MATERIAL PERTURBATION OF A RECTANGULAR CAVITY A rectangular cavity operating in the TE101 mode is perturbed by the insertion of a thin dielectric slab into the bottom of the cavity, as shown in Figure 6.24. Use the perturbational result of (6.100) to derive an expression for the change in resonant

Question

MATERIAL PERTURBATION OF A RECTANGULAR CAVITY A rectangular cavity operating in the [latex]\ TE_{101}[/latex] mode is perturbed by the insertion of a thin dielectric slab into the bottom of the cavity, as shown in Figure 6.24. Use the perturbational result of (6.100)[latex]\ \frac{\omega -\omega _{0}}{\omega _{0}} \simeq \frac{-\int_{V_{0} }^{}{\left(\Delta \epsilon \left|\bar{E_{0}} \right|^{2}+\Delta \mu \left|\bar{H_{0}} \right|^{2} \right)}dv }{\int_{V_{0} }^{}{\left(\epsilon \left|\bar{E_{0}} \right|^{2}+ \mu \left|\bar{H_{0}} \right|^{2} \right)}dv }[/latex]to derive an expression for the change in resonant frequency.

Accepted Answer

From (6.42a)[latex]\ E_{y}=E_{0}\sin \frac{\pi x}{a} \sin \frac{\ell \pi z}{d}[/latex]–

(6.42c)[latex]\ H_{z}=\frac{j\pi E_{0}}{k\eta a} \cos \frac{\pi x}{a} \sin \frac{\ell \pi z}{d}[/latex], the fields for the unperturbed TE101 cavity mode can be written as

[latex]\ E_{y}=A\sin \frac{\pi x}{a} \sin \frac{\pi z}{d}[/latex]

[latex]\ H_{x}=\frac{-jA}{Z_{TE}} \sin \frac{\pi x}{a} \cos \frac{\pi z}{d} ,[/latex]

[latex]\ H_{z}=\frac{j\pi A}{k\eta a} \cos \frac{\pi x}{a} \sin \frac{\pi z}{d} ,[/latex]
In the numerator of (6.100),[latex]\ \frac{\omega -\omega _{0}}{\omega _{0}} \simeq \frac{-\int_{V_{0} }^{}{\left(\Delta \epsilon \left|\bar{E_{0}} \right|^{2}+\Delta \mu \left|\bar{H_{0}} \right|^{2} \right)}dv }{\int_{V_{0} }^{}{\left(\epsilon \left|\bar{E_{0}} \right|^{2}+ \mu \left|\bar{H_{0}} \right|^{2} \right)}dv } [/latex]

[latex]\ \Delta \epsilon =\left(\epsilon _{r}-1\right)\epsilon _{0} for 0\leq y\leq t [/latex] and zero elsewhere.

The integral can then be evaluated as

[latex]\ \int_{V}^{}{\left(\Delta \epsilon \left|\bar{E_{0}} \right|^{2} +\Delta \mu \left|\bar{H_{0}} \right|^{2} \right) }dv=\left(\epsilon _{r}-1\right) \epsilon _{0}\int_{x=0}^{a}{\int_{y=0}^{t}{\int_{z=0}^{d}{\left|E_{y}\right|^{2} }\;dz }\;dy }\;dx[/latex]

[latex]\ =\frac{\left(\epsilon _{r}-1\right) \epsilon _{0}A^{2}atd}{4} .[/latex]

The denominator of (6.100) is proportional to the total energy in the unperturbed cavity, which was evaluated in (6.43); thus,

[latex]\ \int_{V}^{}{\left(\epsilon \left|\bar{E_{0}} \right|^{2} + \mu \left|\bar{H_{0}} \right|^{2} \right) }dv=\frac{adb\epsilon _{0} }{2}A^{2} .[/latex]

Then (6.100) gives the fractional change (decrease) in resonant frequency as

[latex]\ \frac{\omega -\omega _{0}}{\omega _{0}} =\frac{-\left(\epsilon _{r}-1\right)t }{2b}[/latex].

Question 6.7: MATERIAL PERTURBATION OF A RECTANGULAR CAVITY A rectangular ...

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