Background on acoustic transversal filters

Figure 1(a) shows the transversal topology of a 6th-order filter, whose acoustic resonator-equivalent circuit corresponds to a Butterworth Van Dyke (BVD) [Reference Rosenbaum10]. Analogously, Fig. 1(b) outlines the transversal topology for a multiplexing configuration [Reference Perea-Robles, Mateu and Collado5]. These figures reveal how the resonators are arranged in a transversal manner and all of them are electrically connected from input to output. The filter performance is created through the contribution of each individual path. Basically, resonators are grouped into two kinds of branches: positive and negative, where the resonators in the negative branches need to experience a 180° phase shift. The synthesis procedure to extract the acoustic transversal topology has been fully detailed in [Reference Mateu, Yusuf, Perea-Robles, Collado, Gimenez and Aigner9], but it mainly relies on the extraction of the conventional transversal network [Reference Cameron, Kudsia and Mansour11] in which a set of cross-couplings between input and output is added in order to convert the conventional LC resonators into BVD circuits which reproduce the behavior of piezoelectric crystals and thin film resonators [Reference Rosenbaum10]. The way these cross-couplings are added, and the nature of the characteristic polynomials offers full flexibility on selecting the electro-acoustic coupling coefficient. Mateu et al. [Reference Mateu, Yusuf, Perea-Robles, Collado, Gimenez and Aigner9] also demonstrate how circuit transformations can be applied to introduce control over other critical parameters on the filter implementation, such as the impedance of the resonators and their resonant frequencies. This allows the synthesis of any filter response, in terms of bandwidth and position of transmission zeros.

Fig. 1.

(a) Transversal network for a single acoustic filter. (b) Transversal network for a multiplexing filtering function.

Despite the great benefits, the requirement of an unbalanced network attained by the inclusion of an external BALUN at the input or output of the filter might lead to a poor performance of the filter due to, for instance, unbalancing effects [Reference Mateu, Collado and Aigner6].

Transversal topology for N-plexer responses

As per interest of the present study, transversal topologies allow for direct connection of different transversal filters without the inclusion of external matching networks between the standalone filters and without much effect on the individual single channel performance. This allows for the synthesis and design of multiplexer responses just by directly connecting each filter, which has been synthesized as a standalone filter. Figure 1(b) illustrates this concept.

This approach was successfully used for the synthesis and design of a 9-plexer, where each individual filter was formed by six resonators, making a filtering stage with a total of 54 resonators [Reference Perea-Robles, Mateu and Collado5]. The great advantage of this topology is that the out-of-band (OoB) response of the standalone filters behaves as an open circuit or close to it in a wideband range.

This section illustrates the feasibility of the direct connection of transversal filters by a simple analysis of the circuit shown in Fig. 2. This circuit, and for the sake of simplicity, accounts for a 2nd-order symmetric standalone acoustic transversal filter. For this analysis, the shunt inductances at each branch are defined as admittances Y ₁ and $Y_1^{\prime}$ and the BVD resonators as admittances Y ₂ and $Y_2^{\prime}$. The LC series circuits at the motional branches of the BVD model are defined as admittances Y _r and $Y_r^{\prime}$ and the static capacitances as admittances Y _p and $Y_p^{\prime}$. The admittance matrices of each branch can be respectively read as:

(1)$$\eqalign{&[ Y ] = \left[{\matrix{ {Y_1 + Y_2} & {-Y_2} \cr {-Y_2} & {Y_1 + Y_2} \cr } } \right]; \cr & \quad [ {Y^{\prime}} ] = \left[{\matrix{ {Y_1^{\prime} + Y_2^{\prime}} & {Y_2^{\prime}} \cr {Y_2^{\prime}} & {Y_1^{\prime} + Y_2^{\prime}} \cr } } \right]}$$

where the parallel admittances Y _p and $Y_p^{\prime}$ are equal at all frequencies and the same happens with Y ₁ and $Y_1^{\prime}$. In addition, at resonant frequency Y _p = −Y ₁ and $Y_p^{\prime} = {-}Y_1^{\prime}$. Moreover, the admittances forming the motional branches Y _r and $Y_r^{\prime}$ become smaller out of resonance.

Fig. 2.

Circuital model of a 2nd-order acoustic transversal filter.

Also note that the parameters Y ₂₁ and Y ₁₂ of both matrices differ 180° due to the inherent phase shift in the negative branches of the filter. By considering these assumptions, the OoB admittance matrices can be written as

(2)$$\eqalign{&[ Y_{OoB}] = \left[{\matrix{ {Y_1 + Y_p + \xi } & {-Y_p-\xi } \cr {-Y_p-\xi } & {Y_1 + Y_p + \xi } \cr } } \right]; \cr & \quad [ Y_{OoB}^{\prime} ] = \left[{\matrix{ {Y_1^{\prime} + Y_p^{\prime} + \eta } & {Y_p^{\prime} + \eta } \cr {Y_p^{\prime} + \eta } & {Y_1^{\prime} + Y_p^{\prime} + \eta } \cr } } \right]}$$

where ξ and η are phenomenological terms added to account for the effects of the motional branches Y _r and $Y_r^{\prime}$ into the OoB response. The overall response is the combination of both matrices which could lead into a phenomenological description as follows:

(3)$$[ Y_{Overall}] = [ Y_{OoB}] + [ Y_{OoB}^{\prime} ] = \left[{\matrix{ {\delta + {\rm \Delta }} & \nu \cr \nu & {\delta + {\rm \Delta }} \cr } } \right]$$

where δ = 2(Y ₁ + Y _p), Δ = η + ξ, and ν = η − ξ. Note that this is for the ideal situation, when both static capacitances are equal, and they compensate each other. Whenever this does not happen, another residual parameter appears on ν accounting for the non-compensated error. At this point, it is easy to see that as far as δ, Δ, and ν are equal or close to 0, the better is the filter performance.

Table 1 outlines the value of the phenomenological parameters δ, Δ, and ν at the central frequency of a 4-plexer of 2nd-order filters (centered at f ₁, f ₂, f ₃, and f ₄), all with 80 MHz bandwidth and for different guard bands (GB = 1.25, 2, 4, and 7),whose frequency spacing at the band edges are 15, 60, 180, and 360 MHz, respectively.

Table 1.

Error parameter of the 2nd-order acoustic transversal 4-plexer

The worst case occurs when the bands are very close (GB = 1.25), because the admittance of the motional branch Y _r of each filter is not far enough in frequency to become small, so the accumulated errors at Δ are larger. Note that the value of Δ converges to 0 as the frequency spacing between bands increases. The same occurs with the parameters ν since it accounts for the difference of the errors associated with the positive (ξ) and the negative branches (η). The best performances occur at GB = 2 and GB = 4, where all the parameters of the matrices are almost similar and closest to 0, whose results are outlined in Fig. 3. When the bands have a large separation (GB = 7), the worsening of the responses arise from the different behavior of Y ₁ and Y _p. Recall that Y _p accounts for the static capacitance, synthesized from the added π-network admittance inverters, and Y ₁ corresponds to the inductance transformed from the negative capacitance of value −Y _p. Then, the equivalences in magnitude only occur at frequencies close to resonant frequencies. Then for large GB, the value of δ diverges from 0.

Fig. 3.

4-plexer response of (a) GB = 2 and (b) GB = 4.

The following section applies the transversal topology [Reference Perea-Robles, Mateu and Collado5,Reference Mateu, Yusuf, Perea-Robles, Collado, Gimenez and Aigner9] to a CRF configuration of the acoustic resonator. The major asset of this new approach is the possibility of getting rid of the BALUN, by using the properties of the CRF configuration on selecting the phase.

Transversal filters based on acoustically coupled resonator filters

Figures 4(a) and 4(b) outline two CRF sections where two resonators are coupled through a coupling layers configuration (green and purple) [Reference Lakin12,Reference Lakin13]. Black layers are used to indicate the electrodes sandwiching the piezoelectric (white). The thickness and materials forming the coupling layers set the strength of the coupling between resonators. Note as well that for a proper design these need to be set to emulate an admittance inverter. This behaves as a dual-mode resonator. The difference between the CRF sections of Figs. 4(a) and 4(b) relies on the way the ground planes are interconnected and the way the signal is extracted from the dual-mode resonator, this results in a 180° phase shift between the two resonators [Reference Fattinger, Aigner and Nessler14].

Fig. 4.

Cross-section of a CRF stage, (a) and (b) correspond to a 180° phase difference [Reference Fattinger, Aigner and Nessler14].

Taking therefore the nature of the CRF configuration the synthesis procedure starts by synthesizing a conventional transversal network as indicated in Fig. 5(a) [Reference Cameron, Kudsia and Mansour11], where each transversal path is formed by a conventional shunted LC resonator sandwiched between two admittance inverters. By considering symmetric responses in frequency, the resonators can be grouped in pairs in such a way that when circuit transformation is applied, the filter results in a transversal topology where each path is formed by two coupled resonators [Reference Cameron, Kudsia and Mansour11], as depicted in Fig. 5(b).

Fig. 5.

Nodal description of (a) conventional transversal network, (b) dual-mode transversal network, and (c) acoustic CRF transversal configuration.

As it occurs in the conventional transversal network, Fig. 5(b), half of the paths have a negative coupling, which in Fig. 1(a) claimed the need of a BALUN. However, in that case the phase of the signal path can be selected as indicated in Fig. 4 and whose implementation in a transversal filter would correspond to the topology of Fig. 5(c).

The next step is to transform the network of Fig. 5(b) in CRF configurations similar to the one in Fig. 5(c). To do that, let us consider a single path of the transversal network of Fig. 5(b). The equivalent circuit of each path can be presented as in Fig. 6(a). Node scaling [Reference Cameron, Kudsia and Mansour11] at each path can be applied to obtain unitary coupling between the resonators to the source and load, respectively. This affects the impedance of the resonators now emulated by $L_p^{\prime}$ and $C_p^{\prime}$. Although not indicated in Fig. 6(b), one of the unitary couplings might include a 180° phase shift. Note that the coupling between resonators, which is acoustic, is also modified. At this point, the LC resonators are transformed from shunt to series configuration, as in Fig. 6(c). This implies to change the circuit values being now $L_s = C_p^{\prime}$ and $C_s = L_p^{\prime}$. Now, the coupling in the middle is also transformed from an admittance inverter into an impedance inverter whose value is $K_{AC} = 1/J_{AC}^{\prime} \;$.

Fig. 6.

Circuit transformation of synthesized network: (a) LC shunt resonators, (b) LC shunt resonators with normalized input/output coupling, and (c) LC series.

To model the CRF section of Fig. 4, the equivalent circuit needs to include the effect of the electrostatic capacitance, C ₀. This can be done as indicated in Fig. 7(a) [Reference Lakin12]. The value of the electrostatic capacitance depends on the electro-acoustic coupling $k_e^2 {\boldsymbol \;}$ and the capacitance accounting for the acoustic resonance C _s, which is obtained from the synthesis, as:

(4)$$C_0 = \displaystyle{{C_s} \over {{( {1-( 4/\pi^2) k_e^2 ( {1 + ( 4/\pi^2) k_e^2 } ) } ) }^{{-}2}-1}}_\;$$

Fig. 7.

Circuital transformations: (a) including C ₀ effect, (b) acoustic transmission lines, (c) asymmetrical transmission lines, and (d) Mason model.

The circuit of Fig. 7(a) reveals that this electrostatic capacitance appears as a shunt capacitance at the input (and output) of the resonator and as −C ₀ in series with the resonator [Reference Lakin12]. The effect of the negative electrostatic capacitance requires to include a new series capacitance C _n to obtain an overall series capacitance C _s, as defined by the synthesis. The resonant frequency of the resonators is modified since now there is a new value for the capacitance forming the resonator (being $f_0 = 1/2\pi \sqrt {L_sC_n}$). A similar process is done to compensate the shunt electrostatic capacitance, by means of a compensating inductance L ₀, see Fig. 7(a). The value of the compensating inductances is calculated as follows: $L_0 = 1/\omega _0^2 C_0$. Note that the different frequency dependence impedances from C ₀ and L ₀ will result in discrepancies between the synthesized and the design filter response whose effect will be more noticeable for wide bandwidths.

Then the series LC resonators are transformed into acoustic transmission lines by applying circuit transformations of common microwave theory [Reference Hong and Lancaster15,Reference Pozar16] giving rise to the circuit Fig. 7(b), where Z ₂ corresponds to the impedance at the input of the electro-acoustic transformer in the Mason model and Z ₁ to the impedances facing the acoustic boundaries, such as electrodes, Bragg mirror or even air [Reference Lakin12]:

(5)$$Z_1 = jZ_p{\kern 1pt} {\rm tan}\left({\displaystyle{{\beta d} \over 2}} \right); \;\quad Z_2 = \displaystyle{{Z_p} \over {\,j{\kern 1pt} {\rm sin( }\beta d/2) }}$$

In (5), Z _p is the characteristic impedance of the piezoelectric material, which is defined by the area of the device, the propagation velocity v _p and density ρ of the material. β is the propagation constant and d is the thickness of the piezoelectric plate which can be directly calculated through v _p and the resonant frequency f ₀ as d =  v _p/2f ₀ [Reference Mason17].

Although this circuit is formed by transmission lines, it is not representing the circuit of an acoustic CRF, note that the transmission lines are symmetric and the acoustic impedances Z ₁, which are the impedances facing the electrodes or reflectors in the Mason model, are connected to the electrical part, thereby facing the electrostatic capacitance. At this point, impedance (node) scaling is applied to go to an asymmetric transmission line circuit (Fig. 7(c)). The impedances are reduced by a factor of 4 [Reference Perea-Robles, Mateu, Collado and Aigner18]. The last step is to add the electro-acoustic transformer to achieve the actual Mason model, which implies that both, the impedance of the resonators and the acoustic coupling are scaled by the turn ratio of the transformer (Fig. 7(d)) [Reference Perea-Robles, Mateu, Collado and Aigner18].

The procedure above has been used to synthesize the transversal network based on CRF resonators for the case of a filter centered at 3.85 GHz and 80 MHz bandwidth. The filter response considering the full distributed model of Fig. 7(d) for each path of the transversal network is outlined in Fig. 8(a), in black and magenta. The results perfectly overlap the synthesized response, indicated in blue and red. Now the overall resulting network does not need any BALUN since the required 180° can be obtained by the way the input or output ports are connected, as indicated in Fig. 5(c) [Reference Fattinger, Aigner and Nessler14].

Fig. 8.

Performance of acoustic transversal CRF topology: (a) comparison acoustic transversal CRF and BAW transversal with ideal BALUN and (b) acoustic transversal CRF 9-plexer.

Before moving to the design of a multiplexer configuration using such an approach, the effects of the losses are carefully evaluated for the circuit configuration of Fig. 7, where in addition to the limited Q of the acoustic resonators, the losses of the always existing input and output compensating inductances are also considered.

Evaluation of the losses in acoustic transversal CRF

The effect of the losses in electro-acoustic devices may be adequately described by a viscous damping [Reference Rosenbaum10, Reference Auld19]. If we only consider the wave propagation in the longitudinal direction (piston mode), the impedance's equations in (5) can be modified and transformed into their hyperbolic form as

(6)$$\eqalign{& Z_{1, \;Lossy} = Z_p{\kern 1pt} {\rm tanh}\left({( \alpha + j\beta ) \displaystyle{d \over 2}} \right); \cr & \quad Z_{2, \;Lossy} = \displaystyle{{Z_p} \over {{\rm sinh}( {( \alpha + j\beta ) d} ) }}}$$

where α is the attenuation factor and is given by

(7)$$\alpha = \displaystyle{{\omega ^2} \over 2}\displaystyle{{\eta _{44}} \over {c_{44}}}\sqrt {\displaystyle{\rho \over {c_{44}}}} \;\;\;$$

with η ₄₄ and c ₄₄ being the viscosity and elastic constants, respectively, in the longitudinal direction. However, in the CRF configuration depicted in Fig. 7(d), there exists another source of electrical losses, the compensating inductances (L ₀), whose value depends on the FBW of the filter and the electro-acoustic coupling (4) and has a finite quality factor (QL = 30 in this analysis).

Taking these effects under consideration, Fig. 9 shows the evolution of the compensating inductance and the insertion losses of the filter as a function of the electro-acoustic coupling and for three different FBWs (6%, 12%, and 24%) for a CRF transversal filter operating at 3.85 GHz. As depicted in Fig. 9(a), the compensating inductances increases with the electro-acoustic coupling and decreases for wider bandwidths. The inductance values directly affect their losses whose effect on the insertion losses is outlined, in dashed lines in Fig. 9(b), revealing that the effects of the inductance are more harmful for wide bandwidths and small electro-acoustic couplings.

Fig. 9.

Evolution of (a) the compensating inductance and (b) the insertion losses due to viscous damping (bold) and inductance size (dash) over the electro-acoustic coupling coefficient for FBWs of (red) 6%, (black) 12%, and (blue) 24%.

Along with the losses coming from the compensating inductances, Fig. 9(b) also depicts the insertion losses due to the acoustic propagation occurring in the piezoelectric material. These results reveal an opposite behavior and confirm that the acoustic losses increase for larger electro-acoustic coupling and decreases for larger bandwidths.