SURFACE WAVE PROPAGATION CONSTANTS Calculate and plot the propagation constants of the first three propagating surface wave modes of a grounded dielectric sheet with εr = 2.55, for d/λ0 = 0 to 1.2.

Question

SURFACE WAVE PROPAGATION CONSTANTS

Calculate and plot the propagation constants of the first three propagating surface wave modes of a grounded dielectric sheet with [latex]\ \epsilon _{r}=2.55[/latex], for [latex]\ d/\lambda _{0}=0[/latex] to 1.2.

Accepted Answer

The first three propagating surface wave modes are the [latex]TM_{0}[/latex], [latex]TE_{1}[/latex], and [latex]TM_{1}[/latex]
modes. The cutoff frequencies for these modes can be found from (3.167) [latex]\ f_{c}=\frac{nc}{2d\sqrt{\epsilon _{r}-1} } n=0,1,2,.....[/latex]and
(3.174) [latex]\ f_{c}=\frac{\left(2n-1 ight)c }{2d\sqrt{\epsilon _{r}-1} } for n=1,2,3,...[/latex]as

[latex]\ TM_{0}: f_{c}=0\Longrightarrow \frac{d}{\lambda _{0}} =0,[/latex]

[latex]\ TE_{1}:f_{c}=\frac{c}{4d\sqrt{\epsilon _{r}-1} }\Longrightarrow \frac{d}{\lambda _{0}}=\frac{1}{\left(4\sqrt{\epsilon _{r}-1} ight) }[/latex]

[latex]\ TM_{1}:f_{c}=\frac{c}{2d\sqrt{\epsilon _{r}-1} }\Longrightarrow \frac{d}{\lambda _{0}}=\frac{1}{\left(2\sqrt{\epsilon _{r}-1} ight) }[/latex]

The propagation constants can be found from the numerical solution of (3.165)[latex]\ k_{c} an k_{c} d=\epsilon _{r}h[/latex]
and (3.166)[latex]\ k^{2}_{c}+h^{2}=\left(\epsilon _{r}-1 ight)k^{2}_{0}[/latex] for the TM modes and (3.172)[latex]\ -k_{c}\cot k_{c}d=h[/latex] and (3.173)[latex]\ k^{2}_{c}+h^{2}=\left(\epsilon _{r}-1 ight)k^{2}_{0}[/latex] for the TE modes. This can be done with a relatively simple root-finding algorithm (see the Point of Interest on root-finding algorithms); the results are shown in Figure 3.21.

Question 3.4: SURFACE WAVE PROPAGATION CONSTANTS Calculate and plot the pr...

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