ELEC 241 Lab

Background

The Communication Equation

Free Wave Propagation.

Suppose we have a source (transmitter) of power

that is radiating isotropically (i.e. uniformly in all directions). Then at a distance

the power is spread uniformly over the surface of a sphere and the power density (in $\rm W/m^2$ ) is $\displaystyle S_{ISO}(r)=\frac{P_T}{4\pi r^2}$

Transmitter Antenna Gain.

If the transmitter antena could focus all of the radiated power into a beam, then the power outside the beam would be zero, but the same total power would be concentrated into a smaller angle, so the power density inside the beam will be higher. In particular, if the beam is uniform, of solid angle $\Omega$ , then $\displaystyle S(r)=\frac{P_T}{\Omega r^2}$ Define the transmitter antenna gain to be $\displaystyle G_{TA}=\frac{4\pi}{\Omega}$ . Then $\displaystyle S(r)=\frac{P_T G_{TA}}{4\pi r^2}$

Receiver Antenna Area.

What is the power available at the receiver? If the power density at the receiver is $S\: \rm W/m^2$ and the receiver antenna has an effective area of $A_{eff}\: \rm m^2$ then $\displaystyle P_{rec}= S\cdot A_{eff} = \frac{P_T G_{TA}}{4\pi r^2} A_{eff}$

The Friis Equation.

The previous equation is asymmetric in that it represents the transmitter antenna by its gain and the receiver antenna by its effective area. With the help of a little E\&M theory, we can show that the gain of the receiver antenna is $\displaystyle G_{RA}=\frac{4\pi}{\lambda^2}A_{eff}$ . Plugging this into the previous equations gives us the Friis equation: $\displaystyle P_{rec}= G_{TA}G_{RA}\left[\frac{\lambda}{4\pi r}\right]^2 P_T$

Light Waves vs. Radio Waves

At the beginning of the semester, our Grand Plan called for a wireless communication system based on optical signals. However, the factor of $\lambda^2$ in the Friis equation suggests that light waves, with wavelengths of a few hundred nanometers, would be at a considerable disadvantage with respect to radio waves, with wavelengths of several meters. Indeed, our first attempt at optical communication in Part 3 of Experiment 2.3 had rather poor range.

However, the fact that antenna gain increases inversely with $\lambda^2$ suggests that a receiver antenna (i.e. a lens) of a few centimeters diameter would put us back in the running. But remember that high gain means narrow beamwidth, so such an antenna would require very careful alignment of the transmitter and receiver.

Experiment 2.3 pointed out another potential problem with an optical channel: our optoelectronic devices are nonlinear in their relationship between voltage and intensity of light. In Experiment 4.3 we were able to linearize the photodiode by using a transresistance amplifier, and we can linearize the LED with a transconductance amplifier.