6 002 CIRCUITS AND ELECTRONICS The Impedance Model 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 17 Review Sinusoidal Steady State SSS Reading 13 1 13 2 vI Vi cos t C vO Focus on steady state only care about vP as vH dies away Focus on sinusoids SSS R Sinusoidal Steady State SSS Reading 13 1 13 2 Reading Section 13 3 from course notes 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 17 vP Review Vi cos t 1 usual circuit model sneak in Vi e j t drive set up DE complex algebra V p cos t V p nightmare trig Vp 2 The Sneaky Path 3 vH take 4 real total part V p e j t Vi 1 j RC Vp contains all the information we need Vp V p Amplitude of output cosine phase 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 17 Review vO V p cos t V p Vp Vi Vp 1 H j transfer function 1 j RC remember demo 1 Vi 1 2 1 1 RC 1 2 R 2C 2 Bode plot Vp Vi 1 RC break frequency 0 1 RC RC tan 1 1 4 2 The Frequency View 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 17 Is there an even simpler way to get Vp Vi Vp 1 j RC Divide numerator and denominator by j C 1 V p Vi j C 1 R j C Hmmm looks like a voltage divider relationship ZC V p Vi ZC R Let s explore further 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 17 The Impedance Model Is there an even simpler way to get Vp Consider vR vC iR i R I r e j t vR RiR R vR Vr e j t Vr e j t RI r e j t Resistor iC C iC I C e j t vC VC e j t Vr RI r dvC iC C dt I C e j t CVC j e j t 1 IC j C ZC di vL L L dt Capacitor vL iL i L I l e j t L vL Vl e j t VC Vl e j t LI l j e j t Inductor Vl j L I l ZL 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 17 The Impedance Model In other words capacitor Ic Vc ZC Vc Z C I c 1 ZC j C impedance inductor resistor Il Vl Vr ZL Ir ZR Vl Z l I l Z l j L Vr Z r I r Zr R For a drive of the form Vc e j t complex amplitude Vc is related to the complex amplitude Ic algebraically by a generalization of Ohm s Law 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 17 Back to RC example R C vC vI Impedance model ZR R Ic Vc Vi 1 ZC j C 1 ZC j C Vc Vi Vi 1 ZC Z R R j C Vc 1 Vi 1 j RC Done All our old friends apply KVL KCL superposition 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 17 Another example recall series RLC Remember we want only the steady state response to sinusoid Ir L Vi e j t Vi Vi cos t C R Vi Z R Vr Z L ZC Z R Vr Vr e j t Vr cos t Vr Vi R Vr 1 j L R j C Vr Vi j CR 2 LC 1 j CR We will study this and other functions in more detail in the next lecture 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 17 The Big Picture V p cos t V p Vi cos t usual circuit model set up DE nightmare trig 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 17 The Big Picture V p cos t V p Vi cos t usual circuit model Vi e j t drive set up DE nightmare trig complex algebra take real part 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 17 The Big Picture V p cos t V p Vi cos t usual circuit model Vi e j t drive set up DE nightmare trig complex algebra impedance based circuit model take real part complex algebra No D E s no trig 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 17 Back to Ir Vr j RC Vi 1 j RC 2 LC Vi L C R Vr Let s study this transfer function Vr j RC Vi 1 j RC 2 LC j RC 1 2 LC j RC 2 1 LC j RC 1 2 LC j RC Vr Vi 1 RC 2 LC RC 2 2 Observe Low RC R High L LC 1 1 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 17 Graphically Vr Vi RC 1 2 LC RC 2 2 Low RC …
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