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LSU PHYS 2102 - Electrical Oscillations

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Physics 2102Jonathan DowlingLecture 21: THU 01 APRLecture 21: THU 01 APR 20102010Ch. 31.4Ch. 31.4––7: Electrical Oscillations, LC7: Electrical Oscillations, LCCircuits, Alternating CurrentCircuits, Alternating CurrentLC Circuit: At t=0 1/3 Of Energy Utotal is on Capacitor C andTwo Thirds On Inductor L. Find Everything! (Phase ϕ0=?) UB(0)UE0( )=12L q0!sin("0)#$%&212Cq0cos("0)#$%&2=Utotal/ 32Utotal/ 3UB(t) =12L q0!sin(!t +"0)[ ]2UEt( )=12Cq0cos(!t +"0)[ ]2UB(0) =12L q0!sin("0)[ ]2= Utotal/ 3UE0( )=12Cq0cos("0)[ ]2= 2Utotal/ 3 LC q0!sin("0)#$%&2q0cos("0)#$%&2=12 != 1 / LCq0= VC tan(!0) =12!0= arctan 1 / 2( )= 35.3°q = q0cos(!t +"0)i(t) = #q0!sin(!t +"0)$i (t) = #!2q0cos(!t +"0)VL(t) = #q0Ccos(!t +"0)VC(t) =q0Ccos(!t +"0)Damped LCR OscillatorDamped LCR OscillatorIdeal LC circuit without resistance: oscillations goon forever; ω = (LC)–1/2Real circuit has resistance, dissipates energy:oscillations die out, or are “damped”Math is complicated! Important points:– Frequency of oscillator shifts away fromω = (LC)-1/2– Peak CHARGE decays with time constant =– τQLCR=2L/R– For small damping, peak ENERGY decays withtime constant– τULCR= L/RCRL0 4 8 12 16 200.00.20.40.60.81.0Etime (s)Umax=Q22Ce!RtLU22If we add a resistor in an circuit (see figure) we mustmodify the energy equation, because now energy isbeing dissipated on the resistor: .E BRLdUi RdtqU U U= != + =Damped Oscillations in an CircuitRCL222 2Li dU q dq diLi i RC dt C dt dt+ " = + = !( )222/222 210. This is the same equation as thatof the damped harmonics osc 0, which has theillator: The angular f solution re( ) c que:os .bt mmd x dxm b kxdt dtxdq di d q d q dqi L R qdt dt dt dt dtt x e tC! "#= $ = $ + + =+ + =%= +( )2/2222ncy For the damped circuit the solution is:The angular freque1 ( ) cos . .4ncy . 4Rt LRqk bRCt Qe tmLLC Lm! " !!#% %##+== =%(31-6)/ 2Rt LQe!/ 2Rt LQe!( )q tQQ!( )q t( )/ 2( ) cosRt Lq t Qe t! "#$= +2214RLC L!"= #/222The equations above describe a harmonic oscillator with an exponentially decayingamplitude . The angular frequency of the damped oscillator1 is always smaller than the angular 4Rt LQeRLC L!"#= "221frequency of the 1undamped oscillator. If the term we can use the approximation .4LCRL LC!! !=#$!!RC= RC !!!!!!!!!!!!!!!!RL= L / R!!!!!!!!!!!!!RCL= 2L / RSummarySummary• Capacitor and inductor combination producesan electrical oscillator, natural frequencyof oscillator is ω=1/√LC• Total energy in circuit is conserved:switches between capacitor (electric field)and inductor (magnetic field).• If a resistor is included in the circuit, thetotal energy decays (is dissipated by R).Alternating Current:To keep oscillations going we need todrive the circuit with an external emfthat produces a current that goesback and forth.Notice that there are two frequenciesinvolved: one at which the circuit wouldoscillate “naturally”. The other is the frequency at which wedrive theoscillation.However, the “natural” oscillation usually dies offquickly (exponentially) with time. Therefore in thelong run, circuits actually oscillate with the frequencyat which they are driven. (All this is true for thegentleman trying to make the lady swing back andforth in the picture too).We have studied that a loop of wire,spinning in a constant magnetic fieldwill have an induced emf that oscillates with time, E = Emsin(!dt)That is, it is an AC generator.Alternating Current:AC’s are very easy to generate, they are also easy to amplify anddecrease in voltage. This in turn makes them easy to send in distributiongrids like the ones that power our homes. Because the interplay of AC and oscillating circuits can be quitecomplex, we will start by steps, studying how currents and voltagesrespond in various simple circuits to AC’s.AC Driven Circuits:1) A Resistor:0=!Rvemf vR= emf = Emsin(!dt) iR=vRR=EmRsin(!dt)Resistors behave in AC very much as in DC, current and voltage are proportional (as functions of time in the case of AC),that is, they are “in phase”.For time dependent periodic situations it is useful torepresent magnitudes using Steinmetz “phasors”. Theseare vectors that rotate at a frequency ωd , theirmagnitude is equal to the amplitude of the quantity inquestion and their projection on the vertical axisrepresents the instantaneous value of the quantity.CharlesSteinmetzAC Driven Circuits:2) Capacitors: vC= emf = Emsin(!dt) qC= C emf = CEmsin(!dt) iC=dqCdt=!dCEmcos(!dt) iC=!dCEmsin(!dt + 900) iC=EmXsin(!dt + 900)reactance"" 1 whereCXd!= im=EmX looks like i =VRCapacitors “oppose a resistance” to AC (reactance) of frequency-dependent magnitude 1/ωd C (this idea is true only for maximum amplitudes, the instantaneous story is more complex).AC Driven Circuits:3) Inductors: vL= emf = Emsin(!dt)LdtvidtidLvLLLL!="= iL= !EmL"dcos("dt) =EmL!dsin(!dt " 900) iL=EmXsin(!dt " 900) im=EmXdLX!= whereInductors “oppose a resistance” to AC (reactance) of frequency-dependent magnitude ωd L(this idea is true only for maximum amplitudes, the instantaneous story is more complex).Power Station Transmission linesErms =735 kV , I rms = 500 AHome110 VT1T2Step-uptransformerStep-downtransformerR = 220Ω1000 km=!2heat rmsThe resistance of the power line . is fixed (220 in our example). Heating of power lines . This parameter is also fixed (55 MW in our exR RAP I R!= "=Energy Transmission Requirements!trans rms rmsheat transheat rmsample).Power transmitted (368 MW in our example).In our example is almost 15 % of and is acceptable.To keep we must keep as low as possible. The onlP IP PP I= Erms rmsy way to accomplish thisis by . In our example 735 kV. To do that we need a device that can change the amplitude of any ac voltage (either increase or decrease).=increasing E E(31-24)Ptrans=iV= “big”Pheat=i2R= “small”Solution:Big V!Thomas Edison pushedfor the development ofa DC power network.George Westinghousebacked Tesla’s developmentof an AC power network.The DC vs. AC Current WarsNikola Tesla wasinstrumental in developingAC networks.Edison was a brute-force experimenter, but was no mathematician. AC cannot beproperly understood or exploited without a substantial understanding of mathematicsand mathematical physics, which Tesla possessed.The most common example is the Tesla three-phase power system used forindustrial applications and for power transmission. The most obvious advantageof three phase power transmission


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