Slide 1Slide 2The CapacitorSlide 4Voltage in Terms of Current; Capacitor UsesSlide 6Slide 7Stored EnergyA more rigorous derivationExample: Current, Power & Energy for a CapacitorSlide 11Capacitors in ParallelCapacitors in SeriesCapacitive Voltage DividerSlide 15Application Example: MEMS Accelerometer to deploy the airbag in a vehicle collisionSensing the Differential CapacitanceSlide 18Practical CapacitorsThe InductorSlide 21Slide 22Slide 23Inductors in SeriesInductors in ParallelSummarySlide 27Slide 28High-Voltage Direct-Current Power TransmissionRelative advantages of HVDC over HVAC power transmissionSlide 31Slide 32Week 3bEE 42 and 100, Fall 2005 1New topics – energy storage elements Capacitors InductorsWeek 3bEE 42 and 100, Fall 2005 2Books on Reserve for EE 42 and 100 in the Bechtel Engineering Library“The Art of Electronics” by Horowitz and Hill (2nd edition) -- A terrific source book on practical electronics“Electrical Engineering Uncovered” by White and Doering (2nd edition) – Freshman intro to aspects of engineering and EE in particular”Newton’s Telecom Dictionary: The authoritative resource for Telecommunications” by Newton (“18th edition – he updates it annually) – A place to find definitions of all terms and acronyms connected with telecommunications. TK5102.N486 Shelved withdictionaries to right of entry gate.Week 3bEE 42 and 100, Fall 2005 3The CapacitorTwo conductors (a,b) separated by an insulator:difference in potential = Vab=> equal & opposite charges Q on conductorsQ = CVabwhere C is the capacitance of the structure, positive (+) charge is on the conductor at higher potentialParallel-plate capacitor:• area of the plates = A (m2)• separation between plates = d (m) • dielectric permittivity of insulator = (F/m)=> capacitancedAC(stored charge in terms of voltage)F(F)Vab+-+Q-QWeek 3bEE 42 and 100, Fall 2005 4Symbol:Units: Farads (Coulombs/Volt)Current-Voltage relationship:orNote: Q (vc) must be a continuous function of time+vc–icdtdCvdtdvCdtdQicccC C(typical range of values: 1 pF to 1 F; for “supercapa-citors” up to a few F!)+Electrolytic (polarized) capacitorCIf C (geometry) is unchanging, iC = dvC/dtWeek 3bEE 42 and 100, Fall 2005 5Voltage in Terms of Current; Capacitor Uses)0()(1)0()(1)()0()()(000ctctcctcvdttiCCQdttiCtvQdttitQUses: 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 performanceWeek 3bEE 42 and 100, Fall 2005 6Week 3bEE 42 and 100, Fall 2005 7Schematic Symbol and Water Model for a CapacitorWeek 3bEE 42 and 100, Fall 2005 8You might think the energy stored on a capacitor is QV = CV2, which has the dimension of Joules. But during charging, the average voltage across the capacitor was only half the final value of V for a linear capacitor.Thus, energy 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 5 volts stores 63 J; if it discharged at a constant rate in 1 ms energy is discharged at a 63 kW rate!)Stored EnergyCAPACITORS STORE ELECTRIC ENERGYWeek 3bEE 42 and 100, Fall 2005 9FinalInitialcFinalInitialFinalInitialcccVvVvdQ vdttttt dtdQVvVvvdt ivw2CV212CV21VvVvdv CvwInitialFinalFinalInitialcc+vc–icA more rigorous derivationWeek 3bEE 42 and 100, Fall 2005 10Example: Current, Power & Energy for a CapacitordtdvCi –+v(t)10 Fi(t)t (s)v (V)0 2 3 4 51t (s)02 3 4 511i (A)vc and q must be continuousfunctions of time; however,ic can be discontinuous.)0()(1)(0vdiCtvtNote: In “steady state”(dc operation), timederivatives are zero C is an open circuitWeek 3bEE 42 and 100, Fall 2005 11vip 0 2 3 4 51w (J)–+v(t)10 Fi(t)t (s)02 3 4 51p (W)t (s)2021CvpdwtWeek 3bEE 42 and 100, Fall 2005 12Capacitors in Parallel21CCCeqi(t)+v(t)–C1C2i1(t) i2(t)i(t)+v(t)–CeqEquivalent capacitance of capacitors in parallel is the sumdtdvCieqWeek 3bEE 42 and 100, Fall 2005 13Capacitors in Seriesi(t)C1+ v1(t) –i(t)+v(t)=v1(t)+v2(t)–CeqC2+ v2(t) – 21111 CCCeqWeek 3bEE 42 and 100, Fall 2005 14Capacitive Voltage DividerQ: Suppose the voltage applied across a series combination of capacitors is changed by v. How will this affect the voltage across each individual capacitor?21vvv v+vC1C2+ v2(t)+v2–+ v1+v1–+–Note that no net charge cancan be introduced to this node.Therefore, Q1+Q2=0Q1+Q1-Q1Q1Q2+Q2Q2Q2Q1=C1v1Q2=C2v22211 vCvC vCCCv 2112Note: Capacitors in series have the same incremental charge.Week 3bEE 42 and 100, Fall 2005 15MEMS Airbag Deployment AccelerometerChip about 1 cm2 holding in themiddle an electromechanicalaccelerometer around which areelectronic test and calibrationcircuits (Analog Devices, Inc.) Hundreds of millions have beensold.Airbag of car that crashed into the back of a stopped Mercedes. Within 0.3 seconds after deceleration the bag is supposed to be empty. Driver was not hurt in any way; chassis distortion meant that this car was written off.Week 3bEE 42 and 100, Fall 2005 16Application Example: MEMS Accelerometerto deploy the airbag in a vehicle collision•Capacitive MEMS position sensor used to measure acceleration (by measuring force on a proof mass) MEMS = micro-• electro-mechanical systemsFIXED OUTER PLATESg1g2Week 3bEE 42 and 100, Fall 2005 17Sensing the Differential Capacitance–Begin with capacitances electrically discharged–Fixed electrodes are then charged to +Vs and –Vs–Movable electrode (proof mass) is then charged to VoconstgggggggAgAgAgAVVVCCCCVCCCVVsossso12121221212121211)2(C1C2Vs–VsVoCircuit modelWeek 3bEE 42 and 100, Fall 2005 18Flexible conducting diaphragmSound wavesCylindrical air-filled cavityConducting rigid cupCondenser microphone Electret microphoneElectret: insulator(e.g., teflon) that wasbombarded with electronsthat remain imbeddedin it to “bias” thecondenser.Widely used in tele-phone handsets;available at RadioShackGX1EconstxVoutVout ~ xx EconstEconstX1GGGVoutVout ~ x EconstApplication: Condenser
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