Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.002 Fall 2000 Lecture 176.002CIRCUITS ANDELECTRONICSThe Impedance ModelCite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.002 Fall 2000 Lecture 17 Sinusoidal Steady State (SSS)Reading 13.1, 13.2+–OvtVviIωcos=+–RC Focus on steady state, only careabout vPas vHdies away. Focus on sinusoids.Reading: Section 13.3 from course notes.SSSReview Sinusoidal Steady State (SSS)Reading 13.1, 13.2Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.002 Fall 2000 Lecture 1734HvtotalReviewVpcontains all the information we need:ppVV∠Amplitude of output cosinephasesneakinVi ejωtdrivecomplexalgebratakerealpartThe Sneaky PathpVtViωcos[]ppVtV∠+ωcossetupDEusualcircuitmodelnightmaretrig.1vPtjpeVωRCjViω+12Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.002 Fall 2000 Lecture 17ipVVtransferfunction()ωωjHRCjVVip=+=11()ppOVtVv∠+=ωcos22211CRω+break frequencyBode plotωRC1=ω1RC1ω21rememberdemoωRC1=ω4π−2π−0ipVV∠⎟⎠⎞⎜⎝⎛−−1RCtan1ωThe Frequency ViewReviewCite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.002 Fall 2000 Lecture 17Is there an even simpler wayto get Vp?RCjVVipω+=1Divide numerator and denominator by jωC.RCjCjVVip+=ωω11Hmmm… looks like a voltage divider relationship.Let’s explore further…RZZVVCCip+=Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.002 Fall 2000 Lecture 17The Impedance ModelIs there an even simpler way to get Vp?Consider:tjrReIiω=tjrReVvω=RRRiv=tjrtjreRIeVωω=rrRIV=RRi+–RvResistortjCCeIiω=tjCCeVvω=CCi+–CvCapacitorCCICj1Vω=dtdvCiCC=tjCtjCejCVeIωωω=CZLLi+–LvtjlLeIiω=tjlLeVvω=dtdiLvLL=tjltjlejLIeVωωω=InductorllILjVω=LZCite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.002 Fall 2000 Lecture 17In other words,inductorLFor a drive of the form Vcejωt ,complex amplitude Vcis related to the complex amplitude Icalgebraically,by a generalization of Ohm’s Law. jZlω=lllIZV=lI+–lVLZresistorrrrIZV=RZr=RZrI+–rVcapacitorCj1ZCω=cCcIZV=impedancecI+–cVCZThe Impedance ModelCite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.002 Fall 2000 Lecture 17Impedance model: All our old friends apply!KVL, KCL, superposition…Back to RC example…iRCCicVZZZVRCj1Cj1V+=+=ωωicVRCj11Vω+=Done!+–CvIv+–RC+–cViV+–RZR=CjZCω1=cICite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.002 Fall 2000 Lecture 17Another example, recall series RLC:We will study this and other functionsin more detail in the next lecture.RCjLjRVVir++=ωω1RCLRirZZZZVV++=CRjLCCRjVVirωωω++−=12+–LrICR+–rViVtjreVω()rrVtV∠+ωcostjieVωtViωcosRemember, we want only the steady-state response to sinusoidCite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.002 Fall 2000 Lecture 17The Big Picture…tViωcos[]ppVtV∠+ωcossetupDEusualcircuitmodelnightmaretrig.Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.002 Fall 2000 Lecture 17The Big Picture…tViωcos[]ppVtV∠+ωcossetupDEusualcircuitmodelnightmaretrig.Vi ejωtdrivecomplexalgebratakerealpartCite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.002 Fall 2000 Lecture 17The Big Picture…No D.E.s, no trig!tViωcos[]ppVtV∠+ωcossetupDEusualcircuitmodelnightmaretrig.Vi ejωtdrivecomplexalgebratakerealpartcomplexalgebraimpedance-basedcircuit modelCite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.002 Fall 2000 Lecture 17Back toLCRCj1RCjVV2irωωω−+=()()()RCjLC1RCjLC1RCjLC1RCj222ωωωωωωω−−−−⋅+−=()()222irRCLC1RCVVωωω+−=:LowωRCω≈:HighωLRω≈:1LC=ω1≈Let’s study this transfer function+–LrIC+–rViVRLCRCj1RCjVV2irωωω−+=ObserveCite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.002 Fall 2000 Lecture 17Graphically()()2221 RCLCRCVVirωωω+−=More next week…:LowωRCω≈:HighωLRω≈:1LC=ω1≈irVVLC1ωLRωRCω1 “Band Pass”Remember this trick to sketch the form of transfer functions
View Full Document
Unlocking...