Question

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)].

Short answer

Short Answer: The second derivative of the group dispersion parameter is found by differentiating the given first derivative expression with respect to the radian frequency 𝜔. By applying the chain rule and the product rule, we derive the expression: $$ \frac{d^2\beta}{d\omega^2}=-\frac{n}{\omega c}\left(\frac{\omega_c}{\omega}\right)^2\left[1-\left(\frac{\omega_c}{\omega}\right)^2\right]^{-3/2} $$ This expression is equivalent to the given expression, confirming our derivation is correct.

Step-by-step solution

The given first derivative expression is: $$ \frac{d \beta}{d \omega}=-\frac{n}{c}\frac{\omega_{c}}{\omega}\left[1-\left(\frac{\omega_{c}}{\omega}\right)^{2}\right]^{-1 / 2} $$ #Step 2: Differentiate the first derivative expression with respect to \(\omega\)#

To find the second derivative expression, we differentiate the given first derivative expression with respect to \(\omega\). This requires applying the chain rule and the product rule: $$ \frac{d^2\beta}{d\omega^2}=\frac{d}{d\omega}\left[\frac{d\beta}{d\omega}\right] $$ Using the chain rule and the product rule, we get: $$ \frac{d^2\beta}{d\omega^2}=\frac{d}{d\omega}\left[-\frac{n}{c}\frac{\omega_c}{\omega}\left[1-\left(\frac{\omega_c}{\omega}\right)^2\right]^{-1/2}\right] $$ #Step 3: Simplify the expression#

Now we will simplify the expression and express the result in terms of \(\omega\), \(\omega_c\), \(n\), and \(c\). This will help us identify whether our result matches the given expression. We start by differentiating the inside function, which contains the term \(\left[1-\left(\frac{\omega_c}{\omega}\right)^2\right]^{-1/2}\), with respect to \(\omega\): $$ \frac{d}{d\omega}\left[\left[1-\left(\frac{\omega_c}{\omega}\right)^2\right]^{-1/2}\right] = -\frac{1}{2}\left[1-\left(\frac{\omega_c}{\omega}\right)^2\right]^{-3/2}\left(-2\frac{\omega_c^2}{\omega^3}\right) $$ Now we differentiate the whole expression and simplify: $$ \frac{d^2\beta}{d\omega^2}=-\frac{n}{c}\frac{\omega_c}{\omega^2}\left[-\frac{1}{2}\left[1-\left(\frac{\omega_c}{\omega}\right)^2\right]^{-3/2}\left(-2\frac{\omega_c^2}{\omega^3}\right)\right] $$ $$ \frac{d^2\beta}{d\omega^2}=-\frac{n}{\omega c}\left(\frac{\omega_c}{\omega}\right)^2\left[1-\left(\frac{\omega_c}{\omega}\right)^2\right]^{-3/2} $$ #Step 4: Compare the derived expression with the given expression#

Our derived expression is: $$ \frac{d^2\beta}{d\omega^2}=-\frac{n}{\omega c}\left(\frac{\omega_c}{\omega}\right)^2\left[1-\left(\frac{\omega_c}{\omega}\right)^2\right]^{-3/2} $$ The given expression is: $$ \frac{d^{2} \beta}{d \omega^{2}}=-\frac{n}{\omega c}\left(\frac{\omega_{c}}{\omega}\right)^{2}\left[1-\left(\frac{\omega_{c}}{\omega}\right)^{2}\right]^{-3 / 2} $$ Since both expressions are the same, we have successfully derived the expression for the group dispersion parameter.

Key concepts

Waveguide Modes

Waveguide modes are specific patterns of electromagnetic fields that can propagate through a waveguide without attenuation, as long as the operating frequency is above a certain threshold. These modes arise due to the boundary conditions imposed on the electromagnetic fields by the waveguide structure.

In a rectangular waveguide, modes are classified into transverse electric (TE) and transverse magnetic (TM) based on the orientation of the electric and magnetic fields. TE modes have no electric field component in the direction of wave propagation, while TM modes have no magnetic field component in that direction. There's also a hybrid mode known as transverse electromagnetic (TEM), but it can only exist in waveguides that support a continuous electrical path, like coaxial cables.

Importance of Understanding Modes

Knowing about waveguide modes is crucial for engineers and physicists working with waveguides, as each mode has specific characteristics and dispersion properties that can impact the performance of communication systems and other applications using waveguides.

Radian Cutoff Frequency

The radian cutoff frequency, denoted as \(\theta_{c}\), is a fundamental concept when dealing with waveguide modes. It represents the minimum angular frequency at which a particular mode can propagate through a waveguide. Below this frequency, the mode becomes evanescent and attenuates rapidly. The cutoff frequency is dependent on the waveguide's dimensions and the mode in question.

The equation for the radian cutoff frequency of the TE modes in a rectangular waveguide is given by:
\[\theta_{c} = (m^2\frac{\pi^2}{a^2} + n^2\frac{\pi^2}{b^2})^{1/2}\frac{c}{\epsilon_r}\]
where \( m \) and \( n \) are the mode numbers, \( a \) and \(b\) are the dimensions of the waveguide, \( c \) is the speed of light, and \( \epsilon_r \) is the relative permittivity of the medium inside the waveguide.

Applications in Design

The knowledge of the cutoff frequency is essential for the design of waveguides, as it determines the frequency bands that the waveguide can support. It also has implications for the allowable bandwidth and the design of filters and multiplexers in telecommunication systems.

Differentiation in Electromagnetics

Differentiation plays a critical role in developing equations and solutions in electromagnetics. For example, the process of finding the second derivative of the propagation constant \(\beta\) with respect to angular frequency \(\theta\), as shown in the exercise, involves complex differentiation rules.

Differentiating the propagation constant is significant as it allows us to uncover the behavior of group velocity dispersion in waveguides. This dispersion can lead to signal distortion over long distances, which is particularly critical to consider in high-speed communication systems.

Complexities and Techniques

In the example provided, we see the application of the product rule and the chain rule — essential differentiation techniques for handling the product of two functions and the composition of functions, respectively. Mastering these techniques provides a strong foundation for anyone diving into the field of electromagnetics, as they will encounter similar mathematical challenges frequently.