Propagation of plane EM wave in conducting medium (or) lossy dielectrics

A lossy dielectric medium is one which an EM wave as it propagates losses power owing to imperfect dielectric,that is a lossy dielectric is an imperfect conductor that is a partially conducting medium (\sigma \neq 0) . 

where as a lossless dielectric is a  (\sigma =0) perfect dielectric,then wave equations for conductors are also holds good here 

i.e, \bigtriangledown ^{2}\overrightarrow{E}=\mu \sigma \frac{\partial \overrightarrow{E}}{\partial t} +\mu \epsilon \frac{\partial ^{2}\overrightarrow{E}}{\partial t^{2}}

\frac{\partial }{\partial t}= jw

then \bigtriangledown ^{2}\overrightarrow{E}= j\omega \mu \sigma \overrightarrow{E}+\mu \epsilon(j\omega ) ^{2}\overrightarrow{E}

\bigtriangledown ^{2}\overrightarrow{E}= (\sigma + j \omega \epsilon )j\omega \mu \overrightarrow{E}

\bigtriangledown ^{2}\overrightarrow{E}= \gamma ^{2} \overrightarrow{E}

\bigtriangledown ^{2}\overrightarrow{E}- \gamma ^{2} \overrightarrow{E}=0---------Equation (1)

Equation (1) is called helm holtz equation and \gamma is  called propagation constant.

\gamma ^{2} =j\omega \mu (\sigma +j\omega \epsilon )

\gamma ^{2}= j\omega \mu \sigma -\omega ^{2}\mu \epsilon

Since \gamma is a complex quantity it can be expressed as \gamma = \alpha +j\beta

\alpha– is attenuation constant measured in Nepers/meter.

\beta-is phase constant measured in radians/meter.

(\alpha +j\beta ) ^{2}= j\omega \mu \sigma -\omega ^{2}\mu \epsilon

\alpha^{2} +2j\alpha \beta-\beta ^{2} = j\omega \mu \sigma -\omega ^{2}\mu \epsilon

by equating real and imaginary parts separately \alpha ^{2}-\beta ^{2}= -\omega ^{2}\mu \epsilon------Equation(2)

 and 2\alpha \beta =\omega \mu \sigma

\alpha =\frac{\omega \mu \sigma}{2\beta }

 by substituting  \alpha value in the equation (2)   \frac{\omega ^{2}\mu ^{2}\sigma ^{2}}{4\beta ^{2}}-\beta ^{2}=-\omega ^{2}\mu \epsilon

{\omega ^{2}\mu ^{2}\sigma ^{2}}-4\beta ^{4}=-4\omega ^{2}\beta ^{2}\mu \epsilon

4\beta ^{4}-4\omega ^{2}\beta ^{2}\mu \epsilon -{\omega ^{2}\mu ^{2}\sigma ^{2}}=0

let \beta ^{2}=t 

4t^{2}-4\omega ^{2}t\mu \epsilon -{\omega ^{2}\mu ^{2}\sigma ^{2}}=0

t^{2}-\omega ^{2}t\mu \epsilon -\frac{\omega ^{2}\mu ^{2}\sigma ^{2}}{4}=0

the roots of the above quadratic expression are

t=\frac{\omega ^{2}\mu \epsilon \pm \sqrt{\omega ^{4}\mu ^{2}\epsilon ^{2}-4(-\frac{\omega ^{2}\mu ^{2}\sigma ^{2}}{4})}}{2}

t=\frac{\omega ^{2}\mu \epsilon \pm \sqrt{\omega ^{4}\mu ^{2}\epsilon ^{2}(1+\frac{\sigma }{\omega \epsilon })^{2}}}{2}

\beta ^{2}=\frac{\omega ^{2}\mu \epsilon \pm \sqrt{\omega ^{4}\mu ^{2}\epsilon ^{2}(1+\frac{\sigma }{\omega \epsilon })^{2}}}{2}

\beta =\sqrt{\frac{\omega ^{2}\mu \epsilon \pm \sqrt{\omega ^{4}\mu ^{2}\epsilon ^{2}(1+\frac{\sigma }{\omega \epsilon })^{2}}}{2}}

\beta =\sqrt{\frac{\omega ^{2}\mu \epsilon (1+ \sqrt{(1+\frac{\sigma }{\omega \epsilon })^{2}})}{2}}

similarly,

 

 

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Author: Lakshmi Prasanna Ponnala

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