L Oscillatios Kirchoff s loop rule I di Q VL V L dt ++++ - - - - L 3/27/28
, r Q.. 2 4 6 x 6.28 I. f( x) f( x).. r.. 2 4 6 x 6.28 di dt f( x) Q Q cos( t) I Q si( t) di dt Q cos( t) 2 o x, r.. V. x f( ) r. 2 4 6. x 6.28 V f( x) L. L Oscillatios 2 4 6 t x 6.28. 2 4 6 3/27/28 t 2 x 6.28
Exaple t= t=t + + Q - - = L = L Q o Q At t=, the capacitor i the L circuit show has a total charge Q. At t = t, the capacitor is ucharged. What is the value of V ab, the voltage across the iductor at tie t? (a) V ab < (b) V ab = (c) V ab > 3/27/28 3
Exaple 2 At t= the capacitor has charge Q ; the resultig oscillatios have frequecy. The axiu curret i the circuit durig these oscillatios has value I. What is the relatio betwee I ad I 2, the axiu curret i the circuit whe the iitial charge = 2Q? + + Q = - - Q o t= L (a) I 2 = I (b) I 2 = 2I (c) I 2 = 4I 3/27/28 4
Exaple 3 At t= the capacitor has charge Q ; the resultig oscillatios have frequecy. The axiu curret i the circuit durig these oscillatios has value I. What is the relatio betwee ad 2, the frequecy of oscillatios whe the iitial charge = 2Q? + + Q = - - Q o t= L (a) 2 = /2 (b) 2 = (c) 2 = 2 3/27/28 5
L Oscillatios: Eergy heck Oscillatio frequecy fro the loop equatio. has bee foud The other ukows ( Q, ) are foud fro the iitial coditios. e.g. i our origial exaple we took as give, iitial values for the charge (Q i ) ad curret (). For these values: Q = Q i, =. Questio: Does this solutio coserve eergy? L 3/27/28 6
Eergy heck x, r.. r Eergy i apacitor 2 2 U E ( t) Q cos ( t ) 2 Eergy i Iductor 2 2 2 U B( t) L Q si ( t ) 2 L 2 2 U B( t) Q si ( t ) 2 Therefore, U ( t) U ( t) E B Q U E x, r.. r 2 2 3/27/28 7 f( x) f( x).5 U B.5 t 2 4 6 x 2 4 6 t
U B versus U E 3/27/28 8
L Oscillatios with Fiite R, r If L has fiite R, the eergy will be dissipated i R ad the oscillatios will becoe daped... r x, r.. r r Q Q f( x) f( x) t x R = 5 R t 5 x 3/27/28
Drive Oscillatios A L circuit is a atural oscillator. resoace L i abseceof resistive loss I a real L circuit, we ust accout for the resistace of the iductor. This resistace will dap out the oscillatios. + + - - x, r.. r Q R r L f( x) 5 x t 3/27/28
A ircuits: Series LR R Stateet of proble: Give = sit, fid I(t). Everythig else will follow. L 3/27/28 2
Phasors: LR Give: sit Assue: I I si( ) t Q di dt I I V si( ) R RI RI t Q V Icos( t ) di V L LI cos( t ) L dt Fro these equatios, we ca draw the phasor diagra at the right. cos( t ) cos( t ) I X 3/27/28 3 I X L I R
Phasors: LR I X I X L I R I (X L -X ) XL X ta R I R X L X L 2 Z R X L X 2 3/27/28 4 I R X X I 2 2 2 L 2 R X L X 2 Z 2
Phasors: LR XL X ta R I R X X 2 2 2 L 2 3/27/28 5
Resoace For fixed R,,L the curret I will be a axiu at the resoat frequecy which akes the ipedace Z purely resistive. ie: I Z 2 R X 2 L X reaches a axiu whe: X X the frequecy at which this coditio is obtaied is give fro: ol o L o Note that the resoat frequecy is idetical to the atural frequecy of the L circuit by itself! At this frequecy, the curret ad the drivig voltage are i phase! XL X ta R 3/27/28 8 L
Power i LR ircuit The power supplied by the ef i a series LR circuit depeds o the frequecy. The axiu power is supplied at the resoat frequecy. The istataeous power (for soe frequecy, ) delivered at tie t is give by: 2 P( t) ( t) I( t) sit I si( t ) I ( t) R The ost useful quatity to cosider here is ot the istataeous power but rather the average power delivered i a cycle. P( t) I sit si( t ) P( t) I cos 2 3/27/28 9
Power i LR ircuit This result is ofte rewritte i ters of rs values: rs 2 I rs I 2 P ( t ) r s I r s co s Power delivered depeds o the phase,, the power factor Phase depeds o the values of L,, R, ad ad therefore... X X ta L R cos 3/27/28 2 R Z
Maxwell Equatios: Electroagetic Waves Maxwell s Equatios cotai the wave equatio The velocity of electroagetic waves: c = 2.99792458 x 8 /s The relatioship betwee E ad B i a EM wave Eergy i EM waves: the Poytig vector x z y 3/27/28 22
The equatios so far... Gauss Law for E Fields E da S Q iside Gauss Law for B Fields B da S Faraday s Law d E dl dt S B da Apere s Law B dl I 3/27/28 23
A proble with Apere s Law osider a wire ad a capacitor. is a loop. Tie depedet situatio: curret flows i the wire as the capacitor charges up or dow. 3/27/28 24
Maxwell s Displaceet urret, I d I d de d E da dt dt Puttig ito chagig electric flux just as d dt B dl I I d dt e, this eas that a results i a agetic field, gives rise to a electric field. d 3/27/28 25
alculatig Displaceet urret osider a parallel plate capacitor with circular plates of radius R. If charge is flowig oto oe plate ad off the other plate at a rate I = dq/dt what is I d? The displaceet curret is ot a curret. It represets agetic fields geerated by tie varyig electric fields. 3/27/28 26
alculatig the B field A 2 B B 4 B 2 T de dt 4 V / V 5.s s A de 2r 7 dt Exaple N A 2 8.85 2 2 N 2 2 5 V s 3/27/28 27
Maxwell s Equatios (865) S S E B E dl B dl I da da Q :soeties d dt I S iside S B I d E t : Gauss's law da called I S B t Gauss's law for agetis da: Faraday's law da: Apere- Maxwell law i Systee Iteratioal (SI or ks) uits 3/27/28 28 d dt S E da