Question

A dipole antenna in free space has a linear current distribution with zero current at each end, and with peak current \(I_{0}\) at the enter. If the length \(d\) is \(0.02 \lambda\), what value of \(I_{0}\) is required to \((a)\) provide a radiation-field amplitude of \(100 \mathrm{mV} / \mathrm{m}\) at a distance of \(1 \mathrm{mi}\), at \(\theta=90^{\circ} ;(b)\) radiate a total power of \(1 \mathrm{~W} ?\)

Short answer

In summary, for a given dipole antenna in free space with a linear current distribution and length \(0.02 \lambda\), the peak current \(I_{0}\) required to provide a radiation field amplitude of 100mV/m at a distance of 1mi and angle \(\theta=90^{\circ}\) is approximately 5.09 A, and to radiate a total power of 1 W is approximately 9.01 A.

Step-by-step solution

Find the electric field strength equation

From the Electric Field Equation for a dipole antenna, the Electric field at a distance `\(r\)` in the far field is given by: $$ E_{\theta}=\frac{j \cdot \eta \cdot I_{0} \cdot d \cdot \sin \theta}{8 \pi r}e^{-j k r} $$ Where, \(E_{\theta}\) is the electric field strength, \(\eta\) is the intrinsic impedance of free space (\(\approx 377 \Omega\)), \(I_{0}\) is the peak current, \(d\) is the length of the dipole antenna, \(\theta\) is the angle measured from the axis of the antenna, \(r\) is the distance from the antenna, \(k = 2\pi/\lambda\) is the wave number.

Calculate the peak current for given electric field strength

Given \(E_{\theta} = 100mV/m = 100\times10^{-3} V/m\), \(\theta = 90^{\circ}\), and \(r = 1mi = 1609.34m\), we can calculate \(I_{0}\) by isolating \(I_{0}\) in the equation and substituting the given values: $$ I_{0} = \frac{8 \pi r E_{\theta} e^{jk r}}{j \eta d \sin \theta} $$ Substitute the given values: $$ I_{0} = \frac{8 \pi (1609.34) (100\times10^{-3}) e^{j k (1609.34)}}{j 377 (0.02 \lambda) \sin 90^{\circ}} $$ \(I_{0} \approx 5.09 A\)

Calculate radiated power from peak current and radiation resistance

The total radiated power, \(P_r\), is given by the formula: $$ P_{r}=\frac{I_{0}^{2} R_{r}}{2} $$ Where \(R_{r}\) is the radiation resistance of the dipole antenna, and it can be calculated as: $$ R_{r} \approx 20 \pi^{2}\left(\frac{d}{\lambda}\right)^{2} $$ Substitute the given length of the dipole antenna: $$ R_{r} \approx 20 \pi^{2}\left(\frac{0.02 \lambda}{\lambda}\right)^{2} = 20 \pi^{2}(0.02)^2 \approx 0.025 \Omega $$

Calculate peak current for the given radiated power

For a desired radiated power of \(P_r = 1W\), we can calculate the required peak current \(I_{0}\) using the derived equations: $$ I_{0} = \sqrt{2\frac{P_r}{R_r}} $$ Substitute the given values: $$ I_{0} = \sqrt{2\frac{1}{0.025}} \approx 9.01 A $$ So, the values of \(I_{0}\) required to provide a radiation-field amplitude of 100mV/m at a distance of 1mi at \(\theta = 90^{\circ}\) is approximately 5.09 A, and to radiate a total power of 1 W is approximately 9.01 A.

Key concepts

Understanding Electric Field Strength in Dipole Antennas

Electric field strength is a measure of the force that an electric charge would feel in the presence of an electromagnetic field. In the context of a dipole antenna, it essentially provides us with information about the intensity of the electromagnetic field at a given point in space.

The electric field strength, represented by the symbol \( E_{\theta} \), is particularly important for understanding the impact of an antenna's radiation at a specific location. It is determined by several factors, including the peak current \( I_0 \), the length of the antenna \( d \), the intrinsic impedance of free space \( \eta \), and the distance from the antenna \( r \).

Using the provided formula and values, it's possible to elucidate the concept by calculating the required peak current to achieve a specific electric field strength at a certain distance. This can help students understand the relationship between the physical properties of the antenna and the resulting electromagnetic field.

Radiation Resistance: A Key Factor in Antenna Design

Radiation resistance \( R_r \) is an essential parameter in antenna design. It represents the resistance that an antenna appears to have when it radiates energy into free space, which pertains to the part of the input resistance that is responsible for radiated power. It's an imaginary resistance since the energy isn't lost as heat but instead emitted into the surrounding area as radio waves.

In the equation \( R_r \approx 20 \pi^2(\frac{d}{\lambda})^2 \), where \( d \) is the length of the antenna and \( \lambda \) is the wavelength, we see that radiation resistance depends on the size of the antenna in relation to the wavelength. This concept becomes critical when designing antennas for efficient radiation; a higher radiation resistance implies better efficiency in converting input power into radiated energy.

For instance, in the exercise, substituting the given values into the formula allows students to calculate the radiation resistance and thereby understand how to determine the amount of power an antenna can efficiently radiate.

Calculating Radiated Power from an Antenna

Radiated power, denoted as \( P_r \), is the amount of power that an antenna emits into space. It's a crucial aspect because it directly affects the range and strength of the signal that the antenna can broadcast.

The formula \( P_r = \frac{I_0^2 R_r}{2} \) reveals that radiated power depends on two major factors: the square of the peak current \( I_0 \) and the radiation resistance \( R_r \) of the antenna. This quadratic relationship with current indicates that small increases in current can lead to significant increases in radiated power. Moreover, understanding how to tweak \( I_0 \) and \( R_r \) enables engineers to manipulate the power output to suit specific applications.

By applying this knowledge to the sample problem, we can derive the peak current required to achieve a desired power output. Through these calculations, students not only practice their mathematical skills but also learn to appreciate the nuanced relationships between various antenna parameters and their practical implications.