2.
INTEGRAL EQUATIONS AND METHOD OF MOMENTS
2.1. Introduction
The advent of high-speed digital computers over the last decades has lead to the development of efficient numerical techniques applicable to a wide variety of real electromagnetic radiation and scattering problems. For these reasons, there has been a growing interest in the use and development of computer codes for analyzing arbitrarily shaped bodies.
The key to the solution of any antenna or scattering problem is the knowledge of the physical or equivalent current density distributions in the device volume or surface. The main objective then of any solution method is to be able to predict accurately the current density over the antenna or scatterer.
One of the most notable approaches, referred herein as the Integral Equation (IE) technique, casts the solution for the induced current in the form of an integral equation where the unknown quantity density is part of the integrand. Numerical technique, like the Method of Moments, can then be used to solve for the current density. Once this step is accomplished, the field scattered by the object can be evaluated using the traditional radiation integrals [1], [21]-[25].
In general, there are many different forms of IE. Two of the most popular for time-harmonic electromagnetics are the Magnetic Field Integral Equation (MFIE) and the Electric Field Integral Equation (EFIE). The MFIE enforces the boundary condition on the magnetic field tangential component while the EFIE performs the same operation dealing with the electric field. In arbitrary shaped bodies, the EFIE presents the advantage to be applicable both to open and closed bodies, whereas the MFIE is applicable only to the second type of problem. On the other hand, the EFIE is more difficult to solve respect to the MFIE due to the presence of derivates in conjunction with a singular kernel appearing in the IE.
2.2. The Method of Moments
As mentioned above, one of the most employed numerical techniques for the solution of electromagnetic Integral Equations is the Method of Moments. Let us consider a general IE that can be express as:
( )
λ β ,where
Θ
is an integro-differential linear operator, β is the known excitation vector and λ is the unknown vector to be determined. If a solution exists, it will be unique and given by:( )
1
,
λ= Θ− β (2.2)
where Θ−1is the inverse integro-differential operator.
Since in practice is very difficult to find an analytical solution for the (2.1), which therefore exists only for some simple cases, it more convenient to solve the proposed IE through numerical techniques. For these reasons, it is worthwhile to introduce the inner product notation ,⋅⋅ which will be used to cast the IE (2.1) into a set on linear equations that can be solved numerically. A typical, but not unique, inner product definition is: * 1, 2 1 2 , S dS ξ ξ =
∫∫
ξ ξ⋅ (2.3)where S in the common support of the two vectorial functions ξ1 and ξ2,
while * indicates the complex conjugate operator.
The mentioned inner product satisfies the following scalar properties:
( )
( )
1 2 2 1 1 2 3 1 3 2 3 * 1 1 1 * 1 1 1 , , , , , , , 0 0 0,1, , , 0 0 0,1, , if n n N if n n N ξ ξ ξ ξ αξ βξ ξ α ξ ξ β ξ ξ ξ ξ ξ ξ ξ ξ = + = + > ≠ = = = = (2.4)where α and β are scalar quantities, while ξ1, ξ2 and ξ3 are functions
belonging to a N-dimensional vectorial space.
By using the previous relations, two vectorial functions ξ1 and ξ2 are
defined orthogonal if their inner product is equal to zero:
1, 2 0 .
ξ ξ = (2.5)
member of the set. If the set is both complete and orthogonal, it can be used to express any function belonging to the inner product space in the sense that:
( )
( )
1 1 1 0 , N m m m m m n m n n λ ∞ θ ξ λ ∞ θ ξ = = = −∑
=∑
−∑
= (2.6)where θm are scalar unique coefficients given by:
, . , m m m m ξ λ θ ξ ξ = (2.7)
In practice, one is forced to project the chosen function set onto a finite-dimension subspace of the original one expressing the final solution λΜ as:
1 , M M m m m λ θ ξ = =
∑
(2.8)where ξm are the so-called basis functions defined on the
Θ
support, whileθm are the unknown complex weights to be determined. The complex
coefficients θm are chosen in order to minimize the distance between λ
and its truncated counterpart λM:
(
M)
M .d λ λ− = λ λ− (2.9)
Substituting the (2.8) into (2.1), one obtains:
( )
1 , M m m m θ ξ β ε = Θ − =∑
(2.10)where ε is the residual error vector. In order to obtain a linear set of equations, ε is forced to be orthogonal to a set of testing functions
ω
lwhich means:
, 0 0,1, , .
l l L
ω ε = = (2.11)
( )
1 , , 0,1, , , M m l m l m l L θ ω ξ ω β = Θ = =∑
(2.12)which can be written in a matrix form as: ,
θ β
Λ ⋅ = (2.13) where:( )
( )
( )
( )
( )
( )
( )
( )
( )
1 1 1 2 1 2 1 2 2 2 1 2 1 1 2 2 , , , , , , , , , . M M M M M M M M ω ξ ω ξ ω ξ ω ξ ω ξ ω ξ ω ξ ω ξ ω ξ θ β θ β θ β θ β Θ Θ Θ Θ Θ Θ Λ = Θ Θ Θ = = (2.14)If the matrix (2.14) is non-singular, the complex weights θm can be
directly determined inverting the matrix: 1
,
θ = Λ ⋅− β (2.15) where the super-script -1 denotes the inverse matrix.
It is worthwhile to remark that, since the described process involves the projection of a continuous integro-differential space onto a finite-dimension subspace, the obtained result is always an approximation that can be made reasonably accurate with a proper choice of the basis and testing functions.
2.3. Basis Functions
One important step in any MoM numerical procedure is a proper choice of the basis functions, which, in general, should accurately represent the
unknown function minimizing at the same time the computational effort due to their employment.
The basis function set can be divided into three general classes: i. Subdomain basis functions;
ii. Entire domain basis functions; iii. High level basis functions.
2.3.1. Subdomain Basis Functions
The subdomain basis function class are the most common one. This kind of functions is non-zero only over a section of the unknown function domain and do not require any a-priori knowledge of the function they want to represent. The subdomain approach involves the division of the structure into N non overlapping segments defining each base in conjunction with the limits of one or more subintervals.
The most common subdomain basis function is the piecewise constant (Fig. 2.1) which is defined by:
( )
1 1 . 0 n n n x x x x otherwise ξ = − ≤ ≤ (2.16)Once the unknown coefficients are determined, these kind of bases produce a staircase representation of the unknown function (Fig. 2.2).
Other important subdomain basis functions are the piecewise linear functions (Fig. 2.3), which are seen to cover two contiguous segments with an overlap between adjacent functions. The final piecewise linear representation is seen to be smother with respect to the piecewise constant one but at the cost of an increased computational complexity (Fig. 2.4). The piecewise linear subdomain basis functions are defined by:
( )
1 1 1 1 1 1 . 0 n n n n n n n n n n n x x x x x x x x x x x x x x x otherwise ξ − − − + + + − ≤ ≤ − = − ≤ ≤ − (2.17)-0,5 0 0,5 1 1,5 x 0 x1 x2 x3 x4 x5 ξ n (x)
Figure 2.1. Piecewise constant subdomain basis function.
Nevertheless a further increase in the basis complexity with respect to the linear case may not lead to an improvement in the representation accuracy, if proper meshing approaches are used, advantage in computational time and resistance to errors can be gained by using the piecewise sinusoidal subdomain basis functions (Figs. 2.5-2.6).
( )
(
)
(
)
(
)
(
)
1 1 1 1 1 1 . 0 n n n n n n n n n n n sin x x x x x sin x x x sin x x x x x sin x x otherwise β β ξ β β − − − + + + − ≤ ≤ − = − ≤ ≤ − (2.18)2.3.2. Entire Domain Basis Functions
The entire domain basis functions are defined on the whole structure domain to be analyzed, thus no segmentation is required.
A common entire domain basis function is the sinusoidal function which reads like:
( )
(
2)
1 1 . 2 2 n n a x x cos l x l l π ξ = − − ≤ ≤ (2.19) 0 0,2 0,4 0,6 0,8 1 x 0 x1 x2 x3 x4 x5 λ λM λ( x)Figure 2.2. Representation of an unknown function with piecewise constant
subdomain functions. -0,5 0 0,5 1 1,5 x 0 x1 x2 x3 x4 x5 ξ n (x)
0 0,2 0,4 0,6 0,8 1 x 0 x1 x2 x3 x4 x5 λ λΜ λ( x)
Figure 2.4. Representation of an unknown function with piecewise linear
subdomain functions. -0,5 0 0,5 1 1,5 x 0 x1 x2 x3 x4 x5 ξ n (x)
0 0,4 0,8 1,2 x 0 x1 x2 x3 x4 x5 λ λM λ( x)
Figure 2.6. Representation of an unknown function with piecewise sinusoidal
subdomain functions.
The entire domain basis functions are particularly suitable for the analysis of electromagnetic problems where the current distribution is known to follow a certain pattern like the case of a wire dipole. This kind of basis functions, which can be generated by using Legendre, Hermite, Maclaurin or Tschebyscheff polynomials, lead to a bad convergence of the numerical methods and present difficulties to model arbitrary or complicated structures but produce a smaller matrix system with respect to the subdomain ones.
2.3.3. High Level Basis Functions
The high level basis functions are defined on wider supports, called macro-domains, which can be several wavelengths in size. This kind of bases includes a large number of triangular or quadrilateral subdomain elements leading to a significant reduction in the MoM number of unknowns without generating difficulties in the modeling of complicated devices.
2.4. Testing Functions
The expansion (2.10) leads to one equation with N unknowns. It alone is not sufficient to determine all the N θn coefficients since N linearly
independent equations are needed. This can be accomplished by using the previous inner product definition (2.3) in conjunction with a set of testing functions ωm. There are different approaches for the choice of the testing
functions like the Last Square Method or the Point Matching, but among them, the Galerkin’s procedure, which involves equal basis and testing functions, leads to more accurate and fast convergent numerical solutions.
2.5. The Rao-Wilton-Glisson Basis (RWG) Functions
In this section, a set of subdomain bases introduced by Glisson [23]-[26] will be discussed for an efficient solution of the Electric Field Integral Equation. The Rao-Wilton-Glisson (RWG) basis functions adopt a triangular patch modeling of the scatterer since this kind of approach is capable of accurately describe any geometrical surface or boundary (Fig. 2.7).
Figure 2.7. Double patch array modeled with RWG basis functions.
It is worthwhile to remark that the MoM technique requires a λ/10-λ/20 discretization which means that the side of the triangle cannot be longer than λ/10 at the highest operating frequency.
Each RWG basis is associated with an interior edge of the triangle model and vanishes outside the two adjacent patches. In particular, the two triangles related to the nth edge are indicated herein as T + or T - where the
superscript + or – is determined by the choice of the positive current reference direction for that edge.
Any point belonging to Tn+ can be express by r with respect to the origin
of the Cartesian reference system or by ρn+ defined with respect to the
free-vertex of the triangle. Similar concepts apply to the position vector ρn
-except that is directed towards the free-vertex of Tn- (Fig. 2.8).
The vector basis function associated to the nth edge is given by:
( )
2 , 2 n n n n n n n n n l r T A r l r T A ρ ξ ρ + + + − − − ∈ = ∈ (2.20)where ln is the length of the common edge, while An+ and An- are the areas
of the triangles Tn+ and Tn- respectively.
Figure 2.8. Triangle patches and geometrical parameters associated with an
interior edge.
In the MoM procedure, the RWG basis functions are used to approximate the induced surface current distribution and presents some properties which make them uniquely suited to this role:
i. The current has no component normal to the boundary (except the common edge) of the surface formed by the triangle pair Tn+ and
Tn- and hence no line charges exist along this boundary;
ii. The normal current component to the nth edge is constant and continuous across the edge (Fig. 2.9). In fact, the normal component of ρn± is just the height of the triangle Tn± with the
common edge as the base. The height can be express as 2An±/ln and
this factor normalizes the basis function in a way that the flux density normal to the edge n is unity. This result ensures, with i), that the function ξn is free of line charges;
iii. The surface divergence of ξn, which is proportional to the surface
charge density associated with the basis element, can be express as: , n n n s n n n n l r T A l r T A ξ + + − − ∈ ∇ ⋅ = − ∈ (2.21)
where the surface divergence operator in Tn± is defined as:
1 . s n n
ρ
±ρ
± ∂ ∇ = ∂ (2.22)The total charge density associated with each RWG basis function is zero and the charge density in each triangle is constant;
iv. The moment of ξn is given by;
(
)
(
)
, 2 n n c c n n n n n n n T T l A A ξ ξ dS ρ ρ + − + − + + + + =∫∫
= + (2.23)where ρnc± is the vector going from the free-vertex to the centroids
of the triangles.
Since the current density on the surface scatterer S can be express as reported in (2.8), where M is the total number of non-boundary edges, up to three bases may have nonzero values within each triangle face, but only
the base associated with an edge presents a normal component to it. Furthermore, the RWG normal component at each edge is unity, allowing the θm interpretation as the current density value flowing through the mth
edge.
Figure 2.9. Geometry for the construction of the basis function normal
component.
Because of the considerable variation of the basis function flow line direction, it is not obvious that a linear composition of the RWG’s can represent a constant current density in an arbitrary direction within each triangle. That can be seen with the aid of Fig. 2.10 which shows a triangle
Tq with the edge labeled 1, 2 and 3. Considering the vector r
1, r2 and r3 as
indicated, the basis functions in the triangle are: 1, 2,3 , 2 n n n n l n A ξ = ρ = (2.24)
where Aq is the triangle area and for simplicity, the current reference directions are assumed to be out of the triangle for each edge. It is apparent from the figure that the linear combination l2ξ1-l1ξ2 and l3ξ1
-l1ξ3 are constant vectors for every point in Tq and are parallel to the side 2
Since the two expressions are linearly independent, every constant vector r in the triangle can be generated through a proper linear combination of the proposed basis elements.
2.6. Solution of the EFIE via the Method of Moments
Let us denote as S the surface of a perfectly conducting object with a unit normal
ˆn
. An incident field Ei, defined as the field generated by the impressed sources in absence of the object, is impinging on S and induces a surface current density J. The Electric Field Integral Equation is based on the boundary condition imposing that the total tangential electric field Etot on Perfectly Electric Conducting (PEC) surface vanishes.Figure 2.10. Local coordinates and edges for the triangle Tq.
(
)
ˆ ˆ 0 , ˆ ˆ tot s i s i n E n E E on S n E n E on S × = × + = × = − × (2.25)where Es is the electric scattered field. The vector Es can be computed from the surface current by using the magnetic vector potential A and the scalar potential φ defined as (Fig. 2.11):
,
S
( )
( ) ( )
( )
( ) ( )
' ' ' ' ' ' , 4 , 1 , 4 S S A r J r G r r dS r r G r r dS µ π φ σ πε = =∫∫
∫∫
(2.27)( )
' ' ' , , jk r r jkR e e G r r R r r k ω εµ − − − = = − = (2.28)where µ and ε are the permeability and permittivity of the surrounding medium, G(r,r’) is the free-space Green's functions, while r and r’ are the generic observation and source position vector respectively.
Figure 2.11. Vector used to solve the radiation problem.
The surface charge density σ is related to the surface divergence of the induced current J by the equation continuity:
.
s
J
j
ωσ
∇ ⋅ = −
(2.29)( )
' '( )
' ' ˆ . ˆ 4 i jkR jkR s S S n E n e e j J r dS J r dS R R ωµ π ε − − − × = ∇ − × + ∇ ⋅ ∫∫
∫∫
(2.30)The next step in the EFIE solution via the Method of Moments is the selection of the testing procedure. Choosing the Galerkin’s approach and the RWG basis functions, one obtains [23]-[27]:
, , , ,
i
n n n
E ξ = jω Aξ + ∇φ ξ (2.31)
where ξn is the nth testing function.
Making use of a surface vector calculus identity [28] (Appendix C) and the properties of the RWG basis functions, the last term in (2.31) can be written as: , n s n . S dS φ ξ φ ξ ∇ = −
∫∫
∇ ⋅ (2.32) By applying the (2.21), the integral (2.32) can be approximated with the reported expression:( ) ( )
1 1 , m m s n n n n S T T c c n n n dS l dS dS A A l r r φ ξ φ φ φ φ + − + − + − ∇ ⋅ = − ≈ − ∫∫
∫∫
∫∫
(2.33)where φ(rnc±) is the value of φ at the triangle centroids.
With a similar approach applied to the vector potential and the incident field term in (2.31), it follows:
( )
( )
( )
( )
1 1 , 2 2 , 2 m m i i i n n n n n T n T i c i c n c n c n m m c c n n E E E l dS dS A A A A A E r E r l A r A r ξ ρ ρ ρ ρ + − + − + − + − + − + − = ⋅ − ⋅ ≈ ⋅ + ⋅ ∫∫
∫∫
(2.34)where the integral over the triangle surface is eliminated by approximating
Ei or A with their value at the centroids. Applying the (2.32)-(2.34), the EFIE
becomes:
( ) ( )
( )
( )
( )
( )
, 2 2 c c n n n c c c c n n m n m i c c i c c n n m n m l r r l A r A r l E r E r φ φ ρ ρ ρ ρ + − + + − − + + − − − + + ⋅ + ⋅ = = ⋅ + ⋅ (2.35)which represents the equation enforced on each patch edge.
The purpose of the (2.33)-(2.35) in the numerical computation of the MoM matrix entities, is to eliminate the potential integration approximating a double surface integral with a quantity expressed as a single surface integral.
The proposed approach is justified by the smooth nature of the involved quantities on the basis subdomains thanks to their integral definition [29].
Substituting the current expansion (2.8) into the (2.35), one obtaines a linear system on N x N equations which can be express as:
,
Z I⋅ =V (2.36)
where
Z
is a N x N matrix while I and V are column vectors of length N.Usually I and V are defined as “generalized current and voltage” vector while
Z
is referred as “generalized impedance matrix” whose genericelement Zmn can be physically interpreted as the electromagnetic reaction
of the nth base on the mth one.
The term Zmn and Vm are given by:
, 2 2 c c m m mn m mn mn mn mn Z l jω A ρ A ρ φ φ + − + − + − = ⋅ + ⋅ + − (2.37(a)) , 2 2 c c m m m m m m V l E ρ E ρ + − + − = ⋅ + ⋅ (2.37(b)) where
( )
, i c m m E± =E r ± (2.38)( )
( )
' ' ' ' ' 4 1 . 4 mn mn jkR mn n mn S jkR mn s n mn S c mn m e A r dS R e r dS j R R r r µ ξ π φ ξ π ωε ± ± − ± ± − ± ± ± ± = = − ∇ ⋅ = −∫∫
∫∫
(2.39)Once the impedance matrix and the forcing vector are evaluated, the resulting system of linear equations can be solved for the unknown vector
I. The evaluation of the Zmn impedance matrix element involves the
integration over the source triangles Tn± with an observation point located
in the triangles Tm±. It is easy to see that some of the integrations required
for the term Zmn are also needed for the term Zpq with p ≠ m and q ≠ n if p
is an edge of Tm± and q is an edge of Tn±. However, focusing on the
face-pairs indeed on the edge-face-pairs, one can notice that the same integrals associated to the vector and scalar potential evaluation are involved in all the terms Zmn having the edge n as interior edge of the source triangle and
the edge m as non-boundary edge in the observation triangle. For this reason, it is more convenient to evaluate the required potential integrals by face-pair rather than by edge-pair combinations. For each face-pair combination, the potential integrals will be scaled by the appropriate coefficients and their contribution added in the proper matrix entities.
In accordance with the previous discussion, consider a general combination of source and test triangle (Fig. 2.12). Each of the three bases which can exist on the triangle Tq is proportional to the vectors:
(
')
1, 2, 3 ,i r ri i
ρ = ± − = (2.40)
where the positive sign is used if the positive current reference direction is out the triangle Tq and the negative sign otherwise. The vector magnetic and scalar electric potential associated to the ith basis of the triangle p observed at the centroids of the face q are given by:
' , 4 2 p q jkR pq i i q i p T l e A dS A R µ ρ π − =
∫∫
(2.41)' 1 , 4 p i q jkR pq i q p T l e dS j A R φ π ωε − = −
∫∫
(2.42) where ' . p cp R = r −r (2.43)The integrals (2.41)-(2.42) can be evaluated efficiently using a local coordinate system defined within Tq noting that the vectors ρi (Fig. 2.12)
divide the patch into three subtriangles of areas A1, A2 and A3 with l1, l2
and l3 as one of their sides respectively. These areas are linearly dependent
since:
Figure 2.12. Electromagnetic interaction between the triangle Tq and Tp.
1 2 3 q .
A +A +A =A (2.44)
The “normalized areas coordinates” can be introduced as follow [30]: 3 1 , 2 , , q q q A A A A A A ψ = τ = ν = (2.45)
which must satisfy:
1 .
Note that:
(
)
12 3 0 1, 0 1, 0 1 (1, 0, 0) . , , (0,1, 0) (0, 0,1) r r r r r r ψ τ ν ψ τ ν ≤ ≤ ≤ ≤ ≤ ≤ = = = = (2.47)The inverse transformation, from local coordinate system to Cartesian coordinates, is given by:
'
1 2 3 .
r =
ψ
r +τ
r +ν
r (2.48)By using the previous definitions, the surface integrals can be written as:
( )
1 1(
1 2 3)
0 0 . q T r dS r r r d d τ η = − η ψ +τ +ν ψ τ∫∫
∫ ∫
(2.49)With (2.40)-(2.43) and (2.48), it follows that:
(
1 2 3)
4 , 2 pq i pq pq pq pq i i pq i pq i l A r I r I r I r I l I j ψ τ ν µ π φ πωε = ± + + − = (2.50) where 1 1 0 0 , p jkR pq p e I d d R τ ψ τ − =∫ ∫
(2.51) 1 1 0 0 , p jkR pq p e I d d R τ ψ ψ ψ τ − =∫ ∫
(2.52) 1 1 0 0 , p jkR pq p e I d d R τ τ τ ψ τ − =∫ ∫
(2.53) . pq pq pq pq Iν =I −Iψ −Iτ (2.54)It is obvious that only three independent integrals, which contribute up to nine elements in the impedance matrix, have to be numerically evaluated for each face-pair combination. For a closed object with N RWG basis, the number of independent integrals computed with the face-pair approach is 4N2/3 while with the edge-pair one is 12 N2. The numerical
integration (2.51)-(2.54) can be accomplished with proper quadrature techniques developed for triangular supports (Appendix D) [31]. However, for the self term (p = q), the integrand function is singular and for these cases the kernel singular portion must be removed and integrated analytically [27], [32].
2.7. Testing Procedure with the DGFs
As in the previous section, the MoM analysis of a PEC object embedded in layered media requires the evaluation of the magnetic vector potential A and electric scalar potential φ. Following the same procedure and adopting the DGF definition reported in the first chapter, it follows that:
(
)
(
)
, 1 , 1 , , , ˆ ˆ ˆ , n n n n n n n A A x y z T M A A x xx x m xx m x m T T M A A y yy y m yy m y m T T A A A z zx x zy y zz z T A A A m zx m x zy m y zz m z T A A x A y A z G J G JdS A G J dS G dS A G J dS G dS A G J G J G J dS G G G dS θ ξ θ ξ θ ξ ξ ξ ± ± ± ± ± ± = = = + + = = ⋅ = ⋅ = ⋅ = ⋅ = ⋅ = ⋅ + ⋅ + ⋅ = ⋅ + ⋅ + ⋅∫∫
∑
∫∫
∫∫
∑
∫∫
∫∫
∫∫
1 , M m= ± ∑ ∫∫
(2.55)where Tn± is the source base, M is total number of interior edges and θm
are the unknown complex coefficients to be determined. In a similar way, the scalar potential and the correction factor (1.40) can be express as:
, 1 ˆ, ˆ , n n M m m z m T T C z Jφ C z JdSφ θ Cφξ dS ± = ± = ⋅ =
∑
∫∫
∫∫
(2.56)(
)
(
)
(
)
' ' ' 1 ' ' 1 1 , . n n n n M m m m T T M M m m m m m m T T K J K J dS K dS K dS K dS φ φ φ φ φ θ ξ θ ξ θ ξ ± ± ± ± = = = ∇ ⋅ = ⋅ ∇ ⋅ = ⋅ ∇ ⋅ = ⋅ ∇ ⋅ = ∇ ⋅ ∑
∫∫
∫∫
∑
∑
∫∫
∫∫
(2.57)By using the (2.55)-(2.57), the general impedance matrix Zmn element
and the associated excitation vector Vm can be express as:
(
)
(
)
0 0 1 , 2 2 c c mn m mn m mn m mn mn c c m m m m m m Z l j A A j V l E E ωµ ρ ρ φ φ ωε ρ ρ + + − − + − + − + − = − ⋅ + ⋅ + − = ⋅ + ⋅ (2.58)2.8. Numerical Results
In this section some examples of microstrip circuits etched in layered media will be presented in order to validate the results of the developed code. In particular an open-end microstrip, a T-junction and a two-stub filter will be analyzed comparing the frequency responses with a commercial code, Ansoft Designer, which implements the same numerical procedure (Figs. 2.13-2.21).
Figure 2.13. Geometry of the open-end microstrip line.
2.0 GHz to 6.0 GHz for the T-Junction and from 2.5GHz to 4.5GHz for the two-stub filter are shown in Figs. 2.14-2.15, 2.17-2.18, 2.20-2.21 respectively. 0 0.2 0.4 0.6 0.8 1 -180 -120 -60 0 60 120 180 10 20 30 40 50 60 70 80 90 MoM Reference MoM Reference M ag ni tud e P h as e( deg ree s ) Position(mm) Phase Magnitude
Figure 2.14. Microstrip normalized current distribution at 2.5 GHz.
0 0,2 0,4 0,6 0,8 1 -180 -120 -60 0 60 120 180 10 20 30 40 50 60 70 80 90 MoM Reference MoM Reference M ag ni tud e P h as e( deg ree s ) Position(mm) Phase Magnitude
Figure 2.16. Geometry of the open-end T-junction microstrip problem. -50 -40 -30 -20 -10 0 -50 -40 -30 -20 -10 0 2 2.5 3 3.5 4 4.5 5 5.5 6 MoM Reference MoM Reference S 11 (d B ) 21 S (d B) Frequency(GHz) S 11 S 21
-180 -120 -60 0 60 120 180 -180 -120 -60 0 60 120 180 2 2.5 3 3.5 4 4.5 5 5.5 6 MoM Reference MoM Reference P h as e( S 11 ) Ph as e( S 21 ) Frequency(GHz) S 11 S 21
Figure 2.18. Comparison of the S-parameter phase for the T-junction problem.
Figure 2.19. Dielectric environment and geometrical parameters of the two stub
-60 -50 -40 -30 -20 -10 0 -60 -50 -40 -30 -20 -10 0 2.5 3 3.5 4 4.5 MoM Reference MoM Reference S 11 (d B ) 21 S (d B) Frequency(GHz) S 11 S 21
Figure 2.20. Comparison of the S-parameter magnitude for the two stub problem.
-180 -120 -60 0 60 120 180 -180 -120 -60 0 60 120 180 2.5 3 3.5 4 4.5 MoM Reference MoM Reference P h as e( S 11 ) Ph as e( S 21 ) Frequency(GHz) S 11 S 21
