Example Problems in Electro Magnetic Theory Wave propagation

1. A medium like Copper conductor which is characterized by the parameters $\bg_black&space;\sigma&space;=&space;5.8&space;X&space;10^{7}&space;Mho's/meter$ and $\epsilon&space;_{r}=1,\mu&space;_{r}=1$ uniform plane wave of frequency 50 Hz. Find $\alpha&space;,\beta&space;,v,\eta$  and $\lambda$.

Ans.  Given $\bg_black&space;\bg_black&space;\sigma&space;=&space;5.8&space;X&space;10^{7}&space;Mho's/meter$  ,     $\bg_black&space;\epsilon&space;_{r}=1,\mu&space;_{r}=1$    and $\bg_white&space;f=&space;50&space;Hz$

$\bg_white&space;\alpha&space;=?&space;,\beta&space;=?&space;,v&space;=&space;?,\eta&space;=?$ and $\bg_white&space;\lambda&space;=?$

Find the Loss tangent $\bg_white&space;\frac{\sigma&space;}{\omega&space;\epsilon&space;}&space;=&space;\frac{5.8X&space;10^{7}}{2&space;\pi&space;X50X\epsilon&space;_{o}\epsilon&space;_{r}}$

$\bg_white&space;\bg_white&space;\frac{\sigma&space;}{\omega&space;\epsilon&space;}&space;=&space;\frac{5.8X&space;10^{7}}{100\pi&space;X\epsilon&space;_{o}}$

$\bg_white&space;\bg_white&space;\frac{\sigma&space;}{\omega&space;\epsilon&space;}&space;=&space;2.08&space;X&space;10&space;^{16}>&space;>&space;1$

So given medium is a Conductor (Copper)

then $\bg_white&space;\alpha&space;(or)&space;\beta&space;=\sqrt{\frac{\omega&space;\mu&space;\sigma&space;}{2}}$

$\bg_white&space;=\sqrt{\frac{5.8X10^{7}X2\pi&space;X&space;50X\mu&space;_{o}}{2}}$

$\bg_white&space;\alpha&space;=&space;106.99$  , $\bg_white&space;\beta&space;=106.99$.

$\bg_white&space;v_{p}=\frac{\omega&space;}{\beta&space;}$  $\bg_white&space;=\frac{2\pi&space;X50}{106.99}$$\bg_white&space;=2.936&space;meters/Sec$.

$\bg_white&space;\lambda&space;=\frac{2\pi&space;}{\beta&space;}=\frac{2\pi&space;}{106.99}=0.0587&space;meters.$

$\bg_white&space;\eta&space;=\sqrt{\frac{j\omega&space;\mu&space;}{(\sigma&space;+j\omega&space;\epsilon&space;)}}$

$\bg_white&space;=\sqrt{\frac{jX2\pi&space;X50X\mu&space;_{o}}{(5.8X10^{7}+j2\pi&space;X50X\epsilon&space;_{o})}}$

$\bg_white&space;=&space;\sqrt{\frac{j&space;3.947&space;X10^{-4}}{(5.8X10^{7}+j&space;2.78&space;X10^{-9})}}$

$\bg_white&space;=&space;\sqrt{\frac{&space;3.947&space;X10^{-4}\angle&space;90^{o}}{(5.8X10^{7}\angle&space;-2.74&space;X&space;10^{-15})}}$

$\bg_white&space;=&space;\sqrt{0.68&space;X10&space;^{-11}}\angle&space;\frac{90-(2.74&space;X&space;10^{-5})}{2}$

$\eta&space;=&space;2.6&space;X&space;10^{-6}\angle&space;45^{o}$.

2. If $\bg_white&space;\epsilon&space;_{r}=9,\mu&space;=\mu&space;_{o}$ for a medium in which a wave with a frequency of $\bg_white&space;f=&space;0.3&space;GHz$ is propagating . Determine the propagation constant and intrinsic impedance of the medium when $\bg_white&space;\sigma&space;=0.$

Ans: Given $\bg_white&space;\epsilon&space;_{r}=9$,  $\bg_white&space;\mu&space;=\mu&space;_{o}$ , $\bg_white&space;f=0.3GHz$ and $\bg_white&space;\sigma&space;=0$.

$\bg_white&space;\gamma&space;=?,\eta&space;=?$

Since $\bg_white&space;\sigma&space;=0$, the given medium is a lossless Di-electric.

which implies $\bg_white&space;\alpha&space;=&space;\frac{\sigma&space;}{2}\sqrt{\frac{\mu&space;}{\epsilon&space;}}&space;=0.$

$\bg_white&space;\beta&space;=&space;\omega&space;\sqrt{\mu&space;\epsilon&space;}$

$\bg_white&space;=2\pi&space;X&space;o.3X10^{9}\sqrt{\mu&space;_{o}X9\epsilon&space;_{o}}$

$\bg_white&space;=&space;18.86$.

$\bg_white&space;\eta&space;=&space;\sqrt{\frac{\mu&space;}{\epsilon&space;}}$

$\bg_white&space;\eta&space;=&space;\sqrt{\frac{\mu_{o}\mu&space;_{r}&space;}{\epsilon_{o}\epsilon&space;_{r}&space;}}$

$\bg_white&space;\eta&space;=&space;\sqrt{\frac{\mu_{o}&space;}{9\epsilon_{o}&space;}}$

$\bg_white&space;\eta&space;=&space;\frac{120\pi&space;}{3}$

$\bg_white&space;\eta&space;=&space;40$ Ω.

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Author: vikramarka

Completed M.Tech in Digital Electronics and Communication Systems and currently working as a faculty.