3. A FREQUENCY INTERPOLATION ALGORITHM FOR PLANAR MICROSTRIP STRUCTURES 3.1. Introduction

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3. A FREQUENCY INTERPOLATION ALGORITHM FOR

PLANAR MICROSTRIP STRUCTURES

3.1. Introduction

The Method of Moments is usually the numerical procedure adopted for the analysis of microstrip circuits and antennas etched in layered media. The MoM approach, in fact, reduces the problem domain only to regions where the surface currents are defined lowering significantly the total number of unknowns when compared to Finite Methods [1], [21], [35]-[38]. Even so, increasing the problem electrical dimensions, the impedance matrix computation time consumes a considerable part of the total one and must be repeated at each frequency, leading to evident numerical inefficiencies. One promising approach to speed up the MoM solution time is the impedance matrix interpolation over the frequency range of interest which was first proposed by Newman and Forrai for the scattering analysis of microstrip patch, straight dipole antennas and flat square plates [39]-[40]. Virga and Rahmat-Samii have also applied a similar technique to analyze the electromagnetic behavior of complex antennas designed for personal communication applications and to predict the response of a Frequency Selective Surface (FSS) [41]-[44]. In this chapter a novel interpolation algorithm, based on the Green’s function frequency variation, will be presented in order to predict the MoM impedance matrix behavior changing the operating frequency. The proposed approach relies on cubic spline polynomials and extracts analytically the Green’s function singular behavior when the distance between source and observation point is extremely small. To prove the accuracy and the numerical efficiency of the presented scheme, the algorithm has been applied to some common planar microstrip problems like an edge coupled microstrip filter, a stepped filter and a three-ring resonator problem.

3.2. Spline Fitting Approach

In order to understand the MoM impedance matrix frequency behavior, let us consider the formulation proposed in the previous chapters. Each MoM matrix element can be separated into two terms: the contribution of the magnetic vector potential A related to the magnetic vector Green’s function A

G and the contribution of the electric scalar potential φ related

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frequency variation is intimately related to the Green’s function one allowing us to postulate a proper fitting scheme for the matrix element interpolation procedure. The behavior of the Green’s function for a planar microstrip structure has been investigated by many researchers [45]-[47]. It has been demonstrated that the vector Green’s function is slightly related to the operating frequency while the corresponding scalar term variation is determined by the distance between source and observation point. In the near region, the scalar Green’s function varies little with the frequency, while its magnitude is proportional to f2 and f4 in the intermediate and far zone respectively. For these reasons, is useful to employ different fitting schemes depending on the distance between source and testing functions.

When dealing with the free-space Green’s function, the normalized distance k0r has been used to distinguish its behavior, where k0 is the propagation constant while r is the distance between the source and field point [39]-[47]. However in [48], has been shown that, for the analysis of microstrip devices, k0r must be replaced with keρ where ke is the effective wavenumber of the microstrip structure and ρ is the radial distance in a cylindrical reference system. Mittra etc. have identified the following distance criteria on the basis of extensive numerical experiments with microstrip structures: ρ = 0.45 λe, where λe is the effective wavelength, for the transition between near and intermediate region and ρ = 0.75 λe for moving in the far zone. A microstrip line, shown in Fig. 3.1, has been considered to illustrate the frequency behavior of the Green’s function and of the related MoM impedance matrix.

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Figures 3.2-3.7 show a typical DGFs and MoM matrix behavior as a function of the operating frequency ranging from 3.0 GHz to 5.0 GHz. The radial distance ρ between source and observation point has been progressively increased considering the near region case (ρ = 1.0 µm), the intermediate region case (ρ = 50.0 mm) and the far region one (ρ = 100.0 mm) respectively. 7,5 104 7,75 104 8 104 8,25 104 8,5 104 -0,02 -0,01 0 0,01 3 3,5 4 4,5 5 ρ = 1.0 µm ρ = 50.0 mm ρ = 100.0 mm Re a l( G x x ) Re a l( G x x ₎ Frequency(GHz)

Figure 3.2. Frequency variation of the vector potential DGFs real part.

4,5 104 4,75 104 5 104 5,25 104 5,5 104 -0,04 -0,02 0 0,02 3 3,5 4 4,5 5 ρ = 1.0µm ρ = 50.0mm ρ = 100.0mm Re a l( K φ ) Re a l( K φ ) Frequency(GHz)

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-0,2 -0,15 -0,1 -0,05 0 0,05 3 3,5 4 4,5 5 ρ = 1.0 µm ρ = 50.0 mm ρ = 100.0 mm Im ag( G x x ) Frequency(GHz)

Figure 3.4. Frequency variation of the vector potential DGFs imaginary part.

-0,05 0 0,05 0,1 0,15 3 3,5 4 4,5 5 ρ = 1.0 µm ρ = 50.0 mm ρ = 100.0 mm Im ag( K φ ) Frequency(GHz)

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-1 10-6 0 1 10-6 2 10-6 3 10-6 3 3,5 4 4,5 5 ρ = 0.0 mm ρ = 50.0 mm ρ = 100.0 mm Re a l( Z ) Frequency(GHz)

Figure 3.6. Real part of an impedance matrix element for different radial

distances. -0,02 -0,015 -0,01 -0,005 -1 10-7 -5 10-8 0 5 10-8 1 10-7 1,5 10-7 3 3,5 4 4,5 5 ρ = 0.0 mm ρ = 50.0 mm ρ = 100.0 mm Im ag( Z ) Ima g (Z ) Frequency(GHz)

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The current distribution on the microstrip line has been modeled by using fifty RWG basis functions keeping constant the total number of unknowns on the entire frequency band. In the near region, the imaginary part of the impedance matrix element dominates on the real one showing a 1/f frequency behavior with a small phase variation. In the intermediate zone, the real part starts to rise and becomes comparable to the imaginary one in the last region where the phase presents rapid transitions with the frequency.

The proposed approach ensures a considerable time saving and accurately approximates the elements of the Dyadic Green’s Functions, as well as, the MoM impedance matrix elements, by directly computing only a small subset of the required samples within the entire range of interest. In particular the fitting scheme is based on cubic spline polynomials that employ a set of (n+1) known samples at frequencies {fn}:

( )

(

)

( )

(

)

( )

(

)

0 0 1 1 1 2 1 , , , , n n n S f f f f S f f f f S f S f f f f ₊  ∈  _∈  =    _∈   (3.1) where:

( )

3 0 1 0,1, , 1 1, 2 . 0,1, , 1 0,1, , i j i i i i j k j k i i S f f j n i S f S f j n f f k n α = − = = − = ∂ ∂ = = − ∂ ∂ =

∑

   (3.2)

Since the elements of the DGFs diverge when the distance between source and observation point is extremely small, the fitting scheme S(f) has been multiplied, in the near region, with the correction term fsingular(ρ,z,z’,f) reported below:

(

'

)

min 2 2 , , , , singular f z z f z λ ρ ρ = + ∆ (3.3)

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when approximating the magnitude of the Green’s function or the magnitude of the impedance matrix elements. In (3.3), λmin is the wavelength in the dielectric layer with the highest value of εr, while ρ and ∆z represent the horizontal and vertical distances between source and observation points. Moreover, this regularization of the function behavior avoids the need to deal with a large number of samples (Figs. 3.8-3.9).

At this point, the frequency step size has to be chosen. By using the effective wavenumber, the corresponding step size ∆ke = 2π/λe has been obtained. The fitting scheme requires that the interpolation step size does not introduce a phase change more than π, hence ∆keρ ≤ π [48]. From this reason, the maximum interpolation step size can be express as:

, 2 H M max e f f ρ λ ∆ =       (3.4)

where fH is the upper limit of the frequency band and ρmax is the maximum distance between source and testing functions.

3.3. Numerical Results

The spline fitting algorithm described above has been applied to several planar microstrip problems in order to prove its accuracy and numerical efficiency. The chosen examples are:

i. an edge coupled microstrip filter; ii. a stepped bandpass filter;

iii. a three-ring resonator filter.

In order to evaluate the algorithm performance, the parameters introduced in [48] have been employed:

( )

2 2 , i i i DGFs i i Fitting f DCIM f Normalized Error f DCIM f − =

∑

(3.5)

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( )

2 2 , Fitted Direct ij ij i j Z Direct ij i j Z f Z f Normalized Error f Z f − =

∑∑

(3.6)

( )

% 100*Fitting Time Sample Time ,

Relative Time

Direct Time

+

= _(3.7)

where Fittingi(f) is the Green’s function value obtained with the proposed approach for ρi, while the elements of DCIMi(f) are given by the direct DCIM calculation. Also ZijFitted and ZijDirect represent the impedance matrix element obtained by using the fitting algorithm and the direct procedure respectively. In particular, the Relative Time(%) has been defined as the ratio between the sum of the CPU fitting time and the CPU time required to obtain the input samples for the fitting procedure and the CPU time required by the direct approach. The performance of the spline algorithm has been compared to that described in [48], which has been taken here as the reference approach.

10-3 10-2 10-1 1 10 100 1000 104 105 10-3 10-2 10-1 1 10 102 DCIM f singular DCIM / f singular M agni tu de (G x x ) ρ(mm)

Figure 3.8. Regularization process applied to the Green’s function vector

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10-3 10-2 10-1 1 10 100 1000 104 105 10-3 10-2 10-1 1 10 102 DCIM f singular DCIM / f singular M agn it u d e( K φ ) ρ(mm)

Figure 3.9. Regularization process applied to the Green’s function scalar potential.

The reference algorithm has been applied both to the impedance matrix and directly to the Green’s function since the MoM matrix frequency variation is intimately related to that of the dyadic Green’s function [48].

3.3.1. Edge coupled microstrip filter

The first example analyzed with the cubic spline interpolation scheme is an edge coupled microstrip filter as shown in Fig. 3.10. The filter model at the highest frequency (13.0 GHz) requires 156 unknowns when a RWG meshing approach is applied and remains unchanged over the entire frequency range of interest. The filter is characterized by two resonance frequencies, the former around 9.5 GHz and the latter at 11.0 GHz. The fitting scheme has been implemented from 9.0 GHz to 13.0 GHz in order to evaluate both the DGFs and the MoM impedance matrix every 80.0 MHz The total number of employed samples is 6 and the entire frequency band has been divided into two equal sub-bands.

The results given by direct calculation and by the cubic spline frequency fitting applied both to the dyadic Green’s function and to the MoM impedance matrix are compared in Fig. 3.11-3.14 at 11.0 GHz.

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Figure 3.10. Geometry of the edge coupled microstrip filter. 10-1 1 10 102 103 104 105 -20 -15 -10 -5 0 10-3 10-2 10-1 1 10 Fitted Solution Direct Solution Fitted Solution Direct Solution M ag ni tud e( G x x ) P ha s e( G x x ₎ ρ(mm) Phase Magnitude

Figure 3.11. Comparison between the direct solution and the fitted results for the

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1 10 102 103 104 105 -10 0 10 20 30 40 10-3 10-2 10-1 1 10 Fitted Solution Direct Solution Fitted Solution Direct Solution M a gni tu d e( K φ ) P has e( K φ ) ρ(mm) Phase Magnitude

scalar potential DGFs at 11.0 GHz. 10-9 10-8 10-7 10-6 10-5 10-4 10-3 0 20 40 60 80 100 120 140 160 Fitted Solution Direct Solution M ag ni tu d e( Z ) Element

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-300 -200 -100 0 100 200 300 0 20 40 60 80 100 120 140 160 Fitted Solution Direct Solution P h as e( Z ) Element

impedance matrix 78th_{row phase.}

Figs. 3.15-3.17 show the DGFs normalized error and the impedance matrix normalized error for the analyzed device obtained by using the spline algorithm and the reference one.

The results for the scattering and impedance parameters are shown in Figs. 3.18-3.21. It is evident that the presented technique generates accurate results even at the frequencies near the resonances. The relative time (RT) for the DGFs and the impedance matrix calculation has been reported in Table 3.1 comparing the spline performance with the reference algorithm ones. The employed personal computer is an Intel Core 2 Duo with a 3.0 GHz CPU and 4 GB RAM.

Table 3.1. Relative Time for the spline algorithm and the reference one.

Algorithm _{Potential DGFs}Vector Scalar Potential _DGFs Impedance _Matrix Cubic Spline 21.81 19.51 22.49

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10-8 10-7 10-6 10-5 9 9,5 10 10,5 11 11,5 12 12,5 13 Spline Algorithm Reference Algorithm N o rm al iz ed E rr or DG F s (f) Frequency(GHz)

Figure 3.15. Green’s function vector potential normalized error for the spline

algorithm and the reference one.

10-6 10-5 10-4 10-3 10-2 10-1 9 9,5 10 10,5 11 11,5 12 12,5 13 Spline Algorithm Reference Algorithm N o rm al iz ed E rr or DG F s (f) Frequency(GHz)

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10-6 10-5 10-4 10-3 10-2 9 9,5 10 10,5 11 11,5 12 12,5 13 Spline Algorithm Reference Algorithm N o rm al iz ed E rr o r Z (f) Frequency(GHz)

Figure 3.17. Impedance matrix normalized error for the spline algorithm and the

reference one. -30 -25 -20 -15 -10 -5 0 -30 -25 -20 -15 -10 -5 0 9 10 11 12 13 Fitted Solution Direct Solution Fitted Solution Direct Solution S 11 (d B ) 21 S (d B) Frequency(GHz) S 21 S 11

Figure 3.18. Comparison of the S-parameter magnitude for the edge coupled

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-180 -120 -60 0 60 120 180 -180 -120 -60 0 60 120 180 9 10 11 12 13 Fitted Solution Direct Solution Fitted Solution Direct Solution P h as e( S 11 ) P_h as e( S 21 ) Frequency(GHz) S 21 S 11

Figure 3.19. Comparison of the S-parameter phase for the edge coupled

microstrip filter. 10 102 103 104 105 1 102 100 103 104 105 9 10 11 12 13 Fitted Solution Direct Solution Fitted Solution Direct Solution M ag ni tud e( Z 11 ) M ag ni tud e( Z 21 ) Frequency(GHz) Z 21 Z 11

Figure 3.20. Comparison of the Z-parameter magnitude for the coupled edge

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-180 -120 -60 0 60 120 180 -180 -120 -60 0 60 120 180 9 10 11 12 13 Fitted Solution Direct Solution Fitted Solution Direct Solution P h as e( Z 11 ) P h as e( Z 21 ) Frequency(GHz) Z 21 Z 11

Figure 3.21. Comparison of the Z-parameter phase for the edge coupled

microstrip filter.

3.3.2. Bandpass Stepped Filter

The second example considered is a bandpass stepped filter shown in Fig. 3.22. The total number of unknowns is 183 when the filter is modeled by using RWG basis functions.

The filter presents a resonance frequency around 5.38 GHz and the interpolation scheme has been implemented from 4.0 GHz to 6.0 GHz with a number of employed samples equal to 6. The results given by the cubic spline frequency fitting applied both to the dyadic Green’s function and to the MoM impedance matrix are reported in Fig. 3.23-3.26 at 5.0 GHz.

Figs. 3.27-3.33 show the DGFs normalized error, the impedance matrix normalized error, the results for the scattering and impedance parameters obtained by using the spline fitting approach and the direct calculation respectively.

It is worthwhile to mention that the presented technique generates accurate results on the entire frequency band.

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3.22. Geometry of the stepped filter. 0,1 1 10 102 103 104 105 -20 -10 0 10 10-3 10-2 10-1 1 10 Fitted Solution Direct Solution Fitted Solution Direct Solution M ag ni tud e( G x x ) P h as e( G x x ₎ ρ(mm) Phase Magnitude

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1 10 102 103 104 105 -20 -10 0 10 10-3 10-2 10-1 1 10 Fitted Solution Direct Solution Fitted Solution Direct Solution M a gni tu d e( K φ ) _P has e( K φ ) ρ(mm) Phase Magnitude

scalar potential DGFs at 5.0 GHz. 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 0 50 100 150 200 Fitted Solution Direct Solution M ag ni tu d e( Z ) Element

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-1500 -1000 -500 0 500 0 50 100 150 200 Fitted Solution Direct Solution P h as e( Z ) Element

impedance matrix 92th_{row phase.}

10-9 10-8 10-7 10-6 10-5 4 4,5 5 5,5 6 Spline Algorithm Reference Algorithm N o rm al iz ed E rr or DG F s (f) Frequency(GHz)

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10-7 10-6 10-5 10-4 10-3 10-2 4 4,5 5 5,5 6 Spline Algorithm Reference Algorithm N o rm al iz ed E rr or DG F s (f) Frequency(GHz)

Figure 3.28. Green’s function scalar potential normalized error for the spline

10-7 10-6 10-5 10-4 10-3 10-2 4 4,5 5 5,5 6 Spline Algorithm Reference Algorithm N o rm al iz ed E rr or Z (f) Frequency(GHz)

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-40 -30 -20 -10 0 -40 -30 -20 -10 0 4 4,5 5 5,5 6 Fitted Solution Direct Solution Fitted Solution Direct Solution S 11 (d B ) ₂₁ S (d B) Frequency(GHz) S 21 S 11

Figure 3.30. Comparison of the S-parameter magnitude for the stepped filter.

-180 -120 -60 0 60 120 180 -180 -120 -60 0 60 120 180 4 4,5 5 5,5 6 Fitted Solution Direct Solution Fitted Solution Direct Solution P h as e( S 11 ) P_h as e( S 21 ) Frequency(GHz) S 21 S 11

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10-1 1 10 102 103 104 1 10 102 103 104 4 4,5 5 5,5 6 Fitted Solution Direct Solution Fitted Solution Direct Solution M ag ni tud e( Z 11 ) M ag ni tud e( Z 21 ) Frequency(GHz) Z 21 Z 11

Figure 3.32. Comparison of the Z-parameter magnitude for the stepped filter.

-100 -50 0 50 100 -180 -120 -60 0 60 120 180 4 4,5 5 5,5 6 Fitted Solution Direct Solution Fitted Solution Direct Solution P h as e( Z 11 ) P_h as e( Z 21 ) Frequency(GHz) Z 21 Z 11

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The relative time (RT) for the DGFs and the impedance matrix calculation has been reported in Table 3.2 both for the spline algorithm and the reference one.

Algorithm _{Potential DGFs}Vector Scalar Potential _DGFs Impedance _Matrix Cubic Spline 23.17 20.50 20.40

Reference 35.37 32.26 32.82

3.3.3. Three-Ring Resonator Filter

Fig. 3.34 shows a filter composed by three ring resonators which is the next candidate in order to validate the cubic spline performance.

Figure 3.34. Geometry of the three-ring resonator filter.

For this structure, the substrate is characterized by a relative dielectric constant and a thickness equal to 2.2 and 0.5 mm respectively. The filter is

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with 212 RWG basis functions at the highest frequency of interest. The interpolation scheme is implemented from 10.0 GHz to 15.0 GHz with 6 input frequency samples. Figs. 3.35-3.38 show the comparison between the direct calculation and the spline fitting algorithm applied to the DGFs and of the MoM impedance matrix. The relative time (RT) for the DGFs and the impedance matrix calculation has been reported in Table 3.3 both the spline algorithm and the reference one.

The DGFs error, the impedance matrix error and the frequency filter behavior have been shown in Figs. 3.39-3.45.

Algorithm _{Potential DGFs}Vector Scalar Potential _DGFs Impedance _Matrix Cubic Spline 21.23 19.34 20.64 Reference 33.10 29.02 33.59 0,1 1 10 102 103 104 105 -30 -20 -10 0 10 10-3 10-2 10-1 1 10 Fitted Solution Direct Solution Fitted Solution Direct Solution M ag ni tu d e( G x x ) P h as e( G x x ₎ ρ(mm) Phase Magnitude

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103 104 105 -10 -5 0 5 10 15 20 10-3 10-2 10-1 1 10 Fitted Solution Direct Solution Fitted Solution Direct Solution M a gni tu d e( K φ ) _P has e( K φ ) ρ(mm) Phase Magnitude

scalar potential DGFs at 12.5 GHz. 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 0 50 100 150 200 250 Fitted Solution Direct Solution M ag ni tu d e( Z ) Element

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-1500 -1000 -500 0 500 1000 1500 0 50 100 150 200 250 Fitted Solution Direct Solution P h as e( Z ) Element

impedance matrix 106th row phase.

10-10 10-9 10-8 10-7 10-6 10-5 10 11 12 13 14 15 Spline Algorithm Reference Algorithm N o rm al iz ed E rr or DG F s (f) Frequency(GHz)

Figure 3.39. Green’s function vector potential normalized error for the spline

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10-3 10-2 10-1 1 10 11 12 13 14 15 Spline Algorithm Reference Algorithm N o rm al iz ed E rr or DG F s (f) Frequency(GHz)

Figure 3.40. Green’s function scalar potential normalized error for the spline

10-8 10-7 10-6 10-5 10-4 10-3 10-2 10 11 12 13 14 15 Spline Algorithm Reference Algorithm N o rm al iz ed E rr or Z (f) Frequency(GHz)

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-40 -30 -20 -10 0 -40 -30 -20 -10 0 10 11 12 13 14 15 Fitted Solution Direct Solution Fitted Solution Direct Solution S 11 (d B) 21 S (d B ) Frequency(GHz) S 21 S 11

Figure 3.42. Comparison of the S-parameter magnitude for the three-ring

resonator filter. -180 -120 -60 0 60 120 180 -180 -120 -60 0 60 120 180 10 11 12 13 14 15 Fitted Solution Direct Solution Fitted Solution Direct Solution P h as e( S 11 ) P_h as e( S 21 ) Frequency(GHz) S 21 S 11

Figure 3.43. Comparison of the S-parameter phase for the three-ring resonator

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0,1 1 10 100 1000 104 10 100 1000 104 10 11 12 13 14 15 Fitted Solution Direct Solution Fitted Solution Direct Solution M ag ni tu d e( Z 11 ) M_ag ni tu d e( Z 21 ) Frequency(GHz) Z 21 Z 11

Figure 3.44. Comparison of the Z-parameter magnitude for the three-ring

resonator filter. -180 -120 -60 0 60 120 -180 -120 -60 0 60 120 180 10 11 12 13 14 15 Fitted Solution Direct Solution Fitted Solution Direct Solution P h as e( Z 11 ) P_h as e( Z 21 ) Frequency(GHz) Z 21 Z 11

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