Miniaturized High-Performance Bandpass Hairpin Filter Circuit and Filter Based on Hybrid Coupling

Abstract

Miniaturized high-performance band-pass hairpin filter circuit and filter based on hybrid coupling, belonging to the technical field of microwave filters, which solves the problem of how to design a miniaturized band-pass hairpin filter circuit and filter with high out-of-band rejection performance and small return loss in the passband; the first and second signal paths of the filter circuit are hybrid-coupled to generate a transmission zero point respectively outside the lower-sideband cut-off frequency and outside the upper-sideband cut-off frequency. The transmission zero point enhances the out-of-band rejection performance of the filter circuit and plays a great role in suppressing high-order harmonics, avoiding the influence of noise on communication performance; the filter covering dielectric layer reduces the insertion loss in the passband, the T-shaped slot is set at the coupling end of the mouth-shaped resonator to achieve the suppression of high-order harmonics, the arm length of the mouth-shaped resonator is adjusted to change the resonance frequency, and the coupling distance of the mouth-shaped resonator is adjusted to achieve the bandwidth and reach the matching effect; it has the characteristics of miniaturization, high stability, low loss, high out-of-band rejection, etc.

Description

Technical Field

[0001] The present invention belongs to the technical field of microwave filters, and relates to a miniaturized high-performance band-pass hairpin filter circuit and filter based on hybrid coupling. Background Art

[0002] With the development of radio frequency passive devices and the widespread use of 5G, in order to achieve the goal of connecting all things, higher requirements are put forward for radio frequency microwave modules and communication base stations involved in transmitting signals, making it necessary to continuously optimize and improve performance and size. This makes various devices in the radio frequency front-end the objects we need to focus on. Therefore, the radio frequency front-end will develop towards the direction of miniaturization and integration.

[0003] Among radio frequency front-end passive devices, filters play a crucial role. Their performance and miniaturization have a huge impact on the quality of the entire system. Therefore, the research on high-performance, miniaturized and integrated radio frequency passive filters is particularly important. With the improvement of living standards, while demanding high performance of terminal products, there is a great interest in delicate and small intelligent products, which also promotes the development of various radio frequency devices including filters towards the direction of miniaturization, low insertion loss, high rejection, high power handling capacity, and low cost.

[0004] The microstrip hairpin filter has a simple structure, is easy to achieve the transmission characteristics of the Sub6 frequency band, and is convenient for processing. However, traditional microstrip hairpin filters, such as Figure 8 shown, have the disadvantages of too large size and excessive insertion loss due to electromagnetic radiation; waveguide filters, due to their high quality factor characteristics, can achieve very low in-band insertion loss while realizing a high rectangularity coefficient, but dielectric waveguide filters have cumbersome processing procedures and too large device sizes. Summary of the Invention

[0005] The purpose of the present invention is to design a miniaturized band-pass hairpin filter circuit and filter with high out-of-band rejection performance and small in-band return loss.

[0006] The present invention solves the above technical problems through the following technical solutions:

[0007] A miniaturized high-performance band-pass hairpin filter circuit based on hybrid coupling, comprising: a first feeding port (121), a second feeding port (122), a first square resonator (123), a second square resonator (124), a third square resonator (125), and a fourth square resonator (126);

[0008] The first feeding port (121), the first rectangular resonator (123), the second rectangular resonator (124), and the second feeding port (122) form a first signal path of the filter circuit. The first rectangular resonator (123) and the second rectangular resonator (124) in the first signal path are connected by inductive coupling, presenting a -90° phase difference.

[0009] The first feeding port (121), the first rectangular resonator (123), the third rectangular resonator (125), the fourth rectangular resonator (126), the second rectangular resonator (124), and the second feeding port (122) form a second signal path of the filter circuit. Between the first rectangular resonator (123) and the third rectangular resonator (125) in the second signal path, a +90° phase difference is formed through hybrid coupling, between the third rectangular resonator (125) and the fourth rectangular resonator (126), a +90° phase difference is formed through capacitive coupling, and between the fourth rectangular resonator (126) and the second rectangular resonator (124), a +90° phase difference is formed through hybrid coupling.

[0010] The first signal path and the second signal path each generate a transmission zero outside the lower sideband cut-off frequency and outside the upper sideband cut-off frequency.

[0011] A notch (127) is provided on one side of each rectangular resonator, and a T-shaped slot (128) with the same size is etched on the other side. The T-shaped slot (128) is arranged at the coupling end of the rectangular resonator. The first feeding port (121) is connected to the first rectangular resonator (123). The third rectangular resonator (125) is arranged on the side of the first rectangular resonator (123) far from the first feeding port (121). The first rectangular resonator (123) and the third rectangular resonator (125) are arranged staggeredly, with a gap left between the first rectangular resonator (123) and the third rectangular resonator (125). The T-shaped slots (128) of the first rectangular resonator (123) and the third rectangular resonator (125) form an interleaved I-shaped slot. The notch (127) of the first rectangular resonator (123) faces downward, and the notch (127) of the third rectangular resonator (125) faces upward.

[0012] The structural settings of the first feeding port (121), the first rectangular resonator (123), and the third rectangular resonator (125) are axisymmetric about the x-axis with the structural settings of the second feeding port (122), the second rectangular resonator (124), and the fourth rectangular resonator (126).

[0013] A filter using the above-mentioned band-pass hairpin filter circuit includes: a first shielding layer (10), a lid (11), a dielectric layer (13), a second shielding layer (14), and a metal post via array (15); the first shielding layer (10), the lid (11), the filter circuit (12), the dielectric layer (13), and the second shielding layer (14) are stacked on top of each other from top to bottom in sequence; the metal post via array (15) is uniformly arranged on the four peripheral edges of the band-pass hairpin filter except for the feeding end, and the metal post via array (15) penetrates through the entire band-pass hairpin filter to connect the first shielding layer (10) and the second shielding layer (14).

[0014] The design formula for the order of the filter is as follows:

[0015]

[0016] Wherein, R is the normalized frequency, FBW is the relative bandwidth of the filter, f is the cut-off frequency, f0 is the center frequency, and the calculation formula for the quality factor of the filter is as follows:

[0017]

[0018]

[0019] Wherein, Q1 and Q2 are the quality factors of the filter, and g0, g1, g n , g n+1 are the normalized element parameters.

[0020] The calculation formula for the coupling coefficient between the resonators of the filter is as follows:

[0021]

[0022] Wherein, g i is the normalized element parameter, and K i,i+1 is the coupling coefficient between the resonators.

[0023] The calculation formula for the length of the resonator of the filter is as follows:

[0024]

[0025] Wherein, c is the speed of light, and ε eff0 is the effective dielectric constant.

[0026] The calculation formula for the tap position of the filter is as follows:

[0027]

[0028] Among them, L is the length of the resonator, that is, a quarter wavelength, Z0 is the impedance of the feeding port, Zr is the characteristic impedance of the filter, and Q is the quality factor of the filter.

[0029] The advantages of the present invention are as follows:

[0030] On the one hand, the filtering circuit of the present invention generates a transmission zero point outside the lower sideband cut-off frequency and outside the upper sideband cut-off frequency through hybrid coupling in the first signal path and the second signal path. The transmission zero point enhances the out-of-band rejection performance of the filtering circuit, plays a great role in suppressing high-order harmonics, and avoids the influence of noise on communication performance. On the other hand, the filter of the present invention sets a T-shaped slot (128) at the coupling end of the square resonator (123), adjusts the size of the T-shaped slot (128) to achieve the suppression of high-order harmonics, adjusts the arm length of the square resonator (123) to change the resonance frequency; adjusts the coupling distance of the square resonator (123) to achieve the required bandwidth and achieve a matching effect; reduces the insertion loss in the passband by covering the dielectric layer (13), has the characteristics of miniaturization, high stability, low loss, high out-of-band rejection, etc., is convenient for design and debugging, and is applicable to Sub6 frequency band communication. Description of the Drawings

[0031] Figure 1 is an exploded view of the structure of a miniaturized high-performance band-pass hairpin filter based on hybrid coupling according to an embodiment of the present invention;

[0032] Figure 2 is a top perspective view of a miniaturized high-performance band-pass hairpin filter based on hybrid coupling according to an embodiment of the present invention;

[0033] Figure 3 is a schematic structural diagram of the filtering circuit of a miniaturized high-performance band-pass hairpin filter based on hybrid coupling according to an embodiment of the present invention;

[0034] Figure 4 is an equivalent circuit diagram of a miniaturized high-performance band-pass hairpin filter based on hybrid coupling according to an embodiment of the present invention;

[0035] Figure 5 is a size design diagram of the filtering circuit of a miniaturized high-performance band-pass hairpin filter based on hybrid coupling according to an embodiment of the present invention;

[0036] Figure 6 is a relationship diagram of the coupling coefficient and the coupling distance of a miniaturized high-performance band-pass hairpin filter based on hybrid coupling according to an embodiment of the present invention;

[0037] Figure 7 is an S-parameter diagram of a miniaturized high-performance band-pass hairpin filter based on hybrid coupling according to an embodiment of the present invention;

[0038] Figure 8 It is a schematic structural diagram of a traditional hairpin filter. Specific embodiments

[0039] For the purpose of making the objectives, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the embodiments of the present invention. Apparently, the described embodiments are some, but not all, of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the scope of protection of the present invention.

[0040] The technical solutions of the present invention will be further described below in conjunction with the accompanying drawings of the specification and specific embodiments:

[0041] Embodiment 1

[0042] 1. Structural composition of the filter

[0043] As Figure 1 and Figure 2 shown, the miniaturized high-performance bandpass hairpin filter based on hybrid coupling in the embodiment of the present invention includes: a first shielding layer (10), a lid (11), a filter circuit (12), a dielectric layer (13), a second shielding layer (14), and a metal post via array (15); the first shielding layer (10), the lid (11), the filter circuit (12), the dielectric layer (13), and the second shielding layer (14) are stacked together in sequence from top to bottom; the metal post via array (15) is uniformly arranged on the four peripheral edges of the bandpass hairpin filter except for the feeding end, and the metal post via array (15) vertically penetrates the entire bandpass hairpin filter to connect the first shielding layer (10) and the second shielding layer (14).

[0044] The thickness of the dielectric layer (13) is 1.6 mm, the dielectric constant is 6.15, and the loss tangent is 0.002. The overall thickness of the filter is 3.3 mm; the thickness of the lid (11) is 1.6 mm.

[0045] The function of the dielectric layer (13) is to provide dielectric support, isolate the microstrip structure, and reduce transmission loss. The dielectric board can serve as a medium for signal transmission, providing an environment for signal transmission; the dielectric constant and thickness of the dielectric board determine the capacitance value and operating frequency of the capacitor, and play a supporting role for the circuit structure; the lid (11) is made of the same material as the dielectric layer (13). The lid (11) is used to enclose the filter and confine electromagnetic waves, thereby reducing the size of the filter and the insertion loss, reducing the insertion loss in the passband, and thus improving the transmission performance of the filter; the function of the metal post via array (15) is to form a boundary condition, thereby restricting the propagation range of electromagnetic waves in the microstrip structure, creating a closed environment, which can effectively shield external interference and leakage, ensuring that the electromagnetic waves in the filter can exist and be transmitted in the form of a high quality factor. The metal post via array (15) can also serve as a support structure for the filter to support the installation and wiring of other components and connectors. The first shielding layer (10) and the second shielding layer (14) are used to minimize electromagnetic radiation and interference, reduce crosstalk and mutual interference in the circuit, and improve the isolation and anti-interference performance of the circuit.

[0046] As Figure 3 shown, the filter circuit (12) includes: a first feed port (121), a second feed port (122), a first square resonator (123), a second square resonator (124), a third square resonator (125), and a fourth square resonator (126); a notch (127) is formed on one side of each square resonator, and a T-shaped slot (128) with the same size is etched on the other side. The T-shaped slot (128) is arranged at the coupling end of the square resonator; the T-shaped slot (128) enhances the harmonic suppression effect and reduces the crosstalk influence of harmonic signals on the transmitted signal.

[0047] The first feed port (121) is connected to the first square resonator (123). The third square resonator (125) is arranged on the side of the first square resonator (123) away from the first feed port (121). The first square resonator (123) and the third square resonator (125) are arranged in an interleaved manner, with a gap left between the first square resonator (123) and the third square resonator (125). The T-shaped slots (128) of the first square resonator (123) and the third square resonator (125) form an interleaved I-shaped slot. The interleaved I-shaped slot introduces circuit stubs to form transmission zeros, achieving the effect of suppressing high-order harmonics, forming an equivalent circuit of a series capacitor and inductor to ground to introduce transmission zeros and thus achieving the effect of suppressing high-order harmonics; the notch (127) of the first square resonator (123) faces downward, and the notch (127) of the third square resonator (125) faces upward.

[0048] The structural settings of the first feeding port (121), the first square resonator (123), and the third square resonator (125) are axisymmetric about the x-axis with the structural settings of the second feeding port (122), the second square resonator (124), and the fourth square resonator (126).

[0049] Refer to Figure 8 Or the specification appendix of the Chinese invention patent application document "A Hairpin-Type Bandpass Filter" with the application publication date of December 11, 2020 and the application publication number of CN112072238A Figure 1 In this embodiment of the present invention, the square resonator is formed by folding the two resonant arms of the traditional hairpin resonator inward and leaving a notch (127) with a certain distance, and a T-shaped groove (128) with the same size is etched at the middle position of the coupling end of the square resonator in this embodiment of the present invention; folding the two resonant arms of the traditional hairpin resonator inward can greatly reduce the size of a single resonator, and placing the resonators symmetrically up and down to form an annular signal transmission path can reduce the overall size of the filter, and according to the annular placement, a 180° phase difference can be formed through hybrid coupling and capacitive coupling, thereby introducing a transmission zero point and enhancing the out-of-band rejection performance of the filter.

[0050] 2. Signal path of the filtering circuit

[0051] As Figure 3 shown, taking the example of the signal input from the first feeding port (121) and output from the second feeding port (122) for illustration, the signal can also be input from the second feeding port (122) and output from the first feeding port (121).

[0052] The first signal path is: the first feeding port (121) → the first square resonator (123) → the second square resonator (124) → the second feeding port (122); between the first square resonator (123) and the second square resonator (124) in the first signal path, they are connected by inductive coupling, presenting a -90° phase difference.

[0053] The second signal path is: the first power feeding port (121) → the first square resonator (123) → the third square resonator (125) → the fourth square resonator (126) → the second square resonator (124) → the second power feeding port (122); between the first square resonator (123) and the third square resonator (125) of the second signal path, a +90° phase difference is formed through hybrid coupling connection, between the third square resonator (125) and the fourth square resonator (126), a +90° phase difference is formed through capacitive coupling connection, and between the fourth square resonator (126) and the second square resonator (124), a +90° phase difference is formed through hybrid coupling connection. Combining with the comparison and analysis of the magnitude of the center frequency f0, a transmission zero is generated outside the lower sideband cut-off frequency and outside the upper sideband cut-off frequency respectively. The specific phase change is shown in Table 1.

[0054] Table 1 Phase change in the signal path

[0055]

[0056] As Figure 4 shown, it is the equivalent circuit diagram of the filter. A transmission zero is generated outside the lower sideband cut-off frequency and outside the upper sideband cut-off frequency respectively. The introduction of the two transmission zeros enhances the out-of-band rejection ability of the filter, plays a great role in suppressing high-order harmonics, avoids the influence of noise on the communication performance, and improves the signal transmission performance of the filter.

[0057] 3. Size design of the filter

[0058] First of all, the quality factor of the filter is related to the position of the tap and has a great influence on the impedance matching of the filter; secondly, the theoretical distance between the resonators is obtained from the relationship between the calculated coupling coefficient and the distance. The distance not only affects the matching but also affects the bandwidth.

[0059] (1) The design formula for the order of the filter is as follows:

[0060]

[0061] where R is the normalized frequency, FBW is the relative bandwidth of the filter, f is the cut-off frequency, and f0 is the center frequency.

[0062] (2) The calculation formula for the quality factor of the filter is as follows:

[0063]

[0064]

[0065] where Q1 and Q2 are the quality factors of the filter, g0, g1, g n 、gn+1 is the normalized element parameter.

[0066] (3) The calculation formula for the coupling coefficient between resonators is as follows:

[0067]

[0068] where, g i is the normalized element parameter, and K i,i+1 is the coupling coefficient between resonators.

[0069] (4) The length of the resonator

[0070] The center frequency of the entire filter is determined by the length of the resonator. Therefore, an appropriate length L of the resonator needs to be calculated according to the requirements of the designed center frequency. Generally, the length of the U-shaped hairpin resonator is taken as one-quarter wavelength. The calculation formula for the length of the resonator is as follows:

[0071]

[0072] where, c is the speed of light, and ε eff0 is the effective dielectric constant;

[0073] Substitute the quality factors calculated from formulas (2) and (3) into formula (6) to further calculate the tap position. The calculation formula for the tap position is as follows:

[0074]

[0075] where, L is the length of the resonator, that is, one-quarter wavelength, Z0 is the impedance of the feeding port, and Zr is the characteristic impedance of the filter.

[0076] Such as Figure 5As shown, the dimensions of the first feeding port (121) and the second feeding port (122) are the same, with a width wf = 2 mm and a length l3 = 3 mm; the dimensions of the first square resonator (123) are the same as those of the second square resonator (124), and the branch length l1 of the second square resonator (124) is 5.6 mm; the dimensions of the third square resonator (125) are the same as those of the fourth square resonator (126), and the branch length l2 of the fourth square resonator (126) is 6.2 mm; the central branch lengths of the first square resonator (123), the second square resonator (124), the third square resonator (125), and the fourth square resonator (126) are the same, and the central branch length g2 = 5.4 mm; the dimensions of the notches (127) of the first square resonator (123), the second square resonator (124), the third square resonator (125), and the fourth square resonator (126) are the same, and the width w2 of the notch (127) is 0.5 mm; the dimensions of the T-shaped slots (128) of the first square resonator (123), the second square resonator (124), the third square resonator (125), and the fourth square resonator (126) are the same, the width sw of the T-shaped slot (128) is 0.25 mm, the length sl is 2.6 mm, and the opening width sw2 of the T-shaped slot (128) is 0.5 mm.

[0077] The coupling coefficients and time delays without the influence of parasitic effects are calculated according to the filter-designer theory, as shown in Table 2 below.

[0078] Table 2 Coupling Coefficients and Time Delays Calculated Theoretically for the Filter

[0079] Delay 2.03578 nS <![CDATA[K S1 = K 1S > 0.0958215 <![CDATA[K 12 = K 21 > 0.0804536 <![CDATA[K 14 = K 41 > -0.0202513 <![CDATA[K 23 = K 32 > 0.0736337 <![CDATA[K 34 = K 43 > 0.0804536 <![CDATA[K 4L = K L4 > 0.0768396

[0080] As Figure 6 shown, according to the eigenmode of the HFSS software and using the formula to calculate the relationship between the coupling coefficients and coupling distances between the square resonators, where f1 and f2 are the resonance frequencies of different eigenmodes, and by comparing the parameters calculated according to the filter-designer theory, the coupling distance s1 between the first square resonator (123) and the third square resonator (125) is determined to be 0.24 mm, the coupling distance s2 between the third square resonator (125) and the fourth square resonator (126) is 0.24 mm, and the coupling distance s3 between the first square resonator (123) and the second square resonator (124) is 2.5 mm.

[0081] 4. S-parameter Diagram of the Filter

[0082] As Figure 7As shown, it is the S-parameter diagram of the miniaturized high-performance bandpass hairpin filter based on hybrid coupling according to the embodiment of the present invention. It can be seen from the figure that the passband range of the filter is 3.05 - 3.35 GHz, the in-band insertion loss is -0.8 dB, the insertion loss at the cut-off frequency is -1.1 dB, and the return loss in the passband is less than -15 dB; there is a transmission zero at 600 MHz outside the lower sideband cut-off frequency and at 450 MHz outside the upper sideband cut-off frequency respectively, and the out-of-band attenuation reaches -65 dB and -52 dB respectively. The attenuation within 3.85f0 outside the upper sideband is greater than 30 dB.

[0083] The above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that: they can still modify the technical solutions recorded in the foregoing embodiments, or perform equivalent replacements on some of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A miniaturized high-performance bandpass hairpin filter circuit based on hybrid coupling, characterized in that include: A first feeding port (121), a second feeding port (122), a first U-shaped resonator (123), a second U-shaped resonator (124), a third U-shaped resonator (125), and a fourth U-shaped resonator (126); The first feeding port (121), the first U-shaped resonator (123), the second U-shaped resonator (124) and the second feeding port (122) form a first signal path of the filter circuit; the first U-shaped resonator (123) and the second U-shaped resonator (124) in the first signal path are connected via inductive coupling, presenting a phase difference of -90°; The first feeding port (121), the first U-shaped resonator (123), the third U-shaped resonator (125), the fourth U-shaped resonator (126), the second U-shaped resonator (124), and the second feeding port (122) form a second signal path of the filter circuit; the first U-shaped resonator (123) and the third U-shaped resonator (125) in the second signal path are connected by hybrid coupling to form a phase difference of +90°, the third U-shaped resonator (125) and the fourth U-shaped resonator (126) are connected by capacitive coupling to form a phase difference of +90°, and the fourth U-shaped resonator (126) and the second U-shaped resonator (124) are connected by hybrid coupling to form a phase difference of +90°; The first signal path and the second signal path generate a transmission zero point outside the lower sideband cutoff frequency and outside the upper sideband cutoff frequency respectively through hybrid coupling.

2. The miniaturized high-performance bandpass hairpin filter circuit based on hybrid coupling according to claim 1, wherein A notch (127) is provided on one side of each U-shaped resonator, and a T-shaped groove (128) of the same size is etched on the other side, and the T-shaped groove (128) is arranged at the coupling end of the U-shaped resonator; the first feeding port (121) is connected to the first U-shaped resonator (123), the third U-shaped resonator (125) is arranged on a side of the first U-shaped resonator (123) away from the first feeding port (121), and the first U-shaped resonator (123) is connected to the first U-shaped resonator (123). ) and the third U-shaped resonator (125) are arranged alternately, a gap is left between the first U-shaped resonator (123) and the third U-shaped resonator (125), the T-shaped groove (128) of the first U-shaped resonator (123) and the T-shaped groove (128) of the third U-shaped resonator (125) form a staggered I-shaped groove; the notch (127) of the first U-shaped resonator (123) is downward, and the notch (127) of the third U-shaped resonator (125) is upward.

3. The miniaturized high-performance bandpass hairpin filter circuit based on hybrid coupling according to claim 2, wherein The structural arrangement of the first feeding port (121), the first U-shaped resonator (123), and the third U-shaped resonator (125) is axially symmetrical with the structural arrangement of the second feeding port (122), the second U-shaped resonator (124), and the fourth U-shaped resonator (126) about the x-axis.

4. A filter using the band-pass hairpin filter circuit according to any one of claims 1-3, characterized in that, include: The first shielding layer (10), the lid (11), the dielectric layer (13), the second shielding layer (14), and the metal post via array (15); the first shielding layer (10), the lid (11), the filter circuit (12), the dielectric layer (13), and the second shielding layer (14) are stacked on top of each other from top to bottom; the metal post via array (15) is uniformly arranged on the four peripheral edges of the bandpass hairpin filter except for the feeding end, and the metal post via array (15) penetrates through the entire bandpass hairpin filter to connect the first shielding layer (10) and the second shielding layer (14).

5. The filter according to claim 4, wherein The design formula for the order of the filter is as follows: where R is the normalized frequency, FBW is the relative bandwidth of the filter, f is the cut-off frequency, and f0 is the center frequency.

6. The filter according to claim 5, characterized in that, The calculation formula for the quality factor of the filter is as follows: where Q1 and Q2 are the quality factors of the filter, and g0, g1, g n , g n+1 are normalized element parameters.

7. The filter according to claim 6, characterized in that, The calculation formula for the coupling coefficient between the resonators of the filter is as follows: Among them, g i is the normalized element parameter, and K i,i+1 is the coupling coefficient between resonators.

8. The filter according to claim 7, characterized in that, The calculation formula for the length of the resonator of the filter is as follows: where c is the speed of light and ε eff0 is the effective dielectric constant.

9. The filter according to claim 8, wherein, The calculation formula for the tap position of the filter is as follows: where L is the length of the resonator, i.e., a quarter wavelength, Z0 is the impedance of the feeding port, Zr is the characteristic impedance of the filter, and Q is the quality factor of the filter.

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