CIRCUITS ETCHED IN LAYERED MEDIA
4.1. Introduction
The competitive communication market environment has led to the development of more and more efficient microwave CAD tools. One approach in order to reduce the computation burden is to use circuit simulators, in which the microstrip discontinuities are modeled as lumped elements by employing a theory based algorithm. The network-type simulators reduce the CPU time to be much less than those of full-wave electromagnetic counterparts but produce accurate results only at low-frequencies when the parasitic coupling effects are negligible. Thus, it is almost always necessary to validate a design based on the network-theory-based circuit simulator by using a full-wave electromagnetic solver.
One of the most widely employed full-wave algorithm is the Method of Moments, which numerically solves integral equations formulated by using the Dyadic Green’s Functions for the induced currents. Although more accurate than the circuit approach, the MoM technique requires much greater CPU and memory resources than the circuit-based techniques, which is a cause for concern when incorporating the MoM solver in the prototyping and optimization process. This is because the MoM associated matrix grows rapidly as the problem electrical dimensions become large or finite mesh is used to model complex structures in order to guarantee a reasonable solution accuracy. In particular, if N is the total number of unknowns, the memory allocation and the CPU time increase as O(N2) and O(N3) respectively.
In views of this, different approaches to speed up the MoM process have been deeply researched over the last decades. For instance, the Fast Multipole Method (FMM) [49] and the Impedance Matrix Location (IML) [50] have been developed in the context of iterative solvers. Even if the FMM realizes a saving in the memory allocation procedure by storing only the near field interaction of the large solving matrix and employs an efficient spherical harmonic expansion technique for speeding up the solution of the related linear system, the FMM is bounded by a discretization size ranging from λ/10 to λ/20, which makes the MoM matrix grow rapidly as the geometry becomes electrically large. The IML transforms the source/testing basis functions into ones resembling
traveling or standing waves, but its accuracy increases rapidly keeping more matrix elements and its application is limited to sufficiently smooth objects.
Wavelet expansions, based on the theory of multiresolution analysis, have been proposed in order to render the MoM matrix sparse by using a thresholding procedure [51].
Recently, a technique called MNM has been introduced in [52]-[53] for frequency sweeping analysis and successfully applied to a wide variety of problems.
Another emerging approach for efficient MoM analysis of microstrip structure is based on the concept of segmentation or domain decomposition and several techniques have been proposed to implement this concept. For instance, in [54], the modified diakoptic theory [55] originally proposed for antenna problems, has been applied to microstrip structures even if its use has been relatively limited. The same is true for the diakoptic-theory-based Multilevel Moments Method (MMM) [56], which carries out an iterative basis function refinement to solve passive planar structure problems. The Subdomain Multilevel Approach (SMA), which utilizes the so-called Macro Basis Functions (MBFs) [57], is a novel technique for reducing the matrix size associated with large planar antenna array problems.
More recently, the Characteristic Basis Function Method (CBFM) has been successfully applied for the analysis of Monolithic Microwave Integrate Circuits (MMICs) [58] and for scattering problems involving three-dimensional objects [59]-[60].
In this chapter the conventional CBFM and two new CBFM-based techniques will be introduced for an efficient simulation of microstrip circuits etched in layered media. The first one, referred to herein as the CBFM-EMA (CBFM Equivalent Medium Approach), is well-suited for accurate and fast calculation of the S-parameters of microstrip circuits. A second approach, the Spectral Domain CBFM (CBFM-SP), aims at further improving the efficiency of the conventional CBFM and is suitable for a rigorous analysis of problems involving a large number of unknowns.
4.2. The Characteristic Basis Function Method
In this section, the CBFM will be introduced by applying the presented procedure to a 2x1 patch array (Fig. 4.1). Each antenna consists of a rectangular patch with length and width equal to 50.0 mm and 45.0 mm
respectively. The array elements are feed by a 50.0 Ω microstrip line with a 5.0 mm x 12.5 mm feed inset.
Figure 4.1. Geometry of the 2x1 patch array fed by microstrip lines.
The array is placed on a substrate whose thickness is equal to 1.6 mm and the relative dielectric constant is 2.2. The center-to-center separation distance is 60.0 mm.
The first step in the CBFM is to divide the original problem geometry into several smaller regions, commonly referred to as “blocks” (Fig. 4.2). In this example, the array elements are geometrically isolated and each of them is identified as a separate block, hence the total number of subdomains is 2.
Next, two types of Characteristic Basis Functions (CBFs) are determined for each block. The first of these, the Primary CBFs, are defined as the solution for the induced current in isolated blocks when they are directly
excited at one of their input ports (Figs. 4.3-4.4). The generation of the Primary CBFs starts by defining appropriate interfaces between adjacent blocks such that the coupling effect between the blocks is minimized and the number of unknowns in each subdomain is relatively small.
Figure 4.2. Geometry of the 2x1 patch array divided in two blocks.
The number of Primary CBFs for each block is usually equal to the number of interfaces that the block shares with the others plus its original excitation ports. To avoid a truncation effect in the current distribution generated by fictitious sources, a small extension is introduced for the “internal” interfaces such that these ports are slightly shifted away from the block area. The impedance matrix associated with the extended blocks is evaluated, to determine the induced current distributions, by solving the matrix equation: , , 1, 2 , 1, 2 , . ii i n i n i i M n N Z J V = = ⋅ = (4.1)
In the above equation, Zii is the impedance matrix of the i
th extended
is the relative block source, M is the block number and Ni is the number of
Primary CBFs for the ith block. In particular, considering the 2x1 patch problem, M=2 and Ni=1 respectively. The CBFs are then generated by
discarding the basis elements that belong to the extended regions.
The second type of CBFs, called the Secondary CBF, is introduced to account for the coupling effects between the blocks (Figs. 4.5-4.6). It represents the current distribution induced on a particular block due to the Primary CBFs defined on the other blocks. The Secondary CBFs are generated inverting the linear system of equations:
, , , 1, 2 , , 1, 2, , 1, 2 tot
,
i i i j ii i n ij j m i j M i j n N N N m NZ
J
Z
J
= ≠ = = + +⋅
=
⋅
(4.2) where Zii is the impedance block matrix associated with the section ith,
ij
Z is the coupling matrix describing the electromagnetic interaction between blocks i and j, and Nitot is the total number of CBFs for the ith subdomain.
The final current distribution J induced on the microstrip circuit can be expressed as a linear combination of the CBFs as follows:
1 2 1, 1, 2, 2, , , 1 1 1
,
M N N N n n n n M n M n n n n J α J α J α J = = = =∑
+∑
+ +∑
(4.3)where the αi,j are the unknown complex expansion coefficients to be
determined and Jij is the jth CBF of the ith block.
After generating the CBFs, a thresholding procedure can be employed in order to reduce the number of CBFs and the reduced matrix size. One possible approach is to discard a secondary CBF if its norm ratio to the corresponding primary CBF for the block falls below a certain level:
, Sec Pry Sec Pry Use if I I Discard if I I ε ε ≥ < (4.4)
where ISec is the vector-2 norm of the secondary CBF induced by IPry , Pry
I
Figure 4.3. X component of the current density distribution for the Primary CBF of the first block.
Figure 4.4. Y component of the current density distribution for the Primary CBF of the first block.
Figure 4.5. X component of the current density distribution for the Secondary CBF of the first block.
Figure 4.6. Y component of the current density distribution for the Secondary CBF of the first block.
In this case, the number of Secondary CBFs varies with the operating frequency since the mutual coupling effect changes.
Another approach to reduce the total number of CBFs is to apply distance criteria on the separation between the Primary and Secondary CBF blocks. The number of Secondary CBFs remains fixed with the frequency in this case but some a priori information about the mutual coupling effect have to be used.
The final step to be performed is the generation of the NtotxNtot reduced matrix (where Ntot is the total number of CBFs on each block), that can be
evaluated by applying the Galerkin testing procedure implemented by using the same set of CBFs as basis and testing functions:
,
R R R T R T Z V Z J Z J V J Vα
⋅ = = ⋅ ⋅ = ⋅ (4.5) where RZ is the reduced matrix, J is the matrix form of all the CBFs and VR is the Right-Hand-Side vector related to the circuit excitation.
Typically the dimension of the reduced matrix is much smaller compared to that which would be generated by employing the conventional Moment Method formulation and can be solved directly by applying the LU decomposition technique.
4.3. Numerical Results
The CBFM method described above has been applied to different planar microstrip antenna arrays and the obtained results have been compared to that derived by using the direct calculation. The chosen examples are:
i. 2x1 planar patch array; ii. 2x2 planar patch array; iii. 4x2 planar patch array. 4.3.1. 2x1 Planar Patch Array
The first example analyzed with the conventional CBFM method is a 2x1 planar patch array fed by 50 Ω microstrip lines (Fig. 4.1). The S-parameters and Z-parameters for the patch array have been reported in Figs. 4.7-4.10.
The radiation patterns have been compared to those obtained by using the direct solution on the principal planes (φ = 0° and 90°) in Figs.4.11-4.12.
-60 -50 -40 -30 -20 -10 0 -60 -50 -40 -30 -20 -10 0 2 2,1 2,2 2,3 2,4 2,5 CBMoM MoM CBFM MoM S 11 (d B ) 21 S (d B ) Frequrncy(GHz) S 21 S 11
Figure 4.7. Comparison of the S-parameter magnitude for the 2x1 patch array.
-180 -160 -140 -120 -100 -80 -60 -180 -120 -60 0 60 120 180 2 2,1 2,2 2,3 2,4 2,5 CBMoM MoM CBFM MoM P h as e( S 11 ) Phas e( S 21 ) Frequency(GHz) S 21 S 11
1 10 100 0,01 0,1 1 10 100 2 2,1 2,2 2,3 2,4 2,5 CBMoM MoM CBFM MoM M agni tu de( Z 11 ) M agni tu de (Z 21 ) Frequency(GHz) Z 21 Z 11
Figure 4.9. Comparison of the Z-parameter magnitude for the 2x1 patch array.
-150 -120 -90 -60 -30 0 30 60 -180 -120 -60 0 60 120 180 2 2,1 2,2 2,3 2,4 2,5 CBMoM MoM CBFM MoM P h as e( Z 11 ) Phas e( Z 21 ) Frequency(GHz) Z 21 Z 11
-50 -40 -30 -20 -10 0 10 20 -50 -40 -30 -20 -10 0 10 20 -90 -60 -30 0 30 60 90 CBFM MoM E θ (d B ) E φ (d B ) Theta(Degree) E θ E φ
Figure 4.11. Radiation pattern of the electric field magnitude on φ = 0° plane at 2.275 GHz. -180 -120 -60 0 60 120 180 -180 -120 -60 0 60 120 180 -90 -60 -30 0 30 60 90 CBMoM MoM CBFM MoM P h as e( E θ ) P has e( E φ ) Theta(Degree) Eθ E φ
Figure 4.12. Radiation pattern of the electric field phase on φ = 0° plane at 2.275 GHz.
0 5 10 15 -90 -60 -30 0 -90 -60 -30 0 30 60 90 CBMoM MoM CBFM MoM E θ (d B ) E φ (d B ) Theta(Degree) Eφ Eθ
Figure 4.13. Radiation pattern of the electric field magnitude on φ = 90° plane at 2.275 GHz. -180 -120 -60 0 60 120 180 -180 -120 -60 0 60 120 180 -90 -60 -30 0 30 60 90 CBMoM MoM CBFM MoM P h as e( E θ ) Phas e( E φ ) Theta(Degree) Eθ E φ
A conventional RWG meshing approach has been applied with a total number of unknowns equal to 1018 since each block presents 509 basis functions. The CBF method has been implemented over a frequency band ranging from 2.0 GHz to 2.5 GHz by using 4 CBFs including Primary and Secondary Characteristic Basis Functions. It is obvious from the presented results that all of the array parameters obtained by using the CBFM matched well with the one given by the direct calculation. The employed personal computer is an Intel Core 2 Duo with a 3.0 GHz CPU and 4 GB RAM.
4.3.2. 2x2 Planar Patch Array
The second example considered is a 2x2 microstrip patch array antenna shown in Fig. 4.15. The total number of unknowns is 2036 when the antenna is modeled by using RWG basis functions.
The radiation patterns obtained by using the CBFM and the MoM procedure have been reported in Figs.4.16-4.19.
-60 -45 -30 -15 0 15 30 -60 -45 -30 -15 0 15 30 -90 -60 -30 0 30 60 90 CBMoM MoM CBFM MoM M agni tu de (E θ ) Magni tu de (E φ ) Theta(Degree) E θ Eφ
Figure 4.16. Radiation pattern of the electric field magnitude on φ = 0° plane at 2.275 GHz. -180 -120 -60 0 60 120 180 -180 -120 -60 0 60 120 180 -90 -60 -30 0 30 60 90 CBMoM MoM CBFM MoM P h as e( E θ ) Phas e( E φ ) Theta(Degree) Eθ E φ
-60 -45 -30 -15 0 15 30 -90 -75 -60 -45 -30 -15 0 -90 -60 -30 0 30 60 90 CBMoM MoM CBFM MoM E θ (d B ) φ E (d B ) Theta(Degree) E φ Eθ
Figure 4.18. Radiation pattern of the electric field magnitude on φ = 90° plane at 2.275 GHz. -180 -120 -60 0 60 120 180 -180 -120 -60 0 60 120 180 -90 -60 -30 0 30 60 90 CBMoM MoM CBFM MoM P h as e( E θ ) Phas e( E φ ) Theta(Degree) Eθ E φ
Figure 4.19. Radiation pattern of the electric field phase on φ = 90° plane at 2.275 GHz.
The current distribution at 2.275 GHz obtained by using the CBFM and the MoM procedure have been reported in Figures 4.20-4.23.
Figure 4.20. X component current distribution obtained by using the CBFM procedure at 2.275 GHz.
Figure 4.22. Y component current distribution obtained by using the CBFM procedure at 2.275 GHz.
Figure 4.23. Y component current distribution obtained by using the conventional MoM procedure at 2.275 GHz.
4.3.3. 4x2 Planar Patch Array
The last example, proposed in order to show the CBFM accuracy and the numerical efficiency, is a 4x2 microstrip patch array antenna shown in Fig. 4.24. The total number of unknowns is 4072 when the antenna is modeled by using RWG basis functions.
Figure 4.24. Geometry of the 4x2 patch array fed by microstrip lines.
The radiation patterns on the principal planes (φ = 0° and φ = 90°) at 2.275 GHz obtained by using the CBFM and the MoM procedure have been
-40 -30 -20 -10 0 10 20 30 -40 -30 -20 -10 0 10 20 30 -90 -60 -30 0 30 60 90 CBFM MoM CBFM MoM E θ (d B ) φ E(d B ) Theta(Degree) E φ E θ
Figure 4.25. Radiation pattern of the electric field magnitude on φ = 0° plane at 2.275 GHz. -180 -120 -60 0 60 120 180 -180 -120 -60 0 60 120 180 -90 -45 0 45 90 CBFM MoM CBFM MoM P h as e( E θ ) Phas e( E φ ) Theta(Degree) E θ E φ
Figure 4.26. Radiation pattern of the electric field phase on φ = 0° plane at 2.275 GHz.
-20 -10 0 10 20 30 -80 -60 -40 -20 0 20 -90 -60 -30 0 30 60 90 CBFM MoM CBFM MoM E θ (d B ) φ E (d B ) Theta(Degree) E θ E φ
Figure 4.27. Radiation pattern of the electric field magnitude on φ = 90° plane at 2.275 GHz. -180 -120 -60 0 60 120 180 -180 -120 -60 0 60 120 180 -90 -60 -30 0 30 60 90 CBFM MoM CBFM MoM P h as e( E θ ) Phas e( E φ ) Theta(Degree) Eθ Eφ
Figure 4.28. Radiation pattern of the electric field phase on φ = 90° plane at 2.275 GHz.
Table 4.1 provides a measure of the CBFM numerical efficiency with respect to the classical MoM solution by reporting the Relative Time parameter defined as the ratio between the CBFM CPU time and the MoM CPU time as a function of the total number of unknowns.
Table 4.1. Relative time (%) for the analyzed patch array example.
Number of Patches Unknowns Relative Time(%)
2 1018 53.06
4 2036 28.066
8 4072 15.61
4.4. A New Technique for Efficient Simulation of Microstrip
Circuits
In this section a CBFM-based method has been introduced for a fast and efficient design of microstrip circuits etched in layered media. More in detail, the process starts by exploiting the CBFM-EMA for a fast calculation of the S-parameters of the preliminary microstrip circuit structure. Next, a further refinement of the device is accomplished by using CBFM-SP, which guarantees a more rigorous analysis of problems involving a large number of unknowns.
Let us assume that the analyzed structure is stratified along the z-axis and that the multilayered stack is grounded (Fig. 4.29).
As reported in the first chapter, an element of the spatial domain dyadic Green’s functions f(ρ;z|z’) is related to its spectral domain counterpart
F(kρ;z|z’) via the Sommerfeld Integral as follows [6]:
(
')
(
')
( )
0 0 1 ; | ; | , 2 f ρ z z F kρ z z J kρρ k dkρ ρ π ∞ =∫
(4.6)where J0 is the Bessel function of the first kind of order 0, ρ is the
horizontal distance in a cylindrical coordinate system and kρ is the spectral
variable corresponding to the transverse direction.
To face the difficulty with the numerical computation of the SI due the oscillatory and slowly decaying nature of its integrand [7], the Discrete Complex Image Method has been introduced [12], [17] and [19]. Then, the Sommerfeld Identity can be used to cast the spatial domain Green’s function in a closed form. Though the DCIM definitely represents an improvement over the SI numerical evaluation, it can become relatively inefficient when the Green’s function has to be evaluated either at different frequencies, for different combinations of the two vertical coordinates z and z’ or for dielectric environments characterized by a large number of images. This is because the GPOF procedure must be carried out anew each time either the frequency or the combination of the vertical coordinates of the source and observation points are changed, and a new set of complex exponentials must be derived for each new combination of the above parameters. Recently, there have been some other significant contributions for the analysis of general printed structures at a fixed frequency and planar microstrip devices at multiple frequencies [48], [61]-[67].
To overcome these difficulties, the CBFM-EMA has been employed, which is well suited for a fast prototyping of the circuit dimensions and electrical parameters with a good approximation of the device response [68]. The CBFM-EMA replaces the stratified medium with a semi-infinite homogeneous grounded slab (Fig. 4.30) whose DGFs can be derived analytically.
The crucial step in the proposed procedure is an accurate evaluation of the effective dielectric constant εeff which can either be computed
analytically for some simple cases or be numerically evaluated for complex environments. For single-slab problems the effective dielectric constant
Figure 4.30. Equivalent grounded homogeneous medium.
( )
1 2 1 2 2 , , 1.5 1 0 , 2 1 10 1 1 1 12 1 2 2 1 1 1 12 1 2 2 1 4 4 1 1 1 2 log 1 2 0.04 1 r r r r r eff stat eff stat r eff stat eff t w Factor w h t w w h Factor F hf w c t w t f F ε ε ε ε ε ε ε ε ε ε − − − + − + + + ≤ + − + + > − + + − + + + = = − = = 2 , (4.7)where w represents the width of the microstrip, t the thickness of the dielectric slab and c0 the speed of light in vacuum.
The Green’s function for this equivalent medium, backed by a ground plane, can be determined analytically simply by using the image theory (Fig. 4.31). A comparison between the DGFs scalar potential term, computed analytically and numerically, is reported in Fig. 4.32. The results in this figure are relative to a semi-infinite grounded dielectric medium, with an effective dielectric constant equal to 2.0. The source is located at a height t = 1.0 mm above the ground plane.
The basic CBFM-EMA concept consists on evaluating both the CBFs and the unknown expansion coefficients in an equivalent homogeneous domain, employing the effective dielectric constant of the source section. The proposed approach leads to a considerable time-saving when compared to the conventional approach. This can be accomplished thanks to the analytical nature of the equivalent Green’s Function which is composed only of the direct ray and a reflected one due to the ground plane image for this case. The CBFM-EMA technique, despite the fact it is approximate, has proven to yield quite accurate results since large variations in the ratio among the metal trace width have relatively small effect on the effective dielectric constant.
Figure 4.31. Real and image source in the equivalent medium.
Once this first step has produced the initial design, a more accurate approach, the CBFM-SP, is involved for a more accurate analysis. Even if the CBFM-SP is faster than classical techniques and suitable for analyzing problems with a large number of unknowns, the use of the CBFM-EMA has to be preferred for the preliminary circuit prototyping. The CBFM-SP employs the same approach exploited in the first step to evaluate the CBFs but uses the rigorous spectral domain DGFs (which can be evaluated analytically) to determine the expansion coefficients as reported in (4.8):
( )
2(
) (
)
, 1 , , , 2 x y ij i x y j x y x y k k Z J k k E k k dk dk π =∫∫
⋅ (4.8)where the Zij is the generic reduced matrix element in the wave-number
domain, Ji and Ej represent the Fourier transform of the i
th CBF and of
the electric field generated by the jth CBF, respectively.
10-2 10-1 1 10 102 103 104 105 -20 -10 0 10 10-3 10-2 10-1 1 10 Analytical DGFs Numerical DGFs Analytical DGFs Numerical DGFs M a gni tu d e( K φ ) P has e( K φ ) ρ(mm) Phase Magnitude
Figure 4.32. Comparison between the analytical and numerical DGF scalar term for the homogeneous grounded medium.
The computation of the integral (4.8) can be quickly accomplished since the CBFs are high level basis functions defined on a wide spatial domain leading to a confined transform that can be determined via a FFT algorithm. An example is reported in Figs. 4.34-4.37 for a Primary and a Secondary CBF of the stepped filter reported in Fig. 4.33. It is important to underline that for the CBF case the basis functions can be adapted to each block while for the MoM case, the entire domain basis functions do not provide the same degree of freedom. In order to reduce the integration interval, the Green’s Function direct ray and the ground plane image have been extracted and integrated in the spatial domain as reported in (4.9) for the case of a single dielectric slab (Figs. 4.38):
( ) ( )
(
2)
Reduced 1 z , jk t static z F k F k e k ρ = ρ −α − − (4.9)where F k
( )
ρ can either be the vector or the scalar potential DGFs,( )
Reduced
coefficient associated to the direct ray between the source and the observation points, kzi is the z-component of the wavenumber, t is the
thickness of the dielectric slab.
Figure 4.33. Geometry of the stepped filter.
Figure 4.34. Example of Primary CBF spatial current density distribution for the previous stepped filter.
Figure 4.35. Example of Primary CBF spectral current density distribution relative to the example in Fig. 4.33.
Figure 4.36. Example of Secondary CBF spatial current density distribution for the previous stepped filter.
Figure 4.37. Example of Secondary CBF spectral current density distribution relative to the example in Fig. 4.33.
10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 -60 -30 0 30 60 Kφ Kφ Reduced M agni tu de Kρ/K 0
It is worthwhile to point out that a set of basis obtained via the CBFM-EMA approach, which is obviously approximate, can still be employed to accurately generate the final circuit solution by using the Spectral Domain CBFM, which is a rigorous method.
4.5. Numerical Results
In this section some examples comparing the numerical results obtained with the classical MoM, the CBFM-EMA and the Spectral Domain CBFM are presented to illustrate the accuracy and numerical efficiency of the proposed method at each step. First a two-stub filter problem has been analyzed (Fig. 4.39). The results for the S-parameters within the frequency range 2.5 - 5.5 GHz are shown in Figs. 4.40-4.43.
We note that the S-parameter plots, derived by using the CBFM-EMA and the direct MoM solution are in good agreement but they are further refined by the CBFM-SP, both in magnitude and phase.
-60 -50 -40 -30 -20 -10 0 -60 -50 -40 -30 -20 -10 0 2,5 3 3,5 4 4,5 5 5,5 CBMoM-EMA MoM CBFM-EMA MoM S 11 (d B ) 21 S (d B ) Frequency(GHz) S 21 S 11
Figure 4.40. Comparison between the S-parameter magnitude of the two-stub filter calculated by using the CBFM-EMA and the MoM approach.
-180 -120 -60 0 60 120 180 -180 -120 -60 0 60 120 180 2,5 3 3,5 4 4,5 5 5,5 CBMoM-EMA MoM CBFM-EMA MoM P h as e( S 11 ) Phas e( S 21 ) Frequency(GHz) S 11 S 21
-60 -50 -40 -30 -20 -10 0 -60 -50 -40 -30 -20 -10 0 2,5 3 3,5 4 4,5 5 5,5 CBMoM-SP MoM CBFM-SP MoM S 11 (d B ) 21 S (d B ) Frequency(GHz) S 21 S 11
Figure 4.42. Comparison between the S-parameter magnitude of the two-stub filter calculated by using the CBFM-SP and the MoM approach.
-180 -120 -60 0 60 120 180 -180 -120 -60 0 60 120 180 2,5 3 3,5 4 4,5 5 5,5 CBMoM-SP MoM CBFM-SP MoM P h as e( S 11 ) Phas e( S 21 ) Frequency(GHz) S 11 S 21
Figure 4.43. Comparison between the S-parameter phase of the two-stub filter calculated by using the CBFM-SP and the MoM approach.
The next test example comprises a one-stub filter (Fig. 4.44). The corresponding S-parameters for this test case are illustrated in Figs. 4.45-4.58 from 3.0 to 5.0 GHz.
Figure 4.44. Geometry of a one-stub microstrip filter.
The final example addresses a bandpass stepped filter (Fig. 4.49). The S-parameters calculated from 3.0 GHz to 5.0 GHz are reported in Figs. 4.50-4.53.
It is evident that both steps generate good results even if the CBFM-SP is more accurate at frequencies near the resonances and for microstrip structures characterized by different trace widths.
In order to provide a measure of the computational efficiency of the proposed approaches, the Relative Time (RT) has been defined as the ratio between the CPU time required by the CBFM-EMA, or the CBFM-SP, and that needed to generate the results by using the conventional CBFM, which
-50 -40 -30 -20 -10 0 -50 -40 -30 -20 -10 0 3 3,5 4 4,5 5 CBMoM-EMA MoM CBFM-EMA MoM S 11 (d B ) 21 S (d B ) Frequency(GHz) S 21 S11
Figure 4.45. Comparison between the S-parameter magnitude for the one-stub filter calculated by using the CBFM-EMA and the MoM approach.
-180 -120 -60 0 60 120 180 -180 -120 -60 0 60 120 180 3 3,5 4 4,5 5 CBMoM-EMA MoM CBFM-EMA MoM P h as e( S 11 ) Phas e( S 21 ) Frequency(GHz) S 21 S 11
Figure 4.46. Comparison between the S-parameter phase for the one-stub filter calculated by using the CBFM-EMA and the MoM approach.
-50 -40 -30 -20 -10 0 -50 -40 -30 -20 -10 0 3 3,5 4 4,5 5 CBMoM-SP MoM CBFM-SP MoM S 11 (d B ) 21 S (d B ) Frequency(GHz) S 21 S11
Figure 4.47. Comparison between the S-parameter magnitude for the one-stub filter evaluated by using the CBFM-SP and the MoM approach.
-180 -120 -60 0 60 120 180 -180 -120 -60 0 60 120 180 3 3,5 4 4,5 5 CBMoM-SP MoM CBFM-SP MoM P h as e( S 11 ) Phas e( S 21 ) Frequency(GHz) S 21 S 11
Figure 4.49. Geometry of the proposed bandpass stepped filter. -30 -20 -10 0 -30 -20 -10 0 4 4,5 5 5,5 6 CBMoM-EMA MoM CBFM-EMA MoM S 11 (d B ) 21 S (d B ) Frequency(GHz) -40 -40 S 21 S 11
Figure 4.50. Comparison between the S-parameter magnitude for the stepped filter calculated by using the CBFM-EMA and the MoM approach.
-180 -120 -60 0 60 120 180 -180 -120 -60 0 60 120 180 4 4,5 5 5,5 6 CBMoM-EMA MoM CBFM-EMA MoM P h as e( S 11 ) Phas e( S 21 ) Frequency(GHz) S 21 S 11
Figure 4.51. Comparison between the S-parameter phase for the stepped filter calculated by using the CBFM-EMA and MoM approach.
-30 -20 -10 0 -30 -20 -10 0 4 4,5 5 5,5 6 CBMoM-SP MoM CBFM-SP MoM S 11 (d B ) S 21 (d B ) Frequency (GHz) -40 S 21 S 11 -40
-180 -120 -60 0 60 120 180 -180 -120 -60 0 60 120 180 4 4,5 5 5,5 6 CBMoM-SP MoM CBFM-SP MoM P h as e( S 11 ) Phas e( S 21 ) Frequency (GHz) S 21 S 11
Figure 4.53. Comparison between the S-parameter phase for the stepped filter evaluated by using the CBFM-SP and the MoM approach.
Table 4.2. Relative Time for the microstrip analyzed problems.
Approach Example 1 Example 2 Example 3
CBFM-EMA 50.82 53.98 47.72
CBFM-SP 81.89 85.73 86.79
For the two-stub filter example, then, the CBFM-SP time performance has been analyzed as the highest frequency of interest increases (Fig. 4.54).
In particular, Total refers to the ratio between the CBFM-SP total solving time and the conventional CBFM. Impedance Matrix refers to the ratio among the CBFM-SP impedance matrix filling time and the CBFM counterpart.
As it is apparent, the CBFM-SP solution time is larger with respect to the conventional CBFM only when a small number of unknowns are involved. For larger problems, the CPU time necessary to fill the reduced matrix becomes negligible in comparison to that required to build the MoM
impedance matrix and this, in turn, leads to a considerable time-saving with respect to the conventional approach.
70 75 80 85 90 95 100 105 2,5 3,5 4,5 5,5 6,5 7,5 8,5 Total Impedance Matrix Re la tiv e T im e (% ) Frequency(GHz)
Figure 4.54. Relative Time for the CBFM-SP approach as the highest frequency of interest changes and consequently the number of unknowns for the two-stub filter problem increases.
