Analysis of Thin Wire Antennas
Author:
Rahul Gladwin.
Advisor: Prof. J. Jin
Department of Electrical and Computer
Engineering, UIUC.
Introduction
When we design antennas, it is vital
to be able to estimate the current
distribution on its surface.
 From the current distribution, we
can calculate the input impedance,
gain and the far-field pattern for the
antenna.

Introduction
(cont…)
Theoretical calculations may be used
to analyze antennas with simple
geometry, however, as we begin to
analyze coupled antennas, the work
becomes more tedious.
 It becomes necessary to numerically
model the antenna to determine its
current distribution.

Introduction
(cont…)
In this project, I have written a
MATLAB program to model the
current distribution on thin-wire
single and coupled one-dimensional
antennas.
 The algorithm used to evaluate the
integrals is based on the Method of
Moments and Hallén-Pocklington
equations

Introduction

(cont…)
To test my program, I have the
modeled current distributions on a
single dipole antenna and due to
the mutual coupling between closely
spaced linear antennas like those in
a three-element Yagi-Uda array
antenna.
Theoretical Formulation

This purpose of this program is to
determine the electric charge
density and electric current density
that result when an impressed
electric field acts on a one
dimensional thin wire antenna.
Theoretical Formulation
(cont…)
Until now, we’ve assumed that J is
known and we have solved for E. I
have now turned this around and
solved for J using a known E.
 Where J is the current density and E
is the electric field intensity.

Theoretical Formulation
(cont…)

Obviously, E is not known anywhere,
but we do know that Etan = 0 on the
surface of the PEC (Perfect Electric
Conductor).

My further derivations take off from
here.
Theoretical Formulation



(cont…)
When studying antennas, we can run into
two situations: receiving and transmitting
antennas.
A successful program should consider
both these situations and the differences,
if any, should be incorporated in the
algorithm.
I started by drawing the two scenarios
and writing out the respective equations.
Transmitting Antennas
Receiving Antennas
Theoretical Formulation
(cont…)
The important thing to realize is that
whether we’re dealing with either of
the two situations, the impressed
Electric field induces a current J to
flow on and in the wire.
 J(r), in turn produces a scattered
field Es. The total electric field
produced is E=Ei+Es.

Theoretical Formulation
(cont…)
For simplicity, I assumed the wires
to be perfectly conducting. Thus, the
tangential component of the total
field must be zero on perfectly
conducting wires.
 This leads to an integral equation
that can be solved for J. Once J is
known, it can be used to find the
required current distribution.

Theoretical Formulation
(cont…)

Now, all I needed is the expression
for Etan on the wire surface that is in
terms of the unknown J.

The first step is to find an equation
relating E to J. It can be derived as
follows.
Derivation for Escattered
Derivation for Escattered
Derivation for Escattered
Derivation for Escattered
Derivation for Escattered
The above is the exact solution
Simplifying assumptions
The wire is a PEC (Perfect Electric
Conductor).
 Current flows only on the surface of
the wire.
 Current only flows on the + axis and
is uniformly distributed over the wire
surface.

Simplifying assumptions
(cont…)
The wire is thin. I assumed this to
enforce that Etan= 0 on PEC wire.
 Both Ez and Ephi are tangential
components but the thin wire
assumptions don’t allow a Ephi
component. So I only cared about
the z-component of Etan.

Simplifying assumptions

(cont…)
From the previous assumptions, I
can write the current density within
the wire as:
Simplifying assumptions
(cont…)

After enforcing Etan = 0; by
symmetry, Etan = Ephi

Again, putting down only the zcomponent of the scattered electric
field equation, we get:
Simplifying assumptions
(cont…)
Simplifying assumptions
This is my main equation for
evaluating Ez. However, there is a
problem with the above equation: a
singularity exists.
(cont…)
Simplifying assumptions
(cont…)
Simplifying assumptions
(cont…)
Integral Equation of the
First Kind is found:
Note that the previous equation can be
changed to n equations and n unknowns.
That is, the above equation can be solved
by discretizing, that is, writing I(z’) as,
Integral equation of the first kind (above)
Basis function: Triangular
Weighting function: Delta

The Delta function amounts to
forcing Etan=0 only at a discrete
set of points. The weighting
function is equal to the delta
function.
Weighting function: Delta

In other words, I’m forcing the
weighted average of Etan=0 within
each interval to be zero. The
system becomes:
The above is the nth equation
Final system of equations
Once the above equation is inverted, you
can find the current distribution for a
whole range of antenna excitations.
The Final Equation:
Once In is known, we can calculate
patterns, gain, etc. of the antenna
Analysis of a Straight
Dipole
An example follows.
The Straight Dipole
This section provides results from
simulations of a 47 cm straight
dipole.
 The straight dipole is analyzed at
resonance (300Mhz) in order to
validate the model.

The Straight Dipole

(cont…)
In order to test my program, I
simulated the following 47 cm long
dipole because it should exhibit
resonance at around 300Mhz.
The Straight Dipole
(cont…)
For analyzing a straight dipole, the
program prompts for the number of
moments used and the frequency as
initial inputs.
 Based on the derivations, the
program then plots the current
magnitude as the functions of
positions on the dipole.

The Straight Dipole
(cont…)
I used a moment density of 55
moments per wavelength and
frequencies of 300Mhz, 600Mhz and
900Mhz.
 The output follows:

Analyzing coupled
antennas
Numerical Solutions
to the HallénPocklington
equations for
coupled dipoles
Theoretical Formulation

Here, I discuss their numerical
solution. For K antennas in line, on
the pth antenna, we have:
Theoretical Formulation

(cont…)
In the above equation, Vp(z) is
defined to be the sum of the
(scaled) vector potentials due to the
currents on all antennas:
Theoretical Formulation

The Impedance kernel is:
(cont…)
Theoretical Formulation

(cont…)
For the Pth antenna, the solution for
V(p) is of the form:
Theoretical Formulation

(cont…)
Assuming that all the antennas are
center driven, we obtain the coupled
system of Hallén equations, for p =
1, 2, . . . , K:
Theoretical Formulation

(cont…)
To solve the previous system of
equations, I applied a pulse-function
expansion of the form:
…and took N = 2M + 1 sampling
points on each antenna.
Theoretical Formulation

(cont…)
On the qth antenna, we have:
Therefore, the pulse-function
expansion for the qth current must
use a square pulse of width delta
zq.
Theoretical Formulation

(cont…)
Therefore, the current expansion on
the qth element should be:
Theoretical Formulation

(cont…)
I used the previous equation and
sampled along the pth antenna
…and obtained the discretized
Hallén system as:
Theoretical Formulation

(cont…)
The previous equation can be
written in a more compact form
since p=1,2,3…K. The new form is:
Theoretical Formulation

(cont…)
Now, n-dimensional vectors can me
defined:
Theoretical Formulation

(cont…)
This system provides k coupled
matrix equations by which we can
determined the k sampled currents
on the antennas. For example, if
k=3, we have:
Theoretical Formulation

(cont…)
This matrix can also be written in
the form:
Theoretical Formulation
(cont…)
The solution to this equation is of
the form:
I have written a MATLAB script that
solves the above equation for
currents on coupled antennas.
Analysis of a Three
element Yagi-Uda array
An Example follows
Yagi-Uda array
The three-element Yagi I simulated
consisted of one reflector, one driven
element and one director.
 The corresponding antenna lengths,
radii and locations on the x-axis
(with the driven element at origin)
were in units of ‘lambda’
(wavelengths - meter)

Yagi-Uda array
(cont…)
My program prompts the values of
‘L’, ‘a’ and ‘d’ as inputs.
 L=antenna length, a=radius and
d=distance along the horizontal
axis. Here is the data I used:

Reference
1.
Harrington, Roger F., Field Computation by Moment
Methods. New York: IEEE Press, 1993.
2.
Micheilssen, Eric. ECE 354 Lecture Notes on Antennas. The
University of Illinois at Urbana-Champaign, 2003.
3.
Janaswamy, Ramakrishna. Radiowave Propagation and
Smart Antennas for wireless communications, Kluwe
Academic Publishers, Boston, 1999.
4.
IEEE Antennas and Propagation Magazine, Vol 44, No. 4,
August 2002.
5. Pozar, David. Microstrip Antennas : The Analysis and Design
of Microstrip Antennas and Arrays, Wiley-IEEE Press, July
1995.
Special Thanks To
Prof. J. Jin
and
his research group