Part 7: apacitanc And apacitors 7. Elctric harg And Elctric Filds onsidr a pair of flat, conducting plats, arrangd paralll to ach othr (as in figur 7.) and sparatd by an insulator, which may simply b air. Each plat, as an lctrical conductor, will contain larg numbrs of mobil ngativly chargd lctrons. If th plats ar connctd to a D supply ngativ lctrons will b attractd from th uppr plat to th positiv pol of th supply, but for vry lctron that dos this anothr must lav th ngativ pol of th supply and mov to th bottom plat. Th uppr plat will bcom positivly chargd owing to its shortag of lctrons, whras th surplus lctrons in th lowr plat will giv it a ngativ charg. Th diffrnc in th polarity of charg btwn th plats mans that a potntial diffrnc (PD) xists btwn thm, th flow of lctrons dying away and casing whn th PD btwn th plats is th sam as th supply voltag. Onc this has occurrd th supply can no longr supply nough nrgy to rmov lctrons from th uppr plat or push lctrons onto th lowr plat and th plats ar said to b 'chargd'. Figur 7.: A chargd plat systm. An lctric fild xists btwn th two chargd plats. Lik a magntic fild, an lctric fild can Figur 7.: Th lctric fild btwn two chargd plats. not b sn but can b dscribd in trms of a charg particl moving though it. Also, lik magntic filds, lctric filds ar imagind as lins of lctric flux. If th plats ar disconnctd from th supply and connctd togthr through a rsistor, lctrons will flow th ngativly chargd plat to th positivly chargd plat. This currnt will di away as th chargs on th plats ar rducd. As th currnt flows it is libratd as hat, thus it can b sn that nrgy is stord in a chargd capacitor. Th charg stord in an lctric fild can b masurd in coulombs (), that is th currnt which flows to charg th plats multiplid by th tim for which th flow occurs. It should b notd howvr, that with a stady supply voltag th currnt will rduc as th charg on th plats incrass, so that th plats ar not chargd by a stady currnt. In practic th total charg that can b stord by paralll-plats dpnds on th plat ara, th distanc btwn th plats, th PD btwn thm and th natur of th insulating matrial that sparats thm. 7. apacitanc For any systm of paralll plats th ratio of th quantity of charg stord to th PD btwn th
plats is a constant. This constant is calld capacitanc () and a systm of charg storing plats is calld a capacitor (somtims rronously calld a condnsr). Th circuit symbol for a capacitor is shown in figur 7.3. Figur 7.3: Th circuit symbol for a capacitor. Th unit of capacitanc is th farad (F) and: U whr: = capacitanc (F) = stord charg in th capacitanc () U = PD btwn th capacitor's plats (V) Sinc, in Part w notd that = It, w can say that: It U whr: I = avrag charging currnt to capacitor () t = tim for which charging currnt passs (s) A capacitor has a capacitanc of on farad if it can stor on coulomb whn th PD btwn its plats is on volt. Howvr, sinc in practic on farad turns out to b quit a larg valu of capacitanc, most capacitancs ar quot in microfarads (μf) which is qual to -6 F. Not that if a D voltag is connctd to a capacitor, th capacitor will charg until th PD across it is th sam as th supply voltag. It will rmain chargd until a path is providd for th currnt to flow btwn th plats. If howvr, th capacitor is connctd to an A supply, whr th currnt dirction is continually rvrsing, it will continually charg and discharg. 7.3 Dilctric Brakdown And Prmittivity Th insulating matrial which occupis th spac btwn th plats of a capacitor is calld th dilctric. Insulating matrials do not normal conduct lctricity bcaus thy do not hav fr lctron that can flow as a currnt. Howvr, if insulators ar subjctd to a high nough lctric fild, lctrons can b torn fr and th insulating proprtis ar lost in a procss calld dilctric brakdown. This phnomnon can b vry srious if cabl insulation fails. Th strngth of an lctric fild can b xprssd in trms of th PD applid across a crtain thicknss of insulator, this valu is calld th avrag potntial gradint and usually masurs in kilovolts pr millimtr (kv/mm). Th potntial gradint at which th insulator fails is calld its dilctric strngth and this can b usd as a masur of how good an insulator is. Tabl 7. contains th dilctric strngth of svral insulators. Exampl
An imprgnatd-papr capacitor of a paralll-plat powr capacitor is.5mm thick, and has a dilctric strngth of 5kV/mm. If th capacitor concrnd has an applid PD of.8kv, what is th avrag potntial gradint in th papr? At what applid voltag could th capacitor b xpctd to brak down? avrag potntial avrag potntial applid voltag (kv) gradint dilctric thicknss (mm).8 gradint 3.6kV / mm.5 Likly brakdown voltag (kv) = dilctric strngth (kv/mm) dilctric thicknss (mm) = 5.5 =.5kV Matrial Dilctric Strngth (kv/mm) Air 3 Gnral Baklit -5 Plugs, machinry tc. Bitumn 4 abl-saling boxs Glass 5- Ovrhad-lin insulators Application Mica 4-5 ommutators, hot lmnts, capacitors Micanit 3 Machins Imprgnatd Papr 4- Powr cabls, capacitors Paraffin Wax 8 apacitors Porclain 9- Fus carrirs, ovrhad-lin insulators Tabl 7.: Th dilctric strngth of svral common insulators. Dilctric matrials can also b charactrisd in trms of thir rlativ prmittivity ( r ). Th absolut prmittivity of a matrial is dfind as th ratio of th charg pr unit ara on a surfac of that matrial, to th intnsity of th lctric fild producd. Rlativ prmittivity is dfind as th ratio of th capacitanc of a capacitor with th matrial in qustion btwn its plats, to th capacitanc of th sam capacitor with a vacuum btwn its plat. This ratio is also qual to th prmittivity of th dilctric to th prmittivity of fr spac (i.. a vacuum). Thus: r whr: r = rlativ prmittivity of th dilctric matrial, no units = th capacitanc of a capacitor with th matrial in qustion (F) = th capacitanc of th sam capacitor with a vacuum btwn its plat (F) = absolut prmittivity of th matrial (F/m) = th prmittivity of fr spac (F/m) Not that th prmittivity of fr spac is a constant qual to 8.85 - F/m (or /Nm ), and that by dfinition th rlativ prmittivity of a vacuum is. Not also that prmittivity () is vry similar in concpt and nam to th prmability (μ) of magntic matrials howvr, th two should not b confusd. Tabl 7. contains th rlativ prmability of som common dilctric matrials.
Dilctric Matrial Rlativ Prmittivity ( r) Air Baklit 4.5-5.5 Glass 5 - Mica 3-7 Imprgnatd Papr Polystyrn.5 Porclain 6-7 Tabl 7.: Rlativ prmability of som common dilctric matrials. 7.4 apacitors In Th Paralll And In Sris onsidr thr capacitors in paralll, as shown in figur 7.4, ach will hav th sam voltag (V) across it and th total charg ( T ) will b th sum of th individual chargs. Thrfor: T = + + 3 Figur 7.4: Thr capacitors in paralll. But = V and if T is th quivalnt capacitanc of th thr capacitors: Thrfor: T U = U + U + 3 U T = + + 3 Now considr th thr capacitors connctd in sris, lik thos in figur 7.5. Th supply voltag is split across th thr capacitors and sinc th sam charging currnt will flow through ach capacitor thy will ach acquir th sam charg in th sam tim. Thus: U = U + U + U 3 Figur 7.5: Thr capacitors in sris. But U = /, so if T is th quivalnt valu of thr sris-connctd capacitors: Thrfor: T T 3 3
Exampl alculat th quivalnt capacitanc of 5μF, μf and 3μF capacitors connctd in sris. T 5 3 6 3 3 3 3μF Ths capacitors ar connctd to a 4V D supply. alculat th charg on ach capacitor, and th potntial diffrnc across ach. Total stord charg, = T V = 3-3 4 = 7μ Sinc th capacitors ar connctd in sris, th charg on ach is th sam as th total charg, i.. 7μ. Th PD across th 5μF capacitor, Th PD across th μf capacitor, Th PD across th 3μF capacitor, hck: 44 + 7 + 4 = 4V 7 5 6 US 6 7 6 US 6 7 3 6 US 6 44V 7V 4V Not that th capacitor with th lowst valu has th largst voltag across it and if th capacitors ar similarly constructd thy must all b ratd at th highst voltag. For this rason, capacitors ar not oftn connctd in sris unlss thy ar idntical. 7.5 apacitanc Of A Paralll apacitor Th capacitanc of a paralll plat capacitor dpnds on: Plat ara (A in m ), so that α A Th distanc btwn th plats (d in m ), so that α /d Absolut prmittivity of th dilctric ( in F/m), so that α Numbr of plats (N), capacitors can b constructd from a sris of intrlocking plats (figur 7.6), so that α (N-). ombining th abov factors givs: A(N ) d Th absolut prmittivity of th dilctric is rally usd and sinc = r th quation bcoms: r A(N ) d
Figur 7.6: An intrlavd-plat capacitor, containing plats it is dfctivly capacitors in paralll. 7.6 Th Enrgy Stord In A apacitor Lik inductors which stor nrgy in a magntic fild, capacitors stor nrgy in an lctric fild. Howvr, unlik inductors, capacitors can kp th nrgy stord onc th supply has bn disconnctd although it will dissipat ovr tim. Th nrgy stord in a capacitor can b givn by: V W Th amount of nrgy stord in most capacitors is small howvr, it is still nough to giv a prson a shock. Although th charg, and thus th nrgy, will lak away ovr tim a discharg rsistor is usually placd across th trminals of th capacitor. Such a rsistor must hav a valu high nough to prvnt apprciabl currnts flowing through it from th charging sourc but low nough to discharg th capacitor within a rasonabl tim. A typical valu would b MΩ. 7.7 Growth And Dcay urvs As mntiond in sction 7. th currnt charging two paralll plats is not a constant but dis away as th as th charg builds up and th PD across th capacitors trminals incrass. onsidr th arrangmnt in figur 7.7 whr a capacitor is connctd in sris with a rsistor to a D supply with an EMF of E volts. Th instantanous PD across th capacitor (i.. th PD at any givn tim, t) Figur 7.7: A capacitor charg and discharg circuit. is dnotd by v, th instantanous currnt flowing to th capacitor is i and th instantanous charg on th capacitor is q. If th capacitor is initially unchargd, whn th switch is in position, th initial currnt (I ) will b givn by I = V/R. As th charg on th capacitor builds up, so dos th PD (v) across its plats. This voltag opposs th charg flowing to th capacitor and th ffctiv circuit voltag bcoms V-v and th instantanous currnt will b lss than th initial currnt. Thus:
V v i R Thrfor, th currnt which is initially larg, falls away as charging progrsss, whras th voltag across th capacitor and th charg on th capacitor both incras (figur 7.8). Whn th capacitor is fully charg v = E and q =, whr is th maximum charg on th capacitor ( = V). Whn th switch is movd to position th capacitor will discharg. Initially v = E and I = E/R howvr, as discharging continus v rducs and i = v/r. Both th voltag and currnt fall off following a similar curv, and as thy do so th charg hld by th capacitor also rducs (figur 7.9). Figur 7.8: Th charging curv for a capacitor. Figur 7.9: Th discharging curv for a capacitor. Th tim takn for charging and discharging dpnds on th tim constant (τ) of th circuit which is givn by: τ = R and a capacitor will normally b dischargd within 5 tim constants. Th quitations for th charging and discharging of capacitors ar vry similar to th quations for th growth and dcay curvs for inductors. Thy ar includd hr for compltnss. Th instantanous charg whn charging: q = E( -t/τ ) = ( -t/τ ) Th instantanous currnt whn charging: i E R t I t Th instantanous charg whn discharging: Th instantanous currnt whn discharging: q = E -t/τ = -t/τ i R t I t