# Impedance on thin wire dipole using method of moments

1. Aug 18, 2012

### Cram

1. The problem statement, all variables and given/known data
Write code to calculate the input impedance on a thin wire antenna using the below form of Pocklington's integral equation. Use Method of moments.
Use a pulse as the basis function, point matching and delta-gap generator as the 'source'.
Assume the wire is broken up into 7 segments (N=7), the wire is 0.47λ long with radius 0.005λ.

2. Relevant equations
Pocklington's integral equation was given as
$E^{i}_{z}=\frac{j}{ωε}∫^{L/2}_{-L/2}I_{z}(z')\frac{e^{-jkR}}{4 \pi R^{5}}[(1+jkR)(2R^{2}-3a^{2})+(kaR)^{2}]dz'$ where $R=\sqrt{a^{2}+(z-z')^{2}}$

Note that the source point (vertical z) is on the outside of the wire and the field/observation point (z') is on the axis of the wire.

3. The attempt at a solution
I have broken down pocklington's equation to try and fit it into the matrix expression $= [Z]^{-1}[E]$. I ignored the pulse function as it was a constant 1 over the integral for each segment.  is the set of weights representing the current on each segment which ends up out the front of the integral since they don't have dependence on z'.

For the E column, I used all zeroes except for the 'gap' (relating to middle segment N=4) where it is V/Δz.
In this case V=1. I multipiled [E] by constant -j4∏ωε as I moved them from the RHS to the LHS in the above equation.

For the Z matrix, I am using the integral of the Green's function, which is the everything under the integral to the right of $I_{z}(z')$. I am numerically integrating it over each segment using quadgk() in Matlab.

To get , multiply inverse of [Z] by [E].
Then finally to get the input impedance, I divide V (=1) by the centre segment current N=4.

The result I am getting (1.5541e+01 + 4.8658e+02j) doesn't match up with what I am expecting (closer to 100 + j100).

The gist of my program is shown by pseudocode:
N=7; freq=3e8; omega=2*pi*freq; c=3e8; lambda=c/freq; k=2*pi/lambda; a=0.005*lambda
epsilon=8.85e-12; dz=L/N;
<<
for(m=1:N) {
zm=m*dz-(dz/2);

for(n=1:N) {​

zn=n*dz-(dz/2);

//This integrates over nth segment length
}; ​
};

E=zeros(N,1);
E(4,1) = -j*4*pi*omega*epsilon/dz;

I=inv(Z)*E;

Zin=1/I(4,1);
>>

Any assistance on the general approach would be greatly appreciated. I have tried different integrators which has yielded different but silly results.
I believe the integration of the Green's function is the problem.

Thanks,
Cram

Last edited: Aug 18, 2012
2. Aug 25, 2012

### Cram

Bumping as thread has changed categories.