Two-ray model advantages and path loss derivation

Advantages of the two-ray model over free space model

• Free space assumes only a direct path; in real mobile links there is almost always a ground-reflected wave as well. The two-ray model includes it.
• It is accurate for predicting large-scale signal strength over distances of several km for systems with tall towers (above about 50 m), and also for LOS microcells in urban areas.
• It shows that beyond a break distance power falls as 1/d⁴ (40 dB/decade), which matches measurements better than the 1/d² of free space.
• It shows the effect of antenna heights (loss decreases as ht and hr increase) and the alternating peaks and nulls close to the transmitter.
• Path loss becomes independent of frequency at large d.

Derivation of path loss

  Tx
   |\  E_LOS (d')
ht | \_______________________ Rx
   |  \                    _/|
   |   \  E_g (d'')     _/   | hr
 --+----\____________/-------+-- ground
   |<----------- d --------->|

Path difference (method of images, image of Tx at −ht):

d'  = √[d² + (ht − hr)²],  d'' = √[d² + (ht + hr)²]
Δ = d'' − d' ≈ 2ht·hr/d   (for d ≫ ht + hr)
θΔ = 2πΔ/λ = 4π·ht·hr/(λd)

Total field. With free space field E0 at reference d0, LOS field E0d0/d', and ground reflection coefficient Γ ≈ −1 (grazing incidence):

ETOT = E_LOS + E_g
|ETOT| = (E0d0/d)·|1 − e^(−jθΔ)|
       = (E0d0/d)·√[(1 − cosθΔ)² + sin²θΔ]
       = (E0d0/d)·√(2 − 2cosθΔ)
       = 2(E0d0/d)·sin(θΔ/2)

For large d, θΔ/2 < 0.3 rad, so sin(θΔ/2) ≈ θΔ/2:

|ETOT| ≈ 2(E0d0/d)·(2π·ht·hr/(λd)) = k/d²

Received power. Since Pr ∝ |E|², and using the Friis equation at the reference:

Pr = Pt·Gt·Gr·ht²·hr² / d⁴

Path loss in dB:

PL(dB) = 40 log d − (10 log Gt + 10 log Gr
                     + 20 log ht + 20 log hr)

Thus at large distances, path loss rises 40 dB per decade of distance, decreases by 6 dB when either antenna height doubles, and does not depend on frequency.