# [Physics] Radio Transmission distance (FM)

I am trying to find a formula (if one exists, and this is the right place) to try and calculate how far a radio station can transmit, under ideal conditions. I have searched Google, and I can not find a solid answer.

• Effective Radiated Power: 910 Watts
• Transmission Power: 936 Watts
• Transmitter Height: ~39m
• Terrain: Flat

Thanks for the help.

You can make some assumptions like a typical FM receiver needs -110 dBm to work. Then assume you have an isotropic antenna in both cases because you didn't say anything about the antennas so we'll ignore the gain.

Next take a look at the path loss based on the 910 W (+59.6 dBm) power.

Your path loss can not exceed 59.6 + 110 = 169.6 dB (with loss dB and dBm are the same, but dB and dBm to watts is different).

The free space loss model is $L_{fs}=32.45 + 20*log(d_{km}) + 20*log(f_{MHz})$

You'll need to solve this for $D_{km}$ and set $L_{fs}=169.6$ and $f_{MHz}=89.7$.

There are many other path loss models. NIST suggests several

• Free Space Model (shown above...least accurate)
• CCIR
• Hata
• Walfisch-Ikegami (WIM)

Having the antenna height will reduce your path loss some. In the popular WIM model in a line of sight (LOS) situation like yours where the base antenna is >30m high with no obstructions in the direct path then you can use a more realistic estimate:

$L_{wim-los}=42.64+26*log(d_{km})+20*Log(f_{MHz})$

Also the ITU has a specification ITU 1546 Method for point-to-area predictions for terrestrial propagation (pdf here). You can also find MatLab solutions and a lot of literature on their technique and field test results for broadcast FM. (FYI the models are all very similar in structure to NIST's models).