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Analysis of Thin Wire Antennas

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					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 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.
 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

				
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posted:11/11/2011
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