Electricity and Magnetism Lecture 13 - Physics 121 Electromagnetic Oscillations in LC & LCR Circuits,

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1 lectrcty and Magnetsm ecture 3 - Physcs lectromagnetc Oscllatons n & rcuts, Y&F hapter 3, Sec. 5-6 Alternatng urrent rcuts, Y&F hapter 3, Sec. - Summary: and crcuts Mechancal Harmonc Oscllator rcut Oscllatons amped Oscllatons n an rcut A rcuts, Phasors, Forced Oscllatons Phase elatons for urrent and oltage n Smple esstve, apactve, Inductve rcuts. Summary

2 ecap: and crcuts wth constant MF - Tme dependent effects (t) nf (t) c capactv e tme / e t t / (t) e nf (t) constant Growth Phase ecay Phase / (t) nf e t / t / (t) e nductv e tme ( t) constant nf d Now n same crcut, tme varyng MF > New effects Generalzed esstances: eactances, Impedance /( w ) w [ Ohms ] Z New behavor: esonant Oscllaton n rcut w / res () / New behavor: amped Oscllaton n rcut xternal A can drve crcut, frequency w =pf

3 efnton of an oscllatng system: ecall: esonant mechancal oscllatons Perodc, repettve behavor System state ( t ) = state( t + T ) = = state( t + NT ) T = perod = tme to complete one complete cycle State can mean: poston and velocty, electrc and magnetc felds, Mechancal example: Sprng oscllator (smple harmonc moton) Hooke' s Systems that oscllate obey equatons lke ths... aw restorng mech constant Kblock Usp mv d mech Wth solutons lke ths... x(t) x cos( w t ) What oscllates for a sprng n SHM? poston & velocty, sprng force, nergy oscllates between % knetc and % potental: K max = U max Substtutons can convert mechancal to equatons: x v m k / w k /m w / Uel kx U K mv block force : F kx ma O kx d x dx dx dx no frcton m kx v d x w x w k/m opyrght U. Janow Fall 3

4 lectrcal Oscllatons n an crcut, zero resstance a harge capactor fully to = then swtch to b Krchoff loop equaton: - Substtute: U and d d d (t) (t) An oscllator / w equaton where soluton: (t) cos( w t ) d(t) (t) sn( wt ) w What oscllates? harge, current, B & felds, U B, U Peaks of current and charge are out of phase by 9 Show: Total potental energy s constant (t) cos ( wt ) Peak alues Are qual U B w U B (t) sn U tot ( w U t ) U (t) + where U B (t) w c c, b / U U B U tot s constant

5 etals: use energy conservaton to deduce oscllatons The total energy: It s constant so: (t) U UB U (t) du d d d d The defntons and mply that: ( ) d ther (no current ever flows) or: Oscllator soluton: ) d cos( wt d d d oscllator equaton To evaluate w : plug the frst and second dervatves of the soluton nto the dfferental equaton. d w sn( wt ) d w cos( wt ) The resonant oscllaton frequency w s: d cos( w w t ) w

6 Oscllatons Forever? 3 : What do you thnk wll happen to the oscllatons n a true crcut (versus a real crcut) over a long tme? A.They wll stop after one complete cycle. B.They wll contnue forever..they wll contnue for awhle, and then suddenly stop..they wll contnue for awhle, but eventually de away..there s not enough nformaton to tell what wll happen.

7 Potental nergy alternates between all electrostatc and all magnetc two reversals per perod s fully charged twce each cycle (opposte polarty)

8 xample: A 4 µf capactor s charged to = 5., and then dscharged through a.3 Henry nductance n an crcut Use precedng solutons wth = a) Fnd the oscllaton perod and frequency f = ω / π ω (.3)( 4 x 6 ) f ω π 45 Hz 93 rad/s T Perod f π w 6.9 ms b) Fnd the maxmum (peak) current (dfferentate (t) ) (t) cos( w t) (t) max d w d cos( w w t) 4x - w sn 6 x 5 x 93 ( w 8 t) ma c) When does the frst current maxmum occur? When sn(w t) = Maxma of (t): All energy s n feld Maxma of (t): All energy s n B feld urrent maxma at T/4, 3T/4, (n+)t/4 Frst One: T f 6. 9 ms T 4. 7 Others at: 3.4 ms ncr ements ms

9 xample a) Fnd the voltage across the capactor n the crcut as a functon of tme. = 3 mh, = mf The capactor s charged to =. oul. at tme t =. The resonant frequency s: w rad/s 4 (3 H)( F) The voltage across the capactor has the same tme dependence as the charge: (t) cos( wt ) (t) At tme t =, =, so choose phase angle =. 3 (t) cos( 577t) cos( 577t) 4 F volts b) What s the expresson for the current n the crcut? The current s: d 3 w sn( wt) ( )(577rad/s) sn(577t).577sn(577t) amps c) How long untl the capactor charge s reversed? That happens every ½ perod, gven by: T p w 5.44 ms

10 Whch urrent s Greatest? 3 : The expressons below could be used to represent the charge on a capactor n an crcut. Whch one has the greatest maxmum current magntude? A. (t) = sn(5t) B. (t) = cos(4t). (t) = cos(4t+p/). (t) = sn(t). (t) = 4 cos(t) (t) cos( wt ) d(t)

11 Tme needed to dscharge the capactor n crcut 3 3: The three crcuts below have dentcal nductors and capactors. Order the crcuts accordng to ther oscllaton frequency n ascendng order. A. I, II, III. B. II, I, III.. III, I, II.. III, II, I.. II, III, I. I. II. III. w / p T para ser

12 crcuts: Add seres resstance rcuts stll oscllate but oscllaton s damped harge capactor fully to = then swtch to b Stored energy decays wth tme due to resstance U tot U (t) U Underdamped: ω ( ) B (t) esstance dsspates stored energy. The power s: Oscllator equaton results, but wth dampng (decay) term Soluton s cosne wth exponental decay (weak or under-damped case) Shfted resonant frequency w (t) (t) du tot (t) e cos( w't ) rtcally damped ' ω t/ ω' ω can be real or magnary ( / ) (t) d (t) d w(t) w ω ( ) + / ' ω a b Overdamped s magnary / ω ( ) (t) ω ' t - ωt e complcated exponental π t opyrght. Janow Fall t 3

13 esonant frequency wth dampng 3 4: How does the resonant frequency w for an deal crcut (no resstance) compare wth w for an under-damped crcut whose resstance cannot be gnored? A. The resonant frequency for the non-deal, damped crcut s hgher than for the deal one (w > w ). B. The resonant frequency for the damped crcut s lower than for the deal one (w < w ).. The resstance n the crcut does not affect the resonant frequency they are the same (w = w ).. The damped crcut has an magnary value of w. ω' / ω ( / )

14 Alternatng urrent (A) - MF A s easer to transmt than A transmsson voltage can be changed by usng a transformer. ommercal electrc power (home or offce) s A, not. The U.S., the A frequency s 6 Hz. Most other countres use 5 Hz. Sketch: a crude A generator. MF appears n a rotatng a col of wre n a magnetc feld (Faraday s aw) Slp rngs and brushes allow the MF to be taken off the col wthout twstng the wres. Generators convert mechancal energy to electrcal energy. Power to rotate the col can come from a water or steam turbne, wndmll, or turbojet engne. epresent outputs as snusodal functons: nd (t) m snw sn( w ) d t

15 xternal A MF drvng a crcut xternal, snusodal, nstantaneous MF appled to load: (t ) max sn(ωt ) max ampltude The potental across the load ( t ) (t) load ω the drvng frequency ω resonant frequency w, n general urrent n load flows wth same frequency w...but may be retarded or advanced (relatve to ) by phase angle F (due to nerta and stffness /) urrent has the same ampltude and phase everywhere n a branch xample: I( t ) I max sn ( ωt Seres rcut Φ ) (t) I(t) verythng oscllates at drvng frequency w load At resonance : ω ω / F s zero at resonance crcut acts purely resstvely. Otherwse F s + or (current leads or lags appled MF)

16 Instantaneous, peak, average, and MS quanttes for A crcuts max (t ) sn(ωt ) ( t ) sn( ω max Instantaneous voltages and currents: depend on tme through argument: wt perodc, repettve, oscllatory possbly advanced or retarded relatve to each other by phase angles represented by rotatng phasors see below Peak voltage and current ampltudes are just the coeffcents out front max max Smple tme averages of perodc quanttes are zero (and useless). xample: Integrate over a whole number of perods one s enough (wt=p) sn( wt F) Integrand s odd av av t Φ ) (t ) max sn(ωt ) (t ) sn(ω t ) max F MS averages are used the way nstantaneous quanttes were n crcuts MS means root, mean, squared. / / max rms (t ) av sn ( wt F) / / rms (t ) Integrand s even av / max (t ) opyrght. Janow Fall 3

17 max Phasor Pcture: y max F (t ) sn(ωt ) I max w t- F I( t ) max sn ( ωt Φ ) x Show current and potentals as vectors rotatng at frequency w The measured nstantaneous values of ( t ) and ( t ) are projectons of the phasors on the y-axs. The lengths of the vectors are the peak ampltudes. Φ phase angle measures when peaks pass Φ and w t are ndependent urrent s the same (phase ncluded) everywhere n a sngle branch of any crcut. MF (t) appled to the crcut can lead or lag the current by a phase angle F n the range [-p/, +p/]. Seres crcut: elate nternal voltage drops to phase of the current oltage across s n phase wth the current. oltage across lags the current by 9. oltage across leads the current by 9.

18 A crcut, resstance only current and voltage n phase ε m ( t ) Φ Krchoff loop rule: urrent: (t ) (t ) ε M oltage drop across : max ε - (t ) ε(t) sn(ω (t) sn(ω t - Φ) (t) t ) (defnton of resstance) (t ) Peak current and peak voltage are n phase n a purely resstve part of a crcut, rotatng at the drvng frequency w I (t ) I ω t xample ε t 4 εm t, f 6 Hz τ f 6 appled to a Ω esstor. Fnd the current (t ) at t 4 ε M (t) w = pf = p = /6 s. t = /4 max sn(ω t ) π (t ) sn A m sn(p./4) s

19 apactance-only A crcut: current leads voltage by p/ Krchoff loop rule: harge: urrent: Note: max (t ) (t ) cos ( ω t ) ε(t) (t) (t ) max sn(ω d(t ) ω π sn(ωt ) ) max t ); cos(ω urrent leads the voltage by p/ n a pure capactve part of a crcut (F negatve) max 9 Phasor Pcture (t ) c max (t ) (t ) ( max recall max ω t sn(ω eactances are ratos of peak values t ) ; max m t ) ε max sn( w t F) ( t ) so...phase angle F p/ for capactor efnton: capactve reactance (t ) max sn(ω π apactve eactance lmtng cases w : Infnte reactance. blocked. acts lke broken wre. w nfnty: eactance s zero. apactor acts lke a smple wre ω (Ohms ) t )

20 Inductance-only A crcut: current lags voltage by p/ Krchoff loop rule: (t ) cos (ω t ) ( t) (t ) m sn(ω sn(ω t t ) ; π ) m max d d (t) Faraday aw: (t ) max max urrent: (t ) d sn(ω t) cos(ωt) ω Note: max max eactances are ratos of peak values ε max sn( w t F) so...phase angle F p/ for nductor efnton: nductve reactance ω (Ohms ) urrent lags the voltage by p/ n a pure nductve part of a crcut (F postve) Phasor Pcture (t ) (t ) ω t 9 max max max (t ) max sn(ω t π Inductve eactance lmtng cases w : Zero reactance. Inductance acts lke a wre. w nfnty: Infnte reactance. Inductance acts lke a broken wre. )

21 urrent & voltage phases n pure,, and crcuts urrent s the same everywhere n a sngle branch (ncludng phase) Phases of voltages n elements are referenced to the current phasor Apply snusodal voltage (t) = m sn(w t) For pure,, or loads, phase angles are, p/, -p/ eactance means rato of peak voltage to peak current (generalzed resstances). & n phase esstance / max lags by p/ apactve eactance / max w leads by p/ Inductve eactance / w max

22 Seres crcut drven by an external A voltage Apply MF: t) max sn( w ( t) w s the drvng frequency The current (t) s the same everywhere n the crcut (t) max sn( wt F) Same frequency dependance as (t) but... urrent leads or lags (t) by a constant phase angle F Same phase for the current n,,, & m F w t m F m w t-f m w t-f Phasors all rotate W at frequency w engths of phasors are the peak values (ampltudes) The y components are the measured values. Plot voltages n components wth phases relatve to current phasor m : has same phase as m m lags m by p/ mx leads m by p/ X Krchoff oop rule for potentals (measured along y) t) (t) (t) (t) ( m ( ) lags by 8 m along m perpendcular to m

23 Summary: ecture 3/4 hapter 3 - & A rcuts, Oscllatons

24 Summary: and crcut results