EE40 Lecture 11 Josh HugLogistical ThingsLogisticsiClicker Question #1iClicker Question #2Easy Method for AC CircuitsSlide 7Memory Circuits with Exponential SourceInverse SuperpositionReal Part of ExpressionSlide 11Slide 12Slide 13Slide 14ImpedanceMethod of Impedance Analysis (without Phasors)Phasors (not in the book!)PhasorsWhy are phasors useful?Slide 20Method of Impedance Analysis (with Phasors)ExampleSlide 23Slide 24Slide 25Slide 26Harder ExampleFiltersExample FilterTransfer FunctionsUsing a Transfer FunctionUsing a Transfer Function (general)Bode Magnitude PlotBode Magnitude Plot in Context of CircuitBode Phase PlotBode Phase Plot in Context of CircuitFrequency vs. Time DomainFourier Transform ExampleSlide 39Slide 40Types of FiltersFilter ExamplesManually PlottingSlide 44Extra SlidesWhy do equivalent Impedances work?1EE40 Summer 2010HugEE40Lecture 11Josh Hug7/19/20102EE40 Summer 2010HugLogistical Things•Lab 4 tomorrow•Lab 5 (active filter lab) on Wednesday–Prototype for future lab for EE40–Prelab is very short, sorry.–Please give us our feedback–Google docs for labs and general class comments now available (link shared via email)–Bring a music player if you have one (if not, you can use the signal generator in the lab)•HW5 due tomorrow at 2PM•HW6 due Friday at 5PM (also short)•Midterm next Wednesday 7/28–Focus is heavily on HW4, 5, 6, and Labs P1, 4, 5–Will reuse concepts from HW 1,2,33EE40 Summer 2010HugLogistics•No lunch today•Slightly shorter lecture today•Midterm regrade requests due today•Office hours Cory 240 2:30-4PM or so4EE40 Summer 2010HugiClicker Question #15EE40 Summer 2010HugiClicker Question #26EE40 Summer 2010HugEasy Method for AC Circuits•We want to find the voltage across the capacitor for the following circuit•Homogenous solution is easy, since source is irrelevant•Finding particular solution the usual way (plugging in a guess, finding coefficients that cancel) is painful7EE40 Summer 2010HugEasy Method for AC CircuitsPlug into ODEPlug into ODE8EE40 Summer 2010HugMemory Circuits with Exponential Source9EE40 Summer 2010HugInverse Superposition•Just find real part and we’re done!10EE40 Summer 2010HugReal Part of Expression•Finding the real part of the expression is easy, it just involves some old school math that you’ve probably forgotten (HW5 has complex number exercises)11EE40 Summer 2010HugReal Part of Expression•What we have is basically the product of two complex numbers•Let’s convert the left one to polar form12EE40 Summer 2010HugReal Part of Expression13EE40 Summer 2010HugReal Part of Expression•Thus, particular solution (forced response) of original cosine source is just the real part14EE40 Summer 2010HugEasy Method for AC CircuitsPlug into ODEWrite ODEJust as actually writing the ODE isn’t necessary for DC sources, we can avoid the ODE again in AC circuits: Impedance Analysis15EE40 Summer 2010HugImpedanceFor a complex exponential source:Rewrite as:Looks a lot like…voltage divider16EE40 Summer 2010HugMethod of Impedance Analysis (without Phasors)Lugging these complex exponential functions is algebraically annoying17EE40 Summer 2010HugPhasors (not in the book!)18EE40 Summer 2010HugPhasors19EE40 Summer 2010HugWhy are phasors useful?20EE40 Summer 2010HugWhy are phasors useful?21EE40 Summer 2010HugMethod of Impedance Analysis (with Phasors)22EE40 Summer 2010HugExample23EE40 Summer 2010HugExample24EE40 Summer 2010HugExample25EE40 Summer 2010HugExample26EE40 Summer 2010HugExampleNot to scale27EE40 Summer 2010HugHarder Example•On board28EE40 Summer 2010HugFilters•Often, we’ll want to build circuits which react differently based on different signal frequencies, e.g.–Splitting audio signals into low and high portions (for delivery to tweeter and subwoofer)–Removing noise of a particular frequency (e.g. 60 Hz noise or vuvuzela sound)–Removing signals except those at a certain frequency29EE40 Summer 2010HugExample Filter30EE40 Summer 2010HugTransfer Functions31EE40 Summer 2010HugUsing a Transfer Function32EE40 Summer 2010HugUsing a Transfer Function (general)33EE40 Summer 2010HugBode Magnitude PlotLinear Scale Log Scale34EE40 Summer 2010HugBode Magnitude Plot in Context of Circuit35EE40 Summer 2010HugBode Phase PlotLinear Scale Semilog Scale36EE40 Summer 2010HugBode Phase Plot in Context of Circuit37EE40 Summer 2010HugFrequency vs. Time Domain•Almost always, our signals consist of multiple frequencies•Examples:–Sound made when you press a buttons on a phone is two pure sine waves added together (DTMF)–Antennas on radio theoretically pick up ALL frequencies of ALL transmissions•Using a technique known as the Fourier Transform, we can convert any signal into a sum of sinusoids–See EE20 for more details38EE40 Summer 2010HugFourier Transform Example39EE40 Summer 2010HugFourier Transform Example40EE40 Summer 2010HugFourier Transform Example•If we apply a filter with the frequency response on the left to the signal on the rightThen we’ll get:41EE40 Summer 2010HugTypes of Filters•Passive Filters–Filters with no sources (i.e. just R, L, and C)–Don’t require power source–Scale to larger signals (no op-amp saturation)–Cheap•Active Filters–Filters with active elements, e.g. op-amps–More complex transfer function•No need for inductors (can be large and expensive, hard to make in integrated circuits)•More easily tunable–Response more independent of load (buffering)42EE40 Summer 2010HugFilter Examples•On board43EE40 Summer 2010HugManually Plotting•In this day and age, it is rarely necessary to make our Bode plots manually•However, learning how to do this will build your intuition for what a transfer function means•Manual plotting of bode plots is essentially a set of tricks for manually plotting curves on a loglog axis•We will only teach a subset of the full method (see EE20 for a more thorough treatment)44EE40 Summer 2010HugExample Filter45EE40 Summer 2010HugExtra Slides46EE40 Summer 2010HugWhy do equivalent Impedances
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