1EE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu1Announcements HW #1 Due today at 6pm. HW #2 posted online today and due next Tuesday at 6pm. Due to scheduling conflicts with some students, classes will resume normally this week and next. Midterm tentatively 7/12.EE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu2Review Mesh and Nodal Analysis Superposition Equivalent Circuits Thevenin Norton Measuring Voltages and Currents2EE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu3Review: Thevenin Equivalent ExampleFind the Thevenin equivalent with respect to the terminals a,b:EE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu4Lecture #4OUTLINE The capacitor The inductor 1stOrder Circuits Transient and Steady-State responseReadingChapter 3, Chap 4.1-4.53EE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu5The 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)EE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu6Symbol:Units: Farads (Coulombs/Volt)Current-Voltage relationship:orNote: Q (vc) must be a continuous function of timeCapacitor+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= C dvC/dt4EE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu7Voltage 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 performanceEE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu8You 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 ENERGY5EE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu9∫===∫==∫===⋅=FinalInitialcFinalInitialFinalInitialcccVvVvdQ vdttttt dtdQVvVvvdt ivw2CV212CV21VvVvdv CvwInitialFinalFinalInitialcc−∫====+vc–icA more rigorous derivationEE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu10Example: 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 circuit6EE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu11vip=0 23451w (J)–+v(t)10 μFi(t)t (μs)023451p (W)t (μs)2021Cvpdwt∫==τExample: Current, Power & Energy for a CapacitorEE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu12Capacitors in Seriesi(t)C1+ v1(t) –i(t)+v(t)=v1(t)+v2(t)–CeqC2+ v2(t) –21111CCCeq+=Proof:7EE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu13Capacitors in Parallel+v(t)_C1i(t)=i1(t)+i2(t)CeqC2+v(t)_i1(t)i2(t)Ceq= C1+C2Proof:EE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu14 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 Capacitors8EE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu15Symbol:Units: Henrys (Volts • second / Ampere)Current in terms of voltage:Note: iLmust be a continuous function of timeInductor+vL–iL∫+==ttLLLLtidvLtidttvLdi0)()(1)()(10ττL(typical range of values: μH to 10 H)EE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu16Stored EnergyConsider an inductor having an initial current i(t0) = i02022121)()()()()()(0LiLitwdptwtitvtptt−=====∫ττINDUCTORS STORE MAGNETIC ENERGY9EE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu17Inductors in Seriesi(t)L1+ v1(t) –i(t)+v(t)=v1(t)+v2(t)–LeqL2+ v2(t) –Leq= L1+L2EE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu18Inductors in Parallel+v(t)_L1i(t)=i1(t)+i2(t)LeqL2+v(t)_i1(t)i2(t)21111LLLeq+=10EE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu19First-Order Circuits A circuit that contains only sources, resistors and an inductor is called an RL circuit. A circuit that contains only sources, resistors and a capacitor is called an RC circuit. RL and RC circuits are called first-order circuits because their voltages and currents are described by first-order differential equations.–+vs LR–+vs CRi iEE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu20Transient vs. Steady-State Response The momentary behavior of a circuit (in response to a change in stimulation) is referred to as its transient response. The behavior of a circuit a long time (many time constants) after the change in voltage or current is called the steady-state response.11EE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu21Review (Conceptual) Any* first-order circuit can be reduced to a Thévenin (or Norton) equivalent connected to either a single equivalent inductor or capacitor. In steady state, an inductor behaves like a short circuit In steady state, a capacitor behaves like an open circuit–+VThCRThLRThIThEE40 Summer 2005:
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