HMSIW cavity

HMSIW cavity is extracted by cutting the SIW cavity along a factitious magnetic wall, as shown in Fig. 1(a). The resonant frequency of TM _m−0.5,n,0 of HMSIW cavity is given by

(1)$$f_{{TM}_{{m-0.5\comma n\comma 0}}} = {c\over 2 \sqrt{\mu_{r} \varepsilon_{r}}} \sqrt{\left({m-0.5\over W}\right)^{2} + \left({n\over L}\right)^{2}}$$

where c is the speed of light, W and L are the dimensions of the cavity, and ɛ_r and μ_r are dielectric substrate parameters.

Fig. 1.

(a) HMSIW cavity. The E-field distribution of first five modes of the cavity: (b) TM ₁₁₀, (c) TM ₁₂₀, (d) TM ₃₁₀, (e) TM ₁₃₀, and (f) TM ₃₂₀ (W = 8.5 mm, L = 15.0 mm, S = 1.0 mm, and D=0.6 mm).

The first step is to determine the dimensions of the HMSIW cavity. The initial size of the HMSIW cavity could be calculated with prescribed resonant frequencies of TM ₁₁₀ and TM ₃₁₀ modes by the following equation, according to [Reference Guo, Zhu and Wu22]:

(2)$$\eqalign{& W = \displaystyle{c \over {2\sqrt {\mu _r\varepsilon _r} }}\sqrt {\displaystyle{2 \over {\,f_{TM_{1.5,1,0}}^2 -f_{TM_{0.5,1,0}}^2 }}} + \displaystyle{{D^2} \over {0.95\cdot S}} \cr & L = \displaystyle{c \over {2\sqrt {\mu _r\varepsilon _r} }}\sqrt {\displaystyle{8 \over {9f_{TM_{0.5,1,0}}^2 -f_{TM_{1.5,1,0}}^2 }}} + \displaystyle{{D^2} \over {0.95\cdot S}}} $$

The HMSIW cavity is made using an RT/duroid 5880 substrate (0.508 mm thickness) with ɛ_r = 2.2 and tanδ = 0.0009. The parameters are (in mm): diameter of vias D = 0.6, pitch S =1.0, L = 15, and W = 8.5. According to the synthesized design parameters, TM ₁₁₀ and TM ₃₁₀ modes of cavity are expected to be located at 8.6 and 18.3 GHz. This structure is simulated with a high-frequency structure simulator (HFSS) eigenmode simulator, and the electric field distribution of the first five modes of HMSIW cavity is shown in Figs 1(b)–1(f). The first five modes are TM ₁₁₀, TM ₁₂₀, TM ₃₁₀, TM ₁₃₀, and TM ₃₂₀, their resonant frequencies are 8.6, 13.6, 18.3, 19.3, and 21.16 GHz. The unloaded quality factor (Q _u) of all the modes is above 400.

HMSIW cavity with back-to-back connected DGS unit-cell

HMSIW cavity is loaded with the half-mode DGS unit-cell to generate a passband at low frequency. The half-mode DGS is extracted by cutting back-to-back connected E-shaped DGS along the magnetic wall, as shown in Fig. 2. The proposed novel half-mode DGS size is half, and its resonant characteristics are almost similar to the original back-to-back connected E-shaped DGS [Reference Cai, Qian, Cao, Zhang and Yang26]. From Figs 1(a) and 2(b), both the HMSIW cavity and half-mode DGS are sharing the magnetic wall. Therefore, the half-mode DGS unit-cell is loaded into the HMSIW cavity, as shown in Fig. 3. This structure is simulated with the HFSS EM-simulator.

Fig. 2.

(a) Back-to-back connected E-shaped DGS unit-cell. (b) Half-mode DGS unit-cell.

Fig. 3.

HMSIW cavity with half-mode DGS unit-cell (W = 8.5 mm, L = 15 mm, S = 1.0 mm, D = 0.6 mm, W _s = 2.0 mm, W ₁ = 0.8 mm, L ₁₁ = 4.0 mm, L ₂ = 4.1 mm, L ₃ = 3.7 mm, L ₄ = 1.35 mm, S ₁ = 0.4 mm, t = 6 mm, and S ₂ = 0.3 mm).

The equivalent circuit of HMSIW cavity with half-mode DGS is shown in Fig. 4. The HMSIW cavity is like an ordinary two-wire transmission line loaded with via-walls which provide a shunt inductor (L _d). Half-mode DGS is constituted as the shunt connected resonant tanks (L _r and C _r). The shunt connection of L _z and C _z models the mixed-coupling between DGS resonators. L _c and C _c indicate magnetic and electric coupling, respectively at feedlines. It is noted that this circuit model is a simplified version, and it is simulated by using an advanced design system circuit simulator. In Fig. 5, the electromagnetic (EM) and circuit simulations of fig. 3 and fig. 4, respectively, are compared. Therefore, this circuit is correct and fully capable of explaining the transmission characteristics. The even- and odd-mode analysis method has been used to find the resonant frequencies (f _odd and f _even) and two TZs f _zp1 and f _zp1 [Reference Xu, Ma, Meng and Yeo4]. The even and odd mode circuits are shown in Fig. 6. The even and odd mode frequencies are obtained by finding the input admittance and solving the equation of Im(Y _in)=0 using respective equivalent circuits:

(3)$$\eqalign{& f_{odd} = \displaystyle{{\sqrt {L_rL_z + L_cL_z + 2L_cL_r} } \over {2\pi \sqrt {L_cL_rL_z\left( {C_c + C_r + 2C_z} \right)} }} \cr & f_{even} = \displaystyle{{\sqrt {L_r + L_c} } \over {2\pi \sqrt {L_cL_r\left( {C_c + C_r} \right)} }}} $$

Fig. 4.

Equivalent circuit of the half-mode DGS unit-cell. (The element values are: L_c = 1.26 nH, C_c = 0.77 pF, L_d = 3.1 nH, L_z = 1.04 nH, C_z = 0.33 pF, L_r = 0.58 nH, and C_r = 0.87 pF.)

Fig. 5.

EM simulation and circuit simulation of Figs 3 and 4, respectively.

Fig. 6.

Equivalent circuits of Fig. 4: (a) odd mode and (b) even mode.

Fig. 7.

The coupling coefficient (M₁₂) varied with S _1.

On the other hand, Y _in,even = Y _in,odd condition gives the TZs:

(4)$$f_{{ZP} 1} = {1\over 2 \pi \sqrt{L_{{c}} C_{{c}}}}\comma\; \quad f_{{ZP} 2} = {1\over 2 \pi \sqrt{L_{{z}} C_{{z}}}}$$

As is well known, the coupling coefficient (K) between two resonators and external quality factor (Q _ext) can be evaluated by [Reference Hong and Lancaster27]

(5)$$K = {\vert f_{odd}^{2}-f_{even}^{2}\vert \over f_{odd}^{2} + f_{even}^{2}}$$

(6)$$Q_{ext} = {\,f_{0}\over \Delta f_{3-dB}}$$

where f ₀ is the center frequency (CF) and $f_{3\hbox {-dB}}$ is 3-dB bandwidth. Here, the DGS length should be set to a quarter wavelength at the CF of passband. Then, the design parameters could be calculated using Q _ext and M _ij from prescribed specifications of the passband using the synthesis procedure described in [Reference Hong and Lancaster27].

The coupling coefficient (M ₁₂) = 0.0802 and external quality factor (Q _ext) = 13.76 are calculated using lowpass prototype element values [Reference Hong and Lancaster27] for 20 dB return loss bandwidth 4.83% and CF 6.2 GHz. Figures 7 and 8 show the extraction of coupling coefficient with respect to S ₁ using equation (5) and the extraction of an external quality factor with respect to t using equation (6), respectively. The parameters S ₁ and t are chosen based on M ₁₂ and Q _ext values. As shown in Fig. 5, the odd and even mode frequencies of the proposed DGS are 6.1 and 6.3 GHz, respectively. The TZs are at 4 and 8 GHz, which increase roll-off characteristics of the first passband.

Fig. 8.

The variation of external quality factor (Q_ext) with t.

HMSIW cavity with slot perturbation

HMSIW cavity's first four mode frequencies are pushed up to 11, 13, 17.5, and 21 GHz, due to restricts of the fields to a smaller region and change of current distribution by DGS unit-cell. Moreover, TM ₁₃₀ mode current distribution is not supported by DGS unit-cell. In order to make a dual passband using a mode shift technique, a slot line is etched along with half-mode DGS on the top layer of HMSIW cavity, as shown in Fig. 9. Fig. 10 shows the impact of slot line on resonant modes of the cavity. The slot line shifts TM ₁₂₀ and TM ₃₂₀ modes resonant frequency effectively, while the resonant frequencies of TM ₁₁₀ and TM ₃₁₀ modes are almost unchanged. The reason is, the etched slot does not affect the current distribution of TM ₁₁₀ and TM ₃₁₀ modes, whereas the etched slot perturbs the current distribution of TM ₁₂₀ and TM ₃₂₀ modes. Therefore, the slot line length (L ₁) and the feed position (t) are fixed to prescribed fractional bandwidths with return loss better than 20 dB. However, the DGS modes are constant for slot length. Therefore, the DGS modes give passband at low-frequency, TM ₁₁₀ and TM ₁₂₀ make second passband and TM ₃₁₀ and TM ₃₂₀ generate third transmission band.

Fig. 9.

HMSIW cavity with slot perturbation (W = 8.5 mm, L = 15 mm, S = 1.0 mm, D=0.6 mm, L ₁ = 3.1 mm, L ₂ = 4.1 mm, L ₃ = 3.7 mm, L ₄ = 1.35 mm, S ₁ = 0.4 mm, and S ₂ = 0.3 mm).

Fig. 10.

The variation of frequencies of the first four resonant modes versus L ₁.

Triple-band filter

The proposed triple-band filter configuration is shown in Fig. 11. It consists of half-mode DGS unit-cell and HMSIW cavity with slot perturbation. The DGS modes give the first band, and the HMSIW cavity with slot line generates the second and third transmission bands. The proposed HMSIW cavity with DGS unit-cell is excited with step impedance I/O feedlines.

Fig. 11.

Layout of the proposed triple-band structure.

Design procedure

The design procedure of triple-band BPF based on the proposed multi-mode HMSIW cavity can be epitomized as follows:

1. (1) Determine the initial dimensions of the HMSIW cavity, as discussed in section “HMSIW cavity” according to the CFs of the second and third passbands.

2. (2) Determine the DGS length (L ₁₁ + L ₂/2 + L ₃ + L ₄), which is nearly a quarter-wavelength at first passband CF.

3. (3) Determine the design parameters consisting of Q _ext and M _ij from prescribed specifications, including the CF and the bandwidth of the first passband using the synthesis procedure described in [Reference Hong and Lancaster27].

4. (4) Extract the curves of the external quality factor Q _ext and the coupling coefficients M _ij in the first passband versus the positions of the feeding ports t and the coupling slot S ₁, respectively. The parameters S ₁ and t selection depends on M ₁₂ and Q _ext values.

5. (5) Adjust the length of the etched slot (L ₁) and the feed position (t) to achieve prescribed fractional bandwidths of second and third passbands with return loss better than 20 dB.

6. (6) Tune filter parameters to optimize the performance.

As an example, a tri-band filter is designed with the targeted specifications as follows:

1. (1) passband 1: 6.2 GHz (FBW: 0.09);

2. (2) passband 2: 12.5 GHz (FBW: 0.062);

3. (3) passband 3: 18.5 GHz (FBW: 0.05);

4. (4) in-band return loss: 20.0 dB.

First, the initial dimensions of the HMSIW cavity are determined according to TM ₁₁₀ and TM ₃₁₀ modes resonant frequencies at 8.6 and 18.3 GHz, respectively, using equation (2). The parameters are (in mm): diameter of vias D = 0.6, pitch S =1.0, L = 15, and W = 8.5. Second, the initial length of half-DGS is calculated at the first CF which is nearly a quarter-wavelength. The calculated DGS length is L ₁₁+L ₂/2+L ₃+L ₄ = 36 mm. Then, the coupling coefficient (M ₁₂)=0.0802 and external quality factor (Q _ext)=13.76 are calculated using lowpass prototype element values [Reference Hong and Lancaster27] of 20 dB return loss bandwidth 4.83% and CF 6.2 GHz. The parameters S ₁ and t are chosen to be 0.3 and 6 mm, respectively based on M ₁₂ and Q _ext values from Figs 7 and 8. Next, the etched slot length (L ₁) and the feed position (t) are chosen as 6.8 and 7 mm, respectively based on fractional bandwidths of second and third passbands with return loss better than 20 dB.

All dimensions are optimized to obtain better characteristics in three bands using the HFSS electromagnetic (EM) simulator. The dimensions are given in Table 1. The EM-simulated S-parameters of Fig. 11 are shown in Fig. 12. The first, second, and third passband CFs are 6.2, 13.0, and 19.1 GHz, respectively. There are two TZs at 4 and 8 GHz due to half-mode DGS, which provide attenuation of 30 dB from DC to 5 GHz and 20 dB from 7 to 10.4 GHz, respectively. The HMSIW cavity does not support the modes which do not satisfy magnetic wall conditions. Therefore, the significant separation is obtained between the second and third passbands. The first passband can be tuned by changing L ₄, while the second and third passbands are almost constant, as shown in Fig. 13. In Fig. 14, the vias are placed where they show the effect on second passband modes and do not affect third passband modes. By changing the position of vias (via), the second passband is tuned, and the first and third passbands are almost constant. Therefore, the bands can be tuned independently.

Fig. 12.

EM-simulated S-parameters of Fig. 11.

Fig. 13.

Frequency response of the BPF with varied L ₄.

Fig. 14.

Frequency response of the BPF with varied via.

Table 1.

Dimensions of the proposed structure

As shown in Fig. 12, the attenuation between the second and third passbands is around 15 dB. To improve this attenuation, a notch band is needed between the second and third passbands. The well-known techniques are conventional open-circuit stub, spur line, and embedded open-circuited stub [Reference Hong and Lancaster27]. The embedded open-circuited stub shows good performance and does not affect the main transmission line characteristics because the majority of the current flows around the edges of the microstrip line. In Fig. 15, T-shaped embedded open-circuited stubs are loaded in input/output (I/O) feedlines to get wideband notch characteristics. Here, the stub line width should be small to get high characteristic impedance, and the length of the stub line depends on the notch band CF. The simulated S-parameters are shown in Fig. 16, which shows that the embedded T-shaped open-circuited stub provided 45 dB attenuation between the second and third passbands.

Fig. 15.

Embedded T-shaped open-circuited stub in feedlines (C ₁ = 1.2 mm, C ₂ = 0.3 mm, C ₃ = 3.0 mm, C ₄ = 0.25 mm, and C ₅ = 0.7 mm).

Fig. 16.

Measured and simulated S-parameters.

Fig. 17.

Photo of fabricated triple-band BPF.