Slide 1Logistics and Lab ReminderHW ClarificationTo the board…RLC CircuitsLet’s step back a secondApplication to Digital Integrated Circuits (ICs)Digital SignalsCircuit Model for a Logic GateLogic Level TransitionsSequential SwitchingPulse DistortionExampleSlide 14Parasitic CapacitancesAC InputsSolving Circuits with AC SourcesSolving Circuits with AC SourcesSolving Circuits with AC SourcesPhasorsTwo PathsBasic Idea and Derivation of ImpedancesNew Voltage Source ProblemNew Voltage Source ProblemTo RecapThe Inverse Superposition TrickInverse SuperpositionReal Part of ExpressionReal Part of ExpressionReal Part of ExpressionReal Part of ExpressionWait…. That was easier?ImpedanceImpedance Method for Solving AC CircuitsImpedance AnalysisImpedance Analysis ExampleExtra Slides1EE40 Summer 2010HugEE40Lecture 10Josh Hug7/17/20102EE40 Summer 2010HugLogistics and Lab Reminder•If you have not submitted a spec and want to do a custom Project 2, talk to me right after class•HW4 due today at 5•HW5 due Tuesday at 2PM (it will be short, and up by 5 PM today)•As requested, all reading assignments for next week will be posted tonight•We expect you to understand lab concepts. For example, the Schmitt Trigger:–Do you know what they are and what they do?3EE40 Summer 2010HugHW Clarification•There are a bunch of hints on the bspace forums•“Zero state response” and “zero input response” are terms that I haven’t used in lecture, but they’re really easy and they’re in the book–Zero input response: The response you get with f(t)=0 [same as homogeneous solution]–Zero state response: The response you get with y(0)=0 [complete response with initial condition equal to zero]4EE40 Summer 2010HugTo the board…•For LC and RLC circuits5EE40 Summer 2010HugRLC Circuits•They are important, but not so much for digital integrated circuit design•They do play a role in the world of analog circuits, but that’s a bit specialized for us to spend a great deal of time–Usually care more about “frequency response” than the actual shape of the response in time•If you want to learn more about analog circuit design (it is hard and probably awesome), see EE6EE40 Summer 2010HugLet’s step back a second•Earlier this week, I said capacitors are good for–Storing energy–Filtering–Modeling unwanted capacitances in digital circuits•We’ve discussed the first case pretty heavily now, and filtering will come in great detail next week•For now, let’s talk about delay modeling7EE40 Summer 2010HugWhen we perform a sequence of computations using a digital circuit, we switch the input voltages between logic 0 (e.g. 0 Volts) and logic 1 (e.g. 5 Volts).The output of the digital circuit changes between logic 0 and logic 1 as computations are performed.Application to Digital Integrated Circuits (ICs)8EE40 Summer 2010Hug•Every node in a real circuit has capacitance; it’s the charging of these capacitances that limits circuit performance (speed)We compute with pulses. We send beautiful pulses in:But we receive lousy-looking pulses at the output:Capacitor charging effects are responsible!timevoltagetimevoltageDigital Signals9EE40 Summer 2010HugCircuit Model for a Logic Gate•As we’ll discuss in a couple of weeks, electronic building blocks referred to as “logic gates” are used to implement logical functions (NAND, NOR, NOT) in digital ICs–Any logical function can be implemented using these gates.•A logic gate can be modeled as a simple RC circuit:+Vout–RVin(t) +Cswitches between “low” (logic 0) and “high” (logic 1) voltage states10EE40 Summer 2010HugTransition from “0” to “1”(capacitor charging)timeVout0VhighRC0.63VhighVoutVhightimeRC0.37VhighTransition from “1” to “0”(capacitor discharging)(Vhigh is the logic 1 voltage level)Logic Level Transitions0 RCthighouteVtV/1)(RCthighouteVtV/)(11EE40 Summer 2010HugWhat if we step up the input,wait for the output to respond,then bring the input back down? timeVin00timeVin00VouttimeVin00VoutSequential Switching12EE40 Summer 2010HugThe input voltage pulse width must be long enough; otherwise the output pulse doesn’t make it.(We need to wait for the output to reach a recognizable logic level, before changing the input again.)01234560 1 2 3 4 5TimeVoutPulse width = 0.1RC01234560 1 2 3 4 5TimeVout01234560 5 10 15 20 25TimeVoutPulse Distortion+Vout–RVin(t)C+–Pulse width = 10RCPulse width = RC13EE40 Summer 2010HugVinRVoutCSuppose a voltage pulse of width5 ms and height 4 V is applied to theinput of this circuit beginning at t = 0:R = 2.5 kΩC = 1 nF• First, Vout will increase exponentially toward 4 V.• When Vin goes back down, Vout will decrease exponentially back down to 0 V.What is the peak value of Vout?The output increases for 5 ms, or 2 time constants. It reaches 1-e-2 or 86% of the final value.0.86 x 4 V = 3.44 V is the peak valueExamplet = RC = 2.5 ms14EE40 Summer 2010Hug00.511.522.533.540 2 4 6 8 10Vout(t) =4-4e-t/2.5ms for 0 ≤ t ≤ 5 ms3.44e-(t-5ms)/2.5ms for t > 5 ms{15EE40 Summer 2010HugParasitic Capacitances•We’ll discuss these parasitic capacitances in the context of digital integrated circuits right after midterm 216EE40 Summer 2010HugAC Inputs•We’ve discussed to this point how we deal with constant and weird mathematically ideal inputs •Next we’ll discuss sinusoidal inputs or AC inputs, useful for, in order of increasing generality:–Finding 60 Hz wall voltage response–Finding response to inputs that can be approximated by a sum of sinusoids (e.g. square waves)–Finding “frequency response”•17EE40 Summer 2010HugSolving Circuits with AC Sources•In principle, we can use the MPHS to solve the circuit below:•Will finding the homogeneous solution be difficult?18EE40 Summer 2010HugSolving Circuits with AC Sources•Will finding the particular solution be difficult?19EE40 Summer 2010HugSolving Circuits with AC Sources•Will finding the particular solution be difficult?��= ��−���+√2 �� ��2+�2�2cos (� � +5 �4)20EE40 Summer 2010HugPhasors•Solving simple resistive circuits–Hard way (kitchen sink method)–Easy way (node voltage)•Op-amp circuits–Hard way (taking limits as )–Easy way (summing point constraint)•Requires negative feedback, which can be hard to identify•Circuits with memory –Hard way (solving ODE)–Easy way (intuitive method)•Requires DC sources–Next will come an
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