With the location of the poles of the Green’s function obtained in Chapter 3 and based on the dyadic Green’s function in spectral domain derived in Chapter 2, a Matlab program was written to analyze the planar structures using the Method of Moments.
We can evaluate the Green’s function accurately using DCIM with the location of the poles. The Green’s function is then applied to MoM to analyze planar structures such as antennas and filters. In this section, a brief introduction of MoM is given and a simple microstrip bandpass filter is designed, fabricated and measured for comparison.
The precise location of the poles’ of the Green’s function was used in the DCIM method for accurate evaluating the Green’s function in spatial domain. The DCIM divides the Green’s function into three main parts [34]. They are extraction of the quasi-static terms, contributions of the surface wave poles (SWP) which are obtained in the previous sections, and the remaining portion. The quasi-static contribution is inverse transformed analytically through the Sommerfeld identity. The surface wave contribution comes from the poles in the spectral Green’s functions. The surface wave poles are extracted first, subsequently the contribution of surface wave is evaluated through the residue calculus. The remaining portion in the spectral representation is expanded into a series of exponentials using either the Prony or GPOF methods.
With the appropriate Green’s function and boundary conditions, the surface current of a specified problem can be calculated. With an appropriate integral equation, we need to expand the unknown function in terms of known basis functions with unknown coefficients. The basis function for MoM in this thesis is chosen to be RWG (Rao- Wilton-Glisson) basis function. The surface of the metal filter is divided into separate triangles. Each pair of triangles, having a common edge, constitutes the corresponding RWG edge element as shown in Fig. 12. One of the triangles has a plus sign and the other a minus sign. A vector function (or basis function)
( )
( ) ( )
( ) ( )
/ 2 , in / 2 , in 0, otherwise
l A T
f l A T
ρ ρ
+ + +
− −
⎧⎪
= ⎨⎪
⎪⎪
⎩
r r
r r r − (4.1)
is assigned to the edge element. Here l is the edge length and A is the area of triangleT . Vectors
±
± ρ± are shown in Fig. 12. Vector ρ+connects the free vertex of the plus triangle to the observation point r. Vector ρ−connects observation point to the free vertex of the minus triangle.
r
C+
r C r −
T+
T− l
ρ+
ρ− d
O
The surface electric current on the metal surface (a vector) is a sum of the contributions of eqn. (4.1) over all edge elements, with unknown coefficients. These coefficients are found from the moment equations. The moment equations are a linear system of equations with the impedance matrixZ. There are a number of methods for the calculation of the impedance matrix [59], [128]-[131]. These methods employ different ways of integrating over surface triangles. Analytical approaches (line integrals and potential integrals) are accurate and fast but require extensive preliminary mathematical work. An alternative is to use a numerical integration over a triangle [131]. If the quadrature points do not coincide with the triangle’s midpoint, no separate calculations for the diagonal elements of the impedance matrix are necessary.
All elements of the impedance matrix can be calculated straightforwardly, using the same formula.
Fig. 13 Barycentric subdivision of the primary triangle. The triangle’s midpoint is shown by a white circle.
Fig. 13 shows the so-called barycentric subdivision of an arbitrary triangle [132]. Any primary triangle can be divided into 9 equally small subtriangles by the use of the
“one-third” rule. Further we assume that the integrand is constant within each small triangle. Then the integral of a function g over the primary triangle Tm is equal to
( ) 9 ( )
9 1
m
m c
k T k
g dS A g
=
= ∑
∫ r r (4.2)
where represents -th primary triangle and points , m m rkc k=1,...9 are the midpoints of nine subtriangles shown in Fig. 13 by black circle. is the area of the primary triangle.
Am
The impedance matrix determines electromagnetic interaction between different edge elements. If the edge elements m and are treated as small but finite electric dipoles, the matrix element
n
Zmn describes the contribution of dipole n (through the radiated field) to the electric current of dipole , and vice versa. The size of the impedance matrix is equal to the number of edge elements. The impedance matrix of the electric field integral equation is given by
m
( C / 2 C / 2)
mn m mn m mn m mn mn
Z =l ⎡⎣jω A+ ⋅ρ + +A− ⋅ρ − + Φ −− Φ+ ⎤⎦ (4.3) where index m and correspond to two edge elements, the term with the bracket n ( )⋅
denotes the dot product. is the edge length of element . lm m ρCm± are vectors between the free vertex point, , and the centroid point vm± rmC±, of the two triangles of the edge element m, respectively.
Tm± C
ρm+ is directed away from the vertex of triangle Tm+, whereas ρCm− is directed toward the vertex of triangle Tm−. The expressions for vector
and the scalar can be found in [59].
A Φ
The surface current density on a surface S of the plate or on other perfectly electrically conducting (PEC) structures is given by an expansion into RWG basis functions over M edge elements [59]. The current coefficients Im can be obtained by solving the impedance equation
Z I⋅ =V (4.4)
where V is a voltage excitation vector.
To demonstrate the accuracy and reliability of our DCIM algorithm with the new pole extraction method in the previous chapter, a bandpass filter using the Stepped Impedance Resonator (SIR) was designed and fabricated, which is shown in Fig. 14.
The filter is fabricated on a substrate with dielectric permittivity of εr =10 and thickness of 62 mils. As shown in Fig. 15, the simulated and measured centre frequency of the filter obtained is around 2.65 GHz. Relatively good agreement is observed between the results obtained using IE3D and the MoM alrogithm. The discrepancy is attributed to the different ways of approximating the Green’s function, the meshing of the filter structure, the zero metal thickness assumption in the MoM code. In general, the limitation of the simulation software may lead to discrepancy between the simulated and the measured results. Both our MoM code and the IE3D assume infinite ground and substrate. The MoM code, on the other hand, assumes zero metal thickness and lossless substrate. The variation of the substrate, i.e. permittivity and thickness, can cause the difference between the actual results and the expected data. There also exist fabrication tolerance and measurement errors which cause the discrepancy between simulation and measurement, and these include the variation of the width and the length of the microstrip lines, the soldering between SMA
connectors and the circuit, the measurement calibration while using the network analyzer.
Fig. 14 Photograph of the bandpass filter with SIR
(a) Comparison of measured and simulated S11 using MoM and IE3D
(b) Comparison of measured and simulated S21 using MoM and IE3D Fig. 15 Comparison of measured and simulated results