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Eliminating a common integral in the expressions,
it is readily seen that
4 n eos4 Viel»
o =
o
J*
k2
eos4 rp
S i2
(5)
Equation (5) relates the back-scattering cross
section to the quantity S, which in turn is invariant
for small variations of I(z).
Evaluation of the back-scattering cross section
As in the case of the unloaded antenna, the
remaining part consists of the determination of S by
assuming an approximate current distribution which
contains one or more free parameters An (n = 1; 2;
3;...). The condition that S be stationary
95
3 An
= 0
(6)
automatically adjusts I(z) to an n-th order
approximation of the true value. Since the equation for S
contains the load impedance which may change in
magnitude from 0 to oo, the current distribution will
vary greatly with Z. This, however, will not
introduce serious complications (compared to the
unloaded case) in the calculation of 5 and hence o.
By analogy with the work of Tai, and in order to
make use of the integrals already solved, the trial
function for the current distribution considered is
taken to be
I (z) = Io (eos Ax z + A eos M z) (7)
with
Åt =
2 h
Å2 =
3 71
27T
which fulfills the boundary conditions of vanishing
currents at the ends of the dipole
1(h) =I(-h) =0.
I0 is an arbitrary constant with the dimension of a
current. The parameter A is to be determined by
eq. (6).
The final result gives the broadside back-scattering
of arbitrarily centre-loaded dipoles as a function of
kh, ka and Z, as follows
256•(kh)*
jj = eos4 ip ■
2 + | y" + \yl
4 nZ 16
Zo ’ 9
(y11 y2i - y12*) + ^ (yn + y23 - 2 y12)
Lo
where the y:s are the functions evaluated by Tai7
(8)
The back-scattering cross section was computed for
this trial function for ka = 0.066 and different dipole
lengths as a function of a reactive load Z = jX. Some
results are plotted in fig. 5 and discussed together
with data taken from experiments.
In addition to these computations the
back-scattering cross section has been evaluated using
I(z) = Io[coskz + A] (9)
as a trial function for the current distribution. The
agreement with the experimental results was much
poorer in this case.
Computations. Poles, zeros, maxima and minima
The expression for the back-scattering cross section
obtained above has the form
= const. IS |l
a + bZ
c + dZ
(10)
where a, b, c, d, and the constant are functions of
the length and diameter of the antenna and Z is the
loading impedance, R + j X.
The expression (10) is an Analytical function of
the complex variable Z in the form of the common
linear fractional transformation. Some general
properties of the behavior of o/A2 can thus be obtained
from a study of the positions of the pole and the
zero of (10). This method is particularly useful in
determining the influence of the losses in the
loading line when comparing theoretical and
experimental results.
Restricting calculations to reactive loads ranging
from X — — oo to oo, a/A2 has been evaluated for
eight A\h-values. X — 0 corresponds to a straight
unloaded wire.
Of special interest of course, are the maxima and
minima of a/A2 = F (X). The X-values giving extrema
can be found by differentiating | S | with respect
to X. This yields a second degree equation in X
X2 [d2 (a"b’ - ab") - c"d | b |2] + X [d21 a |2 - | c |2 | b |2] +
+ c"da2 - c2(a"b’ - a’b") = 0 (11)
where a = a’ -f j a", b = b’ + j b", c = c’ + j c", d
being real. The roots of this equation for different
kh-Yalues are tabulated in table 1, together with the
values of a/A2 at these points and at zero (unloaded
case) and infinite reactance. Three of the
theoretically computed curves are shown together with the
corresponding experimental results in the next
section.
Table 1. The roots of eq. (11) for different kh-valnes and the values of a/A’ at these points and at zero and
infinite reactance.
kh
Xr,
Xr,
o/A2
unloaded
X = 0
inf. load
X — oo
minimum
X = Xmin
maximum
X — Xmax
1 28.7 4.3 0.057 1.4 • 10"3 10-io 0.86
jr/2 23.4 — 2.1 0.635 0.015 2 • 10"9 0.93
3^/4 16.2 — 15.0 0.280 0.248 3 • 10"8 1.23
JT 9.57 — 340 0.285 2.14 6 • 10"4 2.14
5ji/4 3.17 13.5 0.261 1.81 0.060 3.30
3W2 — 3.62 2.26 1.40 1.29 0.87 1.51
2ji — 23.6 — 8.2 1.28 1.18 1.13 1.33
5 JT/2 — 37.5 — 12.7 1.77 1.43 1.27 1.97
.72 ELTEKNIK 1959
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