Coupling-routing diagram and design examples

The generalized normalized coupling-routing diagrams of the proposed dual- and triple-band BPF networks together with their single-band prototype counterpart for $N$th-order architectures are depicted in Figure 1. As shown, the baseline single-band prototype is formed by $N-2$ resonating nodes in an in-line topology with its respective input and output feeding sections. Each feeding section comprises a zero-susceptance non-resonating node with two dangling resonating nodes directly coupled to it. These input and output feeding sections contribute two out-of-band TZs at the natural frequencies of their dangling resonating nodes to the overall filtering response – i.e., at $\Omega_{z1}=\Omega_1$ and $\Omega_{z2}=\Omega_2$ and at $\Omega_{z3}=\Omega_3$ and $\Omega_{z4}=\Omega_4$ for the input and output feeding sections, respectively – by creating a short circuit in the in-line path at these frequencies, as well as one pole located between the two TZs. The formulas for the normalized frequencies of these poles corresponding to the input and output feeding sections, $\Omega_{pS}$ and $\Omega_{pL}$, respectively, which are obtained by imposing the mutual cancellation of the equivalent normalized admittances of their two constituent dangling branches, are as follows:

(1)\begin{equation} \Omega_{pS}=\frac{M_{S,N-1}\Omega_2+M_{S,N}\Omega_1}{M_{S,N-1}^2+M_{S,N}^2} \end{equation}

(2)\begin{equation} \Omega_{pL}=\frac{M_{N+1,L}\Omega_4+M_{N+2,L}\Omega_3}{M_{N+1,L}^2+M_{N+2,L}^2}. \end{equation}

Figure 1.

Normalized coupling-routing diagrams for $N$th-order realizations of the dual- and triple-band BPF networks and its initial single-band prototype counterpart to which the frequency transformations $\Omega \rightarrow \mathcal{F}_{M=2}\left(\Omega \right)$ in (3) and $\Omega \rightarrow \mathcal{F}_{M=3}\left(\Omega \right)$ and (4), respectively, are applied (black circles: resonating nodes with normalized natural frequencies $\Omega_0,\Omega_1,\Omega_2,\Omega_3,\Omega_4,\Omega_0+\Omega_A,\Omega_1+\Omega_A,\Omega_2+\Omega_A,\Omega_3+\Omega_A,\Omega_4+\Omega_A,$ $\Omega_B,$ and $\Omega_C$; white circles: unitary source [S] and load [L]; continuous lines: couplings; $M$, $\alpha_{1},$ and $\alpha_{2}$ variables: normalized coupling coefficients or inversion constants of the normalized admittance inverters; $Y_0 = 0.02$ $\mho$: reference $/$port admittance).

Figure 2.

Effect of the normalized-frequency transformations $\Omega \rightarrow \mathcal{F}_{M=2}\left(\Omega \right)$ in (3) and $\Omega \rightarrow \mathcal{F}_{M=3}\left(\Omega \right)$ in (4) in terms of power transmission $(|S_{21}|)$ and input-reflection $(|S_{11}|)$ responses – i.e., from single-band prototype to dual- and triple-band BPFs, respectively – using the normalized coupling-routing diagrams in Figure 1 for fourth-order realizations – i.e., $N=4$; dual-band BPF: $\alpha_{1}=3.464$; triple-band BPF: $\alpha_{1}=2.58$ and $\alpha_{2}=2.12$; $M_{S,1} = M_{S,3} = M_{S,4} = M_{2,L} = M_{5,L} = M_{6,L} = 1$, $M_{1,2} = 0.92$, $\Omega_0 =$ $\Omega_A =$ $\Omega_B =$ $\Omega_C = 0$, and $\Omega_1=-\Omega_2 = \Omega_3=-\Omega_4 = -2$ in both cases.

Figure 3.

Power transmission ( $|S_{21}|$) and input-reflection ( $|S_{11}|$) responses of frequency-symmetrical and asymmetrical theoretical design examples of the normalized coupling-routing diagrams of the dual- and triple-band BPFs in Figure 1 for fourth- and sixth-order realizations – i.e., $N=4$ and $6$, respectively – together with the $|S_{21}|$ and $|S_{11}|$ responses of its initial single-band prototype counterpart in the frequency-symmetrical cases. (a) Example 1: $M_{S,1} = M_{S,3} = M_{S,4} = M_{2,L} = M_{5,L} = M_{6,L} = 1$, $M_{1,2} = 0.92$, $\alpha_{1}=3.46$, $\Omega_0 = \Omega_A = \Omega_B = 0$, and $\Omega_1=-\Omega_2 = \Omega_3=-\Omega_4 = -2$. (b) Examples 2 ( $\Omega_A = -3$) and 3 ( $\Omega_A = 3$): $M_{S,1} = M_{S,3} = M_{S,4} = M_{1,2}= M_{2,L} = M_{5,L} = M_{6,L} = 1$, $\alpha_{1}=3.46$, $\Omega_0 = \Omega_B = 0$, $\Omega_1=-\Omega_2 = -2$, and $\Omega_3=-\Omega_4 = -2.75$. (c) Examples 4: $M_{S,1} = M_{S,3} = M_{S,4} = M_{2,L} = M_{5,L} = M_{6,L} = 1$, $M_{1,2} = 0.92$, $\alpha_{1}=1.304$, $\Omega_0 = \Omega_A = \Omega_B = 0$, and $\Omega_1=-\Omega_2 = \Omega_3=-\Omega_4 = -2$. (d) Example 5: $\alpha_{1}=3.46$, $\alpha_2=2.574$, $\Omega_0=$ $\Omega_A=$ $\Omega_B=$ $\Omega_C=0$, and $\Omega_1=$ $-\Omega_2=$ $\Omega_3=$ $-\Omega_4=-2$. (e) Examples 6 ( $\Omega_A=3$) and 7 ( $\Omega_A=-3$): $\alpha_{1}=3.464$, $\alpha_2=2.034$, $\Omega_0=$ $\Omega_B=$ $\Omega_C=0$, and $\Omega_1=$ $-\Omega_2=$ $\Omega_3=$ $-\Omega_4=-2$. (f) Examples 8 ( $\Omega_B=-1.6$ and $\Omega_C=-2.4$) and 9 ( $\Omega_B=1.6$ and $\Omega_C=2.4$): $\alpha_{1}=2.664$, $\alpha_2=1.914$, $\Omega_0=$ $\Omega_A=0$, and $\Omega_1=$ $-\Omega_2$ = $\Omega_3=$ $-\Omega_4=-2$. (g) Examples 10 ( $\alpha_{1}=1.304$ and $\alpha_{2}=1.074$) and 11 ( $\alpha_{1}=4.424$ and $\alpha_2=3.174$): $\Omega_0=$ $\Omega_A$ = $\Omega_B$ = $\Omega_C=0$, $\Omega_1=-\Omega_2 = \Omega_3=-\Omega_4 = -2$. Examples (h) ( $\alpha_{2}=2.154$) and (i) ( $\alpha_{2}=0.414$): $\alpha_{1}=1.064$, $\Omega_0$ = $\Omega_A$ = $\Omega_B$ = $\Omega_C=0$, $\Omega_1=-\Omega_2 = \Omega_3=-\Omega_4 = -2$; $M_{S,1} = M_{4,L}=0.94$, $M_{S,5} = M_{S,6} = M_{7,L} = M_{8,L} = 1$, $M_{1,2}=M_{3,4}=0.81$, and $M_{2,3}=0.72$ for all triple-band BPF examples in (d)–(i).

Consequently, the resulting single-band filtering response exhibits up to $N$ in-band reflection zeros (RZs) and up to four finite-frequency stopband TZs at distinct spectral locations – which may be grouped in pairs to yield double-multiplicity TZs, i.e., $\Omega_{z1}=\Omega_{z3}$ and $\Omega_{z2}=\Omega_{z4}$ – without the need for cross-coupling mechanisms in the single-band prototype.

The dual- and triple-band BPFs are derived by applying split-type single-to-dual-band and single-to-triple-band frequency transformations, $\Omega \rightarrow \mathcal{F}_{M=2}(\Omega)$ and $\Omega \rightarrow \mathcal{F}_{M=3}(\Omega)$, respectively, to the initial single-band prototype in the normalized domain. The analytical expressions of these frequency transformations are as follows:

(3)\begin{equation} \mathcal{F}_{M=2}(\Omega) = \Omega - \Omega_A - \dfrac{\alpha_1^2}{ \Omega - \Omega_B } \end{equation}

(4)\begin{equation} \mathcal{F}_{M=3}(\Omega) = \Omega - \Omega_A - \dfrac{\alpha_1^2}{ \Omega - \Omega_B - \dfrac{\alpha_2^2}{ \Omega - \Omega_C }} \end{equation}

where $\Omega_A,\Omega_B, \Omega_C,\alpha_{1}$, and $\alpha_{2}\in \mathbb{R}$. Due to these frequency transformations, additional resonating nodes appear coupled to each resonating node of the single-band prototype – namely, one and two extra resonating nodes for the dual- and triple-band cases, respectively. This equivalently results in the frequency mapping of the single-band prototype response into dual- and triple-band responses with up to five and eight distinct finite-frequency stopband TZs, respectively. The middle TZ(s) between passbands have multiplicity $N-2$ and come from the TZ at $\pm\infty$ of the core single-band prototype after frequency transformation. The TZs closer to the passbands come from their counterparts in the baseline single-band prototype after frequency mapping. Furthermore, the in-band characteristics of the single-band prototype are preserved and transferred to the passbands of the dual- and triple-band BPF responses in terms of filter order and input-power-matching profile. Note that the frequency transformations in (3) and (4) correspond to the particular cases of the generalized split-type single-to- $M$-band frequency transformation $\Omega \rightarrow \mathcal{F}_{M}(\Omega)$ in (5) for $M=2$ – dual-band response – and $M=3$ – triple-band response – with $\Omega_B\equiv\Omega_{A,1}$ and $\Omega_C\equiv\Omega_{A,2}$ for notation simplicity.

(5)\begin{align} & \Omega \;\longrightarrow\; \mathcal{F}_M(\Omega)\nonumber\\ & \quad= \Omega - \Omega_A - \dfrac{\alpha_1^2}{ \Omega - \Omega_{A,1} - \dfrac{\alpha_2^2}{\Omega - \Omega_{A,2} - \dfrac{\alpha_3^2}{\ddots - \dfrac{\alpha_{M-1}^2}{\Omega - \Omega_{A,M-1}}}}} \end{align}

For illustrative purposes, Figure 2 presents examples of the theoretical power transmission and input-reflection responses for the dual- and triple-band BPF cases to show the effect of the normalized-frequency transformations in (3) and (4), respectively. As observed, the sharp-rejection transmission band of the initial single-band prototype employing input $/$output extracted cavities in Figure 1 is split into two and three narrower passbands for the dual- and triple-band cases, respectively. All resonating nodes contribute equally to shaping these transmission bands. Simultaneously, five and eight stopband TZs are obtained in the dual- and triple-band responses, with trios of TZs located between passbands and additional TZs appearing below the lowest and above the highest passbands. In particular, the spectral locations of these TZs closest to the passbands are determined by their homologous counterparts in the initial single-band prototype in Figure 1, and by the parameters $\alpha_{1}$ and $\alpha_{2}$ of the frequency transformations. Consequently, the steep-selectivity characteristics of the original single-band response are retained in the resulting dual- and triple-band BPF networks, along with the pre-designed minimum in-band power-matching level of 21.5 dB for all transmission bands, each exhibiting equirriple-type in-band behavior.

For $\Omega_0=0$ – i.e., normalized single-band prototype in Figure 1 centered at $\Omega=0$ – the normalized center frequencies of the passbands of the $M$-band BPF derived from the frequency transformation in (5) can be obtained as follows:

(6)\begin{equation} \left.\mathcal{F}_{M}(\Omega)\right|_{\Omega_{01},\Omega_{02},\ldots,\Omega_{0M}}=0 \end{equation}

where $\Omega_{01} \lt \Omega_{02} \lt \cdots \lt \Omega_{0M}$ denote the normalized center frequencies from the lowest to the highest passband. Their formulas for the resulting dual-, triple-, and quadruple-band responses – i.e., $M=2,3,$ and $4$ – assuming that $\Omega_{A,1}=\Omega_{A,2}=\Omega_{A,3}=0$ for simplicity, which are frequency symmetrical with regard to $\Omega=0$, are given below.

• • Dual-band BPF – i.e., $M=2$:

(7)\begin{equation} \Omega_{01},\Omega_{02}= \pm |\alpha_1|. \end{equation}

• • Triple-band BPF – i.e., $M=3$:

(8)\begin{equation} \Omega_{01},\Omega_{03}= \pm \sqrt{\alpha_1^2+\alpha_2^2} \qquad \Omega_{02} = 0. \end{equation}

• • Quadruple-band BPF – i.e., $M=4$:

(9)\begin{gather}\Omega_{01},\Omega_{02},\Omega_{03},\Omega_{04}= \cr \pm\sqrt{\frac{\alpha_1^2+\alpha_2^2+\alpha_3^2\pm\sqrt{\left(\alpha_1^2+\alpha_2^2+\alpha_3^2\right)^2-4\alpha_1^2\alpha_3^2}}{2}}.\end{gather}

As demonstration, various theoretical design examples are presented in Figure 3, illustrating the power-transmission and input-reflection responses associated with the normalized coupling-routing diagrams of the dual- and triple-band BPF networks in Figure 1 for fourth- and sixth-order – i.e., $N = 4$ and $6$ – schemes, respectively. All examples have in-band input-power-matching levels above 21.5 dB. Specifically, Figure 3(a) shows a frequency-symmetrical dual-band BPF with four in-band RZs and five distinct out-of-band TZs, along with its baseline single-band prototype. Two examples of spectrally asymmetrical dual-band BPFs exhibiting three in-band RZs and nine out-of-band TZs at nonuniform finite frequencies are depicted in Figure 3(b). On the other hand, Figure 3(c) shows the flexibility to control the frequency spacing between the two passbands by simply adjusting $\alpha_{1}$, which defines the coupling between the extra resonators owing the single-to-dual-band frequency transformation and the resonating nodes of the initial single-band prototype; particularly, a stronger coupling $\alpha_{1}$ leads to a wider inter-passband separation, and vice versa. Additionally, Figure 3(d)–(i) illustrates the sixth-order triple-band BPF case. As in the fourth-order dual-band BPFs, the obtained responses reveal high degree of design versatility. Specifically, asymmetric-passband configurations can be synthesized, in which two transmission bands exhibit identical bandwidths while the third one differs (Figure 3(e)). Moreover, fully independent control of the three passband bandwidths is also achievable (Figure 3(f)). In both cases – Figure 3(e) and (f) – the locations of the center frequencies of the passbands and out-of-band TZs remain adjustable. In addition, the spacing between adjacent passbands can be reduced or increased by just tuning the coupling between the extra resonators and the resonating nodes of the initial single-band prototype – i.e., $\alpha_{2}$ – (Figure 3(g)). Finally, further flexibility is obtained by enabling either independent or simultaneous control of the bandwidths of all three passbands without modifying their center frequencies or stopband TZ positions, as illustrated in Figure 3(h) and (i).

Figure 4.

Dispersion diagram of the GGW and dimensions of the constituent pin ( $h=6.475$ mm, $a_{p}=1.3$ mm, $p/2=1.53$ mm, and $g=0.125$ mm).

Core GGW

Since the GGW multi-extracted-cavity dual-band BPF prototype later introduced in the Experimental results section is developed for Ku-band operation, the careful selection of the periodic pin surface that forms the GGW core is essential. The geometrical parameters of the metallic pins to be employed in the referred filter design are detailed in the caption of Figure 4. This figure presents the dispersion characteristics of the periodic structure, which were computed using the full-wave EM solver CST Studio Suite. The analyzed pin array exhibits the properties of an AMC, preventing EM-field propagation over a defined frequency interval due to the presence of an EBG. As can be observed from the dispersion diagram in Figure 4, the forbidden EM-propagation band spans from 6.7 to 22.57 GHz, thereby fully encompassing the Ku band. Consequently, this periodic-pin-lattice configuration effectively suppresses unwanted parallel-plate modes within the operational frequency range, ensuring a proper GGW behavior and making the proposed pin geometry well suited for the intended Ku-band dual-band filter realization – whose lower and upper passbands are specified to be centered at 14 and 14.4 GHz, respectively.

Fabrication and measurement

To demonstrate the experimental viability of the proposed approach of GGW sharp-rejection multi-band BPF based on input $/$output multi-extracted cavities and the split-type single-to-multi-band frequency transformation, a proof-of-concept prototype of Ku-band fourth-order dual-band BPF has been EM simulated, manufactured, and characterized. Its theoretical design starts from the normalized coupling-routing diagram introduced in the Coupling-routing diagram and design examples section – see the bottom part of Figure 1 – for the conceived high-selectivity dual-passband filter, while its physical realization relies on the GGW structure designed in the Core GGW section – see Figure 4 – serving as the core transmission medium. The prefixed specifications for the lower and upper passbands are as follows: center frequencies approximately equal to 14 and 14.4 GHz, respectively, and identical 3-dB absolute bandwidths of 200 MHz for both transmission bands – which correspond to 3-dB fractional bandwidths of 1.44% and 1.3%, respectively. The top/bottom, lateral, and 3-D views of the geometrical layout of the developed Ku-band multi-extracted-cavity fourth-order dual-band BPF circuit are depicted in Figure 5. As shown, a symmetrical configuration was adopted, in which the two halves of the structure are implemented on different layers in a vertically stacked arrangement to enhance circuit compactness.

Figure 5.

Geometrical layout and associated normalized coupling-routing diagram of the designed prototype of Ku-band multi-extracted-cavity fourth-order dual-band BPF in GGW technology; non-redundant physical dimensions in mm are indicated ( $a_{p}$ and $p$ are the variables designated in Figure 4, $w_{1} = 15.79$ mm, $w_{2} = 8.95$ mm, $w_{a} = 2.5$ mm, $w_{p1} = 2.91$ mm, $w_{p2} = 2.63$ mm, $w_{p3} = 3.62$ mm, $w_{p4} = 3.53$ mm, $w_{s1} = 1.04$ mm, $w_{s2} = 2.75$ mm, $w_{s3} = 3.75$ mm, $l_{1} = 11.12$ mm, $l_{2} = 13.14$ mm, $l_{3} = 12.11$ mm, $l_{4} = 10.91$ mm, $l_{5} = 13.14$ mm, $l_{6} = 37.94$ mm, $l_{a} = 7.25$ mm, $s_{1} = 7.87$ mm, $s_{2} = 6.45$ mm, $s_{3} = 8.43$ mm, and all the pins are of the same height [ $h$]). (a) 3-D view of the geometrical layout. (b) Top and bottom views of the geometrical layout – left-hand side – illustrating the cavity numbering and the corresponding couplings as labeled in its associated normalized coupling-routing diagram – right-hand side. (c) Lateral view of the geometrical layout.

Figure 6.

EM-simulated and theoretical – i.e., normalized coupling-routing diagram in Figure 5(b) translated to the bandpass domain with the normalized-lowpass-to-bandpass frequency transformation $\Omega \rightarrow \Omega(f)$ in (10) – power transmission $(|S_{21}|)$ and input-reflection $(|S_{11}|)$ responses of the designed Ku-band multi-extracted-cavity fourth-order dual-band BPF in GGW technology ( $M_{S,1}$= $M_{2,L}=0.89$, $M_{S,3}$ = $M_{S,4}$= $M_{5,L}$ = $M_{6,L} = 1$, $M_{1,2} = 0.75$, $\alpha_1=0.9$, $\Omega_0$ = $\Omega_A$ = $\Omega_B = 0$, and $\Omega_1=-\Omega_2 = \Omega_3=-\Omega_4 = -1.79$).

Figure 7.

EM-simulation-based sensitivity analyses of the designed Ku-band multi-extracted-cavity fourth-order dual-band BPF in GGW technology (dimensions in mm). (a) Variation of the lenght $l_{a}$ of the coupling aperture: step of 0.05 mm. (b) Variation of the width $w_{a}$ of the coupling aperture: step of 0.05 mm. (c) Variation of the window width $s_{1}$ of the inductive-type coupling in the top and bottom layer: step of 0.05 mm. (d) Variation of the window width $s_{2}$ of the inductive-type coupling in the top and bottom layer: step of 0.2 mm.

For the implementation of the resonating nodes, GGW rectangular cavities operating in their fundamental TE $_{101}$ resonance mode were employed. The inter-resonator couplings were realized by means of inductive-type irises. The two adjacent GGW cavities interacting across layers – namely, those associated to resonating nodes 1 and 2 in the corresponding normalized coupling-routing diagram of Figure 5(b) – were coupled through an aperture or coupling window etched in the intermediate metallic plane. As a result, an all-inductive-coupling dual-band BPF configuration was obtained, in which finite-frequency out-of-band TZs are generated on both sides of the passbands without requiring capacitive-type couplings. Due to geometrical constraints, the two dangling GGW cavities adjacent to the input and output feeding sections were placed at different positions on opposite sides of the structure. The WR-62-to-coaxial input and output transitions were co-integrated into a single metallic block, as illustrated in Figure 5(a) and (c). Two different types of top and bottom lids were fabricated: one without tuning elements and another incorporating tuning screws located both inside the cavity resonators and within the inductive coupling regions, as shown in Figure 5(a). These tuning elements were included to compensate for fabrication and assembly tolerances that may degrade the measured RF performance of the manufactured GGW dual-band BPF prototype with respect to the EM simulations for the final values of the geometrical design parameters. This choice is justified by the sensitivity analysis presented later in this section as supporting study.

The EM-simulated power-transmission and input-reflection responses of the designed Ku-band GGW dual-band BPF, together with the theoretical responses obtained from the associated coupling-routing diagram shown in Figure 5(b) – i.e., particular case of the $N$th-order dual-band coupling-routing diagram at the bottom of Figure 1 for $N=4$ – are plotted in Figure 6. The EM simulations were performed with the commercial EM software Ansys HFSS. For comparison purposes, the theoretical responses were translated from the normalized lowpass domain by applying the following frequency transformation $\Omega \rightarrow \Omega(f)$:

(10)\begin{equation} \Omega \rightarrow \Omega(f)=\left(\frac{f_d}{b}\right)\left(\frac{f}{f_d}-\frac{f_d}{f}\right) \end{equation}

with $f_d=14.17$ GHz and $b=0.33$ GHz as inter-band frequency and scaling-factor parameters, respectively. These theoretical responses were matched to the EM-simulated results through an optimization procedure carried out using Matlab $^{\text{TM}}$. As observed, two sharp-rejection transmission bands are achieved at the target center-frequency positions, together with the five expected finite-frequency out-of-band TZs distributed at both sides of the passbands, three of which are located between the two transmission bands.

Figure 8.

Photographs – different views – of the manufactured prototype of Ku-band multi-extracted-cavity fourth-order dual-band BPF in GGW technology (overall dimensions without the input $/$output connector block: $99.42 \times 59.31 \times 25.34$ mm $^{3}$). (a) Internal view. (b) View with top $/$bottom lids without tuning screws. (c) View with top $/$bottom lids with tuning screws.

Figure 9.

EM-simulated and measured power transmission $(|S_{21}|)$, input-reflection $(|S_{11}|)$, and in-band group-delay $(\tau_{g}^{21})$ responses of the manufactured prototype of $Ku$-band multi-extracted-cavity fourth-order dual-band BPF in GGW technology. (a) Measurements realized using top $/$bottom lids without tuning screws: $|S_{21}|$ and $|S_{11}|$. (b) Measurements in (a) represented in a broader frequency range. (c) Measurements realized using top $/$bottom lids with tuning screws after adjustment of screws: $|S_{21}|$ and $|S_{11}|$. (d) Measurements realized using top $/$bottom lids with tuning screws after adjustment of screws: $\tau_{g}^{21}$ for lower – i.e., 1st band – and upper – i.e., 2nd band – passbands.

To further evaluate robustness against fabrication tolerances, EM-simulation-based sensitivity analyses were conducted for the designed GGW $Ku$-band multi-extracted-cavity fourth-order dual-band BPF shown in Figure 5. Figure 7 illustrates the impact on its EM-simulated power transmission and input-reflection responses caused by variations in several representative physical dimensions of its structure. Specifically, perturbations of the following geometrical parameters were considered in this study – the considered variations were chosen to match or exceed tolerances of the manufacturing process employed for the RF prototype of this work: (i) the aperture length $l_{a}$ (Figure 7(a)), (ii) the aperture width $w_{a}$ (Figure 7(b)), (iii) the window width $s_{1}$ of the inductive-type coupling in the top and bottom layers (Figure 7(c)), and (iv) the window width $s_{2}$ of the inductive-type coupling in the top and bottom layers (Figure 7(d)). As observed, depending on which parameter is being altered, even small variations in these dimensions produce perceptible modifications in the frequency responses of the GGW dual-band BPF prototype. These effects manifest in changes in the in-band input-power-matching levels, bandwidths, out-of-band rejection profile, and $/$or the location of the TZs, as well as the frequency spacing between the two passbands. In particular, the inter-band spacing either increases or decreases depending on the effective coupling strength between pairs of cavities derived from the split-type dual-band frequency transformation, which is mainly controlled by the window width $s_{2}$. These observations highlight the importance of a precise fabrication and justify the inclusion of tuning screws to mitigate $/$compensate such undesired manufacturing imperfections in practical implementations.

A photograph of the developed prototype of GGW $Ku$-band multi-extracted-cavity fourth-order dual-band BPF is shown in Figure 8. The structure was manufactured using a computer numerical control (CNC) milling machine with a mechanical precision of 0.02 mm, employing aluminum 6061 alloy with a conductive oxidation surface treatment. The measured power transmission, input-reflection, and in-band group-delay responses – obtained using both types of lids, i.e., without and with tuning screws – of this prototype are compared with the EM-simulated results in Figure 9. The measurements were performed using a Keysight 5224A vector network analyzer. As illustrated in Figure 9(a), a reasonable overall agreement between simulations and measurements is observed for the untuned case except for a degradation in the in-band input-power-matching levels, which is attributed to fabrication and assembly tolerances – as revealed by the sensitivity analysis shown in Figure 7. These discrepancies are effectively corrected through tuning, as confirmed by the results in Figure 9(c) and (d). After tuning, the main measured characteristics of the fabricated GGW dual-band BPF are as follows. For the lower and upper passbands, the center frequencies are 14.04 and 14.43 GHz, respectively, with 3-dB absolute bandwidths of 211 and 209 MHz – corresponding to 3-dB fractional bandwidths of 1.5 $\%$ and 1.45 $\%$. The minimum in-band power-insertion-loss levels are 0.9 and 0.76 dB, while the in-band return-loss levels exceed 15.1 and 15 dB, respectively. The maximum in-band group-delay variations are 7.7 ns for the lower band and 10.3 ns for the upper band. Finally, the measured finite-frequency TZs are located at 13.86, 14.17, 14.24, 14.3, and 14.6 GHz, confirming the sharp-rejection characteristics of the proposed filter design. Finally, note that narrow-band spurious spikes attributed to higher-order modes can be observed in Figure 9(b) within the frequency range 18–20 GHz of the measured 10–20-GHz interval. The partial $/$total mitigation of these undesired transmission peaks or their displacement to higher frequencies using different well-known design strategies from classic rectangular-waveguide filters – e.g., mode-selective coupling, cavity-shape optimization including ridged- or stepped-cavity geometries, or undesired-mode absorption by means of deliberately embedded RF lossy elements – remains as further research work.

Comparison with the state-of-the-art

A comparison with the most relevant prior-art works in the field of GGW RF filtering components is summarized in Table 1. The values therein contained have been estimated as best as possible from measured or EM-simulated data when not directly provided. Most of these implementations correspond to single-band BPFs or to combinations of two BPFs forming diplexers, highlighting the scarcity of multi-band BPF devices based on GGW technology.

Table 1.

Comparison with prior-art RF single- and dual-band BPFs and RF diplexers in GGW technology

BPF = bandpass filter; FBW = fractional bandwidth; GGW = groove gap waveguide; TZ = transmission zero; N.A. = not applicable or available; T.W. = this work; $^{\dagger}$ = estimated from graphs or data from simulations;

# = For TZs, only those observed in the corresponding figure are reported, although additional TZs may exist outside the shown frequency range; for the circuit size, the values are taken directly from the corresponding reference; $^{\ast}$ = performance metrics for differential-mode operation; $\lambda_0$ = wavelength at vacuum at the center frequency for single-band BPFs, at the center frequency of the lower passband for dual-band BPFs, and at the center frequency of the lower channel for diplexers.

From Table 1, it can be concluded that the $Ku$-band GGW dual-band BPF prototype of this work exhibits competitive RF performance compared with the previously reported GGW dual-band BPFs in [Reference Malki, Yang and Gómez-García29, Reference Huang, Zhang, Zhou, Lu, Tam, Zhang and Tang43–Reference Malki, Yang and Gómez-García45]. Specifically, the GGW dual-passband filters in [Reference Huang, Zhang, Zhou, Lu, Tam, Zhang and Tang43] and [Reference Wu, Wu, Yao, Huang, Xu and Wang44] are restricted to second-order experimental validations and rely on more sensitive cross-coupling schemes for TZ generation. In addition, they feature fewer finite-frequency stopband TZs than the proposed filter. The dual-band BPFs reported in [Reference Malki, Yang and Gómez-García29], in both reflective and reflectionless configurations, are based on quadruplet networks to achieve quasi-elliptic-type dual-passband responses. Consequently, their physical implementations require both inductive- and capacitive-type couplings, increasing structural complexity. In contrast, the GGW dual-band BPF in [Reference Malki, Yang and Gómez-García45] employs a reactive load or oversized GGW cavity connected at the input port of a low-selectivity GGW dual-band BPF unit to produce multiple out-of-band TZs without cross coupling. However, the overall transfer-function envelope is still governed by the low-order dual-band BPF unit, resulting in reduced attenuation levels in some out-of-band regions but in-band low-loss characteristics as main advantage. Furthermore, the lower and upper cut-off slopes of the first and second passbands – measured from the 3-dB cut-off frequency to the nearest 40-dB stopband frequency relative to the minimum insertion-loss level – are 207.2 and 93.9 dB $/$GHz for the filter in [Reference Malki, Yang and Gómez-García45], respectively. These values are significantly lower than those achieved by the developed GGW $Ku$-band multi-extracted-cavity dual-band BPF prototype, which are equal to 787.2 and 743 dB $/$GHz for the referred cut-off slopes, respectively.

Finally, it is worth noting that the experimental validation of the proposed GGW sharp-rejection multi-band BPF concept for implementations with more than two passbands, or operating above the Ku-band, can be readily achieved by incorporating additional inductively coupled cavities or by using higher-precision fabrication processes. Consequently, these extensions do not introduce any fundamental or additional technological challenges beyond those already encountered in the present GGW filter prototype.