Fig. 12 Considered cases with three a rotated inserts or b inclined inserts in the waveguide Þlter

this is expected to be a real phenomenon during actual device operation. Figure 12a depicts the considered case with three rotated inserts, which has been shown to exhibit the most signiÞcant response degradation among these models. The rotation angle was set to be the same for all three inserts, either positive or negative. The maximum rotation angle for which the amplitude characteristic can be considered acceptable for arbitrary position of all three inserts was θ = 8◦ (case 1 is an example with the most signiÞcant response deviation).

Some of the considered positions with three inclined inserts are depicted in Fig. 12b. Similarly as for the rotated inserts, these models are highlighted as the most critical ones. The inclination angle is the same for all three inserts, either positive or negative. It has been shown that case 2 results in the most signiÞcant response deviation, even for small angles. However, the most probable is case 1, which may occur if the supporting structures attached to the top and bottom walls are slightly shifted, so all three inserts are inclined by the same angle and in the same direction. Thus, for perfectly fabricated inserts with inclination angle α ≈ 13◦, there is a deviation of the 3-dB bandwidth by around 5 % compared with the reference bandwidth of the original Þlter. On the other hand, for inserts fabricated with slightly shorter height (b1 ≈ 10.1 mm) and with inclination angle α = 8◦ for all three inserts, the Þlter response is preserved and the parameters of the amplitude characteristic meet the aforementioned set of conditions.

For the case when the Þlter inserts are simultaneously rotated and inclined, we found that the adopted criteria would be met if the rotation angle was θ ≤ 7◦ and the inclination angle was α ≤ 5◦.

4 Conclusions

The research presented in this paper considers various phenomena signiÞcant for investigation of the robustness of waveguide Þlters and the deviation of their frequency response. The Þlter response was investigated in terms of various parameters, including the implementation technology, the tolerance of the machine used for fabrication, and positioning of the inserts inside the waveguide.

Detailed analysis was performed for a waveguide resonator with a single insert and for a third-order Þlter, to obtain good insight into the operation of these structures and to identify crucial parameters affecting their performance.

WIPL-D software was used for the analysis, so precise models of the waveguide structures could be made, taking into account all the considered phenomena and effects. We found that the obtained results can be trusted, since they were

veriÞed experimentally. Measurement results are also provided for chosen examples, as an additional contribution to this research, conÞrming that the proposed approach can be used in practice.

Regarding substrate parameters, we found that the major inßuence on the Þlter response was from the dielectric permittivity, while variation of the losses and substrate thickness did not cause signiÞcant deviation, since the Þlter response remained within the tolerance limits. However, if the permittivity does not exceed the values provided by the manufacturer, there is no concern that its deviation will signiÞcantly degrade the desired response. We present a closed-form expression, based on a linear Þt, relating the resonance frequency and substrate permittivity, which can be used as a design formula for Þlter implementation.

Furthermore, we discovered that increase of the digging depth during the milling process has the most signiÞcant effect on the Þlter response. The value of the digging depth should be below 10 µm, if possible, to avoid Þlter response degradation. The other possible issues related to machine tolerance (inaccuracy of traces and dimensions of the printedcircuit inserts) practically do not degrade the Þlter response.

Finally, changing the position of the inserts, which can occur during the measurement procedure as well as in regular operation, did not introduce degradation of the Þlter response in the passband, for the critical angles we have determined, even though the length of the inverter may be slightly changed by this effect. This applies to the resonator with a single insert, as well as to the Þlter with three arbitrarily rotated or inclined inserts.

We adopted the following criteria and measures for Þlter performance degradation: the Þlter performance is not signiÞcantly degraded if (1) the relative change of the center frequency is less than 1 %, (2) the relative change of the bandwidth is less than 2 %, and (3) the absolute change of the passband attenuation is less than 0.3 dB. We found that these criteria will be met if the rotation angle is θ ≤ 7◦ and the inclination angle is α ≤ 5◦, simultaneously.

According to the obtained results, for both the resonator and Þlter, the frequency response changes in the same manner when the same parameter is varied. In summary, the major impact on the frequency response was from the dielectric permittivity of the printed-circuit board, a parameter that cannot be inßuenced, and the digging depth into the substrate in the milling process, which can be controlled.

Also, the simulation results show very good agreement with the experimental results obtained by measurements on a fabricated Þlter. This conclusion is important, as it conÞrms the accuracy of the models and the applicability of the proposed approach. The presented method may be applicable for similar practical problems regarding waveguide Þlters, since the presented results can highlight the most signiÞcant phenomena for the fabrication process.

The advantage of the proposed approach is its ability to improve and shorten the design process, since it enables the majority of settings and analyses to be performed in software, without unnecessary fabrication.

The Þndings of this work can be generalized to other types of waveguide Þlters that use metamaterial-inspired resonating inserts, and possibly to waveguide Þlters operating at higher frequencies.

Our future work in this area might include reliability study of the considered Þlter structure.

Acknowledgments This work was supported by the Ministry of Education, Science, and Technological Development of the Republic of Serbia under Grant TR32005.

References

1.Bonache, J., Gil, I., Garcia-Garcia, J., Martin, F.: Complementary split rings resonators (CSRRs): towards the miniaturization of microwave device design. J. Comput. Electron. 5(2), 193Ð197 (2006)

2.Moyra, T., Parui, S.K., Das, S.: Modeling and validation of high selective harmonic reduced bandpass Þlter using coupled resonators and defected ground structures. J Comput. Electron. 11(4), 330Ð335 (2012)

3.Moyra, T.: Modeling of bandpass Þlter using defected ground structure and defected microstrip structure. J Comput. Electron. 12(2), 287Ð296 (2013)

4.Ortiz, N., Baena, J.D., Beruete, M., Falcone, F., Laso, M.A.G., Lopetegi, T., Marques, R., Martin, F., Garcia-Garcia, J., Sorolla, M.: Complementary split ring resonator for compact waveguide Þlter design. Microw. Opt. Technol. Lett. 46(1), 88Ð92 (2005)

5.Bahrami, H., Hakkak, M., Pirhadi, A.: Using complementary split ring resonators (CSRR) to design bandpass waveguide Þlters. In: Proceedings Asia-PaciÞc Microwave Conference (APMC), Bangkok, pp. 1Ð4 (2007)

6.Potrebi«c, M.M., Toši«c, D.V., Cvetkovi«c, Z.Ž., Radosavljevi«c, N.: WIPL-D modeling and results for waveguide Þlters with printedcircuit inserts. In: Proceedings 28th International Conference on Microelectronics (MIEL), Niš, pp. 309Ð312 (2012)

7.ul Haq, T., Khan, M.F., Siddiqui, O.F.: Design and implementation of waveguide bandpass Þlter using complementary metaresonator. Appl. Phys. A 122(1), 34Ð38 (2016)

8. Stefanovski, S., Potrebi«c, M., Toši«c, D.: Design and analysis of bandpass waveguide Þlters using novel complementary split ring resonators. In: Proceedings 11th International Conference on Telecommunication in Modern Satellite, Cable and Broadcasting Services (TELSIKS), Niš, pp. 257Ð260 (2013)

9.Bahrami, H., Hakkak, M., Pirhadi, A.: Analysis and design of highly compact bandpass waveguide Þlter using complementary split ring resonators (CSRR). Prog. Electromagn. Res. 80, 107Ð 122 (2008)

10.Stefanovski, S., Potrebi«c, M., Toši«c, D., Stamenkovi«c, Z.: A novel compact dual-band bandpass waveguide Þlter. In: Proceedings IEEE 18th International Symposium on Design and Diagnostics of Electronic Circuits and Systems (DDECS), Belgrade, pp. 51Ð56 (2015)

11.Stefanovski, S., Potrebi«c, M., Toši«c, D., Stamenkovi«c, Z.: Compact dual-band bandpass waveguide Þlter with H-plane inserts. J. Circuit Syst. Comput. 25(3), 1640015 (2016). doi:10.1142/ S0218126616400156

12.Fallahzadeh, S., Bahrami, H., Tayarani, M.: A novel dual-band bandstop waveguide Þlter using split ring resonators. Prog. Electromagn. Res. Lett. 12, 133Ð139 (2009)

13.Stefanovski, S., Potrebi«c, M., Toši«c, D.: Novel realization of bandstop waveguide Þlters. Technics (special edition), pp. 69Ð76 (2013)

14.Stefanovski, S.Lj, Potrebi«c, M.M., Toši«c, D.V.: A novel design of dual-band bandstop waveguide Þlter using split ring resonators. J. Optoelectron. Adv. Mater. 16(3Ð4), 486Ð493 (2014)

15.Islam, S.S., Faruque, M.R.I., Islam, M.T.: A new double negative metamaterial for multi-band microwave applications. Appl. Phys. A 116(2), 723Ð733 (2014)

16.Cameron, R.J., Kudsia, C.M., Mansour, R.R.: Microwave Filters for Communication Systems: Fundamentals, Design, and Applications, 1st edn. Wiley, New Jersey (2007)

17.Lutovac, M.D., Toši«c, D.V., Evans, B.L.: Filter Design for Signal Processing Using MATLAB and Mathematica, 1st edn. Prentice Hall, Upper Saddle River, NJ (2001)

18.OÕReilly, S., Duffy, M., Ott, T., OÕDonnell, T., McCloskey, P., Mathuna, C.O.: Characterization of embedded Þlters in advanced printed wiring boards. Microelectron. Reliab. 41(6), 781Ð788 (2001)

19.Rane, S., Puri, V.: Thin Þlm, thick Þlm microstrip band pass Þlter: a comparison and effect of bulk overlay. Microelectron. Reliab. 42(12), 1953Ð1958 (2002)

20.Kim, H.H., Ju, B.K., Lee, Y.H., Lee, S.H., Lee, J.K., Kim, S.W.: Fabrication of suspended thin Þlm resonator for application of RF bandpass Þlter. Microelectron. Reliab. 44(2), 237Ð243 (2004)

21.Gao, S.C., Li, L.W., Yeo, T.S., Leong, M.S.: A dual-frequency compact microstrip patch antenna. Radio Sci. 36(6), 1669Ð1682 (2001)

22.Albooyeh, M., LotÞ Neyestanak, A.A., Mirzapour, B.: Wideband dual posts waveguide band pass Þlter. Int. J. Microw. Opt. Technol. 2(3), 203Ð209 (2007)

23.Choi, N.S., Kim, D.H., Jeung, G., Park, J.G., Byun, J.K.: Design optimization of waveguide Þlters using continuum design sensitivity analysis. IEEE T. Magn. 46(8), 2771Ð2774 (2010)

24.Villaroya, R.L.: E-plane parallel coupled resonators for waveguide bandpass Þlter applications. PhD dissertation, Heriot-Watt University, Edinburgh (2012)

25.Soto, P., de Llanos, D., Boria, V.E., Tarin, E., Gimeno, B., Onoro, A., Hidalgo, I., Padilla, M.J.: Performance analysis and comparison of symmetrical and asymmetrical conÞgurations of evanescent mode ridge waveguide Þlters. Radio Sci. 44(6), RS6010 (2009). doi:10.1029/2008RS004034

26.Bornemann, J., Uher, J.: Design of waveguide Þlters without tuning elements for production-efÞcient fabrication by milling. In: Proceedings Asia-PaciÞc Microwave Conference (APMC), Taipei, pp. 759Ð762 (2001)

27.Zhao, C., Kaufmann, T., Zhu, Y., Lim, C.C.: EfÞcient approaches to eliminate inßuence caused by micro-machining in fabricating H-plane iris band-pass Þlters. In: Proceedings Asia-PaciÞc Microwave Conference (APMC), Sendai, pp. 1306Ð1308 (2014)

28.Kolundžija, B.M., Djordjevi«c, A.R.: Electromagnetic Modeling of Composite Metallic and Dielectric Structures, 1st edn. Artech House, Norwood, MA (2002)

29.Djordjevi«c, A.R., Ol«can, D.I., Zaji«c, A.G.: Modeling and design of milled microwave printed circuit boards. Microw. Opt. Technol. Lett. 53(2), 264Ð270 (2011)

30.Stefanovski, S., Potrebi«c, M., Toši«c, D.: Structure for precise positioning of inserts in waveguide Þlters. In: Proceedings 21st Telecommunications Forum (TELFOR), Belgrade, pp. 689Ð692 (2013)