Analysis and design of the tri-band filter

The printed circuit board layout of the proposed tri-band filter is illustrated in Fig. 1. The filter consists of three different resonators named SLRRR, SSLSIR, and SIR as shown in Fig. 2. The symmetrical SSLSIRs are arranged inside the SLRRR with coupled gap g ₁ and g ₂, two identical SIRs with 0° tapped feed structure are loaded on both sides of SLRRR with coupled gap g ₃. In the following section, SLRRR is described as an example to illustrate the analysis method of the resonator as SIR and SSLSIR are well documented in [Reference Jiang, Shen, Wang, Huang, Peng and Wang2, Reference Sachan and Chauhan3], respectively.

Fig. 1. Layout of the proposed tri-band filter with dimensions.

Fig. 2. Layout of (a) the SLRRR, (b) the SIR, and (c) the SSLSIR.

The SLRRR consists of a rectangular ring resonator and two open stubs with electrical length θ _i, i = 1, 2, 3 and characteristic impedance Z _i, i = 1, 2, 3 which is depicted in Fig. 2(a). The open stubs are loaded at the symmetrical plane on both sides of the ring. The SLRRR is placed between SSLSIR and SIR to provide a tight couple for each band. Since SLRRR is a symmetrical structure, even-odd mode equivalent circuit can be obtained by short and open symmetrical line respectively, as shown in Fig. 3. For simplicity, assume ${Z}_ 1 = {Z}_ 2 = {Z}_{c}$ and${\theta }_ 1 + {\theta }_ 2 = {\theta }_{c}$, the input admittance of odd-even mode from the input port can be formulated by equations (1) and (2):

(1)$${Y}_{{in\_odd}} = -{\,j}2{/}{Z}_{c}{\tan}{\theta }_{c}, \;$$

(2)$${Y}_{{in\_even}} = \displaystyle{{-j 2( \tan {\theta }_ 3 + R\tan {\theta }_{c}) } \over {{Z}_{c}( \tan {\theta }_ 3\tan {\theta }_{c}-R) }}, \;$$

where Z _e is the characteristic impedance corresponding to half-width of W4, ${R} = {Z}_{e}{/}{Z}_{c}$ is the electrical length ratio of the two sections in even mode. The resonant frequencies of the odd and even modes can be derived by making the input admittance ${Y}_{{in\_odd}}$ and ${Y}_{{in\_even}}$ equal to 0, which can be expressed as (3) and (4):

(3)$${\tan}\theta _c = \infty , \;$$

(4)$$\tan {\theta }_{e} + {R}\cdot \tan {\theta }_{c} = 0.$$

Fig. 3. Equivalent circuit of SLRRR in odd-mode analysis (a) and even-mode analysis (b).

Figure 4 shows the curves of the first order odd-even mode f _odd1 and f _even1 normalized by the fundamental mode f _odd0 and f _even0 with a different combination of electrical length ratios and impedance ratio R, respectively. It can be found that fundamental odd-mode and high order odd-mode are fixed multiples. By changing Z ₃ and θ ₃, even resonant modes can be distributed without affecting odd resonant modes. Thus, the design flexibility of the multi-band filter is improved.

Fig. 4. The first order odd-mode f_{odd 1} normalized by f_{odd 0} and first order even-mode f_{even 1} normalized by f_{even 0}.

Similarly, the SSLSIR and SIR can be analyzed by the same odd-even mode method. The symmetrical SSLSIRs are arranged inside the SLRRR and two identical SIRs are loaded on both side of SLRRR. The structure not only leads to a compact size but also achieves a novel cross-coupling as denoted in Fig. 5(a). For optimal parameters, W ₄ = 1.9 mm and L ₄ = 7.9 mm, the fundamental modes f _odd0 and f _even0 of SLRRR are constructed to the desired second passband. By tuning physical length L ₅ = 5.3 mm, L ₆ = 0.5 mm, L ₇ = 1.3 mm, the fundamental modes f _odd0 and f _even0 of SSLSIR can be adjusted to the second passband and the third passband. Also, using the optimized parameter L ₁ = 9.72 mm, L ₂ = 3 mm, W ₂ = 2.7 mm, the first three resonant modes f _odd0, f _even0, and f _odd1 of SIR could be adjusted to first passband, second passband, and third passband, respectively. Figure 5(b) shows the distribution of all modes in the spectrum which constitutes the foundation response of the controllable tri-band.

Fig. 5. (a) Coupling diagram of the tri-band BPF. (b) Resonant distribution of SLRRR, SIRLLL, and SIR.

In order to generate controllable bandwidth, it is necessary to determine the external quality factor Q _e for each band. The external quality factor can be calculated by (5):

(5)$${Q}_{e} = \displaystyle{{{\,f}_ 0} \over {{\Delta }{\,f}_{{\pm }90^\circ }}}, \;$$

here ${\Delta }{f}_{{\pm }90^\circ }$ is the frequency difference of the phase up and down by 90^° based on the center frequency. The optimal external quality factor Q _e of the three passbands can be obtained by adjusting the gap g ₅, g ₂, g ₃, respectively. Figure 6(a) shows the simulated curve of Q _e as a function of the respective gap for the three bands with fixed length and width L ₄ = 7.9 mm, L ₃ = 9 mm, L ₅ = 5.2 mm，W ₄ = 7.9 mm, W ₃ = 0.5 mm, W ₅ = 1 mm. A larger coupled gap results in a higher external quality factor but a smaller coupling coefficient and bandwidth, in other words, a controllable bandwidth can be reached by a different gap. The surface electric field distribution at the three center frequencies is shown in Fig. 6(b). The signals travel through different paths, which depict the ability to control the bandwidth of each band independently.

Fig. 6. (a) Simulated fractional bandwidth and Q _e versus g ₅, g ₂, g ₃ for tri-band. (b) Surface electric field distribution of the filter at the three center frequencies.

By the configuration of 0° feed structure and two quarter-wavelength lines loaded at the I/O ports, four TZs are introduced to improve the out-of-band rejection, as shown in Fig. 7. Two quarter-wavelength lines are constructed from the tapped position indicated by d ₁ to the end of the SIR, respectively, thus signals are short-circuited at the corresponding frequency to obtain TZ ₁ and TZ ₂. Similarly, the other two quarter-wavelength lines which are denoted by L ₈ and L ₉ lead to the formation of TZ ₃ and TZ ₄. The positions of TZ ₂ and TZ ₃ can be optimized between the second and third bands to achieve a sharp passband skirt. Obviously, the four TZs can be controlled independently by the generation principles.

Fig. 7. The simulated and measured results of the proposed tri-band filter with a photograph.

Measurement and discussion

In order to verify the principle, an example of the tri-band filter is optimized and fabricated by employing double-sided microwave board Rogers 4350B with permittivity of 3.48, thickness of 0.5 mm and loss tangent of 0.0035. The photograph of the fabricated tri-band filter including the simulated and measured S parameter curves is given in Fig. 7. The overall size of the filter is 26.3 × 12.3 mm². The dimension parameters are optimized using HFSS 15.0, which can be summarized as follows: L ₁ = 9.72 mm, L ₂ = 3 mm, L ₃ = 9 mm, L ₄ = 7.9 mm, L ₅ = 5.3 mm, L ₆ = 0.5 mm, L ₇ = 1.3 mm, L ₈ = 5.75 mm, L ₉ = 5.45 mm, W ₁ = 1 mm, W ₂ = 2.7 mm, W ₃ = 0.5 mm, W ₄ = 1.9 mm, W ₅ = 1 mm, W ₆ = 2 mm, W ₇ = 1 mm, W ₈ = 0.5 mm, d ₁ = 4.1 mm, d ₂ = 1.8 mm, g ₃ = g ₄ = g ₅ = 0.2 mm, and the radius r of the via which is short-circuited to the reference ground is 0.5 mm.

The center frequencies of the optimized filter are located at 2.76, 5.7, and 7.63 GHz with fractional bandwidths of 13%, 5.7%, and 3.7%. The minimum insertion loss of each passband is −0.9 dB, −1.7 dB, and −2.2 dB, respectively. Due to the fabrication tolerance and slight variation of the substrate's permittivity, TZ ₂ and TZ ₃ are almost coincident at 6.6 GHz, however, a sharp upper passband skirt is still achieved. Four TZs located at 3.2, 6.6, 6.7, and 8.56 GHz are close to the edge of the passbands, which can improve the out-of-band rejection significantly. The slopes of the upper edge of the three passbands are 161, 80, and 41 dB/GHz, which indicate that sharp upper skirts are achieved. For performance comparison, Table 1 lists the results of recently reported works that are similar to the proposed tri-band filter. In our work, the filter has the characteristics of miniaturization and a wide bandwidth of −30 dB attenuation between passbands due to the sharp shirts.

Table 1. Performance comparison with recently reported works.