Experiment NE05 Inductor -Resistor-Capacitor (LRC) Series Circuit Laboratory Manual Department of Physics The University of Hong Kong

Transcription

1 Expermen NE05 Inucr -essr-apacr () Seres rcu abrary Manual Deparmen Physcs The Unversy Hng Kng Ams T 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: Thery - Backgrun Inrman Summary: rcu Impenence, Z Phase erence wh Phasr agram 0 leas by 90 lags n by 90 leas lags n by 90 an by 90 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. The 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 (b) Terms. Per, requency an angular requency O sn P - O P T 4 T 3T 4 Fgure 3: sne uncn me graph An alernang curren r e.m.. vares percally wh me n magnue an recn. The smples a.c. uncn s a sne uncn an s shwn n Fgure 3. The general equan a sne uncn s shwn belw. y Amplue Asn Inal Angular phase requency Tme One cmplee alernan s calle a cycle an he number cycles ccurrng n ne secn s erme he requency ( ) he alernang quany. The SI un requency s he herz (Hz) an was prevusly he cycle per secn. The requency he elecrcy supply n Hng Kng s 50Hz whch means ha he uran ne cycle, knwn as he per (T ), s /50 = 0.0s. In general. T elanshps beween per, requency an angular requency: T () r () T (3) T. Smples rm a.c. The 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. T Page

3 a. Peak value, 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 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 T ) -mean-square value (r.m.s. value) a. Denn: The -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,. The average rae a whch supples elecrcal energy he lamp equals he seay rae supply by he.c. ( c..) an n pracce s hs aspec whch s en mpran. b. Pr: In general, cnserng he energy supple a ressr wh ressance, T.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 Graphcally, he area Pac.. curve s equal ha Pc.. 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, sn? Meh : Meh : T / 0 sn 0 sn T sn 0 cs / ( ) ( ) cs ( ) ecr agram r Phasr agram / sn ( ) 0 P y P ' sn cs 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 wh unrm angular velcy, he prjecn OP; ' ' OP n Oy ' a me (measure rm he me when OP passes hrugh OO ' ) s sn. I OP s rece as shwn Fgure 6 by he arrw n, hen we can say ha he prjecn n Y Oy ' he Page 4

5 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. (Smple harmnc mn, y esulan phasr y y Fgure 7: Phase erence beween w alernang quanes The 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 ) The wave rms w such an he crrespnng vecr agram are shwn n Fgure 7 r me. The 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 The ressance, ene by, measures he ably a ressr ppse he passage curren hrugh. The rmula r he ressance a cnucr s l (3) A 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 The curren I hrugh a ressr 3 s n phase wh he vlage apple. 3 Symbl ressr cul be IE-syle ( ) r Amercan-syle ( ). Page 5

6 Fgure 8: Schemac agram pure ressance crcu eerrng he Fgure 8, he a.c. surce s a snusal vlage..e. e apple penal erence acrss ressr, sn (4) Where s he peak value penal erence acrss he ressr. 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 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. The 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. 4 Symbl capacr cul be ( ). Page 6

7 apacance s a measure he capacy sre charge an hereby elecrcal energy 5. The capacance s an nrnsc prpery ha cul be calculae by knwng he mensn an maeral use. Fr parallel plaes nly, he penal erence an elecrc el acrss he capacr wh separan an surace area are: A surace E By elmnang E (0) Q Q Q E 4 r 4 r A () E surace 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. The 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. Ths 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 , sn O A T cs erence Fgure 0: Phase erence beween an nserng a pure capacance crcu as shwn n Fgure, curren are u sep because curren an apple penal lw s a maxmum mmeaely an uncharge capacr s cnnece an a.c. supply. There s as ye n charge n he capacr ppse he arrval charge. Thus 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 n he capacr s ncreasng less. A A he penal erence he apple penal erence s ncreasng bu a a ecreasng rae, he charge Q bu less quckly, whch means ha he chargng curren s a maxmum an r a bre mmen s cnsan. The charge n he capacr wll als be a maxmum an cnsan. The rae lw charge s herere zer, Q.e. he curren s zer. The 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 The charge Q n he capacance a me s Q (4) Fr he curren lwng hrugh he capacr we can wre Page 8

9 = rae change charge Q Q (5) (Denn capacance) v u ( uv) u v (quen rule) where u an v are uncn (6) ( ) 0 as capacance s a cnsan an nepenen (7), sn (8), sn (sn ) sn where an are cnsan an nepenen, sn (9) 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 (35) The 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. Ths expressn resembles whch enes ressance, replacng. The quany s aken as a measure he ppsn a capacr a.c. an s calle capacve reacance r. m. s.. Hence, (38) The hm s he SI un r. m. s. he un s - s (herz) an ha s The 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). The 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 N A Fr an nnely lng slen, he nucance s (4) l, 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. The 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 T 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. Therere 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. Ther 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 ). The 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 () Mahemacal reamen 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 sn (4) where, s s peak value an where s he requency he supply. The 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 sn cs an where s a cnsan an 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) The 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) The ra r.m.s. penal erence ha curren equal he ra peak value penal erence ha curren. r. m. s.,.e. (5) r. m. s. Subsung equan (50) n equan (5) an Page

13 Ths expressn resembles r. m. s., (5) r. m. s. whch enes ressance, replacng. The quany s aken as a measure he ppsn an nucr a.c. an s calle nucve reacance. Hence, r. m. s. (53) r. m. s. Phasr agrams r pure ressance, pure capacance an pure nucance crcu The vecr agram r a pure ressance n an a.c. crcu. The curren s n phase wh apple penal erence (.e. phase erence = 0 r 360 ) as shwn n Fgure 4. The vecr agram r a pure capacance (.e.. nne elecrc ressance) n an a.c. crcu. The 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. The penal erence acrss s n phase wh The vecr agram s as shwn n Fgure 8., an, ha acrss, lags n by 90 (r raans) eerence Phasr Fgure 7: Schemac seres crcu. Fgure 8: Phasr agram seres crcu The vecr sum an equals he apple penal erence S hence Page 3

14 Bu an The quany where (54) S s he capacve reacance S (55) (56) S s calle he mpeance Z 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 han 90 (r less han raans) an s gven by an (58) a phase angle whch s less seres A.. crcu The analyss n Fgure 9 s smlar as seres A.. crcu bu n hs case he penal erence acrss leas n he curren an he penal erence acrss s agan n phase wh. As bere he apple penal erence equals he vecr sum (59) S an, an 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 (60) S 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 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 The crcu rms a smple harmnc scllar r curren an resnance ccurs uner specc cnns. 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 s n phase wh..e. n anphase r u phase. I s greaer han,r n her wrs, recn Bu he reacance. The vecr sum an an are herere 80 (hal a cycle) u phase, s greaer han, her resul equals he apple penal erence (65) S, an where an equals Hence he al mpeance, hence S S s he reacance (66) (67) Z Tal s gven by S ZTal (68) r ZTal (69) an equals s n he, herere an s Page 5

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

17 ressance snce he capacve reacance an nucve reacance cancel each her an he curren has a maxmum value (gven by ). The phase angle (gven by an ) s zer, he apple penal erence an he curren are n phase an here s as sa be resnance. Accrng equan(68), s bane rm Z Tal (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 Expermen : Fnng he resnan requency hrugh curves ng Seup Prceures. Se up he PASO 850 Unversal Inerace an he cmpuer.. nnec banana plug pach crs n he OUTPUTS pr n he PASO 850 Unversal Inerace. Fgure 4. nnecng pwer surce he Inerace he rcu bar as seres crcu 3. nnec he llwng 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 hree vlage sensrs wh each elecrnc cmpnens Page 8

19 Expermenal Prceure Fgure 8. nguran he crcu Fgure 9(a) Oupu requency Panel sgnal generar Fgure 9(b) Sample rae Panel sgnal generar. Open he apsne wrkng le EP_UID.cap n he NE05 ler. lck Save Expermen As uncn save yur wn le whch UID shul be yur/yur parner Unversy n.(e.g. EP5_ cap). lck he upper sgn r lwer sgn nex sample rae g n he Sgnal Generar as shwn n Fgure 9(b). Ajus he Sample ae 0000 Hz. 3. An hen change he Sgnal Generar Au me as shwn n Fg.9 (a), s wll sar an sp aumacally when yu sar an sp measurng aa. 4. Se he Amplue an crrespnng requency as shwn n he wrkshee (Table.) by clckng he upper sgn r lwer sgn as shwn n Fg.9(a) respecvely. 5. lck ecr sar me uran measuremen an hen clck Sp aer 5 secns laer. 6. Wach he Graphs penal erence acrss nucr me respecvely. All curves are shul be a sne uncn., ressr an capacr versus 7. lck he F l an all curves as sne uncn 8. By clckng Save Expermen As save yur wrkng le as EP_UID_0Hz.cap. e.g. Exp_ _0Hz.cap. Page 9

20 Fgure 30 Save Acvy As 9. ecr he Amplue an Phase sne ng curve vlage acrss capacr, ressr an nucr versus me respecvely n he wrkshee (Table.). 0. lck Delee as un an hen selec Delee All uns elee all prevus recre aa. Fgure 3 Delee All/as Daa uns Panel. epea sep (4) sep () wh changng he requency n sep (4) an he le name n sep (9) accrngly. The all measuremen requences are lse n he rs clumn Table. n he wrkshee. Page 0

21 Expermen : Daa Analyss expermen Expermenal Prceure. Open he apsne wrkng le EP_UID.cap. lck Save Acvy As uncn save yur wn le where UID shul be yur/yur parner Unversy n.(e.g. EP_ cap). In he wnws, w ables are shwn.(.e. Table he penal erence acrss he nucr versus requency an he penal erence acrss he capacr versus requency) Fgure 3. Tables 3. Ener he Amplue vs. requency an an vs. requency wh eren requences recre n he Table. wrkshee as he p.. acrss nucr an p.. acrss capacr n he ables shwn n he apsne. 4. Aer enerng he aa, w se aa pns wll shw n he graph paper accrngly. F he aa pns p.. acrss nucr by F Tl an chse prprnal ; an he aa pns p.. acrss capacr by F Tl an chse nverse as shwn n gure 33. Fgure 33. Fng cnguran Fgure 34. rnae Tl Page

22 5. Tw bes curves are shwn an hey nersec a a pn. Zm n he graph n he eal he nersec pn. Use he rnae Tl rea he x an y crnaes he nersecn pn (.e. (x, y)) as shwn n Fgure By clckng Save Expermen As save yur grup le as Exp_UID.cap. (e.g. Exp_ cap) 7. mplee he Table. Table.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 rm Table.3, an hen recr all prperes nucr, ressr an capacr uner resnan requency n he Table 3.. 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