Week 3bEECS 42, Spring 2005New topics – energy storage elementsCapacitorsInductorsWeek 3bEECS 42, Spring 2005Books on Reserve for EECS 42 in Engineering Library“The Art of Electronics” by Horowitz and Hill (1stand 2ndeditions) -- A terrific source book on electronics“Electrical Engineering Uncovered” by White and Doering (2ndedition) – Freshman intro to aspects of engineering and EE in particular”Newton’s Telecom Dictionary: The authoritative resource for Telecommunications” by Newton (“18thedition– he updates it annually) – A place to find definitions of all terms and acronyms connected with telecommunications“Electrical Engineering: Principles and Applications” by Hambley (3rdedition) – Backup copy of text for EECS 42Week 3bEECS 42, Spring 2005The EECS 42 Supplementary Reader is now availableat Copy Central, 2483 Hearst Avenue (price: $12.99)It contains selections from two textbooks thatwe will use when studying semiconductor devices:Microelectronics: An Integrated Approach(by Roger Howe and Charles Sodini)Digital Integrated Circuits: A Design Perspective(by Jan Rabaey et al.) ReaderWeek 3bEECS 42, Spring 2005The 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 potentialParallel-plate capacitor:• area of the plates = A (m2)• separation between plates = d (m)• dielectric permittivity of insulator = ε(F/m)=> capacitancedACε=(stored charge in terms of voltage)F(F)Week 3bEECS 42, Spring 2005Symbol:Units: Farads (Coulombs/Volt)Current-Voltage relationship:orNote: Q (vc) must be a continuous function of time+vc–icdtdCvdtdvCdtdQiccc+==C C(typical range of values: 1 pF to 1 µF; for “supercapa-citors” up to a few F!)+Electrolytic (polarized)capacitorCIf C (geometry) is unchanging, iC= dvC/dtWeek 3bEECS 42, Spring 2005Voltage in Terms of Current; Capacitor Uses)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 performanceWeek 3bEECS 42, Spring 2005Week 3bEECS 42, Spring 2005Schematic Symbol and Water Model for a CapacitorWeek 3bEECS 42, Spring 2005You 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 5volts stores 63 J; if it discharged at aconstant rate in 1 ms energy isdischarged at a 63 kW rate!)Stored EnergyCAPACITORS STORE ELECTRIC ENERGYWeek 3bEECS 42, Spring 2005∫===∫==∫===⋅=FinalInitialcFinalInitialFinalInitialcccVvVvdQ vdttttt dtdQVvVvvdt ivw2CV212CV21VvVvdv CvwInitialFinalFinalInitialcc−∫====+vc–icA more rigorous derivationWeek 3bEECS 42, Spring 2005Example: Current, Power & Energy for a CapacitordtdvCi =–+v(t)10 µFi(t)t (µs)v (V)0 23451t (µs)0234511i (µA)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 circuitWeek 3bEECS 42, Spring 2005vip=0 23451w (J)–+v(t)10 µFi(t)t (µs)023451p (W)t (µs)2021Cvpdwt∫==τWeek 3bEECS 42, Spring 2005Capacitors in Parallel21CCCeq+=i(t)+v(t)–C1C2i1(t) i2(t)i(t)+v(t)–CeqEquivalent capacitance of capacitors in parallel is the sumdtdvCieq=Week 3bEECS 42, Spring 2005Capacitors in Seriesi(t)C1+ v1(t) –i(t)+v(t)=v1(t)+v2(t)–CeqC2+ v2(t) –21111CCCeq+=Week 3bEECS 42, Spring 2005Capacitive 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-Q1−∆Q1Q2+∆Q2−Q2−∆Q2∆Q1=C1∆v1∆Q2=C2∆v22211 vCvC∆=∆⇒vCCCv ∆+=∆2112Note: Capacitors in series have the same incremental charge.Week 3bEECS 42, Spring 2005Application 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 3bEECS 42, Spring 2005Sensing the Differential Capacitance– Begin with capacitances electrically discharged– Fixed electrodes are then charged to +Vsand –Vs– Movable electrode (proof mass) is then charged to VoconstgggggggAgAgAgAVVVCCCCVCCCVVsossso12121221212121211)2(−=+−=+−=+−=++−=εεεεC1C2Vs–VsVoCircuit modelWeek 3bEECS 42, Spring 2005• A capacitor can be constructed by interleaving the plates with two dielectric layers and rolling them up, to achieve a compact size.• To achieve a small volume, a very thin dielectric with a high dielectric constant is desirable. However, dielectric materials break down and become conductors when the electric field (units: V/cm) is too high.– Real capacitors have maximum voltage ratings– An engineering trade-off exists between compact size and high voltage ratingPractical CapacitorsWeek 3bEECS 42, Spring 2005The Inductor• An inductor is constructed by coiling a wire around some type of form.• Current flowing through the coil creates a magnetic field and a magnetic flux that links the coil: LiL• When the current changes, the magnetic flux changes Æ a voltage across the coil is induced:iLvL(t)dtdiLtvLL=)(+_Note: In “steady state” (dc operation), timederivatives are zero Æ L is a short circuitWeek 3bEECS 42, Spring 2005Symbol:Units: Henrys (Volts • second / Ampere)Current in terms of voltage:Note: iLmust be a continuous function of time+vL–iL∫+==ttLLLLtidvLtidttvLdi0)()(1)()(10ττL(typical range of values: µH to 10 H)Week 3bEECS 42, Spring 2005Schematic Symbol and Water Model of an InductorWeek 3bEECS 42, Spring 2005Stored EnergyConsider an inductor having an initial current i(t0) = i02022121)()()()()()(0LiLitwdptwtitvtptt−=====∫ττINDUCTORS STORE MAGNETIC ENERGYWeek 3bEECS 42, Spring 2005Inductors in
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