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12.3 Transmission Losses The [EIRP] may be thought of as the power input to one end of the transmission link, and the problem is to find the power received at the other end. Losses will occur along the way, some of which are constant.
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Other losses can only be estimated from statistical data, and some of these are dependent on weather conditions, especially on rainfall. The first step in the calculations is to determine the losses for clearweather or clear-sky conditions. These calculations take into account the losses, including those calculated on a statistical basis, which do not vary significantly with time. Losses which are weather-related, and other losses which fluctuate with time, are then allowed for by introducing appropriate fade margins into the transmission equation.
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12.3.1 Free-space transmission
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As a first step in the loss calculations, the power loss resulting from the spreading of the signal in space must be determined. This calculation is similar for the uplink and the downlink of a satellite circuit. Using Eqs. (12.1) and (12.2) gives the power-flux density at the receiving antenna as
EIRP 4 r2
(12.6)
The power delivered to a matched receiver is this power-flux density multiplied by the effective aperture of the receiving antenna, given by Eq. (6.15). The received power is therefore PR
MAeff 2 EIRP l GR 4 r2 4
(12.7) l b 4 r
(EIRP)(GR)a
Recall that r is the distance, or range, between the transmit and receive antennas and GR is the isotropic power gain of the receiving antenna. The subscript R is used to identify the receiving antenna. The right-hand side of Eq. (12.7) is separated into three terms associated with the transmitter, receiver, and free space, respectively. In decibel notation, the equation becomes [PR] [EIRP] [GR] 10 loga 4 r b l
(12.8)
The received power in dBW is therefore given as the sum of the transmitted EIRP in dBW plus the receiver antenna gain in dB minus a third term, which represents the free-space loss in decibels. The free-space loss component in decibels is given by [FSL] 10 loga 4 r b l
(12.9)
Twelve
Normally, the frequency rather than wavelength will be known, and the substitution l c/f can be made, where c 108 m/s. With frequency in megahertz and distance in kilometers, it is left as an exercise for the student to show that the free-space loss is given by [FSL] 32.4 20 log r 20 log f (12.10)
Equation (12.8) can then be written as [PR] [EIRP] [GR] [FSL] (12.11)
The received power [PR] will be in dBW when the [EIRP] is in dBW, and [FSL] in dB. Equation (12.9) is applicable to both the uplink and the downlink of a satellite circuit, as will be shown in more detail shortly.
Example 12.3 The range between a ground station and a satellite is 42,000 km. Calculate the free-space loss at a frequency of 6 GHz.
Solution
[FSL]
20 log 42,000
20 log 6000
200.4 dB
This is a very large loss. Suppose that the [EIRP] is 56 dBW (as calculated in Example 12.1 for a radiated power of 6 W) and the receive antenna gain is 50 dB. The receive power would be 56 50 200.4 94.4 dBW. This is 355 pW. It also may be expressed as 64.4 dBm, which is 64.4 dB below the 1-mW reference level. Equation (12.11) shows that the received power is increased by increasing antenna gain as expected, and Eq. (6.32) shows that antenna gain is inversely proportional to the square of the wavelength. Hence, it might be thought that increasing the frequency of operation (and therefore decreasing wavelength) would increase the received power. However, Eq. (12.9) shows that the free-space loss is also inversely proportional to the square of the wavelength, so these two effects cancel. It follows, therefore, that for a constant EIRP, the received power is independent of frequency of operation. If the transmit power is a specified constant, rather than the EIRP, then the received power will increase with increasing frequency for given antenna dish sizes at the transmitter and receiver. It is left as an exercise for the student to show that under these conditions the received power is directly proportional to the square of the frequency.
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