Question

Signals are transmitted at a $1-\mathrm{m}$ carrier wavelength between two identical half-wave dipole antennas spaced by $1\mathrm{km}$. The antennas are oriented such that they are exactly parallel to each other. ( $a$ ) If the transmitting antenna radiates 100 watts, how much power is dissipated by a matched load at the receiving antenna? ( $b$ ) Suppose the receiving antenna is rotated by ${45}^{\circ }$ while the two antennas remain in the same plane. What is the received power in this case?

Short answer

Answer: The power dissipated by a matched load at the receiving antenna when both antennas are parallel to each other is approximately $9.54\times {10}^{-9}$ watts. The received power when the receiving antenna is rotated by 45 degrees is approximately $4.77\times {10}^{-9}$ watts.

Step-by-step solution

Use the Friis Transmission Equation

The Friis transmission equation relates the received power ($P_r$) to the transmitted power ($P_t$), the gain of transmitting and receiving antennas ($G_t$ and $G_r$), the distance between antennas ($d$), and the wavelength ($\lambda$) where $P_r=P_t\frac{G_t{G}_r{\lambda }^2}{(4\pi d{)}^2}$. Given that we have the transmitting power, distance between antennas, and carrier wavelength, we can use this equation to determine the received power for both cases. Note that the gains for the antennas depend on the positions and orientations of the antennas.

Find the gains of antennas and received power in the parallel case

For half-wave dipole antennas, the gain in any direction can be calculated using the following formula: $G(\theta )=\frac{3}{2}(\cos (\theta ){)}^2$, where $\theta$ is the angle between the direction of maximum gain and the direction an antenna is pointing. Since the antennas are parallel to each other and exactly aligned in this case, the angle $\theta$ for each antenna is 0. Hence, $G_t=G_r=G(0)=\frac{3}{2}(\cos (0){)}^2=\frac{3}{2}$. Now using the Friis transmission equation, we have $P_r=100\frac{\frac{3}{2}\cdot \frac{3}{2}(1{)}^2}{(4\pi 1000{)}^2}\approx 9.54\times {10}^{-9}\text{W}$. So, the power dissipated by a matched load at the receiving antenna when both antennas are parallel to each other is approximately $9.54\times {10}^{-9}$ watts.

Find the gains of antennas and received power with a 45-degree rotation

Now the receiving antenna is rotated by 45 degrees while remaining in the same plane. We will recalculate the gain for receiving the antenna ($G_r$) with $\theta ={45}^{\circ }$. The gain for the transmitting antenna ($G_t$) stays the same as before. $G_r=G(45)=\frac{3}{2}(\cos ({45}^{\circ }){)}^2=\frac{3}{4}$. Using the Friis transmission equation with the new $G_r$, we have $P_r=100\frac{\frac{3}{2}\cdot \frac{3}{4}(1{)}^2}{(4\pi 1000{)}^2}\approx 4.77\times {10}^{-9}\text{W}$. So, the received power when the receiving antenna is rotated by 45 degrees is approximately $4.77\times {10}^{-9}$ watts.

Key concepts

Antenna Gain

Antenna gain is a measure of an antenna's ability to direct or concentrate radio frequency energy in a particular direction when transmitting, or to collect energy more efficiently when receiving, as compared to a hypothetical ideal antenna, known as an isotropic radiator. This parameter is critical in telecommunication systems because it significantly impacts the range and quality of the wireless communication link.

The gain of an antenna is usually expressed in decibels (dB) over an isotropic radiator (dBi) or sometimes over a dipole (dBd). It's important to note that the gain does not increase the total amount of energy transmitted; rather, it redistributes the energy in a way that can send or receive signals more efficiently in certain directions. This means that an antenna with a higher gain can potentially transmit a signal farther in its preferred direction, assuming all other factors are constant.

Despite the importance of high gain values, a trade-off exists because antennas with high gain focus energy into narrower beams, which can make system design more sensitive to antenna orientation and alignment.

Dipole Antennas

Dipole antennas are a simple type of antenna that are widely used in various wireless communication systems. They consist of two identical conductive elements, such as metal rods, arranged end to end with a small gap at the center for the feedline.

A half-wave dipole antenna, which is the type mentioned in the exercise, is designed to be half the wavelength of the frequency it's intended to receive or transmit. This type of antenna is popular due to its simplicity and relatively broad bandwidth. The radiation pattern of a dipole antenna is doughnut-shaped, with the strongest radiation at the sides and the weakest at the ends.

Since dipole antennas radiate radio waves with efficiency and predictability, they serve as a reference point for calculating the gain of other types of antennas. They are also commonly used in theoretical analysis and practical applications because of their straightforward design and well-understood characteristics.

Electromagnetic Wave Propagation

Electromagnetic wave propagation refers to how radio waves travel from one point to another through different media such as air, vacuum, or various materials. Understanding the principles of wave propagation is essential for designing effective communication systems. The waves can be affected by several factors, including frequency, distance, environmental conditions, and obstacles.

Radio waves are a form of electromagnetic radiation, and they can propagate through space in a manner similar to the way light waves do. They can travel in straight lines, reflect off surfaces, refract through materials, and diffract around obstacles. Additionally, they can be absorbed or scattered by materials they encounter, leading to signal loss.

When discussing propagation, the Friis Transmission Equation used in the exercise becomes very pertinent. It's a foundational formula used to predict the received signal strength when the transmitter and receiver have a clear line-of-sight path. The equation takes into account the transmitted power, antenna gains, distance between antennas, and wavelength of the signal to estimate the power arriving at the receiving antenna.

Antenna Orientation

Antenna orientation refers to the position and the direction in which an antenna is set up to transmit or receive signals. This aspect is vital in communication systems because the alignment between the transmitting and receiving antennas can greatly influence the signal strength and quality of reception.

Correct orientation ensures maximum gain from the antenna's radiation pattern, which typically varies with the angle. If antennas are misaligned, it can result in significant loss of signal. For instance, in the exercise, orientation plays a crucial role in determining the received power. When the antennas are exactly parallel, as they are in the initial scenario, they can efficiently transmit and receive signals due to the optimal alignment of their radiation patterns. However, if one antenna, such as the receiver, is rotated by 45 degrees, the alignment is no longer optimal, causing a drop in received power as evidenced by the calculations.

Therefore, when installing antennas for communication systems, the orientation must be carefully considered and precisely adjusted to ensure the best possible performance. This can involve both the azimuthal orientation (rotation around a vertical axis) and the elevation angle (tilt above or below the horizon).