Novel Evanescent-Mode Cavity Filter With Reconfigurable Rat-Race Coupler, Quadrature Coupler and Multi-Pole Filtering Functions

A novel evanescent-mode cavity filter with multiple reconfigurable functions including filtering rat-race coupler, filtering quadrature coupler and multi-pole filtering is proposed in this paper. It simply consists of four substrate integrated waveguide (SIW) evanescent-mode cavity resonators which are arranged in a ring loop. By loading piezoelectric actuators on top of each resonator and inserting varactors between adjacent resonators, the structure can achieve multiple reconfigurable functions among filtering rat-race coupler, filtering quadrature coupler and single-ended multi-pole filters, in addition to continuously frequency, independent bandwidth, and power-division tuning. Measurement shows that the filtering quadrature coupler corves a frequency range of 2.08-3.00 GHz with insertion loss of 2.99 ± 0.31dB, while the filtering rat-race coupler corves a frequency range of 2.31-2.71 GHz with insertion loss of 2.97 ± 0.13dB; The single-ended filter can be switched among two-pole, three-pole, and four-pole with frequency range of 2.38 to 3.01 GHz, 2.47 to 3.08 GHz and 2.53 to 3.00 GHz, respectively. Good agreement is obtained between simulated and measured results.


I. INTRODUCTION
Quadrature coupler and rat-race coupler are essential components and have been widely used in microwave circuits and systems [1]- [4]. Many techniques have been proposed to realize miniaturized couplers. For instances, in [5]- [7], folded microstrip lines were used to obtain compact couplers; in [8], co-planar waveguide structure was used to design coupler with broadband performance and compact size; in [9], laterally offset dual ring structure was used to realize quadband couplers. As the demand for multi-band applications increases, reconfigurable couplers with controllable center frequency, bandwidth and power-division ratio were also proposed [10]- [13]. In [14] and [15], couplers with continuously frequency tuning have been proposed, good tuning performance was obtained, however, all of them can afford only one coupler. In [16], rat-race or branch-line coupler was demonstrated in one single unit by electrically switching, however, the capability of frequency agility is not provided.
The associate editor coordinating the review of this manuscript and approving it for publication was Feng Lin. On the other hand, filters are also important components in microwave circuit and systems. To reduce system component counts, a great number of studies have been made to integrate coupler and filter together. In [17]- [24], different techniques have been proposed to realize the filtering couplers, however, these works only afford fixed band applications. In [25] and [26], reconfigurable filtering couplers were proposed. Even though flexible frequency tunability in these filtering couplers was achieved by controlling tunable corresponding varactors, these designs only have a simple function as either a filtering quadrature coupler or a filtering rat-race coupler, and did not allow filtering rat-race coupler and filtering quadrature coupler integrated in one single unit. In [27], a reconfigurable filter with rat-race coupler and quadrature coupler functions has been proposed, good performance has been obtained. However, the design in [27] is based on conventional microstrip line technology which has limitations of low quality factor, low power handling capability, and relatively narrow frequency tuning range.
In this paper, a novel reconfigurable multi-functional filtering coupler is proposed. It is constructed by four substrate integrated waveguide (SIW) evanescent-mode cavity resonators. By loading piezoelectric actuators on top of each resonator and inserting varactors between adjacent resonators, the structure can achieve multiple reconfigurable functions among filtering rat-race coupler, filtering quadrature coupler, single-ended two-pole, three-pole, and four-pole filters, in addition to continuously frequency, independent bandwidth, and power-division tuning. Throughout analysis is given to demonstrate the proposed technology. Good agreement between simulation and measurement has been obtained. As compared to the work in [27], this proposed design is an independent work that has demonstrated reconfigurable multifunction filter with good performance using EVA resonator technology. By using the EVA resonator technology, it could bring advantages of low insertion loss and high power handling. Fig. 1(a) shows the 3-D view of the proposed multi-functional four-port coupler. It is fabricated using conventional single-layer printed-circuit and consists of four (evanescentmode) EVA cavity resonators. Each EVA cavity resonator is constructed by an inner capacitive post and a circular outer via-wall. The inner capacitive post is composed of a circular path in the middle plane which is connected to bottom plane by six metallic vias. Above the circular path, a small cavity is fabricated and piezoelectric disk actuators are placed on top of the cavity for frequency tuning [28]. The adjacent evanescent mode cavities are connected by tunable varactors placing on the bottom plane, as shown in Fig. 1(b). The input/output coupling at each port is realized by opening an arc-slot which is underneath each EVA cavity on the bottom metallic plane.  It is well known that an ideal J-inverter can be implemented using one quarter-wave or three quarter-wave transmission line [29]. Thus, for analysis convenience, J-inverters with −90 • phase shift are replaced by quarter-wavelength lines, while the J-inverters with 90 • phase shift are represented by three quarters-wavelength lines, as shown in fig. 2(b).

II. DESIGN
The proposed four-port coupler structure can work under different function modes by reconfiguring the coupling coefficients between adjacent resonators. These function modes include the filtering quadrature coupler mode, filtering ratrace coupler mode and the multi-pole bandpass filtering mode, which will be discussed below in details.

A. FILTERING QUADRATURE COUPLER MODE
In filtering quadrature coupler mode, the coupling coefficients between adjacent resonators are configured as that J 12 and J 34 have 90 • phase delay and J 23 and J 14 have -90 • phase delay. Thus, the equivalent circuit in Fig. 2 can be represented by the circuit given in Fig. 3(a) where the J-inverters with −90 • phase shift are replaced by a quarter-wavelength lines and the J-inverters with 90 • phase shift are represented by three quarters-wavelength lines. The equivalent relation between J-inverters and the characteristic admittance of the transmission lines are given as, where Y 1 and Y 2 are normalized with respect to 50 source impedance (Y 1 > 0, Y 2 > 0), the ''+'' sign denotes the 90 • J-invertor, and the ''−'' sign denotes −90 • J-inverter. Since the structure is symmetrical with to X-X' plane, the proposed coupler can be analyzed with the even-and odd-mode analysis method, as shown in Fig. 3(b) and (c). Under even-mode excitation, a virtual open is assigned along the symmetrical plane, thus, the transmission matrix ([ABCD] e ) of the even mode half circuit in Fig. 3 (b) can be calculated as, where Y LC is the input admittance of the LC resonant tank. Similarly, under odd-mode excitation, the transmission matrix ([ABCD] o ) of the odd mode half circuit in Fig. 3 (c) can be calculated as, Assuming the working frequency is near the frequency of the resonant tank ω 0 , and it has Y LC ≈ 0. Converting the [ABCD] matrix in (2) and (3) to scattering matrix, the corresponding reflection and transmission coefficients under even and odd mode half circuits are obtained as, Based on (4)- (7), the reflection coefficient and transmission coefficient of the four-port coupler in Fig.3 (a) can be obtained as in the following equations: Assume port 1 and 4 in the proposed quadrature coupler serve as the input and isolation port, respectively, it has, Combing (8), (11), (12) and (13), Y 1 and Y 2 is found to have the following relation, Combining (9), (10) and (14), transmission coefficient from port1 to port2 and from port1 to port3 are obtained as, From (15) and (16), it can be found that output port 2 and 3 have 90 • phase difference, indicating an ideal quadrature coupler around the working frequency ω 0 . When the frequency is away from ω 0 , the input admittance of the L-C resonant tank increases, and provides a low-impedance path to ground, thus shorting the signal to ground and exhibiting a filtering coupler performance. The power division ratio k pr between output port 2 and 3 is found using (15) and (16) as, Combine (1), (12), (13), (15), (16) and (17), it has, Thus, it is concluded that once the power ratio between output port 2 and 3 of the quadrature coupler is specified, the value of the J-invertor between resonators can be determined to obtain a quadrature coupler with given power-division ratio.
It should be noted that equations (18) and (19) only guarantee the quadrature coupler performance with 90 • phase difference and power-division ratio of k pr at the outputs, they do not guarantee the desired filter performance with specific fractional bandwidth and ripple level. In order to obtain desired filter quadrature coupler performance for given filter specifications, the coupling between resonators in the proposed filter coupler needs to map with the conventional coupled resonators [26]. Fig. 3(d) shows the structure of conventional coupled resonators with J-invertor in between. Assuming the characteristic admittance of the J-invertor is J c , the coupling coefficient k c between the coupled resonators can be calculated from the low-pass prototype filter as, where FBW is the fractional bandwidth, g 1 and g 2 are element value in the corresponding low-pass prototype filter, ω 0 is the angular resonant frequency, C is the capacitance of the L-C resonant tank. The mapping of the coupling coefficients between the proposed filter coupler and the conventional coupled resonators can be approximately obtained as [30], where k ij is coupling coefficient between adjacent resonators i and j in the proposed filter quadrature coupler for given filter bandwidth and ripple level. Since k c can be calculated from the prototype filter for given filter bandwidth and ripple level [29], the k ij can then obtained from (21) and (22) following the same filter requirement. The external quality factor of the proposed filter coupler stays the same with the external quality factor calculated from the coupled resonator theory for given filter requirement.
In filtering quadrature coupler mode, it is noted from (21) and (22) that the coupling coefficients k 12 and k 34 are positive while k 23 and k 14 are negative. The coupling sign depends on the physical coupling type of the coupled resonators, which usually refers to the magnetic (inductive) coupling or the electric (capacitive) coupling. If the magnetic coupling is assigned to be negative coupling, then the electric coupling will be positive coupling.
For the idea 3dB filter quadrature coupler (k pr = 1), the required coupling coefficients between resonators are calculated as,

B. FILTERING RAT-RACE COUPLER MODE
In filtering rat-race coupler mode, the coupling between adjacent resonators are configured as that J 12 , J 14 , and J 34 have 90 • phase delay while and J 23 have −90 • phase delay. Thus, the equivalent circuit in Fig. 2 can be represented by the circuit given in Fig. 4. The J-inverters with −90 • phase shift are replaced by a quarter-wavelength line while the J-inverters with 90 • phase shift are represented by three quarters-wavelength line. The equivalent relation between Jinverters and transmission lines are given as, Port 1 and 3 are the sum and difference ports, respectively. With symmetrical plane along the X-X' plane, the proposed rat-race coupler can be analyzed using the same even-and odd-mode method as given in the previous section, and the scattering parameters of the proposed rat-race coupler can be obtained as, From (25), it can be seen that port 2 and 4 are 0 • in phase when port 1 servers as the input, and are 180 • out of phase when port 3 servers as the input. To obtain ideal match at port 1 and ideal match at port 3, it has Combine (25) and (26), the relation between Y 1 and Y 2 can be obtained, The power division ratio between port 2 and 4 can be calculated as, Combine (24), (25), (26), and (28), it has, Thus, it is again concluded that once the power division ratio between output port 2 and 4 of the rat-race coupler is specified, the value of the J-invertor between resonators can be determined to obtain a rat-race coupler with given powerdivision ratio. VOLUME 8, 2020 Similarly, in order to obtain desired filtering rat-race coupler performance for given filter specifications, the mapping of the coupling coefficients between the proposed filter ratrace coupler and the conventional coupled resonators can be approximately obtained as, For the idea 3dB rat-race coupler (k pr = 1), the required coupling coefficients between resonators are calculated as,

C. SINGLE-ENDED SECOND-ORDER/THIRD-ORDER/FOURTH-ORDER BANDPASS FILTERS MODE
In single-ended bandpass filtering mode, some ports in the coupler are left open and the coupling coefficients between adjacent resonators are reconfigured to realize second-order, third-order and fourth-order filters.

1) SECOND-ORDER BANDPASS FILTERING MODE
In this mode, port 3 and 4 are left open and coupling coefficients between resonator 1 and 2, between resonator 2 and 3, and between 3 and 4 are made to be zero as k 12 = k 23 = k 34 = 0. Since k 12 = k 23 = k 34 = 0, it is straightforward that the signal can only go from port 1 to port 4 through resonator 1 and resonator 4, forming a second-order bandpass filter.

2) THIRD-ORDER BANDPASS FILTERING MODE
In this mode, port 2 and 4 are left open and coupling coefficients between resonator 1 and 2, between resonator 2 and 3, are made to be zero as k 12 = k 23 = 0. Thus, the signal can only go from port 1 to port 3 through resonator 1, resonator 4 and resonator 3 in sequence, forming a third-order bandpass filter.

3) FOURTH-ORDER FILTERING MODE
In this mode, port 3 and 4 are left open and coupling coefficients between resonator 1 and 2 is made to be zero as k 12 = 0. Thus, the signal can only go from port 1 to port 2 through resonator 1, 4, 3 and 2 in sequence, forming a fourthorder bandpass filter. In Summary, the proposed four resonators structure can work under different function modes by reconfiguring the coupling coefficients between adjacent resonators.    [30] while the coupling from the coupling capacitator C c mainly contributed to the electric coupling [31]. Fig. 5 (c) shows equivalent mixed coupling circuit of the proposed structure. An impedance inverter K = ωL m and an admittance inverter J = ωC m was used to represent the magnetic coupling and the electric coupling, respectively [29]. The coupling coefficient k ij is mixed coupling of magnetic coupling k m and electric coupling k e as [29],

D. ANALYSIS OF COUPLING COEFFICIENT
When the magnetic coupling k m is larger than the electric coupling k e , the total coupling k ij is dominated by the magnetic coupling and verse vice. The magnetic coupling k m can be controlled by the width window w and distance l between two adjacent resonators while the electric coupling k e can be controlled by the value of the coupling capacitance. Fig. 6(a) shows the extracted coupling coefficients versus coupling capacitance at different width w and distance l using the method given in [29]. As can be observed, the coupling k ij can be tuned from a negative value to zero and from zero to positive. When the coupling capacitance is week, the coupling is mainly dominated the magnetic coupling in nature (k e < k m ); When the coupling capacitance increases, the electrical coupling between adjacent resonators increases. When it reaches a certain value C t , the electric coupling cancels out the magnetic coupling (k e = k m ), resulting in zero coupling. As the coupling capacitance increases from C t , the electrical coupling keeps increasing and becomes larger than the magnetic coupling. The total coupling is dominated by electric coupling (k e > k m ).
Therefore, by using the proposed the coupling structure, the coupling coefficients between adjacent resonators can be easily controlled to be either magnetic coupling or electric coupling or zero, so as to satisfy the requirements for filtering rat-race coupler mode, filtering quadrature coupler mode, and single-ended filtering mode. The required inter-stage coupling between adjacent resonators under different filter modes can be approximated read from Fig. 6(a).
The input/output coupling of the four ports is determined by size of the opening slots on the top plane, as shown in Fig. 5(d). To introduce more tuning flexibility, a shunt capacitance to ground is added at the input/output lines. Fig. 6(b) shows the extracted external coupling versus the shunt capacitance value. As can be observed, the Q e values can be controlled by changing the opening angle size of the slot φ and varactor C e . The required input/output coupling for different filtering modes can be approximated read from Fig. 6(b). A flowchart of this process is shown in Fig. 7.

III. SIMULATIONS AND MEASUREMENTS
The filtering crossover is designed to have quasi-Chebyshev response with a 3-dB FBW of 5% with ripple level of 0.1 dB, and the center frequency of 2.64 GHz is designed for the filtering coupler. The design parameters corresponding to these specifications can be obtained in Fig. 6 and Table 1.
The proposed reconfigurable multifunctional filter was fabricated on Rogers 4350B with dielectric constant   The inter-resonator and external coupling control network is on the backside of the substrate as shown in Fig. 1(b). GaAs varactors MA46H201 (C j = 0.3 ∼ 23 pF) from Macom Inc. were used to control the inter-stage coupling and external coupling control, respectively. The photograph of the fabricated filter are shown in Fig. 8 and it occupies a footprint size of 0.31λ g × 0.32λ g , where λ g is the guided wavelength at the lowest frequency (2.08 GHz).

A. SIMULATED AND MEASURED RESULTS OF QUADRATURE FILTERING COUPLER
In filtering quadrature coupler mode, the coupling between resonators is tuned to satisfy the equation (21) and (22) by controlling the coupling varactors. Fig. 9 presents the measured and simulated S 21 and S 31 of the proposed quadrature filtering coupler with k pr = 1. Measurement agrees well with the simulation. The center frequency of the filtering coupler can be tuned from 2.08 to 3.00 GHz with insertion loss from 2.68 to 3.30 dB. The amplitude imbalance between S 21 and S 31 is less than 0.3 dB across the passband. The measured return loss at port 1 and isolation at port 4. The measured reflection and isolation are better than 10.22 and 15.13 dB, respectively, for all tuning states. The measured phase imblance is less than 5 • within 30MHz bandwidth.
By controlling the coupling amount between the resonators, the power-division ratio k pr between port 2 and 3 can be also adjusted. Fig. 10 presents the measured and simulated S-parameters under different power-division ratios of 1 dB, 3 dB and 5 dB in filtering quadrature coupler mode. The measures insertion losses (S 21 and S 31 ) for the cases of 1dB, 3dB and 5dB power-division ratios at 2.64 GHz are (5.56dB and 6.49dB), (4.46dB and 7.46dB) and (3.58dB and 8.54dB), respectively. Note that the ideal insert loss in lossless case for the cases of 1dB, 3dB and 5dB powerdivision ratios are (3.53 and 2.53 dB), (4.77 and 1.77 dB), and (6.19 and 1.19 dB). The measured phase imbalance is   less than 5 • within 40MHz bandwidth under different powerdivision ratios. Fig. 11 presents the measured and simulated S-parameters of proposed quadrature filtering coupler for bandwidth control. The 1-dB bandwidth can vary in the range of 60 -75 MHz. The measured insertion loss decreases with the increasing bandwidth. The phase imbalance is less than 5 • within 30MHz bandwidth.

B. SIMULATED AND MEASURED RESULTS OF RAT-RACE FILTERING COUPLER
In filtering rat-race coupling mode, the coupling between resonators is reconfigured to satisfy the equation (28) and (32). Fig. 12 shows the measured and simulated S-parameters  in filtering rat-race coupling mode with k pr = 1 (0 dB). When excited from port 1 (sum port), it has an equal power dividing and 0 • phase difference between ports 2 and 4, and the passband can be tuned from 2.31 to 2.71 GHz. The measured insertion loss of S 21 and S 41 varies from 2.84 to 3.1 dB within the full tuning range. The reflection at port 1 and isolation at port 3 is better than 15.29 and 17.19 dB, respectively. The measured phase imbalance and amplitude imbalance between S 21 and S 41 are better than 8 • and 0.4 dB, respectively. When excited from port 3 (differential port), it has an equal power dividing and 180 • phase difference between ports 2 and 4; the passband can be also tuned from 2.31 to 2.71 GHz. The measured insertion loss of S 23 and S 43 varies from 2.36 to 3.79 dB within the full tuning range. The reflection at port 3 and isolation at port 1 is better than 10.42 and 16.32dB. The measured phase imbalance and amplitude imbalance between S 21 and S 41 are better than 8.5 • and 0.4 dB, respectively. Fig. 13 presents the measured and simulated S-parameters in filtering rat-race coupling mode with k pr = 1 dB, 3 dB and 5 dB at 2.65 GHz. When excited from port 1 (sum port), the measured insertion loss of (S 21 and S 41 ) for k pr = 1, 3 and 5 dB are (6.79 and 5.77 dB), (7.73 and 4.794 dB), and (8.567 and 3.65 dB), respectively. Note that the ideal insert loss in lossless case of (S 21 and S 41 ) for k pr = 1, 3 and 5 dB are (3.53 and 2.53 dB), (4.77 and 1.77 dB), and (6.19 and 1.19 dB), respectively. Thus, the average insertion loss for S 21 and S 41 is less than 2.8 dB. The phase imbalance for S 21 and S 41 is again less than 5 • within 50MHz bandwidth. When excited from port 3 (differential port), the measured insertion loss of (S 23 and S 43 ) for k pr = 1, 3 and 5 dB are (6.42 and 5.62 dB), (7.01 and 4.50 dB), and (7.97 and 3.85dB), respectively, with a maximum phase error of ±5 • within 50MHz bandwidth. The average insertion loss for S 23 and S 43 is less than 3 dB. The phase imbalance for S 23 and S 43 is again less than 5 • within 50MHz bandwidth.
Fig. 14 presents the measured and simulated S-parameters of proposed rat-race filtering coupler for bandwidth control. As can be observed, the −1 dB bandwidth for the passband can be tuned from 70 to 90 MHz, while keeping good phase and amplitude imbalance.

C. SIMULATED AND MEASURED RESULTS OF SECOND-ORDER, THIRD-ORDER AND FOURTH-ORDER FILTERS
In second-order bandpass filtering mode, port 3 and 4 are left open and coupling coefficients between resonator 1 and 2, between resonator 2 and 3, and between 3 and 4 are made to be zero as k 12 = k 23 = k 34 = 0. Fig. 15 (a) presents the measured and simulated S-parameters in second-order bandpass filtering mode between port 1 and 2. Good agreement is observed between simulation and measurement. The center frequency can be tuned from 2.38 to 3.01 GHz with insertion loss varying from 1.1 to 2.64 dB and constant absolute-bandwidth of 25-MHz. The stopband suppression at frequency of 100MHz away from center frequency is about 10 dB. By floating port 2 and 4 and making coupling coefficients k 12 = k 23 = 0, the proposed coupler can work as a conventional third-order bandpass filter between port 1 and 3. Fig. 15 (b) presents the measured and simulated S-parameters in third-order bandpass filtering mode. Again, good agreement is observed between simulation and measurement. The center frequency can be tuned from 2.47 to 3.08 GHz with insertion loss varying from 1.79 to 2.67 dB. The stop band suppression at frequency of 100MHz away from center frequency is about 20 dB.
Similarly, when port 3 and 4 are left open and coupling coefficients between resonator 1 and 2 are made to be zero as k 12 = 0, the proposed coupler can work as a conventional fourth-order bandpass filter between port 1 and 2. Fig. 15 (c) presents the measured and simulated S-parameters in fourthorder bandpass filtering mode. The center frequency can be tuned from 2.53 to 3 GHz. The measured insertion loss varys from 3.2 to 3.6 dB. The stopband suppression at frequency of 100MHz away from center frequency is better than 30 dB. Note than the tuning range for the four-order bandpass filter is narrower than the third-and second-order counterparts, this is because more varactors are involved in the tuning process for four-order bandpass filter than for the third-and second-order counterparts, and some varactors may have smaller tuning ranges than others, thus, limiting the full tuning range for the fourth-order bandpass filter. Table 2 gives the comparison between this proposed work and other state-of-the-art reconfigurable filtering couplers. The proposed reconfigurable filtering coupler has successfully realized three reconfigurable functions in one single unit using the EVA resonator technology. As compared to the work in [27], this proposed work exhibits lower insertion loss due to the higher quality factor in EVA resonators. It should be noted that the insertion loss of this proposed work is still limited by the varactors loaded between resonators. The insertion loss may be significantly improved by replacing the current varactors with MEMS varactors, and is expected be much better than that of the work in [27] with the same MEMS varactors. Meanwhile, the power handling is also expected to be better than the work in [27].

IV. CONCLUSION
A novel reconfigurable filtering coupler utilizing EVA resonators is presented. It can achieve rat-race filtering coupler, quadrature filtering coupler and multiple-order filters with frequency, bandwidth and power-dividing ratio controls, demonstrating excellent tuning capabilities. Good agreement between the measured and simulated results has been obtained.