### First-order wideband filter

The filter construction starts with the MEMS resonator. The electrical response of a MEMS resonator is often represented with the modified Butterworth–Van Dyke (MBVD) model, in which *C*_{0} is the static capacitance of the resonator and *L*_{m}, *C*_{m}, and *R*_{m} are the motional inductance, motional capacitance, and motional resistance, respectively. Extraction of the MBVD parameters can be found in many literature reports^{24,25,26,27,28,29}. The motional branch forms the mechanical resonance equivalently in the electrical domain via piezoelectricity. As an example, the measured response of a MEMS resonator is plotted and fitted in Fig. 1a. The demonstrated resonator has a typical resonant frequency *f*_{s} and an antiresonant frequency *f*_{p}. The quality factor and the piezoelectric coupling factor are two important parameters for the performance of the resonators, which can be related to the MBVD model parameters. Therefore, the MBVD model is extracted from the measured response and will be used as the basis for the following filter analysis.

As shown in Fig. 1d, the proposed first-order filter is obtained from the combination of the topology in Fig. 1b, c, which consists of a MEMS resonator, a parallel inductor *L*_{0}, and a pair of three-lumped-element matching networks, which includes a series inductor *L*_{s}, a shunt inductor *L*_{p}, and a shunt capacitor *C*_{p}. When the MEMS resonator parallel with the inductor *L*_{0}, two symmetric transmission zeros *f*_{a1} and *f*_{a2} are generated, as indicated in Fig. 1b. The filter is centered at the same resonant frequency (*f*_{s}) as the MEMS resonator, and it features a large bandwidth and two deep transmission zeros (TZs).

The series resonance frequency *f*_{s} is the frequency where *L*_{m} and *C*_{m} cancel each other out, while the antiresonance frequency *f*_{p} arises when *C*_{0} and *C*_{m} collectively cancel out *L*_{m}. The parallel *L*_{0} is added to decouple the antiresonance and generate two symmetric TZs: *f*_{a1} and *f*_{a2}. The two TZs are determined by the MBVD model and *L*_{0}, as shown in Fig. 1b and computed as

$$f_{{rm{a}}1,{rm{a}}2} = frac{1}{{2pi }}sqrt {frac{{omega _{rm{s}}^2} mp {sqrt {t_1t_2} } + {omega _{{rm{s}}0}^2} + {C_{rm{m}}L_0omega _{rm{s}}^2omega _{{rm{s}}0}^2}}{2}}$$

(1)

where

$$t_1 = C_{rm{m}}L_0omega _{rm{s}}^2omega _{{rm{s}}0}^2 + omega _{rm{s}}^2 – 2omega _{rm{s}}omega _{{rm{s}}0} + omega _{rm{s}}^2$$

$$t_2 = C_{rm{m}}L_0omega _{rm{s}}^2omega _{{rm{s}}0}^2 + omega _{rm{s}}^2 + 2omega _{rm{s}}omega _{{rm{s}}0} + omega _{rm{s}}^2$$

$$omega _{rm{s}} = 1/sqrt {C_{rm{m}}L_{rm{m}}}$$

$$omega _{{mathrm{{{s}}}}0} = 1/sqrt {C_0L_0}$$

If we set, *ω*_{s} = *ω*_{s0}, then

$$L_0 = frac{{L_{rm{m}}C_{rm{m}}}}{{C_0}} = frac{1}{{omega _{rm{s}}^2C_0}} = frac{1}{{(2pi f_{rm{s}})^2C_0}}$$

(2)

Then, Eq. (1) can be rewritten as

$$f_{{rm{a}}1,{rm{a}}2} = frac{{omega _{rm{s}}}}{{2pi }}sqrt {frac{{2 + C_{rm{m}}/C_0 pm sqrt {(2 + C_{rm{m}}/C_0)^2 – 4} }}{2}}$$

(3)

Since the piezoelectric coupling factor of the resonator device is proportional to *C*_{m}/*C*_{0}^{3} and the TZs determine the maximum bandwidth of the filter, Eq. (3) implies that a higher bandwidth widening capability can be obtained by using resonators with a larger piezoelectric coupling factor.

As demonstrated in Fig. 1c, the lumped-element matching network consists of a series inductor *L*_{s}, a shunt inductor *L*_{p}, and a shunt capacitor *C*_{p}. The purpose of the matching networks is to widen the bandwidth of the filter. The three-lumped elements resonate at *f*_{s} and are regulated by

$$left( {frac{1}{{L_{rm{p}}}} + frac{1}{{L_{rm{s}}}}} right) cdot frac{1}{{C_{rm{p}}}} = omega _{rm{s}}^2$$

(4)

Only when *L*_{s}, *L*_{p}, and *C*_{p} obey Eq. (4) can we obtain a functional matching network. If any two of three are determined, then the third one can be acquired by Eq. (4). Therefore, only two parameters are independent, and the left parameter is dependent. In later discussion, *L*_{s} and *L*_{p} will be chosen as independent variables and *C*_{p} as the dependent variable.

Finally, as shown in Fig. 1d, the proposed filter is formed by combining the two topologies in Fig. 1b, c. In this filtering topology, the MEMS resonator and *L*_{0} provide sharp roll-off, TZs, and set the center frequency of the filter, while the matching network, which in essence is an LC bandpass filter, offers the wide bandwidth for the filter. The filtering topology enjoys high-performance flexibility by tuning the lumped elements in the matching networks. As shown in Fig. 2b, when *L*_{s} and *L*_{p} are given a different value while *C*_{p} is determined by Eq. (4) *L*_{s} and *L*_{p} have different effects on the filter bandwidth and out-of-band rejection. In general, larger *L*_{s} and *L*_{p} lead to a greater fractional bandwidth (FBW), while *L*_{p} has a greater impact on out-of-band rejection than *L*_{s}, and a smaller *L*_{p} corresponds to greater out-of-band rejection. *Q*_{s} of the resonator and the lumped elements together determine the IL of the filter. By leveraging *L*_{s} and *L*_{p}, it is possible to achieve wide bandwidth and excellent out-of-band rejection simultaneously.

As an example of the tunability, Fig. 2a demonstrates three filters (named Filters A, B, and C) with different matching networks (for quantification purposes, we assume *f*_{s} = 229 MHz and *f*_{p } = 231 MHz based on a measured MEMS resonator; *k*_{t}^{2} is then computed to be 1.7%). The values of the three-lumped elements *L*_{s}, *L*_{p}, and *C*_{p} are listed in Fig. 2c. Among the three filters, Filter A has the largest *L*_{s} and largest *L*_{p}. *C*_{p} is therefore the smallest. As explained above, a larger *L*_{s} and *L*_{p} produce a wider BW, and a larger *L*_{p} leads to smaller out-of-band rejection. Therefore, Filter A is assumed to have the largest BW and the smallest out-of-band rejection, which is verified by the simulation in Fig. 2a. Filter C has the smallest *L*_{s} and *L*_{p}, so it has the smallest BW and the largest out-of-band rejection. The comparison of the performance of the three filters is concluded in Fig. 2c. Figure 2b illustrates the simulated filtering FBW performance versus different *L*_{s} and *L*_{p} values; the dashed lines show the out-of-band rejection values for guidance.

The proposed first-order filters stand out for their wide FBWs. To justify their bandwidth widening capabilities, the three filters (A, B, and C) are compared with existing widely used ladder and lattice filtering topologies. The bandwidth widening factor (BWF), defined as the quotient of the FBW and the resonator’s electromechanical coupling, is adopted for evaluation. Conventional ladder and lattice topologies^{27}, which are shown in Fig. 1e, f, typically achieve FBWs of 1/3–1/2 of the resonator’s coupling^{27,28}. Taking the assumed MEMS resonator that has an electromechanical coupling of 1.7%, for example, the FBWs of the ladder and lattice topologies are 0.78% and 0.95%, respectively. Figure 2d compares the BWFs of the three first-order filters and the ladder and lattice topologies. Apparently, the proposed first-order filters have the largest BWF of 4.2, which is 9.1 and 7.5 times higher than the ladder and lattice topologies, respectively.

Though showing a large FBW, the roll-off and out-of-band rejection of the first-order wideband filter still needs enhancement. For further improvement, a second-order wideband filter is then proposed.

### Second-order wideband filter

As shown in Fig. 3a, the second-order wideband filter is composed of two series of first-order wideband filters. The two first-order wideband filters are designed to be image symmetrical for simultaneous matching of the two ports. In order to have more freedom of matching impedance, one side of the matching network is changed to be *L*_{s2}, *L*_{p2}, and *C*_{p2}. The roles of *L*_{0}, *L*_{s1}, *L*_{p1}, *C*_{p1}, *L*_{s2}, *L*_{p2}, and *C*_{p2} are the same as those described in the first-order wideband filter, and they are also regulated by Eqs. (2) and (4). The second-order wideband filter features a much better performance of out-of-band rejection and roll-off. In addition, the FBW has also been improved. The simulated response of the second-order wideband filter in Fig. 3b demonstrates a large FBW of 6.7% and an extremely high out-of-band rejection of over 60 dB. Moreover, the roll-off is also significantly enhanced, and the -30dB shape factor is only 1.05.

The second-order wideband filter enjoys favorable performance flexibility since it has four tuning elements (*L*_{s1}, *L*_{p1}, *L*_{s2}, and *L*_{p2}), while the first-order filter has only two tuning elements (*L*_{s} and *L*_{p}). Similarly, to demonstrate the tuning capability, another three second-order wideband filters (Filters D, E, and F) are constructed using different matching networks. The values of the lumped elements of each filter are listed in Fig. 3c. The functions of *L*_{s1}, *L*_{s2}, *L*_{p1}, and *L*_{p2} are similar to those in the first-order case. Filter D has the largest *L*_{s1}, *L*_{s2}, *L*_{p1}, and *L*_{p2}, so it has the widest FBW but the smallest out-of-band rejection (Fig. 3b). Filter F has the smallest *L*_{s1}, *L*_{s2}, *L*_{p1}, and *L*_{p2}, which corresponds to the smallest FBW but the highest out-of-band rejection. Filter E has intermediate values of *L*_{s1}, *L*_{s2}, *L*_{p1}, and *L*_{p2}, and its FBW and out-of-band rejection are between Filter Ds and Filter Fs. Overall, the second-order filters exhibit much better out-of-band rejection and roll-off than the first-order filters.

The bandwidth widening effect of the second-order wideband filters is also researched and compared in Fig. 3d. Filter D has the highest BWF of 4.6 because it uses the largest *L*_{s1}, *L*_{s2}, *L*_{p1}, and *L*_{p2}. This BWF could be further increased if higher *L*_{s1}, *L*_{s2}, *L*_{p1}, and *L*_{p2} were applied. Of course, the compromise will be the degradation of out-of-band rejection. Filter F has the smallest BWF of 2.2, but its out-of-band rejection is as high as 76 dB. Compared with the first-order wideband filter C, which has the same BWF of 2.2, the second-order wideband filters achieve much better out-of-band rejection performance.

The MEMS resonators in the proposed first-order and second-order filters are constructed by the MBVD model. There are no specific requirements of the types of resonators. The MEMS resonators could be various MEMS resonators, such as AlN Lamb wave resonators, SAW resonators, FBARs, or LiNbO_{3} A1 resonators. To validate the feasibility and achieve multiple-frequency filters, the following section will replace the general MEMS resonator with a concrete AlN Lamb wave resonator.