EE40Lecture 10Venkat AnantharamChapter 3The CapacitorCapacitorVoltage in Terms of CurrentStored EnergyA more rigorous derivationExample: Current, Power & Energy for a CapacitorCapacitors in series and parallelInductorStored EnergyInductors in Series and ParallelSummarySlide 1EE40 Spring 08 Venkat AnantharamEE40Lecture 10Venkat Anantharam2/13/08 Reading: Chap. 3Slide 2EE40 Spring 08 Venkat AnantharamChapter 3• Outline– The capacitor– The inductorSlide 3EE40 Spring 08 Venkat AnantharamThe CapacitorTwo conductors (a,b) separated by an insulator:difference in potential = Vab=> equal & opposite charge Q on conductorsQ = CVabwhere C is the capacitance of the structure, ¾ positive (+) charge is on the conductor at higher potential. ¾ the net charge is zero.(stored charge on each plate in terms of voltage)Parallel-plate capacitor:• area of the plates = A (m2)• separation between plates = d (m)• dielectric permittivity of insulator = ε(F/m)=> capacitancedACε=F(F)Slide 4EE40 Spring 08 Venkat AnantharamCapacitorSymbol:Units: Farads (Coulombs/Volt)Current-Voltage relationship:or+vc–icdtdvCdtdQicc==C C(typical range of values: 1 pF to 1 µF; for “supercapa-citors” up to a few F!)+Electrolytic (polarized)capacitorCThese have high capacitance and cannotsupport voltage drops of the wrong polarityTo write this it is important to have use a passive convention, otherwise you need a minus sign.Note: vcmust be a continuous function of time since thecharge stored on each plate cannot change suddenlySlide 5EE40 Spring 08 Venkat AnantharamVoltage in Terms of Current)0()(1)0()(1)()0()()(000ctctcctcvdttiCCQdttiCtvQdttitQ+=+=+=∫∫∫Uses: Capacitors are used to store energy for camera flashbulbs,in filters that separate various frequency signals, andthey appear as undesired “parasitic” elements in circuits wherethey usually degrade circuit performanceAt higher frequencies capacitors become increasingly like short circuitsSlide 6EE40 Spring 08 Venkat AnantharamStored EnergyCAPACITORS STORE ELECTRIC ENERGYDuring charging, the average voltage across the capacitor was only half the final value of V for a linear capacitor.Thus, the energy needed tobuild up the charge is.221 21CVQV =Example: A 1 pF capacitance charged to 5 Volts has ½(5V)2(1pF) = 12.5 pJ(A 5F supercapacitor charged to 5volts stores 63 J; if it discharged at aconstant rate in 1 ms energy isdischarged at a 63 kW rate!)Slide 7EE40 Spring 08 Venkat AnantharamA more rigorous derivation∫===∫==∫===⋅=FinalInitialcFinalInitialFinalInitialcccVvVvdQ vdttttt dtdQVvVvvdt ivw2CV212CV21VvVvdv CvwInitialFinalFinalInitialcc−∫====icThis derivation holds independent of the circuit!+vc–Slide 8EE40 Spring 08 Venkat AnantharamExample: Current, Power & Energy for a CapacitordtdvCi =–+v(t)i(t)10 µFt (µs)v (V)0 23451t (µs)0234511i (A)vcand q must be continuousfunctions of time; however,iccan be discontinuous.)0()(1)(0vdiCtvt+=∫ττNote: In “steady state”(dc operation), timederivatives are zeroÆ C is an open circuitSlide 9EE40 Spring 08 Venkat Anantharamvip=0w (J)23451–+v(t)10 µFi(t)01p (W)t (µs)2345t (µs)2021Cvpdwt∫==τSlide 10EE40 Spring 08 Venkat AnantharamCapacitors in series and parallel+ v1(t) –C2+ v2(t) –+v(t)=v1(t)+v2(t)–C1i(t)i(t)Ceq21111CCCeq+=Similarly, for capacitors in parallel, the capacitance adds.Slide 11EE40 Spring 08 Venkat AnantharamInductorSymbol:Units: Henrys (Volts • second / Ampere)Current in terms of voltage:Note: iLmust be a continuous function of timebecause magnetic flux cannot change suddenly+vL–iL∫+==ttLLLLtidvLtidttvLdi0)()(1)()(10ττL(typical range of values: µH to 10 H)To write this it is important to usethe passive configuration.Slide 12EE40 Spring 08 Venkat AnantharamStored EnergyConsider an inductor having an initial current i(t0) = i02022121)()()()()()(0LiLitwdptwtitvtptt−=====∫ττINDUCTORS STORE MAGNETIC ENERGYAt higher frequencies inductors behave increasingly like open circuits.Slide 13EE40 Spring 08 Venkat AnantharamInductors in Series and ParallelCommonCurrentCommonVoltageSlide 14EE40 Spring 08 Venkat AnantharamSummaryCapacitorv cannot change instantaneouslyi can change instantaneouslyDo not short-circuit a chargedcapacitor (-> infinite current!)n cap.’s in series:n cap.’s in parallel:In steady state (not time-varying), a capacitor behaves like an open circuit.Inductori cannot change instantaneouslyv can change instantaneouslyDo not open-circuit an inductor with current (-> infinite voltage!)n ind.’s in series:n ind.’s in parallel:In steady state, an inductor behaves like a short circuit.∑∑====niieqniieqCCCC111121;2dviC w Cvdt==21;2divL w
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