Filter analysis and design

Figure 1(a) shows the structure of a conventional two-pole SIW bandpass filter. Two SIW cavities are coupled through an iris-window to produce a passband. Its simulated S-parameters are presented in Fig. 1(b). It is seen that the TE₁₀₁ modes of the cavity generate a passband at 10 GHz while the second passband is created at 15.6 GHz due to the first higher-order mode TE₁₀₂ of the cavity. The simulated E-field distributions in the filter at 10 GHz (f ₁₀₁) and 15.6 GHz (f ₁₀₂) are plotted in Figs 2(a) and 2(b), respectively. It can be seen that the first passband is created by TE₁₀₁ mode of the cavities while the second passband is generated by the TE₁₀₂ mode of the cavities which degrades the stopband performance of the filter. The stopband performance can be improved by suppressing the passband constituted by TE₁₀₂ modes.

Fig. 1.

(a) Structure of a conventional two-pole SIW bandpass filter (a = 11.5 mm, b = 11.2 mm, and g = 5.1 mm) and (b) simulated S-parameters.

Fig. 2.

Simulated E-field distributions in the filter at (a) 10 GHz (f ₁₀₁) and (b) 15.6 GHz (f ₁₀₂).

The configuration of the proposed compact SIW bandpass filter with a wide-stopband is illustrated in Fig. 3. A short-circuited coplanar line is introduced between two SIW cavities [Reference Azad and Mohan15]. The iris-window between the two SIW cavities produces magnetic coupling of TE₁₀₁ and TE₁₀₂ modes of the cavity, whereas the electric coupling of these two modes is generated by the short-circuited coplanar line, as discussed in [Reference Azad and Mohan15]. Thus, the overall couplings of TE₁₀₁ and TE₁₀₂ modes are mixed coupling, and the coupling of the two modes can be controlled by adjusting the width of the iris-window (g), the length (l), and the width (w ₄) of the short-circuited coplanar line. The passband created by the TE₁₀₂ mode of the cavity can be suppressed by minimizing the coupling of TE₁₀₂ mode.

Fig. 3.

Configuration of the proposed compact SIW bandpass filter with wide-stopband performance.

In this work, the short-circuited coplanar line is not only used to control the coupling of the two modes of the cavity, but also acts as a resonator. The resonant frequencies of the TE₁₀₁ mode of SIW cavities (f ₁ and f ₂) and the fundamental resonant frequency of the short-circuited coplanar line (f ₃) can be adjusted to constitute a third-order filter response. The initial dimensions of the SIW cavity resonating at its fundamental resonant mode TE₁₀₁ can be evaluated by using the relations:

(1)$$\eqalign{&f_{{\rm TE}101} = \displaystyle{c \over {2\sqrt {\varepsilon _r}}} \sqrt {{\left( {\displaystyle{1 \over {a_e}}} \right)}^2 + {\left( {\displaystyle{1 \over {b_e}}} \right)}^2}; \cr &a_e = a-\displaystyle{{d^2} \over {0.95p}},\quad b_e = b-\displaystyle{{d^2} \over {0.95p}}$$

where c is the speed of light in vacuum, ɛ _r is the dielectric constant of the substrate, d is the diameter of the metalized via-holes and p is the center-to-center spacing between the via-holes. Furthermore, the fundamental resonant frequency of the short-circuited coplanar line can be approximately determined by using the relation:

(2)$$f_3\simeq \displaystyle{c \over {2l\sqrt {\varepsilon _{re}}}} $$

where l is the length of the short-circuited coplanar line resonator and ɛ _re is the effective dielectric constant of the substrate which depends on the width of the short-circuited coplanar line and the height of the substrate. The resonant frequencies f ₁, f ₂, and f ₃ are determined for desired CF and bandwidth of the filter. The size of the proposed filter is very compact compared to conventional third-order SIW filters. The filter is designed on the Rogers RO4003C substrate with ɛ _r = 3.55, h = 0.508 mm, and tan δ = 0.0027. The metalized via-holes have d = 1 mm and p = 1.6 mm. Full-wave EM simulator Ansoft HFSS is used to perform the analysis and design of the filter.

The simulated S-parameters of the proposed filter under the condition of weak I/O coupling are shown in Fig. 4. The dimensions of the filter in the simulation are: a = 11.25, b = 11.2, wf = 1.2, lin = 1, lf = 3, t = 0.4, g = 4.8, l = 10.25, w ₁ = 1.8, w ₂ = 1.4, w ₃ = 1, and w ₄ = 0.2 (unit: mm). As can be seen, the resonance due to the short-circuited coplanar line (f ₃) occurs at 9.29 GHz while that of the TE₁₀₁ mode of the SIW cavities (f ₁ and f ₂) occur at 10.5 and 10.99 GHz, which illustrate that a passband can be created by adjusting the resonant frequencies f ₁, f ₂, and f ₃ appropriately. At the same time, the passband created at 15.93 GHz due to TE₁₀₂ mode of the cavities degrades the stopband attenuation of the filter, and it needs to be eliminated in order to improve the stopband performance of the filter. Figure 5 presents the simulated E-field distributions in the proposed filter at 9.29 GHz (f ₃), 10.5 GHz (f ₁), 10.99 GHz (f ₂), and 15.93 GHz (f ₁₀₂). It can be clearly observed that the first resonance is created by the short-circuited coplanar line introduced between the two SIW cavities while the second and third resonances are generated by the TE₁₀₁ mode of the cavities. On the other hand, the undesired passband is generated by the TE₁₀₂ mode of the cavities, and the stopband performance of the filter is degraded.

Fig. 4.

Simulated S-parameters of the filter under weak coupling condition (a) |S ₂₁| and (b) |S ₁₁|.

Fig. 5.

Simulated E-field distributions in the proposed filter at (a) 9.29 GHz (f ₃), (b) 10.5 GHz (f ₁), (c) 10.99 GHz (f ₂), and (d) 15.93 GHz (f ₁₀₂).

Figure 6 presents the simulated S-parameters of the filter against variation in g while keeping l and w ₄ fixed under the condition of weak I/O coupling. The overall coupling of TE₁₀₁ mode is electric dominant. The magnetic coupling decreases with a decrease in g, and the overall coupling of TE₁₀₁ mode increases. This phenomenon can be verified by the graphs plotted in Fig. 6. The overall coupling of TE₁₀₁ mode increases since the separation between f ₁ and f ₂ increases [Reference Hong and Lancaster16] as g decreases from 5.5 to 3 mm, as shown in Figs 6(a) and 6(b). On the other hand, the overall coupling of TE₁₀₂ mode is magnetic dominant, and it decreases with a decrease in g. As can be seen, for g = 5.5 mm, a passband is generated at nearly 16 GHz due to TE₁₀₂ mode, and it is suppressed below −18 dB when g = 3 mm, as shown in Fig. 6(a). Since f ₂ increases while f ₁ and f ₃ remain almost constant with a decrease in g, the CF of the filter slightly shifts towards higher frequency, and the bandwidth of the filter shows a small increase. In addition, two transmission zeros (TZs) are observed at the upper stopband. The location of the first TZ shifts towards higher frequency while that of the second TZ moves towards lower frequency if g is decreased, as depicted in Fig. 6(a).

Fig. 6.

Simulated S-parameters of the filter against variation in g (l = 10.25 mm and w ₄ = 0.2 mm) (a) |S ₂₁| and (b) |S ₁₁|.

The simulated S-parameters of the filter against variation in l while maintaining constant g and w ₄ are presented in Fig. 7. The overall coupling of TE₁₀₁ mode is electric dominant. The magnetic coupling increases as l is increased, and the overall coupling of TE₁₀₁ mode decreases. This behavior is verified by the curves presented in Fig. 7. The overall coupling of TE₁₀₁ mode decreases since the separation between f ₁ and f ₂ decreases when l is increased from 10 to 10.5 mm, as depicted in Figs 7(a) and 7(b). However, the overall coupling of TE₁₀₂ mode is magnetic dominant, and it increases with increasing l. As a result, the suppression level of the passband produced by the TE₁₀₂ modes decreases slightly, still keeping below −16 dB, as shown in Fig. 7(a). Since the length of the short-circuited coplanar line l increases, f ₃ decreases largely. At the same time, since f ₂ slightly decreases while f ₁ remains constant with an increase in l, the CF of the filter slightly moves towards lower frequency, and the bandwidth of the filter is slightly modified, as shown in Figs 7(a) and 7(b). Moreover, the positions of the TZs shift towards lower frequency as l increases, as depicted in Fig. 7(a).

Fig. 7.

Simulated S-parameters of the filter against variation in l (g = 3 mm and w ₄ = 0.2 mm) (a) |S ₂₁| and (b) |S ₁₁|.

Figure 8 depicts the simulated S-parameters of the filter against variation in w ₄ while keeping fixed g and l. The overall coupling of TE₁₀₁ mode is electric dominant, and it increases as w ₄ is increased because the electric coupling increases with an increase in w ₄. The graphs plotted in Fig. 8 verify this phenomenon. The overall coupling of TE₁₀₁ mode increases slightly since the separation between f ₁ and f ₂ shows a slight increase as w ₄ is increased from 0.2 to 0.4 mm, as presented in Figs 8(a) and 8(b). Furthermore, the suppression level of the passband generated by the TE₁₀₂ modes increases from −18 to −22.5 dB with an increase in w ₄ because the overall coupling of TE₁₀₂ mode is magnetic dominant, and it decreases with increasing w ₄. Since the overall dimension of the short-circuited coplanar line increases as w ₄ is increased, f ₃ decreases largely. At the same time, since f ₂ slightly increases while f ₁ remains constant with increasing w ₄, the CF of the filter slightly shifts towards lower frequency, and the bandwidth of the filter increases, as shown in Figs 8(a) and 8(b). Moreover, the position of the first TZ shifts towards higher frequency while that of the second TZ moves towards lower frequency as w ₄ is increased, as depicted in Fig. 8(a).

Fig. 8.

Simulated S-parameters of the filter against variation in w ₄ (g = 3 mm and l = 10 mm) (a) |S ₂₁| and (b) |S ₁₁|.

In order to illustrate the effect of I/O coupling on the frequency response of the filter, the simulated S-parameters against variation in lf are presented in Fig. 9. It is found that the CF of the filter slightly shifts towards lower frequency, and the bandwidth of the filter shows a small decrease as lf is increased from 5.8 to 7.4 mm, as depicted in Figs 9(a) and 9(b). Moreover, the in-band insertion loss shows a small variation with a change in lf, as presented in Fig. 9(a). At the same time, the passband return loss greatly changes with variation in lf. It can be observed that lf = 6.6 mm exhibits better return loss in the passband for the chosen filter dimensions, as shown in Fig. 9(b). Furthermore, the locations of the TZs remain almost unchanged as lf varies while the suppression level of the TE₁₀₂ mode is slightly improved with increasing lf.

Fig. 9.

Simulated S-parameters of the filter against variation in lf (g = 2.85 mm, l = 10.3 mm, and w ₄ = 0.2 mm) (a) |S ₂₁| and (b) |S ₁₁|.

In order to compare the proposed filter with the conventional SIW filters, a SIW third-order inline filter is designed. The two filters are designed for identical CF and bandwidth. Figure 10(a) presents the structure of a conventional SIW third-order inline filter. The dimensions of the SIW inline filter are: a ₁ = 11.3 mm, a ₂ = 10.45 mm, and b = 11.2 mm, and the dimensions of the proposed filter are: a = 11.25 mm and b = 11.2 mm. The overall size of the SIW inline filter excluding microstrip feedlines is 34.6 mm × 12.3 mm (including side via-holes) while that of the proposed filter excluding microstrip feedlines is 23.4 mm × 12.25 mm (including side via-holes). It can be observed that the filter size is reduced by a factor of 1/3 in comparison with SIW counterpart. The comparison of simulated |S₂₁| between the conventional SIW third-order inline filter and the proposed filter is presented in Fig. 10(b). It can be seen that the proposed filter introduces an insertion loss of only 0.2 dB compared to SIW counterpart. Since the Q-factor of the short-circuited coplanar line resonator is lower than that of the SIW cavity resonator, the insertion loss of the proposed filter is slightly higher and the power-handling capability is slightly lower compared to that of the SIW filter. However, TZs can be generated using the proposed structure while the conventional SIW inline filters do not produce any TZs.

Fig. 10.

(a) Structure of the conventional SIW third-order inline filter and (b) comparison of simulated |S ₂₁| between conventional and proposed SIW filters.

Above results show that a compact third-order bandpass filter can be designed using two SIW cavities and a short-circuited coplanar line. The CF and the bandwidth of the filter can be controlled by adjusting the structural parameters of the filter. In addition, wide-stopband performance can be obtained by suppressing the passband constituted by the TE₁₀₂ modes. The design steps of the filter can be summarized as follows:

1. (1) Determine the desired specifications of the filter such as the CF (f ₀) and FBW (Δ).

2. (2) The initial values of the structural parameters a, b, and l are determined for desired f ₀.

3. (3) The geometrical parameters g, l, and w ₄ can be determined for appropriate f ₁, f ₂, and f ₃ that satisfy the desired f ₀ and Δ. It is suggested to choose a small value of g in order to suppress the passband created by TE₁₀₂ mode.

4. (4) Next, the lf is adjusted to achieve better return loss in the passband. It should be noted that the CF and the bandwidth are slightly modified when lf is adjusted.

5. (5) Finally, the dimensions of the filter are fine tuned to obtain the desired filter specifications.

Fabrication and measurement

A compact third-order bandpass filter with wide-stopband performance is designed, fabricated, and measured to validate the proposed approach. The specifications of the filter are: CF of f ₀ = 10 GHz and 3-dB FBW of Δ = 17.5%. The final dimensions of the filter are: a = 11.25, b = 11.2, wf = 1.2, lin = 1, lf = 6.51, t = 0.4, g = 2.85, l = 10.34, w ₁ = 1.8, w ₂ = 1.4, w ₃ = 1, and w ₄ = 0.2 (unit: mm). Figure 11(a) shows the photograph of the fabricated filter, and the simulation and measurement results are presented in Fig. 11(b). The measurement results show that the CF of the filter is f ₀ = 9.97 GHz and the 3-dB FBW is Δ = 14.74%. The measured minimum in-band insertion loss is around 1.65 dB, and the passband return loss is higher than 10 dB. In addition, two TZs are observed at 15.25 and 18.74 GHz in the measured response of the filter. Moreover, the first higher-order mode TE₁₀₂ is suppressed below −25 dB, and the rejection level is below −20 dB from 12.08 to 19.35 GHz (1.94f ₀). The stopband rejection of the filter is limited due to the presence of the second resonant mode of the short-circuited coplanar line at around 19.64 GHz. Furthermore, the simulated radiation loss R _r is extracted using the relation R _r = 1 − |S ₁₁|² − |S ₂₁|² under the conditions of the lossless substrate and conductor. It can be observed that the radiation loss is below 5% in the frequency range of 6–19.6 GHz, as shown in Fig. 11(b), which implies that the proposed structure has very small radiation loss. The overall filter size excluding microstrip feedlines is 23.4 mm × 12.25 mm, which is equivalent to 1.47λ _g × 0.77λ _g, where λ _g is the guided wavelength at f ₀. Moreover, the simulation and measurement results are in close agreement. The discrepancies in the measurement results can be attributed to fabrication tolerances and losses caused by SubMiniature version A connectors.

Fig. 11.

(a) Photograph of the fabricated filter and (b) simulated and measured results.

Figure 12 shows the simulated E-field distributions in the proposed filter at 10 GHz (f ₀) and 16 GHz (f ₁₀₂). It can be seen that TE₁₀₁ mode in the cavities and the short-circuited coplanar line constitute the passband while the TE₁₀₂ mode is successfully suppressed, and the passband due to TE₁₀₂ mode is not generated. Moreover, Table 1 summarizes the comparison of the proposed filter with reported compact filters using SIW and planar resonators. As can be observed, the proposed filter meets given filter specifications, and shows the advantages of wide bandwidth and wide-stopband rejection while maintaining compact size.

Fig. 12.

Simulated E-field distributions in the proposed filter at (a) 10 GHz (f ₀) and (b) 16 GHz (f ₁₀₂).

Table 1.

Comparison of the proposed filter with reported compact filters using SIW and planar resonators

f ₀, center frequency; BW, bandwidth; IL, insertion loss.

^a Estimated data.