7 12 2010 Lecture 8 Capacitors and Inductors Congratulations to Paul the Octopus on getting 8 8 world cup predictions correct EE40 Summer 2010 Hug 1 So 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 EE40 Summer 2010 Hug 2 For 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 sources EE40 Summer 2010 Hug 3 Announcements Midterm 2 will be on the 28th Elements with memory Make sure you do pre lab before lab tomorrow Who has finished it EE40 Summer 2010 Hug 4 The 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 EE40 Summer 2010 Hug 5 The 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 EE40 Summer 2010 Hug 6 The 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 C EE40 Summer 2010 Hug 7 The Capacitor Thus if you connect a voltage source to the plates Like charges will move to get away from the source EE40 Summer 2010 Charge flow is current Current will stop once charges reach equilibrium with voltage source i e Energy has been stored Hug 8 The Capacitor Zero VC EE40 Summer 2010 Zero current Lots of current VC VS Lots of current the other way Zero current VC VS Zero VC Hug 9 iClicker Lots of current Zero current High VC EE40 Summer 2010 Acts like a A Short circuit B Open circuit Zero VC VC VS C Resistor D Voltage source E Current source Lots of current Zero current Zero VC Hug 10 Extreme 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 EE40 Summer 2010 Hug 11 How 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 EE40 Summer 2010 Hug 12 Practical 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 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 rating EE40 Summer 2010 Hug 13 Capacitors Useful for Storing Energy Filtering Modeling unwanted capacitive effects particularly delay EE40 Summer 2010 Hug 14 Capacitor or Symbol C C Units Farads Coulombs Volt C Electrolytic polarized capacitor These have high capacitance and cannot support voltage drops of the wrong polarity typical range of values 1 pF to 1 mF for supercapacitors up to a few F Current Voltage relationship dQ dvc ic C dt dt ic vc Note vc must be a continuous function of time since the charge stored on each plate cannot change suddenly EE40 Summer 2010 Hug 15 Node Voltage with Capacitors ic vc dvc dQ ic C dt dt At the top right node we write KCL Current to the left through resistor Current down through the capacitor EE40 Summer 2010 Hug 16 Node Voltage with Capacitors ic vc dvc dQ ic C dt dt Or in other words Or more compactly EE40 Summer 2010 Hug 17 ODEs 0 Later today we ll talk about how to solve ODEs For now let s talk about inductors EE40 Summer 2010 Hug 18 Inductors 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 EE40 Summer 2010 Hug 19 Two 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 B t v t Wikipedia EE40 Summer 2010 Hug 20 Inductor 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 B t EE40 Summer 2010 v t Hug 21 Inductor Basics 2 B t v t Current in a wire causes induces a voltage in any nearby circuit EE40 Summer 2010 Hug 22 Inductors Basics 3 If we make a loop the entire loop of wire will all contribute to the magnetic field through the loop What s more this field will go through the loop producing the current Self induced voltage Self inductance EE40 Summer 2010 From Dr Richard F W Bader Professor of Chemistry McMaster University Hug 23 Inductors Basics 4 More loops More magnetic field generated More circuit to receive magnetic field Inductors are literally just loops of wire Just like capacitors are just two conductors separated by an insulator or a gap Just like resistors are just stuff with wires stuck to the ends EE40 Summer 2010 University of Surrey http personal ee surrey ac uk Personal H M UGLabs components inductors htm Hug 24 Inductors Capacitors hold a voltage in the form of stored charge Inductors hold a current in the form of stored magnetic field dude For webcast viewers see drawings on board notes for more comparison to capacitor EE40 Summer 2010 Hug 25 Symbol L Units Henrys Volts second Ampere typical range of values mH to 10 H Current in terms of voltage 1 diL vL t dt L t 1 iL t vL d i t0 L t0 iL vL Note iL must be a continuous function of time
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