4.1 Wire dipoleBasic theoryIn this chapter, we introduce the reader to the
momentmethod
computing of parameters of a
wire dipole.
We present all the information in English in this layer so that the reader can become familiar with English terms, which are used in the area of antennas
and computational electromagnetics. The Dutch translation of this chapter can be found in the layer B.
All the technical parameters of antennas
(gain,
input impedance,
directivity pattern)
can be computed if
current distribution
on the antenna surface is known. Unfortunately, computation of current distribution makes troubles because integral equations have to be solved.
There are two basic approaches to the solution of integral equations – iterative and moment ones. Iterative methods are based on rough
approximation of the current distribution (e.g. sinus one) that is iteratively made more accurate. On the other hand,
moment methods
transform integral equations to a set of simultaneous linear equations, which are solved by matrix operations.
In this chapter, our attention is turned to the moment analysis of
wire antennas.
In all the cases, antennas are supposed to be circular cylinders of the radius a and of the length 2h. The antenna
axis is situated to the axis z (fig. 4.1A.1) of the
cylindrical coordinate system
(r, ρ, z). The antenna is placed to the vacuum (μ = μ_{0},
ε = ε_{0}, σ = 0) and no losses are considered.

Fig. 4.1A.2  Exciting electrical field between dipole terminals 

In the center of the cylinder (z=0), there is a short gap. In the
gap, we assume a hypothetical harmonic generator, which produces an azimuthally
symmetric exciting electrical field (fig. 4.1A.2). The voltage across the
gap

$V={\displaystyle {\int}_{gap}{E}_{z}dz}$

( 4.1A.1 )

is supposed to be 1V. In (4.1A.1), E_{z} is the zcomponent of the exciting electrical field
intensity on the interpolated antenna surface (fig. 4.1A.2). Outside the gap, E_{z} is zero due to the perfect
conductivity> of the cylinder.
I. Methods of Moments
Let us to consider a general integral equation

$\underset{a}{\overset{b}{\int}}f\left(z,\xi \right)d\xi =g\left(z\right)$,

( 4.1A.2 )

where f is an unknown function, <a,b> is the integration interval and g is a known function describing sources. The
moment solution
of (4.1A.2) can be reached within 3 steps:
 The unknown function f is approximated by a
linear combination
of known
basis functions
f_{n} and unknown coefficients c_{n}

$f\approx \tilde{f}={\displaystyle \sum _{n=1}^{N}{c}_{n}{f}_{n}}$.

( 4.1A.3 )

 The approximation of the unknown function f^{~} is substituted back into the solved equation (4.1A.2). After
that, the summation and the integration are swapped. This yields

$\sum _{n=1}^{N}{c}_{n}{\displaystyle \underset{a}{\overset{b}{\int}}{f}_{n}\left(z,\xi \right)d\xi}}=g\left(z\right)+R\left(z\right)$.

( 4.1A.4 )

Here, R(z) is the
residuum
which expresses the fact that the approximation f^{~} does not meet (4.1A.2) exactly.
Equation (4.1A.4) is one equation for N unknown coefficients c_{n}.
 The approximation f^{~} is as accurate as possible if the
residuum
R is minimal. Hence, the residuum is minimized by the
method of weighted residua
: product of a
weighting function
w and the residuum R integrated over the region of interest <a, b> has to be zero
[5]. If N weighting functions are used then the set of N
simultaneous linear equations for N unknown coefficients c_{n} is obtained

$\underset{a}{\overset{b}{\int}}{w}_{m}\left(z\right)R\left(z\right)dz}=0\text{\hspace{1em}}m=\mathrm{0,1,...}\text{\hspace{0.05em}}\text{\hspace{0.05em}}N$,

( 4.1A.5a )


$\sum _{n=1}^{N}{c}_{n}{\displaystyle \underset{a}{\overset{b}{\int}}{w}_{m}\left(z\right){\displaystyle \underset{a}{\overset{b}{\int}}{f}_{n}\left(z,\xi \right)d\xi}dz}}={\displaystyle \underset{a}{\overset{b}{\int}}{w}_{m}\left(z\right)g\left(z\right)dz$.

( 4.1A.5b )

Both
basis functions
and
weighting functions
have to be linearly independent on the interval <a,b>.
II. Basis Functions
Basis functions
can be global or local ones.
Global basis functions
are defined on all the region of interest <a,b>. E.g., system of functions

${f}_{n}\left(z\right)=\mathrm{cos}\left(\frac{\pi nz}{h}\right)$

( 4.1A.6 )


Fig. 4.1A.3  Multibasis approximations a) piecewise constant, b) piecewise linear 

is on <a,b> linearly independent and coefficients c_{n} in the approximation

$f\left(z\right)\approx \tilde{f}\left(z\right)={\displaystyle \sum _{n=1}^{N}{c}_{n}{f}_{n}}={\displaystyle \sum _{n=1}^{N}{c}_{n}\mathrm{cos}\left(\frac{\pi nz}{h}\right)}$

( 4.1A.7 )

have got then meaning of Fourier coefficients of the current distribution.
Approximation based on the global basis functions is called the
singlebasis approximation.
Local basis functions
are defined on all the region too but each of them is nonzero only on a subregion of the interval of interest <a,b> as can be seen in
fig. 4.1A.3. If
basis functions
are normalized then coefficients c_{n} have got the meaning of
nodal values
(samples) of the computed function f (fig. 4.1A.3). Approximation based on the local basis functions is called
the multibasis approximation.
III. Weighting functions
Point matching
and
Galerkin’s methods
are the most common ways of the residuum minimization.
Point matching
(or collocation) uses
Dirac pulses,
which are place to points where the values of unknown current distribution are computed, as weighting functions

${w}_{m}\left(z\right)=\delta \left(z{z}_{m}\right)$.

( 4.1A.8 )

Point matching
method exhibits very low computational requirements because one integration is eliminated in 4.1A.5b) thanks to
the filtering property of
Dirac pulses

$\sum _{n=1}^{N}{c}_{n}{\displaystyle \underset{a}{\overset{b}{\int}}{f}_{n}\left({z}_{m},\xi \right)d\xi}}=g\left({z}_{m}\right)$.

( 4.1A.9 )

On the other hand, the residuum minimization is related to the matching points z_{m} only.
In
Galerkin’s method,
weighting functions are identical with basis ones

${w}_{m}\left(z\right)={f}_{m}\left(z\right)$.

( 4.1A.10 )

Galerkin’s method
exhibits higher computational requirements in comparison with
point matching
because one of integrations is not eliminated in this case. On the other hand, the residuum minimization is performed with all the points z
∈ <a,b>.
IV. Wire antennas
Assume the cylindrical antenna of fig. 4.1A.1. Then, the radiated electromagnetic field can be expressed in terms of
vector potentials
A and
scalar potentials
φ, respectively. Potentials have to meet
inhomogeneous wave equations
[2]

$\frac{{\partial}^{2}{A}_{z}\left(z\right)}{\partial {z}^{2}}+{k}^{2}{A}_{z}\left(z\right)={\mu}_{0}{J}_{z}\left(z\right)$,

( 4.1A.11a )


$\frac{{\partial}^{2}\phi \left(z\right)}{\partial {z}^{2}}+{k}^{2}\phi \left(z\right)=\frac{\rho \left(z\right)}{{\epsilon}_{0}}$.

( 4.1A.11b )

Here, J_{z} is the zcomponent of the current density [A.m^{2}] impressed to the antenna by the
source, ρ is the volume charge density [C.m^{3}] on the antenna, A_{z} is the zcomponent of the
vector potential
and φ is the
scalar potential,
k=2π/λ is the
wavenumber
and λ is the
wavelength.
The current flowing on the antenna causes charge accumulation at the antenna cylinder. This fact can be described by the
continuity equation
[2]

$\frac{\partial {J}_{z}\left(z\right)}{\partial z}+j\omega \rho \left(z\right)=0$.

( 4.1A.12a )

If radius of the antenna cylinder is much smaller than the wavelength a << λ then the current and charge can be
assumed to be concentrated in the axis of the cylinder
[5], and solving (4.1A.11) yields
[2]

${A}_{z}\left(z\right)=\frac{\mu}{4\pi}{\displaystyle {\int}_{2h}{I}_{z}\left(\xi \right)\frac{\mathrm{exp}\left[jkR\left(z,\xi \right)\right]}{R\left(z,\xi \right)}d\xi}$,

( 4.1A.12b )


$\phi \left(z\right)=\frac{1}{4\pi \epsilon}{\displaystyle {\int}_{2h}\sigma \left(\xi \right)\frac{\mathrm{exp}\left[jkR\left(z,\xi \right)\right]}{R\left(z,\xi \right)}d\xi}$.

( 4.1A.12c )

Here, I_{z}(ξ) is the current [A] flowing in the axis of wire, σ(ξ) denotes the length charge density
[C.m^{1}] on the axis of wire, R(z,ξ) is the distance between the location ξ of electromagnetic
field sources I_{z}(ξ) and σ(ξ) and the location z potentials A(z) and
φ(z).
On the basis of A(z) and φ(z), electrical intensity of the field radiated by the antenna can be computed
[2]

${E}_{z}^{s}\left(z\right)=j\omega {A}_{z}\left(z\right)\frac{\partial \phi \left(z\right)}{\partial z}$.

( 4.1A.12d )

Electrical intensity has to meet the
boundary condition
on the antenna surface S

${E}_{z}^{i}+{E}_{z}^{s}=0$ on S

( 4.1A.12e )

E_{z}^{i} denotes electrical intensity of the incident wave. In the case of an usual transmitting antenna,
E_{z}^{i} is the intensity by the feeding source (on the antenna surface), i.e. the intensity in the exciting gap (fig.
4.1A.2).
If the current distribution on the antenna is to be computed then the set of equations (4.1A.12) has to be solved.
In order to meet the
boundary condition
4.1A.12e), electrical intensity (and consequently potentials) has to be computed on the surface of the wire.
That is why the distance R is described by the equation

$R\left(z,\xi \right)=\sqrt{{a}^{2}+{\left(z\xi \right)}^{2}}$.

( 4.1A.13 )

In the following paragraphs, piecewise constant
basis functions
and Dirac
weighting functions
are used to solve (4.1A.12).

Fig. 4.1A.4  Piecewise constant approximation 

In the first step, the analyzed structure has to be discretized. Segmentation of the antenna is depicted in fig. 4.1A.4.
Lower bounds of segments are signed by ””, upper ones by ”+”. Lower bound of the first segment and upper bound
of the last one are shifted from the ends of the antenna in order to meet the condition I(h)=I(h)=0.
Segments’ lengths are Δ = 2α.
Substitung piecewise constant approximation to the integral equations 4.1A.12b,c) yields

${A}_{z}\left(z\right)\approx \frac{\mu}{4\pi}{\displaystyle \sum _{n=1}^{N}{I}_{n}{\displaystyle \underset{h+\left(n\mathrm{0,5}\right)\text{\Delta}}{\overset{h+\left(n+\mathrm{0,5}\right)\text{\Delta}}{\int}}\frac{\mathrm{exp}\left[jkR\left(z,\xi \right)\right]}{R\left(z,\xi \right)}d\xi}}$,

( 4.1A.14b )


$\phi \left(z\right)\approx \frac{1}{4\pi \epsilon}{\displaystyle \sum _{n=1}^{N}{\sigma}_{n}{\displaystyle \underset{h+\left(n\mathrm{0,5}\right)\text{\Delta}}{\overset{h+\left(n+\mathrm{0,5}\right)\text{\Delta}}{\int}}\frac{\mathrm{exp}\left[jkR\left(z,\xi \right)\right]}{R\left(z,\xi \right)}d\xi}}$.

( 4.1A.14c )

Here, I_{n} and σ_{n} are
nodal values
of current and charge density distributions.
Since the first derivative of the piecewise constant approximation is zero on the constant sections and does not exist on their borders,
4.1A.12a) and 4.1A.12d) are rewritten in terms of
finite differences.
If the fact that I_{n} = I_{z}(h+nΔ) is considered then the
continuity equation
can be expressed as

$\frac{{I}_{z}\left(h+\left(n+1\right)\text{\Delta}\right){I}_{z}\left(h+n\text{\Delta}\right)}{\text{\Delta}}+j\omega \sigma \left(h+\left(n+\mathrm{0,5}\right)\text{\Delta}\right)\approx 0$

( 4.1A.15a )

and the relation for computing electrical intensity is then of the form

${E}_{z}^{s}\left(h+n\text{\Delta}\right)\approx j\omega {A}_{z}\left(h+n\text{\Delta}\right)\frac{\phi \left[h+\left(n+\mathrm{0,5}\right)\text{\Delta}\right]\phi \left[h+\left(n\mathrm{0,5}\right)\text{\Delta}\right]}{\text{\Delta}}$.

( 4.1A.15d )

Relations 4.1A.15a) and 4.1A.15d) show that
Dirac pulses
for point matching have to be placed to the center of segments for the
vector potential

${A}_{z}\left(h+m\text{\Delta}\right)\approx \frac{\mu}{4\pi}{\displaystyle \sum _{n=1}^{N}{I}_{n}{\displaystyle \underset{h+\left(n\mathrm{0,5}\right)\text{\Delta}}{\overset{h+\left(n+\mathrm{0,5}\right)\text{\Delta}}{\int}}\frac{\mathrm{exp}\left[jkR\left(h+m\text{\Delta},\xi \right)\right]}{R\left(h+m\text{\Delta},\xi \right)}d\xi}}$

( 4.1A.15b )

and to borders of segments for the
scalar potential

$\phi \left[h+\left(m+\mathrm{0,5}\right)\text{\Delta}\right]\approx \frac{1}{4\pi \epsilon}{\displaystyle \sum _{n=1}^{N}{\sigma}_{{n}^{+}}{\displaystyle \underset{h+n\text{\Delta}}{\overset{h+\left(n+1\right)\text{\Delta}}{\int}}\frac{\mathrm{exp}\left\{jkR\left[h+\left(m+\mathrm{0,5}\right)\text{\Delta},\xi \right]\right\}}{R\left[h+\left(m+\mathrm{0,5}\right)\text{\Delta},\xi \right]}d\xi}}$.

( 4.1A.15c )

In 4.1A.15c), σ_{n+} = σ [h+(n+0.5)Δ].
Now, (4.1A.15) can be rewritten into a more compact form

${\sigma}_{{n}^{+}}\approx \frac{1}{j\omega}\left[\frac{{I}_{n+1}{I}_{n}}{\text{\Delta}}\right]$,

( 4.1A.16a )


${A}_{z}\left(m\right)\approx \frac{\mu}{4\pi}{\displaystyle \sum _{n=1}^{N}{I}_{n}{\displaystyle \underset{{\text{\Delta}}_{n}}{\int}\frac{\mathrm{exp}\left[jkR\left(m,\xi \right)\right]}{R\left(m,\xi \right)}d\xi}}$,

( 4.1A.16b )


$\phi \left({m}^{+}\right)\approx \frac{1}{4\pi \epsilon}{\displaystyle \sum _{n=1}^{N}{\sigma}_{{n}^{+}}{\displaystyle \underset{{\text{\Delta}}_{{n}^{+}}}{\int}\frac{\mathrm{exp}\left[jkR\left({m}^{+},\xi \right)\right]}{R\left({m}^{+},\xi \right)}d\xi}}$,

( 4.1A.16c )


${E}_{z}^{i}\left(m\right)\approx j\omega {A}_{z}\left(m\right)\frac{\phi \left({m}^{+}\right)\phi \left({m}^{}\right)}{\text{\Delta}}$.

( 4.1A.16d )

In 4.1A.16d), the
boundary condition
4.1A.12e) is included.
Now, let’s have a look at the
continuity theorem
4.1A.16a); it expresses the fact that segments of the antenna can be replaced by
elementary electrical dipoles
(fig. 4.1A.5). Taking this idea in mind, submission of n^{th} segment to
the scalar potential can be computed on the basis of 4.1A.16c) as

$\phi \left({m}^{+}\right)=\frac{1}{j\omega \epsilon}\left[{I}_{n}{\displaystyle \underset{{\text{\Delta}}_{{n}^{+}}}{\int}\frac{\mathrm{exp}\left(jkR\right)}{4\pi R}}d\xi {I}_{n}{\displaystyle \underset{{\text{\Delta}}_{{n}^{}}}{\int}\frac{\mathrm{exp}\left(jkR\right)}{4\pi R}d\xi}\right]\frac{1}{\text{\Delta}}$.

( 4.1A.17 )

Substituting (4.1A.17) and 4.1A.16b) to 4.1A.16d) and multiplying both
sides by Δ yields

${\mathbf{E}}_{z}^{i}\text{\Delta}=\mathbf{Z}\mathbf{I}$,

( 4.1A.18 )

where

${Z}_{mn}=j\omega \mu \text{\Delta}{\displaystyle \underset{{\text{\Delta}}_{n}}{\int}\frac{\mathrm{exp}\left[jkR\left(m,\xi \right)\right]}{4\pi R\left(m,\xi \right)}d\xi}+$
$+\frac{1}{j\omega \epsilon}\left[{\displaystyle \underset{{\text{\Delta}}_{{n}^{+}}}{\int}\frac{\mathrm{exp}\left[jkR\left({m}^{+},\xi \right)\right]}{4\pi R\left({m}^{+},\xi \right)}d\xi}{\displaystyle \underset{{\text{\Delta}}_{{n}^{}}}{\int}\frac{\mathrm{exp}\left[jkR\left({m}^{+},\xi \right)\right]}{4\pi R\left({m}^{+},\xi \right)}d\xi}\right]\frac{1}{\text{\Delta}}$
$\frac{1}{j\omega \epsilon}\left[{\displaystyle \underset{{\text{\Delta}}_{{n}^{+}}}{\int}\frac{\mathrm{exp}\left[jkR\left({m}^{},\xi \right)\right]}{4\pi R\left({m}^{},\xi \right)}d\xi}{\displaystyle \underset{{\text{\Delta}}_{{n}^{}}}{\int}\frac{\mathrm{exp}\left[jkR\left({m}^{},\xi \right)\right]}{4\pi R\left({m}^{},\xi \right)}d\xi}\right]\frac{1}{\text{\Delta}}$,

( 4.1A.19 )


Fig. 4.1A.5  Antenna as a set of elementary electrical dipoles 

denotes submission of current and charge on n^{th} segment
to the voltage induced on the m^{th} segment.
Since electrical intensity is zero on all the segments except of the source
gap, elements of voltage vector are zero except of the gapsegment corresponding
element that equals 1. Then, (4.1A.18) provides the current distribution
I. Ratio of input voltage and input current gives then the input impedance
of the analyzed antenna.
An example of the analysis results is depicted on fig. 4.1A.6; module
and phase of the
current distribution
of the dipole h = λ and a = 0.001588 λ is plotted there.
The presented results (fig. 4.1A.6) can be obtained using a computer program,
whose user’s guide is presented in the layer C.
In the layer D, the program is described from the programmer’s point of view.

Fig. 4.1A.6  Current distribution on the symetrical dipole. Piecewise constant approximation, point matching. Length of dipole 2λ, diameter 0.001588λ, number of segments 64. 

