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Finding the amplitude of the sum vector

  1. Sep 12, 2013 #1
    1. The problem statement, all variables and given/known data
    A signal E(t) is made up of three terms, each having the same frequency but differing in phase:

    E(t) = E0cos(ωt) + E0cos(ωt + δ) + E0cos(ωt + 2δ)

    It is possible to find the amplitude of the sum vector by summing each vector described as a magnitude multiplied by a phase. The sum will therefore contain three terms. You can simplify this to express the sum as a real number times a phase. The real number is the amplitude of the sum vector. Make a plot of the amplitude of the sum vector as a function of δ as δ varies from 0 to 2∏.


    2. Relevant equations

    A signal can be represented as the real part of a complex number z,
    z = Aei(wt+∅) = A[cos(wt+∅) + jsin(wt+∅)]


    3. The attempt at a solution

    E(t) = E0cos(ωt) + E0cos(ωt + δ) + E0cos(ωt + 2δ)
    E(t) = E0exp(iωt) + E0exp(i(ωt + δ)) + E0exp(i(ωt + 2δ))
    E(t) = E0exp(iωt) * (1 + exp(iδ) + exp(i2δ))
    E(t) = E0cos(ωt) * (1 + exp(iδ) + exp(i2δ))

    I can't think of a way to simplify it any further to put it in the form real number * a phase term. Am I missing something?
     
  2. jcsd
  3. Sep 13, 2013 #2
    First of all, this equation really makes no sense:
    because you are using two different notations for the imaginary unit in the same expression. If you use i to denote the imaginary unit, you should have:
    Aei(wt+∅) = A[cos(wt+∅) + i sin(wt+∅)]

    Second, between the first and second steps in your attempt at a solution, you incorrectly applied that formula. You have changed a real expression into an expression with a real part and a non-zero imaginary part.

    If you do some simple algebra you should see that you can represent cos(x) as
    cos(x) = (eix+e-ix)/2
    Alternatively, you could write cos(x) as the real part of eix, e.g.
    cos(x)+cos(y) = Re[eix + eiy]

    Try applying those formulae and see if you can make any more progress. And feel free to post again if you get stuck. But you should make a better attempt before I give you any more hints.
     
    Last edited: Sep 13, 2013
  4. Sep 14, 2013 #3

    ehild

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    You can not use the same notation for the real E(t) and the complex one. Write E(t)=Re(Z) where Z=E0exp(iωt) + E0exp(i(ωt + δ)) + E0exp(i(ωt + 2δ)), and make it equal to Aexp(i(ωt+θ)), where A and θ ( both real) are to be determined.

    You need to transform the factor 1 + exp(iδ) + exp(i2δ) into the exponential form, 1 + exp(iδ) + exp(i2δ)=Cexp(iψ). For that, expand the exponentials exp(iδ)=cosδ+isinδ. Note that exp(i2δ)=[exp(iδ)]2. Collect the real and imaginary parts of the sum, and find the amplitude and phase.

    ehild
     
    Last edited: Sep 14, 2013
  5. Sep 15, 2013 #4

    ehild

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    the solution is much easier if you factor out E0exp(iωt)exp(iδ):

    [tex]\hat Z(t)=E_0 e^{i \omega t}+E_0 e^{i (\omega t+\delta)}+E_0 e^{i (\omega t+2\delta)}=E_0 e^{i (\omega t+\delta)} (e^{-i\delta}+1+e^{i\delta})[/tex]

    But [itex] e^{-i\delta}+1+e^{i\delta}=1+2\cos \delta[/itex], a real quantity. That makes

    [tex]\hat Z(t)=E_0 (1+2\cos \delta)e^{i (\omega t+\delta)} [/tex]




    ehild
     
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