Improved Designs for Highly Integrated Lowpass–Bandpass Filters

This study presents three dual-band lowpass–bandpass filters. The first handles lowpass and bandpass bands through its fifth-order and third-order Chebyshev filter responses, respectively. This filter uses three circuits to achieve flexibility in cutoff frequencies and fractional bandwidths. The second is a higher-order filter: it is a dual-band filter with ninth-order lowpass and fifth-order bandpass responses. In general, an extra 50-<inline-formula> <tex-math notation="LaTeX">$\Omega $ </tex-math></inline-formula> transmission line might be required to connect each port conveniently to an external circuit for the first or second proposed filter, which could increase the circuit area. The third is a dual-band filter with a <inline-formula> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula>/4 impedance transformer near each port to facilitate connections with external circuits. Each of the proposed dual-band lowpass–bandpass filters has a systematic design procedure when the lowpass and bandpass band responses are given separately. In addition, the proposed filters can provide rapid prediction through the use of ideal circuit simulations instead of full-wave simulations.


I. INTRODUCTION
Dual-band filters with bandpass responses in each passband were widely used [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. To meet compact size requirements, [1] and [7] used ceramic lamination technique and printed circuit board to develop multilayered structures and [8] used meandering stepped-impedance resonators to reduce the sizes. Coupled-feed structures were used by [2] for dual-band external quality factors, and [3] designed dual-band transformer as a dual-band impedance match for each presented filter input or output port. A previous study [4] proposed dual-band coupling coefficients that could be independently designed, but external quality factors resulted in a complex disjointed design. Another study [6] designed dualband filters with a properly arranged feeding structure that can achieve good spurious suppression response, but they needed extra impedance transformers. To avoid the additional transformers, dual-band external quality factors that can be satisfied through systematic direct-feed filter design were The associate editor coordinating the review of this manuscript and approving it for publication was Giovanni Angiulli . first proposed by [9], [10], [11], [12], and [13] introduced reconfigurable dual-band filters that changed their frequency responses depending on the biased circuit conditions.
For example, when one LPF or BPF response was indicated, the other response was determined under the indicated response; this design offered little flexibility. Furthermore, the f 0 (center frequency of the BPF response) to f c (cutoff frequency of the LPF response) ratio was greater than 5, which might be unsuitable when the LPF band is close to the bandpass band. A previous study [24] used quasi-lumped elements to achieve the required capacitances and inductances for DB-LBFs. However, the device's equivalent circuit was complex, the parasitic effect of the lumped elements was insufficiently low, and extra chip capacitors for [24] were required, resulting in a complex design or possibly increasing circuit costs. Moreover, the LBF and BPF passbands featured the use of transmission zero locations in the design of their responses, which made it unsuitable for one to use familiar methods of synthesis, such as the Chebyshev method, to rapidly predict results; this approach failed to achieve a higher-order filter design. The synthesis issue was also faced in quasi-lumped circuits such as LPF [26], reconfigurable bandpass/lowpass filter [27], or triple-band lowpassbandpass filter (TB-LBF) [29] design. In general, filters using lumped elements to approach their design responses can meet compact circuit size results. However, they usually need time-consuming optimization processes because of producing massive undesired parasitic effects. [30] used planner circuit design, but its equivalent circuit was complicated and it failed to provide the useful synthesis method such as the Chebyshev approach for designing higher-order DB-LBF responses.
To the best of our knowledge, little research has been conducted on DB-LBFs or reconfigurable DB-LBFs, and studies such as [15], [16], [21], [24], [27], and [30] have low circuit integration, parasitic effect issue, or complicated design issue. This study designed three types of DB-LBFs (LBF A, LBF B, and LBF C), each achieving the complete integration of LPF and BPF. The proposed DB-LBFs are also systematically designed when the specifications of LPF and BPF are independently indicated. Fig. 1(a) presents the proposed LBF A structure. It is composed of eight transmission line sections, X A11 -X A32 , X A4 , and X A5 , with electric lengths of θ A11 -θ A32 , θ A4 , and θ A5 , respectively; the characteristic impedances are Z A11 -Z A32 , where Z Ai1 = Z Ai2 = Z Ai (i = 13), Z A4 , and Z A5 , respectively. Y inA1 -Y inA3 are input admittances. Fig. 1(a) can be redrawn to feature a bilateral symmetrical structure [ Fig. 1(b)]. Fig. 2(a) is the fifth-order LPF equivalent circuit of LBF A with the same eight transmission lines (X A11 -X A32 , X A4 , and X A5 ) in the design of its response. In Fig. 2(a), C 1 , C 3 , and C 5 are capacitances and L 2 and L 4 are inductances. The design equations of the LPF capacitances and inductances are as follows [31]:

II. DESIGN OF LBF A
where g k , R 0 , and f c are the prototype element value, source impedance, and cutoff frequency, respectively. The coupled-resonator BPF response of the center frequency f 0 is formed using three resonators (R A1 , which comprises X A11 and X A12 ; R A2 , which comprises X A21 and X A22 ; and R A3 , which comprises X A31 and X A32 ) and two inverters (X A4 and X A5 ). The n-order coupled-resonator BPF design equations [31] are where Q ei /Q eo is the external quality factor Q L of the input or output resonator, n is the filter order number, g 0 g n+1 denotes lowpass filter prototype lumped-element values, FBW is the fractional bandwidth variable, and M ii+1 is the coupling coefficient between two adjacent resonators. The λ/2 uniform impedance resonator (UIR) resonators R A1 R A3 resonate at f 0 and are equivalent to three shunt-to-ground capacitors with the capacitances of C 1 , C 3 , and C 5 , respectively, when their frequencies are at f c . Furthermore, the feed-line locations of R A1 and R A3 are required to satisfy the external quality factors of (3) and (4), respectively. The external quality factor Q L of a lossless resonator is also noted by [32] and written as In (6), ω is the angular frequency variable, ω 0 = 2πf 0 is the center angular frequency, R L is the input impedance from the resonator looking into the load, and B is the susceptance of the input admittance Y in from the feed point to the resonator. Fig. 2(b) illustrates the input or output λ/2 UIR structure that forms the basis of the external quality factor calculation, where Z s and θ s are the characteristic impedance and electrical length, respectively. By using (6), one can derive the external quality factor of Fig. 2(b) as follows: Because R A1 must satisfy the requisite Q L through the use of (7) at f 0 and because its input admittance Y inA equals the admittance of C 1 in Fig. 2(a) at ω c (ω c = 2πf c ), the related design equations [23] can be written as In (8) and (9), θ A11 is the electrical length at f 0 and Z 0 is the system impedance. The remaining two unknown variables, Z A1 and θ A11 , can be solved by (8) and (9) when the specifications of LPF and BPF are determined. LBF A is designed for fifth-order-LPF and third-order-BPF dual-band Chebyshev responses; LBF A has a bilateral symmetrical circuit and R A1 has the same design parameters as R A3 . Line section X A4 serves as a λ/4 inverter between R A1 and R A2 at f 0 , and it approaches L 2 in Fig. 2(a) at f c ; this design is the same as that of line section X A5 because of the symmetrical property of LBF A. In practice, f c < f 0 is optimal, f c should not be too close to f 0 , and the characteristic impedance of X A4 should not be low. The characteristic impedance Z λ/4 of a λ/4 inverter [31] can be approached as Let L k = L 2 in (10); this yields the characteristic impedance Z A4 = Z λ/4 of X A4 . Furthermore, the connected-coupling technique of either [33] or [34] can be applied to make the coupling coefficients M 12 be between R A1 and R A2 and M 23 be between R A2 and R A3 ; these are achieved using two connected lines: the first is X A4 and the second is X A5 . Fig. 2(c) demonstrates a coupling structure for R A1 and R A2 , where the capacitance C W = 0.001pF models a weak coupling. For giving the LPF and BPF specifications, the required element values of Fig. 2(a) and Q L in (3) and (4) are determined.
To facilitate the design, the following steps are executed: Step I A : The design of R A1 must satisfy the conditions of (8) and (9). Thus, R A1 can meet the required Q L at f 0 and C 1 of Fig. 2(a) at f c . The design parameters of R A3 are the same as those of R A1 because LBF A has a bilateral symmetrical structure. Thus, Q L at f 0 and C 5 of Fig. 2  of R A1 , R A2 , and R A3 is λ/2 at f 0 , enabling each of them to reach resonance condition at f 0 .
Step II A : The characteristic impedances Z A4 of X A4 or Z A5 of X A5 can be designed using (10). Thus, L 2 and L 4 of LPF are met.
Step III A : R A2 is λ/2 at f 0 to satisfy the BPF resonance condition, but it must still satisfy C 3 of Fig. 2(a) at f c . The design equation is similar to that of (9) and can be written as where θ A21 is the electrical length of f 0 . All the circuit parameters of R A1 have been determined in Steps I A and II A . C 3 , f c , and f 0 are identified when the response specifications of LPF and BPF are given. The remaining two design variables θ A21 and Z A2 are arbitrarily indicated one variable and the other variable can be solved using (11). The coupling coefficient between R A1 and R A2 is M 12 , which can be designed by varying θ A21 and Z A2 in Fig. 2(c) to meet BPF specification using (5). Thus, LBF A can achieve bilateral symmetry. Therefore, the design parameters of X A5 and R A3 are designed to be similar to those of X A4 and R A1 , respectively; this allows M 23 to simultaneously meet the required value when M 12 designed through the adjustment of the length of θ A21 . By systematically following the procedure in Steps I A -III A , one can design the DB-LBF response of LBF A without undertaking the time-consuming task of  optimizing the LPF and BPF responses. In this study, all the DB-LBFs were implemented on an RO4003C substrate with a thickness of 0.508 mm, loss tangent of 0.006, and dielectric constant of 3.58. LBF A has three circuits: Case A 1 , Case A 2 , and Case A 3 . Table 1 summarizes the specifications of the three circuits.
Based on the design described in this section, the related design parameters can be obtained from Table 2 and the remaining parameters are also known because LBF A is a bilateral symmetrical circuit and R A1 -R A3 are UIRs. Fig. 3 illustrates simulated instances of Cases A 1 -A 3 for comparison. A comparison of Case A 1 with Case A 2 [ Fig. 3(a)      circuit area layouts and theses lines ignore their meander effects in circuit designs. For lowpass band, ideal lumped LPF simulation meets exactly required 0.1 dB equal-ripple value in Case A 1 or Case A 3 ; 0.5 dB equal-ripple value in Case A 2 . Minor equal-ripple value errors for other lowpass or bandpass band responses in Figs. 7-9, which are caused by the approaching circuit designs or measured errors. Note that each BPF band has three ripples for each |S 11 | corresponding to third-order equal-ripple filter response. Table 3 lists the approximated measurement results.

III. DESIGN OF LBF B
The LBF B structure [ Fig. 10(a)] is an example of a higherorder DB-LBF response; it is composed of 14 transmission line sections: X B11 -X B52 and X B6 -X B9 , with electrical VOLUME 11, 2023 lengths of θ B11 -θ B52 and θ B6 -θ B9 , respectively. The characteristic impedances are Z B11 -Z B52 , where Z Bi1 = Z Bi2 = Z Bi (i = 15) and Z B6 -Z B9 , respectively. Y inB1 -Y inB3 are input admittances. Fig. 10(a) can be redrawn to feature a bilateral symmetrical structure [ Fig. 10(b)]. The design of the ninth-order LPF Chebyshev response is based on these 14 transmission lines, with an equivalent circuit presented in Fig. 10(c). The coupled-resonator BPF response of the center frequency f 0 [31] is formed by five resonators R B1 -R B5 , with R Bi comprising X Bi1 and X Bi2 , and four λ/4 inverters X B6 -X B9 . At f c , R B1 -R B5 are also equivalent to the five shunt-to-ground capacitances C 1 -C 9 of Fig. 10(c), respectively. Each of the λ/4 inverters X B6 -X B9 has a similar design to the λ/4 inverter X A4 in LPF A. The design procedure is summarized as follows: Step I B : The design of R B1 or R B5 is similar to that of R A1 in Step I A .
Step II B : At f c , the four λ/4 inverters X B6 -X B9 are designed to satisfy the four series inductances L 2 -L 8 of Fig. 10(c), respectively. A similar design as that in Step II A is used for this step.
Step III B : R B2 must simultaneously resonate at f 0 and achieve the shunt-to-ground capacitance C 3 [in Fig. 10(c)] at f c , which has a similar design to R A2 in Step III A . Because the designs of R B1 and X B6 are completed in Steps I B and II B , their circuit parameters are fixed. C 3 , f c , and f 0 are identified when the specifications of the LPF and BPF responses are determined. The corresponding Z B2 is obtained through adjustment of θ B21 , which can be used to design the required coupling coefficient M 12 between R B1 and R B2 . Based on a similar design concept, R B3 resonates at f 0 and is also equivalent to the shunt-to-ground capacitance C 5 of Fig. 10(c). The parameters of R B2 and X B7 cannot be changed after the M 12 and Step II B designs are completed, respectively. The coupling coefficient M 23 between R B2 and R B3 is designed by adjusting θ B31 given the corresponding Z B3 . Because LBF B has a bilateral symmetrical configuration, the design of the remaining parts is obtained. LBF B, which has a Chebyshev ninth-order lowpass filter and fifth-order bandpass dual-band filter, is designed in its entirety by following Steps I B -III B systematically; this process does away with the need for the time-consuming task of optimizing the two desired bands. Compared with LBF A, LBF B achieves a higher-order DB-LBF response and can be extended to higher-order designs for DB-LBFs.
LBF B is designed for f c = 1.2 GHz for a 0.1-dB equalripple LPF response and f 0 = 2.4 GHz and FBW = 9% for a 0.1-dB equal-ripple BPF response. Based on the   19 , Z B3 = 47.54 , Z B6 = Z B9 = 87.07 , and Z B7 = Z B8 = 97.58 . Fig. 11 details the layout and presents a photograph of LBF B, and the simulated and measured results are presented in Fig. 12. In the lowpass band, the measured passband maximum insertion loss and cutoff frequency are approximately 0.769 dB and 1.2 GHz, respectively; in the bandpass band, the measured passband minimum insertion loss, 3 dB FBW, and center frequency are approximately 2.98 dB, 8.3%, and 2.415 GHz, respectively. For lowpass band, ideal lumped LPF simulation meets exactly required 0.1 dB equal-ripple value. Minor equal-ripple value errors for other lowpass or bandpass band responses in Fig.12, which  are caused by the approaching circuit design or measured errors. Fig. 13(a) presents the structure of LBF C with a DB-LBF response that improves the input or output circuit layout flexibility of the previously proposed filters in this study. LBF C has ten transmission line sections: X C11 -X C32 and X C4 -X C7 , with electrical lengths of θ C11 -θ C32 and θ C4 -θ C7 , respectively; the characteristic impedances are Z C11 -Z C32 , where Z Ci1 = Z Ci2 = Z Ci (i = 13) and Z C4 -Z C7 , respectively. Y inC1 and Y inC2 are input admittances, and Z inC4 is the input impedance. Fig. 13(a) can be redrawn to depict a bilateral symmetrical structure [ Fig. 13(b)]. The ten transmission line sections with an equivalent circuit illustrated in Fig. 13(c) are used to design a seventh-order LPF Chebyshev response, where L 1 -L 7 and C 2 -C 6 are inductances and capacitances, respectively. Resonators R C1 -R C3 (R Ci comprises X Ci1 and X Ci2 ), two λ/4 inverters (X C5 and X C6 ), and two λ/4 impedance transformers (X C4 and X C7 ) are used to design the coupled-resonator BPF response [27] with center frequency f 0 . The LBF C design procedure is as follows:

IV. DESIGN OF LBF C
Step I C : The length of X C4 -X C7 equals λ/4 at f 0 and satisfies the inductances L 1 -L 7 of Fig. 13(c) at f c when the specification of the dual-band Chebyshev lowpass and bandpass response is given. The characteristic impedances of X C4 -X C7 can be obtained using (10) and the specifications of theL 1 -L 7 inductances.
Step II C : Z C4 has been determined in Step I C , and X C4 serves as the λ/4 impedance transformer at f 0 . Thus, where Z 0 is the system or load impedance of Port 1; Z in4 is solved using (12); and we then let R L = Z in4 , θ s = θ C11 , and Z S = Z C1 in (7). Therefore, the Q L of R C1 at f 0 can be written as The design of (13) is similar to that presented in [28]. Z C4 has been solved, Z 0 = 50 is the system impedance, and Q L is identified using the given bandpass response specification. R C1 is equivalent to C 2 in Fig. 13(c) at f C . Y inC1 at f C is as follows: The two remaining unknown variables, Z C1 and θ C11 , can be obtained using (13) and (14) when the DB-LBF response specification is indicated.
Step III C : R C2 simultaneously resonates at f 0 and must be equivalent to the shunt-to-ground capacitance C 4 of Fig. 13(c), which has the same design as R A2 in Step III A . Because R C1 and X C5 are designed in Steps I C and II C , respectively, their parameters are fixed. For the DB-LBF response specification, C 3 , f c , and f 0 have been identified. Similar to what is done in Step III A , the required coupling coefficient M 12 between R C1 and R C2 can be achieved by changing the adjusted θ C21 to obtain the corresponding Z C2 value. Finally, the remaining design of LBF C is completed by virtue of its bilateral symmetry.
Each LBF A and LBF B port is at an input or output resonator. Thus, an additional Z 0 = 50 transmission line may be required at each port for ease of connection to other external circuits. X C4 or X C7 are included in LBF C and have a λ/4 length at f 0 to increase port layout flexibility and facilitate the connection to another circuit without the need for an additional Z 0 transmission line. By systematically following the LBF C design, one can achieve the required DB-LBF response without having to undertake any timeconsuming processes.
LBF C is designed for f c = 1.2GHz for a 0.1-dB equalripple LPF response and f 0 = 2.4GHz and FBW = 9% for a 0.1-dB equal-ripple BPF response. Based on the specifications, the related design parameters of f 0 are θ C11 = θ C31 = 65   presented in Fig. 15. In the lowpass band, the measured passband maximum insertion loss and cutoff frequency are approximately 1.1 dB and 1.2 GHz, respectively; in the bandpass band, the measured passband minimum insertion loss, 3 dB FBW, and center frequency are approximately 1.684 dB, 12.2%, and 2.423 GHz, respectively. For lowpass band, ideal lumped LPF simulation meets exactly required 0.1 dB for the first or second equal-ripple value, but its third ripple is shifted approximately from 1.075 GHz (0.1 dB value) to 0.993 GHz (0.375 dB value), which error results from input or output transformer doesn't use enough high characteristic impedance to approach the required lumped inductance of LPF. However, it still achieves a satisfied initial LPF response design. Moreover, minor equal-ripple value errors for other lowpass or bandpass band responses in Fig.15, which are caused by the approaching circuit design or measured errors.  Table 4 lists the results for the proposed DB-LBFs and those from other studies for comparison. The DB-LBF in [21] could be designed using a synthesized method, such as in a Chebyshev response design; however, the lowpass and bandpass band specifications could not be independently indicated. Although the specification of the lowpass band was given and that of the bandpass band was determined, the proposed DB-LBFs provided separate dual-band specifications and could be systematically designed using the synthesized method. Even if Cases A 1 -A 3 have higher filter orders, Cases A 1 -A 3 were more compact than those in [16] and [15].

V. COMPARISON BETWEEN THE PROPOSED DB-LBFS AND THOSE IN THE LITERATURE
Compared with the proposed DB-LBFs, those in [24] were smaller; however, the circuits in [24] had fewer filter orders than those in the proposed circuits. Furthermore, [24] required a chip capacitor for the DB-LBF, which might increase circuit cost. The figure of merit (FOM) [26] for LPF response is defined as follows.
In (14), RSB is relative stopband bandwidth, ξ is roll-offrate parameter, SF is suppression factor, and NSS is normalized structure size. The related FOM comparisons are also included in Table 4.

VI. CONCLUSION
This paper presents three DB-LBFs (LBF A, LBF B, and LBF C). The structure of LBF A boasts a flexible design, as is evident in the cutoff frequency for a lowpass band and the fractional bandwidth for a bandpass band. LBF B is also proposed as a higher-order DB-LBF. Finally, the LBF C structure can provide a flexible input or output layout to facilitate connection to an external circuit. These proposed DB-LBFs comprise transmission lines without any capacitive coupling structure; thus, the proposed circuits can provide rapid predictions through the use of ideal circuit simulations. Furthermore, six transmission zeros in LBF A or LBF C and ten transmission zeros in LBF B, wherein each zero is produced by the corresponding open stub. The transmission zeros can improve the selectivity and stopband response of each proposed DB-LBF. Overall, each layout has an acceptable DB-LBF response below |±2%| variation in the proposed fabricated circuit layout substrate. Specifically, LBF C has high characteristic impedance (126.55 )λ/4 inverter line causing its manufacturing sensitivity issue, which can be relaxed by using a thicker substrate. The performance of the circuits formulated in this study was also validated using simulations and empirical measurements.