Revision: June 12, E Main Suite D Pullman, WA (509) Voice and Fax

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1 .: apacors Reson: June, 5 E Man Sue D Pullman, WA Voce an Fax Oerew We begn our suy of energy sorage elemens wh a scusson of capacors. apacors, lke ressors, are passe wo-ermnal crcu elemens. Tha s, no exernal power supply s necessary o make hem funcon. apacors conss of a non-conuce maeral or elecrc whch separaes wo elecrcal conucors; capacors sore energy n he form of an elecrc fel se up n he elecrc maeral. In hs chaper, we escrbe physcal properes of capacors an proe a mahemacal moel for an eal capacor. Usng hs eal capacor moel, we wll eelop mahemacal relaonshps for he energy sore n a capacor an goernng relaons for seres an parallel connecons of capacors. The chaper wll conclue wh a bref scusson of praccal non-eal capacors. Before begnnng hs chaper, you shoul be able o: Ealuae negral an fferenal relaons Skech boh he negral an erae of a gen me funcon Defne olage an curren n erms of elecrcal charge haper. Deermne equalen ressance of seres an parallel combnaons of ressors haper.5 Afer compleng hs chaper, you shoul be able o: Wre he crcu symbol for a capacor Sae he mechansm by whch a capacor sores energy Sae from memory he olage-curren relaonshp for a capacor n boh fferenal an negral form Sae from memory he response of a capacor o consan olages an nsananeous olage changes Wre he mahemacal expresson escrbng energy sorage n a capacor Deermne he equalen capacance of seres an parallel combnaons of capacors Skech a crcu escrbng a non-eal capacor Ths chaper requres: /A Doc: XXX-YYY page of opyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners.

2 .: apacors apacors: Two elecrcally conuce boes, when separae by a non-conuce or nsulang maeral, wll form a capacor. Fgure llusraes he specal case of a parallel plae capacor. The nonconuce maeral beween he plaes s calle a elecrc; he maeral propery of he elecrc whch s currenly mporan o us s s permy, ε. When a olage poenal fference s apple across he wo plaes, as shown n Fgure, charge accumulaes on he plaes he plae wh he hgher olage poenal wll accumulae pose charge q, whle he plae wh he lower olage poenal wll accumulae negae charge, -q. The charge fference beween he plaes nuces an elecrc fel n he elecrc maeral; he capacor sores energy n hs elecrc fel. The capacance of he capacor s a quany whch ells us, essenally, how much energy can be sore by he capacor. Hgher capacance means ha more energy can be sore by he capacor. apacance has uns of Faras, abbreae F. The amoun of capacance a capacor has s goerne by he geomery of he capacor he shape of he conucors an her orenaon relae o one anoher an he permy of he elecrc beween he conucors. These effecs can be complex an ffcul o quanfy mahemacally; raher han aemp a comprehense scusson of hese effecs, we wll smply clam ha, n general, capacance s epenen upon he followng parameers: The spacng beween he conuce boes he sance n Fgure. As he separaon beween he boes ncreases, he capacance ecreases. The surface area of he conuce boes. As he surface area of he conucors ncreases, he capacance ncreases. The surface area referre o here s he area oer whch boh he conucors an he elecrc oerlap. The permy of he elecrc. As he permy ncreases, he capacance ncreases. The parallel-plae capacor shown n Fgure, for example, has capacance A ε Fgure. Parallel plae capacor wh apple olage across conucors. page of opyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners.

3 .: apacors Mahemacally, he capacance of he ece relaes he olage fference beween he plaes an he charge accumulaon assocae wh hs olage: q V apacors whch obey he relaonshp of equaon are lnear capacors, snce he poenal fference beween he conuce surfaces s lnearly relae o he charge on he surfaces. Please noe ha he charges on he upper an lower plae of he capacor n Fgure are equal an oppose hus, f we ncrease he charge on one plae, he charge on he oher plae mus ecrease by he same amoun. Ths s conssen wh our preous assumpon elecrcal crcu elemens canno accumulae charge, an curren enerng one ermnal of a capacor mus leae he oher ermnal of he capacor. Snce curren s efne as he me rae of change of charge, q, equaon can be re- wren n erms of he curren hrough he capacor: [ ] Snce he capacance of a gen capacor s consan, equaon can be wren as 3 The crcu symbol for a capacor s shown n Fgure, along wh he sgn conenons for he olage-curren relaonshp of equaon 3. We use our passe sgn conenon for he olagecurren relaonshp pose curren s assume o ener he ermnal wh pose olage polary. Fgure. apacor crcu symbol an olage-curren sgn conenon. Inegrang boh ses of equaon 3 resuls n he followng form for he capacor s olage-curren relaonshp: 4 where s a known olage a some nal me,. We use a ummy arable of negraon,, o emphasze ha he only whch sures he negraon process s he upper lm of he negral. page 3 of opyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners.

4 .: apacors Imporan resul: The olage-curren relaonshp for an eal capacor can be sae n eher fferenal or negral form, as follows: Example : If he olage as a funcon of me across a capacor wh capacance µf s as shown below, eermne he curren as a funcon of me hrough he capacor. <<: The olage rae of change s V/s. Thus, -6 F V/s µa. <<: The olage s consan. Thus, A. <<3: The olage rae of change s -5 V/s. Thus, -6 F-5 V/s -5 µa. 3<<4: The olage s consan. Thus, A. A plo of he curren hrough he capacor as a funcon of me s shown below. urren, A page 4 of opyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners.

5 .: apacors Example : If he curren as a funcon of me hrough a capacor wh capacance mf s as shown below, eermne he olage as a funcon of me across he capacor. Assume ha he olage across he capacor s V a me. A me, he olage s gen o be V. In he me pero << secon, he curren ncreases lnearly an he olage wll ncrease quaracally. The oal olage change urng hs me pero s he negral of he curren, whch s smply he area uner he curren cure e by he capacance: 3 Asec.5V..F In he me pero << secons, he curren s consan a ma. The olage change s he area uner he curren cure e by he capacance: -3 A sec/.f V. The oal olage a secons s, hen,.5v V.5V. In he me pero <<3 secons, he curren s consan a - ma. The olage change s he negae of he olage change from << sec. The oal olage a 3 secons s, hen,.5v V.5V. In he me pero 3<<4 secons, he curren s zero. The negral of zero oer any me pero s zero, so here s no change n olage urng hs me range an he olage remans consan a.5v. A plo of he olage across he capacor as a funcon of me s shown below. page 5 of opyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners.

6 .: apacors I s ofen useful, when analyzng crcus conanng capacors, o examne he crcu s response o consan operang conons an o nsananeous changes n operang conon. We examne he capacor s response o each of hese operang conons below: apacor response o consan olage: If he olage across he capacor s consan, equaon 3 ncaes ha he curren hrough he capacor s zero. Thus, f he olage across he capacor s consan, he capacor s equalen o a open crcu. Ths propery can be exremely useful n eermnng a crcu s seay-sae response o consan npus. If he npus o a crcu change from one consan alue o anoher, he ransen componens of he response wll eenually e ou an all crcu parameers wll become consan. Uner hese conons, capacors can be replace wh open crcus an he crcu analyze relaely easly. As we wll see laer, hs operang conon can be useful n eermnng he response of crcus conanng capacors an n ouble-checkng resuls obane usng oher mehos. apacor response o nsananeous olage changes: If he olage across he capacor changes nsananeously, he rae of change of olage s nfne. Thus, by equaon 3, f we wsh o change he olage across a capacor nsananeously, we mus supply nfne curren o he capacor. Ths mples ha nfne power s aalable, whch s no physcally possble. Thus, n any praccal crcu, he olage across a capacor canno change nsananeously. Any crcu whch allows an nsananeous change n he olage across an eal capacor s no physcally realzable. We may somemes assume, for mahemacal conenence, ha an eal capacor s olage changes suenly; howeer, mus be emphasze ha hs assumpon requres an unerlyng assumpon ha nfne power s aalable an s hus no an allowable operang conon n any physcal crcu. Imporan apacor Properes: apacors can be replace by open-crcus, uner crcumsances when all operang conons are consan. Volages across capacors canno change nsananeously. o such requremen s place on currens. Energy Sorage: The power sspae by a capacor s p 5 Snce boh olage an curren are funcons of me, he power sspaon wll also be a funcon of me. The power as a funcon of me s calle he nsananeous power, snce proes he power sspaon a any nsan n me. page 6 of opyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners.

7 .: apacors Subsung equaon 3 no equaon 5 resuls n: p 6 Snce power s, by efnon, he rae of change of energy, he energy s he me negral of power. Inegrang equaon 6 wh respec o me ges he followng expresson for he energy sore n a capacor: W where we hae se our lower lms of negraon a - o ao ssues relae o nal conons. We assume ha no energy s sore n he capacor a me - so ha W 7 From equaon 7 we see ha he energy sore n a capacor s always a non-negae quany, so W. Ieal capacors o no sspae energy, as ressors o. apacors sore energy when s proe o hem from he crcu; hs energy can laer be recoere an reurne o he crcu. Example: onser he crcu shown below. The olage apple o he capacor by he source s as shown. Plo he power absorbe by he capacor an he energy sore n he capacor as funcons of me. Power s mos realy compue by akng he prouc of olage an curren. The curren can be eermne from equaon 3. The curren as a funcon of me s ploe below. page 7 of opyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners.

8 .: apacors The power absorbe by he capacor s eermne by akng a pon-by-pon prouc beween he olage an curren. Recall ha power s absorbe or generae base on he passe sgn conenon. If he relae sgns beween olage an curren agree wh he passe sgn conenon, he crcu elemen s absorbng power. If he relae sgns beween olage an curren are oppose o he passe sgn conenon, he elemen s generang power. Thus, he capacor n hs example s absorbng power for he frs mcrosecon. I generaes power reurns power o he olage source urng he secon mcrosecon. Afer he secon mcrosecon, he curren s zero an he capacor neher absorbs nor generaes power. The energy sore n he capacor can be eermne eher from negrang he power or from applcaon of equaon 7 o he olage cure proe n he problem saemen. The energy n he capacor as a funcon of me s shown below: Durng he frs mcrosecon, whle he capacor s absorbng power, he energy n he capacor s ncreasng. The maxmum energy n he capacor s 5 µj, a µs. Durng he secon mcrosecon, he capacor s releasng power back o he crcu an he energy n he capacor s ecreasng. A µs, he capacor sll has.5 µj of sore energy. Afer µs, he capacor neher absorbs nor generaes energy an he energy sore n he capacor remans a.5µj. page 8 of opyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners.

9 .: apacors page 9 of opyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners. apacors n Seres: onser he seres connecon of capacors shown n Fgure 3. Fgure 3. Seres connecon of capacors. Applyng Krchoff s olage law aroun he loop resuls n: L 8 Usng equaon 4 o wre he capacor olage rops n erms of he curren hrough he loop ges: [ ] L L L L Ths can be re-wren usng summaon noaon as k k 9 Ths s he same equaon ha goerns he crcu of Fgure 4, f k k eq Thus, he crcus of Fgure 3 an Fgure 4 are equalen crcus, f he equalen ressance s chosen accorng o equaon.

10 .: apacors Fgure 4. Equalen crcu o Fgure 3. For he specal case of wo capacors an n seres, equaon smplfes o eq Equaons an are analogous o he equaons whch proe he equalen ressance of parallel combnaons of ressors. apacors n Parallel: onser he parallel combnaon of capacorsl, as shown n Fgure 5. Fgure 5. Seres connecon of capacors. Applyng Krchoff s curren law a he upper noe resuls n: L Usng equaon 4 o wre he capacor currens n erms of her olage rop ges: L L Usng summaon noaon resuls n page of opyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners.

11 .: apacors k k 3 Ths s he same equaon ha goerns he crcu of Fgure 6, f eq k k 4 Thus, he equalen capacance of a parallel combnaon of capacors s smply he sum of he nual capacances. Ths resul s analogous o he equaons whch proe he equalen ressance of a seres combnaon of ressors. Fgure 6. Equalen crcu o Fgure 5. Summary: Seres an Parallel apacors The equalen capacance of a seres combnaon of capacors,,, s goerne by a relaon whch s analogous o ha prong he equalen ressance of a parallel combnaon of ressors: eq k k The equalen capacance of a parallel combnaon of capacors,,, s goerne by a relaon whch s analogous o ha prong he equalen ressance of a seres combnaon of ressors: eq k k page of opyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners.

12 .: apacors Praccal apacors: ommercally aalable capacors are manufacure n a we range of boh conucor an elecrc maerals an are aalable n a we range of capacances an olage rangs. The olage rang of he ece s he maxmum olage whch can be safely apple o he capacor; usng olages hgher han he rae alue may amage he capacor. The capacance of commercally aalable capacors s commonly measure n mcro-faras µf; one mcrofara s -6 of a Fara or pcofaras pf; one pcofara s - of a Fara. Large capacors are aalable, bu are relaely nfrequenly use. These are generally calle super-capacors or ulra-capacors an are aalable n capacances up o ens of Faras. For mos applcaons, howeer, usng one woul be comparable o buyng a car wh a gallon gas ank. Seeral approaches are use for labelng a capacor wh s capacance alue. Large capacors ofen hae her alue prne planly on hem, such as " uf" for mcrofaras. Smaller capacors, appearng as small sks or wafers, ofen hae her alues prne on hem n an encoe manner. For hese capacors, a hree g number ncaes he capacor alue n pco-faras. The frs wo gs proes he "base" number, an he hr g proes an exponen of so, for example, "4" prne on a capacor ncaes a capacance alue of x 4 or pf. Occasonally, a capacor wll only show a wo g number, n whch case ha number s smply he capacor alue n pf. For compleeness, f a capacor shows a hree g number an he hr g s 8 or 9, hen he frs wo gs are mulple by. an. respecely. apacors are generally classfe accorng o he elecrc maeral use. ommon capacor ypes nclue mca, ceramc, Mylar, paper, Teflon an polysyrene. An mporan class of capacors whch requre specal menon are elecrolyc capacors. Elecrolyc capacors hae relaely large capacances relae o oher ypes of capacors of smlar sze. Howeer, some care mus be exercse when usng elecrolyc capacors hey are polarze an mus be connece o a crcu wh he correc polary. The pose lea of he capacor mus be connece o he pose lea of he crcu. onnecng he pose lea of he capacor o he negae lea of a crcu can esroy he capacor. Polarze capacors eher hae a ark srpe near he pn ha mus be kep a he hgher olage, or a "-" near he pn ha mus be kep a a lower olage. Praccal capacors, unlke eal capacors, wll sspae some power. Ths power loss s prmarly ue o leakage currens. These currens are ue o he fac ha real elecrc maerals are no perfec nsulaors some small curren wll en o flow hrough hem. The oerall effec s comparable o placng a hgh ressance n parallel wh an eal capacor, as shown n Fgure 7. Dfferen ypes of capacors hae fferen leakage currens. Mca capacors en o hae low leakage currens, he leakage currens of ceramc capacors ary accorng o he ype of capacor, an elecrolyc capacors hae hgh leakage currens. Fgure 7. Moel of praccal capacor nclung leakage curren pah. page of opyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners.