91Ó°ÊÓ

Consider the formula for the reflection coefficient as

$$\begin{gathered} R=\frac{I_R}{I_I} \\ =\frac{{\left({\mathbf{Î\mu }}_{\mathbf{0}\mathbf{R}}\right)}^{\mathbf{2}}}{{\left({\mathbf{Î\mu }}_{\mathbf{0}\mathbf{I}}\right)}^{\mathbf{2}}} \end{gathered}$$ …. (1)

Consider the formula for the transmission medium permittivity as

${\stackrel{¯}{\mathrm{Î\mu }}}_{0T}=\frac{2}{1+\mathrm{β}}{\stackrel{¯}{\mathrm{Î\mu }}}_{oI}$

Write the expression for the permittivity of the reflected region as

$$\begin{gathered} {\stackrel{¯}{\mathrm{Î\mu }}}_{0R}={\left(\frac{1-\mathrm{β}}{1+\mathrm{β}}\right)}^2{\stackrel{¯}{\mathrm{Î\mu }}}_{oI}\left\{\mathrm{β}=\frac{{\mathrm{μ}}_1{V}_1}{{\mathrm{μ}}_2{V}_2}\right\} \\ \frac{{\stackrel{¯}{\mathrm{Î\mu }}}_{0R}}{{\stackrel{¯}{\mathrm{Î\mu }}}_{0R}}={\left(\frac{1-\mathrm{β}}{1+\mathrm{β}}\right)}^2 \end{gathered}$$

From above and from the equation (1) write the expression for the reflection coefficient as

$R={\left(\frac{1-\mathrm{β}}{1+\mathrm{β}}\right)}^2$$R={\left(\frac{1-\mathrm{β}}{1+\mathrm{β}}\right)}^2$

Write the expression for the transmission coefficients as

$T=\frac{{\stackrel{¯}{\mathrm{Î\mu }}}_2{v}_2}{{\stackrel{¯}{\mathrm{Î\mu }}}_1{v}_1}{\left(\frac{{\stackrel{¯}{\mathrm{Î\mu }}}_{0R}}{{\stackrel{¯}{\mathrm{Î\mu }}}_{0R}}\right)}^2$

Simplify as:

$$\begin{gathered} \frac{{\stackrel{¯}{\mathrm{Î\mu }}}_2{v}_2}{{\stackrel{¯}{\mathrm{Î\mu }}}_1{v}_1}=\frac{{\mathrm{μ}}_1}{{\mathrm{μ}}_2}\frac{{\stackrel{¯}{\mathrm{Î\mu }}}_2{\mathrm{μ}}_2}{{\stackrel{¯}{\mathrm{Î\mu }}}_1{\mathrm{μ}}_1}\frac{v_2}{v_1} \\ \frac{{\stackrel{¯}{\mathrm{Î\mu }}}_2{v}_2}{{\stackrel{¯}{\mathrm{Î\mu }}}_1{v}_1}=\frac{{\mathrm{μ}}_1}{{\mathrm{μ}}_2}{\left(\frac{v_1}{v_2}\right)}^2\frac{v_2}{v_1} \\ \frac{{\stackrel{¯}{\mathrm{Î\mu }}}_2{v}_2}{{\stackrel{¯}{\mathrm{Î\mu }}}_1{v}_1}=\frac{{\mathrm{μ}}_1}{{\mathrm{μ}}_2}\frac{v_2}{v_1} \\ \mathrm{β}=\frac{{\mathrm{μ}}_1}{{\mathrm{μ}}_2}\frac{v_2}{v_1} \end{gathered}$$

Rewrite the expression for the transmission coefficient as

$T=\mathrm{β}\frac{{\mathrm{Î\mu }}_{or}^2}{{\mathrm{Î\mu }}_I^2}$

Rewrite the expression for the transmission line coefficient as

$T=\mathrm{β}{\left(\frac{2}{1+\mathrm{β}}\right)}^2$

Add the expression for the reflection and transmission coefficient as

$$\begin{gathered} R+T={\left(\frac{1-\mathrm{β}}{1+\mathrm{β}}\right)}^2+\mathrm{β}{\left(\frac{2}{1+\mathrm{β}}\right)}^2 \\ =\frac{1}{{\left(1+\mathrm{β}\right)}^2}\left[4\mathrm{β}+{\left(1-\mathrm{β}\right)}^2\right] \\ =\frac{1}{{\left(1+\mathrm{β}\right)}^2}\left[4\mathrm{β}+\left(1+{\mathrm{β}}^2-2\mathrm{β}\right)\right] \\ =1 \end{gathered}$$

Therefore, the expression for the reflection coefficient is $R={\left(\frac{1-\mathrm{β}}{1+\mathrm{β}}\right)}^2$and transmission coefficient is$T=\mathrm{β}\frac{{\mathrm{Î\mu }}_{or}^2}{{\mathrm{Î\mu }}_I^2}$ . The proof for is shown R + T = 1.