Assuming two identical antennas, with one transmitting [a sine wave] and one receiving, spaced apart by some whole number multiple of the wavelength, and both probed identically and fed to an oscilloscope, would the waves be in phase, or 180 degrees out (or something else?!)?
This is actually a very interesting question, which gets right to the heart of how dipoles work.
The short answer is that if they are thin, parallel, side-by-side, half-wave dipoles, spaced a whole number of wavelengths apart, and with the receive dipole connected to a conjugate matched load, then the voltage across the receive dipole terminals will lead the voltage across the transmit dipole terminals by about 45
o.
This in itself is perhaps not so surprising, because of the phase shift introduced by the reactive part of the dipole impedances. It is rather more surprising if we look at the phase relationship between the currents going in to the terminals of the two dipoles. It turns out that the current going in to the receive dipole will lag the current going in to the transmit dipole by 90
o, unless the dipoles are less than a couple of wavelengths or so apart. Or to put it another way, compared to a straight length of hypothetical, light-speed, coaxial cable. The electromagnetic wave will appear to have covered the distance a quarter of a cycle quicker!
One way to analyse this system is to look at the mutual impedance Z
21 between the parallel side-by-side half-wave dipoles for any given separation.
Figure 10-12 on page 426 of "Antennas" (second edition) by John D. Kraus shows almost what we need. R
21 and X
21 are the real and imaginary components of Z
21. Unfortunately, the graph in Kraus only goes up to two wavelengths, where there are still significant near-field effects. So I wrote a few lines of code to carry out the numerical integration of
equation 10-5-3 on page 424 of Kraus, so that I could extend the graph to five wavelengths. I have attached the mutual impedance graph to the end of this post.
This graph shows that beyond the near-field, the mutual impedance is purely reactive when the dipoles are separated by a whole number of wavelengths. As expected, at zero separation, the mutual impedance is equal to the self impedance of the dipoles (Z
11 and Z
22).
If this two-port network is conjugate matched terminated, then the current going in to port 2 will be in-phase with the current going in to port 1 when the mutual impedance is negative real. As shown on the mutual impedance graph, for larger distances, this occurs at (N-0.25) wavelengths where N is an integer. So at N wavelengths, the current going in to the receive dipole will lag the current going in to the transmit dipole by 90
o.
The relevant solutions to Maxwell's equations for the E and H fields, in the vicinity of a dipole at a particular frequency, contain three coupled modes. These modes are akin to the vibrational modes of a bell.
- One mode, whose amplitude falls off with distance as 1/r3, that is coupled directly to the charge density along the dipole.
- One mode, whose amplitude falls off with distance as 1/r2, that is coupled directly to the current density along the dipole. For a short electric dipole, this mode leads the 1/r3 mode by 90o.
- One mode, whose amplitude falls off with distance as 1/r, that is not coupled directly to either the charge or current densities along the dipole. This 1/r mode is the only one that extends into the far-field.
In terms of Maxwell's equations, the 1/r
2 mode couples to the 1/r mode through the action of additional terms in the curl partial differentials, which arise because of the extreme rate of change of amplitude with distance close to the dipole.
The consequence of this coupling is that the 1/r mode leads the 1/r
2 mode by 90
o. There is a diagram showing these couplings at the end of this post.
At a receiving dipole, the 1/r mode from the transmitter can directly couple to the charge and current densities along the dipole, with no inherent change of phase (other than the current being of opposite sign). However, if the dipole is resonant at a different frequency, the current could lag or lead by up to an additional +/- 90
o.
I have redone my comment on parasitic elements, because my first attempt was misleading. The lesson being that maths is good, maths plus intuition is better, but intuition on its own can easily lead one astray. To get anywhere near to an additional +/- 90
o would require almost complete detuning of the parasitic element, rendering it worthless.
Section 11-9 starting on page 476 of Kraus analyses parasitic elements in terms of their mutual and self impedances. Based on the equations presented by Kraus, the most that can be achieved when detuning a half wavelength dipole is about +/- 45
o, which would result in a 3dB reduction in the reflected/directed power. If a 6dB reduction is acceptable, then about +/- 60
o could be achieved. So with respect to its 1/r mode, a practical director element (shorter than the resonant length) could get to within -30
o to -45
o of being in phase with the received signal. Similarly, a practical reflector element could get to within +30
o to +45
o of being in anti-phase with the received signal.