Second-Order Circuits Cont�d by LA4c64


									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
         d x(t )         dx(t )
                  2 0          0 x(t )  0

           dt 2           dt
• To find the natural solution, we solve the
  characteristic equation:
              s  2 0 s    0
               2                      2
• Which has two roots: s1 and s2.

                     ECE201 Lect-22               4
          Step-by-Step Approach
1. Assume solution (only dc sources allowed):
   i.    x(t) = K1 + K2 e-t/
   ii.   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         5
             Step-by-Step Approach
5.        Find time constant (), or characteristic roots (s)
     i.     Looking across the terminals of the C or L element,
            form Thevenin equivalent circuit; =RThC or =L/RTh
     ii.    Write ODE at t>0; find s from characteristic equation
6.        Finish up
     i.     Simply put the answer together.
     ii.    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                       6
           Class Examples
• Learning Extension E7.10
• Learning Extension E7.11

                  ECE201 Lect-22   7

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