EECS 105 Fall 2003 Lecture 5 Lecture 5 2nd Order Circuits in the Time Domain Physics of Conduction Prof Niknejad Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 5 Prof A Niknejad Lecture Outline l Second order circuits l Time Domain Response Physics of Conduction Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 5 Prof A Niknejad Series LCR Step Response l l l Consider the transient response of the following circuit when we apply a step at input Without inductor the cap charges with RC time constant EECS 40 Where does the inductor come from Intentional inductor placed in series Every physical loop has inductance parasitic Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 5 Prof A Niknejad LCR Step Response L Small l l We know the steady state response is a constant voltage of Vdd across capacitor inductor is short cap is open For the case of zero inductance we know solution is of the following form v0 t Vdd v0 t Vdd 1 e t t Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 5 Prof A Niknejad LCR Circuit ODE l Apply KVL to derive governing dynamic equations vs t vC t vR t vL t l Inductor and capacitor currents voltages take the dvC di form i i C v L C L dt dt d 2 vC d dvC vL L C LC 2 dt dt dt dvC vR iR RC dt l We have the following 2nd order ODE dvC d 2 vC vs t vC t RC LC 2 dt dt Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 5 Prof A Niknejad Initial Conditions l For the solution of a second order circuit we need to specify to initial conditions IC v0 0 vC 0 0 V i 0 iL 0 0 V l l For t 0 the source voltage is Vdd We can now solve for the following non homogeneous equation subject to above IC dvC d 2 vC Vdd vC t RC LC 2 dt dt d Steady state 0 dt Vdd vC Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 5 Prof A Niknejad Guess Solution l Let s subtract out the steady state solution vC t Vdd v t d 2v Vdd Vdd v t RC LC 2 dt dt dv d 2v 0 v t RC LC 2 dt dt dv l Guess solution is of the following form v t Ae st 1 RCs LCs 0 Ae st RC sAe st LC s 2 Ae st 0 Ae st 2 0 1 RCs LCs 2 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 5 Prof A Niknejad Again We re Back to Algebra l Our guess is valid if we can find values of s that satisfy this equation 1 Q 0 1 RCs LCs l l l 2 1 s 2 s 0 2 2 1 0 The solutions are s 2 1 This is the same equation we solved last lecture There we found three interesting cases 1 1 1 Department of EECS Underdamped Critically Damped Overdamped University of California Berkeley EECS 105 Fall 2003 Lecture 5 Prof A Niknejad General Case l Solutions are real or complex conj depending on if 1 or 1 s1 1 2 s 1 s2 vC t Vdd A exp s1t B exp s2t vC 0 Vdd A B 0 dvC t i 0 C 0 As1 exp s1t Bs2 exp s2t t 0 0 dt t 0 As1 Bs2 0 A B Vdd Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 5 Prof A Niknejad Final Solution General Case l Solve for A and B A s1 A Vdd s2 s1 s1 Vdd 1 Vdd Vdd Vdd s2 s2 A B Vdd A s1 s1 s1 1 1 1 s2 s2 s2 s1 Vdd Vdd s2 vC t Vdd exp s1t exp s2t s1 s1 1 1 s2 s2 1 s1t s1 s2t e e vC t Vdd 1 s1 s 2 1 s 2 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 5 Prof A Niknejad Overdamped Case l 1 Time constants are real and negative s1 1 2 s 1 0 s2 1 Vdd 1 2 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 5 Prof A Niknejad Critically Damped l 1 Time constants are real and equal 1 1 2 s 1 lim vC t Vdd 1 e t te t 1 1 Vdd 1 1 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 5 Prof A Niknejad Underdamped l Now the s values are complex conjugate s1 a jb s2 a jb vC t Vdd A exp a jb t B exp a jb t vC t Vdd eat A exp jbt B exp jbt s1 Vdd Vdd Vdd s2 A B s1 s2 1 1 s1 1 s2 s2 s1 vC t Vdd eat A exp jbt A exp jbt Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 5 Prof A Niknejad Underdamped cont l So we have vC t Vdd eat A exp jbt A exp jbt vC t Vdd eat 2 Re A exp jbt vC t Vdd eat 2 A cos t Vdd A s1 1 s2 Department of EECS Vdd s1 1 s2 University of California Berkeley EECS 105 Fall 2003 Lecture 5 Prof A Niknejad Underdamped Peaking l For 1 the step response overshoots 1 Vdd 1 5 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 5 Prof A Niknejad Extremely Underdamped 1 Vdd 1 01 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 5 Prof A Niknejad Ohm s Law l One of the first things we learn as EECS majors is V I R l Is this trivial Maybe what s really going on is the following V f I f 0 f 0 I f 0 I 2 2 f 0 I l l In the above Taylor exansion if the voltage is zero for zero current then this is generally valid The range of validity radius of convergence is the important question It turns out to be VERY large Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 5 Prof A Niknejad Ohm s Law Revisited l In Physics we learned J E l Is this also trivial Well it s the same as Ohm s law so the questions are related For a rectangular solid I V L I RI J V A L A l l Isn t it strange that current velocity is proportional to Force Where does conductivity come from Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 5 Prof A Niknejad Conductivity of a Gas l l Electrical conduction is due to the motion of positive and negative charges For water with pH 7 the concentration of hydrogen H ions and OH …
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