Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 13: Problem 20

Two rectangular waveguides are joined end-to-end. The guides have identical dimensions, where \(a=2 b\). One guide is air-filled; the other is filled with a lossless dielectric characterized by \(\epsilon_{r}^{\prime} .(a)\) Determine the maximum allowable value of \(\epsilon_{r}^{\prime}\) such that single-mode operation can be simultaneously assured in both guides at some frequency. \((b)\) Write an expression for the frequency range over which single-mode operation will occur in both guides; your answer should be in terms of \(\epsilon_{r}^{\prime}\), guide dimensions as needed, and other known constants.

Short Answer

Expert verified
A: The maximum value of the relative permittivity \(\epsilon_{r}^{\prime}\) that ensures single-mode operation in both waveguides is 0.25.

Step by step solution

01

Determining the cutoff frequency for air-filled waveguide

To determine the cutoff frequency for the air-filled waveguide, we use the expression for the cutoff frequency, \(f_c\) for the TE\(_{10}\) mode in rectangular waveguides, $$f_c = \frac{c}{2a\sqrt{\epsilon_r}}$$ where \(c\) is the speed of light in vacuum, \(a\) is the width of the waveguide, and \(\epsilon_r\) is the relative dielectric constant. For air, \(\epsilon_r = 1\). Therefore, the cutoff frequency for the air-filled waveguide is: $$f_c = \frac{c}{2a}$$
02

Determining the cutoff frequency for the dielectric-filled waveguide

Now, we want to find the cutoff frequency for the dielectric-filled waveguide operating in TE\(_{10}\) mode. In this case, we use the same formula as above with \(\epsilon_r = \epsilon_{r}^{\prime}\). $$f_c^{\prime} = \frac{c}{2a\sqrt{\epsilon_{r}^{\prime}}}$$
03

Ensuring single-mode operation

To ensure single-mode operation, the actual operating frequency, \(f_0\), must be higher than the cutoff frequency for both waveguides. Thus, we must ensure that $$f_c^{\prime} \leq f_0 \leq f_c$$ Using expressions of \(f_c\) and \(f_c^{\prime}\), we have: $$\frac{c}{2a\sqrt{\epsilon_{r}^{\prime}}} \leq f_0 \leq \frac{c}{2a}$$
04

Finding the maximum allowable value of \(\epsilon_{r}^{\prime}\)

To find the maximum value of \(\epsilon_{r}^{\prime}\), we must equate \(f_c^{\prime}\) and \(f_c\): $$\frac{c}{2a\sqrt{\epsilon_{r}^{\prime}}} = \frac{c}{2a}$$ Solve for \(\epsilon_{r}^{\prime}\): $$\epsilon_{r}^{\prime} = \frac{1}{4}$$ Thus, the maximum allowable value of \(\epsilon_{r}^{\prime}\) is 0.25.
05

Writing the expression for the frequency range

To ensure single-mode operation in both waveguides, the operating frequency must lie between \(f_c^{\prime}\) and \(f_c\) given by: $$\frac{c}{2a\sqrt{\epsilon_{r}^{\prime}}} \leq f_0 \leq \frac{c}{2a}$$ which gives us the frequency range for single-mode operation in terms of \(\epsilon_{r}^{\prime}\), guide dimensions and other known constants.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Using the relation \(\langle S\rangle=\frac{1}{2} \operatorname{Re}\left\\{\mathbf{E}_{s} \times \mathbf{H}_{s}^{*}\right\\}\) and Eqs. (106) through (108), show that the average power density in the \(\mathrm{TE}_{10}\) mode in a rectangular waveguide is given by $$ \langle S\rangle=\frac{\beta_{10}}{2 \omega \mu} E_{0}^{2} \sin ^{2}\left(\kappa_{10} x\right) \mathbf{a}_{z} \mathrm{~W} / \mathrm{m}^{2} $$

The mode field radius of a step index fiber is measured as \(4.5 \mu \mathrm{m}\) at free-space wavelength \(\lambda=1.30 \mu \mathrm{m}\). If the cutoff wavelength is specified as \(\lambda_{c}=1.20 \mu \mathrm{m}\), find the expected mode field radius at \(\lambda=1.55 \mu \mathrm{m}\).

Is the mode field radius greater than or less than the fiber core radius in single-mode step index fiber?

Two microstrip lines are fabricated end-to-end on a \(2-\mathrm{mm}\) -thick wafer of lithium niobate \(\left(\epsilon_{r}^{\prime}=4.8\right)\). Line 1 is of \(4 \mathrm{~mm}\) width; line 2 (unfortunately) has been fabricated with a \(5 \mathrm{~mm}\) width. Determine the power loss in \(\mathrm{dB}\) for waves transmitted through the junction.

Show that the group dispersion parameter, \(d^{2} \beta / d \omega^{2}\), for a given mode in a parallel-plate or rectangular waveguide is given by $$ \frac{d^{2} \beta}{d \omega^{2}}=-\frac{n}{\omega c}\left(\frac{\omega_{c}}{\omega}\right)^{2}\left[1-\left(\frac{\omega_{c}}{\omega}\right)^{2}\right]^{-3 / 2} $$ where \(\omega_{c}\) is the radian cutoff frequency for the mode in question [note that the first derivative form was already found, resulting in Eq. (57)].
See all solutions

Recommended explanations on Physics Textbooks

Circular Motion and Gravitation

Read Explanation

Force

Read Explanation

Nuclear Physics

Read Explanation

Magnetism

Read Explanation

Electric Charge Field and Potential

Read Explanation

Fields in Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free