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1 Impdanc Ths worksht and all rlatd fls ar lcnsd undr th Cratv Commons Attrbuton Lcns, vrson 1.0. To vw a copy of ths lcns, vst or snd a lttr to Cratv Commons, 559 Nathan Abbott Way, Stanford, Calforna 94305, USA. Th trms and condtons of ths lcns allow for fr copyng, dstrbuton, and/or modfcaton of all lcnsd works by th gnral publc. Rsourcs and mthods for larnng about ths subjcts (lst a fw hr, n prparaton for your rsarch): 1

2 Quston 1 In ths AC crcut, th rsstor offrs 300 Ω of rsstanc, and th nductor offrs 400 Ω of ractanc. Togthr, thr srs opposton to altrnatng currnt rsults n a currnt of 10 ma from th 5 volt sourc: X L = 400 Ω R = 300 Ω 5 VAC I = 10 ma How many ohms of opposton dos th srs combnaton of rsstor and nductor offr? What nam do w gv to ths quantty, and how do w symbolz t, bng that t s composd of both rsstanc (R) and ractanc (X)? fl Answr 1 Z total = 500 Ω. Follow-up quston: suppos that th nductor suffrs a falur n ts wr wndng, causng t to opn. Explan what ffct ths would hav on crcut currnt and voltag drops. Nots 1 Studnts may xprnc dffculty arrvng at th sam quantty for mpdanc shown n th answr. If ths s th cas, hlp thm problm-solv by suggstng thy smplfy th problm: short past on of th load componnts and calculat th nw crcut currnt. Soon thy wll undrstand th rlatonshp btwn total crcut opposton and total crcut currnt, and b abl to apply ths concpt to th orgnal problm. Ask your studnts why th quantts of 300 Ω and 400 Ω do not add up to 700 Ω lk thy would f thy wr both rsstors. Dos ths scnaro rmnd thm of anothr mathmatcal problm whr 3+4 = 5? Whr hav w sn ths bfor, spcally n th contxt of lctrc crcuts? Onc your studnts mak th cogntv conncton to trgonomtry, ask thm th sgnfcanc of ths numbrs addton. Is t nough that w say a componnt has an opposton to AC of 400 Ω, or s thr mor to ths quantty than a sngl, scalar valu? What typ of numbr would b sutabl for rprsntng such a quantty, and how mght t b wrttn? 2

3 Quston 2 In ths AC crcut, th rsstor offrs 3 kω of rsstanc, and th capactor offrs 4 kω of ractanc. Togthr, thr srs opposton to altrnatng currnt rsults n a currnt of 1 ma from th 5 volt sourc: X C = 4 kω R = 3 kω 5 VAC I = 1 ma How many ohms of opposton dos th srs combnaton of rsstor and capactor offr? What nam do w gv to ths quantty, and how do w symbolz t, bng that t s composd of both rsstanc (R) and ractanc (X)? fl Answr 2 Nots 2 Z total = 5 kω. Studnts may xprnc dffculty arrvng at th sam quantty for mpdanc shown n th answr. If ths s th cas, hlp thm problm-solv by suggstng thy smplfy th problm: short past on of th load componnts and calculat th nw crcut currnt. Soon thy wll undrstand th rlatonshp btwn total crcut opposton and total crcut currnt, and b abl to apply ths concpt to th orgnal problm. Ask your studnts why th quantts of 3 kω and 4 kω do not add up to 7 kω lk thy would f thy wr both rsstors. Dos ths scnaro rmnd thm of anothr mathmatcal problm whr = 5? Whr hav w sn ths bfor, spcally n th contxt of lctrc crcuts? Onc your studnts mak th cogntv conncton to trgonomtry, ask thm th sgnfcanc of ths numbrs addton. Is t nough that w say a componnt has an opposton to AC of 4 kω, or s thr mor to ths quantty than a sngl, scalar valu? What typ of numbr would b sutabl for rprsntng such a quantty, and how mght t b wrttn? 3

4 Quston 3 Whl studyng DC crcut thory, you larnd that rsstanc was an xprsson of a componnt s opposton to lctrc currnt. Thn, whn studyng AC crcut thory, you larnd that ractanc was anothr typ of opposton to currnt. Now, a thrd trm s ntroducd: mpdanc. Lk rsstanc and ractanc, mpdanc s also a form of opposton to lctrc currnt. Explan th dffrnc btwn ths thr quantts (rsstanc, ractanc, and mpdanc) usng your own words. fl Answr 3 Th fundamntal dstncton btwn ths trms s on of abstracton: mpdanc s th most gnral trm, ncompassng both rsstanc and ractanc. Hr s an xplanaton gvn n trms of logcal sts (usng a Vnn dagram), along wth an analogy from anmal taxonomy: Impdanc Z Mammal Rsstanc R Ractanc X Hors Rabbt Rsstanc s a typ of mpdanc, and so s ractanc. Th dffrnc btwn th two has to do wth nrgy xchang. Nots 3 Th gvn answr s far from complt. I v shown th smantc rlatonshp btwn th trms rsstanc, ractanc, and mpdanc, but I hav only hntd at th concptual dstnctons btwn thm. B sur to dscuss wth your studnts what th fundamntal dffrnc s btwn rsstanc and ractanc, n trms of lctrcal nrgy xchang. 4

5 Quston 4 In DC crcuts, w hav Ohm s Law to rlat voltag, currnt, and rsstanc togthr: E = IR In AC crcuts, w smlarly nd a formula to rlat voltag, currnt, and mpdanc togthr. Wrt thr quatons, on solvng for ach of ths thr varabls: a st of Ohm s Law formula for AC crcuts. B prpard to show how you may us algbra to manpulat on of ths quatons nto th othr two forms. fl Answr 4 E = IZ I = E Z Z = E I If usng phasor quantts (complx numbrs) for voltag, currnt, and mpdanc, th propr way to wrt ths quatons s as follows: E = IZ I = E Z Z = E I Nots 4 Bold-facd typ s a common way of dnotng vctor quantts n mathmatcs. Although th us of phasor quantts for voltag, currnt, and mpdanc n th AC form of Ohm s Law ylds crtan dstnct advantags ovr scalar calculatons, ths dos not man on cannot us scalar quantts. Oftn t s approprat to xprss an AC voltag, currnt, or mpdanc as a smpl scalar numbr. 5

6 Quston 5 It s oftn ncssary to rprsnt AC crcut quantts as complx numbrs rathr than as scalar numbrs, bcaus both magntud and phas angl ar ncssary to consdr n crtan calculatons. Whn rprsntng AC voltags and currnts n polar form, th angl gvn rfrs to th phas shft btwn th gvn voltag or currnt, and a rfrnc voltag or currnt at th sam frquncy somwhr ls n th crcut. So, a voltag of 3.5 V 45 o mans a voltag of 3.5 volts magntud, phas-shftd 45 dgrs bhnd (laggng) th rfrnc voltag (or currnt), whch s dfnd to b at an angl of 0 dgrs. But what about mpdanc (Z)? Dos mpdanc hav a phas angl, too, or s t a smpl scalar numbr lk rsstanc or ractanc? Calculat th amount of currnt that would go through a 100 mh nductor wth 36 volts RMS appld to t at a frquncy of 400 Hz. Thn, basd on Ohm s Law for AC crcuts and what you know of th phas rlatonshp btwn voltag and currnt for an nductor, calculat th mpdanc of ths nductor n polar form. Dos a dfnt angl mrg from ths calculaton for th nductor s mpdanc? Explan why or why not. fl Answr 5 Z L = Ω 90 o Nots 5 Ths s a challngng quston, bcaus t asks th studnt to dfnd th applcaton of phas angls to a typ of quantty that dos not rally possss a wav-shap lk AC voltags and currnts do. Concptually, ths s dffcult to grasp. Howvr, th answr s qut clar through th Ohm s Law calculaton (Z = E I ). Although t s natural to assgn a phas angl of 0 o to th 36 volt supply, makng t th rfrnc wavform, ths s not actually ncssary. Work through ths calculaton wth your studnts, assumng dffrnt angls for th voltag n ach nstanc. You should fnd that th mpdanc computs to b th sam xact quantty vry tm. 6

7 Quston 6 Exprss th mpdanc (Z) n both polar and rctangular forms for ach of th followng componnts: A rsstor wth 500 Ω of rsstanc An nductor wth 1.2 kω of ractanc A capactor wth 950 Ω of ractanc A rsstor wth 22 kω of rsstanc A capactor wth 50 kω of ractanc An nductor wth 133 Ω of ractanc fl Answr 6 A rsstor wth 500 Ω of rsstanc: 500 Ω 0 o or j0 Ω An nductor wth 1.2 kω of ractanc: 1.2 kω 90 o or 0 + j1.2k Ω A capactor wth 950 Ω of ractanc: 950 Ω -90 o or 0 - j950 Ω A rsstor wth 22 kω of rsstanc: 22 kω 0 o or 22k + j0 Ω A capactor wth 50 kω of ractanc: 50 kω -90 o or 0 - j50k Ω An nductor wth 133 Ω of ractanc: 133 Ω 90 o or 0 + j133 Ω Nots 6 Follow-up quston: what would th phasors look lk for rsstv, nductv, and capactv mpdancs? In your dscusson wth studnts, mphasz th consstnt natur of phas angls for mpdancs of pur componnts. 7

8 Quston 7 Ral nductors and capactors ar nvr purly ractv. Thr wll nvtably b som rsstanc ntrnsc to ths dvcs as wll. Suppos an nductor has 57 Ω of wndng rsstanc, and 1500 Ω of ractanc at a partcular frquncy. How would ths combnaton b xprssd as a sngl mpdanc? Stat your answr n both polar and rctangular forms. fl Answr 7 Z L = 1501 Ω 87.8 o = 57 + j1500 Ω Nots 7 Mnton to your studnts that ral componnts such as ths may b modld n a dagram as a combnaton of two pur componnts, n ths cas a rsstor and an nductor. Dscuss wth thm th bnfts of modlng componnt charactrstcs n ths mannr, snc t s a vry common practc n ngnrng. Ths s a vry mportant concpt to undrstand: that ractv componnts ar nvr purly ractv. Parastc rsstanc s mpossbl to avod short of usng suprconductors. Evn thn, nductors ar bound to hav som parastc capactanc, and capactors ar bound to hav som parastc nductanc! 8

9 Quston 8 Not only do ractv componnts unavodably possss som parastc ( stray ) rsstanc, but thy also xhbt parastc ractanc of th oppost knd. For nstanc, nductors ar bound to hav a small amount of capactanc bult-n, and capactors ar bound to hav a small amount of nductanc bult-n. Ths ffcts ar not ntntonal, but thy xst anyway. Dscrb how a small amount of capactanc coms to xst wthn an nductor, and how a small amount of nductanc coms to xst wthn a capactor. Explan what t s about th constructon of ths two ractv componnts that allows th xstnc of oppost charactrstcs. fl Answr 8 Capactanc xsts any tm thr ar two conductors sparatd by an nsulatng mdum. Inductanc xsts any tm a magntc fld s prmttd to xst around a currnt-carryng conductor. Look for ach of ths condtons wthn th rspctv structurs of nductors and capactors to dtrmn whr th parastc ffcts orgnat. Nots 8 Onc studnts hav dntfd th mchansms of parastc ractancs, challng thm wth nvntng mans of mnmzng ths ffcts. Ths s an spcally practcal xrcs for undrstandng parastc nductanc n capactors, whch s vry undsrabl n dcouplng capactors usd to stablz powr supply voltags nar ntgratd crcut chps on prntd crcut boards. Fortunatly, most of th stray nductanc n a dcouplng capactor s du to how t s mountd to th board, rathr than anythng wthn th structur of th capactor tslf. 9

10 Quston 9 Suppos you wr gvn a componnt and told t was thr a rsstor, an nductor, or a capactor. Th componnt s unmarkd, and mpossbl to vsually dntfy. Explan what stps you would tak to lctrcally dntfy what typ of componnt t was, and what ts valu was, wthout th us of any tst qupmnt xcpt a sgnal gnrator, a multmtr (capabl of masurng nothng but voltag, currnt, and rsstanc), and som mscllanous passv componnts (rsstors, capactors, nductors, swtchs, tc.). Dmonstrat your tchnqu f possbl. fl Answr 9 Nots 9 Dd you rally thnk I would gv you th answr to ths? Ths s an xcllnt opportunty to branstorm as a group and xprmnt on ral componnts. Thr s obvously mor than on way to mak th dtrmnatons of dntty and valu! Us th class tm to ngag your studnts n lvly dscusson and dbat ovr how to approach ths practcal problm. 10

11 Quston 10 Suppos you wr gvn two componnts and told on was an nductor whl th othr was a capactor. Both componnts ar unmarkd, and mpossbl to vsually dstngush or dntfy. Explan how you could us an ohmmtr to dstngush on from th othr, basd on ach componnt s rspons to drct currnt (DC). Thn, xplan how you could approxmatly masur th valu of ach componnt usng nothng mor than a sn-wav sgnal gnrator and an AC mtr capabl only of prcs AC voltag and currnt masurmnts across a wd frquncy rang (no drct capactanc or nductanc masurmnt capablty), and show how th ractanc quaton for ach componnt (L and C) would b usd n your calculatons. fl Answr 10 Dd you rally thnk I would gv you th answrs to a quston lk ths? Challng quston: suppos th only tst qupmnt you had avalabl was a 6-volt battry and an old analog volt-mllammtr (wth no rsstanc chck functon). How could you us ths prmtv gar to dntfy whch componnt was th nductor and whch was th capactor? Nots 10 Ths s an xcllnt opportunty to branstorm as a group and xprmnt on ral componnts. Th purpos of ths quston s to mak th ractanc quatons mor ral to studnts by havng thm apply th quatons to a ralstc scnaro. Th ohmmtr tst s basd on DC componnt rspons, whch may b thought of n trms of ractanc at a frquncy at or nar zro. Th multmtr/gnrator tst s basd on AC rspons, and wll rqur algbrac manpulaton to convrt th canoncal forms of ths quatons to vrsons approprat for calculatng L and C. 11

12 Quston 11 If a snusodal voltag s appld to an mpdanc wth a phas angl of 0 o, th rsultng voltag and currnt wavforms wll look lk ths: Tm Gvn that powr s th product of voltag and currnt (p = ), plot th wavform for powr n ths crcut. fl Answr 11 p Tm Nots 11 Ask your studnts to obsrv th wavform shown n th answr closly, and dtrmn what sgn th powr valus always ar. Not how th voltag and currnt wavforms altrnat btwn postv and ngatv, but powr dos not. Of what sgnfcanc s ths to us? What dos ths ndcat about th natur of a load wth an mpdanc phas angl of 0 o? 12

13 Quston 12 If a snusodal voltag s appld to an mpdanc wth a phas angl of 90 o, th rsultng voltag and currnt wavforms wll look lk ths: Tm Gvn that powr s th product of voltag and currnt (p = ), plot th wavform for powr n ths crcut. Also, xplan how th mnmonc phras ELI th ICE man appls to ths wavforms. fl Answr 12 p Tm Th mnmonc phras, ELI th ICE man ndcats that ths phas shft s du to an nductanc rathr than a capactanc. 13

14 Nots 12 Ask your studnts to obsrv th wavform shown n th answr closly, and dtrmn what sgn th powr valus ar. Not how th powr wavform altrnats btwn postv and ngatv valus, just as th voltag and currnt wavforms do. Ask your studnts to xplan what ngatv powr could possbly man. Of what sgnfcanc s ths to us? What dos ths ndcat about th natur of a load wth an mpdanc phas angl of 90 o? Th phras, ELI th ICE man has bn usd b gnratons of tchncans to rmmbr th phas rlatonshps btwn voltag and currnt for nductors and capactors, rspctvly. On ara of troubl I v notd wth studnts, though, s bng abl to ntrprt whch wavform s ladng and whch on s laggng, from a tm-doman plot such as ths. 14

15 Quston 13 If a snusodal voltag s appld to an mpdanc wth a phas angl of -90 o, th rsultng voltag and currnt wavforms wll look lk ths: Tm Gvn that powr s th product of voltag and currnt (p = ), plot th wavform for powr n ths crcut. Also, xplan how th mnmonc phras ELI th ICE man appls to ths wavforms. fl Answr 13 p Tm Th mnmonc phras, ELI th ICE man ndcats that ths phas shft s du to a capactanc rathr than an nductanc. 15

16 Nots 13 Ask your studnts to obsrv th wavform shown n th answr closly, and dtrmn what sgn th powr valus ar. Not how th powr wavform altrnats btwn postv and ngatv valus, just as th voltag and currnt wavforms do. Ask your studnts to xplan what ngatv powr could possbly man. Of what sgnfcanc s ths to us? What dos ths ndcat about th natur of a load wth an mpdanc phas angl of -90 o? Th phras, ELI th ICE man has bn usd b gnratons of tchncans to rmmbr th phas rlatonshps btwn voltag and currnt for nductors and capactors, rspctvly. On ara of troubl I v notd wth studnts, though, s bng abl to ntrprt whch wavform s ladng and whch on s laggng, from a tm-doman plot such as ths. 16

17 Quston 14 Spakrs usd for audo rproducton systms (stros, publc addrss systms, tc.) act as powr loads to th amplfrs whch drv thm. Ths dvcs convrt lctrcal nrgy nto sound nrgy, whch thn dsspats nto th surroundng ar. In ths mannr, a spakr acts much lk a rsstor: convrtng on form of nrgy (lctrcal) nto anothr, and thn dsspatng that nrgy nto th surroundng nvronmnt. Naturally, t maks sns to dscrb th natur of such loads n unts of ohms (Ω), so that thy may b mathmatcally analyzd n a mannr smlar to rsstors. Yt, dspt th dsspatv natur of audo spakrs, thr ohms ratng s spcfd as an mpdanc rathr than a rsstanc or a ractanc. Explan why ths s. fl Answr 14 Th trm rsstanc rfrs to th vry spcfc phnomnon of lctrcal frcton, convrtng lctrcal nrgy nto thrmal nrgy. Th trm ractanc rfrs to lctrc currnt opposton rsultng from a nondsspatv xchang of nrgy btwn th componnt and th rst of th crcut. Th trm mpdanc rfrs to any form of opposton to lctrc currnt, whthr that opposton b dsspatv or non-dsspatv n natur. Whl spakrs ar prmarly dsspatv dvcs, most of th nrgy dsspatd by a spakr s not n th form of hat. Nots 14 In a sns, rsstanc may b though of as a spcal (lmtng) cas of mpdanc, just as ractanc s a spcal cas of mpdanc. Dscuss ths concpt wth your studnts, spcally wth rfrnc to dvcs such as spakrs whch ar dsspatv n natur (thy dsspat nrgy) but yt not rsstv n th strct sns of th trm. For ths rason, th word mpdanc fnds broad applcaton n th world of lctroncs, and vn n som scncs outsd of lctrcty/lctroncs! 17

18 Quston 15 Engnrs oftn wrt th capactv and nductv ractanc formula n a dffrnt way from what you may hav sn: X L = ωl X C = 1 ωc Ths quatons should look famlar to you, from havng sn smlar quatons contanng a trm for frquncy (f). Gvn ths quatons forms, what s th mathmatcal dfnton of ω? In othr words, what combnaton of varabls and constants comprs ω, and what unt s t proprly xprssd n? fl Answr 15 Nots 15 ω = 2πf, and t s xprssd n unts of radans pr scond. Studnts who hav takn trgonomtry should rcognz th radan as a unt for masurng angls. Dscuss wth your studnts why multplyng frquncy (f, cycls pr scond) by th constant 2π rsults n th unt changng to radans pr scond. Engnrs oftn rfr to ω as th angular vlocty of an AC systm. Dscuss why th trm vlocty s approprat for ω. 18

19 Quston 16 Engnrs oftn calculat th mpdanc of pur capactancs and pur nductancs n a way that drctly provds rsults n rctangular (complx) form: Z L = jωl Z C = j 1 ωc Th bold-facd typ (Z nstad of Z) sgnfs th calculatd mpdanc as a complx rathr than a scalar quantty. Gvn ths quatons forms, what s th mathmatcal dfnton of ω? In othr words, what combnaton of varabls and constants comprs ω, and what unt s t proprly xprssd n? Also, dtrmn what th quatons would look lk for calculatng th mpdanc of ths srs ntworks: Z =??? Z =??? R R L C fl Answr 16 ω = 2πf s calld th angular vlocty of th crcut, and t s xprssd n unts of radans pr scond. Th mpdanc quatons for th srs LR and RC ntworks ar as follows: Z LR = R + jωl Z RC = R j 1 ωc Nots 16 Studnts who hav takn trgonomtry should rcognz th radan as a unt for masurng angls. Dscuss wth your studnts why multplyng frquncy (f, cycls pr scond) by th constant 2π rsults n th unt changng to radans pr scond. Engnrs oftn rfr to ω as th angular vlocty of an AC systm. Dscuss why th trm vlocty s approprat for ω. 19

20 Quston 17 Th mathmatcal nvrs, or rcprocal, of rsstanc (R) s a quantty calld conductanc (G). G = 1 R Is thr an quvalnt quantty for mpdanc (Z)? What s th rcprocal of mpdanc, and what unt of masurmnt s t xprssd n? Hnt: ts symbol s Y. Is thr an quvalnt quantty for ractanc (X)? What s th rcprocal of ractanc, and what unt of masurmnt s t xprssd n? Hnt: ts symbol s B. fl Answr 17 Y = admttanc, whch s th rcprocal of mpdanc. Admttanc s xprssd n th unt of smns. Y = 1 Z B = suscptanc, whch s th rcprocal of ractanc. Nots 17 B = 1 X Suscptanc s also xprssd n th unt of smns. Ask your studnts whr thy obtand ths nformaton. Also ask thm what th old (pr-smns) unt of masurmnt was. Whr would such quantts b usful n AC crcut calculatons? Ask your studnts whr th quantty of conductanc (G) s usful n DC crcut calculatons. 20

21 Quston 18 Th mathmatcal nvrs, or rcprocal, of rsstanc (R) s a quantty calld conductanc (G). G = 1 R Is thr an quvalnt quantty for ractanc? What s th rcprocal of ractanc, and what unt of masurmnt s t xprssd n? Hnt: ts symbol s B. fl Answr 18 B = suscptanc, whch s th rcprocal of ractanc. Nots 18 B = 1 X Suscptanc, lk conductanc (G) and admttanc (Y ) s xprssd n th unt of smns. Ask your studnts whr thy obtand ths nformaton. Also ask thm what th old (pr-smns) unt of masurmnt was. Whr would such a quantty b usful n AC crcut calculatons? Ask your studnts whr th quantty of conductanc (G) s usful n DC crcut calculatons. 21

22 Quston 19 Don t just st thr! Buld somthng!! Larnng to mathmatcally analyz crcuts rqurs much study and practc. Typcally, studnts practc by workng through lots of sampl problms and chckng thr answrs aganst thos provdd by th txtbook or th nstructor. Whl ths s good, thr s a much bttr way. You wll larn much mor by actually buldng and analyzng ral crcuts, lttng your tst qupmnt provd th answrs nstad of a book or anothr prson. For succssful crcut-buldng xrcss, follow ths stps: 1. Carfully masur and rcord all componnt valus pror to crcut constructon. 2. Draw th schmatc dagram for th crcut to b analyzd. 3. Carfully buld ths crcut on a bradboard or othr convnnt mdum. 4. Chck th accuracy of th crcut s constructon, followng ach wr to ach conncton pont, and vrfyng ths lmnts on-by-on on th dagram. 5. Mathmatcally analyz th crcut, solvng for all voltag and currnt valus. 6. Carfully masur all voltags and currnts, to vrfy th accuracy of your analyss. 7. If thr ar any substantal rrors (gratr than a fw prcnt), carfully chck your crcut s constructon aganst th dagram, thn carfully r-calculat th valus and r-masur. For AC crcuts whr nductv and capactv ractancs (mpdancs) ar a sgnfcant lmnt n th calculatons, I rcommnd hgh qualty (hgh-q) nductors and capactors, and powrng your crcut wth low frquncy voltag (powr-ln frquncy works wll) to mnmz parastc ffcts. If you ar on a rstrctd budgt, I hav found that nxpnsv lctronc muscal kyboards srv wll as functon gnrators for producng a wd rang of audo-frquncy AC sgnals. B sur to choos a kyboard voc that closly mmcs a sn wav (th panflut voc s typcally good), f snusodal wavforms ar an mportant assumpton n your calculatons. As usual, avod vry hgh and vry low rsstor valus, to avod masurmnt rrors causd by mtr loadng. I rcommnd rsstor valus btwn 1 kω and 100 kω. On way you can sav tm and rduc th possblty of rror s to bgn wth a vry smpl crcut and ncrmntally add componnts to ncras ts complxty aftr ach analyss, rathr than buldng a whol nw crcut for ach practc problm. Anothr tm-savng tchnqu s to r-us th sam componnts n a varty of dffrnt crcut confguratons. Ths way, you won t hav to masur any componnt s valu mor than onc. fl Answr 19 Lt th lctrons thmslvs gv you th answrs to your own practc problms! 22

23 Nots 19 It has bn my xprnc that studnts rqur much practc wth crcut analyss to bcom profcnt. To ths nd, nstructors usually provd thr studnts wth lots of practc problms to work through, and provd answrs for studnts to chck thr work aganst. Whl ths approach maks studnts profcnt n crcut thory, t fals to fully ducat thm. Studnts don t just nd mathmatcal practc. Thy also nd ral, hands-on practc buldng crcuts and usng tst qupmnt. So, I suggst th followng altrnatv approach: studnts should buld thr own practc problms wth ral componnts, and try to mathmatcally prdct th varous voltag and currnt valus. Ths way, th mathmatcal thory coms alv, and studnts gan practcal profcncy thy wouldn t gan mrly by solvng quatons. Anothr rason for followng ths mthod of practc s to tach studnts scntfc mthod: th procss of tstng a hypothss (n ths cas, mathmatcal prdctons) by prformng a ral xprmnt. Studnts wll also dvlop ral troublshootng sklls as thy occasonally mak crcut constructon rrors. Spnd a fw momnts of tm wth your class to rvw som of th ruls for buldng crcuts bfor thy bgn. Dscuss ths ssus wth your studnts n th sam Socratc mannr you would normally dscuss th worksht qustons, rathr than smply tllng thm what thy should and should not do. I nvr cas to b amazd at how poorly studnts grasp nstructons whn prsntd n a typcal lctur (nstructor monologu) format! An xcllnt way to ntroduc studnts to th mathmatcal analyss of ral crcuts s to hav thm frst dtrmn componnt valus (L and C) from masurmnts of AC voltag and currnt. Th smplst crcut, of cours, s a sngl componnt connctd to a powr sourc! Not only wll ths tach studnts how to st up AC crcuts proprly and safly, but t wll also tach thm how to masur capactanc and nductanc wthout spcalzd tst qupmnt. A not on ractv componnts: us hgh-qualty capactors and nductors, and try to us low frquncs for th powr supply. Small stp-down powr transformrs work wll for nductors (at last two nductors n on packag!), so long as th voltag appld to any transformr wndng s lss than that transformr s ratd voltag for that wndng (n ordr to avod saturaton of th cor). A not to thos nstructors who may complan about th wastd tm rqurd to hav studnts buld ral crcuts nstad of just mathmatcally analyzng thortcal crcuts: What s th purpos of studnts takng your cours? If your studnts wll b workng wth ral crcuts, thn thy should larn on ral crcuts whnvr possbl. If your goal s to ducat thortcal physcsts, thn stck wth abstract analyss, by all mans! But most of us plan for our studnts to do somthng n th ral world wth th ducaton w gv thm. Th wastd tm spnt buldng ral crcuts wll pay hug dvdnds whn t coms tm for thm to apply thr knowldg to practcal problms. Furthrmor, havng studnts buld thr own practc problms tachs thm how to prform prmary rsarch, thus mpowrng thm to contnu thr lctrcal/lctroncs ducaton autonomously. In most scncs, ralstc xprmnts ar much mor dffcult and xpnsv to st up than lctrcal crcuts. Nuclar physcs, bology, gology, and chmstry profssors would just lov to b abl to hav thr studnts apply advancd mathmatcs to ral xprmnts posng no safty hazard and costng lss than a txtbook. Thy can t, but you can. Explot th convnnc nhrnt to your scnc, and gt thos studnts of yours practcng thr math on lots of ral crcuts! 23