.3: Inucors Reson: June, 5 E Man Sue D Pullman, WA 9963 59 334 636 Voce an Fax Oerew We connue our suy of energy sorage elemens wh a scusson of nucors. Inucors, lke ressors an capacors, are passe wo-ermnal crcu elemens. Tha s, no exernal power supply s necessary o make hem funcon. Inucors commonly conss of a conuce wre wrappe aroun a core maeral; nucors sore energy n he form of a magnec fel se up aroun he curren-carryng wre. In hs chaper, we escrbe physcal properes of nucors an proe a mahemacal moel for an eal nucor. Usng hs eal nucor moel, we wll eelop mahemacal relaonshps for he energy sore n an nucor an goernng relaons for seres an parallel connecons of nucors. The chaper wll conclue wh a bref scusson of praccal non-eal nucors. 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 Chaper. Deermne equalen ressance of seres an parallel combnaons of ressors Chaper.5 Afer compleng hs chaper, you shoul be able o: Wre he crcu symbol for an nucor Sae he mechansm by whch an nucor sores energy Sae from memory he olage-curren relaonshp for an nucor n boh fferenal an negral form Sae from memory he response of an nucor o consan olages an nsananeous curren changes Wre he mahemacal expresson escrbng energy sorage n an nucor Deermne he equalen nucance of seres an parallel combnaons of nucors Skech a crcu escrbng a non-eal nucor Ths chaper requres: /A Doc: XXX-YYY page of Copyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners.
.3: Inucors Inucors: Passng a curren hrough a conuce wre wll creae a magnec fel aroun he wre. Ths magnec fel s generally hough of n erms of as formng close loops of magnec flux aroun he curren-carryng elemen. Ths physcal process s use o creae nucors. Fgure llusraes a common ype of nucor, conssng of a cole wre wrappe aroun a core maeral. Passng a curren hrough he conucng wre ses up lnes of magnec flux, as shown n Fgure ; he nucor sores energy n hs magnec fel. The nucance of he nucor s a quany whch ells us how much energy can be sore by he nucor. Hgher nucance means ha more energy can be sore by he nucor. Inucance has uns of Henrys, abbreae H. The amoun of nucance an nucor has s goerne by he geomery of he nucor an he properes of he core maeral. These effecs can be complex; raher han aemp a comprehense scusson of hese effecs, we wll smply clam ha, n general, nucance s epenen upon he followng parameers: The number of mes he wre s wrappe aroun he core. More cols of wre resuls n a hgher nucance. The core maeral s ype an shape. Core maerals are commonly ferromagnec maerals, snce hey resul n hgher magnec flux an corresponngly hgher energy sorage. Ar, howeer, s a farly commonly use core maeral presumably because of s reay aalably. The spacng beween urns of he wre aroun he core. Fgure. Wre-wrappe nucor wh apple curren hrough conuce wre. We wll enoe he oal magnec flux creae by he nucor by ψ, as shown n Fgure. For a lnear nucor, he flux s proporonal o he curren passng hrough he woun wres. The consan of proporonaly s he nucance, : ψ www.glennc.com page of Copyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners.
.3: Inucors Volage s he me rae of change of magnec flux, so ψ Combnng equaons an resuls n he olage-curren relaonshp for an eal nucor: 3 The crcu symbol for an nucor s shown n Fgure, along wh he sgn conenons for he olage-curren relaonshp of equaon 3. The passe sgn conenon s use n he olagecurren relaonshp, so pose curren s assume o ener he ermnal wh pose olage polary. Fgure. Inucor crcu symbol an olage-curren sgn conenon. Inegrang boh ses of equaon 3 resuls n he followng form for he nucor s olage-curren relaonshp: 4 In equaon 4, s a known curren a some nal me an s use as a ummy arable of negraon o emphasze ha he only whch sures he negraon process s he upper lm of he negral. Imporan resul: The olage-curren relaonshp for an eal nucor can be sae n eher fferenal or negral form, as follows: www.glennc.com page 3 of Copyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners.
.3: Inucors Example : A crcu conans a mh nucor. The curren as a funcon of me hrough he nucor s measure an shown below. Plo he olage across he nucor as a funcon of me. In he me range <<ms, he rae of change of curren s A/sec. Thus, from equaon 3, he olage s.h A / s V. In he me range ms<<ms, he rae of change of curren s -5A/sec. The olage s -.5V. In he me range ms<<3ms, he curren s consan an here s no olage across he nucor. In he me range 3ms<<5ms, he rae of change of curren s -5A/sec. The olage s -.5V. The plo of olage s. me s shown below: Power s he prouc of olage an curren. If he sgns of olage an curren are he same accorng o he passe sgn conenon, he crcu elemen absorbs power. If he sgns of olage an curren are no he same, he crcu elemen generaes power. From he aboe olage an curren cures, he nucor s absorbng power from he crcu urng he mes <<ms an 4ms<<5ms. The nucor reurns power o he crcu urng he mes ms<<ms an 3ms<<4ms. www.glennc.com page 4 of Copyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners.
.3: Inucors Example : If he olage as a funcon of me across an nucor wh nucance mh s as shown below, eermne he curren as a funcon of me hrough he capacor. Assume ha he curren hrough he capacor s A a me. Volage, V A me, he curren s gen o be A. In he me pero << msec, he olage s consan an pose so he curren wll ncrease lnearly. The oal curren change urng hs me pero s he area uner he olage cure cure, e by he nucance: 3 V sec A.. In he me pero << msec, he olage s ecreasng lnearly. The curren urng hs me pero s a quarac cure, concae ownwar. The maxmum alue of curren s.5a, a.5 msec. The curren a he en of hs me pero s A. In he me pero <<3 secons, he olage s consan a -V. The curren change urng hs me pero s he area uner he olage cure, e by he nucance: 3 V sec -A. The oal curren a 3 secons s, hen, A A A.. In he me pero 3<<4 secons, he olage s zero. The negral of zero oer any me pero s zero, so here s no change n curren urng hs me range an he curren remans consan a A. A plo of he hrough he nucor as a funcon of me s shown below. Curren, A www.glennc.com page 5 of Copyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners.
.3: Inucors I s ofen useful, when analyzng crcus conanng nucors, o examne he crcu s response o consan operang conons an o nsananeous changes n operang conon. We examne he nucor s response o each of hese operang conons below: Inucor response o consan olage: If he curren hrough he nucor s consan, equaon 3 ncaes ha he olage across he nucor s zero. Thus, f he curren hrough he nucor s consan, he nucor s equalen o a shor crcu. Inucor response o nsananeous curren changes: If he curren hrough he nucor changes nsananeously, he rae of change of curren s nfne. Thus, by equaon 3, f we wsh o change he curren hrough an nucor nsananeously, we mus supply nfne olage o he nucor. Ths mples ha nfne power s aalable, whch s no physcally possble. Thus, n any praccal crcu, he curren hrough an nucor canno change nsananeously. Any crcu whch allows an nsananeous change n he curren hrough an eal nucor s no physcally realzable. We may somemes assume, for mahemacal conenence, ha an eal nucor s curren 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 Inucor Properes: Inucors can be replace by shor-crcus, uner crcumsances when all operang conons are consan. Currens hrough nucors canno change nsananeously. o such requremen s place on olages. Energy Sorage: The nsananeous power sspae by an elecrcal crcu elemen s he prouc of he olage an curren: p 5 Usng equaon 3 o wre he olage n equaon 5 n erms of he nucor s curren: p 6 As was preously one for capacors, we negrae he power wh respec o me o ge he energy sore n he nucor: www.glennc.com page 6 of Copyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners.
.3: Inucors W Whch, afer some manpulaon comparable o he approach aken when we calculae energy sorage n capacors, resuls n he followng expresson for he energy sore n an nucor: W 7 Inucors n Seres: Conser he seres connecon of nucors shown n Fgure 3. Fgure 3. Seres connecon of nucors. Applyng Krchoff s olage law aroun he loop resuls n: 8 Usng equaon 3 o wre he nucor olage rops n erms of he curren hrough he loop ges: Usng summaon noaon resuls n k k 9 Ths s he same equaon ha goerns he crcu of Fgure 4, f www.glennc.com page 7 of Copyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners.
.3: Inucors www.glennc.com page 8 of Copyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners. k eq k Thus, he equalen nucance of a seres combnaon of nucors s smply he sum of he nual nucances. Ths resul s analogous o he equaons whch proe he equalen ressance of a seres combnaon of ressors. Fgure 4. Equalen crcu o Fgure 3. Inucors n Parallel: Conser he parallel combnaon of nucors, as shown n Fgure 5. - Fgure 5. Parallel combnaon of I nucors. Applyng Krchoff s curren law a he upper noe resuls n: Usng equaon 4 o wre he nucor currens n erms of her olage rops ges: [ ]
.3: Inucors Ths can be re-wren usng summaon noaon as k k Ths s he same equaon ha goerns he crcu of Fgure 4, f eq k k 3 For he specal case of wo nucors an n seres, equaon 3 smplfes o eq 4 Equaons 3 an 4 are analogous o he equaons whch proe he equalen ressance of seres combnaons of ressors. Summary: Seres an Parallel Inucors The equalen nucance of a seres combnaon of nucors,,, s goerne by a relaon whch s analogous o ha prong he equalen ressance of a seres combnaon of ressors: eq k k The equalen nucance of a parallel combnaon of nucors,,, s goerne by a relaon whch s analogous o ha prong he equalen ressance of a parallel combnaon of ressors: eq k k Praccal Inucors: Mos commercally aalable nucors are manufacure by wnng wre n arous col confguraons aroun a core. Cores can be a arey of shapes; Fgure n hs chaper shows a core whch s bascally a cylnrcal bar. Toroal cores are also farly common a closely woun oroal core has he aanage ha he magnec fel s confne nearly enrely o he space nse he wnng. Inucors are aalable wh alues from less han mcro-henry µh -6 Henres up o ens of Henres. A H nucor s ery large; nucances of mos commercally aalable nucors are measure n mllhenres mh -3 Henres or mcrohenres. arger nucors are generally use for low-frequency applcaons n whch he sgnals ary slowly wh me. www.glennc.com page 9 of Copyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners.
.3: Inucors Aemps a creang nucors n negrae-crcu form hae been largely unsuccessful; herefore many crcus whch are o mplemene as negrae crcus o no nclue nucors. Incluson of nucance n he analyss sage of hese crcus may howeer, be mporan. Snce any currencarryng conucor wll creae a magnec fel, he sray nucance of supposely non-nuce crcu elemens can become an mporan conseraon n he analyss an esgn of a crcu. Praccal nucors, unlke he eal nucors scusse n hs chaper, sspae power. An equalen crcu moel for a praccal nucor s generally creae by placng a ressance n seres wh an eal nucor, as shown n Fgure 6. Fgure 6. Equalen crcu moel for a praccal nucor. www.glennc.com page of Copyrgh Dglen, Inc. All rghs resere. Oher prouc an company names menone may be raemarks of her respece owners.