Isotropic antenna Transmit and Receive power

[SPYazdani]

Homework Statement

Plot and compare the path loss (dB) for the free-space and plane-Earth models at 800MHz vs distance on a logarithmic scale for distances from 1m to 40Km. Assume that the antennas are isotropic and have a height of 10m

Homework Equations

Free space: $P_R=\frac{P_T G_T G_R}{L_P}$

Plane Earth: $P_R=P_TG_TG_R(\frac{h_Th_R}{R^2})^2$

Two isotropic antennas separated by a distance $R\epsilon[1m,40km]$ at frequency $f=800MHz$.

The Attempt at a Solution

Isotropic antennae have $G_T=G_R=1$. So That simplifies $P_R=\frac{P_T G_T G_R}{L_P}$ = $P_R=\frac{P_T}{L_P}$
$L_P=(\frac{R4\pi}{\lambda})^2$.

I'm solving the question for 1m for the free space model, then once I have that, plotting it is easy in Excel.

I'm stuck on finding $P_T$. I tried deriving an equation for $P_T$by substituting $L_P=(\frac{R4\pi}{\lambda})^2$ into $P_R=\frac{P_T}{L_P}$ but that lead me nowhere. At least I don't know what the answer means.

Here's what happened.

$P_R=\frac{P_T}{R^24\pi}A_e$
$P_R=\frac{P_T}{L_P}$
$L_P=(\frac{R4\pi}{\lambda})^2$
$\frac{P_T}{R^24\pi}A_e=\frac{P_T}{(\frac{R4\pi}{λ})^2}$
Then a bunch of cancellation on both sides and finally
$Ae 4\pi = \lambda^2$

Help! I don't know how to find $P_T$

 

[marcusl]

What is Lp? How does it relate to what you are asked for?

 

[SPYazdani]

What is Lp? How does it relate to what you are asked for?

For reliable communication, Lp is the minimum signal level required at the receiving antenna. It's a ratio of $\frac{P_T (mW)}{P_R(mW)}$. The distance $R = \frac{\lambda\sqrt{L_P}}{4\pi}$. Rearranging and solving for $L_P = (\frac{4R\pi}{\lambda})^2$ implies the loss is related to the distance separated by the antennas as well as the wavelength of the transmitted signal.

Thanks for pointing that out. I can now solve my problem :D

 

[marcusl]

You're welcome!