NEC-LIST: How does a wire antenna radiate?

Dear Harry and NEC-LISTers,

I very much enjoyed your posting to NEC-LIST regarding radiation, as I
have our private exchanges on the subject. There are a couple of
points that I'd like to elaborate on and/or clarify with respect
specifically to radiation from a thin wire, on which the current
becomes nearly sinusoidal, and radiation from a true sinusoidal
current filament (SCF). So, I'll add some comments after excerpted
paragraphs from your note. I'll apologize in advance to those on
NEC-LIST receiving this message if it seems to go into detail that's
not interesting to you, but would welcome commments from anyone who
has questions, objections, . . .

> As the impulse passes along the conductor, the charges are first
> accelerated in one direction and then immediately equally and
> oppositely in the other, so there is no NET acceleration. Two
> mutually cancellatory kinks are produced and there is therefore no
> net radiation field. This will be true everywhere except at the ends
> where the impulse comes to a halt and is reversed (reflected). Here
> there is net acceleration and therefore radiation. The travelling
> impulse therefore radiates only from the end of the wire. >

The above sounds reasonable, but I am uncertain about the conclusion.
I discuss this uncertainty further below in connection with radiation
from the SCF.

> . . . Moreover, as the supposedly sinusoidal current distribution
> on a wire excited at its centre can be resolved as a pair of
> travelling waves (W Scott Bennett, "A Basic Theorem that Simplifies
> the Analysis of Wire Antennas", APS Magazine, Vol. 40, No. 1,
> February 1998, pp 22-30), it must be equally true that radiation
> from it occurs at its ends and, if the presence of the generator
> there leads to a further discontinuity at which net acceleration of
> charge is possible, at its centre as well. Well almost!

It took me a long time to appreciate the significance of this fact,
i.e., that a SCF can be represented as a sum of two
counter-propagating traveling waves. This is something that I
discussed wrt to the SCF in the October 1996 PCs for AP column. More
about that below.

> It is useful to pause for a moment to consider what antenna geometry
> is needed to support a uniform travelling wave of current. It turns
> out that a biconical antenna will do this as it has a characteristic
> impedance which is constant along its length. When we have a pair of
> uniform cylinders rather than co-apecial, coaxial cones, this will
> not be so and characteristic impedance will vary with distance from
> the centre of the antenna.
>
> Under these circumstances the current standing wave cannot be purely
> sinusoidal and it is to be expected that there will be some
> radiation along the length of the antenna instead of merely at its
> ends and centre (That we should then expect radiation at
> intermediate points would seem also to follow from the work of Simon
> and Biggi - I do not have this reference to hand but am sure that it
> can be found in C H Walter's "Traveling Wave Antennas", for
> example). However for slender antennas this is likely to be second
> order compared with end radiation.

My interpretation of the constant-radius cylinder is that indeed, the
fact its wave impedance varies with length causes a partial reflection
of the Q/I wave, and this produces radiation. But, whether that
length-wise radiation is second order is not clear.

I liked your discussion of mode number and complexity of the radiation
pattern. That is certainly a useful way to think about the problem.

I will confine my remaining comments to how the SCF and an actual,
well perfectly conducting, wire antenna might compare. First, its
relevant to note that the total power radiated by a SCF grows as
Log(kL), with L the length, oscillating between two asymptotes. The
expression for the radiated power is

Prad = (h/4¼)|Io|2{C + Log(kL) - Ci(kL) + 0.5sin(kL)[Si(2kL) - 2Si(kL)]
 + 0.5cos(kL)[C + Log(kL/2) + Ci(2kL) - 2Ci(kL)]}

where C is Euler's constant, Ci and Si are the cosine and sine
integrals and h is the medium impedance [Balanis (1982)]. But, how
can the power grow with increasing length if the SCF is made up of two
traveling waves that exhibit no reflection?

The answer seems to come in part from the expression for the far E-field
which is given by

 E(q) = K1Io{cos[kLcos(q)/2] - cos(kL/2)]}/sin(q)

where q here is the observation angle measured from the filament axis.
Without the sin(q) term in the denominator, the far field is that of
three point sources, two of equal, constant strengths at either end
and one in the center, the strength of the latter varying as cos(kL/2)
and the total power oscillates between two fixed values. With the
sin(q) term, the radiated power is as given by the above equation and
exhibits the Log(kL) trend previously mentioned. It's also worth
observing that for the SCF, the charge at the filament ends is
independent of L and that at the center feedpoint varies in a fixed,
oscillatory manner, with increasing L, so the contributions from these
two locations to the radiated power should not be expected to produce
that Log(kL) trend.

What might be concluded from this? That the sin(q) term effectively
increases the directivity of the SCF as L increases, leading to the
Log(kL) trend in total radiated power. Where does this increased
power come from? Well, FARS (described in a previous posting) shows
that there is a radiation contribution from all along the filament,
falling in amplitude from the ends as 1/distance, and also from the
center feedpoint, the amount of the latter depending on L.
Integrating a term that varies as 1/L yields Log(L), so numerically,
the Log(kL) trend seems to be associated with this interior radiation,
assuming of course that FARS does actually represent local radiation.

When a contsant-radius (in wavelengths) is evaluated as a function of
L using NEC, and the results are normalized to a 1-A maximum current
magnitude, its radiated power varies very similarly to that of the
SCF. This result is shown in Fig. 1 of the April 1999 PCs for AP
column. The difference is that the maxima in total power is slightly
less than that of the SCF. As the NEC radius is decreased the
imaginary component of the wire current approaches the SCF, the real
part approaches zero and the total power becomes even closer to the
SCF. There are several aspects of this that I don't understand. But,
I can't think of any other explanation for the SCF, or that of the NEC
dipole either, for both exhibiting a Log(kL) trend in their total
radiated powers, unless radiation does indeed come from along their
lengths.

Best wishes,

Ed

PS--I just logged on to Jos R Bergervoet's web site,
http://www.iae.nl/users/bergervo/gouy/dipole.html, with the movies he
mentioned in his posting today to NEC-LIST. I'm not sure how to fully
interpret the results there, but the movies are fascinating. EKM

--
Dr. Edmund K. Miller
3225 Calle Celestial
Santa Fe, NM 87501-9613
505-820-7371 (Voice & FAX)
e.miller_at_ieee.org