A) Spatial operator

In the propagation medium (dielectric layer), the metallic via-hole is directed along the z-direction. We define the incident A _z and reflected B _z waves as a linear combination of electric field E _z and the current density vector J _z , (1).

(1) $$\left\{ {\matrix{ {{A_z} = \displaystyle{1 \over {2\sqrt {{z_0}}}} \left( {{E_z} + {z_0}{J_z}} \right),} \cr {{B_z} = \displaystyle{1 \over {2\sqrt {{z_0}}}} \left( {{E_z} - {z_0}{J_z}} \right),} \cr}} \right.$$

where z ₀ is the characteristic impedance of the medium.

The discretization process for a SIW cavity example is illustrated in Fig. 2, we distingue the three sub-domains: sources cells, metallic via-hole cell, and without-metallic-via-hole cell, where H _δ are their projections matrices, are defined as follows:

$$\eqalign{{\left[ {{H_\delta}} \right]_{{N_x},{N_y}}} & = \left\{ {\matrix{ 1 & {\hbox{on the sub-domain }\! "\delta\," } \cr {\rm 0} & {{\rm elsewhere}} \cr}} \right.;\quad \cr \delta & = \left\{ {\matrix{ s & {\hbox{ for the source sub-domain,}} \cr v & {\hbox{for the metallic via-hole sub-domain,}} \cr {nv} & {\hbox {for the without-via-hole sub-domain,}} \cr}} \right.} $$

where N _x is the number of cells according to the x-direction, and N _y is the number of cells according to the y-direction.

Fig. 2.

Discretization of the spatial domain of a substrate-integrated cavity.

The boundary conditions for each point of the structure are presented by a spatial coefficient.

The spatial operator Ω is used to calculate the reflected waves from the incident waves in the spatial domain, as in (2). Its calculation requires the application of the appropriate boundary conditions.

(2) $${A_z}\left( {i,j} \right) = \left[ {\Omega \left( {i,j} \right)} \right]{\rm} {B_z}\left( {i,j} \right) + {A_0},$$

where (i, j) are the coordinates of the cell (in the total structure) according to the x-direction and y-direction and A ₀ is the excitation defined on the source domain. The boundary conditions in the all sub-domains are is defined in the following equations (3)

(3) $$\left\{ {\matrix{ {\hbox{On the sub-domain}} & {{H_v}{\rm :} \left\{ {\matrix{ {{E_z} = 0,} \cr {{J_z} \ne 0,} \cr}} \right.} \cr {\hbox{On the sub-domain}} & {{H_{nv}}{\rm :} \left\{ {\matrix{ {{E_z} \ne 0,} \cr {{J_z} = 0,} \cr}} \right.} \cr {\hbox{On the sub-domain}} & {{H_s}{\rm :} \left\{ {\matrix{ {{E_z} = {E_0},} \cr {{J_z} = {J_0} = \displaystyle{{{E_0}} \over {{z_0}}},} \cr}} \right.} \cr}} \right.$$

where E ₀ is the excitation source of the circuit.

The use of (1) and (3), allows one to establish the relationship between the incidents and reflected waves, as in (4).

(4) $$\left\{ {\matrix{ {{H_v} \Rightarrow A_z^{k + 1} = - B_z^k ,} \hfill \cr {{H_{nv}} \Rightarrow A_z^{k + 1} = B_z^k ,} \hfill \cr {{H_s} \Rightarrow A_z^{k + 1} = 0 \cdot B_z^{k + 1} + {A_0},} \cr } } \right. $$

where “k” is the iteration number.

The general expression for the spatial operator Ω is given by (5):

(5) $$\Omega = 0.{H_s} + {H_{nv}} - {H_v}.$$

B) Spectral study

The medium is modeled by a spectral operator, which permit to take into account of the scattering behavior. This operator permits to calculate the reflected waves from the incident waves in the spectral domain, as in (6), [Reference Hrizi and Sboui14].

(6) $$\left[ {\tilde B} \right] = \left[ \Gamma \right] \cdot \left[ {\tilde A} \right].$$

The connection between the spatial domain and the spectral domain is ensured by a fast transformation in mode using the bi-dimensional fast Fourier transform 2D-FFT and its inverse transform.

$$\left \vert {\matrix{ {\tilde A} \cr {\tilde B} \cr}} \right \vert = FMT\left \vert {\matrix{ A \cr B \cr}} \right \vert. $$

In order to solve the scattering problem, we assume that the structure is 2D-periodic. We consider a bi-periodic array in SIW structure, as depicted in Fig. 3(a). The dimensions of the structure according to O _x and O _y are D _x and D _y . The periods according to O _x and O _y are d _x and d _y . The effective domain of the metal in the basic cell is d ₀, as illustrated in Fig. 3(b). The height of the dielectric layer is h.

Fig. 3.

The cross-section of the bi-periodic SIW schema in case of square metallic via-hole (a) and the configuration of the basic cell (b).

The spectral impedance is obtained from the Helmholtz equation is given by (7), where Δ _T is the transverse Laplace operator

(7) $$\left( {{\Delta _T} + {k^2}} \right){E_z} = j\omega \mu {J_z}, $$

where E is the electric field and $J = H \times \vec n$ is the current density vector, and $\vec n$ is the normal vector.

We consider the modal basis functions given by (8), deduced from the Floquet–Bloch theorem as in [Reference Hamdi, Aguili and Baudrand18–Reference Islam, Zedler and Eleftheriades20].

(8) $$\eqalign{\left \vert {f{}_{pq,mn}} \right\rangle & = {a_{pq,mn}}.\exp \left( {j\left( {{\alpha _p}x + {\beta _q}y} \right)} \right) \cr & \quad \times\exp \left( {j\left( {\displaystyle{{2\pi m} \over {{d_x}}}x + \displaystyle{{2\pi n} \over {{d_y}}}y} \right)} \right) = {f_{pq}}{f_{mn}},} $$

where ${\alpha _p} = 2\pi p/{D_x}$ , ${\beta _q} = 2\pi q/{D_y}$ are the Floquet–Bloch states,  and ${a_{pq,mn}} = \left\langle {{f{}_{pq,mn}}} \mathrel{\left\vert {\vphantom {{f{}_{pq,mn}} {f{}_{pq,mn}}}} \right.} {{f{}_{pq,mn}}} \right\rangle $ are the amplitude of the mode f _pq,mn .

The electric field vector and the current density, using the global modal basis functions (Floquet–Bloch states), are given by (9)

(9) $$\eqalign{E & = \sum\limits_{pq} {\left \vert {{f_{pq}}} \right\rangle {{\tilde E}_{pq}}\left\langle {{f_{pq}}} \right \vert}\, \;{\rm and\;} \; J = \sum\limits_{pq} {\left \vert {{f_{pq}}} \right\rangle {{\tilde J}_{pq}}\left\langle {{f_{pq}}} \right \vert},} $$

The structure is centered at the origin of the coordinate system (the basic cell is located in the center of the structure). So $ - {N_x}/2 \le p \le {N_x}/2 - 1$ and $ - {N_y}/2 \le q \le {N_y}/2 - 1.$

We can write the operator Δ _T in the modal basis f _pq,mn as in (7).

(10) $${\Delta _T}f{}_{pq,mn} = - {\left( {{\alpha _p} + \displaystyle{{2\pi m} \over {{d_x}}}{\rm}} \right)^2} - {\left( {{\beta _q} + \displaystyle{{2\pi n} \over {{d_y}}}} \right)^{\rm 2}}{\rm} {\rm.} $$

The current density is equal to zero in the dielectric area. Thus electric field vector and the current density on metal sub-domain are given by (11):

(11) $${E_{pq}} = \left\langle {{{{\tilde E}_{pq}}}} \mathrel{\left \vert {\vphantom {{{{\tilde E}_{pq}}} {{h_m}}}} \right.} {{{h_m}}} \right\rangle ;\quad {\rm} {J_{pq}} = \left\langle {{{{\tilde J}_{pq}}}} \mathrel{\left \vert {\vphantom {{{{\tilde J}_{pq}}} {{h_m}}}} \right.} {{{h_m}}} \right\rangle, $$

where h _m is the effective domain of the metal sub-domain in the basic cell, as illustrated by Fig. 3(b), given by:

$${h_m} = \left\{ {\matrix{ {\matrix{ 1 & {{\rm si} \left( {x,y} \right) \in \left[ { - \displaystyle{{{{\rm d}_{\rm 0}}} \over {\rm 2}},\displaystyle{{{d_0}} \over 2}} \right] \times {\rm} \left[ { - \displaystyle{{{{\rm d}_{\rm 0}}} \over {\rm 2}},\displaystyle{{{d_0}} \over 2}} \right]} \cr 0 & {{\rm elsewhere}{\rm.}} \cr} {\rm,}} \hfill \cr {{\rm}} \hfill \cr}} \right.$$

The spectral impedance z _pq connects the components J _pq and E _pq as in (12):

(12) $${E_{pq}} = z{}_{pq} \cdot {J_{pq}}.$$

We can write the impedance operator using the modal base $\left\langle {{f_{pq}}} \right\vert$ as in (13).

(13) $$z = \sum\limits_{p,q} {\left \vert {{f_{pq}}} \right\rangle {z_{pq}}\left\langle {{f_{pq}}} \right \vert} {\rm} {\rm.} $$

To take into account the configuration of the basic cell we can decompose the impedance operator on the local modal base: $\left\langle {{f_{pq}}{f_{mn}}} \right\vert$ , as in (14).

(14) $${z_{pq}} = \sum\limits_{m,n} {\left \vert {{f_{pq}}{f_{mn}}} \right\rangle {z_{pq,mn}}\left\langle {{f_{pq}}{f_{mn}}} \right \vert} {\rm} {\rm.} $$

Coupling (7–10–12–14); the impedance operator z _pq is given by (15).

(15) $${z_{pq}} = \sum\limits_{m,n} {\left\langle {{{H_m}}} \mathrel{\left \vert {\vphantom {{{H_m}} {{f_{pq}}{f_{mn}}}}} \right.} {{{f_{pq}}{f_{mn}}}} \right\rangle} {\rm} j\omega \mu \displaystyle{1 \over {{k^2} - \alpha _{p,m}^2 - \beta _{q,n}^2}} \left\langle {{{f_{pq}}{f_{mn}}}} \mathrel{\left \vert {\vphantom {{{f_{pq}}{f_{mn}}} {{H_m}}}} \right.} {{{H_m}}} \right\rangle, $$

where ${\alpha _{p,m}} = {\alpha _p} + (2\pi m/{d_x})$ and ${\beta _{q,n}} = {\beta _q} + (2\pi n/{d_y}){\rm } {\rm.} $ The final expression of the spectral impedance is given by (16)

(16) $${z_{pq}} = \sum\limits_{mn} {j\omega \mu \displaystyle{{{{\left \vert {\left\langle {{H_m}{f_{pq}}{f_{mn}}} \right\rangle} \right \vert} ^2}} \over {{k^2} - \alpha _{p,m}^2 - \beta _{q,n}^2}}} {\rm,} $$

where ${H_m} = (1/\sqrt s ){h_m}$ and S is the cross-section of metallic via-hole

The propagation constant is given by the following expression: ${\gamma_{pq}} = \sqrt {( {{k^2} - \alpha_{p,m}^2 - \beta_{q,n}^2})}$ In the case of shorted-circuited structure with a ground plane at a height h, the spectral operator Γ _pq is given by (17).

(17) $${\Gamma _{pq}} = \displaystyle{{{z_0} - {z_{pq}}\coth \big( {{\gamma _{pq}}h} \big)} \over {{z_0} + {z_{pq}}\coth \big ( {{\gamma _{pq}}h} \big)}}{\rm} {\rm.} $$

After the convergence value-iteration of the iterative process, the output parameters are extracted. The distribution of the electric field and the current is given by (18).

(18) $$\eqalign{& {E_z}\left( {i,j} \right) = \sqrt {{Z_0}} \left( {{A_z}\left( {i,j} \right) + {B_z}\left( {i,j} \right)} \right){\rm ;} \cr & {\rm} {J_z}\left( {i,j} \right) = \displaystyle{1 \over {\sqrt {{Z_0}}}} \left( {{A_z}\left( {i,j} \right) - {B_z}\left( {i,j} \right)} \right).} $$

The matrix S of the studied structure is given by (19).

(19) $$\left[ S \right] = \displaystyle{{\left[ {{I_d}} \right] - {z_0}\left[ Y \right]} \over {\left[ {{I_d}} \right] - {z_0}\left[ Y \right]}},$$

where I _d is the identity matrix and Y is the admittance matrix of circuit.

A) Integrated substrate cavity

The structure presented in Fig. 4 is a single substrate-integrated cavity first presented in [Reference Amendola, Arnieri and Boccia12], the lateral walls are realized by rows of metallic via-holes. The used substrate is Arlon 25FR (tm) with dielectric constant of 3.58 and thickness of 0.787 mm. The total structure dimensions are 15.17 × 24 mm². The metallic via-holes have the same diameter d ₀ = 0.2 mm and with a period p = 2 mm.

Fig. 4.

SIW cavity.

To verify the stability of the proposed approach, we have performed a study of the convergence behavior of iterations and of the number of modal bases (m,n). Figure 5 illustrate the convergence curves value- iterations of the reflection and transmission coefficients taken at a frequency equal to 10 GHz. The convergence is reached around 200 iterations (with a ripple rate of 10⁻³).

Fig. 5.

Convergence curves value-iterations of S-parameters (Frequency = 10 GHz; Number of iterations = 250 iterations).

On the other hand, the developed approach is based on the superposition of infinite series of modes. Considering, the limitation of the computational resources which increase considerably with this number, it is necessary to stop this series at a finite number of modes (M,N). In this part, we studied the convergence as function of the number of modes (M,N). The curves in Fig. 6(a) present the reflexion coefficient (S11) for different numbers (M,N).

Fig. 6.

(a) Reflection coefficient for different number of the modes (M,N). (b) Convergence curve of the resonance frequency depending of the number of modes (M,N).

Figure 6(b) represents the convergence curve of the resonant frequency of the cavity versus (M,N). The convergence value-modes achieved around 30 modes. Therefore, it is unnecessary to take a number of modes higher than 30.

The convergence value-iterations and value-modes show that the proposed method is numerically stable.

Using the same example of the SIW cavity, depicted in the Fig. 4, we study the effect of the diameter of the metallic via-hole on the frequency response. Figure 7(a) depicts the reflection coefficient of different values of the diameter d ₀. Figure 7(b) presents the behavior of the resonance frequency depending on the diameter of metallic via-hole. The increase of the diameter has effect to decrease the resonance frequency of the cavity.

Fig. 7.

(a) Effect of the diameter of the metallic via-hole. (b) Resonance frequency versus the via-hole diameter.

The results obtained are compared with both measurements that are available in [Reference Amendola, Arnieri and Boccia12] and with data obtained by the FEM (commercial software HFSS). Figures 8(a) and 8(b) display the transmission and reflection coefficients for the frequency range 8–10 GHz. A satisfactory agreement is observed. This allows validating the accuracy of the present method. Moreover, the HFSS software present a frequency shift of 55 MHz, this shift is close to zero for our method.

Fig. 8.

(a) Reflection coefficient S11 versus frequency. (b) Transmission coefficient S21 versus frequency.

All simulations are carried out using a CPU Intel (R) Core (TM) i5 CPU 650@3.20 GHz with a RAM equal to 6 GB. The error and the computation CPU time are recorded in Table 1. The proposed method has the most precise results and the average error does not exceed 2%. Also, the proposed method allows a reduction of the computing time of more than 80%. In fact, the HFSS software becomes very slow if we introduce the metallic via-holes.

Table 1.

Computation performance comparison of the proposed method and the FEM method for the SIW cavity example.

B) Quasi-elliptic V-band filter

The structure showed in Fig. 9 is a quasi-elliptic filter [Reference Zelenchuk and Fusco21].This circuit is used in high-speed wireless system, in the V-band, with 9 GHz bandwidth. In [Reference Guglielmi, Jarry, Kerherver, Roquebrun and Schmitt22], the multi-mode cavity filter techniques are used to design this circuit. The via-holes have the dimension of d ₀ = 0.25 mm and spaced of 0.4 mm and the other dimensions of the filter are:

$$\eqalign{{b_1} & = 2.55\;{\rm mm},{b_2} = 1.56\,{\rm mm},\;{b_3} = 6.32\,{\rm mm},\;{b_4} = 1.71\,{\rm mm}, \cr & \quad {b_5} = 2.11\,{\rm mm},\;{b_6} = 6.71\,{\rm mm,}\;{b_7} = 2.25\,{\rm mm}, \cr {a_1} & = 1.84\,{\rm mm},\;{a_2} = 1.84\,{\rm mm},\;{a_3} = 1.65\,{\rm mm},\;{a_4} = 2.5\,{\rm mm}, \cr & \quad {a_5} = 1.12\,{\rm mm},\;{\varepsilon _r} = 2.1,\;h = 0.2\,{\rm mm}{\rm.}} $$

Fig. 9.

Top view of the SIW quasi-elliptic filter.

Figures 10(a) and 10(b) present the reflection and transmission coefficients obtained by both the proposed method and by the commercial software HFSS, and compared with the measurement data available in [Reference Zelenchuk and Fusco21]. A satisfactory agreement is observed.

Fig. 10.

(a) Reflection coefficient of the SIW quasi-elliptic filter. (b) Transmission coefficient of the SIW quasi-elliptic filter.