1Copyright 2001, Regents of University of CaliforniaHandout on RC Circuits. A.R. NeureutherVersion Date 09/08/03EECS 42 Intro. Digital Electronic, Fall 2003Charging and Discharging RC Circuits Handout for EECS 42 Lectures 6 & 7Developed by Professor W.G. Oldham to provide understanding of transient issues in computer logic. Extensions by Professor A.R. Neureuther in Spring 2003 to include sequential switching of logic gates as occurs in the EECS 43 logic gate experiment.Schwarz & Oldham 8.1 Pulse ShapesCopyright 2001, Regents of University of CaliforniaHandout on RC Circuits. A.R. NeureutherVersion Date 09/08/03EECS 42 Intro. Digital Electronic, Fall 2003Charging and discharging in RC Circuits(an enlightened approach)• Before we analyze real electronic circuits - lets study RC circuits• Rationale: Every node in a circuit has capacitance to ground, like it or not, and it’s the charging of these capacitances thatlimits real circuit performance (speed)RC charging effects are responsible …. So lets review them.Relevance to digital circuits: We communicate with pulses We send beautiful pulses outtimevoltageBut we receive lousy-looking pulses and must restore themtimevoltageCopyright 2001, Regents of University of CaliforniaHandout on RC Circuits. A.R. NeureutherVersion Date 09/08/03EECS 42 Intro. Digital Electronic, Fall 2003LOGIC GATE DELAY τDTime delay τDoccurs between input and output: “computation” is not instantaneous Value of input at t = 0+determines value of output at later time t = τDABF0110Logic StatettτD00Input (A and B tied together)Output (Ideal delayed step-function)Actual exponential voltage versus time.Capacitance to GroundFCopyright 2001, Regents of University of CaliforniaHandout on RC Circuits. A.R. NeureutherVersion Date 09/08/03EECS 42 Intro. Digital Electronic, Fall 2003tttLogic stateτ2τ0τSIGNAL DELAY: TIMING DIAGRAMSShow transitions of variables vs time10tτ2τ3τNote that C changes two gate delays after A switches.Note B changes one gate delay after A switchesABCDABDCNote that D changes three gate delays after A switches.Oscilloscope ProbeTiming Diagrams are an extension of the logic diagrams in O&S Ch 11 & 12Copyright 2001, Regents of University of CaliforniaHandout on RC Circuits. A.R. NeureutherVersion Date 09/08/03EECS 42 Intro. Digital Electronic, Fall 2003Simplification for time behavior of RC CircuitsBefore any input change occurs we have a dc circuit problem (that is we can use dc circuit analysis to relate the output to the input).We call the time period during which the output changes the transientWe can predict a lot about the transient behavior from the pre- and post-transient dc solutionstimevoltageinputtimevoltageoutputLong after the input change occurs things “settle down” …. Nothing is changing …. So again we have a dc circuit problem.Copyright 2001, Regents of University of CaliforniaHandout on RC Circuits. A.R. NeureutherVersion Date 09/08/03EECS 42 Intro. Digital Electronic, Fall 2003What environment do pulses face?• Every real wire in a circuit has resistance.• Every junction (node) has capacitance to ground and to other nodes.• The active circuit elements (transistors) add additional resistance in series with the wires, and additional capacitance in parallel with the node capacitance.A pulse originating at node I will arrive delayed and distorted at node O because it takes time to charge C through RInput node Output nodegroundRCVin+-IOIf we focus on the circuit which distorts the pulses produced by Vin, its most simple form consists simply of an R and a C. (Vinrepresents the time-varying source which produces the input pulse.)Thus the most basic model circuit for studying transients consists of a resistor driving a capacitor.2Copyright 2001, Regents of University of CaliforniaHandout on RC Circuits. A.R. NeureutherVersion Date 09/08/03EECS 42 Intro. Digital Electronic, Fall 2003The RC Circuit to Study(All single-capacitor circuits reduce to this one)• R represents total resistance (wire plus whatever drives the input node)Input node Output nodegroundRC• C represents the total capacitance from node to the outside world (from devices, nearby wires, ground etc)Copyright 2001, Regents of University of CaliforniaHandout on RC Circuits. A.R. NeureutherVersion Date 09/08/03EECS 42 Intro. Digital Electronic, Fall 2003RC RESPONSE•Vin“jumps” at t=0, but Voutcannot “jump” like Vin. Why not?Case 1 – Rising voltage. Capacitor uncharged: Apply + voltage step) Because an instantaneous change in a capacitor voltage would require instantaneous increase in energy stored (1/2CV²), that is, infinite power. (Mathematically, V must be differentiable: I=CdV/dt)Input node Output nodegroundRCVinVout+-V does not “jump” at t=0 , i.e. V(t=0+) = V(t=0-) timeVin00V1VoutTherefore the dc solution before the transient tells us the capacitor voltage at the beginning of the transient. Copyright 2001, Regents of University of CaliforniaHandout on RC Circuits. A.R. NeureutherVersion Date 09/08/03EECS 42 Intro. Digital Electronic, Fall 2003RC RESPONSECase 1 Continued – Capacitor uncharged: Apply voltage stepAfter the transient is over (nothing changing anymore) it means d(V)/dt = 0 ; that is all currents must be zero. From Ohm’s law, the voltage across R must be zero, i.e. Vin= Vout.•Voutapproaches its final value asymptotically (It never actually gets exactly to V1, but it gets arbitrarily close). Why?) That is, Vout→ V1 as t →∞. (Asymptotic behavior)timeVin00V1VoutInput node Output nodegroundRCVinVout+-Again the dc
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