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Partial Fractions Expansion

  1. Apr 16, 2014 #1
    1. The problem statement, all variables and given/known data

    Find the partial fractions expansion in the following form,

    [tex]G(s) = \frac{1}{(s+1)(s^{2}+4)} = \frac{A}{s+1} + \frac{B}{s+j2} + \frac{B^{*}}{s-j2}[/tex]

    2. Relevant equations



    3. The attempt at a solution

    I expanded things out and found the following,

    [tex]1 = A(s^{2} + 4) + B(s^{2} + (1-j2)s -j2) + B^{*}(s^{2} + (1+j2)s + j2)[/tex]

    From this I get the following equations,

    [tex]A + B + B^{*} = 0[/tex]

    [tex]B(1-j2) + B^{*}(1+j2) = 0[/tex]

    [tex]4A - Bj2 + B^{*}j2 = 1[/tex]

    This doesn't seem like a pleasant set of equations to solve.

    I did another partial fractions expansion like so,


    [tex]G(s) = \frac{1}{(s+1)(s^{2}+4)} = \frac{D}{s+1} + \frac{Es + F}{s^{2}+4}[/tex]

    and found,

    [tex]D = \frac{1}{5}, E = - \frac{1}{5}, F = \frac{1}{5}[/tex]

    but this doesn't give me the form I need.

    Any ideas? Any convenient ways to solve this?
     
  2. jcsd
  3. Apr 16, 2014 #2

    jbunniii

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    Hint: if ##B^*## is the complex conjugate of ##B##, then ##B+B^* = 2\text{Re}(B)## and ##B - B^* = 2i\text{Im}(B)##. You can use these facts to simplify the unpleasant equations you obtained.
     
  4. Apr 16, 2014 #3

    lurflurf

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    solve by inspection

    $$\frac{1}{(s+1)(s^{2}+4)} =\frac{1}{(s+1)((-1)^{2}+4)}+\frac{1}{(2j+1)(2j+2j)(s-2j)}$$
    $$+\frac{1}{(-2j+1)(s+2j)(-2j-2j)}= \frac{A}{s+1} + \frac{B}{s+2j} + \frac{B^{*}}{s-2j}$$

    in general

    $$\prod_k \frac{1}{x-a_k}=\sum_l \frac{1}{x-a_l}\prod_{k \ne l} \frac{1}{a_l-a_k}$$
     
    Last edited by a moderator: Apr 16, 2014
  5. Apr 16, 2014 #4

    SammyS

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    or in Engineering terms where usually ##\ j=\sqrt{-1}\ ##, you have

    ##\ B - B^* = 2j\text{ Im}(B) \ ##
     
  6. Apr 16, 2014 #5

    jbunniii

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    Sorry, I just couldn't bring myself to type ##j##. :tongue:
     
  7. Apr 16, 2014 #6

    Mark44

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    I know what you mean. Also, when I saw j2 in the first post, I first thought he meant j2. I like 2j better than j2, and 2i better than 2j.
     
  8. Apr 17, 2014 #7

    Ray Vickson

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    It is probably easier to first expand as
    [tex] \frac{1}{(s+1)(s^2+4)} = \frac{A}{s+1} + \frac{Bs+C}{s^2+4}[/tex]
    then expand
    [tex] \frac{1}{s^2+4} = \frac{E}{s+2i} + \frac{F}{s-2i}[/tex]
    BTW: in TeX (or LaTeX) you do not need to write 's^{2}'; just plain 's^2' will do. You only need the '{.}' if the super-script (or sub-script) is more than one term. (Even a multi-letter single command can do without the '{.}'; for example, look at ##s^\alpha##, which was entered as s^\alpha.)
     
  9. Apr 17, 2014 #8

    Ray Vickson

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    It is probably easier to first expand as
    [tex] \frac{1}{(s+1)(s^2+4)} = \frac{A}{s+1} + \frac{Bs+C}{s^2+4}[/tex]
    then expand
    [tex] \frac{1}{s^2+4} = \frac{E}{s+2i} + \frac{F}{s-2i}[/tex]
    BTW: in TeX (or LaTeX) you do not need to write 's^{2}'; just plain 's^2' will do. You only need the '{.}' if the super-script (or sub-script) is more than one term. (Even a multi-letter single command can do without the '{.}'; for example, look at ##s^\alpha##, which was entered as s^\alpha.
     
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