Question

The cutoff frequency of the \(m=1 \mathrm{TE}\) and TM modes in an air-filled parallel-plate guide is known to be \(f_{c 1}=7.5 \mathrm{GHz}\). The guide is used at wavelength \(\lambda=1.5 \mathrm{~cm}\). Find the group velocity of the \(m=2 \mathrm{TE}\) and \(\mathrm{TM}\) modes.

Short answer

Based on the given information, the cutoff frequency for the m=1 TE and TM modes is 7.5 GHz, and the wavelength is 1.5 cm. By following the step-by-step solution and using the appropriate formulas, we found that the group velocity of the m=2 TE and TM modes is approximately \(1.35\times10^8\) m/s.

Step-by-step solution

Identify known information and find the operating frequency

We are given the cutoff frequency (\(f_{c1}\)) for the m=1 TE and TM modes as 7.5 GHz, and the wavelength (\(\lambda\)) as 1.5 cm. We need to find the operating frequency (\(f\)) using: $$ f = \frac{c}{\lambda} $$ where \(c\) is the speed of light in a vacuum (approximately \(3 \times 10^8\) m/s).

Calculate the operating frequency

Using the given wavelength, we can find the operating frequency as follows: $$ f = \frac{3 \times 10^8}{0.015} = 20 \times 10^9 \ \text{Hz} $$ Now we have the operating frequency (20 GHz).

Find the cutoff frequency for m=2 TE and TM modes

The cutoff frequency (\(f_{c}\)) for the mth mode in a parallel-plate waveguide can be calculated using the following formula: $$ f_{c} = \frac{m}{2d}\sqrt{\mu \epsilon} $$ Since we're dealing with an air-filled waveguide, \(\mu\) and \(\epsilon\) will be equal to their free space values (\(\mu_0\) and \(\epsilon_0\)). Using the given \(f_{c1}\) (7.5 GHz), we can find the value of \(2d\sqrt{\mu_0 \epsilon_0}\) as follows: $$ 2d\sqrt{\mu_0 \epsilon_0} = \frac{1}{7.5 \times 10^9} $$ Now, we can find the \(f_{c2}\) using the same formula by plugging in \(m=2\) and the calculated value of \(2d\sqrt{\mu_0 \epsilon_0}\): $$ f_{c2} = \frac{2}{\frac{1}{7.5 \times 10^9}} = 15 \times 10^9 \ \text{Hz} $$ Now we have the cutoff frequency for the m=2 TE and TM modes (15 GHz).

Calculate the group velocity for the m=2 TE and TM modes

The group velocity (\(v_g\)) can be calculated using the following formula: $$ v_g = c\sqrt{1-\left(\frac{f_c}{f}\right)^2} $$ Now that we have the cutoff frequency for the m=2 TE and TM modes and the operating frequency, we can find the group velocity for the m=2 TE and TM modes: $$ v_g = 3\times10^8\sqrt{1-\left(\frac{15\times10^9}{20\times10^9}\right)^2} $$

Calculate the final group velocity

Evaluating the expression, we obtain the group velocity for the m=2 TE and TM modes: $$ v_g = 3\times10^8\sqrt{1-\left(\frac{3}{4}\right)^2} \approx 1.35\times10^8 \ \text{m/s} $$ Hence, the group velocity of the m=2 TE and TM modes is approximately \(1.35\times10^8\) m/s.

Key concepts

Cutoff Frequency

The cutoff frequency is a fundamental concept in waveguide theory, representing the lowest operating frequency at which a particular mode of a wave can propagate through the waveguide without attenuation. For instance, in the exercise, we encountered the cutoff frequency for the TE and TM modes in a parallel-plate waveguide. This frequency depends on the mode number and the physical dimensions of the waveguide. Specifically, it varies directly with the mode index (e.g., m=1, m=2, etc.) and inversely with twice the distance between the waveguide plates, symbolized as 2d. In mathematical form, the cutoff frequency, f_c, for a particular mode can be expressed as: \[\begin{equation} f_{c} = \frac{m}{2d}\sqrt{\mu \epsilon}\end{equation}\]where m is the mode number, and \(\mu\) and \(\epsilon\) represent the permeability and permittivity of the medium inside the waveguide, respectively.Understanding cutoff frequency is crucial as it defines the operational bandwidth and hence the applications of the waveguide. For example, a waveguide might be designed to operate above the cutoff frequency for lower attenuation and better signal integrity. Operating below the cutoff frequency would lead to signal loss as the wave becomes evanescent (i.e., it decays exponentially along the waveguide).

TE and TM Modes

The TE (transverse electric) and TM (transverse magnetic) modes are two primary types of electromagnetic wave modes that can propagate through a waveguide. In TE modes, the electric field is purely transverse to the direction of propagation, meaning that it has no longitudinal component. Conversely, in TM modes, the magnetic field is transverse, and the electric field has both transverse and longitudinal components.These modes have different field distributions and are identified by mode numbers, like TE_m or TM_m, where m is an integer denoting the mode order. Higher mode numbers correspond to higher cutoff frequencies and more complex field patterns within the waveguide. For design and analysis purposes, it's essential to consider the specific mode because each will behave differently in terms of propagation characteristics, like phase velocity, group velocity, and attenuation.Furthermore, distinct applications may utilize specific modes to achieve desired outcomes—for instance, higher-order modes for higher bandwidth applications but which may require more precise control to avoid mode interference.

Parallel-Plate Waveguide

A parallel-plate waveguide is a simple type of waveguide consisting of two conductive plates separated by a given distance, often filled with a dielectric material. This structure supports the propagation of electromagnetic waves along the direction of the plates. The waveguide's characteristics, such as supported modes and frequency response are dictated by the separation of the plates and the properties of the dielectric.This kind of waveguide is valuable for its simple geometry and ease of analysis. TE and TM modes, as previously discussed, can propagate through it, with the field distribution being influenced by the plate separation. In practical applications, the design of a parallel-plate waveguide is tailored to the specific needs of the system, such as the required operating frequency range and the selection of the dielectric material.

Wavelength

Wavelength (\(\lambda\)) is the distance between successive crests (or any identical points) of a wave. It's inversely related to the frequency of the wave; higher frequencies correspond to shorter wavelengths and vice versa. In the context of the given problem, wavelength is a key element that we use to determine the operating frequency of the waveguide. The relationship between the speed of light (\(c\)), frequency (\(f\)), and wavelength (\(\lambda\)) is described by the equation:\[\begin{equation} f = \frac{c}{\lambda}\end{equation}\]Consequently, when we know the wavelength of the signal and the speed at which it travels, we can calculate the operating frequency. It's imperative to understand this relationship as it lays the groundwork for frequency domain analysis of electromagnetic waves and their application in technologies such as waveguides, antennas, and communication systems.

Operating Frequency

The operating frequency is the actual frequency at which a device or system is functioning. It is central to the operation of waveguides and other RF (radio frequency) devices. This frequency plays a critical role in determining the possible modes of operation and the efficiency of the wave propagation.In the context of our exercise, the operating frequency was found to be crucial for calculating the group velocity of various modes within the waveguide. It's essential to operate a waveguide above the cutoff frequency for the particular mode to ensure efficient transmission. Operating below this frequency results in signal attenuation and thus is avoided. By knowing the operating frequency, we can further evaluate the performance of the system and refine our design and diagnostics for a wide range of electromagnetic applications.