The above equation illustrates the fundamental relationship between transmitter antenna and rectenna sizes. For a given wavelength and transmission distance, the product DtDr is constant. In order to collect all the radiated power within the mainlobe of the antenna, the rectenna diameter must be equal or larger than the footprint diameter. If the receiver antenna is smaller than the footprint the efficiency of the link decreases. This is illustrated in Figure 7.21 where receiving efficiency is plotted as a function of rectenna diameter. Any antenna of finite size will introduce sidelobes in the radiating pattern. The energy can be focused into the mainlobe by introducing a tapering of the illumination pattern towards the edges of the transmitting antenna. The tapering is usually expressed in decibel notation where a 10 dB taper indicates a factor of 10 reduction in intensity from center to edge. This causes a widening of the mainlobe and decreased sidelobes. The amount of tapering used for the antenna is essentially a tradeoff between footprint-size and antenna efficiency. For power beaming applications a Gaussian distribution with a 10 dB taper is found to be the most efficient. [Woodcock, 1980] The power density distribution on the ground is directly related to the radiation pattern of the array. Assuming that the maximum radiation level occurs at boresight from the transmitting antenna, the power density function Sr can be expressed as where Pt is the Power delivered to the antenna and H is the Distance from antenna to rectenna Gt is the Transmitting antenna gain, which can be expressed as Care should be taken when employing this formula. Gt is generally a function of the radiation angle and the formula only applies to the far-field of the antenna. Sr is in this case the maximum electromagnetic flux density incident on the receiver antenna
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