# Characterization of Microstrip Ring with a Narrow Gap  by an Iterative Method

Authors: M. Yeddes; A. Gharsallah; A. Gharbi; H. Baudrand
Journal: International Journal of Microwave Science and Technology
 (2008)

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## Abstract

This paper presents an implementation of an iterative method based on the wave's concept, for analysing a ring resonator. This method includes a two-dimensional fourier modal transformation (2D-FMT) in a wave guide environment. The method has the advantage of simplicity in that it does not involve basis functions and inversion of matrices, as used in other calculation methods. Therefore, it is capable of analyzing larger bodies than other classical techniques. An implementation of the iterative calculation is shown for the extraction ofS parameters of microwave components. The good agreement between simulation results and experimental published data justifies the design procedure and validates the present analysis approach.


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## Body

## 1. Introduction

In the Photonic Band Gap structures (PBG), an artificial lattice of dielectric can give rise to frequency regions where photons cannot propagate in any direction.  Microwave circuits with a PBG structure are difficult to implement in compact form because such a structure requires at least five or six periods of units cells, and the size of one period is directly proportional to the wavelength of the center frequency of the PBG. In this paper, we show that a simple microstrip ring with a narrow gap can exhibit PBG characteristics.An implementation of the iterative calculation is shown for extraction ofSij parameters of a ring resonator. It consists in generating a recursive relationship between a given wave source and reflected waves at the interface containing the circuit. This discontinuity plane is divided into cells and characterized by a scattering operator matrix depending on boundary conditions. Then, a 2D-FMT algorithm is used to pass from spatial to spectral domain for each iteration of the recursive process [1]. The regular ring shows S parameters similar to those of a conventional transmission line; the introduction of a small gap in this regular ring induces two stop bands which can be tuned by a varactor incorporated into the narrow gap; this technique is detailed in the second part of this paper.


## 2. Formulation

The wave concept is introduced by writing the tangential electric fieldEi and surface tangential current density Ji in terms of incident and reflected waves (see Figure 1) [1, 2]. It leads to the following set of equations:
(1)Ai=12Zoi(Ei+ZoiJi),i=1,2,Bi=12Zoi(Ei-ZoiJi),i=1,2,
where J⇀i is the surface tangential current density as H→i⋀n→i, with ni a unit vector normal to the interface Ω; Ai and Bi are the incident and reflected waves associated with the discontinuity interface Ω; Z0i is the characteristic impedance of the middle i, given by Z0i=μ0/ε0εri, where εri, is the relative permittivity
of the region I; Ei and  Hi are the tangential electric field and magnetic field, respectively, on the plan Ω.Figure 1
Definition of wave concept.FMT: fourier modal transform.On each region (region 1, 2), it is possible to definite a scattering
matrix on spatial domain and reflection coefficient on spectral domain.The interfaceΩ (Spatial domain) in which the circuit is defined is divided into pixels [3]; the interface contains three subdomains (dielectric, metal, and source).By using the boundary condition in each subdomain (dielectric, metal,
and source), it is possible to define the scattering matrixSM, who binds the incident waves to the reflected waves on the spatial domain [1, 3]:
(2)SM=(Q′Q′′Q′′′Q′′′′),
where   Q′=−Hm−((−1+n1+n2)/(1+n1+n2))Hs+((1−N2)/(1+N2))Hd, Q′′=(2N/(1+N2))Hd+(2n/(1+n1+n2))Hs, Q′′′=(2N/(1+N2))Hd+(2n/(1+n1+n2))Hs, and Q′′′′=−Hm−((−1−n1+n2)/(1+n1+n2))Hs+((1−N2)/(1+N2))Hd,   where
(3)N=Z01Z02,n=Z0Z01Z02,n1=Z0Z01,n2=Z0Z02,(4)Hj={1on the sub domainj,j=source, dielectric, metal.0elsewhere,j=source, dielectric, metal.The incident waves are reflected on the upper  and lower parts of the
metallic box. So the relationship between waves (Aiα and Biα(α=TE, TM)) in spectral domain is [1, 3]
(5)[BiTEBiTM]=[ΓiTE00ΓiTM][AiTEAiTM],
where
(6)Γiα=1-Z0iYmn,iαCoth(γmn,ihi)1+Z0iYmn,iαCoth(γmn,ihi),Ymn,iTE=γmn,ijωε0μ0,Ymn,iTM=jωε0μ0γmn,i,γmn,i=(mπa)2+(nπb)2−k02εri.hi: the I middle height.This approach could then be considered as alternative between spatial
and spectral domains. From the passage spatial to spectral domain,  we use a 2D-FMT algorithm in wave guide environment
(see AppendixB).The iterative process continues to the nth iteration, when the electric
field and the current density converge; the iterative process is stopped. It is possible to calculateEi and Ji at the interface Ω:
(7)Ei=Zoi(Ai+Bi),Ji=1Zoi(Ai−Bi).Finally, it is possible to determine the admittance matrix, from that, the scattering matrixSij between the  two ports can be obtained by the following equation:
(8)[S]=[1−[Y]][1+[Y]]−1,
[Y] is the admittance matrix.The algorithm of the  iterative process is given by Figure2.Figure 2
Iterative processes algorithm.Nx and Ny: the numbers of pixels in the plan Ω.Am,n and Bm,n: incident wave and reflected wave, respectively, in the spectral domain.Ax,y and Bx,y: incident wave and reflected wave, respectively, in the spatial domain.Figure3 shows the S12 convergence at 5 GHz for the structure given by Figure 4.Figure 3
S12 versus iterations number.Figure 4
Ring resonator geometry.We can see that the iterative process converges at 350 iterations. So we can calculateS12 versus frequency, using at each frequency the same number of the iterations.


## 3. Results and Illustrations

Figure5 shows the simulated and measured spectra of transmitted microwave (S12) of the microstip ring with the directly connected feedline.Figure 5
Transmission coefficient of the ring resonator.The geometry and dimensions of the regular ring are given by Figure4; it is fabricated on a RT/Duriod 6010 substrate.A comparison between the simulation results and measured results [4] is done; a good agreement is observed. The mismatch between the simulation and measurement data is inferior to 6 dB for the majority of frequencies. We note that the regular ring shows S12 parameter similar to those of conventional transmission line.The introduction  of a small gap in the closed loop of the regular ring induces strong attenuations around 6.5 GHz and 20 GHz corresponding ton=1 and n=3 in (9):
(9)fn=nc2πRεeff,
where 2πR=nλ and εeff=6.77 are given by simulation by [4], c is speed of light and λ is wavelength.The new geometry of the ring is given by Figure6; the dimension of the gap is g=0.2 mm. This structure induces multiple reflections with a fixed phase correlation.Figure 6
Ring with gap.The measured and simulated reflected coefficients(S11) are illustrated
in Figure 7, and Figure 8 shows the transmitted coefficient (S21). We note the absence of the stop band around 13 GHz correspond to n=2 in (9), because the microwave propagation along the ring is successively reflected with the fixed
phase correlation when they satisfy 2πR−G=nλ, where G is the gap dimension.Figure 7
Reflection coefficientS11 of the microstrip ring with a narrow gap.Figure 8
Transmission coefficientS12 of the ring with a narrow gap.We see a satisfactory argument between, respectively, measured [4] and simulated reflected coefficient S11 and transmitted coefficient S12.In the frequency range from 12 GHz to 17 GHz,
the measurement data for(S11) show distinct standing wave behavior. This does not have any physical interest. They are disturbances because of the complexity of the structure.Because a ring resonator can be represented by a parallel LC circuit of inductanceL and capacitance C, introduction of a narrow gap in the ring slightly changes the values of C and L from the original ones.  The value of C depends on the gap width G, the line width w, and the circumference 2πR, while the value of L depends on the value of w and 2πR.Thus the attenuations valleys around 6.5 GHz and 20 GHz, witch were explained to be due to multiple reflections occurring at the
gap, correspond to the resonance frequencies of the equivalent LC circuit of the ring.If we change the values ofG, we can change the capacitance C, consequently the resonance frequency corresponding to ω=1/LC takes another values.Figure9 shows the simulated transmitted coefficient (S12) of different values of G; we can observe the shift of the two attenuations valleys.Figure 9
Transmission characteristics of the ring resonator for different value ofG.Another method can be used for varying the capacitanceC; we can mount a varactor diode at the gap of the ring (see Figure 10), when a varactor is connected parallel to the gap, the total capacitance at the discontinuity is increased to the sum of the varactor capacitance and the gap capacitance.Figure 10
Ring with varactor diode.In simulation, the varactor diode is considered an auxiliary source that is used to study in a disjoined way two distinct parts of a circuit [5].The expression of the density  of current according to electric field for the two principal’s source and the auxiliary source will be according to following equations:(10a)J1=Y11E1+Y12E2+Y13E3,(10b)J2=Y21E1+Y22E2+Y23E3,(10c)J3=Y31E1+Y32E2+Y33E3.(J1,E1) and (J2,E2): density of current and electric field in the tow principal’s source.(J3,E3): density of current and electric field in the auxiliary source.Yij are the coupling admittance between different sources determinate with the iterative method.Figure11 shows the equivalent circuit of the varactor diode.Figure 11
Equivalent circuit of the varactor diode.According to this circuit, we can deduce the following relation:(11)ID=(jωCj(0)jωCj(Rs+jωLp)+1−(VD/ϕ)+jωCp)VD.
With VL=E3G and IL=J3W, W and G are
the diode dimensions.IfW=G, we obtained
(12)J3=(jωCj(0)jωCj(Rs+jωLp)+1−E3/ϕ+jωCp)E3.
The value of E3/ϕ tends towards zero, therefore we will have
(13)J3=(jωCj(0)jωCj(Rs+jωLp)+(1−E3/2ϕ)+jωCp)E3.
Using (13) and (10b), we have
(14)Y31E1+Y32E2=(jωCp−Y33)E3+jωCjE3jωCj(Rs+jωLp)+(1−E3/2ϕ).
Well,
(15)jωCjE3jωCj(Rs+jωLp)+(1−E3/2ϕ)≈jωCjE3jωCj(Rs+jωLp)−1.
Consequently, (14) becomes
(16)Y31E1+Y32E2=(jωCp−Y33+jωCjjωCj(Rs+jωLp)−1)E3⇒E3=Y31E1+Y32E2jωCp−Y33+jωCj/(jωCj(Rs+jωLp)−1).
Replacing E3 in (10a) and (10b), we obtain
(17)J1=(Y11+Y13K1)E1+(Y12+Y13K2)E2,J2=(Y21+Y23K1)E1+(Y22+Y23K2)E2.
With
(18)K1=Y31jωCp−Y33+jωCj/(jωCj(Rs+jωLp)−1),K2=Y32jωCp−Y33+jωCj/(jωCj(Rs+jωLp)−1).
The admittance matrix of the ring with varactor diode is given by (19)
(19)Y=[Y11+Y13K1Y12+Y13K2Y21+Y23K1Y22+Y23K2].The simulated results for the transmission coefficient of ring with a narrow gap and the ring with the varactor mounted at the gap are shown in Figure12; we can see the new
valley obtained with the varactor.Figure 12
Transmission coefficient with and without varactor diode.


## 4. Conclusion

In this paper, we have proposed a microstrip ring that exhibits PBG characteristics. The stop band of the ring resonator is caused by multiple reflections with a fixed phase correlation at the gap; we saw different technique for the control of those stopped bands; the results are
determined by using an iterative technique. The comparison of numerical results with the measurement published data verified the validation of the proposed technique. This technique takes the advantage of simplicity and it is in conjunction with the 2D-FMT which allows a high computational speed and memory consumption.



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*Source: 538216-2008-04-16.xml*