Calculating Antenna Range Ratio Based on Height

The range of an antenna for line-of-sight propagation depends on its height above the ground. For practical purposes, considering the curvature of the Earth, the maximum distance or range \( d \) from an antenna of height \( h \) to the horizon is given by the formula:

$$d \approx \sqrt{2Rh}$$

where \( R \) is the radius of the Earth (approximately 6400 km) and \( h \) is the height of the antenna.

This formula shows that the range is directly proportional to the square root of the antenna's height.

We are given the height of a transmitting antenna (\( h_T \)) and a receiving antenna (\( h_R \)).

• Height of the transmitting antenna, \( h_T = 64 \) m
• Height of the receiving antenna, \( h_R = 16 \) m

We need to find the ratio of the range of the transmitting antenna to the range of the receiving antenna. Let \( d_T \) be the range of the transmitting antenna and \( d_R \) be the range of the receiving antenna.

Using the formula for range:

$$d_T = \sqrt{2Rh_T}$$

$$d_R = \sqrt{2Rh_R}$$

The ratio of their ranges is:

$$\frac{d_T}{d_R} = \frac{\sqrt{2Rh_T}}{\sqrt{2Rh_R}}$$

We can simplify this expression:

$$\frac{d_T}{d_R} = \sqrt{\frac{2Rh_T}{2Rh_R}}$$

The \( 2R \) terms cancel out:

$$\frac{d_T}{d_R} = \sqrt{\frac{h_T}{h_R}}$$

Now, substitute the given values for \( h_T \) and \( h_R \):

$$\frac{d_T}{d_R} = \sqrt{\frac{64 \text{ m}}{16 \text{ m}}}$$

Calculate the ratio inside the square root:

$$\frac{h_T}{h_R} = \frac{64}{16} = 4$$

Now, take the square root of the result:

$$\frac{d_T}{d_R} = \sqrt{4} = 2$$

So, the ratio of the range of the transmitting antenna to the range of the receiving antenna is 2:1.

This means the transmitting antenna with height 64 m has a range that is twice the range of the receiving antenna with height 16 m, assuming both ranges are calculated to the horizon individually.

Revision Table: Antenna Range Calculation

Additional Information: Line-of-Sight Propagation and Total Range

Line-of-sight (LOS) propagation is the transmission of electromagnetic waves directly from the transmitting antenna to the receiving antenna. This is the primary mode for signals like FM radio, television broadcasts, and microwave communication.

The range of radio waves in this mode is limited by the Earth's curvature and obstacles. The formula \( d \approx \sqrt{2Rh} \) calculates the distance from an antenna to the radio horizon, which is slightly farther than the visual horizon due to atmospheric refraction. We often use an effective Earth radius \( kR \), where \( k \approx 4/3 \) for standard atmospheric conditions, modifying the formula to \( d \approx \sqrt{2kRh} \). However, for calculating ratios, the \( 2kR \) term cancels out, so using \( \sqrt{2Rh} \) is sufficient for finding the ratio.

The total maximum line-of-sight distance \( D \) between a transmitting antenna of height \( h_T \) and a receiving antenna of height \( h_R \) is the sum of their individual ranges to the horizon:

$$D = d_T + d_R \approx \sqrt{2Rh_T} + \sqrt{2Rh_R}$$

In our specific case, the question asks for the ratio of the range of the transmitting antenna to the receiving antenna, which is interpreted as the ratio of their individual ranges to the horizon (\( d_T : d_R \)). The calculation \( \sqrt{h_T/h_R} \) correctly yields this ratio.