Truncated tesseractic honeycomb

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.

File:Two ray ground reflected model.svg

Ray tracing of 2-ray ground reflected model

Mathematical Derivation

From the figure the received line of sight component may be written as

$r_{los}(t)=Re\{\frac{\lambda \sqrt{G_{los}}}{4\pi }\times \frac{s(t)e^{-j2\pi l/\lambda }}{l}\}$

and the ground reflected component may be written as

$r_{gr}(t)=Re\{\frac{\lambda \Gamma (\theta )\sqrt{G_{gr}}}{4\pi }\times \frac{s(t-\tau )e^{-j2\pi (x+x^{\prime })/\lambda }}{x+x^{\prime }}\}$

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

Ground Reflection $\Gamma (\theta )=\frac{sin\theta -X}{sin\theta +X}$

where $X_v=$$\frac{\sqrt{{\epsilon }_g-{cos}^2\theta }}{{\epsilon }_g}$

$X_h=\sqrt{{\epsilon }_g-{cos}^2\theta }$

From the figure $x+x^{\prime }=\sqrt{(h_t+h_r{)}^2+d^2}$ and $l=\sqrt{(h_t-h_r{)}^2+d^2}$

the path difference between then $\Delta d=x+x^{\prime }-l=\sqrt{(h_t+h_r{)}^2+d^2}-\sqrt{(h_t-h_r{)}^2+d^2}$

the phase difference between the waves $\Delta \varphi =\frac{2\pi \Delta d}{\lambda }$

The power of the signal received is $r_{los}^2+r_{rg}^2$ 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

$|s(t){|}^2(\frac{\lambda }{4\pi }{)}^2\times (\frac{G_{los}{e}^{-j2\pi l/\lambda }}{l}+\Gamma (\theta )G_{gr}\frac{e^{-j2\pi (x+x^{\prime })/\lambda }}{x+x^{\prime }}{)}^2$

$=P_t(\frac{\lambda }{4\pi }{)}^2\times (\frac{G_{los}}{l}+\Gamma (\theta )G_{gr}\frac{e^{-j\Delta \varphi }}{x+x^{\prime }}{)}^2$

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

$x+x^{\prime }-l=\sqrt{(h_t+h_r{)}^2+d^2}-\sqrt{(h_t-h_r{)}^2+d^2}$

$=d\sqrt{\frac{(h_t+h_r{)}^2}{d^2}+1}-\sqrt{\frac{(h_t-h_r{)}^2}{d^2}+1}$

$\approx \frac{d}{2}\ast (\frac{(h_t+h_r{)}^2}{d^2}-\frac{(h_t-h_r{)}^2}{d^2})$

$=\frac{2h_t{h}_r}{d}$

Phase difference may be approximated as

$\Delta \varphi \approx \frac{4\pi h_t{h}_r}{\lambda d}$

When d increases asymptotically $d\approx l\approx x+x^{\prime },\Gamma (\theta )\approx -1,G_{los}approx{G}_{gr}$

File:Reflection co-efficient ground reflection of radio waves.jpg

Reflection co-efficient tends to -1 for large d.

$P_r=P_t(\frac{\lambda G}{4\pi d}{)}^2\times (1-e^{-j\Delta \varphi }{)}^2$

Expanding $e^{-j\Delta \varphi }$using Taylor series an retaining only the first term

$P_r=P_t(\frac{\lambda G}{4\pi d}{)}^2\times (-j\Delta \varphi {)}^2$

$P_r\approx P_t(\frac{\lambda G}{4\pi d}{)}^2\times (\frac{4\pi h_t{h}_r}{\lambda d}{)}^2$

$P_r\approx P_t\frac{G{h}_t^2{h}_r^2}{d^4}$

Power varies with inverse fourth power of distance for large d.

In logarithmic units

In logarithmic units : $P_{r_{dBm}}=P_{t_{dBm}}+10log(G{h}_t^2{h}_r^2)-40log(d)$

Path loss : $PL=P_{t_{dBm}}-P_{r_{dBm}}=40log(d)-10log(G{h}_t^2{h}_r^2)$

Power vs. Distance Characteristics

File:2-ray ground reflected Power vs distance.jpg

Power vs distance plot

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 a local maximum.

As a case of log distance path loss model

The standard expression of Log distance path loss model is

$PL=P_{T_{dBm}}-P_{R_{dBm}}=P{L}_0+10\gamma {\log }_{10}\frac{d}{d_0}+X_g,$

The path loss of 2-ray ground reflected wave is : $PL=P_{t_{dBm}}-P_{r_{dBm}}=40log(d)-10log(G{h}_t^2{h}_r^2)$

where $P{L}_0=40log(d_0)-10log(G{h}_t^2{h}_r^2)$ $X_g=0$ and $\gamma =4ford,d_0>d_c$ 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.

See Also

• Radio propagation model
• Free-space path loss
• Friis transmission equation

References

20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534