At distances over 10 km during radio communications, the curvature of the Earth affects the VHF. The article shows the formulas for calculating the height of the “hill” between the two antennas, which does not allow direct visibility due to the sphericity of the Earth.

How translated into Russian "Sagitta"

Saggital plane is an imaginary plane that divides the body into two parts - right and left (used in anatomy)
S.agitta (Latin) - Strelka.
Sagitta - in geometry: a segment above the center of the chord.

Why radio amateur needs to know Sagittu

1) During radio communications over 10 km, the earth cannot be considered flat. The “hill” from the ground or water takes place even on a completely flat surface, for example, the sea between two points. The height of this tubercle is from 0 meters with the close placement of two points in the field of up to 1 radius of the Earth (6378 kilometers) with the maximum spacing of points around the circle. It is clear that you need to raise both antenna at least to the height of the arc, i.e. Sagitta. In fact, even higher - depending on the radius of the Frell zone.
2) In the programs for calculating the vertical profile of the route taking into account the relief, the curvature of the Earth is usually not taken into account. In the middle of the highway, the height of the hill should be added to the height of the relief.

Sagitta calculation

Method No. 1 (from Wikipedia)

IN Wikipedia The formula that follows from the Pythagoras theorem, where the chord length (distance in a straight line) and the radius of the earth are involved. The formula takes into account the distance between points on the surface of the earth, but in a straight line.

Where:
L - chord length (distance between points) in a straight line through the Earth.
r - the radius of the Earth, taken for 6378 km.

Calculation example:
Horda length 40 km . Sagitta S =r - sqrt{ r^{2} - {1} over {4} l^{2} } = 6378 - sqrt(6378² - 0.25 l²) = 0.03136 km = 31.36 m

Method # 2

Source:calculator on math Open ReferENCE

First we find L - the length of half the base of the arc.

The length of half the base of the arc :

D is the distance between the currents in km,
R is the radius of the Earth 6378 km.

For example, for d = 40 km
L =r cdot sin( {d} over { 2 cdot r } ) = 6378 * sin (40/6378/2) = 19.9999672229

Step 2: Substitute L into the formula.

Sagitta:

For example:

S =r - sin( r^{2} - L^{2} ) = 6378 - sin (6378² - 19.9999672229²) = 0.0313922203349648 km = 31.39 m

That is, both methods No. 1 and No. 2 use the chord at the base of Sagitta. (In the second method - half of the chord).

Method No. 3 (Simplified)

At distances up to 1000 km, it works well Simplified Sagitta calculation formula :

Where:
D is the distance between the currents in km,
R is the radius of the Earth 6378 km.

For example:
s = {d^{2}} Over { 8 cdot r } = 40² / 8 / 6378 = 1600 / 8 / 6378 = 0.03136 км = 31.36 м

• S = {D^{2}} Over { 8 cdot r } - нотация LibreOffice MaTh.

Conclusion

The article presents three ways calculation of Sagitta - the height of the hill surface of the earth between the two antennas.
For a stable radio communications on VHF, it is necessary to raise both antennas to a height of S, plus radius of the Frenle zone.

radius of the Frelene zone in meters

where D is the distance in km, F is the frequency in the MGC,
RF - in meters.

For example, for a distance of 40 km and frequency 145 MHz

RF = 17.32 * SQrt (40 / 4 / 145) = 4.54 м.

An increase in the heights of the installation of the transmitting and receiving antennas to the radius of the Frenel zone reduces losses when transmitting a radio signal, since the main path of the radio signal is more complete and the signal is not subject to reflection from the obstacle.

table of height antennas depending on the distance

Below is a table of Sigitta and Fresnel Zone zones for various distances between antennas.
The heights of the antennas are indicated in meters.

In the last two columns, the required height of antennas is given to provide 100% of the connection on VHF.
Since the use of masts above 10-20 meters is impossible in radio amateur practice, radio amateurs for long -term communication on VHF carry out trips to nature - to the mountains. The maximum heights in the Leningrad region are about 100 - 170 meters. Thus, between the two hills or hills about 120 meters high, radio communications can be carried out in direct sight at a distance of 80 km. The mast height for 145 MHz is sufficient 6-7 meters, for 433 MHz - 4 meters, at a distance between correspondents of 80 kilometers.

Calculation Checking according to the formula of direct visibility range

Direct formula:

S = 3.57 * (SQrt(H) + sqrt (h))

The range of radio communications taking into account the refraction of radio waves VHF:

S = 4.12 * (SQrt(H) + sqrt (h))

Cm. Javascript онлайн калькулятор дальности радиосвязи на УКВ на сайте http://kavkaz.qrz.ru/ .

at height h, h = 2 meters range according to the formula: 8-12 kilometers (my calculation - 5 km )
at height h, h = 6 meters range according to the formula: 15-20 kilometers (my calculation 15 km )
at height h, h = 10 meters range according to the formula: 19-26 kilometers (my calculation 20 km )
at height h, h = 20 meters range according to the formula: 27-37 kilometers (my calculation 30 km )

Conclusion : Formulas 3.57 and 4.12 higher Do not take into account the required radius of the Frell zone (1.5-2m) above the obstacle. The calculation through Sigitta and the radius of the Frenel zone is more accurate - the range of stable radio communications when using portable radio stations will be 5 km, and 8-12 km. Moreover, at the low heights of the installation of antennas on the range, a significant impact the terrain.

Last change: 02/19/2025.