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EECS 105 Fall 2001 Lecture 5 R. T. Howe 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 Dept. of EECS University of California at Berkeley EECS 105 Fall 2001 Lecture 5 R. T. Howe A Second Order System + vL - R vC(t) vS(t) + L iL iR - iC C Where does the inductor come from? Do step response: vS(t) jumps to VDD at t = 0 Dept. of EECS University of California at Berkeley EECS 105 Fall 2001 Lecture 5 R. T. Howe Step Response of L-R-C Circuit Initial conditions: vC(t=0) = 0 V; iL (t=0) = 0 A iL iC dvC 1 t vL t dt C dt L 0 dvC Inductor voltage: vL VDD C R vC dt Dept. of EECS University of California at Berkeley EECS 105 Fall 2001 Lecture 5 R. T. Howe Second-Order Diff. Eqn. for vC(t) Substituting for vL: dvC dvC 1 t R vC dt C VDD C dt L 0 dt Differentiating both sides: d 2vC dvC 1 R vC C 2 VDD C dt L dt Dept. of EECS University of California at Berkeley EECS 105 Fall 2001 Lecture 5 R. T. Howe Solving the 2nd Order ODE 2 d vC dvC LC 2 RC vC VDD dt dt Steady-state solution: vC,ss = VDD (t ) Transient solution: vC,tr = ? … guess vC,tr = aest and substitute: LCs 2 aest RCs aest aest 0 Dept. of EECS University of California at Berkeley EECS 105 Fall 2001 Lecture 5 R. T. Howe Characteristic Equation s 2 L / R s 1 /( LC ) 0 Use quadratic formula to find the roots: 2 R R 1 s1, 2 2L 2 L LC Dept. of EECS University of California at Berkeley EECS 105 Fall 2001 Lecture 5 R. T. Howe Three Cases for s1 and s2 R 1 #1 2 L LC … get two negative, real roots R 1 #2 2L LC … get a single negative root R 1 #3 2L LC … interesting case Dept. of EECS University of California at Berkeley EECS 105 Fall 2001 Lecture 5 R. T. Howe Underdamped Case 2 s1, 2 R R 1 R 1 R j 2L 2 L LC 2L LC 2 L vC (t ) VDD a1e a2e ( R / 2 L )t j (1 / LC ) ( R / 2 L ) 2 t e 2 ( R / 2 L )t j (1 / LC ) ( R / 2 L ) 2 t e d Form of solution … Dept. of EECS University of California at Berkeley EECS 105 Fall 2001 Lecture 5 R. T. Howe Qualitative Underdamped Waveform vC(t) (V) 2.5 2 1.5 1 0.5 0 0 Dept. of EECS 100 200 300 400 500 t University of California at Berkeley EECS 105 Fall 2001 Lecture 5 R. T. Howe Extreme Underdamped Case Exponential decay time is set by R /( 2 L) Small R/L decay takes a long time and oscillation has a frequency that’s nearly 1 /( LC ) Number of cycles during “ringdown” is (1 / ) 2L / R 2 L N (1 / 1 /( LC ) ) LC R C What happens when R = 0 ? Dept. of EECS University of California at Berkeley