6.002 Fall 2000 Lecture 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].6.002CIRCUITS ANDELECTRONICSCapacitorsand First-Order Systems6.002 Fall 2000 Lecture 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].5V0VCAB5VABC505050Reading:Chapters 9 & 10Demo5VExpectedObservedExpect this, right?But observe this!Delay!Motivation6.002 Fall 2000 Lecture 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].The CapacitorGDSn-channel MOSFETsymboln-channelMOSFETn-channelsiliconnmetal++++++oxidedraingatesourceCGSGDSnp6.002 Fall 2000 Lecture 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].Ideal Linear Capacitorobeys DMD!total charge oncapacitor0qq =−+=dEAC =+ ++ + + +-------AEdcoulombs farads voltsvCq=iCq+–v6.002 Fall 2000 Lecture 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].Ideal Linear Capacitordtdqi=()dtCvd=dtdvC=ivCq=Cq+–vA capacitor is an energy storage deviceÆ memory device Æ history matters!⎥⎦⎤⎢⎣⎡=221CvE6.002 Fall 2000 Lecture 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].Apply node method:C+–()tvC()tvI+–RThévenin Equivalent:0=+−dtdvCRvvCICICCvvdtdvCR=+0tt≥()0tvCgivenunitsof timeAnalyzing an RC circuit6.002 Fall 2000 Lecture 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].Let’s do an example:()IIVtv=()00VvC=givenICCVvdtdvCR =+XC+–()tvC()tvI+–R6.002 Fall 2000 Lecture 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].Example…Method of homogeneous and particularsolutions:123Find the particular solution.Find the homogeneous solution.The total solution is the sum ofthe particular and homogeneoussolutions.Use the initial conditions to solvefor the remaining constants.()IIVtv=()()()tvtvtvCPCHC+=totalhomogeneousparticular()00VvC=givenICCVvdtdvCR =+X6.002 Fall 2000 Lecture 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].1Particular solutionICPCPVvdtdvCR =+ICPVv=worksIIIVVdtdVCR =+0In general, use trial and error.vCP: any solution that satisfies theoriginal equationX6.002 Fall 2000 Lecture 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].2Homogeneous solution0=+CHCHvdtdvCRYvCH: solution to the homogeneous equation(set drive to zero)Y0=+ststeAdtedACR0=+ststeAesCARstCHeAv=assume solutionof this form. A, s ?Discard trivial A =0 solution,01=+sCRCharacteristic equationRCs1−=RCtCHAev−=orRCcalled timeconstant τ6.002 Fall 2000 Lecture 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].3Total solutionFind remaining unknown from initialconditions:CHCPCvvv+=RCtICeAVv−+=also()RCtI0CCeRVVdtdvCi−−−==thusGiven,so,or0CVv =at t = 0AVVI0+=I0VVA−=()RCtI0ICeVVVv−−+=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 12tCvIV0V()RCtI0ICeVVVv−−+=RC0Cite 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 12tCvV5V0VVI5=VVO0=50VVI0=VVO5=50tCvV5V0RCte−+
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