Weill, Georges G.

English

Caltech Authors - Main

AFOSR 59-791
CALIFORNIA INSTITUTE OF TECHNOLOGY
Antenna Laboratory
Technical Report No. 20
Georges G. Weill
August 1959
TRANSIENTS ON A LINEAR ANTENNA
Research supported by
the U. S. Air Force Office of Scientific Research
1. Transients on a lossless line
We suppose that we have to deal with a transmission line with dis-
tributed L and C. The line is finite, of length ℓ; the end at 
x = ℓ is open; at x = 0 a signal Aejωt is applied starting at t = 0 .
Both voltage and current intensity satisfy
TRANSIENTS ON A LINEAR ANTENNA
Georges G. Weill
Figure 1.
From the well-known properties of the transmission line the wave front
propagates with a velocity , reflects itself at x = ℓ with
a change of sign, and reaches 0. We assume that no reflection occurs
for x = 0, hence when the wave front reaches 0 for the first time
after t = 0, the steady state is attained.
For the sake of completeness, let us do the complete computation.
The transmission line equations are;
with the following boundary conditions:
Taking the Laplace transform ℒ we get:
hence
Therefore, if
knowing that
one gets:
2.
one gets
Using the boundary conditions
whence,
x = 0,
x = ℓ,
3.
Therefore,
We expand (Re p > 0) the second ratio:
and taking now the inverse Laplace transform, one gets
where , cu(z) = complementary unit step function.
Actually, we are only interested in the behavior of the line between 
t = 0 and t = 2ℓ∕α. Therefore for 0 < t ≤ ℓ∕α we have the first 
term in the series;
For ℓ∕α ≤ t ≤ 2ℓ∕α we have
and that is all, for we suppose there is no reflection at the origin. The 
steady state is attained for t = 2ℓ∕α and is then
2. Radiation pattern of the transient
We now suppose that we open the transmission line, (Figure 2), and 
we assume no substantial change in LC. The magnetic field radiated by
Figure 2.
the line element is:
4.
5.
where [I] is the retarded current intensity. The current on the 
upper wire is then: i(x, t) ;
on the lower wire: i(-x, t) .
for ℓ∕α ≤ t ≤ 2ℓ∕α we get
Thus for we have
Hence for 0 ≤ t ≤ ℓ∕α, we get
, x > 0
, x < 0 .
, x > 0
, x < 0 .
6.
Therefore putting
and noticing that
one gets
We have then to distinguish between several intervals of time
for cos θ > 0.
(For cos θ < 0, 
complete by sym­
metry.)
I
II
III
iv
V
7.
with
Performing the integration one gets
8.
If we now perform in detail the integration, we get the following results
for the periods I - V .
Period I H0 ̸= 0
Period II
Period III
9.
Period IV
10.
Period V
III:
III':
IV :
In Figure 3 we represent in a (t, θ), or more precisely, in a
For cos θ < 0, same results in I', II', IV', and V' . The only change 
occurs in III' where one has to replace cos θ by -cos θ.
The radiation patterns will then be:
I : H0 ̸ = 0
diagram, the regions corresponding to the various periods.
II :
V :
11.
Figure 3.
The five cuts, A,B,C,D,E, give a fair description of the phenomenon.
3. Numerical results
12.
Radiation patterns as shown in Fig. 4 have been drawn for
Diagram A :
Diagram B:
Diagram C :
Diagram D :
Diagram E :
p = 1.2
kℓ = π/2
Diagrams A and B show that the maximum of the field intensity is not in 
the direction perpendicular to the antenna.
Diagram C shows a pattern which is already more of the broadside type. 
Diagrams D and E are completely broadside.
For , diagrams B and C are continuous but all the deriva-
tives are not continuous.
Fig. 4


Transients on a Linear Antenna

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Transients on a Linear Antenna

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