Slide 1So far…For the next 2 weeksAnnouncementsThe CapacitorThe CapacitorThe CapacitorThe CapacitorThe CapacitoriClickerExtreme Corner CaseHow much energy is stored?Practical CapacitorsCapacitorsCapacitorNode Voltage with CapacitorsNode Voltage with CapacitorsODEsInductorsTwo Fundamental PrinciplesInductor Basics (1)Inductor Basics (2)Inductors Basics (3)Inductors Basics (4)InductorsSlide 26SummaryOrdinary Differential EquationsChua’s CircuitChua’s Circuit1EE40 Summer 2010Hug7/12/2010Lecture 8Capacitors and InductorsCongratulations to Paul the Octopus on getting 8/8 world cup predictions correct2EE40 Summer 2010HugSo far…•All circuits we’ve dealt with have reacted instantaneously–Change a resistance, voltage, or current, and everything else reacts instantly•Obviously this isn’t a complete model for electronic device. Why?3EE40 Summer 2010HugFor the next 2 weeks•We’ll be talking about elements with memory–Capacitors–Inductors•Our first 6 lectures taught us how we can take a circuit schematic containing memoryless elements and convert them into algebraic equations•In the next 2 lectures we’ll talk about how to convert circuits with memory into differential equations•The next 3 after that will be about how we can use algebraic equations for circuits with memory if we have AC sources4EE40 Summer 2010HugAnnouncements•Midterm #2 will be on the 28th–Elements with memory•Make sure you do pre-lab before lab tomorrow–Who has finished it?5EE40 Summer 2010HugThe Capacitor•The basic idea is pretty simple–Imagine you have two parallel metal plates, both of which have equal and opposite excess charges–Plates are separated by an insulating layer (air, glass, wood, etc)•The charges would love to balance out•Insulator blocks them (just as the ground blocks you from falling into the center of the earth)6EE40 Summer 2010HugThe Capacitor•If you were to connect a resistive wire to the plates–Charges would flow through the wire•Charge flow is current•Energy is released as heat•7EE40 Summer 2010HugThe Capacitor•Remember that a voltage is the electrical potential between two points in space•Here, we have an imbalance of charge, and thus an electric field, and thus a voltage –Field strength is dependent on number and distribution of charges as well as material properties–Field length is dependent on size of capacitor–Capacitor size and material properties lumped into single “capacitance” C8EE40 Summer 2010HugThe Capacitor•Thus, if you connect a voltage source to the plates–Like charges will move to get away from the source•Charge flow is current•Current will stop once charges reach equilibrium with voltage source, i.e. •Energy has been stored• +-+-+-9EE40 Summer 2010HugThe Capacitor+-+-+-LotsofcurrentZeroVCZerocurrentVC=VSLotsofcurrent[theotherway]VC=VSZerocurrentZeroVC�=�����10EE40 Summer 2010HugiClicker+-+-LotsofcurrentZeroVCZerocurrentVC=VSLotsofcurrentHighVCZerocurrentZeroVCActs like a:A. Short circuitB. Open circuitC. ResistorD. Voltage sourceE. Current source11EE40 Summer 2010HugExtreme Corner Case•What happens if we short a charged capacitor?•Think of it is as the limit as resistance goes to zero:–Infinite current–Lasts only a very short time () until energy is released•Mathematically poorly behaved–Don’t do this•12EE40 Summer 2010HugHow much energy is stored?• • Strictly speaking we shouldn’t use t as our integration variable and also the limit that we’re integrating to, but you know what I mean…13EE40 Summer 2010Hug•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 Capacitors14EE40 Summer 2010HugCapacitors•Useful for–Storing Energy–Filtering–Modeling unwanted capacitive effects, particularly delay15EE40 Summer 2010HugSymbol:Units: Farads (Coulombs/Volt)Current-Voltage relationship:orNote: vc must be a continuous function of time since the charge stored on each plate cannot change suddenlyCapacitor+vc–icC C(typical range of values: 1 pF to 1 mF; for “supercapa-citors” up to a few F!)+Electrolytic (polarized) capacitorCThesehavehighcapacitanceandcannotsupportvoltagedropsofthewrongpolaritydtdvCdtdQicc16EE40 Summer 2010HugNode Voltage with Capacitors+vc–ic•At the top right node, we write KCL•Current to the left through resistor:•Current down through the capacitordtdvCdtdQicc17EE40 Summer 2010HugNode Voltage with Capacitors+vc–ic•Or in other words•Or more compactly:dtdvCdtdQicc18EE40 Summer 2010HugODEs•Later today, we’ll talk about how to solve ODEs…•For now, let’s talk about inductors��−���+���′=019EE40 Summer 2010HugInductors•Capacitors are a piece of cake to understand, just rely on Coulomb’s Law•Inductors, by contrast, involve magnetic fields, and rely instead on Faraday’s Law–Comprehension comes with greater difficulty•Thus, we’ll treat inductors as mathematical objects and leave the derivation to Physics 7B (or page 467 of the book)20EE40 Summer 2010HugTwo Fundamental Principles•The flow of current induces a magnetic field (Ampere’s Law)•A change in magnetic field through a loop of wire induces a voltage (Faraday’s Law)+ (Wikipedia))t(B)t(v21EE40 Summer 2010HugInductor Basics (1)•When we connect a voltage source to a wire, current clearly takes a little time to get moving•Thus, the magnetic field builds to some maximum strength over time+ )t(B)t(v22EE40 Summer 2010HugInductor Basics (2)+ Current in a wire causes induces a voltage in any nearby circuit)t(B)t(v23EE40 Summer 2010HugInductors Basics (3)•If we make a loop, the entire loop of wire will all contribute to the magnetic field through the loopFrom:Dr.RichardF.W.BaderProfessorofChemistry/McMasterUniversity•What’s more, this field will go through the loop producing the current!–Self
View Full Document
Unlocking...