6.002 Fall 2000 Lecture 1 12 6.002 CIRCUITS AND ELECTRONICS Capacitors and First-Order Systems 6.002 Fall 2000 Lecture 2 12 5V 0V C A B 5V A B C 5 0 5 0 5 0 Reading: Chapters 9 & 10 Demo 5V Expected Observed Expect this, right? But observe this! Delay! Motivation 6.002 Fall 2000 Lecture 3 12 The Capacitor G D S n-channel MOSFET symbol n-channel MOSFET n-channel s i l i c o n n m e t a l + + + + + + o x i d e drain gate source C GS G D S n p 6.002 Fall 2000 Lecture 4 12 Ideal Linear Capacitor obeys DMD! total charge on capacitor 0qq =−+= d EA C = + + + + + + -- ----- A E d coulombs farads volts vCq = i C q + – v 6.002 Fall 2000 Lecture 5 12 Ideal Linear Capacitor dt dq i = ( ) dt Cvd = dt dv C = i vCq = C q + – v A capacitor is an energy storage device Æ memory device Æ history matters!       = 2 2 1 CvE 6.002 Fall 2000 Lecture 6 12 Apply node method: C + – () tv C () tv I + – R Thévenin Equivalent: 0=+ − dt dv C R vv CIC IC C vv d t dv CR =+ 0 tt ≥ ( ) 0 tv C given units of time Analyzing an RC circuit 6.002 Fall 2000 Lecture 7 12 Let’s do an example: () II Vtv = () 0 0 Vv C = given IC C Vv d t dv CR =+ X C + – () tv C () tv I + – R 6.002 Fall 2000 Lecture 8 12 Example… Method of homogeneous and particular solutions: 1 2 3 Find the particular solution. Find the homogeneous solution. The total solution is the sum of the particular and homogeneous solutions. Use the initial conditions to solve for the remaining constants. () II Vtv = () ( )() tvtvtv CPCHC += total homogeneous particular () 0 0 Vv C = given IC C Vv d t dv CR =+ X 6.002 Fall 2000 Lecture 9 12 1 Particular solution ICP CP Vv d t dv CR =+ ICP Vv = works II I VV d t dV CR =+ 0 In general, use trial and error. v CP : any solution that satisfies the original equation X 6.002 Fall 2000 Lecture 10 12 2 Homogeneous solution 0 =+ CH CH v d t dv CR Y v CH : solution to the homogeneous equation (set drive to zero) Y 0 =+ st st eA d t edA CR 0=+ stst eAesCAR st C H eAv = assume solution of this form. A , s ? Discard trivial A =0 solution, 01 =+sCR Characteristic equation RC s 1 −= RC t C H Aev − = or RC called time constant τ . 6.002 CIRCUITS AND ELECTRONICS Capacitors and First-Order Systems 6.002 Fall 2000 Lecture 2 12 5V 0V C A B 5V A B C 5 0 5 0 5 0 Reading: Chapters 9 &