Dual-Band Bandpass Coaxial Cavity Filters With Controllable Center Frequencies and Bandwidths Using F-Shaped Resonators

In this study, a novel dual-band bandpass coaxial cavity filter with large design freedoms for center frequencies and bandwidths is proposed by introducing F-shaped dual-mode resonators. Each F-shaped dual-mode resonator is capacitively loaded with two tuning screws for simultaneously tuning two frequencies. A dual-iris structure is implemented for the couplings between the two dual-mode resonators, with two screws applied to enhance the couplings. A dual-disc capacitive coupling is implemented at the input and output ports. By introducing these structures, the frequencies and bandwidths of the two passbands can have large design freedoms. A third-order dual-band filter at 0.845 GHz/1.790 GHz with bandwidths of 12 MHz/26 MHz is also designed, fabricated, and measured. Low insertion losses of 0.4 dB and 0.35 dB are achieved at the passbands. In comparison with planar dual-band bandpass filters, the filter presented in this study offers lower insertion losses and more convenience in filter tuning.


I. INTRODUCTION
Recently, the increasing demand for wireless communication has increased the requirement for dual-mode or multimode transceivers or systems that can support two or more communication standards.Consequently, dual-band bandpass filters have attracted the attention of many researchers.
Industrial dual-band bandpass filters are always realized by two filters that share the same input and output ports [1], but they entail complex structures and large dimensions.Dualmode resonators, such as stepped-impedance resonators [2] and stub-loaded resonators (SLRs) [3] in planar forms, are used to construct dual-band filters, but the bandwidths can hardly be independently assigned due to limited design freedoms.Microstrip and defected ground structure resonators implemented on two layers of a substrate can also be used to realize two passbands, but their development is costly [4].
In contrast to planar dual-band filters, cavity-type dualband bandpass filters offer considerably lower insertion loss.In [5], a dual-capacitively loaded waveguide cavity resonator is proposed for the dual-band filter, but bandwidths cannot be designed independently.In [6], a triple-conductor TEM dual-mode resonator is proposed, and a dual-band bandpass filter is designed with good features (i.e., low insertion loss).In [7], a dual-band bandpass filter is implemented within a waveguide, but high insertion loss is obtained because of the high-loss strip-line resonating modes.A low-loss dielectric loaded cavity dual-band bandpass filter with multiple transmission zeros is previously realized by using degenerate TM modes and by introducing nonresonant modes [8].Doubleground-plane dual-mode coaxial resonators [9] are used to design dual-band bandpass filters with independently tunable frequencies, but the configuration is attained at the expense of a complex structure.Stepped-impedance resonators [10] and SLRs [11] in coaxial form are also used to design dual-band bandpass filters.In [12], a dual-coaxial resonator is proposed to design compact multichannel bandpass filters.In [13], a new class of compact inline dual-band bandpass filters using TM-mode dielectric resonators in planar configuration is proposed, but frequencies cannot be designed independently.In [14], a dual-band bandpass filter is achieved using close dual post configurations that can generate different resonances.A side-coupled configuration is proposed and implemented in an elliptical waveguide dual-band filter [15].In [16], a dual-band filter is proposed, and each individual band is controlled by a dedicated polarization of the dual-mode resonators.
Reconfigurable filters have reconfigurable passbands because of the use of varactors [17] or switches [18], and they can be used in dual-mode or multimode telecommunication systems.However, these filters cannot simultaneously work for two or more passbands and have considerably higher insertion loss due to the introduction of high-loss tuning/switching elements.
In this study, the proposed F-shaped dual-mode resonator is used to simplify the structures and ensure flexibility in assigning bandwidths.A dual-iris coupling structure and a dual-disc capacitive coupling structure are adopted in the dual-band bandpass filter design to simultaneously control the two bandwidths.The designed filter exhibits lower insertion losses compared with planar dual-band bandpass filters.

II. F-SHAPED DUAL-MODE RESONATOR AND FILTER DESIGN METHOD
Fig. 1(a) shows the circuit of an F-shaped dual-mode resonator with three sections of transmission lines (i.e., AO, AB, and AC) and is short-circuited at O. Two capacitive loadings C 1 and C 2 are introduced by inserting metal tuning screws.The characteristic admittance of the sections is represented by Y 0 .Assuming a pathway from Point A toward three sections, when the values of C 1 and C 2 are sufficiently small to be ignored, the input admittance values can be expressed as follows: Resonance can be achieved by calculating for Y in = Y O + Y B + Y C = 0, and the following equation can be obtained after simplification: The transcendental equation can be solved by numerical methods, such as the bisection method.The lowest two zeros are taken as the resonating frequencies of the first and second modes.
The ratio of the center frequency f 2 of the higher frequency to the center frequency f 1 of the lower frequency in the two passbands is defined as R f , where R f = f 2 /f 1 .The length ratios of the branches in the F-shaped resonators are defined as R 21 and R 32 , where R 21 = l 2 /l 1 , and R 32 = l 3 /l 2 .R f is determined by R 21 and R 32 , as shown in Fig. 1(b).By changing the values of R 21 and R 32 , R f can reach a value range of 1.7-3.3.The value range can be expanded by setting larger value ranges for R 21 and R 32 .
With a fixed value of R f , the resonant frequencies f 1 and f 2 are inversely proportional to the section lengths of an Fshaped resonator.Moreover, by tuning the values of R 21 and R 32 , the frequency ratio R f can be efficiently adjusted, as shown in Fig. 1(b).Thus, the central frequencies of the two passbands can be freely adjusted by changing the three section lengths of the resonator.In practical design, the eigenmode solver of the commercial software HFSS [19] can be used to simulate the resonator precisely.Resonators can be coupled in two ways: capacitive and inductive coupling.Capacitive coupling generally occurs in parts with strong electric fields (i.e., here, the electric field is strong at Points C and B of the F-shaped resonators).Inductive coupling occurs in parts with strong magnetic fields.Electric and magnetic field coupling jointly affect the total coupling coefficient.
According to the voltage and electric field distributions of the two modes, the capacitive loading C 1 at Position C can effectively tune the frequency of Mode 1, whereas C 2 at Position B can better tune the second resonating frequency.A combination of C 1 and C 2 is introduced by two screws, which are represented by T 1 and T 2 in Fig. 4(a).These screws can provide full tuning of the two resonating frequencies.
Fig. 4(a) shows the proposed dual-iris coupling structure.Two screws are inserted into the two irises to enhance the coupling between two F-shaped dual-mode resonators.The values of the frequencies of the two modes used in the resonator are 0.845 GHz and 1.790 GHz.Screw S 1 (between Lines AC) introduces more coupling for the lower modes, whereas screw S 2 (between Lines AB) has more effect on Mode 2 coupling.By adjusting the dimension of the two screws, the coupling ratio between the two modes can be effectively tuned.Fig. 4(a) presents a coupled resonator model that uses two F-shaped dual-mode resonators rather than a complete filter.By using this model, coupling coefficients k 1 and k 2 between the two resonators can be calculated.The calculation formula of the coupling coefficient is expressed as follows: where fe 1 , fe 2 , fo 1 , and fo 2 , which represent the even and odd modes of the coupled resonators in Fig. 4(a), can be calculated using the eigenmode solver of the HFSS.In a third-order filter with a symmetrical structure, the third resonator is symmetrical and the same as the first resonator; thus, its parameters are also the same as the first resonator.The necessary coupling coefficient k and external quality factor Q e in the two passbands must be obtained to achieve the required BW 1 and BW 2 , respectively.Coupling coefficient k and the external quality factor Q e can be calculated as follows: ) The realized relative bandwidths of the two passbands are proportional to k 1 and k 2 and inversely proportional to Q e1 and Q e2 , respectively.
The values of k 1 and k 2 are mostly affected by iris heights (H 1 and H 2 ) and screw lengths (L S1 and L S2 ).and the center frequencies of the two passbands are 845 MHz and 1790 MHz, then the value of BW 1 can range from 5.25 MHz to 12 MHz with a relative bandwidth of 0.62%-1.42%,and the value of BW 2 can range from 1.58 MHz to 79.27 MHz, with a relative bandwidth of 0.088%-4.43%,using (4).
The distances between the capacitive discs and the resonators (D 1 and D 2 ) are the most influential parameters in Q e .The adjustable ranges of Q e1 and Q e2 shown in Fig. 5(b) and (c) can be obtained by changing D 1 and D 2 , which are 17-74 and 10-144, respectively.When only one passband is controlled, the value of BW 1 can range from 8.82 MHz to 38.41 MHz, with a relative bandwidth of 1.04%-4.55%,and the value of BW 2 can range from 9.61 MHz to 138.32 MHz, with a relative bandwidth of 0.54%-7.73%,using (5).
The relative values of k 1 and k 2 can also be adjusted.When H 1 is fixed, as illustrated in Fig. 4(b), by altering H 2 , k 1 can be made to be significantly larger than k 2 at specific screw lengths L S1 and L S2 .Thus, the k 1 /k 2 ratio can shift dramatically, with a range of 4-7 (0.016/0.004-0.007/0.001).Similarly, as illustrated in Fig. 4(c), when L S1 is fixed, k 2 can be configured to be significantly larger than k 1 at certain iris heights H 1 and H 2 .The range of k 1 /k 2 ratio is 1.75-3.57(0.007/0.004 -0.05/0.014).Hence, the bandwidth ratio of the two passbands can also be adjusted.
Similarly, the relative values of Q e1 and Q e2 can also be changed.Q e1 increases when either D 1 or D 2 increases, but Q e2 increases when D 2 grows and decreases when D 1 increases.By adjusting the values of D 1 and D 2 , the Q e2 /Q e1 ratio can be drastically shifted, with a range of 0.59-1.95(10/17-144/74), implying that the bandwidth ratio of the two passbands can also be adjusted.
By adjusting the values of k 1 and k 2 and Q e1 and Q e2 , the designed filter can adjust the ranges and ratios of the two passbands over a wide range.
The model of the designed third-order filter is shown in Fig. 6.

III. EXAMPLE FILTER DESIGN AND RESULTS
The 4G network is still used widely at present.The two frequency bands commonly used for mobile phones by China's four major telecom operators are 703-960 MHz and 1710-1880 MHz.Thus, a third-order dual-band bandpass filter is designed with center frequencies of f 1 = 845 MHz and f 2 = 1790 MHz.The designed bandwidths are BW 1 = 12 MHz and BW 2 = 26 MHz.The resonators, tuning screws, and capacitive discs are made of copper with a conductivity of 5.7 × 10 7 S/m.The cavity is made of aluminum with a conductivity of 3.5 × 10 7 S/m.The cavity and all components are silverplated to increase the external Q factor of the resonators.For the two passbands, the coupling coefficients and external Q factors at a passband ripple of 0.027 dB can be formulated as The size of the F-shaped dual-mode resonator is roughly equivalent to the size of the single-mode resonator with the same resonant frequency as the lower resonant frequency of the F-shaped resonator.The resonator dimensions in the aforementioned design are as follows: resonator width W 1 = 5 mm, thickness W = 5 mm, AO section length L 3 = 20 mm, AB section length L 2 = 21.2 mm, AC section length L 4 = 25 mm, and L 1 = 22 mm.The lengths of T 1 and T 2 are L T1 = 1 mm and L T2 = 3.8 mm, respectively.
In the dual-iris coupling structure, the width of the two coupling windows is W i = 9.8 mm, and the heights are H 1 = 24 mm and H 2 = 33 mm.The lengths of the two screws are L S1 = 23 mm and L S2 = 8 mm.When the dimensions are determined, the F-shaped resonators and housing are machined separately, and a screw hole is reserved at Point O of each resonator.Then, three F-shaped resonators are mounted to the housing by three screws.
For the dimensions of the input and output structures, the diameters of the two capacitive discs are 1 = 2 = 6.7 mm, and the distances from Positions C and B of the input or output resonator are D 1 = 2.65 mm and D 2 = 1.15 mm.In practical implementation, suitable assembly fixtures may be used to ensure good assembly precision.For example, Teflon rods with heights of D 1 and D 2 may be manufactured and placed between the discs and the resonators when fixing the discs to achieve precise distances D 1 and D 2 .Nonetheless, the filter must be fine-tuned to achieve good filtering performance.D 1 and D 2 can be changed in actual tuning by varying the bending of the metal lines linked between the SMA connectors and the capacitive discs.There are some techniques that can be used to widen the upper stopband and suppress the first spurious band.The simplest method is to employ a lowpass filter, which is often with low loss and is small in dimension.In a second approach, mixed F-shaped resonators [20] may be used.These resonators may have identical first and second resonating frequencies but vastly different third resonating frequencies.The third resonating frequencies cannot be combined to produce the first spurious band.The lengths and widths of the three transmission line sections in Fig. 1(a) may vary among the three F-shape resonators.For the purpose of suppressing the first spurious band, open stubs at the input and output ports are another option.
Errors are expected to arise during processing and manufacturing, further suggesting that the actual situation will also likely differ from the ideal situation.The fabricated filter should be adjusted according to the parameter values obtained from the simulation.Updating can be achieved by continuously adjusting the tuning screws and the capacitor discs at both ends.Fig. 9 shows the complete power handling testing system.This system consists of two power supplies, a signal generator, a power amplifier (PA), the fabricated filter, an attenuator, and a spectrum analyzer.The power amplifier can work at 0.7-2 GHz, with a maximum output power of 100 W. The result shows that the designed filter can work well at two center frequencies with power of 100 W, proving that the filter can be used in many communication systems.
The filter's contrasted S-parameter responses before, during, and after the power handling test are shown in Fig. 10.Due to the high output power of the PA, calibration on the output port is not possible when measuring the filter during the power handling.Therefore, we determined the gains of the PA (G 1 (dB)) and the cascaded PA and filter circuits (G 2 (dB)).S 21 (dB) of the filter can be roughly calculated as G 2 -G 1 at the two passbands.S 11 is not provided due to calibration issues.The S-parameter responses measured before and after the power handling test are shown to be in strict agreement in Fig. 10. S 21 , which was measured during the power handling test, almost has the same curve as the other plots, but because of a lack of calibration, it has a slightly different passband insertion loss.
Table I presents a comparison between the proposed filter and other reported planar and cavity filters.Similar to other cavity filters, the proposed filter has considerably lower insertion loss than planar dual-band bandpass filters, because of the markedly higher quality factor of the cavity resonators.The filters in [10] use stepped impedance resonators to achieve dual-narrowband and dual-wideband filters, but it lacks the elements in this study that can improve the ability to control the center frequencies and bandwidths, such as screws.In [14], two posts are connected together to produce a set of resonator pairs, which are utilized to achieve two passbands, one in odd mode and one in even mode.However, the design freedom of the coupling structure is limited.Moreover, the greater the coupling between two posts is, the larger the distance between the two passbands will be, but the overall distance will be limited.In comparison, a relatively large frequency ratio can be obtained by modifying the length and characteristic impedance of each transmission line of the F-shaped resonators presented in this study.Moreover, in contrast to the planar filters, the proposed filter has large design freedoms for the center frequencies and bandwidths due to the 3D filter design and the appropriate adoption of tuning elements.By using compact F-shaped resonators, the proposed filter has considerably small dimensions.

IV. CONCLUSION
A novel dual-band coaxial cavity filter is designed and fabricated in this study.By introducing F-shaped dual-mode resonators and applying tuning screws, the designed dualband filter has large design freedoms for the center frequency

FIGURE 2 .
FIGURE 2. Voltage and electric field distributions of two modes: (a) Voltage distribution of mode 1; (b) voltage distribution of mode 2; (c) e-field distribution of mode 1; (d) e-field distribution of mode 2.

Fig. 2 (
a) and (b) show the voltage distributions of the two resonating modes when l 1 = 20 mm, l 2 = 22 mm, and l 3 =

FIGURE 3 .
FIGURE 3. Effect of resonator width on Q-factor Q 1 and Q 2 .

Fig. 4 (
b) and (c) show many curves of k 1 and k 2 , allowing for the viewing of important dimensions, including the heights of the irises H 1 and H 2 and the lengths of the tuning screws L S1 and L S2 .Among these parameters, L S1 and H 2 have larger effects on k 1 and k 2 .Fig.4(b) also shows that k 1 and k 2 increase with the heights of irises H 1 and H 2 .H 2 has a larger effect on coupling coefficient k 1 , whereas H 1 has a larger effect on coupling coefficient k 2 .Furthermore, Fig.4(c) shows that the tuning screws loaded in the irises can effectively increase the coupling coefficients.By increasing L S1 , both coupling coefficients are effectively increased.If L S2 increases, then k 1 remains almost unchanged, whereas k 2 effectively increases.The ratio between k 1 and k 2 can be updated by tuning the four dimensions.Tuning the widths of the irises is expected to have a similar effect as tuning the iris heights.Fig. 5(a) shows the proposed dual-disc capacitive coupling structure used for input/output coupling.Two capacitive discs are loaded at Positions C and B of the input or output resonator.The discs mainly affect the external Q factor of Modes 1 and 2, with C and B corresponding to the maximum voltage points of each mode.By tuning the disc diameters or the spaces between the discs and the resonator rods, external quality factors Q e1 and Q e2 , which represent the external couplings, can be tuned.Fig. 5(b) and (c) show the variations in Q e1 and Q e2 with changing D 1 and D 2 .Q e1 is higher when either D 1 or D 2 increases due to decreased capacitive couplings at Position B or C. By contrast, Q e2 is higher when D 2 increases and lower when D 1 increases.This finding is due to the fact that the voltage at Point C is inverse to the voltage at Point B for Mode 2, as shown in Fig. 2(c) and (d).
Fig. 4(b) and (c) show the adjustable ranges of k 1 and k 2 by changing the iris heights and screw lengths, which are 0.007-0.016and 0.001-0.05,respectively.If only one bandwidth is controlled

FIGURE 5 .
FIGURE 5. Effect of distance between the capacitor disc and resonator on external Q factors Q e1 and Q e2 : (a) Input/output coupling structure; (b) effect on Q e1 ; (c) effect on Q e2 .

FIGURE 6 .
FIGURE 6. Model of the designed third-order filter.

Fig. 7
shows the simulation and measured results of the dual-band bandpass filter.According to the measurement results, the center frequencies of the first and second passbands are 844 MHz and 1794 MHz, with bandwidths of 10 MHz and 24 MHz, respectively.The return loss of both passbands is lower than −20 dB, and the insertion loss values are approximately 0.4 dB and 0.35 dB.A high isolation of 55 dB is achieved between the two passbands.The simulated and measured S-parameters are in good agreement with each other.By using the measured insertion losses, bandwidths, and return losses, the actual Q-factor values of the two modes can be approximated as 2150 for Mode 1 and 3000 for Mode 2. The simulated and calculated approximate actual values of Q-factor are relatively close.The errors between them may be due to ignoring the thickness of the housing, tuning screws, and other practical factors.Fig. 7(d) shows the measured outof-band response of the designed filter.The first spurious band

FIGURE 7 .
FIGURE 7. Simulated and measured responses: (a) Response of the wideband; (b) response of the first passband; (c) response of the second passband; (d) response of out-of-band.

FIGURE 8 .
FIGURE 8. Photograph of the fabricated dual-band bandpass filter.

FIGURE 9 .TABLE 1 .
FIGURE 9. Photograph of the power handling testing system.