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By A. Nony Mous
HTML'd and edited by Mike Morris WA6ILQ
This file showed up in the repeater-builder's email as a word file. I wrote back and asked if he wanted credit, and he declined. Since no credit was asked, none is given. And I just HTML'd it, I've not checked the math.
We often hear VHF and UHF radio frequencies referred to as "line-of-sight". In reality, the calculated horizon for visual line-of-sight (i.e. the visual horizon) is not the same as the horizon for radio wave propagation. Unless there is a mitigating factor (for example, the antenna-to-tower spacing causing a pattern shift) the range and coverage of your repeater will be limited to your radio horizon. This horizon goes beyond the calculated visual horizon due to a combination of factors including direct radiation and reflected ground wave, and occasionally refraction. Note that radio horizon does not care if it is AM or FM or SSB. And it's just a measure of distance, not a guarantee of communications range. Many other factors will have an affect on communications range such as receiver sensitivity, local noise floor, antenna gain, antenna pattern and more... Even though both horizons (i.e. each end of the path) can be calculated, the calculations do not take into account the reality of our planet. Dust, fog (i.e. water vapor) and solid objects (hills, etc) have an effect... And don't forget mankind - EMI (electromagnetic interference) has a considerable effect... be it incidental (i.e. the local noise floor at the radio site), accidental (i.e. an adjacent channel transmitter "bleeding over" on to your channel drowning out the weaker signals, or even deliberate interference on your channel which can totally block the signal you are trying to receive.
To understand the radio horizon you first need to understand visual horizon.
Calculating the Visual Horizon
In Statute (land) Miles
First, measure the height of the viewer above ground level (in feet). Call it "H", for height.
The visual horizon distance in statute miles will be the square root of the result of H divided by 0.5736
Height to center of your eyeball(s) = 5.5ft
5.5ft divided by 0.5736 = 9.588
Square Root of 9.588 = 3.10 statute miles, the maximum distance you could theoretically see if standing on the beach in California looking out across the Pacific Ocean.
Note: If your feet are just in the water you might want to know what the distance to the horizon is in nautical miles... If so, multiply statute miles by 0.869 (or just take 86.9% on your calculator). If you're curious, 3.1 statute miles is about 2.7 nautical miles.
The visual horizon distance in kilometers will be the square root of the result of the height in centimeters divided by 6.752
167.6cm divided by 6.752 = 24.822
Square root of 24.822 = 4.982 km or just a hair under 5km. This is the maximum distance you could theoretically see if standing on the beach in Portugal looking out across the Atlantic Ocean.
The visual horizon of a person standing on the beach of about 3 miles is where the historical 3 mile territorial waters rule came from. The increase in horizon for an increase in height is why the sailing ships had a "crows nest" observers platform as high up on the main mast as was safe, and why even submarines have specially built observers positions as high as they can get them.
Raising the observer from 5.5 feet to 50 feet increases the horizon from about 3 miles to about 9 miles. Some sailing ships had masts over 100 feet (a crows nest at 100 feet gives about 13 miles), and some WW2 warships had observation baloons - an unlucky soul with binoculars and a compass was perched on a seat under a baloon which was towed behind the ship at altitudes up to 500 feet (which gives almost 30 miles). If he saw something he'd write a note giving dexcription and compass bearing and place it in a message canister which was slid down the tow rope. A telephone headset with the cable suspended from the tow rope was implemented in later designs (and let the ships captain talk back to the observer... "You see a WHAT?"...)
Calculating the Radio Horizon
Since radio transmissions involve a transmitting antenna and a receiving antenna, both need to be considered for these calculations. Where the height of your eyeball was the critical measurment in the exercises above, the height of the radiation center of both the receiving and the transmitting antennas is the critical item here. In most antenna designs the vertical center of the antenna is considered to be the radiation center. In the execises below, "H1" is the height of antenna #1 and "H2" is the height of antenna #2.
Note that the calcualtions assume absolutely optimum conditions - quality base stations, zeero-loss feedline, properly installed connectors, a perfectly flat terrain between the two antennas, perfectly circular antenna pattern, perfectly silent noise floor, etc.
An example of what can affect the results is a noisy (RF-wise) vehicle: I can be sitting at a stoplight chatting with a friend across town on 10m SSB and someone pulls along side with a noisy air conditioning fan motor. The local noise can drown out my friend. His signal to me hasn't changed, my receiver and antenna hasn't changed, but I can no longer hear him. As said above, the radio horizon is just an number of theoretical distance, it's not a guarantee of communications range.
Another example: an antenna mounted on a vehicle fender is directional, and if that directional effect is reducing the pattern in the direction of the other antenna then the range in that direction will be reduced (both transmitting and receiving).
Even atmospherics can have an affect: At frequencies where water vapor has an effect you will find that a rain cloud in the path can ruin the communications. Even a flock of large birds flying through a UHF signal path will cause a visible drop in signal strength.
In Statute (land) Miles:
Square root of H1 (in feet) x 1.415 = D1
Square root of H2 (in feet) x 1.415 = D2
The radio horizon (in statute miles) is the sum of D1 and D2.
Antenna #1 height = 100 feet (i.e. a base station antenna on top of a tower)
Antenna #2 height = 8 feet (i..e a roof-mounted antenna on a large truck)
D1 equals the square root of 100 = 10 x 1.415 = 14.15
D2 equals the square root of 8 = 2.828 x 1.415 = 4.00
D1+D2 = 14.15 + 4.00 = 18.15 statute miles (theoretical maximum distance)
So in theory, an antenna with a radiation center 8ft above perfectly flat terrain should be able to receive signals about 18 miles away from an antenna that has its radiation center 100 feet above the same flat terrain.
Square root of H1 (in meters) x 4.124 = D1
Square root of H2 (in meters) x 4.124 = D2
D1 + D2 = Radio Horizon in Kilometers
Antenna #1 height = 15m
Antenna #2 height = 5m
Square root of 15 = 3.873 x 4.124 = 15.972 (D1)
Square root of 5 = 2.236 x 4.124 = 9.221 (D2)
D1+D2 = 15.972 + 9.221 = 25.193 kilometers (theoretical maximum distance)
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This page originally posted on 3-Feb-2006
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