We leave the regime of just working with DC currents and open our view to electronics behaviour over time, using alternating currents (AC).

Transcription

1 Analogue Elecronics 3: A ircuis capaciors We leave he regime of jus working wih D currens and open our view o elecronics behaviour over ime, using alernaing currens (A). apaciors an analogue componen wih memory: A capacior is a break in a circui. In a D circui (saic) you only know his as a swich, wih limied funcionaliy. However, in circuis wih changing he volages and currens i becomes exremely useful, because a capacior also can sore charge and can release he charge again. Using his abiliy capaciors are used as par of circuis which: respond o changes (differeniae), perform averaging (inegrae), selec frequency ranges (filer). Before we jump ino he dynamics we briefly will look again a he saic properies of capaciors. You have already seen his in Physics 2A and probably elsewhere. Saic properies of he capacior: (H&H, 1.12, p. 20) The symbol for a capacior is wo parallel lines wih a gap beween hem. This represens he simples sor of capacior: a pair of parallel plaes wih a small gap beween hem. Generally he same symbol is used for all capacior designs. Only if hey are sensiive o polariy, like elecrolye capaciies are, hen a + is added o he side which has o be wired o he posiive poenial o preven he capacior from becoming conducive or being desroyed ( o blow up in laboraory slang, as i goes wih a bang). Or arrows may be added across he symbol for adjusable capaciies. A capacior can sore a fixed amoun of charge a a paricular volage: Q = (unis: =oulomb, F=Farad and =ol) If you place a fixed volage,, across a capacior curren will flow unil here is a charge of +Q on one surface and Q on he oher surface. Obviously several capaciors can be combined in he same circui. The rules for combining he capaciances are exacly opposie o hose for combining resisances: In series: 1 oal = In parallel: oal = A Farad is a large uni, as i needs large surfaces and small gaps o sore los of charge. In elecric circuis usually he following ranges of capaciies are in use: pf, nf and a mos µf. You may need o combine several capaciors o ge he values you require in he labs. Also noe ha he producion accuracy of capaciies is significanly worse han for oher elemens.

2 Dynamic properies of he capacior: (H&H, 1.13, p. 23) The ineresing properies of capaciors occur in response o volages or currens ha change as a funcion of ime. One way o summarize he behaviour is o say: he larger he curren he faser he volage across he capacior changes. Or mahemaically: d Q = d wih he capaciy consan in ime his simplifies o: I = d (unis: A=Ampere, F=Farad and /s=ola/sec) A curren source ha provides a seady curren o a capacior will generae a seadily increasing volage across he gap, like displayed below (noe again he symbols for he connecion o supply and ground). I + cap cap large ime small The sor of volage ramp ha you can see in he graph is he signaure of a capacior driven by a curren source. The sraigh lines shown here for he volage change wih ime are only approximaions for small imes wha small means will be qualified below. Discharging a capacior hrough a resisor: Imagine ha you could collec posiive and negaive charges on opposie surfaces of a capacior and hen connec i o a resisor. You would have a circui ha looks like his: In pracice you would do his by having a second half of a circui on he righ wih a baery ha you would disconnec wih one swich and hen connec he resisor wih a second swich. This caroon circui is described by: d = I =

3 The soluion of his differenial equaion is: = Ae I describes an exponenial decay in he volage,, wih ime,, across he resisor. The value of he produc is he characerisic ime consan for he decay of he iniial poenial, A, across he capaciy. Tha means ha by choosing he values of and you can conrol he ime dependence of he circui! E.g. a ime consan of =2µs could be achieved by choosing =50Ω and =40nF = % As a reminder: you should know ha in all exponenial decays he ampliude (here he volage across he capaciy) falls by a facor of 1-1/e = 0.63 (or o a remaining facor of 1/e = 0.37) during each ime consan. E.g. afer 5 ime consans he iniial signal will be diminished o 0.7% of is original size. harging a capacior hrough a resisor: This is marginally more complicaed, ha is why we are looking a i second. In he circui below we have baeries ha can be conneced via a swich o a resisor and a capacior. You may noice ha he arrangemen of resisor and capacior is he same as ha of he wo resisors in a volage divider. i We are going o deermine he volage across he capacior as a funcion of ime since he swich was closed (=0). This circui is described by: I = d = i

4 The soluion of his differenial equaion is: = i 1 e Once he swich is closed he volage i is spanning across boh he resisor and capacior ogeher. Wha changes as a funcion of ime is he proporion of he volage drop which is across he wo individual componens. A he momen ha he swich is closed he volage across he capaciy is zero, =0, and he enire volage drop, i, appears across he resisor. As he capacior begins o charge he balance changes unil =i, ha is all of he volage drop appears across he capacior. The ime dependence of his change is again conrolled by he characerisic ime consan. The change in as a funcion of ime is shown on he graph below. =0 is he momen when he swich is closed. In one =1* he capacior has charged up o 63% of is capaciy. In =5*s i has charged up o larger han 99% % 86% 95% 98% 99% small I large I We have now seen he basics operaion: discharging and charging of a capacior. Boh are conrolled by he same ime consan () which can be chosen by selecing componens. Noe ha he exponenial curve can be approximaed by a linear descripion for ime scales small wih respec o he ime consan. Nex we are going o ge a lile more serious, and look a how capaciors can be used o perform calculus operaions.

5 Differeniaors: (H&H, 1.14, p. 25) The defining equaion for a curren hrough a capacior involves a derivaive: I = d Therefore i should be no surprise o discover ha hese componens can hen be harnessed o carry ou calculus operaions. egard he following arrangemen of a uni, in he configuraion of a differeniaor, beween a ime dependen inpu volage signal, in(), and oupu volage signal, (): in () () The volage across is (in ) so: I = d ( in ) The resisor,, acs as a curren sink for he oupu circui: Togeher we have: I = = d ( in ) in is no longer jus a saic volage from a baery. Insead i may be any ime dependen volage which we migh wan o differeniae. The oupu volage as a funcion of ime is: () = d in d Tha is no exacly he derivaive of he inpu signal, bu also depends on is own derivaive. This is like he oupu signal has gained some ineria i canno abruply change is rend anymore. To ease calculaion we can use an approximaion which ofen applies: if he produc is very small hen he capacior will respond quickly and: d d in Thus one can approximae above equaion by neglecing he second erm: () d in

6 This approximaion says: when he ime consan of he elemen is much shorer han he variaion of he inpu volage, in, hen he curren hrough he resisor will be proporional o he change of he inpu volage, wih he capaciy,, being he proporionaliy facor. Below are skeches of he inpu and oupu in boh he ideal case, where is much smaller han he rae of change in he inpu, and in he case where is oo large for he curren o closely follow he rae of change. In he second case he charging and discharging of he capacior smoohes he oupu and makes i resemble he derivaive less. in Square wave Ideal Oupu when is oo large In a few lecures ime, you will see how his performance can be improved on via he use of operaional amplifiers (op-amps), i.e. acive raher han passive componens. Inegraors: (H&H, 1.15, p. 26) Inegraion can also be approximaed in an analogous manner. The relevan circui is picured below. ompared o he differeniaor he inegraor circui has he posiions of he resisor and he capacior swiched: in () () This ime i is he volage across which is (in ) so: I = d = in The capaciy acs here as a charge buffer. Wha we are looking for is o find is he inegral of in(). For ha we need o make a similar approximaion o ha employed previously: if he produc is very large hen he capacior will respond slowly and: in

7 Thus one can approximae above equaion by neglecing he second erm: d in This approximaion says: when he ime consan of he elemen is much larger han he variaion of he inpu volage, in, hen he curren hrough he resisor will be proporional o he value of he inpu volage, wih he inverse capaciy, 1/, being he proporionaliy facor. We can ransform his ino he following inegral: () 1 in() + D If he inpu was a curren I() raher han a volage hen he oupu volage would give an exac inegral via: I = d In oher words: pu a large resisor in series wih a volage source and you will have an approximaion o a curren source. Add a large capaciy o ground o collec he charge and he volage across he capaciy will approximaely be he inegral over he inpu volage, in. Of course, he capaciy will charge up using he exponenial law, as discussed above. Thus, he inegraor only will respond approximaely linearly o he inpu signal for a shor ime scale compared o he ime consan, i.e. a an early poin on he charging up curve discussed before. For inegraion over longer ime scales sauraion effecs will become visible. The naure of he approximaion is picured below. in ou ou In he op graph a square wave inpu, in, is overlaid on he oupu. For a square wave inpu he inegral is a ramp. The arrow o he lower graph shows he naure of he approximaion. If is large hen only he early par of he exponenial charge-up is seen. This early par of he charging curve approximaes a ramp and hence is he inegral of he inpu square wave.

8 In summary: For a differeniaor, we needed fas response, i.e. a small ime consan. For an inegraor we need a slow response, i.e. a large ime consan. ule of humb: A capaciy in series wih he supply wih he resisor connecing as curren sink o ground will block D currens, bu he oupu will reac on changes of he inpu. A resisor in series wih he supply wih he capaciy connecing as a charge buffer o ground will smooh ou fas changes, bu he oupu will reac o slow changes. Properies of an inducor: (H&H, 1.16, p. 28) For compleeness inducors need o be menioned here. Bu hey will remain a bi like he fifh wheel on he car for he res of his lecure series, only employed when needed. An inducor is essenially a wire arranged in he form of a coil, or an elemen which is acing like one. While capaciors sore energy in he elecric field which is induced beween is surfaces, inducors sore energy in he magneic field which is induced in he cenre of he curled up curren. Dynamic properies: In an inducance, L, volage is generaed proporional o he rae of change of he curren in a circui: = L di (unis: =ol, H=Henry and A/s=Ampere/sec) Self inducance in a single circui and muual inducance beween wo circuis, mediaed by a muual magneic field, are disinguished. Essenially inducors ac he opposie way han capaciies. However, hey are rarely used as componens for elecrical circuis in pracice, and if, usually only as componens o efficienly damp oscillaions. Inducors end o be more bulky, more expensive and hey suffer more performance problems han capaciors. Noe however ha oher componens, like wires, resisors and even capaciies, also bear some inducive behaviour in real-world applicaions. Bu he inducive couplings usually are so iny ha hey can safely be negleced and seldom are menioned a all. Only he capaciive couplings of he elemens usually are aken ino accoun, if hey become significan. Self and muual inducance may become a significan correcion for he high frequency behaviour of racks on prined circui boards or in cables (above ~100MHz or 1GHz). Modern programs o design he layou of such racks ake inducive couplings ino accoun (as an addiional correcion o he ypically significanly larger capaciive couplings). Muual inducance becomes he main ool of rade in ransformers, bu hese are no covered in his lecure.