Summary Week 6
    1. Method of Partial Fractions to find inverse Laplace transform

      2.  

       
       
       

      Given:  f(s) = num(s)/den(s)
       

      where num(s)=numerator polynomial in s, den(s)=denominator polynomial in s
       

      Step 1: Find the roots of the denominator polynomial, say  - r1,  ... ,  -rN

      Step 2: Based on these roots separate num(s)/den(s) into partial fractions.
       
       

      Case 1: For each root, say -q, appearing k times, add the following k terms:
       



       


      Case 2: For each complex pair of roots, say -a+ib and -a-ib, add two terms:
       
       



       


      Step 3. If needed, take the inverse Laplace transforms of each terms using the following formulas:
       
       



       



    2. Transfer Functions

      3.  

       

      Definition: a relationship from the input variable, say u, to an output variable, say y, in the Laplace domain

      Function Form: L(y) = G(s) L(u), where G(s) is the transfer function

      Block Diagram Form:
       



       





    3. Equivalent Transfer Functions
a ) Series:
equivalent to


 b ) Parallel Sum:
equivalent to
c) Feedback (Positive):


equivalent to