R. T. HoweEECS 105 Fall 2001 Lecture 5 Dept. of EECS University of California at Berkeley Lecture 5 Last time: –Bode plots for first-order transfer functions (low-pass and high-pass filters) –Rapid sketching techniques Today : –Second order circuits: time domain analysis

R. T. HoweEECS 105 Fall 2001 Lecture 5 Dept. of EECS University of California at Berkeley A Second Order System Where does the inductor come from? Do step response: v S (t) jumps to V DD at t = 0

R. T. HoweEECS 105 Fall 2001 Lecture 5 Dept. of EECS University of California at Berkeley Step Response of L-R-C Circuit Initial conditions: v C (t=0) = 0 V; i L (t=0) = 0 A Inductor voltage:

R. T. HoweEECS 105 Fall 2001 Lecture 5 Dept. of EECS University of California at Berkeley Second-Order Diff. Eqn. for v C (t) Substituting for v L : Differentiating both sides:

R. T. HoweEECS 105 Fall 2001 Lecture 5 Dept. of EECS University of California at Berkeley Solving the 2 nd Order ODE Steady-state solution: v C,ss = V DD Transient solution: v C,tr = ? … guess v C,tr = ae st and substitute:

R. T. HoweEECS 105 Fall 2001 Lecture 5 Dept. of EECS University of California at Berkeley Characteristic Equation Use quadratic formula to find the roots:

R. T. HoweEECS 105 Fall 2001 Lecture 5 Dept. of EECS University of California at Berkeley Three Cases for s 1 and s 2 #1 … get two negative, real roots #2 … get a single negative root #3 … interesting case

R. T. HoweEECS 105 Fall 2001 Lecture 5 Dept. of EECS University of California at Berkeley Underdamped Case dd Form of solution … 

R. T. HoweEECS 105 Fall 2001 Lecture 5 Dept. of EECS University of California at Berkeley Qualitative Underdamped Waveform

R. T. HoweEECS 105 Fall 2001 Lecture 5 Dept. of EECS University of California at Berkeley Extreme Underdamped Case Exponential decay time is set by Small R/L  decay takes a long time and oscillation has a frequency that’s nearly What happens when R = 0  ? Number of cycles during “ringdown” is