(1)Background Theory
by L. A. Dorado
EMMCAP is written in C++ and it was developed with the aim of improving the accuracy in the modeling of wire structures. Curved wire antennas are usually analyzed using codes based on a straight wire approximation to the geometry [1]. This can be a very inefficient method in terms of unknowns or computer memory. By using curved segments, which describe accurately the contour of the geometry, the number of unknowns can be reduced, thus allowing for the solution of bigger problems.
Here is a brief explanation of the theoretical basis for the EMMCAP program system.
Electric Field Integral Equation for curved wires
The current distribution on metallic surfaces with ideal conductivity can be found by solving an Electric Field Integral Equation (EFIE) expressed in the frequency domain:
(1)
where
Ei: Incident Electric Field on the surface S.
n: unit vector on the surface S at the point r.
k: wave number.
J: unknown electric current density flowing on the surface.
G: Green's function, which in free space is defined by:
(2)
The EFIE is an expression of a boundary condition on the surface, namely zero tangential electric field.
When we are dealing with a wire structure, the EFIE reduces to:
(3)
where T is the tangential unit vector describing the contour of the curve G , I(s) is the unknown electric current on the wire, and K(s,s') is the integral equation's kernel defined as:
(4)
The EFIE is averaged about the wire circumference described by the angle f , resulting in the EFIE (3) with the kernel (4), [2]. The current distribution I(s) is then the average value of the current density J in the axial direction; the current in the f direction is neglected. This is a good assumption provided that the wire radius is small with respect to the wavelength.
The wire axis G is defined by its parametric equations that can be written as a vector function:
(5)
which points from the origin to any point on the wire. The parameter s varies over a real interval.
The tangent unit vector can be obtained from the first derivative of (5):
, 
(6)
This parametric description is the key for the accurate modeling of the wire structure. A straight wire approximation to the geometry produces a loss of geometrical information that can't be completely restored. However, this information is not lost if a parametric representation is used to describe the wire locus. It is also possible to improve on the straight wire approximation by using quadratic segments to model the geometry [3], [4].
Thus, the definition of a wire must include its parametric description and its first derivative if an exact representation of the geometry is required, as shown in Fig. 1.
Fig. 1: Parametric description of a curved wire. The tangent
unit vector is obtained from the first derivative of the position vector 
.
The kernel (4) is approximated by the following generalized thin-wire approximation:
, 
(7)
where a is the wire radius.
When the observation point r(s) and the source point r(s') are both in the same straight wire, the distance R reduces to the usual thin-wire approximation:
(8)
Thus, the EFIE and its kernel are also valid for straight wires.
An accurate form of the kernel can be obtained by calculating the integrals in (4), but it requires an extra computational effort for curved wires. A more accurate kernel than the thin-wire approximation for the curved case is under development.
The existence of a PEC ground plane is modeled by means of image currents. This method can be easily implemented by adding an image term to the Green's function, resulting in an additional term for the kernel.
The Method of Moments (MoM) [5] is a technique used to convert the EFIE into a system of linear equations that then can be solved by standard methods.
For simplicity, the integral (linear) operator in (3) will be denoted by L, then the EFIE takes the form:
(9)
where ET is the tangential component of the incident electric field.
The current distribution is approximated by a sum of N basis functions with unknown amplitudes In, giving:
(10)
With this expansion and using the linearity of the operator L, we can write:
(11)
In order to obtain a set of N equations, the functional equation (11) is weighted with a set of N independent testing functions wm, giving:
(12)
where the integrals are calculated over the domain of L. Now we have as many independent equations as unknowns, so (12) can be written as:
(13)
where
: impedance
matrix with dimension 
and
the elements 
.
: current
matrix with dimension 
and
the elements 
.
: voltage
matrix with dimension and
the elements 
.
This fully occupied equation system has to be solved for the unknown currents In. LU decomposition is used in EMMCAP.
The MoM is applied by first dividing the wire structure into N segments, and then defining the basis and testing functions on the segments. Triangular basis and pulse testing functions are used in EMMCAP as shown in Fig. 2.
Fig. 2: Triangular basis and pulse testing functions.
When a curved wire is described parametrically by a vector function (5), the basis and testing functions are curved in the sense that their support is a curved subset of the wire.
In order to fill the impedance matrix ,
an adaptive Gauss-Legendre quadrature rule is applied to compute the involved
integrals.
After having solved the equation system, the currents
are known and other parameters of interest, such as input impedances, voltages,
radiated power and directivity can be computed.
Excitation of the structure
If a discrete voltage source is placed at the i-th segment, the corresponding element in the voltage matrix is simply equal to the voltage of the generator. Thus,
(14)
When an incident plane wave is used as the excitation, each wire segment is excited by the incoming field, which has the form:
(15)
where k is defined by the direction of propagation,
so that 
(wave number), and
r is the evaluation point, Fig. 3. The elements of the voltage matrix
are then defined by:
(16)
where the integration is performed over the m-th segment, and
the vectors and 
are given by (5) and (6), respectively.
Fig. 3: Incident plane wave exciting a wire.
Curved segments vs. Straight segments
Several examples show the advantages of using curved segments with respect to the stability and convergence properties of the solutions. As a consequence of the improved convergence rate, reduced computation time and memory space can be obtained for accurate results.
As an illustration, the figures below show a comparison between EMMCAP, which uses curved segments, and a straight wire approximation to a normal mode helix antenna. The convergence properties of the input impedance and admittance versus the number of unknowns are investigated.
As can be seen from these results, by using curved segments significantly fewer unknowns are needed to predict the input impedance. However, the admittance convergence is questionable for the straight wire case, while it has a notorious convergent behavior for the curved case.
The improvement depends on the geometry and frequency, but generally, if N straight segments are needed to obtain a convergent value, then N/k curved segments are needed to obtain the same value, with k = 2...10. For a straight geometry the improvement factor is k = 1, as can be expected, because there are no curved segments in this case.
Center-fed helical antenna (normal mode) in free space.
Helix radius = 0.0273 wavelengths
Pitch = 0.0363 wavelengths
Number of turns = 10
Wire radius = 0.001 wavelengths
References
