# Analysis of Thin Wire Antennas by liaoqinmei

VIEWS: 13 PAGES: 62

• pg 1
```									Analysis of Thin Wire Antennas

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 z-
component of the scattered electric
field equation, we get:
Simplifying assumptions   (cont…)
Simplifying assumptions               (cont…)

This is my main equation for
evaluating Ez. However, there is a
problem with the above equation: a
singularity exists.
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 Hallén-
Pocklington
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        (cont…)

   The Impedance kernel is:
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.
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
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

```
To top