2-ray Ground Reflected Model is a radio propagation model that predicts path loss when the signal received consists of the line of sight component and multi path component formed predominately by a single ground reflected wave.
2-Ray Ground Reflection diagram including variables for the 2-ray ground reflection propagation algorithm.
From the figure the received line of sight component may be written as
and the ground reflected component may be written as
where s(t) is the transmitted signal Γ(θ) is ground reflection co-efficient and τ is the delay spread of the model and equals (x+x'-l)/c
From the figure and
the path difference between then
the phase difference between the waves
The power of the signal received is
If the signal is narrow band relative to the delay spread τ then s(t)=s(t-τ) the power equation may be simplified as
where Pt is the transmitted power.
When distance between the antennas d is very large relative to the height of the antenna we may expand x+x'-l using Generalized Binomial Theorem
Using the Taylor series of Template:Sqrt:
and taking the first two terms
Phase difference may be approximated as
When d increases asymptotically
Reflection co-efficient tends to -1 for large d.
Expanding using Taylor series
and retaining only the first two terms
Taking the magnitude
Power varies with inverse fourth power of distance for large d.
In logarithmic units
In logarithmic units :
Path loss :
Power vs. Distance Characteristics
When d is small compared to transmitter height two waves add constructively to yield higher power and as d increases these waves add up constructively and destructively giving regions of up-fade and down-fade as d increases beyond the critical distance or first Fresnel zone power drops proportional to inverse fourth power of d. An approximation to critical distance may be obtained by setting Δφ to π as critical distance a local maximum.
As a case of log distance path loss model
The standard expression of Log distance path loss model is
The path loss of 2-ray ground reflected wave is
for the critical distance.
As a case of multi-slope model
The 2-ray ground reflected model may be thought as a case of multi-slope model with break point at critical distance with slope 20 dB/decade before critical distance and slope of 40 dB/decade after the critical distance.