6 002 CIRCUITS AND ELECTRONICS Capacitors and First Order Systems 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 12 Motivation Demo 5V 5V B C A 5V 0V 5 A 0 5 Expect this right But observe this B 0 5 Expected Observed C 0 Reading Chapters 9 10 Delay 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 12 The Capacitor D n channel MOSFET symbol G S drain gate m e t a l n o x i d e source s i l n channel p i MOSFET n channel c o n n D G CGS S 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 12 Ideal Linear Capacitor A E d EA d obeys DMD total charge on capacitor q q 0 C i C q v q C v coulombs farads volts 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 12 Ideal Linear Capacitor i q C q C v v dq i dt d Cv dt dv C dt E 1 Cv 2 2 A capacitor is an energy storage device memory device history matters 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 12 Analyzing an RC circuit Th venin Equivalent vI t R C vC t Apply node method vC vI dvC C 0 R dt dvC vC vI RC dt t t0 vC t0 given units of time 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 12 Let s do an example v I t R C vC t vI t VI vC 0 V0 given dvC RC vC VI dt X 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 12 Example vI t VI vC 0 V0 given RC dvC vC VI dt X vC t vCH t vCP t total homogeneous particular Method of homogeneous and particular solutions 1 Find the particular solution 2 Find the homogeneous solution 3 The total solution is the sum of the particular and homogeneous solutions Use the initial conditions to solve for the remaining constants 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 12 1 Particular solution dvCP vCP VI dt RC vCP VI RC works dVI VI VI dt 0 In general use trial and error vCP any solution that satisfies the original equation X 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 12 2 Homogeneous solution dvCH RC vCH 0 dt Y vCH solution to the homogeneous equation Y set drive to zero vCH A e st assume solution of this form A s dA e st A e st 0 RC dt R CA s e st A e st 0 Discard trivial A 0 solution Characteristic equation R C s 1 0 s or 1 RC vCH Ae t RC RC called time constant 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 12 3 Total solution vC vCP vCH vC VI A e t RC Find remaining unknown from initial conditions at t 0 Given vC V0 so V0 VI A or A V0 VI thus vC VI V0 VI e also iC C t RC dvC V VI e 0 dt R t RC 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 12 vC VI V0 VI e t RC vC VI V0 0 t RC 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 12 Examples vC vC 5V 5V 5 5e t RC 5e t 0V VO 0V VI 5V 5 0 t RC t 0V VO 5V VI 0V 5 0 RC Remember B demo 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 12
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