Scatterers such as rainfall can impose hard-to-predict effects on the propagation of electromagnetic (EM) waves, often making analysis of the near-field and far-field antenna patterns and performance difficult. In order to provide a more meaningful approach to understanding the effects of scatterers on EM propagation, last month Part 1 of this study explored the influence of distributed scatterers on nearfield antenna EM propagation. Part 2 will now be extended to include a precipitation region containing many raindrops, with precipitation contained between the planes z = z₁ and z = z₂ and with the elemental volume dv located at plane z.

From last month, the following relationships need tobe re-established:

SEE EQUATION 20
SEE EQUATION 21
SEE EQUATION 22
SEE EQUATION 23 (Part 1)
SEE EQUATION 24
SEE EQUATION 25

When no precipitation is present, E₃H₃ is identical to E₁H₁. But when precipitation is present, E₃ and E₁ are related by the effective complex refractive index, m^ ,of the precipitation region:

E₃ = E₁ exp[-jk( m^ , – 1)(z – z₁] (26)

where z₁  z  z₂. Equation 26 gives the transmitter field (i.e., the field due to antenna 1 in the presence of precipitation) in the elemental volume dv (Fig. 3). Equation 25 can be used to find the wave detected by the receiving antenna due to scattering by raindrops:

SEE EQUATION 27

where E₁ and E₂ are the complex scalar amplitudes of field vectors E₁ and E₂, respectively.

Assuming single scattering and a sufficient number of drops present in the elemental volume, the wave detected at the receiving antenna due to scattering from all raindrops in the precipitation region can be found by integrating Eq. 27:

SEE EQUATION 28

where vol indicates integration over the precipitation volume, i.e., the region between z = z₁ and z = z₂. Because the beamwidth of the transmit and receive antennas is very small, the volume integral in Eq. 28 may be written

SEE EQUATION 29

and Eq. 28 becomes

SEE EQUATION 30

It is useful at this point to recognize the significance of two equations, Eqs. 9 and 30. It has already been shown that the received wave in the absence of precipitation is given by the relationship for B₁ in Eq. 9. And Eq. 30 gives the received EM wave due to scattering from all the raindrops in the precipitation region. The total wave received in the presence of precipitation is

From Eqs. 9 and 30,

SEE EQUATION 32

Using this in Eq. 31,

SEE EQUATION 33

Defining a and ß as the total attenuation (in Nepers) and phase shift (in rads), respectively, due to the precipitation, then

Comparing Eqs. 33 and 34 yields

SEE EQUATION 35

Parameters m˜ and S(0) can be shown to be related by the simple formula

m˜ = 1 – j(2/k³)NS(0)

Using this relationship in Eq. 35 results in

SEE EQUATION 36

In naturally occurring rainfall, the raindrops will have a drop size distribution, m , given by

where a is the mean drop radius. If the rate of precipitation is uniform throughout, m will be independent of z and Eq. 36 becomes

SEE EQUATION 38

The integral  predicts the received power at antenna 1 due to the transmitted power of antenna 2. The integral

 can be written as

where A is the beamwidth cross section of the common volume of propagation of the two antennas.

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