A plane wave with η=√jwµ/σ=√wµ/σ e^jπ/4=√wπ/2σ+j√wµ/2σ is incident normally on a thick plane conductor lying in the X-Y plane. Its conductivity is 6 × 10^6 S/m and surface impedance is 5 × 10^-4 ∠45°Ω. Determine the propagation • constant and the skin depth in the conductor.

Question

A plane wave with [latex]\eta=\sqrt{\frac{j w \mu}{\sigma}}=\sqrt{\frac{w \mu}{\sigma}} e^{j \pi / 4}=\sqrt{\frac{w \pi}{2 \sigma}}+j \sqrt{\frac{w \mu}{2 \sigma}}[/latex] is incident normally on a thick plane conductor lying in the X-Y plane. Its conductivity is [latex]6 imes 10^{6}[/latex] S/m and surface impedance is [latex]5 imes 10^{-4} \angle 45^{\circ} \Omega[/latex]. Determine the propagation • constant and the skin depth in the conductor.

Accepted Answer

Given, [latex]E_{y}=10 e^{j(\omega t-\beta z)}[/latex] for good conductor, σ >> ωE

[latex]\begin{aligned}&Z_{s}=\sqrt{\frac{j \omega \mu}{ u +j \omega E}} \cong \sqrt{\frac{j \omega \mu}{\sigma}}=R_{s}+j x s \&R_{s}=5 imes 10^{-4} \angle 45^{\circ} \&R_{s}=x_{s}=\sqrt{\frac{\omega / 4}{2 \sigma}}=\frac{5 imes 10^{-4}}{\sqrt{2}}\end{aligned}[/latex]

Propagation constant,

[latex]\begin{aligned}&\alpha=\frac{\sqrt{j \omega \sigma}}{2} \beta=\left(R_{s} ight) \sigma=\frac{5 imes 10^{-4}}{\sqrt{2}} imes 6 imes 10^{6} \&\Rightarrow \alpha=\beta=2|2| \&\gamma=\alpha+ J \beta=2|2|+ j 2|2|\end{aligned}[/latex]

Skin depth, [latex]d=\frac{1}{\alpha}=\frac{1}{2|2|}=0.471 mm[/latex],

Question 2.5.3: A plane wave with η=√jwµ/σ=√wµ/σ e^jπ/4=√wπ/2σ+j√wµ/2σ is in...

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