Question

Small antennas have low efficiencies (as will be seen in Chapter 14 ), and the efficiency increases with size up to the point at which a critical dimension of the antenna is an appreciable fraction of a wavelength, say \(\lambda / 8 .(a) \mathrm{An}\) antenna that is \(12 \mathrm{~cm}\) long is operated in air at \(1 \mathrm{MHz}\). What fraction of a wavelength long is it? \((b)\) The same antenna is embedded in a ferrite material for which \(\epsilon_{r}=20\) and \(\mu_{r}=2,000\). What fraction of a wavelength is it now?

Short answer

Answer: In air, the antenna length is 0.0004 times the wavelength, while in the ferrite material, the antenna length is 0.08 times the wavelength. This means that the fraction occupied by the antenna in the ferrite material is larger compared to when it is in air.

Step-by-step solution

Calculate the Wavelength in Air

To find the wavelength in air at 1 MHz, we first find the speed of light in air. The speed of light in air is approximately equal to the speed of light in a vacuum (\(c=3*10^{8} ms^{-1}\)). The formula to find the wavelength (\(\lambda\)) in terms of the frequency (\(f\)) and the speed of light (\(c\)) is: $$ \lambda = \frac{c}{f} $$ For 1 MHz, the frequency is \(1*10^6 Hz\). Substituting the values, we get: $$ \lambda_{air} = \frac{3*10^{8} ms^{-1}}{1*10^6 Hz} = 300 m $$

Calculate the Antenna Length Fraction in Air

Now that we have the wavelength in air, we can calculate the fraction of a wavelength that the antenna occupies. The antenna length is given as 12 cm, so we will use this value and divide it by the wavelength calculated in the previous step: $$ \frac{Length_{antenna}}{\lambda_{air}} = \frac{0.12 m}{300 m} = 0.0004 $$ So, the antenna length is 0.0004 times the wavelength in air.

Calculate the Wavelength in Ferrite Material

For the ferrite material, we need to find the wavelength considering the given \(\epsilon_{r}\) and \(\mu_{r}\) values. To find the speed of light in the ferrite material (\(v_f\)), we use the formula: $$ v_{f} = \frac{c}{\sqrt{\epsilon_{r} \mu_{r}}} $$ Substituting the known values, we get: $$ v_{f} = \frac{3*10^{8} ms^{-1}}{\sqrt{20*2000}} = \frac{3*10^{8} ms^{-1}}{\sqrt{40000}} = \frac{3*10^{8} ms^{-1}}{200} = 1.5*10^{6} ms^{-1} $$ Now we calculate the wavelength in the ferrite material using the same formula as in Step 1 but with the speed of light in ferrite material ('\(v_{f}\)'): $$ \lambda_{ferrite} = \frac{v_{f}}{f} = \frac{1.5*10^6 ms^{-1}}{1*10^6 Hz} = 1.5 m $$

Calculate the Antenna Length Fraction in Ferrite Material

Now, similarly to Step 2, we find the fraction of a wavelength of the antenna when it's embedded in the ferrite material: $$ \frac{Length_{antenna}}{\lambda_{ferrite}} = \frac{0.12 m}{1.5 m} = 0.08 $$ Hence, the antenna length is 0.08 times the wavelength in the ferrite material. To summarize: (a) In air, the antenna length is 0.0004 times the wavelength. (b) In the ferrite material, the antenna length is 0.08 times the wavelength.

Key concepts

Wavelength Calculation

The concept of wavelength calculation is crucial in understanding how antenna efficiency varies in different environments. The wavelength \( \lambda \) of an electromagnetic wave is the distance over which the wave's shape repeats. It can be calculated using the formula:

• \( \lambda = \frac{c}{f} \)

where \( c \) is the speed of light (approximately \( 3 \times 10^8 \text{ m/s} \)) and \( f \) is the frequency of the wave. For instance, at a frequency of 1 MHz, the wavelength in air is calculated as 300 meters.
This fundamental calculation helps determine how long a particular antenna can be, relative to the wavelength. A key concept here is that antennas are often designed around fractions of a wavelength, such as \( \lambda / 8 \), to optimize their efficiency.
Understanding the relationship between wavelength and antenna length is essential for both theoretical studies and practical antenna design.

Ferrite Material

Ferrite materials are key to enhancing the performance of antennas in specific environments. They are ceramic-like compounds made of iron oxide and other metallic elements.
Ferrites have special properties that are leveraged in electronic components to manage electromagnetic waves. In this context, they greatly impact the effective wavelength when antennas are embedded in them.
The relative permittivity \( \epsilon_r \) and relative permeability \( \mu_r \) of ferrite materials are crucial in this process. These properties determine how electromagnetic waves propagate within the material. For example, a ferrite material with \( \epsilon_r = 20 \) and \( \mu_r = 2000 \) changes the speed of wave propagation significantly.
This alters the calculated wavelength within the material using the formula:

• \( v_{f} = \frac{c}{\sqrt{\epsilon_{r} \mu_{r}}} \)

When we substitute the values, we find that the speed of light in the ferrite becomes much slower, reducing the wavelength inside the ferrite drastically compared to air.

Electromagnetic Properties

Understanding electromagnetic properties is important in antenna design and functionality. Electromagnetic waves consist of electric and magnetic fields propagating through space.
The relative permittivity \( \epsilon_r \) and relative permeability \( \mu_r \) are electromagnetic properties that define how materials interact with these fields. They indicate how much a material can store electrical and magnetic energy, respectively.
For an antenna embedded in a ferrite material:

• High \( \epsilon_r \) means the material can store more electric energy, impacting capacitance.
• High \( \mu_r \) affects the inductive properties, showing how much magnetic energy it can store.

These electromagnetic characteristics explain why the wavelength inside a ferrite material is significantly different from free space, altering antenna behavior and effectiveness.
Such properties are also pivotal in shielding, filtering, and directing electromagnetic fields, thus instrumental in various electromagnetic applications.

Antenna Design

Designing an efficient antenna involves considering the environment it operates in and the materials it's made from. Antenna length, shape, and material all play pivotal roles in its efficiency.
The goal is to ensure that the antenna's size corresponds well with the operating wavelength, as antennas operating at dimensions close to fractions of the wavelength (e.g., \( \lambda/8 \)) tend to be more efficient. In practical terms, this means:

• An antenna too short compared to the wavelength will have poor efficiency and reception.
• Incorporating materials like ferrite adjusts the effective wavelength, allowing for design flexibility in limited spaces.

Antenna design must account for the electromagnetic environment it will function in, whether in air or embedded in different materials such as ferrites. By altering the electromagnetic properties, different design strategies can be utilized to enhance performance in various operational settings.