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Imaginary factor in WAVE guide TE field

  1. Jun 10, 2013 #1
    Hey guys. I'm trying to comprehend the TEmn EM fields in wave guides. I've gone through the derivation, using Pozar's microwave textbook, and for the most part it's straight forward. I am having a hard time though determining what the effect of the imaginary factor in the field equations are.
    Here is the simplest case, a TE10 wave propagating in the z direction, with a picture of the waveguide dimentions and the E field as I would imagine it to be.
    The E and H fields are given as:

    E[itex]_{y}[/itex] = [itex]\frac{-jωμm\pi}{k^{2}a}[/itex] A sin[itex]\frac{mx\pi}{a}[/itex] e[itex]^{-jβz}[/itex]

    H[itex]_{x}[/itex] = [itex]\frac{jβm\pi}{k^{2}a}[/itex] A sin[itex]\frac{mx\pi}{a}[/itex] e[itex]^{-jβz}[/itex]

    k1an3a.jpg

    I understand there is a dependency on z from the e[itex]^{-jβz}[/itex] factor.
    I also understand there is a time and frequency dependancy (not shown) from e[itex]^{jωt}[/itex] factor.
    But what I'm really trying to understand is, how does the factor, [itex]\frac{-jωμm\pi}{k^{2}a}[/itex] , effect the fields?
    I'm not sure how I should tread the imaginary factor in this case.
    Thanks a lot.
     
  2. jcsd
  3. Jun 12, 2013 #2
    When we use complex numbers to describe real quantities, such as amplitudes of fields along a waveguide, the real value is found by multiplying by e^jωt then taking the real part.

    Ey(x,y,z,t) = [itex]Re[\frac{-jωμm\pi}{k^{2}a}[/itex] A sin[itex]\frac{mx\pi}{a}[/itex] e[itex]^{-jβz}[/itex]e[itex]^{jωt}][/itex]:

    These steps are not written explicitly but are understood.
     
  4. Jun 12, 2013 #3
    Thanks emi guy, but I think you missunderstood what part I was asking about. I got that there is an assumed factor e[itex]^{jωt}[/itex], but what I didn't understand was the first factor, colored red, in the Ey field equation below.

    Ey(x,y,z,t) = Re [itex][\frac{-jωμm\pi}{k^{2}a} [/itex] A sin[itex]\frac{mx\pi}{a}[/itex] e[itex]^{-jβz}[/itex]e[itex]^{jωt}][/itex]:

    I've been thinking about it though, about what it's affect on the E field is. If you could tell me if I'm right or not I'd be grateful.
    Using the identity: j = e[itex]^{j\frac{\pi}{2}}[/itex]

    The Ey field becomes:
    Ey(x,y,z,t) = Re [itex][\frac{ωμm\pi}{k^{2}a} [/itex] A sin[itex]\frac{mx\pi}{a}[/itex] e[itex]^{-jβz}[/itex]e[itex]^{jωt}[/itex] e[itex]^{-j\frac{\pi}{2}}[/itex] ]

    which is
    [itex][\frac{ωμm\pi}{k^{2}a} [/itex] A sin[itex]\frac{mx\pi}{a}[/itex] cos([itex]\omega[/itex]t - βz - 90°)

    which will just delay the phase 90° as the wave propagates along the z direction.

    Hope I'm on the right track. Thanks!
     
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