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 C0 is the static capacitance of the resonator and Lm, Cm, and Rm are the motional inductance, motional capacitance, and motional resistance, respectively. Extraction of the MBVD parameters can be found in many literature reports24,25,26,27,28,29. 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 fs and an antiresonant frequency fp. 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.

Fig. 1: The topology and principle of the proposed first-order filter.

a MEMS resonator representation and its MBVD fitting model. The admittance response is fitted with MBVD parameters. b The MEMS resonator and the paralleled inductor (L_0). Two symmetric transmission zeros are generated. The center frequency is the same as the resonant frequency ((f_s)) of the MEMS resonator. c Two three-lumped-element matching networks. Each consists of a series inductor, a shunt inductor, and a shunt capacitor. The two matching networks behave like a wide bandpass filter centered at (f_s). d Proposed first-order wideband filter. It is constructed from the combination of the topologies in (b) and (c). The filter is centered at (f_s). It features a large bandwidth and two deep TZs. e The frequency response and topology of a ladder-type filter. f The frequency response and topology of a lattice-type filter

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 L0, and a pair of three-lumped-element matching networks, which includes a series inductor Ls, a shunt inductor Lp, and a shunt capacitor Cp. When the MEMS resonator parallel with the inductor L0, two symmetric transmission zeros fa1 and fa2 are generated, as indicated in Fig. 1b. The filter is centered at the same resonant frequency (fs) as the MEMS resonator, and it features a large bandwidth and two deep transmission zeros (TZs).

The series resonance frequency fs is the frequency where Lm and Cm cancel each other out, while the antiresonance frequency fp arises when C0 and Cm collectively cancel out Lm. The parallel L0 is added to decouple the antiresonance and generate two symmetric TZs: fa1 and fa2. The two TZs are determined by the MBVD model and L0, as shown in Fig. 1b and computed as

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



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

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

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

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

If we set, ωs = ωs0, then

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


Then, Eq. (1) can be rewritten as

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


Since the piezoelectric coupling factor of the resonator device is proportional to Cm/C03 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 Ls, a shunt inductor Lp, and a shunt capacitor Cp. The purpose of the matching networks is to widen the bandwidth of the filter. The three-lumped elements resonate at fs and are regulated by

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


Only when Ls, Lp, and Cp 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, Ls and Lp will be chosen as independent variables and Cp 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 L0 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 Ls and Lp are given a different value while Cp is determined by Eq. (4) Ls and Lp have different effects on the filter bandwidth and out-of-band rejection. In general, larger Ls and Lp lead to a greater fractional bandwidth (FBW), while Lp has a greater impact on out-of-band rejection than Ls, and a smaller Lp corresponds to greater out-of-band rejection. Qs of the resonator and the lumped elements together determine the IL of the filter. By leveraging Ls and Lp, it is possible to achieve wide bandwidth and excellent out-of-band rejection simultaneously.

Fig. 2: Simulated performance proposed first-order filter.

a Simulated filtering performance of three first-order filters with different matching networks or matching lumped elements. Filter A has the largest Ls and Lp, Filter B has an intermediate Ls and Lp, and Filter C has the smallest Ls and Lp. b Simulated filtering FBW performance versus different Ls and Lp; the colored solid contour lines are the boundaries for the FBW performance of the filter topology, while the dashed lines are boundaries for out-of-band rejection performance. c Parameters and comparison of the three first-order filters. d Comparison of the bandwidth widening effect of three first-order filters and conventional ladder and lattice filtering topologies

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 fs = 229 MHz and fp  = 231 MHz based on a measured MEMS resonator; kt2 is then computed to be 1.7%). The values of the three-lumped elements Ls, Lp, and Cp are listed in Fig. 2c. Among the three filters, Filter A has the largest Ls and largest Lp. Cp is therefore the smallest. As explained above, a larger Ls and Lp produce a wider BW, and a larger Lp 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 Ls and Lp, 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 Ls and Lp 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 topologies27, which are shown in Fig. 1e, f, typically achieve FBWs of 1/3–1/2 of the resonator’s coupling27,28. 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 Ls2, Lp2, and Cp2. The roles of L0, Ls1, Lp1, Cp1, Ls2, Lp2, and Cp2 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.

Fig. 3: The topology and performance of the proposed second-order filter.

a Proposed second-order wideband filter and its simulated filtering response. b Simulated filtering performance of the three second-order wideband filters with different matching networks or matching lumped elements. Filter D has the largest Ls, Ls2, Lp and Lp2, Filter E has an intermediate Ls, Ls2, Lp and Lp2, and Filter F has the smallest Ls, Ls2, Lp and Lp2. c Comparison of the three second-order wideband filters. d Comparison of the bandwidth widening effect of three second-order wideband filters and conventional ladder and lattice filtering topologies

The second-order wideband filter enjoys favorable performance flexibility since it has four tuning elements (Ls1, Lp1, Ls2, and Lp2), while the first-order filter has only two tuning elements (Ls and Lp). 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 Ls1, Ls2, Lp1, and Lp2 are similar to those in the first-order case. Filter D has the largest Ls1, Ls2, Lp1, and Lp2, so it has the widest FBW but the smallest out-of-band rejection (Fig. 3b). Filter F has the smallest Ls1, Ls2, Lp1, and Lp2, which corresponds to the smallest FBW but the highest out-of-band rejection. Filter E has intermediate values of Ls1, Ls2, Lp1, and Lp2, 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 Ls1, Ls2, Lp1, and Lp2. This BWF could be further increased if higher Ls1, Ls2, Lp1, and Lp2 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 LiNbO3 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.

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