Question

An antenna is designed to operate between $4.98\mathrm{GHz}$ and $5.02\mathrm{GHz}$ for a bandwidth of $\mathrm{\Delta }f=0.04\mathrm{GHz}$. Find $\mathrm{\Delta }\lambda ,$ the wavelength range over which the antenna is designed to operate. Hint: The answer is NOT $7.5\mathrm{m}$.

Short answer

The wavelength range is approximately 0.48 mm.

Step-by-step solution

Understand the Relationship Between Frequency and Wavelength

Recall that the speed of light, denoted as $c$, relates wavelength $\lambda$ and frequency $f$ through the equation: $$c=\lambda \times f$$. We will use this relationship to find wavelengths corresponding to the given frequency range.

Convert Frequency to Wavelength for Each Limit

Calculate the wavelength for both the lower and upper frequency limits.1. For the lower frequency $f_1=4.98\text{ GHz}$: ${\lambda }_1=\frac{c}{f_1}=\frac{3\times {10}^8\text{ m/s}}{4.98\times {10}^9\text{ Hz}}$.2. For the upper frequency $f_2=5.02\text{ GHz}$: ${\lambda }_2=\frac{c}{f_2}=\frac{3\times {10}^8\text{ m/s}}{5.02\times {10}^9\text{ Hz}}$.

Calculate Wavelengths

Perform the calculations for ${\lambda }_1$ and ${\lambda }_2$:- ${\lambda }_1=\frac{3\times {10}^8}{4.98\times {10}^9}\approx 0.06024\text{ meters}$.- ${\lambda }_2=\frac{3\times {10}^8}{5.02\times {10}^9}\approx 0.05976\text{ meters}$.

Determine the Wavelength Range $\mathrm{\Delta }\lambda$

Find the difference between the two calculated wavelengths: $\mathrm{\Delta }\lambda ={\lambda }_1-{\lambda }_2$.$\mathrm{\Delta }\lambda =0.06024\text{ m}-0.05976\text{ m}=0.00048\text{ meters}$.

Conclusion

The wavelength range $\mathrm{\Delta }\lambda$ over which the antenna is designed to operate is $0.00048\text{ meters}$ or $0.48\text{ mm}$.

Key concepts

Wavelength Calculation

Understanding how to calculate wavelength is crucial for designing antennas that operate efficiently within a specific frequency range. Wavelength is the distance between consecutive crests of a wave. It is inversely related to the frequency of a wave, which means that as frequency increases, the wavelength decreases.

To calculate wavelength, we often use the formula that relates the speed of light, frequency, and wavelength:

• $$\lambda =\frac{c}{f}$$

Where:

• $\lambda$ is the wavelength in meters.
• $c$ is the speed of light, approximately $3\times {10}^8$ meters per second.
• $f$ is the frequency in hertz (Hz), which is cycles per second.

For instance, if you have a frequency of $5\times {10}^9$ Hz, you can find the wavelength by dividing the speed of light by the frequency. This principle is key in allowing engineers to ensure that antennas efficiently send and receive signals at the desired frequencies.

The range of wavelengths can also be calculated if we know the range of frequencies an antenna operates within, which is achieved by converting the upper and lower frequency limits to their respective wavelengths and finding the difference between them.

Frequency Range

The frequency range is the span of frequencies over which an antenna or other device is designed to operate effectively. It's important to recognize that the frequency range is directly tied to an antenna's capacity to handle different wavelengths. This range encompasses all the frequencies an antenna is capable of transmitting or receiving.

To calculate the bounds of this frequency range, you simply subtract the lower frequency limit from the upper frequency limit. For example, an antenna designed to operate between $4.98\mathrm{GHz}$ and $5.02\mathrm{GHz}$ effectively covers a band of $\mathrm{\Delta }f=0.04\mathrm{GHz}$.

Having a precise understanding of the frequency range enables determination of the appropriate electronic components and materials needed in the antenna design to ensure optimal performance. It is essential for engineers to match these components to specific wavelength requirements, which involves back-and-forth calculations between frequency and wavelength, utilizing the speed of light as a key element.

Speed of Light

The speed of light is a fundamental constant crucial to many calculations in physics, including those in antenna design. Denoted as $c$, the speed of light in a vacuum is approximately $3\times {10}^8$ meters per second. This constant is indispensable when transitioning between the concepts of frequency, wavelength, and speed.

The equation relating these three quantities is:

• $$c=\lambda \times f$$

Here, the speed of light helps to determine how fast electromagnetic waves travel through space. In practical terms, when designing an antenna, it allows for the conversion of frequency to wavelength and vice versa. This conversion is critical because different applications and broadcasting needs specify their functionality based on either factor.

In understanding why the speed of light is so significant in antenna design, consider that it helps maintain synchronization between transmitted and received signals. By ensuring that calculations incorporate this constant properly, engineers can develop precise and effective communication equipment, enabling technologies ranging from simple radio broadcasting to complex satellite communications.