Second-Order Circuits Cont’d
Dr. Holbert
April 24, 2006
ECE201 Lect-22
1
Important Concepts
• The differential equation for the circuit
• Forced (particular) and natural
(complementary) solutions
• Transient and steady-state responses
• 1st order circuits: the time constant ()
• 2nd order circuits: natural frequency (ω0)
and the damping ratio (ζ)
ECE201 Lect-22
2
Building Intuition
• Even though there are an infinite number of
differential equations, they all share
common characteristics that allow intuition
to be developed:
– Particular and complementary solutions
– Effects of initial conditions
– Roots of the characteristic equation
ECE201 Lect-22
3
Second-Order Natural Solution
• The second-order ODE has a form of
2
d x(t )
dx(t )
2

2



0
0 x (t )  0
2
dt
dt
• To find the natural solution, we solve the
characteristic equation:
s  2 0 s    0
2
2
0
• Which has two roots: s1 and s2.
ECE201 Lect-22
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Step-by-Step Approach
1. Assume solution (only dc sources allowed):
i.
ii.
x(t) = K1 + K2 e-t/
x(t) = K1 + K2 es t + K3 es t
1
2
2. At t=0–, draw circuit with C as open circuit and
L as short circuit; find IL(0–) and/or VC(0–)
3. At t=0+, redraw circuit and replace C and/or L
with appropriate source of value obtained in
step #2, and find x(0)=K1+K2 (+K3)
4. At t=, repeat step #2 to find x()=K1
ECE201 Lect-22
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Step-by-Step Approach
5.
Find time constant (), or characteristic roots (s)
i.
ii.
6.
Looking across the terminals of the C or L element,
form Thevenin equivalent circuit; =RThC or =L/RTh
Write ODE at t>0; find s from characteristic equation
Finish up
i.
ii.
Simply put the answer together.
Typically have to use dx(t)/dt│t=0 to generate another
algebraic equation to solve for K2 & K3 (try repeating
the circuit analysis of step #5 at t=0+, which basically
means using the values obtained in step #3)
ECE201 Lect-22
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Class Examples
• Learning Extension E7.10
• Learning Extension E7.11
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