antenna-theory.com

[oz7acs]

Friis transmission formula is basic knowledge for an antenna engineer and most RF engineers. I repeat my prefered version here:
(1) Pr = Srad * Gt * Aer, where
(1a) Srad = Pt / (4*pi*r^2) is the power density at the distance r,
(1b) Aer = Gr * lambda^2 / (4*pi) is the receiver antenna aperture
and Pt, Pr, Gt & Gr the TX- & RX-powers at feed ports and antenna gains.

But here is a problem: Suppose that you want to model the propagation of many different interfering transmitters at various wavelengths at very short distances for a wireless coexistence scenario. The validity of Friis transmission formula is unfortunately limited to far-field conditions, which cannot be held at frequencies below a few hundred MHz at distances of a few cm.
Q1: Is there some way of expanding Friis for validity in the near-field region?
For simplicity we can reduce the formula (1) to one with isotropic antennas, which will not be far from the truth with small terminal antennas of fractions of wavelengths (at least when ignoring their minimal orientations), which gives us:
(2) Pr = Srad * Aer, since Gt = 1 and
(2a) Srad = Pt / (4*pi*r^2), which also should hold in near-field since it only depends on geometry, that is the area of the sphere at distance r sourrounding the transmitter with Pt and
(2b) Aer = lambda^2 / (4*pi) for Gr = 1, BUT this is not true for r less than lambda / (4*pi), since combining these gives us:
(3) Pr = Pt * (lambda / (4*pi*r))^2, where the latter term is the free space path loss that only holds in the far-field!?
Q2: What should be done to expand the validity of Aer and the path loss?

Finally it would be nice to determine the resulting E-field at the reciever from:
(4) Er = sqrt ( Pr * Z0)
Q3: But should Z0 = 377 from free-space be used here or Z0 = 50 ohm from the assumed antenna port impedance?
I hope that somebody can help me with this tricky problem....
Best regards, Peter aka. OZ7ACS

[Schubert]

To answer your first question, gain is a far-field concept. For closely spaced antennas, you don't even know which polarization they are coupling through. For instance, two closely spaced vertically-polarized dipoles could have more energy coupling through the horizontal (and even radially directed E-field that is present in the near field).

For Q2, basically you are trying to generalize the concept of gain to the near field. I don't think you can do it accurately in a simple manner. You could measure the coupling between different antennas at different configurations and try to back out some data, but I don't think it will be very consistent.

For Q3, Z0 should be 50 Ohms or whatever the receiver impedance is. Note that by referring to 377 Ohms, you are referring to the plane-wave impedance, or the ratio of E- to H-field. In the near field, E- and H- are not orthogonal nor related by the 377 Ohm intrinsic impedance. Hence, you can't get the E-field from the received power - it could have come from the H-field!

In summary....the near field is tricky and simple equations won't work. I've tried to do near field measurements in a small chambers, and when averaged over a large number of units that don't see far-field variation, you get unfortunate variance in the near field.