Integral Equation method to solve Dipole Antenna Parameters
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Transcript of Integral Equation method to solve Dipole Antenna Parameters
Current Density, Radiation Pattern, and Input Impedance
Calculation of Dipole Antenna using Integral Equation Method
SAMI UR REHMAN
Abstract: This brief discusses charge density, radiation pattern
and the input impedance of a dipole antenna (0.47λ long and with
a radius of 0.005λ) using Integral Equation(IE) approach and
Method of Moment (MoM) as numerical technique. A complete
mathematical analysis is presented in which MoM is used to
convert the electric field integral equation into a matrix equation.
MATLAB is used to simulate the resultant algebraic equation.
Those simulations are then compared with results obtained by
the commercial software.
Keywords: integral equation, method of moment, radiation
intensity, impedance, current distribution.
INTRODUCTION
Dipole antennas are categorized as straight wire antennas.
Modelling such antennas with MoM approach is well
established field of study. Two methods which have been
proved to be quite preeminent in the analysis of various
antenna problems are Integral Equation (IE) method and
Geometrical Theory of Diffraction (GTD) [1]. The integral
equation method provides the solution of the antenna problems
in the form of an integral where the unknown quantity is a part
of the integrand. Now various numerical techniques, Method
of Moment (MoM) to be the most prominent one can be used
to find the integrand. This method is most convenient for wire-
type antennas and efficient for structures that are small
electrically. The dipole antenna under consideration falls in
this category and therefore we shall use this approach.On the
other hand we use GTD method in situations where the
dimensions of the radiating system are many wavelengths.
High frequency asymptotic techniques can be used to analyze
such problems.
Sami urRehman is a researcher in Analog Mixed Signal Group (AMSG) at School of
Electrical Engineering and Computer Sciences, National University of Sciences and
Technology (NUST). He did his bachelors from NUST and is working towards his MS
degree in Electrical Engineering.
The first step in developingthe solution for the currents is
to derive the appropriate electric field integral equation.
The Method of Moments will then be used to convert the
integral equation into a system of linear equations which
can be solved by various techniques of linear algebra.
The electric field integral equation and Method of
Moments are general relationships, and I shall use these
relations under specific conditions.
ELECTRIC FIELD INTEGRAL EQUATION
The first step in developing the solution for the current on a
wire antenna is determining the appropriate integral equation.
The starting point in deriving the electric field integral
equation (EFIE) is Maxwell’s equations in the:
E jw H M (1)
H jw E J (2)
. eE
(3)
. mH
(4)
along with the continuity equations,
. eJ jw (5)
. mM jw (6)
The boundary conditions on the magnetic and electric
fields at the surface, S, are
2 1( ) sn H H J (7)
SAMI UR REHMAN 2011-NUST-MSEE-53
2 1( ) sn E E M (8)
The electric field outside of S,E2 , may be written as the sum
of an incident electric field, Ei, and a scattered electric field,
Es. The incident electric field induces the surface current
Jswhich in turn creates the scattered field Es. The Equivalence
Principle can be used to remove the PEC giving a
homogeneous free space problem. For the PEC case, the
equivalent current equals the induced current, Js.
Making use of the fact that the divergence of His zero when
m is zero, the magnetic field may be defined as the curl of an
auxillary vector,
H A
Where A is called the magnetic vector potential. Solving the
vector wave equation for the magnetic vector potential due to
Js , the free space solution is found to be
( ) ( )s
s
J r G R ds (9)
where the free space Green’s Function, G , is given by
'exp( )( ) ,
4
j RG R R r r
R
(10)
Now the scattered electric field, derived in [2] can be given by
1( . )s oE jw A j A
(11)
Final step in the derivation of Electric Field Integral Equation
is the application of boundary condition in the above scattered
field equation. Resultantly we get, what as explicitly be
derived in [3], as:
2 ' ' '[ ( ) ( ) ( . ( )) ( )]s S i
o s
jn k J r G R J r G R ds n E
(12)
The above derivation was mostly based on work presented by
Stutzman and Thiele (1987). [4] Provides a helpful insight into
the formulation of MoM over cylindrical structures.
METHOD OF MOMENTS
The Method of Moments (MoM) is a well known technique for
solving linearequations. In antenna analysis, the MoM is used
to convert the electric field integral equation into a matrix
equation or system of linear equations. The basic form of the
equation to be solved by the Method of Moments is
( )L u f (13)
where L is the linear operator, u is the unknown function, and f
is the source or forcing function. In order to create the matrix
equation, the unknown function is defined to be the sum of a
set of known independent functions, un, called basis or
expansion functions with unknown amplitudes, αn,
n n
n
u u (14)
Using the linearity of the operator, L, the unknown amplitudes
can be brought out of the operator giving
( )n n
n
L u f (15)
The unknown amplitudes cannot yet be determined because
there are n unknowns, but one functional equation. A fixed set
of equations are found by defining independent weighting or
testing functions, wm , which are integrated to give m different
linear equations.
, ( ) ,n m n m
n
w L u w f (16)
For antenna problems, the matrix form of above equation is
usually written in a form similar to Ohm’s Law:
,m n n mZ I V (17)
The generalized impedance matrix is given by
, , ( )m n m nZ w L u , the generalized current matrix is
given by n nI , and the generalized voltage matrix is
given by ,m mV w f , .
POCKLINGTON’S IE FOR DIPOLE ANTENNA
One of the most well-known forms of the EFIE was developed
by Pocklington (1897). Let us consider the geometry of Fig. 1
for derivation of Pocklington IE for the case of dipole antenna.
Fig. 1. Simple structure of a dipole antenna
For this geometry, the Lorentz gauge along the wire length, l,
reduces to:
1l
o
Aj l
(18)
Now using this Lorentz gauge the above calculated scattered
field becomes:
l o lE j Al
(19)
where the l subscripts indicate the component of the vectors in
the l direction. After taking the derivative of (18) and
substituting the result into (19), the electric field may be
written as:
22
2
1l l l
o
E k A Aj l
(20)
Making use of the thin wire approximation, which states that
the current around the circumference of the wire is uniform
and the axially directed electric field is to be estimated along
the axis of the current in the l direction, the vector potential, Al,
is given by:
/ 2
' ' ' 2 2
/ 2
( ) ( ) , ( )
L
l
L
A I l G R dl R l l a
(21)
Combining the last two equations and the electric field
boundary condition at the surface of the wire, Pocklington’s
electric field integral equation is given by:
/ 2 2
' 2 '
2
/ 2
( ) ( ) ( )
L
li
o L
jI l G R k G R dl E
l
(22)
Although linear basis functions can be used, curved sinusoidal
basis functions provide better approximation and decrease the
computation time as validated in [5]Now using the Method of
Moments to solve the problem, the current is expanded using
piece-wise sinusoidal basis functions giving:
'
' '11
1
sin ( )( ) ,
sin ( )
nn n n
n n n
k z zI z I z z z
k z z
'' '1
1
1
sin ( )( ) ,
sin ( )
nn n n
n n n
k z zI z I z z z
k z z
(23)
and the weighting functions are pulses one segment wide The
incident electric field is approximated by the delta gap source
model which assumes that the incident electric field is due to
the applied voltage across a small gap in the antenna, of width
d approaching zero, and is confined to that gap. Incorporating
the piece-wise sinusoidal basis functions, the pulse weighting
functions, and the delta gap source model, EFIE becomes:
2'
1
2
[ ( )sin
mn
nm
zz
n n
n o zz
j kI z z
k
' ' '
1( ) 2 ( )cos ] ( )n nz z z z k G R dz dz
2
2
( )
m
m
z
msms
z
Vz z dz
(24)
Where is one segment length, nz is n from the origin from
the z axis, msz is the location of thm source and msV is the
voltage of the thm source.
Lets compare this approach with the approach followed by
Ballanis in “Antenna Theory”. The resulting equation
apparently seems different but conveys almost the same
meaning as conveyed by equation (24).
Pocklingtons IE derived by Ballanis in [1] is given by:
/ 2
' 2 2
2
/ 2
exp( )( ) [(1 )(2 3 )
4
L
L
jkRI z jkR R a
R
2 '( ) ] ( 0)izkaR dz j E (25)
Similarly in [6] authors have derived the following expression
for a generic entry of impedance matrix from the IE:
22 2
exp( 2( ) ( )
8
omn
jZ j rG r
r
22 2{(1 2 )[2 3( ) ] 4 }
aj r a
r
(26)
Both equations (24 and 25) cab be used to calculate the current
distribution and the radiation pattern of the dipole antenna. A
mathematically rigorous approach to work out problems
concerning current distribution on wire antennas is reported in
[7].Current distribution and Impedance obtained from the
above equations is plotted in Fig. 2 and Fig. 3 respectively for
N=79.
Fig. 2 Current distribution on dipole antenna
Fig. 3 Input Impedance of dipole antenna
Now let’s compute the value of impedance by varying the
number of segments N. Fig 4 shows impedance for N=50 and
Fig. 5 for N=100, and we conclude that as we increase the
number of segments the accuracy of the results increases.
The dipole under discussion is a finite length dipole antenna
whose radiation pattern can be plotted once the current
distribution is known since this distribution is used in the E
and H field equations which can be computed to find out the
radiation pattern.
Fig. 4 Impedance for N=50
Fig. 5 Impedance for N=50
[8] Provides the numerical solution for calculating near and far
field Consider the geometry as shown in Fig. 6. Using this
figure we can work out a mathematical expression for the
radiated electric field as shown in (27).
sinexp( ( ))
2
o
o
jI lE j t kr
cr
(27)
Fig. 6 Figure for calculating E far field
Where c is the speed of light and r is the distance from the
dipole to the observation point.For the given parameters, the
radiation pattern is as shown in Fig. 7.
Fig. 7 Radiation pattern of dipole antenna
For the same dipole the 3-D plot is shown in Fig. 6.
Fig.8 3-D radiation pattern of the given dipole
The directivity of such an antenna is shown in Fig. 7.
Fig. 9 Directivity of dipole antenna
But there could be many other ways in which we can find
current distribution over straight wire or cylindrical structures.
Like [9] uses Fourier analysis to approximate the solution of
the Integral Equation to find the current distribution.
SIMULATIONS USING CST
The given dipole antenna is simulated using CST tool and the
results are shown as below:
Fig. 10 Dipole implemented in CST environment
Fig. 11 2-D Radiation pattern of dipole antenna
Fig. 12 Impedance plot of the given dipole
Fig. 13 S11 Parameter of the dipole
Fig. 14 Surface Current Density of the given dipole
CONCLUSION
This paper discusses current distribution, radiation pattern and
the impedance computation of a simple center fed dipole
antenna. A rigorous mathematical modeling has been
performed to compute the aforementioned parameters using
Integral Equation approach. Then MoM was used as a
analytical technique to solve the integral. MATLAB is used to
compute the integral and results are then compared with the
commercial CST tool.
REFERENCES
[1] “Antenna Theory” 2nd Edition by Constantine A. Balanis
[2] “Advanced Engineering Electromagnetics” by Constantine A.
Balanis
[3] Huffman, J.A.; Werner, D.H.; , "Modeling of a cylindrical wire
antenna with flat end caps using a rigorous moment method technique,"
Antennas and Propagation Society International Symposium, 1995. AP-S. Digest , vol.2, no., pp.1258-1261 vol.2, 18-23 Jun 1995
[4] Inagaki, M.; Sawaya, K.; Adachi, S.; , "Numerical analysis of a dipole antenna in the vicinity of conducting circular cylinder with finite
length moment method analysis by using the interior Green's function,"
Antennas and Propagation Society International Symposium, 1998. IEEE , vol.4, no., pp.1918-1921 vol.4, 21-26 Jun 1998
[5] Kubiak, I.; , "The analysis of distribution of the current in the slender cylindrical antenna," MILCOM 97 Proceedings , vol.1, no., pp.247-
251 vol.1, 2-5 Nov 1997
[6] “Antennas for All Applications, 3rd ed. by J. D. Kraus and R. J.
Marhefka
[7] Tsai, L.; , "A numerical solution for the near and far fields of an
annular ring of magnetic current," Antennas and Propagation, IEEE
Transactions on , vol.20, no.5, pp. 569- 576, Sep 1972
[8] Hsieh, H. C.; , "Current Distribution of a Cylindrical Antenna in a
Warm Plasma," Plasma Science, IEEE Transactions on , vol.9, no.2, pp.52-
57, June 1981
[9] Khamas, S.K.; Cook, G.G.; , "Moment-method analysis of printed wire spirals using curved piecewise sinusoidal subdomain basis
and testing functions," Antennas and Propagation, IEEE Transactions on ,
vol.45, no.6, pp.1016-1022, Jun 1997