# Imaginary factor in WAVE guide TE field

1. Jun 10, 2013

### FrankJ777

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$_{y}$ = $\frac{-jωμm\pi}{k^{2}a}$ A sin$\frac{mx\pi}{a}$ e$^{-jβz}$

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

I understand there is a dependency on z from the e$^{-jβz}$ factor.
I also understand there is a time and frequency dependancy (not shown) from e$^{jωt}$ factor.
But what I'm really trying to understand is, how does the factor, $\frac{-jωμm\pi}{k^{2}a}$ , effect the fields?
I'm not sure how I should tread the imaginary factor in this case.
Thanks a lot.

2. Jun 12, 2013

### the_emi_guy

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) = $Re[\frac{-jωμm\pi}{k^{2}a}$ A sin$\frac{mx\pi}{a}$ e$^{-jβz}$e$^{jωt}]$:

These steps are not written explicitly but are understood.

3. Jun 12, 2013

### FrankJ777

Thanks emi guy, but I think you missunderstood what part I was asking about. I got that there is an assumed factor e$^{jωt}$, but what I didn't understand was the first factor, colored red, in the Ey field equation below.

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

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$^{j\frac{\pi}{2}}$

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

which is
$[\frac{ωμm\pi}{k^{2}a}$ A sin$\frac{mx\pi}{a}$ cos($\omega$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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