This is the accepted version of the article: Guerrero Menéndez, Eloi; Verdú Tirado, Jordi; Paco Sánchez, Pedro Antonio de. «Synthesis of Extracted Pole Filters With Transmission Zeros in Both Stopbands and Nonresonant Nodes of the Same Nature». IEEE Microwave and Wireless Components Letters, Vol. 31, Issue 1 (January 2021), p. 17-20. DOI 10.1109/LMWC.2020.3035848 This version is available at https://ddd.uab.cat/record/238011 under the terms of the license 1 Synthesis of Extracted Pole Filters with Transmission Zeros in Both Stopbands and Non-Resonant Nodes of the Same Nature Eloi Guerrero, Graduate Student Member, IEEE, Jordi Verdú, Senior Member, IEEE, and Pedro de Paco, Senior Member, IEEE Abstract—Extracted pole filters alternating transmission zeros b1 b2 bN above and below the passband are intrinsically related to the use of non-resonant nodes of different nature (inductive and Jr1 Jr2 JrN capacitive). This work demonstrates that it is possible to syn- J1 J2 J3 JN+1 thesize an extracted pole filter showing transmission zeros in both sides of the passband but implementing all non-resonant BS B1 B2 BN BL nodes of the same reactive nature, overcoming the limitations imposed by some technologies. The proposed feature is achieved Fig. 1. General lowpass nodal scheme of an extracted pole filter. White dashed by a special consideration of the input reflection phase of the nodes are NRNs represented as frequency invariant susceptances Bi , black generalized Chebyshev filter function, affecting input and output nodes are RNs made of a unit capacitor in parallel to a frequency invariant resonators and placing at most 2 zeros at the opposite band. A susceptance of value bi and black lines are couplings in the form of admittance third order all-capacitive prototype filter has been synthesized inverters. and fabricated to validate the proposed statement. The concept allows to achieve all-capacitive or all-inductive filters. ð ïèð Index Terms—extracted pole, filter synthesis, non-resonant Í ïï ïëð Í ïï Í ïîð nodes. óîð îï çð êð óìð íð ð I. I NTRODUCTION óêð óíð óêð T HE possibility to design filters with arbitrarily placed transmission zeros (TZs) that are implemented by inde- pendent sections of the structure is the main advantage of óèð óïðð óê óì óî ð î ì ê óçð óïîð óïëð óïèð óê óì óî ð î ì ê the extracted pole technique. The contributions in [1], [2] ø®¿¼ñ­÷ ø®¿¼ñ­÷ presented the synthesis method to implement this technique (a) (b) 1.7 -1.97 -2.5 -3 2.7 exploiting the concept of the non-resonant node (NRN), that can be generalized to any technology [3]. An extracted pole Source 0.8192 3.3120 1.6843 5.9465 1.4583 Load section is made up of an NRN coupled to a dangling resonant 1 -1 1 -1 1 -1.5957 node (RN) resonating at the frequency of the TZ, as in Fig. 1. The modularity provided by this technique is fantastic and -1.3429 -0.1420 -7.7672 -1.4456 -13.5101 -0.0783 -2.6373 offers also better sensitivity than more complex structures (c) including cross-couplings. Fig. 2. 5th order filter example with 2 TZs at the lower stopband and all One interesting topic concerning extracted pole sections negative NRNs (all-inductive). (a) Lowpass magnitude response. (b) Lowpass reflection phase response. (c) Nodal lowpass network following the scheme is the sign of the frequency-invariant reactances (FIRs) that in Fig. 1. model NRNs in the lowpass domain. It is commonly accepted that an NRN-RN pair placing a TZ above the passband requires a negative NRN while a TZ below the passband of technologies where only one nature of NRNs can be easily requires a positive one. This influences how these nodes are implemented. [6], [7] propose miniaturized extracted pole filter implemented, either as capacitive or inductive reactances in implementations using asymmetric coupled line structures. the bandpass domain. This might not be an issue in some This configuration can only implement NRNs of inductive technologies such as coaxial cavity where NRNs can be nature since the asymmetric coupled line sections are the result detuned cavities or common microstrip inline where they of the connection of a short-circuited stub (implementing a might be stubs [4], [5]. However, there are some examples negative susceptance), a main line admittance inverter and E. Guerrero, J. Verdú and P. de Paco are with the Department of Telecom- the following short-circuited stub. The authors claim that the munications and Systems Engineering, Universitat Autònoma de Barcelona, technology can only implement transmission zeros in the upper Barcelona, Spain, 08193. E-mail: eloi.guerrero@uab.cat. stopband because of that limitation. This work was supported in part by the Spanish Ministerio de Economı́a y Competitividad under Grant RTI2018-096019-B-C33 Similarly, the E-Plane waveguide technology used in [8] is Manuscript received XXXX YY, ZZZZ; revised XXXXX YY, ZZZZ. limited in that same way. The NRN is formed between septa 2 and has an inductive behaviour. Therefore, apparently only ïð Þï TZs at the upper stopband can be implemented. Moreover, the ë Þî same authors developed a more complex way to locate TZs Þí at the lower stopband in [9] using a longer NRN with the so- ð called dual-mode extracted pole section that locates two poles and a single TZ. However, this approach is more complex and óë looses part of the modularity inherent to pure extracted pole. óïð In this paper we propose how to synthesize filters that óïëð óïðð óëð ð ëð ïðð ïëð locate TZs in both stopbands but keeping all NRNs of the ø¼»¹®»»­÷ ¿¼¼ same nature, that is, having the same sign. This is shown (a) in the example in Fig. 2 where a 5th order filter with TZs Ý¿°ò at −1.7, 1.97, 2.5, 3 and −2.7 has been synthesized and ײ¼ò all NRNs are negative. That is, all of inductive nature when Ý¿°ò implemented in the bandpass domain. Þãï ײ¼ò Þãî Ý¿°ò Þãí II. NRN S AND R EFLECTION P HASE ײ¼ò óïëð óïðð óëð ð ëð ïðð ïëð Achieving filters with single-nature NRNs entails a careful ø¼»¹®»»­÷ consideration of the reflection phase of the filter function. As ¿¼¼ introduced in [10], the S-parameters of a generalized Cheby- (b) shev function are expressed as a function of the characteristic Fig. 3. Sweep on θadd on the third order filter example. (a) Evolution of polynomials P (s), F (s) and E(s). FIR values, (b) Nature of the synthesized NRN in the bandpass domain. Ind. stands for inductive, Cap. stands for capacitive.     S11 (s) S12 (s) 1 F (s)/εr P (s)/ε = S21 (s) S22 (s) E(s) P (s)/ε (−1)N F (s)∗ /εr • Ω1 and ΩN must be of opposite sign to internal TZs. (1) The internal TZs will define the nature of all NRNs since The input reflection phase can be tailored if F (s) is shifted the FIR of their associated NRN-RN pair will maintain its sign by an exponential term of phase θadd . for any value of θadd . Thus, the first and last TZs must be the ones of opposite sign (i.e. allocated in the contrary stopband). F 0 (s) = F (s)ejθadd (2) In this case, up to two TZs can be allocated in the contrary Since during the extraction procedure in [1], [2] NRNs are band. In general terms, for a filter of order N, there will be computed as phase matching elements to the next extracted 2 TZs in one stopband and N-2 in the other. This situation is pole section, modifying the phase has an effect on their value. found in the initial example in Fig. 2. Let us discuss it with a simple example of order N = 3. Notice that due to the input phase modification, the filter Consider the TZ set of Ω = [ 2.6, −2.2, 2.6] and return loss network requires the input and output NRNs, BS and BL RL = 18 dB. In regular conditions we would expect the first that appear whenever you do not meet the phase condition and last FIR elements, let them be B1 and B3 as in Fig. for removing them, that is set in [1]. For example, since the 1, to be negative since they are related to a TZ above the filter in [5] is not limited to NRNs of a single nature, elements passband. The internal FIR B2 should then be positive. Fig. 3 BS and BL can be set to zero. At the same time, [11], [12] shows the value of the three FIR elements with respect to the show that sometimes the input and output NRNs are necessary, additional phase term θadd . It is clearly seen that B1 and B3 in this case directly connected to a dangling resonator due to transition from negative to positive value for extreme values of the triple-mode structure employed. the reflection phase while B2 stays inside the positive region. Finally, it is worth mentioning that a result similar to the At θadd = 180◦ , both B1 and B3 exhibit the same sign as B2 . one we propose can be achieved using the technique in [13] Each element has a wider range of phases for which its sign where phase shifters are extracted instead of unitary inverters. is inverted, however, they are positive at the same time only in In that case, elements B1 and B3 can be set to zero along with the close vicinity of 180◦ . More in detail, the sign transition of BS and BL at expense of implementing phase shifters of an B1 happens when θadd = 180 − 6 S11 (jΩ1 ) and the transition exact value between all extracted pole sections, being B2 the of B3 is when θadd = −180 + 6 S22 (jΩ3 ), where S11 (s) only remaining NRN. and S22 (s) are the original S-parameters of the unmodified Chebyshev function. III. E XPERIMENTAL VALIDATION Clearly, this behaviour can be exploited to design filters that To demonstrate the validity of our proposal, the third order feature zeros in both stopbands only implementing NRNs of filter of the example in section II has been manufactured. Since a single nature. To do so, some conditions must be met: the internal TZ is negative, the sign of all FIRs will be positive ◦ • θadd must be fixed to 180 . what means that all of them will be implemented as capacitive • The order of the filter must be at least 3 (N ≥ 3) reactances. • Internal TZs (Ωk , k ∈ {2, N − 2}) must all have the Let us start by synthesizing the filter setting θadd = 180◦ . same sign. The obtained lowpass values are shown on the nodal scheme 3 -2.6 2.2 -2.6 0.9419 4.6690 0.9419 1 -1 1 -1 2.2587 0.0644 12.8887 0.0644 2.2587 Fig. 4. Lowpass nodal scheme of the N = 3 example filter synthesized with θadd = 180◦ . ð ïèð Í ïï ïëð Í ïï Í ïîð óîð îï çð êð óìð íð ð óêð óíð Fig. 6. Picture of the manufactured all-capacitive third order filter at f0 = óêð 1480 MHz. 50 Ω access lines of 15 mm have been added at input and output óçð óèð óïîð to attach SMA ports. óïëð óïðð óïèð óê óì óî ð î ì ê óê óì óî ð î ì ê ð ø®¿¼ñ­÷ ø®¿¼ñ­÷ (a) (b) Fig. 5. Lowpass response of the third order filter example with all-positive óïð FIRs. (a) Magnitude, (b) Input reflection phase. óîð óíð in Fig. 4, and Fig. 5 shows the lowpass magnitude and phase response of the synthesized filter. Í ïï ³»¿­«®»¼ óìð For fast prototyping, the filter is implemented in microstrip Í ³»¿­«®»¼ îï technology, using quarter-wave lines at f0 for the mainline Í ­·³«´¿¬»¼ óëð ïï couplings and open-circuited quarter-wave resonators at fT Z Í ­·³«´¿¬»¼ (the position of TZs) in parallel with a lumped capacitor to îï ground for the NRN-RN pairs, similar to the case in [4]. óêð ïòî ïòí ïòì ïòë ïòê ïòé ïòè The band specifications are central frequency f0 = 1480 Ú®»¯«»²½§ øÙئ÷ MHz and a bandwidth of 100 MHz. The chosen substrate is RO4003C with 508 µm thickness. In order to adapt the design Fig. 7. Magnitude response of the manufactured third order filter and ADS- Momentum simulation. to easily implementable transmission lines, the coupling matrix elements have been scaled so that the quarter-wave resonators have a characteristic impedance of Zr = 100 Ω. After scaling, IV. C ONCLUSION the resulting characteristic impedance of the main line quarter- This work has covered the synthesis of extracted pole filters wave lines Zml = [15.59, 24.08, 24.08, 15.59] Ω. The that feature transmission zeros in both stopbands implementing lengths of the quarter-wave resonators are 29 mm for the first all NRNs with the same reactive nature. This is achieved and third and 34.8 mm for the second. The length of the thanks to modification of the input reflection phase of the main line admittance inverters is 29.3 mm for the first and filtering function adding a phase shift of 180◦ that is effective fourth, and 30 mm for the second and third. The NRNs are to modify the nature of first and last NRN. This approach implemented with 0603 SMD lumped capacitors by Johanson is specially useful for technologies where only one nature of Technology. The ideal values of the capacitive reactances after the NRNs is easily implementable. Using the reflection phase bandpass transformation are Cin/out = 5 pF for the input and at convenience during the synthesis allows to improve the output elements, C1,3 = 1.469 pF for the first and last NRNs performance of the technology, allowing to implement more and C2 = 11.94 pF for the internal NRN. selective responses. The concept has been demonstrated in a Fig. 6 shows the fabricated filter. The comparison between simple all-capacitive third order microstrip filter. measurement and full-wave simulation with ADS Momen- tum are depicted in Fig. 7 showing a good agreement. The R EFERENCES fabricated filter shows an RL = 15.6 dB. As expected, it [1] S. Amari and G. Macchiarella, “Synthesis of inline filters with arbitrarily implements TZs at both sides of the passband. As known, placed attenuation poles by using nonresonating nodes,” IEEE Trans. mainline quarter-wave microstrip inverters introduce spurious Microw. Theory Techn., vol. 53, no. 10, pp. 3075–3081, Oct. 2005. [2] S. Tamiazzo and G. Macchiarella, “Synthesis of cross-coupled prototype responses close to the band affecting the out-of-band isolation filters including resonant and non-resonant nodes,” IEEE Trans. Microw. as seen in Fig. 7 but do not affect our validation purpose. Theory Techn., vol. 63, no. 10, pp. 3408–3415, Oct. 2015. 4 [3] A. Gimenez, J. Verdu, and P. de Paco, “General synthesis methodology for the design of acoustic wave ladder filters and duplexers,” IEEE Access, vol. 6, pp. 47 969–47 979, 2018. [4] G. Macchiarella and S. Tamiazzo, “Synthesis of microwave duplexers using fully canonical microstrip filters,” in IEEE MTT-S Int. Microw. Symp. Dig. IEEE, Jun. 2009, pp. 721 – 724. [5] G. Macchiarella, S. Tamiazzo, and V. Verri, “A design methodology for fully canonic NRN filters in coaxial technology,” in IEEE MTT-S Int. Microw. Symp. Dig. IEEE, Jun. 2017, pp. 748 – 751. [6] Y. He, G. Wang, X. Song, and L. Sun, “A coupling matrix and admittance function synthesis for mixed topology filters,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 12, pp. 4444–4454, Dec. 2016. [7] J.-H. Park, S. Lee, and Y. Lee, “Extremely miniaturized bandpass filters based on asymmetric coupled lines with equal reactance,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 2, pp. 261–269, Feb. 2012. [8] O. Glubokov and D. Budimir, “Extraction of generalized coupling coefficients for inline extracted pole filters with nonresonating nodes,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 12, pp. 3023–3029, Dec. 2011. [9] ——, “Novel inline waveguide e-plane filters using dual-mode extracted pole section,” in Proc. 41st Eur. Microw. Conf., 2011, pp. 99 – 102. [10] R. J. Cameron, C. M. Kudsia, and R. R. Mansour, Microwave Filters for Communication Systems. Wiley, Mar. 2018. [11] S. Amari, U. Rosenberg, and J. Bornemann, “A novel approach to dual and triple-mode pseudo-elliptic filter design,” in Proc. 34th Eur. Microw. Conf., vol. 2, 2004, pp. 993–996. [12] S. Amari and U. Rosenberg, “New in-line dual- and triple-mode cavity filters with nonresonating nodes,” IEEE Trans. Microw. Theory Techn., vol. 53, no. 4, pp. 1272–1279, 2005. [13] Y. Yang, M. Yu, and Q. Wu, “Advanced synthesis technique for unified extracted pole filters,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 12, pp. 4463–4472, Dec. 2016.