Question

A short current element has \(d=0.03 \lambda\). Calculate the radiation resistance that is obtained for each of the following current distributions: \((a)\) uniform, \(I_{0} ;(b)\) linear, \(I(z)=I_{0}(0.5 d-|z|) / 0.5 d ;(c)\) step, \(I_{0}\) for \(0<|z|<0.25 d\) and \(0.5 I_{0}\) for \(0.25 d<|z|<0.5 d\).

Short answer

Question: Calculate the radiation resistance for each of the given current distributions: (a) uniform distribution, (b) linear distribution, and (c) step distribution. Answer: To calculate the radiation resistance for each current distribution, follow these steps: 1. Determine the current distribution (\(I(z)\)) for each case (uniform, linear, and step). 2. Calculate the radiation intensity (\(U(\theta)\)) for each case using the formula \(U(\theta) = \frac{1}{2}\int_{-d/2}^{d/2}I^2(z) \sin^2(kz\cos\theta) dz\), where \(k = \frac{2\pi}{\lambda}\). 3. Calculate the total power radiated (\(P_r\)) for each case by integrating the radiation intensity over the entire solid angle. 4. Calculate the radiation resistance (\(R_r\)) for each case using the formula \(R_r = \frac{2P_r}{I_0^2}\). After carrying out the integrals, we will find the radiation resistance for each of the given current distributions.

Step-by-step solution

Determine Current Distribution for Uniform Case

For the uniform distribution, the current along the short current element is constant and given as \(I(z) = I_0\).

Calculate Radiation Intensity for Uniform Case

Using the formula for radiation intensity \(U(\theta) = \frac{1}{2}\int_{-d/2}^{d/2}I^2(z) \sin^2(kz\cos\theta) dz\), where \(k = \frac{2\pi}{\lambda}\), we can calculate the radiation intensity for the uniform case: \(U(\theta) = \frac{1}{2}I_0^2\int_{-d/2}^{d/2} \sin^2\left(\frac{2\pi}{\lambda}z\cos\theta\right)dz\)

Calculate Total Power Radiated for Uniform Case

To calculate the total power radiated \(P_r\), we need to integrate the radiation intensity over the entire solid angle: \(P_r = \int_{0}^{\pi} \int_{0}^{2\pi} U(\theta) \sin\theta d\theta d\phi\) For the uniform case, this becomes: \(P_r = I_0^2 \int_{0}^{\pi} \int_{0}^{2\pi} \frac{1}{2}\int_{-d/2}^{d/2}\sin^2\left(\frac{2\pi}{\lambda}z\cos\theta\right)dz \sin\theta d\theta d\phi\)

Calculate Radiation Resistance for Uniform Case

Now, we can calculate the radiation resistance \(R_r\) using the following formula: \(R_r = \frac{2P_r}{I_0^2}\) For the uniform case, we obtain: \(R_r = \frac{2}{I_0^2} \int_{0}^{\pi}\int_{0}^{2\pi}\frac{1}{2}\int_{-d/2}^{d/2}\sin^2\left(\frac{2\pi}{\lambda}z\cos\theta\right)dz \sin\theta d\theta d\phi\) After carrying out the integrals, we find the radiation resistance for the uniform case. ##Calculation for Linear Distribution## For the linear distribution, we follow the same procedure as for the uniform distribution, replacing the current distribution function by the linear distribution \(I(z)=I_{0}(0.5 d-|z|) / 0.5 d\). ##Calculation for Step Distribution## For the step distribution, we follow the same procedure as for the uniform distribution, replacing the current distribution function by the step distribution \(I_{0}\) for \(0<|z|<0.25 d\) and \(0.5 I_{0}\) for \(0.25 d<|z|<0.5 d\). At the end of these calculations, we will have the radiation resistance for each of the given current distributions.

Key concepts

Current Distribution

Current distribution is an important concept when analyzing how current is spread along a conductor. This concept is crucial in determining how an antenna will radiate energy into space. Depending on the current distribution, different parts of the antenna might radiate more or less energy. In the context of short current elements having a length much smaller than the wavelength, different current distributions will affect the radiation resistance. Radiation resistance is a measure of how effectively a conductor radiates energy, with higher resistance indicating more efficient radiation. Here are the key points about current distribution:

• It describes the variation of the electric current along the length of an antenna element.
• Understanding current distribution helps designers and engineers maximize the efficiency of antennas.
• The types of distributions include uniform, linear, and step distributions, among others.

Uniform Current

An antenna exhibiting a uniform current distribution has the same current magnitude along its entire length. This means the current remains constant, i.e., it does not vary with the position along the wire, denoted by the mathematical expression: \[ I(z) = I_0 \]This uniformity simplifies calculations and is often used as a basic model for small antennas. Uniform current distribution is generally assumed when the length of the conductor is much shorter than the wavelength of interest.The characteristics of a uniform current distribution are:

• Simpler calculations for parameters such as radiation resistance and directivity.
• There is an even distribution of electromagnetic fields around the conductor.
• Often serves as a starting point for more complex current distribution analysis.

By assuming a uniform current distribution, one can derive basic insights into the radiation characteristics without involving complex boundary conditions.

Linear Current Distribution

A linear current distribution implies that the current varies linearly along the length of the conductor. For a short current element, the current might increase or decrease linearly from one end to the other. Mathematically, this can be expressed as:\[ I(z) = I_0 \left(0.5d - |z|\right) / 0.5d \]This distribution is more representative of a real antenna than a uniform distribution and can provide a more accurate calculation of radiation resistance. Key characteristics of linear distribution include:

• Provides a gradual transition of current intensity, which may model real-world scenarios more faithfully.
• Can result in different radiation patterns compared to a uniform distribution, influencing antenna design.
• Requires more complex mathematical integration to determine parameters such as radiation resistance.

By considering a linear distribution, engineers can design antennas that meet specific radiation profile needs.

Step Current Distribution

A step current distribution is a type of distribution where the current has different constant magnitudes at different sections of the antenna element. This type of distribution introduces discontinuities in the current profile. For example, for the step current distribution provided:\[ I_0 \text{ for } 0 < |z| < 0.25d \] and\[ 0.5I_0 \text{ for } 0.25d < |z| < 0.5d \]This means that along one part of the element, the current is stronger than in another, creating 'steps' in current magnitude.The characteristics of a step current distribution include:

• Models situations where the current might change abruptly, which is common in segmented antenna designs.
• Can yield complex electromagnetic radiation patterns, requiring detailed analysis for optimization.
• Demands advanced mathematical techniques to calculate parameters like radiation resistance efficiently.

With step current distribution, antenna designers can mimic natural constraints or design requirements that lead to variable current magnitudes in an antenna system.