Yes, as the wire diameter goes to zero, the current distribution
approaches the same on all elements. But in some cases (where the
element height is in the vicinity of a half wavelength) the wires have
to get impossibly thin to achieve good f/b with equal magnitude and
correctly phased base currents. I guess you could categorize needing a
finite diameter wire as a "real world effect" and a zero diameter wire
as "theoretically perfect". As I mentioned, it's not hard to do well at
a quarter wavelength height, but much harder at heights approaching a
half wavelength. For example, I took the EZNEC Cardioid.ez example file
and increased the element heights to 0.4 meter (0.4 wavelength) using 25
segments/element. With wire diameter of 10^-15 mm, the front/back ratio
was still 32 dB. With the original wire diameter of about 0.24 mm, the
front/back was less than 15 dB. And things get worse yet as the elements
get closer to a half wavelength high. But in practice, even at a quarter
wavelength height, people using phased towers might encounter an
unexpectedly low f/b ratio.
For anyone who's interested, I've posted the Technical Correspondence
piece on my web site. You can get it at
http://eznec.com/Amateur/Articles/Current_Dist.pdf.
Roy Lewallen, W7EL
Gene Fuller wrote:
Hi Roy,
I have read many of your articles, and I have no doubt you are correct.
However, in the ideal case, specifically in the limit as the wire
diameter goes to zero, the current perturbation from mutual inductance
vanishes. (The mutual inductance does not vanish, only its impact on
current distribution.)
I just spent a few minutes playing around with EZNEC 3, and I was able
to achieve a null of -52 dBi (-57 dBmax) for two half-wave elements,
with nominal 90 degree spacing and 90 degree phasing. The wire size was
as small as possible. This null was in the symmetry plane and directly
in the anti-end-fire direction of course. I expect with more
computational precision, and perhaps fine tuning frequencies and
dimensions this null could be driven farther. The reported current
imbalance was a maximum of 0.2%, mid-way between the center and the ends
of the wires. The phase imbalance between the wires was a maximum of 0.2
degrees.
I am not trying to say this is practical. I was just pointing out the
Art's use of polygons and canceling phasors was not particularly unique.
We have since learned that what Art is trying to accomplish is to
eliminate all radiation in the back hemisphere. The cardioid example is
obviously moot for his quest.
73,
Gene
W4SZ
Roy Lewallen wrote:
Gene Fuller wrote:
Art,
Why not?
The cardioid pattern from a two-element array was reported back as
least as far as 1937, by the famous George H. Brown. In the ideal
case (free space, no losses, etc.) the radiation directly to the rear
is precisely zero.
If you add various real world effects then the back lobe is not
precisely zero, and this is shown in the ARRL Antenna Book referenced
by Cecil.
. . .
Actually, this isn't quite true. If you manage to get perfectly phased
and equal magnitude currents in two identical elements where the phase
angle equals 180 degrees minus the element spacing (such as the
classic 90-degree fed, 90-degree spaced cardioid), you don't get an
infinite front-back ratio. In the case of the cardioid with typical
diameter quarter wavelength elements, you end up with around a 35 dB
front/back ratio. With longer elements, close to a half wavelength,
the front/back ratio can deteriorate to less than 10 dB when base
currents are identical in magnitude and correctly phased. The reason
is that the mutual coupling between elements alters the current
distribution on the elements. The mutual coupling from element 1 to
element 2 isn't the same as the coupling from element 2 to element 1
(the mutual Z is the same, but the coupled voltage and coupled
impedance aren't). The net result is that the two elements have
different current distributions, so despite having identical magnitude
base currents the two elements don't generate equal magnitude fields.
The overall fields from the two elements end up being imperfectly
phased, also.
This occurs for theoretically perfect and perfectly fed elements, and
isn't due to "real world" effects.
I published some comments about this effect in "Technical
Correspondence" in July 1990 QST ("The Impact of Current Distribution
on Array Patterns"). I'm certainly not the first to have observed it
-- some papers published as early as the '40s are referenced in my
article. But I had never seen its effect on front/back ratio of
cardioids mentioned before. Modern versions of the ARRL Antenna Book
clearly show the small reverse lobe of a typical antenna with quarter
wavelength elements.
I stumbled across it when doing some modeling with ELNEC, the
predecessor of EZNEC, and originally thought it was an error in the
program. You'll see it in a plot from the Cardioid.EZ EZNEC example
file (which is also included with the demo program), and a brief
explanation in the corresponding Antenna Notes file.
A theoretically infinite front/back ratio can be achieved by
modification of the base currents. The amount of modification required
depends on the length and diameter of the elements. Only a small
modification is needed if elements are a quarter wavelength high and
small diameter, but in that case, real world effects will probably
have at least as much and likely more of an effect on the front/back
than the current distribution phenomenon. Rather drastic modification
is required of the base currents of elements approaching a half
wavelength high, however, as elaborated in the "Technical
Correspondence" piece.
Roy Lewallen, W7EL