he consan of inegraion v(0) is he reason for he memory properies of he capacior. Le s now consider he circui shown on Figure 3 where a capacior of capaciance C is conneced o a ime varying volage source v(). i() v() C v Figure 3. Fundamenal capacior circui If he volage v() has he form Then he curren i() becomes v () = Acos( ω) (.8) dv i () = C d = CAωsin( ω) π = CωAcos ω 2 Therefore he curren going hrough a capacior and he volage across he capacior are 90 degrees ou of phase. I is said ha he curren leads he volage by 90 degrees. The general plo of he volage and curren of a capacior is shown on Figure 4. The curren leads he volage by 90 degrees. (.9) 6.07/22.07 Spring 2006, Chanioakis and Cory 3
4 Figure 4 If we ake he raio of he peak volage o he peak curren we obain he quaniy Xc = (.0) Cω Xc has he unis of Vols/Amperes or Ohms and hus i represens some ype of resisance. Noe ha as he frequency ω 0 he quaniy Xc goes o infiniy which implies ha he capacior resembles an open circui. Capaciors do like o pass curren a low frequencies As he frequency becomes very large ω he quaniy Xc goes o zero which implies ha he capacior resembles a shor circui. Capaciors like o pass curren a high frequencies Capaciors conneced in series and in parallel combine o an equivalen capaciance. Le s firs consider he parallel combinaion of capaciors as shown on Figure 5. Noe ha all capaciors have he same volage, v, across hem. i() i i2 i3 in v() v C C2 C3 Cn Figure 5. Parallel combinaion of capaciors. 6.07/22.07 Spring 2006, Chanioakis and Cory 4
5 By applying KCL we obain And since dv ik = Ck we have d i= i i2 i3 in (.) dv dv dv dv i = C C2 C3 Cn d d d d dv = C C2 C3 Cn Ceq d (.2) dv = Ceq d Capaciors conneced in parallel combine like resisors in series Nex le s look a he series combinaion of capaciors as shown on Figure 6. i() C C2 C3 Cn v v2 v3 vn v() Figure 6. Series combinaion of n capaciors. Now by applying KVL around he loop and using Equaion (.7) we have v= v v2 v3 vn = id () v(0) C C2 C3 Cn 0 Ceq = id () v(0) Ceq 0 (.3) Capaciors in series combine like resisors in parallel 6.07/22.07 Spring 2006, Chanioakis and Cory 5
6 By exension we can calculae he volage division rule for capaciors conneced in series. Here le s consider he case of only wo capaciors conneced in series as shown on Figure 7. i() v() v v2 C C2 Figure 7. Series combinaion of wo capaciors The same curren flows hrough boh capaciors and so he volages v and v2 across hem are given by: And KVL around he loop resuls in v = id C (.4) 0 v2 = id C2 (.5) 0 v () = id C C2 (.6) 0 Which in urn gives he volages v and v2 in erms of v and he capaciances: v C2 = v C C 2 v C 2 = v C C 2 (.7) (.8) Similarly in he parallel arrangemen of capaciors (Figure 8) he curren division rule is i C = i C C 2 i C2 2 = i C C 2 (.9) (.20) Assume here ha boh capaciors are iniially uncharged 6.07/22.07 Spring 2006, Chanioakis and Cory 6
7 i() i() v C i C2 i2 Figure 8. Parallel arrangemen of wo capaciors The insananeous power delivered o a capacior is P () = iv () () (.2) The energy sored in a capacior is he inegral of he insananeous power. Assuming ha = v( ) = 0 hen he energy sored he capacior had no charge across is plaes a [ ] in he capacior a ime is E () = P( τ) dτ = v( τ) i( τ) dτ dv( τ ) = v( τ ) C dτ dτ = Cv () 2 2 (.22) 6.07/22.07 Spring 2006, Chanioakis and Cory 7
8 Real Capaciors. If he dielecric maerial beween he plaes of a capacior has a finie resisiviy as compared o infinie resisiviy in he case of an ideal capacior hen here is going o be a small amoun of curren flowing beween he capacior plaes. In addiion here are lead resisance and plae effecs. In general he circui model of a nonideal capacior is shown on Figure 9 i C nonideal = i v C Rs Rp Figure 9. Circui of nonideal capacior The resisance Rp is ypically very large and i represens he resisance of he dielecric maerial. Resisance Rs is ypically small and i corresponds o he lead and plae resisance as well as resisance effecs due o he operaing condiions (for example signal frequency) In pracice we are concerned wih he in series resisance of a capacior called he Equivalen Series Resisance (ESR). ESR is a very imporan capacior characerisic and mus be aken ino consideraion in circui design. Therefore he nonideal capacior model of ineres o us is shown on i R(ESR) C Figure 0. Nonideal capacior wih series resisor. Typical values of ESR are in he mωω range. 6.07/22.07 Spring 2006, Chanioakis and Cory 8
9 A capacior sores energy in he form of an elecric field dv Currenvolage relaionship i = C, v id d = C In DC he capacior acs as an open circui The capaciance C represens he efficiency of soring charge. The uni of capaciance is he Farad (F). Farad=Coulomb/Vol 3 Typical capacior values are in he mf ( 0 2 F) o pf ( 0 F) The energy sored in a capacior is E = Cv 2 2 Large capaciors should always be sored wih shored leads. Example: A 47µF capacior is conneced o a volage which varies in ime as v ( ) = 20sin(200 π) vols. Calculae he curren i() hrough he capacior v() i() C v The curren is given by dv i = C d 6 d 6 = sin(200 π) = π cos(200 π) = 0.59cos(200 π) Amperes d 6.07/22.07 Spring 2006, Chanioakis and Cory 9
10 Example: Calculae he energy sored in he capacior of he circui o he righ under DC condiions. In order o calculae he energy sored in he capacior we mus deermine he volage across i and hen use Equaion (.22). k Ω 8 V uf 2k Ω We know ha under DC condiions he capacior appears as an open circui (no curren flowing hrough i). Therefore he corresponding circui is k Ω v 2k Ω 8 V And from he volage divider formed by he kω and he 2kΩ resisors he volage v is 2Vols. Therefore he energy sored in he capacior is Ec = Cv = = 72µJoule s /22.07 Spring 2006, Chanioakis and Cory 0
11 Example Calculae he energy sored in he capaciors of he following circui under DC condiions. C 50uF 0k Ω 25k Ω 0 V uf C2 C3 50k Ω 0uF Again DC condiions imply ha he capacior behaves like an open circui and he corresponding circui is C v 0 V 0k Ω C2 25k Ω C3 50k Ω v2 From his circui we see ha he volages v and v2 are boh equal o 0 Vols and hus he volage across capacior C is 0 Vols. Therefore he energy sored in he capaciors is: For capacior C: For capacior C2: For capacior C3: 0 Joules EC 2 = C2v = 0 0 = 50µJoules EC 3 = C3v = = 500µJoules /22.07 Spring 2006, Chanioakis and Cory
12 Inducors The inducor is a coil which sores energy in he magneic field Consider a wire of lengh l forming a loop of area A as shown on Figure. A curren i() is flowing hrough he wire as indicaed. This curren generaes a magneic field B which is equal o i () B () = µ (.23) l Where µ is he magneic permeabiliy of he maerial enclosed by he wire. i() B l Loop lengh A Area Figure. Curren loop for he calculaion of inducance The magneic flux, Φ, hrough he loop of area A is Φ = AB() Aµ = i () l = Li() Aµ Where we have defined L. l From Maxwell s equaions we know ha (.24) d v () d (.25) dli () = v () d (.26) And by aking L o be a consan we obain he currenvolage relaionship for his loop of wire also called an inducor. 6.07/22.07 Spring 2006, Chanioakis and Cory 2
13 di v = L (.27) d The parameer L is called he inducance of he inducor. I has he uni of Henry (H). The circui symbol and associaed elecrical variables for he inducor is shown on Figure 2 i L v Figure 2. Circui symbol of inducor. di For DC signals ( 0 d = ) he inducor acs as a shor circui (v=0). Also noe he inducor does no like curren disconinuiies since ha would require ha he volage across i goes o infiniy which is no physically possible. (We should keep his in mind when we design inducive devices. The curren hrough he inducor mus no be allowed o change insananeously.) If we inegrae Equaion (.27) over ime we have di vd = L d (.28) d i = vd L (.29) = vd i(0) L 0 The consan i(0) represens he curren hrough he inducor a ime =0. (Noe ha we have also assumed ha he curren a = was zero.) 6.07/22.07 Spring 2006, Chanioakis and Cory 3
14 Le s now consider he circui shown on Figure 3 where an inducor of inducance L is conneced o a ime varying curren source i(). i() i() v L Figure 3. Fundamenal inducor circui If we assume ha he curren i() has he form Then he volage v() becomes i () = I cos( ω) (.30) o di v () = L d = LI ωsin( ω) o π = LωIo cos ω 2 (.3) Therefore he curren going hrough an inducor and he volage across he inducor are 90 degrees ou of phase. Here he volage leads he curren by 90 degrees. The general plo of he volage and curren of an inducor is shown on Figure 4. Figure /22.07 Spring 2006, Chanioakis and Cory 4
15 Inducor conneced in series and in parallel combine o an equivalen inducance. Le s firs consider he parallel combinaion of inducors as shown on Figure 5. Noe ha all inducors have he same volage across hem. i() i i2 i3 in v() v L L2 L3 Ln By applying KCL we obain Figure 5. Parallel combinaion of inducors. And since i= i i2 i3 in (.32) ik = vd ik(0) Lk we have 0 i = vd i(0) vd i2(0) vd i3(0) vd in(0) L L2 L3 Ln = vd i(0) i2(0) i3(0) in(0) L L2 L3 Ln 0 i(0) Leq = vd i(0) Leq (.33) Inducors in parallel combine like resisors in parallel Nex le s look a he series combinaion of inducors as shown on Figure 6. i() L L2 L3 Ln v v2 v3 vn v() Figure 6. Series combinaion of inducors. 6.07/22.07 Spring 2006, Chanioakis and Cory 5
16 Now by applying KVL around he loop we have v= v v2 v3 vn di = L L2 L3 Ln (.34) Leq d di = Leq d Inducor in series combine like resisor in series The energy sored in an inducor is he inegral of he insananeous power delivered o he inducor. Assuming ha he inducor had no curren flowing hrough i a = i( ) = 0 hen he energy sored in he inducor a ime is [ ] E () = P( τ) dτ = v( τ) i( τ) dτ di( τ ) = L i ( τ ) dτ dτ = Li () 2 2 (.35) 6.07/22.07 Spring 2006, Chanioakis and Cory 6
17 Real Inducors. There are wo conribuions o he nonideal behavior of inducors.. The finie resisance of he wire used o wind he coil 2. The cross urn effecs which become imporan a high frequencies The nonideal inducor may hus be modeled as shown on Figure 7 i L v = v i L Rc resisance of coil (small value) Rf Frequency dependen urn o urn field effecs (imporan a high frequecnies) nonideal Figure 7. Circui momdel of nonideal inducor In addiion o he resisive nonidealiies of inducors here could also be capaciive effecs. These effecs usually become imporan a high frequencies. Unless saed oherwise, hese effecs will be negleced in ou analysis. A inducor sores energy in a magneic field di Currenvolage relaionship v = L, i vd d = L 2 The energy sored in an inducor is E = Li 2 In DC he inducor behaves like a shor circui The inducance L represens he efficiency of soring magneic flux. 6.07/22.07 Spring 2006, Chanioakis and Cory 7
18 Problems: Calculae he equivalen capaciance for he following arrangemens: Ceq F 00 F 5 F 2 F F Ceq C C C C 2C 2C 2C o infiniy Calculae he volage across each capacior and he energy sored in each capacior. 0 V 0uF 2uF 0uF 20uF 5uF In he circui below he curren source provides a curren of i = 0exp( 2 ) ma. Calculae he volage across each capacior and he energy sored in each capacior a ime =2 sec. v i() 0uF 20uF v2 5uF 6.07/22.07 Spring 2006, Chanioakis and Cory 8
19 6.07/22.07 Spring 2006, Chanioakis and Cory 9
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