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Find the phasor representation of an equation

  1. Jan 15, 2008 #1
    1. The problem statement, all variables and given/known data

    Find the phasors of the following time functions:

    (a) [tex]v(t)\,=\,3\,cos\left(\omega\,t\,-\,\frac{\pi}{3}\right)[/tex]

    (b) [tex]v(t)\,=\,12\,sin\left(\omega\,t\,+\,\frac{\pi}{4}\right)[/tex]

    (c) [tex]i(x,\,t)\,=\,2\,e^{-3\,x}\,sin\left(\omega\,t\,+\,\frac{\pi}{6}\right)[/tex]

    (d) [tex]i(t)\,=\,-2\,cos\left(\omega\,t\,+\,\frac{3\pi}{4}\right)[/tex]

    (e) [tex]i(t)\,=\,4\,sin\left(\omega\,t\,+\,\frac{\pi}{3}\right)\,+\,3\,cos\left(\omega\,t\,-\,\frac{\pi}{6}\right)[/tex]

    2. Relevant equations

    A short list of conversions from a larger table in the book. These are conversions from time domain sinusoidal functions on the left to cosine-reference phasor functions on the right.



    3. The attempt at a solution

    (a) [tex]3\,e^{-\frac{\pi}{3}\,j}[/tex]

    (b) [tex]12\,e^{j\,\left(\frac{\pi}{4}\,-\,\frac{\pi}{2}\right)}\,=\,12\,e^{-\frac{\pi}{4}\,j}[/tex]

    (c) [tex]2\,e^{-3\,x}\,e^{j\,\left(\frac{\pi}{6}\,-\,\frac{\pi}{2}\right)}\,=\,2\,e^{-3\,x}\,e^{-\frac{\pi}{3}\,j}\,=\,2\,e^{-3\,x\,-\,\frac{\pi}{3}\,j}[/tex]

    (d) [tex]-2\,e^{\frac{3\pi}{4}\,j}[/tex]

    (e) [tex]4\,e^{j\,\left(\frac{\pi}{3}\,-\,\frac{\pi}{2}\right)}\,+\,3\,e^{j\,\left(-\frac{\pi}{6}\right)}\,=\,7\,e^{j\,\left(-\frac{\pi}{6}\right)}[/tex]

    I have a question especially with (d), the answer is given as...


    I don't understand how they did the last two conversions in the given answer! Can someone please explain, and say whether the others are correct as well?

    Last edited: Jan 15, 2008
  2. jcsd
  3. Jan 16, 2008 #2
    what you have to know is just euler formula: exp(ix)=cos(x)+isin(x).
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