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Homework Help: Complex exponentials & phasors

  1. Sep 15, 2010 #1
    1. The problem statement, all variables and given/known data
    x(t) = 2sin([tex]\omega[/tex]0t + 45o) + cos([tex]\omega[/tex]0t)

    Express x(t) in the form x(t) = Acos([tex]\omega[/tex]0t + [tex]\phi[/tex])

    3. The attempt at a solution

    I don't really know when to begin; I can't find anything about it in the textbook.
     
    Last edited: Sep 15, 2010
  2. jcsd
  3. Sep 15, 2010 #2

    rock.freak667

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    Homework Helper

    well expand out Acos(ω0t+φ) and then equate coefficients.
     
  4. Sep 15, 2010 #3
    I'm not sure what that = 45 is doing. I'm going to assume you meant + 45

    [tex]2sin(\omega_0 t + 45^o) + cos(\omega_0t)[/tex]
    our first step in using phasor notation is to define each sinusoid as either a sine or cosine:
    [tex]2cos(\omega_0 t +45^o - 90^o) + cos(\omega_0 t)[/tex]
    [tex]2cos(\omega_0 t -45^o) + cos(\omega_0 t)[/tex]
    we then define what our phasor is at 0 degrees:
    [tex]cos(\omega_0t) = (1 \angle 0^o)[/tex]
    apply it:
    [tex](2 \angle -45^o) + (1 \angle 0^o)[/tex]
    break up into rectangular coordinates:
    [tex]2cos(-45^o) + 2sin(-45^o)j + 1[/tex]
    use common real and imaginary part arithmetic to bring back into polar form:
    [tex](2.798 \angle -30.3612)[/tex]
    bring out of phasor form:
    [tex]2.798cos(\omega_o t - 30.3612^o)[/tex]
     
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