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EE40 Lec 07Capacitors and InductorsProf. Nathan Cheung09/17/2009Rdi H bl Cht3Reading: Hambley Chapter 3Slide 1EE40 Fall 2009 Prof. CheungThe CapacitorTwo conductors (a,b) separated by an insulator:difference in potential = Vabl& it hQdt=> equal & opposite charge Qon conductorsQ = CVab(stored charge in terms of voltage)abwhere C is the capacitance of the structure, ¾ positive (+) charge is on the conductor at higher potentialParallel-plate capacitor:• area of the plates = Ap• separation between plates = d• dielectric permittivity of insulator = εAεSlide 2EE40 Fall 2009 Prof. Cheung=> capacitancedACε=CapacitorSymbol:orCC+Electrolytic (polarized)CUnits: Farads (Coulombs/Volt)CC(typical range of values: 1 pF to 1µF; for“supercapa-y(p )capacitorThese have high capacitance and cannotsupport voltage drops of the wrong polarityCurrent-Voltage relationship:idd(typical range of values: 1 pF to 1 µF; for supercapacitors” up to a few F!)+vcicdtdvCdtdQicc==–To write this it is important to have use a passive convention, otherwise you need a minus sign.Slide 3EE40 Fall 2009 Prof. CheungNote: vcmust be a continuous function of time since thecharge stored on each plate cannot change suddenlyVoltage in Terms of Current)0()()(tcQdttitQ+=∫)0(1)0()()(0tcQQdttitQ∫∫)0()(1)(0ccCQdttiCtv+=∫)0()(1tdtti+∫)0()(0ccvdttiC+=∫Slide 4EE40 Fall 2009 Prof. CheungTh t d it iQVhi h h thStored EnergyThe energy stored on a capacitor is QV, which has the dimension of Joules. During charging, you might think the average voltageDuring charging, you might think the average voltage across the capacitor was only half the final value of V for a capacitor.2CV1)V1(Q=Th t dCV2 )V2(Q=The stored energyExample: A 1 pF capacitance charged to 5 Volts Stored energy = ½(5V)2=125pJSlide 5EE40 Fall 2009 Prof. CheungStored energy = ½(5V)2= 12.5 pJA more rigorous derivation+icThis derivation holds vc–independent of the circuit!∫==∫∫=⋅=FinalFinalvvdQvdttdQvdtivwFinalv∫=∫∫InitialInitialccvvdQcvdttdtcvdt ivwInitialV2Cv212Cv21vvdv CvInitialFinalFinalcc−∫==Slide 6EE40 Fall 2009 Prof. CheungvInitialExample: Current, Power & Energy for a Capacitor–+v(t)10 µFi(t)v (V))0()(1)(0vdiCtvt+=∫ττ–t (µs)1dvCi=(µ)0 23451i (µA)vcmust be a continuousfunction of time; howeverdtt (µs)023451function of time; however,iccan be discontinuous.(µ)023451Note: In “steady state”(dc operation), timederivatives are zeroSlide 7EE40 Fall 2009 Prof. Cheungderivatives are zeroÆ C is an open circuitCurrent, Power & Energy for a Capacitorvip=p (W)t (µs)023451–+v(t)10 µFi(t)w (J)1tt(µs)2021Cvpdwt∫==τSlide 8EE40 Fall 2009 Prof. Cheung0 23451t (µs)Capacitors in Parallel+i1(t) i2(t)i(t)v(t)–C1C2+21CCCeq+=i(t)+v(t)Ceq–dtdvCieq=Slide 9EE40 Fall 2009 Prof. CheungEquivalent capacitance of capacitors in parallel is the sumdtCapacitors in Series()()C+ v1(t) –+C+ v2(t) –i(t)C1i(t)v(t)=v1(t)+v2(t)CeqC2–111+=Slide 10EE40 Fall 2009 Prof. Cheung21CCCeq+Capacitive 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?voltage across each individual capacitor?21vvv∆+∆=∆∆Q1=C1∆v1+∆C1+v1+∆v1–Note that no net charge cancan be introduced to this node.Therefore, −∆Q1+∆Q2=0Q1+∆Q1-Q1−∆Q1v+∆v++–Q1∆Q1Q2+∆Q22211 vCvC∆=∆⇒C2v2(t)+∆v2–−Q2−∆Q2vCCCv ∆+=∆2112Slide 11EE40 Fall 2009 Prof. Cheung∆Q2=C2∆v221Note: Capacitors in series have the same incremental charge.Comment on Capacitive Voltage DividerQ: Can we always claimvCCCv2112+=CC21+AC1+v1–Q1-Q1Answer:Nov++–Q1Q2No. Divider formula is valid for total voltage only ifC2v2–−Q2gycapacitors are unchargedwhen v is applied.Slide 12EE40 Fall 2009 Prof. CheungApplication Example: MEMS Accelerometer• Capacitive position sensor used to measure acceleration (by measuring force on a(by measuring force on a proof mass)gg1g2Slide 13EE40 Fall 2009 Prof. CheungFIXED OUTER PLATESMEMS Accelerometer :Sensing the Differential Capacitance–Begin with capacitances electrically discharged– Fixed electrodes are then charged to +Vsand –Vs– Movable electrode (proof mass) is then charged to VoVCircuit modelVCCCCVCCCVVssso211)2(+−=++−=C1VsAAVCCCC2121−++εεVoconstggggggAAggVVso12121221−=+−=+=εεC2VSlide 14EE40 Fall 2009 Prof. Cheunggg21–VsOp-Amp Integrator)0()(1)(0CtINovdttvRCtv +−=∫0CRinic– vC+v+–vo++vnvinvp–n–Slide 15EE40 Fall 2009 Prof. CheungPractical Capacitors•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 thematerials break down and become conductors when the electric field (units: V/cm) is too high.– Real capacitors have maximum voltage ratingsAiitdff i t b t t i dSlide 16EE40 Fall 2009 Prof. Cheung–An engineering trade-off exists between compact size and high voltage ratingThe Inductor• An inductor is constructed by coiling a wire around some type of form.ivL(t)+iLvL(t)_• Current flowing through the coil creates a magnetic field and a magnetic flux that links the coil: LiLWh th t h th ti fl h•When the current changes, the magnetic flux changes Æ a voltage across the coil is induced:diLtvL=)(Slide 17EE40 Fall 2009 Prof. CheungdtLtvL=)(Note: In “steady state” (dc operation), timederivatives are zero Æ L is a short circuitInductorSymbol:LUnits: Henrys (Volts • second / Ampere)(typical range of values:µHto10H)Current in terms of voltage:i1(typical range of values: µH to 10 H)+vLiL=tLLdttvLdi )(1To write this it is important to usehi fii–∫+=ttLLtidvLti0)()(1)(0ττthe passive configuration.Slide 18EE40 Fall 2009 Prof. CheungNote: iLmust be a continuous function of timebecause magnetic flux cannot change suddenly0Stored EnergyConsider an inductor having an initial current i(t0) = i0)()()(titvtp==)()(dptwt==∫ττ11)()(0pt∫2022121)( LiLitw−=Slide 19EE40 Fall 2009 Prof. CheungInductors in Seriesdi+ v1(t) –++ v2(t) –dtdiLveq=v(t)L1v(t)+v(t)=v1(t)+v2(t)LeqL2+–+–i(t)i(t)()didididi–()dtdiLdtdiLLdtdiLdtdiLveq=+=+=212121LLLeq+=Slide 20EE40 Fall 2009 Prof. CheungEquivalent inductance of inductors in series is the sumInductors in


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