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1
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

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A Second Order System
Where does the inductor come from?
Do step response vS(t) jumps to VDD at t 0
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Step Response of L-R-C Circuit
Initial conditions vC(t0) 0 V iL (t0) 0
A
Inductor voltage
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Second-Order Diff. Eqn. for vC(t)
Substituting for vL
Differentiating both sides
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Solving the 2nd Order ODE
Steady-state solution vC,ss VDD
Transient solution vC,tr ? guess vC,tr
aest and substitute
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Characteristic Equation
Use quadratic formula to find the roots
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Three Cases for s1 and s2
get two negative, real roots
1
get a single negative root
2
interesting case
3
8
Underdamped Case
?
?d
Form of solution
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Qualitative Underdamped Waveform
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Extreme Underdamped Case
Exponential decay time is set by
Small R/L ? decay takes a long time and
oscillation has a frequency thats nearly
Number of cycles during ringdown is
What happens when R 0 ??