Version 2.1 NE05 - LRC Series Circuit

Transcription

1 ersn. NE05 - Seres rcu Expermen NE05 Inucr -essr-apacr () Seres rcu abrary Manual Deparmen Physcs he Unversy Hng Kng Ams emnsrae varus prperes an seres crcu such as:. Phase erence beween penal erence acrss he crcu elemens an curren he crcu.. esnan requency a seres crcu. 3. Impeance he crcu uner eren alernang crcu (A..) requences. Sel-learnng maeral: hery - Backgrun Inrman Summary: rcu S S S S Impenence, Z Phase erence wh Phasr agram 0 leas by 90 lags n by 90 leas lags n by 90 an by 90 S eerence Phasr Denn: A.. (a) a.c. an.c. In a rec curren (.c.) he r velcy s n ne recn nly. mmn surce.c. s a baery. In an alernang curren (a.c.), he recn he r velcy reverses, usually many mes a secn. A.c. s cmmnly generae by an a.c. generar. he eec a.c. are essenally he same as hse.c. Bh are sasacry r heang an lghng purpses. ac c.. Fgure :.Snusal a.c. Fgure :.cnsan.c. Page

2 ersn. (b) erms. Per, requency an angular requency O sn P - O P NE05 - Seres rcu Fgure 3: sne uncn me graph An alernang curren r e.m.. vares percally wh me n magnue an recn. he smples a.c. uncn s a sne uncn an s shwn n Fgure 3. he general equan a sne uncn s shwn belw. y Amplue Asn Inal Angular phase requency me One cmplee alernan s calle a cycle an he number cycles ccurrng n ne secn s erme he requency ( ) he alernang quany. he SI un requency s he herz (Hz) an was prevusly he cycle per secn. he requency he elecrcy supply n Hng Kng s 50Hz whch means ha he uran ne cycle, knwn as he per ( ), s /50 = 0.0s. In general. elanshps beween per, requency an angular requency: () r () (3). Smples rm a.c. he smples an ms mpran alernang e.m.. can be represene by a sne curve an s sa have a snusal waverm. I can be expresse by he equan: sn 0 (4) where s he e.m.. a me, 0 s he peak r maxmum e.m.. an s a cnsan whch equals where s he requency he e.m.. Smlarly, r a snusal alernang curren we may wre sn 0 (5) 3. 4 mpran values A.. a. Insananeus value, s ene as he e.m.., penal erence, curren an pwer a.c. a ceran me. Is mahemacal expressn a me r penal erence: sn ; r urren: sn. Page

3 ersn. a. Peak value, NE05 - Seres rcu s ene as maxmum r mnmum nsananeus value a.c. Is mahemacal expressn r penal erence an curren are a. Average value/ Mean value, P an Q sn s ene as he mean value a ceran perc quany P s represene by he nan P.I s mahemacal expressn s P 0 P ) -mean-square value (r.m.s. value) a. Denn: he -mean-square (r.m.s.) value an alernang curren (als calle he eecve value) s he seay rec curren whch cnvers elecrcal energy her rms energy n a gven ressance a he same rae as a.c. Alhugh he nsananeus value (curren r e.m..) s varyng,. he average rae a whch supples elecrcal energy he lamp equals he seay rae supply by he.c. ( hs aspec whch s en mpran. b. Pr: In general, cnserng he energy supple a ressr wh ressance, c.. ) an n pracce s.c. P ac.. P c.. Fgure 4: Schemac agrams r a.c. an.c. wh same average pwer upu I he average pwer upu crcus n Fgure 4 s equal,.e. P P c... c. a. c... mean value (6) c.. mean value c (7) square r he mean value he square he curren (8). c. r. m. s. I he a.c. s snusal (.e. he surce s a sne uncn) hen... mean value sn (0) r m s () r. m. s. mean value sn r. m. s. () sn 0 (9) Usually, he symbl mean value r average value a quany x s nae by x r x Usually, he symbl r-mean-square a quany x s nae by x r x Page 3

4 ersn. Graphcally, he area Pac.. curve s equal ha Pc.. NE05 - Seres rcu curve as shwn n Fgure 5. P ac P c P peak P Fgure 5: Pwer generae vs. me graphs r a.c. an.c. surces c. Why mean-square-value sne uncn, 0 cs / ( ) ( ) cs ( ) ecr agram r Phasr agram P y sn P ' Meh : Meh : / 0 sn 0 sn sn / sn ( ) 0? Q sn cs Q sn cs sn cs sn cs sn sn Y O O ' y Y sn Y Fgure 6: ang vecr an alernang quany A snusal alernang quany can be represene by a rang vecr (en calle a Phasr) as shwn n Fgure 6. I s calle a Phasr agram. An alernang quany y Y sn where Y s he peak value he quany an s requency. I he lne OP has lengh Y (.e. raus an magnary crcle) an raes n an anclckwse recn abu O OP; ' ' OP n Oy ' a me (measure rm he me when OP passes hrugh wh unrm angular velcy, he prjecn OO ' ) s I OP s rece as shwn Fgure 6 by he arrw n, hen we can say ha he prjecn n Y sn. Oy ' he Page 4

5 ersn. rang vecr OP gves he value a any nsan he snusal quany beng snusal, can be erve smlarly rm unrm mn n a crcle.) y Y sn y Y sn Y Y y y NE05 - Seres rcu. (Smple harmnc mn, y esulan phasr y y Fgure 7: Phase erence beween w alernang quanes he Phasr agram s very useul r represenng w snusal quanes whch have he same requency r same angular requency bu are n n phase (.e. quanes y an y erence beween hem s, wh 0 ) he wave rms w such an he crrespnng vecr agram are shwn n Fgure 7 r me. he phase y laggng, an hs phase angle s manane beween hem as he vecrs rae. Beng vecrs hey can be ae by he parallelgram law hey represen smlar quanes, e.g. p.s. r currens Algebracally an are expresse by he equans: y Y sn an y Y sn y y essr: (a) essance n a.c. crcu he ressance, ene by, measures he ably a ressr ppse he passage curren hrugh. he rmula r he ressance a cnucr s l A (3) where s he elecrcal ressvy he maeral, l s he lengh, an A s crss-secn area he cnucr. Fr hmc maeral, he curren passng hrugh s recly prprnal penal erence acrss ha maeral..e. Obeys Ohm s law., where s a cnsan. (b) Pure ressve crcu he curren I hrugh a ressr 3 s n phase wh he vlage apple. S Fgure 8: Schemac agram pure ressance crcu 3 Symbl ressr cul be IE-syle ( ) r Amercan-syle ( ). Page 5

6 ersn. eerrng he Fgure 8, he a.c. surce s a snusal vlage..e. e apple penal erence acrss ressr, (4) Where sn s he peak value penal erence acrss he ressr. S NE05 - Seres rcu Accrng Ohm s law, he curren s als a snusal uncn. an n phase wh vlage apple sn (5) e Where Hence an (6) s he peak value curren passng hrugh he ressr. As a resul, are n phase, als sn (7) I, r. m. s (8) r. m. s. ke he case n.c., he peak curren an penal erence acrss. passng hrugh a ressr epens n bh he ressance apacance n a.c. crcus (a) Denn A smple capacr 4 cnsss w parallel recangular cnucng plaes separae by a elecrc (.e. ar, plymers, quarz an glass ec), where a elecrc s a kn nsular whch can be plarze by applyng an elecrc el. When a capacr s cmpleely charge by a baery (.e..c. surce), cnans equal bu ppse charges n he w plaes,.e. ne plae gans elecrns an he her lses he same amun elecrns hrugh he crcu. I he charge sre by a ceran capacr s Q, acually means ha ne plae sres charge Q, an he her plae sres charge Q. Please ne ha here s n curren pass hrugh he capacr by usng.c. surce. Bu here s a vrual curren passng hrugh he capacr by usng a.c. surce. he capacance, ene by, s ene as he charge sre per un vlage apple he capacr. In mahemacal rm, we have Q (9) where Q s he charge sre an s he penal erence acrss he plaes. In SI uns, - capacance s measure n aras ( F ) r J. apacance s a measure he capacy sre charge an hereby elecrcal energy 5. he capacance s an nrnsc prpery ha cul be calculae by knwng he mensn an maeral use. 4 Symbl capacr cul be ( ). Page 6

7 ersn. NE05 - Seres rcu Fr parallel plaes nly, he penal erence an elecrc el acrss he capacr wh separan an surace area A surace are: ur E (0) ur Q Q ur Q E E 4 r 4 r A surace By elmnang E ur n he abve equans, r a parallel-plae capacr, he capacance s () A where s he permvy he elecrc maeral separang he w plaes, beween he plaes, an A s he verlappng area w plaes. (b) Flw a.c. hrugh a capacr ma, r. m. s. ma () s he separan 000F.5 0.3A Fgure 9: Phase erence beween 000F an.5 0.3A As shwn n Fgure 9, a 000F capacr s cnnece n seres wh a.5, 0.3 A lamp, an a.c. supply, he lamp as expece, es n lgh. Is here any curren lw? Wh a r.m.s. 50 Hz supply, s nearly ully l. he a.c. s apparenly lwng hrugh he capacr. In ac, he capacr s beng charge, scharge, charge n he ppse recn an scharge agan, y mes per secn (he requency he a.c.), an he chargng an schargng currens lwng hrugh he lamp lgh. N curren acually passes hrugh he capacr (snce s plaes are separae by an nsular(.e. elecrc maeral) bu appears s an we alk as. A curren wul ceranly be recre by an a.c. mllammeer. (c) Phase relanshps When a.c. lws hrugh an eal ressr (havng n capacance r nucance) he curren an penal erence reach her peak values a he same nsan,.e. hey are n phase. hs s n s r a capacr. Fr a capacr, he curren hrugh s seen lea he penal erence acrss by nequarer a cycle,.e. he curren reaches s maxmum value ne-quarer a cycle bere he penal erence reaches s peak value. Page 7

8 ersn. NE05 - Seres rcu, sn O A cs Fgure 0: Phase erence beween an nserng a pure capacance crcu as shwn n Fgure, curren an apple penal erence are u sep because curren lw s a maxmum mmeaely an uncharge capacr s cnnece an a.c. supply. here s as ye n charge n he capacr ppse he arrval charge. hus a O he apple penal erence, hugh mmenarly zer, s ncreasng a s maxmum rae (he slpe he angen a O he penal erence graph s a maxmum) an s he rae lw charge he curren, s als a maxmum. Beween O an A he penal erence n he capacr s ncreasng Q less. A A Q he apple penal erence s ncreasng bu a a ecreasng rae, he charge bu less quckly, whch means ha he chargng curren s s a maxmum an r a bre mmen s cnsan. he charge n he capacr wll als be a maxmum an cnsan. he rae lw charge s herere zer, Q.e. he curren s zer. he phase erence beween an can hus be explane. () Mahemacal reamen nserng a pure capacance crcu as shwn n Fgure, le a penal erence apple acrss a capacance an le s value a me be gven by where,, sn (3) s s peak value an where s he requency he supply. be S Fgure : Schemac agram pure capacance crcu he charge Q n he capacance a me s Q (4) Fr he curren lwng hrugh he capacr we can wre Page 8

9 ersn. = rae change charge Q (5) NE05 - Seres rcu Q Q (Denn capacance) ( uv) u v v u (6) Q ( ) 0 as capacance s a cnsan an nepenen (7) Q (quen rule) where u an v are uncn Q, sn, sn (8) (sn ) sn Q where an are cnsan an nepenen, sn (9) Q sn cs an, where s a cnsan an nepenen, cs (30), cs (3) Mahemacally, snce cs sn Equan (3) becmes,, cs, sn By cmparng, sn wh, sn, he curren hrugh capacr (a csne uncn) hus leas he apple penal erence (a sne uncn) by ne quarer a cycle r, as s en sae, by raans 6 r 90 ( cycle beng regare as raans r 360 ) as shwn n (3) Fgure 0. e, (33) cs (34) Accrng equan (33), 6 Whu speccally menne, all angle shul be n erms raans raher han egree. Page 9

10 ersn. (35) NE05 - Seres rcu he ra r.m.s. penal erence ha curren equal he ra peak value penal erence ha curren. r. m. s., (36) r. m. s. By subsung equan (35) n equan (36) an r. m. s. (37) r. m. s. hs expressn resembles whch enes ressance, replacng. he quany s aken as a measure he ppsn a capacr a.c. an s calle capacve reacance r. m. s.. Hence, (38) he hm s he SI un r. m. s. he un s - s (herz) an ha s he erm - hen has uns A. I s clear ha ecreases as an ncrease. s eacance s n be cnuse wh ressance; n he laer elecrcal pwer s sspae, whereas s n n a reacance. Inucr (a) Denn An nucr 7 s a evce srng energy n magnec el, r example cls r slens (.e. a seres cls). he nucance, ene by, s he ably ha an nucr sre energy n a magnec el. In parcular, he erm sel-nucance s use escrbe he behavr generang an ppsng elecrmve rce (r calle back e.m..) prprnal he rae change n curren n a crcu. I s Faraay s aw 8,.e. back (39) r back (40) 7 Symbl an nucr/ a slen cul be wh errmagnec maerals ( ) r whu errmagnec maerals ( ). N B N 8 back NA NA where BA l, N B, N s he number urns l n cls n he slen, s he lengh he slen, A s he crss-secnal area slen an s he curren passng hrugh he slen N A N A back an cmparng back l l l Page 0

11 ersn. Fr an nnely lng slen, he nucance s N A (4) l NE05 - Seres rcu, where s he permeably he maeral nse he slen, N s he number urns cls n he slen, A s he crss-secn area he cl, l s he lengh he cl. he SI un nucance s n henrys. (e) Phase relanshps An nucr n an a.c. crcu as shwn n Fgure 3 behaves lke a capacr n ha causes a phase erence beween he apple penal erence an he curren. In hs case, hwever, he curren lags n he penal erence by ne-quarer a cycle (.e. 90 ) as shwn n Fgure., cs sn O A back back back, cs Fgure : Phase erence beween an In Fgure, a O, he curren s zer bu s rae ncrease s a maxmum (as gven by he slpe he angen he curren graph a O r max ) whch means, r an nucr cnsan nucance, ha he rae change lux s als a maxmum ( B A max )9. herere by Faraay s law he back e.m.. (r calle nuce e.m..)s a maxmum ( max ) bu, by enz s law, negave sgn snce acs ppse he curren change. A A he curren an lux are mmenarly a maxmum an cnsan. her rae change s zer (slpe angen curren graph s zer a A r 0 an B A max ) an s he back e.m.. s zer ( nuce 0 ). I he nucr has neglgble ressance, hen a every nsan he apple penal erence mus be nearly equal an ppse he back e.m.. (.e. nuce ). he penal erence acs n he cl whls he e.m.. acs back upn he surce, jus lke w rces acng n eren bes. nuce 9 Fr a lng slen, N B B l Page

12 ersn. () Mahemacal reamen S NE05 - Seres rcu Fgure 3: Schemac agram pure nucance crcu In hs case, s smpler sar wh a pure nucance curren as shwn n Fgure 3. nser an nucr wh nucance hrugh whch curren lws a me where where, sn (4) s s peak value an where s he requency he supply. he back e.m.. (r calle nuce e.m..) n he nucr ue changng curren s back (43) Subsung equan (4) n equan (43) back sn (44) s a cnsan an nepenen back sn (45) sn Q sn cs an where s a cnsan an Q nepenen where s a cnsan an nepenen cs (46) Assumng he nucr has zer ressance, hen r curren lw he apple penal erence mus be equal an ppse he back e.m.., hence (47) Subsung equan (46) n equan (47) cs (48) he apple penal erence s gven by, cs (49) back back where where s he requency he supply an, s s peak value. Accrng equan (48) s gven by, (50) he ra r.m.s. penal erence ha curren equal he ra peak value penal erence ha curren. r. m. s.,.e. Q (5) r. m. s. Subsung equan (50) n equan (5) an Page

13 ersn.. Hence, hs expressn resembles r. m. s., (5) r. m. s. whch enes ressance, NE05 - Seres rcu replacng. he quany s aken as a measure he ppsn an nucr a.c. an s calle nucve reacance r. m. s. (53) r. m. s. Phasr agrams r pure ressance, pure capacance an pure nucance crcu he vecr agram r a pure ressance n an a.c. crcu. he curren s n phase wh apple penal erence (.e. phase erence = 0 r 360 ) as shwn n Fgure 4. he vecr agram r a pure capacance (.e.. nne elecrc ressance) n an a.c. crcu. he curren leas he apple penal erence by 90.(.e. phase erence = 90 ) as shwn n Fgure 5. Fr a pure nucance (.e. zer ressance), n hs case he curren erence by 90.(.e. phase erence = 90 ) as shwn n Fgure 6. Fgure 4: Phasr agram an seres A.. crcu Fgure 5: Phasr agram an Suppse an alernang penal erence lags n he apple penal Fgure 6: Phasr agram an s apple acrss a ressance an a capacance n seres as shwn n Fgure 7. Because n seres arrangemen. he same curren lws hrugh each cmpnen an s he reerence vecr wll be ha represenng. he penal erence acrss s n phase wh shwn n Fgure 8., an, ha acrss, lags n by 90 (r S raans) he vecr agram s as eerence Phasr Fgure 7: Schemac seres crcu. Fgure 8: Phasr agram seres crcu he vecr sum an equals he apple penal erence S hence S Page 3

14 ersn. Bu an he quany where (54) S s he capacve reacance S (55) (56) S s calle he mpeance Z NE05 - Seres rcu an equals, hence he crcu an measures s ppsn a.c. I has ressve an reacve cmpnens an lke bh s measure n hms. Hence (57) S Z Als, rm he vecr agram we see ha he curren leas by S a phase angle whch s less han 90 (r less han raans) an s gven by an (58) seres A.. crcu he analyss n Fgure 9 s smlar as seres A.. crcu bu n hs case he penal erence leas n he curren an he penal erence acrss s agan n phase wh. As acrss bere he apple penal erence S S equals he vecr sum (59) S an, an s S eerence Phasr Fgure 9: Schemac seres crcu. Fgure 0: Phasr agram seres crcu Bu an where s he reacance an equals, hence Hence he mpeance Z S (60) S (6) s gven by S Z (6) Als, rm he vecr agram we see ha he curren lags n S by a phase angle whch s less han 90 (r less han raans) an s gven by an (63) Page 4

15 ersn. NE05 - Seres rcu seres a.c. crcu A crcu as shwn n Fgure cnsss an nucr, a ressr an a capacr s calle an crcu. I can be cnnece n seres r n parallel. In hs expermen, nly he seres crcu s cuse. In any seres crcu, all crcu elemens share he same curren a any pn n he crcu..e. ( ) ( ) ( ) ( ) (64) S he crcu rms a smple harmnc scllar r curren an resnance ccurs uner specc cnns. S S eerence Phasr Fgure : Schemac seres crcu. Fgure : Phasr agram seres crcu Accrng he analyss n, seres crcus, leas he curren (reerence) vecr by 90, lags n by 90, an.e. n anphase r u phase. s n phase wh. I s greaer han,r n her wrs,. he vecr sum recn Bu he reacance an s greaer han are herere 80 (hal a cycle) u phase,, her resul s n he an equals he apple penal erence S, herere (65) S, an where an equals, hence Hence he al mpeance S S s he reacance (66) (67) Z al s gven by S Zal (68) r Zal (69) an equals an s Page 5

16 ersn. Als, rm he vecr agram we see ha he curren lags n less han 90 (r less han Elecrcal resnance Seres resnance Impenence ) raans an s gven by an (70) Z al an S (7) NE05 - Seres rcu by a phase angle whch s urren n a crcu resnance rcu seres resnance cnns: 0 Frequency he a.c. surce Fgure 3: Fgures mpenence vs. requency an curren resnance Zal he equan (68) jus erve r he al mpeance Z al n a seres crcu shws ha Z vares wh he requency he apple penal erence snce an bh epen n.he relanshps are shwn n upper panel Fgure 3. an ncreases wh, ( Q ) an ecreases wh ( Q ), s assume be nepenen (bu can vary). A a ceran requency, calle he resnan requency when an Z al has s mnmum value, beng equal ( Q Zal ). he crcu behaves as pure Page 6

17 ersn. NE05 - Seres rcu ressance snce he capacve reacance an nucve reacance cancel each her an he curren has a maxmum value (gven by ). he phase angle (gven by an ) s zer, he apple penal erence an he curren are n phase an here s as sa be resnance. S Accrng equan (68), s bane rm Zal (7), ha s By subsung equans (38) an (53) n equan (7), (73) r 4 (74) I s n henrys an A resnan requency he a.c. surce s maxmum. n aras, (75) wll be n herz., he physcal sgncan s ha he energy ranser crcu rm Page 7

18 ersn. NE05 - Seres rcu Expermen : Fnng he resnan requency hrugh curves ng Seup Prceures. Se up he PASO 750 Inerace an he cmpuer an sar DaaSu.. nnec banana plug pach crs n he OUPU prs n he PASO 750 Inerace. Fgure 4. nnecng pwer surce he Inerace he rcu bar as seres crcu 3. nnec he lage Sensrs n he nerace. Fgure 5. lage Sensr n hannel A r measurng p.. acrss ressr, Fgure 6. lage Sensr n hannel B r measurng p.. acrss nucr Fgure 7. lage Sensr n hannel r measurng p.. acrss capacr, 4. Ensure he crcu s cnnece he 00 ressr, 00 F capacr, 8. mh nucr n seres an 3 vlage sensrs wh each elecrnc cmpnens Page 8

19 ersn. Expermenal Prceure NE05 - Seres rcu Fgure 8. nguran he Fgure 9. Panel sgnal generar crcu. Open he DaaSu wrkng le NE05_EP_UID.s. lck Save Acvy As uncn creae yur grup le where UID shul be yur/yur parner Unversy n.(e.g. NE05_EP_ s). lck he plus sgn nex Measuremens An Sample ae n he Sgnal Generar as shwn n Fgure 9. Ajus he Sample ae 0000 Hz. he Sgnal Generar s se Au s wll sar an sp aumacally when yu sar an sp measurng aa. 3. Se he Amplue an requency 0 Hz he Sgnal Generar an press ener respecvely 4. lck Sar sar measuremen. 5. Wach he Graphs penal erence acrss nucr, ressr an capacr me respecvely. All curves are shul be a sne uncn. 6. Wa r runnng abu 5 secns. 7. lck Sp sp measuremen. 8. Hghlgh he prn he waverm sarng rm 0 secn, a leas w cmplee cycles. 9. lck he F l an all curves as sne uncn 0. By clckng Save Acvy as, save he le as EP_UID_Hz.s whch means exsng applyng requency. versus Fgure 30 Save Acvy As Fgure 3 Save Acvy As Page 9

20 ersn.. ecr he Amplue an Phase sne ng curve vlage acrss capacr nucr versus me respecvely n he able. he wrkshee. lck Delee All Daa uns elee all he aa pns. NE05 - Seres rcu, ressr an Fgure 3 Delee A Daa uns 3. epea sep (4) sep () wh changng he requency n sep () an he le name n sep (9) accrngly. he measuremen requences are lse n he rs clumn able. n he wrkshee. Expermen : Daa Analyss expermen Expermenal Prceure. Open he DaaSu wrkng le NE05_EP_UID.s. lck Save Acvy As uncn creae yur grup le where UID shul be yur/yur parner Unversy n.(e.g. NE05_EP_ s). In DaaSu, w ables are shwn..e. able he penal erence acrss he nucr versus requency an he penal erence acrss he capacr versus requency are shwn. Fgure 33. ables vs. requency an vs. requency Fgure 34. Inser ws by rgh clck n he able 3. Ener he Amplue an wh eren requences recre n he able. wrkshee as he p.. acrss nucr an p.. acrss capacr n he ables shwn n he DaaSu. Page 0

21 ersn. NE05 - Seres rcu 4. Aer enerng he aa, w se aa pns shwn n he graph he DaaSu. F he aa pns p.. acrss nucr by Fng l an chse prprnal ; an he aa pns p.. acrss capacr by Fng l an chse nverse as shwn n gure 35 an 36 respecvely. Fgure 35. Fng cnguran Prprnal Fgure 36. Fng cnguran Inverse F 5. w bes curves are shwn an hey nersec a a pn. Use he Zm Selec see he eal he nersec pn. Use he Smar l rea he x an y crnaes he nersecn pn (.e. (x, y)). 6. By clckng Save Acvy as, save he le as Exp_UID.s. (e.g. Exp_ s) 7. mplee he able. able.3 n wrkshee. Expermen 3: Knwng he prperes he crcu a resnan requency Expermenal Prceure. epea he prceure llusrae n Expermen by usng he resnan requency un n able.3 n wrkshee, cmplee he able 3. n wrkshee. eerences: apacr:. haper 6, Physcs r Scenss an Engneers wh Mern Physcs 8 h En, Jhn Jewe an aymn Serway Inucr:. haper 3, Physcs r Scenss an Engneers wh Mern Physcs 8 h En, Jhn Jewe an aymn Serway Alernang rcu an crcu: 3. haper 33, Physcs r Scenss an Engneers wh Mern Physcs 8 h En, Jhn Jewe an aymn Serway 4. hp://hyperphyscs.phy-asr.gsu.eu/hbase/elecrc/rlcser.hml Inucance a slen: 5. hp://hyperphyscs.phy-asr.gsu.eu/hbase/magnec/ncur.hml#c Page