Lattice FBAR Filters: Basic Properties and Opportunities for Improving

Journal of Communications Engineering and Networks

Oct. 2014, Vol. 2 Iss. 4, PP. 103-117

Lattice FBAR Filters: Basic Properties and Opportunities for Improving the Frequency Response Ivan Uzunov1, Dobromir Gaydazhiev2, Ventsislav Yantchev3 1

2

SmartLab Bulgaria EOOD; 133, Tsarigr. Chaussee Blvd.; 1784, Sofia, Bulgaria Microelectronic Technologies Dept., Smartcom Bulgaria AD; 133, Tsarigr. Chaussee Blvd.; 1784, Sofia, Bulgaria 3 Dept. of Solid State Electronics, Uppsala University; PO Box 534; S-75121, Uppsala, Sweden *1 [email protected]

Abstract-The paper contains a relatively detailed consideration of lattice filters with bulk acoustic-wave resonators (FBAR): basic theory, investigation and improving of their design. First FBARs and their parameters are introduced briefly. Then the basic structure of lattice filters and its modification with twice lower number of resonators are introduced. A theory of lattice FBAR filters is presented, which allows unified explanation of many of their properties: condition for small or large gain of the filter; existence of maxima of the filter frequency response; emergence of transmission zeros; etc. The effect of using two different resonators instead of a single resonator in the arms of the lattice filter is investigated next. It is demonstrated that this approach suggests extension of the passband bandwidth and increasing the stopband attenuation. The properties of the considered filters are illustrated by comparing their frequency responses. Similar comparison between some good representatives of lattice and ladder filters is also given at the end, accompanied by comments about the benefits of both filter architectures. Keywords- Film Bulk Acoustic-Wave Resonators (FBAR); RF Filters; FBAR Filters; Circuit Theory

I. INTRODUCTION The film bulk acoustic-wave resonators (FBAR) found in the last years fast growing application for realization of high selective filters for the analog front-ends in radio transmitters and receivers. The basic FBAR advantages that have led to their wide application, are very low losses, good temperature stability, appropriate frequency range of operation, ability to handle relatively high power, good compatibility with the existing CMOS technology, etc. [1-4]. FBAR filters are realized usually as double terminated ladder structures (Fig. 1(a)) having a single resonator in each branch. The series resonance frequencies ωsa of the series resonators (marked by subscript “a” in Fig. 1(a)) should be the same as the parallel resonance frequencies ωpb of the shunt resonators (“b” resonators in Fig. 1(a)) for proper operation of the filter. This is the frequency of maximum gain since all resonators “disappear” at this frequency (if their losses are neglected) and the signal is transmitted directly from the source to the load. It is in fact the passband central frequency. The other FBAR resonances create transmission zeros: the parallel resonances ωpa of the series FBARs stop the signal and the series resonances ωsb of shunt FBARs shunt the signal. The distance between FBAR series and parallel resonances is small and the ladder filter has a very sharp cutoff. However the attenuation returns to very small values for frequencies below the lower and above the upper transmission zeros when the filter consists of a small number of resonators (few dB for Г-type filter consisting of a single series and a single shunt resonator). Improvement of stopband attenuation can be achieved by using ladders with many resonators – typically 6-7. Negative side effects of this approach are shrinking of filter passband and increasing of inband losses.

Fig. 1 (a) Ladder FBAR filter with 5 resonators; (b) lattice filter

The distance between series and parallel resonance frequencies of the resonators limits the passband bandwidth. It can be extended technologically by increasing the effective coupling factor of the resonators, which determines the ratio between FBAR series and parallel resonance frequencies [4, 5]. Another approach is to add small inductors (less than 1 nH) in series to the shunt resonators [4, 6]. It provides an extension of the passband by approximately 10% as can be determined by computer simulations [7]. The lattice architecture (Fig. 1(b)) is another possibility for filters with piezoelectric devices [1, 4, 8]. It has two different pairs of FBARs, each one consisting of identical resonators. The theoretical passband of this structure is the frequency band, in

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which the reactances in one of the pairs have a sign opposite to the sign of the reactances in the other pair. This requirement comes from the theory of lattice LC circuits [8]. Its straightforward consequence is a condition similar to that in the ladder filters: the parallel resonance frequencies in one of the pairs must be equal to the series resonance frequencies of the other. It will be assumed in this paper that: (1) The designations ωsa and ωpa will be used for the series and the parallel resonance frequencies of the resonators Xa in Fig. 1(b) and ωsb and ωpb for the corresponding resonances of Xb. The lattice filter transmits directly the signal between the source and the load and has maximum gain at this frequency (ω = ωpa = ωsb), because Xa are equivalent to open circuits and Xb are equivalent to short circuits in this case. However the real passband does not extend in the whole theoretical range from ωsa to ωpb. The attenuation increases smoothly when the frequency moves downwards or upwards from the frequency of maximum gain, achieving undesirable values at the boundaries ωsa and ωpb. Thus the problem for obtaining wider real passband, in which the frequency response is with small variation in prescribed limits, exists in the lattice filters too. The methods, proposed for ladder filters (higher effective coupling factor or small inductors in series with the resonators) can be also applied in lattice filters. Another approach is to use a combination of different resonators in the arms, connected in parallel or in series. This method has been applied earlier for quartz-crystal filters in [8] and for ladder FBAR filters in [9]. The background of the method is the increased order of the impedance of the combined resonators, increasing the transfer function degree as well. Possible consequences of the higher-order transfer function are steeper slopes between the passband and the stopbands and a wider passband. Few questions need to find their answers in a more detailed consideration of this approach, of which the most important are: how the resonance frequencies and the frequency response of parallel or series connected resonators change; how the extra resonators modify the filter frequency response; how to select the parameters of the resonators in order to achieve wider passband and high stopband attenuation; objective comparison between different FBAR filters. Some of these problems were considered by the authors in [10, 11]. The main goal of this paper is to investigate FBAR lattice filters and derive ways for improving their frequency response. It summarizes and extends the results in [10] and [11]. A relatively complete description of the properties of the lattice FBAR filters is also given in the paper. It starts with a short review of FBARs and then a general theory of the lattice filters is presented, which gives an easy explanation of most of their characteristics. It is based on a modified lattice circuit, proposed in [12]. However, practically all results from the theoretical considerations are valid for the basic circuit in Fig. 1(b). This material is given in Section II of the paper. Section III studies parallel and series connection of resonators and shows how the resonance frequencies change after connecting the resonators. The conclusions from these two sections are used in Section IV, where lattice filters with combination of parallel connected resonators in one of the arms (“a” or “b” in Fig. 1(b)) or in both arms are considered. Frequency responses of different types of lattice filters (with single resonators in the arms or with combination of resonators) are compared in the last Section V. The lattice filters are compared also with two samples of ladder filters having relatively wide passband and good stopband attenuation. Some requirements for achieving better objectivity of the comparisons are kept: all characteristics are given in normalized form (usually the central passband frequency is used as the normalization frequency); the resonators in all examples in the paper have the same effective coupling coefficient, i.e. the same ratio between their series and parallel resonance frequencies. II. BASICS OF FBAR LATTICE FILTERS A. FBAR – Structure, Electrical Model and Basic Relationships The FBAR devices consist of a thin layer of piezoelectric material (typically AlN) placed between two metal electrodes. The whole construction is hanging over an air gap and is fixed at the ends to the silicon substrate [1, 4]. This is the membrane type FBAR. Solidly mounted FBAR is also proposed [3, 4]: An acoustic mirror, consisting of few consecutive layers of tungsten (or AlN) and SiO2, is placed between the resonator and the substrate. The FBAR electrical properties are similar to quartz crystal resonators; however there are differences in the parameters. FBAR resonance frequencies are in the GHz range, which makes them very suitable for RF applications [3, 13]. FBAR quality factors are lower than those of quartz crystal resonators (up to few thousands). FBARs have higher effective coupling coefficients ( ), which defines higher relative distance between their series and parallel resonance frequencies. The electrical characteristics of an FBAR are commonly modeled by the modified Butterworth – Van Dyke (mBVD) model shown in Fig. 2(a) [4, 5]. The elements Rm, Lm and Cm in the mBVD model form the “motional tract”. They do not have direct physical meaning and represent the propagation of the acoustic wave in the piezoelectric material. The electrical capacitance formed between the top and bottom FBAR electrodes is represented by the capacitor Cp. The resistors Rm, R0 and Rs represent different sources of power losses in the devices. The FBAR impedance, shown in Fig. 2(b) and (c), has two resonance frequencies – first series and then parallel. The relationship between resonance frequencies and model elements are:

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√

;

√

(

√

,

)

(2)

where ω is the angular frequency (ω = 2πf). Both resonance frequencies are also connected by the effective coupling coefficient , which reflects the relationship between the electrical and acoustic energy in the piezoelectric resonator: (

)

.

(3)

Typically varies between 5% and 7% [14] that corresponds to a variation from 2.07% to 2.92% of the relative distances between ωs and ωp. The ratio between parallel and series resonance frequencies will play an important role in the next considerations and it will be designated by α, i.e. α = ωp/ωs.

Fig. 2 (a) Modified Butterworth - van Dyke (mBVD) model of FBAR; (b) magnitude of FBAR impedance vs. frequency; (c) frequency response of FBAR reactance when the losses are neglected

The following general expression for the FBAR impedance can be derived from Fig. 2(a): (

)[ [

( (

⁄ ⁄

)

(

]

)

]

[

⁄ (

) ⁄

)

]

.

(4)

The time constant τ is very small and the first order term in the numerator of the first expression has negligible effect on the frequencies around the FBAR resonances, so only the approximate expression will be used in the considerations below. The values of Q are usually between 1000 and 4000 [14]. The Q-factor Qs of the series resonance is determined basically from the resistors Rm and Rs, while Qp (Q of the parallel resonance) is defined from Rm and R0. The measured data for these resistors are given in [15], indicating that Rs > R0, thus Qp is bigger than Qs. If the FBAR losses are neglected (Rm = R0 = Rs = 0; τ = 0; Qs = Qp → ∞), then the mBVD model is a lossless LC circuit. Its impedance is pure imaginary, with a frequency response shown in Fig. 2(c). Then the expression for the FBAR impedance is .

(

(5)

)

In fact, four parameters determine FBAR frequency behavior. The resonance frequencies, one of them being usually enough to be specified, define the frequency range of operation. The passband bandwidth of the FBAR filters is basically defined by the distance between resonance frequencies, represented often by the coupling coefficient . The capacitance Cp defines the level of FBAR impedance and it is usually connected with the terminating resistors of the filters. The quality factors determine basically the in-band losses of the FBAR filter. For more generality of the considerations a normalization concerning frequency and impedances will be applied everywhere below. Also, to be able to compare different filter structures, in the next simulations it is assumed that all FBARs have the same relative distance between resonance frequencies, equal to 2.5% (α = 1.025), which corresponds to . B. Basic and Modified Lattice FBAR Filter The basic FBAR lattice filter is shown in Fig. 1(b) and repeated in Fig. 3(a). Its transmission factor is √

(

⁄ )( ⁄

⁄ ) ⁄

,

(6)

where Pm is the maximum power, available from the source E with internal resistance R (Pm = E2/(4R)); Po is the power in the ⁄ ); Za and Zb are the impedances of the resonators Xa and Xb (the definition of the transmission factor of output load ( a lossless passive circuit can be seen in [8]). A modification of the basic filter is proposed in [12] and shown in Fig. 3(b). The first amplifier must have high output impedance and it will be considered as transconductance amplifier with transconductance gm1. The second amplifier is transresistance amplifier with transresistance rm2 (as in Fig. 3(b)) or a current amplifier and must have zero or small input

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impedance. The resistances Ra and Rb can be additional resistors, or the output impedances of the first amplifier, or a combination of both. The transfer function of the circuit in Fig. 3(b) is: ⁄ (

⁄

⁄ )(

⁄

)

.

(7)

Fig. 3 (a) Basic FBAR lattice filter; (b) modified lattice filter from [12]

The conditions gm1rm2 = 1 and Ra = Rb = R make both functions in (6) and (7) identical (T = 1/H), which makes the circuits in Fig. 3 equivalent. The focus here will be on the circuit in Fig 3(b), since it proposes better flexibility in the design and has fewer resonators. The frequency behavior of the circuit can be explained by considering the phasors of the currents. The currents, which flow from the outputs of amplifier gm1, are: (8)

.

These currents split into currents iXa and iXb through resonators and iRa and iRb through resistors. They form the voltages va and vb: and

.

(9)

Two assumptions allow writing these equations: 1) lossless resonators, making the impedances pure imaginary (jXa and jXb); and 2) zero input impedance of the second amplifier. Thus ±90ºphase shift always exists between iRa and iXa, as well between iRb and iXb. The sum iRa + iXa is equal to ia. This means that the sum of their phasors has constant amplitude and it is independent of the frequency. Geometrically, the ends of the phasors iRa and iXa are always on a circle, whose diameter is the phasor ia (Fig. 4(a)), and rotate clockwise along the circle when the frequency increases. Depending on the character of Xa the phasor iXa is in the upper semicircle as in Fig. 4(a) (when Xa is inductive) or in the lower (when Xa is capacitive). Similar conclusions can be done for iRb, iXb and ib (Fig. 4(b)) with the only difference that their directions are opposite.

Fig. 4. (a) Phasors of the currents ia, iXa and iRa; (b) phasors of the currents ib, iXb and iRb; (c) phasors of all currents in the stopband; (d) phasors of all currents in the passband; (e) the phasors at frequency ω = ωpa = ωsb.

The phasor iXa passes the circle one and half times. Its end A starts from point A0 at ω = 0 (then Xa → ∞ and iXa = 0); moves along the upper semicircle when the frequency runs from 0 to ωsa (point A coincides with Am at ω = ωsa); continues along the lower semicircle when ωsa < ω < ωpa (A coincides again with A0 at ω = ωpa); and passes once more the upper semicircle when ω is above ωpa. The motion of the phasor iXb is a mirror-image to that of iXa. Both sets of phasors (“a” and “b”) are united in Fig. 4(c) and (d) with inverting the directions of the currents ib, iXb and iRb. This approach makes it possible to easily demonstrate some properties of the input current io of the second amplifier, which gives in fact the filter output signal. The current io is the sum of iXa and iXb. Since the direction of iXb is inverted in Fig. 4 (c) and (d), io is represented there by the line, connecting the ends A and B of iXa and iXb. Fig. 4(c) shows the line AB for frequencies in the stopband. Then points A and B are in the upper semicircle (Xa and Xb are capacitive) and the distance between them is small.

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The reactances Xa and Xb have opposite character in the passband, points A and B are at opposite sides of the diameter and the distance between them is larger (Fig. 4(d)). The maximum filter gain and the conditions for it can be determined from Fig. 4(d). The line AB cannot exceed the circle diameter, thus the maximum of |io| is equal to |ia| (or |ib|). If a transresistance second amplifier is assumed and taking into account (8), then the maximum voltage gain is gm1rm2. The maximum gain appears once at frequency ω = ωpa = ωsb. Then, due to parallel resonance in Xa and series resonance in Xb, iXa = 0 and io = iXb = ib (Fig. 4(e)). This maximum appears always if condition (1) is satisfied. A condition for existence of a second maximum can be derived by using the angles φa and φb shown in Fig. 4. It follows from Fig. 4(a) and (b): ;

.

When points A and B are ends of a diameter of the circle, then condition

, i.e.

(10) , which gives the (11)

. The expression for XaXb is .

(

)

(12)

It is positive in the region ωsa < ω < ωpb, where it grows monotonically from 0 to infinity. Consequently, the choice of the resistors Ra and Rb (or of the terminating resistors in the basic lattice filter) defines a second frequency ω0 of maximum gain. The filter has a transmission zero when points A and B coincide. When the frequency increases from 0 to ωsa these points move monotonically along the upper semicircle from point O towards point P and point A first reaches point P when ω = ωsa. If point B is before point A at very low frequencies, then A will reach B at a certain frequency, which is the frequency of the transmission zero. This position of points, B before A, means φb < φa, or from (10) (–Xb/Rb) < (–Xa/Ra). When the frequency is much lower, then the FBAR resonance frequencies, ⁄( ) and ⁄( ). Usually the ratio ωp/ωs (i.e. α) is the same for both resonators and then the condition for a transmission zero at frequencies below the resonance frequencies (lower stopband) is .

(13)

Point A begins first its second motion along the upper semicircle above the resonance frequencies. The existence of a transmission zero in this range requires point B to get ahead of A above certain frequency, which is equivalent to have again ⁄( ⁄( φb < φa at higher frequencies. When ω >> ωpb the simplifications for Xa and Xb are ) and ), which means that (13) is also a condition for existence of a transmission zero in the upper stopband. Condition (13) means that transmission zeros in the modified circuit in Fig. 3(b) can be created either by using resonators with different Cp, or by using different resistors Ra and Rb. In contrast, only the first option is available for creating transmission zeros in the basic lattice filter in Fig. 3(a). C. Frequency Response of the Lattice Filter The frequency response is the most important characteristic of the filter, defining its area of application, and here its dependence on a few factors will be studied. The influence of the resistor values is shown first in Fig. 5, obtained by computer simulations. The FBAR parameters, assumed in the simulations, are: ωsa = 0.97561 (1/1.025), ωpa = ωsb = 1, ωpb = 1.025, Cpa = Cpb = 1. Fig. 5(a) shows that the simultaneous change of Ra and Rb moves the passband area downwards or upwards and the motion of the passband center frequency reaches ±1% from the center frequency. The relative passband bandwidth is 2% approximately (Fig. 5(c)), i.e. the center of the passband can be moved by changing Ra and Rb to a distance, approximately equal to one half of the passband bandwidth. The passband bandwidth also changes when Ra and Rb vary. This is illustrated in Fig. 5(b). The values of Ra and Rb are the same as in Fig. 5(a), but the FBAR resonance frequencies are changed in order to achieve identical passband centers of all responses. Numerical data for the relative passband bandwidth are given in Fig. 5(c). Fig. 5(b) shows that the changes of Ra and Rb change also the stopband characteristic of the filter, however small this change is.

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Fig. 5 Frequency responses of the modified lattice filter: (a) frequency responses when Ra and Rb are equal and move simultaneously from 0.5 to 2; (b) the same characteristics, but all centered with respect to unity; (c) relative passband bandwidths, determined at -0.3 dB level, for the filters in (a) and (b); (d) frequency responses at different relative difference δτ between τa and τb (τaτb = 1)

The appearance of transmission zeros, when Ra ≠ Rb, is seen in Fig. 5(d). The parameter δτ, given in this figure, is the | | ⁄√ relative difference between time-constants τa = CpaRa and τb = CpbRb and it is defined as . Fig. 5(d) shows that δτ affects only the position of the zeros (and the stopband behavior) and does not change the filter behavior in the passband. When the capacitances Cpa and Cpb are equal as it is in Fig. 5, δτ is the relative difference between Ra and Rb. An option for passband extension is to violate the condition ωpa = ωsb by letting ωpa < ωsb. The filter frequency response in this case has always two maxima in the passband – one between ωsa and ωpa and a second between ωsb and ωpb. Fig. 6(a) shows the positions of the end points A1 and B1 at frequency ωsa: A1 coincides with the end P of the diameter ia and B1 is on the upper semicircle. Then, when the frequency runs from ωsa to ωpa, point A moves through the whole lower semicircle and its position A2 coincides with point O at ω = ωpa. Point B moves still on the upper semicircle to B2 and it does not reach point P at ω = ωpa, since ωpa < ωsb. Evidently, there is a frequency between ωsa and ωpa, at which the positions A' and B' of these two points form a diameter of the circle (Fig. 6(a)), i.e. the output current, represented by the line A'B', is maximal. Similar is the case when ωsb < ω < ωpb (Fig. 6(b)). Then point A moves on the upper semicircle between points A3 and A4, while point B travels the whole lower semicircle and at a certain frequency in this range their positions A" and B" form a diameter.

Fig. 6 (a) Positions of the end points A and B of the phasors iXa and iXb when ωpa < ωsb: A1 and B1 at ω = ωsa, A2 and B2 at ω = ωpa; (b) the same for this condition at the other resonance frequencies: A3 and B3 at ω = ωsb, A4 and B4 at ω = ωpb; (c) frequency responses of the filter when ωpa and ωsb differ by 0.5%

The distance between ωpa and ωsb is limited by the minimum of the frequency response between both maxima. Fig. 6(c) shows frequency responses of the filter when ωpa and ωsb differ by 0.5%. The relative passband bandwidth (determined at -0.3 dB level) is 2.61% when Ra = Rb = 1. It is increased by more than 40% if compared with the relative bandwidth when ωpa = ωsb and the resistors are the same (1.83% relative bandwidth according to Fig. 5(c)). On the other hand the minimum of the frequency response, existing between its maxima in the passband, limits the possibility for tuning the passband center frequency, since a large change of the resistors increases inadmissibly this minimum. Fig. 6(c) shows that √ -times change of the resistors gives an approximate variation of ±0.5% of the passband center frequency, which can be considered as a boundary for tuning, since the internal passband drop of the frequency response approaches 0.3dB (the conditional limit for passband ripples, accepted here).

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All simulations above are for normalized values of the frequency and of the resistor values. An example to feel better the relationships between the real values: if fpa = fsb = 1.9GHz (this is the filter center frequency), then fsa = 1.9/1.025 = 1.8536GHz and fsb = 1.9×1.025 = 1.9475GHz. The assumption Ra = Rb = 50Ω gives Cpa = Cpb = 1.63pF (normalizing resistance of 50 Ω). III. PROPERTIES OF PARALLEL OR SERIES CONNECTED RESONATORS Before considering circuits with parallel or series connected FBARs, some properties of this connection will be derived. Several parallel connected resonators can be considered as an LC circuit in Foster 2 canonic form [16], if lossless resonators are assumed, represented by their mBVD models, with different resonance frequencies of the particular resonators. This is illustrated in Fig. 7(a), where the capacitors Cp1, Cp2, etc., can be united in a single capacitor. The conclusions for the impedance of this combination are:

Fig. 7 (a) FBARs connected in parallel and their equivalent circuit; (b) frequency response of the impedance of the combined circuit



The series resonance frequencies are the same as those of the particular resonators. They are defined by the pairs Lm1Cm1, Lm2Cm2, etc., in Fig. 7(a), which do not change after the connection of the resonators.



The series and parallel resonance frequencies alternate on the frequency axis. The impedance behaves like an inductor between a series and the next parallel resonance frequency, and like a capacitor between a parallel and the next series resonance (Fig. 7(b)). This is a general property of the lossless LC circuits.



The impedance of the combination has a pole at the origin and a zero at infinity. The pole in the origin is defined by all capacitors, while the zero at infinity is due to parallel capacitances Cp1, Cp2, etc., in the mBVD models. This property, together with the previous, defines the first resonance of the combination to be a series one, if one starts from zero and moves upwards along the frequency axis. Both properties define also the last resonance: it is a parallel one. Thus the number of series and parallel resonances is the same and it is equal to the number of the resonators.



The parallel resonance frequencies move closer to the corresponding series resonances than they are in the individual resonators.

The last property is not as obvious as the previous properties and will be proved. Let us assume that a new resonator with corresponding resonances at ωs' and ωp' is added in parallel to an existing combination and ωs' lies between a parallel ωpk and the next series ωs(k+1) resonance frequencies of that combination (Fig. 8(a)). The frequency ωs' creates new series resonance of the new combination, since it remains the same after the connection. The frequency ωp' could be initially below or above ωs(k+1). In the first case, the interval (ωs', ωp') lies entirely in the interval (ωpk, ωs(k+1)), where the initial combination behaves like a capacitor. This capacitor increases the equivalent Cp of the added resonator, reducing according (2) the distance between its resonance frequencies. Two consecutive series resonances ωs' and ωs(k+1) of the whole combination appear in the second case, when ωp' > ωs(k+1). A parallel resonance must exist between them, thus ωp' moves below ωs(k+1). On the other hand, the added resonator moves all parallel resonance frequencies of the initial combination closer to their series counterparts, since this resonator is equivalent to a capacitor outside of the region (ωs, ωp) and increases the total parallel capacitance of the new combination. The case when ωsk < ωs< ωpk can be considered in a similar way. The frequency ωpk moves between ωsk and ωs to break the sequence of two series resonances and ωp moves closer to ωs due to the extra capacitance added from the initial set of resonators.

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Fig. 8 (a) Positions of the resonance frequencies of the existing set of parallel connected resonators and of the new resonator (b) Dependence of the new parallel resonances on the ratio of the series resonances. The continuous lines give the estimation according to (17)

The case of two parallel resonators is most important and it will be considered in more details, assuming identical coupling factors of both resonators, i.e. identical ratios α. The admittances of the resonators, if the losses are neglected, are: (

)

(

(

)

)

(

)

(14)

.

The condition Y1 + Y2 = 0 determines the parallel resonance frequencies of the parallel connected resonators. It gives: (

)

)(

(

)(

)

This equation is quite general and difficult for investigation. The introduction of the ratios simplifies it: (

)

.

(15)

. ⁄

and

⁄

(16)

The Viète formulas for the polynomial roots give the following connection between its solutions p12 and p22 and the series resonance frequencies: , which is equivalent to . An approximate estimation for the new parallel resonance frequencies follows from these relationships √

√

.

(17)

It gives an idea of the decreasing of the distances between series resonances and the corresponding parallel resonances. Its accuracy is illustrated in Fig. 8(b). The general conclusion from the figure is that formulas (17) give approximately the upper boundary for ωp1 and the lower boundary for ωp2 in most cases.

Fig. 9 (a) Foster 2 (mBVD model) and Foster 1 representation of FBAR; (b) Series connection of FBARs and their equivalent circuit based on Foster 1 form of LC impedances

The series connection of resonators can be analyzed in a similar way. It is more convenient to modify the mBVD FBAR model, using the equivalence between Foster 1 and Foster 2 forms of LC impedances [16]. Fig. 9(a) shows how Foster 2 form (the mBVD model) can be transformed in Foster 1 form. This approach makes it possible to easily demonstrate that the series connection keeps unchanged the parallel resonances of the resonators (Fig. 9(b)). Similar considerations as above prove that the series resonances move closer to their corresponding parallel resonances. It is interesting that the equation for the series resonance frequencies in the case of two series connected FBARs is the same as the equation for parallel resonance frequencies when the resonators are in parallel – equation (15). In fact, equation (15) gives two solutions 1 between s1 and p1 and 2 between s2 and p2. When the resonators are in parallel, 1 and 2 are the new parallel resonance frequencies; when the resonators are in series they are the new series resonance frequencies. Thus the series and parallel connections of the resonators have the same properties concerning the changes of the distances between their resonance frequencies. However, the circuits in the next sections use parallel connection of the resonators. The capacitances Cp, which are proportional to their geometrical sizes, are added when they are connected in parallel, i.e. smaller resonators are necessary for achieving desired total capacitance Cp of the combination. It is opposite in the case of series connected resonators: the total parallel capacitance is less than the parallel capacitances in the individual resonators.

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IV. INVESTIGATION OF LATTICE FILTERS WITH COMBINED RESONATORS IN THE ARMS A. “2+1” Lattice FBAR Filter The effect of using a combination of two parallel resonators instead of a single resonator will be investigated first for the case, when only one of the arm pairs is replaced by such combination. The new circuit of the basic lattice filter is shown in Fig. 10(a) and of the modified lattice filter – in Fig. 10(b). The designation “2+1” lattice filter will be used for this circuit.

Fig. 10 Lattice filters with two parallel resonators in one of the arms (“2+1” lattice filter): (a) basic; (b) modified

The impedance of the parallel connected resonators Xa1 and Xa2 has two series (ωs1a and ωs2a) and two parallel (ωp1a and ωp2a) resonance frequencies, as considered in the previous section. They are arranged as shown in Fig. 11(a). The requirement for opposite character of the impedances in the passband area sets the following relationships between resonance frequencies of the combined resonator in arm “a” and resonance frequencies ωsb and ωpb of Xb: (18) The resonance frequencies of the resonators define the passband and they could be determined in the following way. First, the frequencies ωsb and ωpb are chosen (of course, their ratio must correspond to the effective coupling factor of the device). The frequency ωpb determines also ωs2a, which is the series resonance of Xa1 or Xa2. The other series resonance of Xa1||Xa2 is determined by using the condition ωp1a = ωsb. Equation (15) gives a solution for ωp1a and ωp2a if ωs1a and ωs2a are known. Thus the condition ωp1a = ωsb is in fact a nonlinear equation with a single unknown ωs1a, which can be solved iteratively. The parallel resonance frequencies of Xa1 and Xa2 are defined by their effective coupling factors.

Fig. 11 a) Location of the resonance frequencies of the impedances in the arms of the circuits in Fig.10 (L means inductive character of the corresponding impedance, C means capacitive character) (b) The product XaXb vs. frequency (α = 1.025; keff2 = 5.87%; the normalizing frequency is the geometric mean of ω2 and ω3; the normalization resistance is chosen so that Cpa = Cpb = 1F). (c) Frequency response in the passband at different values of the frequency ω0. They are achieved at the following normalized values of Ra and Rb: ω0 = 0.985 at Ra = Rb = 0.344; ω0 = 1 at Ra = Rb = 0.8245; and ω0 = 1.02 at Ra = Rb = 2.066

For the sake of convenience, the designations ω1, ω2, ω3 and ω4 are introduced in Fig. 11(a) for all resonance frequencies and they will be used hereinafter. Formulas (17) allow the estimation of the ratio ω4/ω1 as approximately equal to α2. It is in fact the ratio between the boundaries of the theoretical passband. The same is the corresponding ratio for a lattice filter with single resonators in the arms. Thus the extra resonators in the filters in Fig. 10 do not extend the theoretical passband. The goal here is to use it more effectively by extending the region where the variations of the frequency response are small. It can be achieved by proper positioning of the frequencies of maximum gain. Two of these frequencies are ω2 and ω3, where the resonators in arms “a” and “b” have opposite resonances. The third frequency of maximum gain is defined by condition (11). In this case, the reactances Xa and Xb are (

)( (

) )(

,

)

(

) (

,

)

(19)

where Cpa is the sum of the parallel capacitances of Xa1 and Xa2. The product XaXb has similar properties as in the case of single resonators in the arms. It increases monotonically from 0 to infinity when ω1 ≤ ω ≤ ω4 (Fig. 11(b)) and the frequency ω0 of the

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third maximum can be positioned everywhere between ω1 and ω4 by proper choice of Ra and Rb. An intuitive guess is to place this frequency between ω2 and ω3. The reason for this is that the interval (ω2, ω3) is the largest from the intervals (ω1, ω2), (ω2, ω3) and (ω3, ω4). It is defined by the resonance frequencies of Xb and takes approximately one half of the area of the theoretical passband. The large distance between ω2 and ω3 can cause large drop of the frequency response, which can be improved by placing ω0 there. This guess is confirmed in Fig. 11(c), where the cases when ω0 is between ω1 and ω2 (ω0 = 0.985); between ω2 and ω3 (ω0 = 1); and between ω3 and ω4 (ω0 = 1.02) are compared. Evidently only the case when ω0 is between ω2 and ω3 has well defined passband with small ripples. The simulations in Fig. 11(c) are done with assumptions for lossless resonators, gm1rm2 = 1 and α = 1.025. They are repeated for different values of α and the conclusion is the same: ω0 must be between ω2 and ω3. Its position can be found more precisely by varying Ra and Rb, aiming to equalize the minima of the frequency response at both sides of ω0. The stopband attenuation depends on how near are the ends A and B of the phasors iXa and iXb (Fig. 4(c)). The ratio r between the tangents of the angles φa and φb can be used as a criterion for nearness of A and B. The expression for r follows from (10) and (19) ⁄

(

)(

(

) (

) )

⁄

( ),

(20)

where a = CpaRa, b = CpbRb and F(ω) is the frequency dependent part of r. This ratio must be closer to 1 for high filter attenuation and there is a transmission zero when it is exactly equal to 1. The function F(ω) is plotted in Fig. 12(a). It is positive in the stopband and negative in the passband. It is close but higher than 1 in the major part of the lower stopband and has a flat maximum, approximately equal to 1.0235 (practically independent of α) and near to the passband. Then it goes sharply downwards when the frequency approaches the lower passband boundary. The behavior of F(ω) in the upper stopband is different. It has values much larger than 1 for frequencies close to the upper passband and decreases smoothly tending to unity when the frequency increases. Fig. 12(a) shows also that F(ω) depends slightly on the parameter α. Thus the effective way to govern the stopband frequency response is the choice of the ratio τa/τb. Four typical cases can be differentiated about this choice: 

τa/τb is significantly larger than 1.0235. A single transmission zero appears in the upper stopband, which is as close to the passband, as τa/τb is larger. The stopband attenuation is small, since r differs significantly from 1 when the frequency is not close to the transmission zero.



τa/τb is between 1.0235 and 1. The filter has three transmission zeros: two in the lower stopband, close to the passband boundary, and one in the upper stopband. The attenuation is good, especially in the lower stopband, where r is very close to unity.



τa/τb = 1. The filter has a single transmission zero in the lower stopband and very good attenuation there, since r is close and tends to 1 when the frequency decreases. The upper stopband also has good monotonically increasing attenuation.



τa/τb is significantly less than 1. There is a single zero in the lower stopband very close to passband boundary. The parameter r differs significantly from 1 in the major part of the stopband and the attenuation is not large there.

Fig. 12 (a) The function F() vs. frequency; (b) and (c) comparison of stopband frequency response of the filter at different values of c = a/b. The normalizing frequency is the center of the passband

These cases are illustrated by the frequency responses in Fig. 12(b) and (c). Fig. 12(b) demonstrates the superiority of the case τa/τb = 1 (τa/τb is marked by c in the plots) over the cases when τa/τb is much more or much less than 1. Frequency responses when τa/τb is around the interval (1, 1.0235) are compared in Fig. 12(c). Evidently, the cases, when τa/τb is within these boundaries, are the best. However the other plots in Fig. 12(c) show that τa/τb can be outside but close to this interval and the attenuation is still good. This observation allows relaxing the stringent requirement for precise tuning to achieve good attenuation. The tuning of the attenuation can be done by proper design of the resonators having prescribed values of the

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capacitors or by varying Ra and Rb independent from each other. The second option is available only for the modified circuit in Fig. 10(b). B. “2+2” Lattice FBAR Filter The next case considered here is the “2 + 2” lattice filter when both arms of the filter consist of pairs of parallel connected resonators (Fig. 13(a)). The condition for displacement of the areas, where combined resonators have opposite character, requires placing the resonance frequencies in the arms as shown in Fig. 13(b). The parallel resonance frequencies of the combined resonators in both arms are functions of the positions of their series resonances. The matching of the resonance frequencies according to the requirement in Fig. 13(b) is more complicated in this case and can be done numerically by the following procedure: First the initial values for ωs1a and ωs2a are chosen and then ωp1a and ωp2a are calculated using equation (16). The second step is to choose ωs1b = ωp1a and ωs2b = ωp2a and to calculate ωp1b and ωp2b again by means of (16). The obtained value for ωp1b should be equal to ωs2a. If they are different, then ωs2a accepts the value of ωp1b and the process is repeated from the beginning. This procedure is not investigated theoretically concerning convergence and uniqueness of the solution. However numerical experiments, done for the practically interesting cases, show that it converges fast on a same solution, which is independent of the initial values. The resonance frequencies, obtained by the proposed procedure, do not extend the region, in which the combined resonators in the arms have an opposite character. For example, if all resonators have identical α = 1.25 and equal parallel capacitances in the mBVD model, then the normalized resonance frequencies, as they are marked in Fig 13(b), are: ω1 = 0.9756 = 1/1.025; ω2 = 0.9802; ω3 = 0.9877; ω4 = 1.007; ω5 = 1.025, i.e. the ratio ω5/ω1 is equal to α2.

Fig. 13 (a) Filter with pairs of resonators connected in parallel in each arm (“2 + 2”); (b) the resonance frequencies of the combined resonators in the arms; (c) passband frequency response at different values of the time-constants τa = CaRa and τb=CbRb; (d) stopband frequency response of this filter when τa = τb (δτ = 0) and when they differ by 20%

The filter frequency response has 3 frequencies of maximum gain, defined by the resonance frequencies of the combined resonators in the arms: ω2, ω3 and ω4. The fourth frequency of the maximum gain is the frequency ω0, at which condition (11) is satisfied. It can be placed everywhere between ω1 and ω5, but it would be better if it is placed between ω3 and ω4. The position of ω0 is governed by the time-constants τa = CaRa and τb=CbRb, where Ca and Cb are the total parallel capacitances of the combined resonators in arms “a” and “b”. Fig. 13(c) shows that the proper choice of ω0 ensures identical and small ripples between ω3 and ω4. There is another sharp peak at ω2, followed by a large drop between ω2 and ω3. This peak exists only theoretically since the FBARs losses remove it. For this reason the passband is between ω3 and ω4. Similarly as in the “1+1” filter, the condition τa = τb gives smooth frequency response in the stopband, while a pair of zeros appear when these values differ (Fig. 13(d)). V. COMPARISON OF THE FREQUENCY RESPONSES OF THE CONSIDERED LATTICE FBAR FILTERS The properties of the considered different lattice FBAR filters can be clarified better if plot together and compare their frequency responses. This is done in Figs. 14-16, where the frequency responses are normalized with respect to passband center frequency. All simulations are done twice: first, assuming lossless resonators and then taking the losses into account. The simulation without losses gives the opportunity to demonstrate the “ideal” frequency response especially in the passband. The FBAR losses are introduced by substituting Qs = 1500 and Qp = 1800 in the expression (4) for the FBAR impedance, which are moderate values for the FBAR Q-factors. Fig. 14 compares the optimal “2+1” lattice filter, when c = 1, with lattice filters with single resonators in the arms (“1+1 filters”). The other comparison in Fig. 15 shows “2+2” filters together with a “2+1” filter and a “1+1” filter without a transmission zero. The resistors Ra and Rb in the filters with single resonators are equal to 1 (i.e. their value is chosen as normalizing impedance), which makes identical both frequencies of maximum gain (ω0 and ωpa = ωsb) and then the passband is relatively narrow. The “1+1” filter with extended passband in Fig. 14 is the filter from Fig. 6, when ωpa and ωsb differ by 0.5%. The values of Ra and Rb for “2+1” and “2+2” filters are chosen in order to ensure approximately equal passband ripples: 0.78 for “2+1” and 0.83 for “2+2”. The normalizing impedances in these cases are defined by the capacitances Cp in the mBVD model: Their values in the single resonators and the total values in the parallel connected resonators must be equal to 1.

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Fig. 14 Comparison of frequency responses of a “2+1” lattice filter and some “1+1” lattice filters: (a) when the FBAR losses are neglected; (b) when the losses in the filters are taken into account

A brief comparison of the frequency responses of lattice and ladder filters is shown in Fig. 16. Filters with 5 and 7 resonators and П-type structure are chosen as representatives of the ladder filters. Their frequency responses are optimized in order to have relatively wide passband and good stopband attenuation. This is achieved by choosing resonators with different parallel capacitances in the mBVD model, which corresponds to resonators with different geometrical sizes (of course, the condition for equality of ωs of the series resonators and ωp of the shunt resonators is satisfied). The normalized parallel capacitances (normalization with respect to passband central frequency and terminating resistors) of the resonators in the filter with 5 resonators are (from left to right): 1; 0.3987; 1.993; 0.3987; 1.258; and for the filter with 7 resonators: 1.584; 0.3168; 3.157; 0.3168; 3.157; 0.3168; 1.993.

Fig. 15 Comparison of frequency responses of “2+2”, “2+1” and “1+1” lattice filters: (a) when the FBAR losses are neglected; (b) when the losses in the filters are taken into account

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Fig. 16 Comparison of frequency responses of lattice and ladder filters: (a) when the FBAR losses are neglected; (b) when the losses in the filters are taken into account

Table I presents some numerical data, obtained from the plots and describing the frequency responses. The passband bandwidth is the frequency band, in which the attenuation increases by not more than 0.3 dB relative to the minimum passband attenuation. This bandwidth is given in percents concerning passband central frequency. The attenuation at ±5% and ±10% relative distance from the passband central frequency is given in Table I, characterizing the stopband attenuation of the filters. TABLE I PARAMETERS OF THE FREQUENCY RESPONSES OF THE SIMULATED FILTERS

Filter “1+1” (Ra=Rb=1) “1+1” with transmission zeros (Ra=0.905, Rb=1.105, δR=0.2) “1+1” with extended passband (Ra=Rb=1) “2+1” (Ra = Rb = 0.78) “2+2”without transmission zeros (Ra=Rb=0.83) “2+2” with transmission zeros (Ra=0.751, Rb=0.917, δR=0.2) Ladder filter with 5 resonators Ladder filter with 7 resonators

Relative passband bandwidth determined at -0.3 dB level without losses with losses 1.84% 1.76% 1.9% 1.82% 2.61% 2.58% 2.97% 2.79% 2.52% 1.98% 2.53% 1.89% 2.05% 1.35% 2.1% 1.13%

Attenuation at some relative distance from passband central frequency* ±5% distance ±10% distance 18.7 / 17.6 dB 31.1 / 29.3 dB 30.9 / 27.4 dB 23 / 23.8 dB 16.9 / 15.8 dB 29.5 / 27.6 dB 38.8 / 22.1 dB 44.6 / 36 dB 26 / 22.3 dB 38 / 34.6 dB 26.9 / 48.5 dB 21.2 / 22.4 dB 35.3 / 32.2 dB 30.3 / 29.4 dB 71.4 / 66.6 dB 63.8 / 62.4 dB

* The first number in the columns with attenuations is the attenuation at -5% or -10% relative distance from the passband central frequency and the second number is the attenuation at +5% or +10% from the central frequency.

Several conclusions can be drawn from the plots in Figs. 14-16 and from the data in Table I, as well as from the previous considerations: 1) The extra resonators added in the arms of the lattice filters allow to extend significantly the passband bandwidth and the stopband attenuation, compared with the filters with single resonators in the arms. However the increased number of resonators requires more accurate adjustment of the filter elements. 2) The losses in the FBARs have effect basically in the passband, introducing small attenuation there, worsening the passband flatness and narrowing the passband bandwidth. However the deterioration of the frequency response of the lattice filters is not significant for the “1+1” and “2+1” lattice filters, due to the small number of resonators, and it is more visible only for the “2+2” lattice filters. This conclusion is valid mostly for modified lattice filters (Fig. 3(b) and Fig. 10(b)). For the basic lattice filters (Fig. 3(a) and Fig. 10(a)), when the resonators are twice more, the losses should have larger effect. 3) An interesting solution is the “2+1” lattice filter. It has the widest passband, compared with the other considered options, and good stopband attenuation, especially in the lower stopband. 4) The comparison of lattice vs. ladder FBAR filters in Fig. 16 and Table I is not exhaustive, however it makes it possible to demonstrate some advantages and disadvantages. The ladder filters have transmission zeros at the series resonance frequencies of the shunt resonators and at the parallel resonances of the series resonators, which ensure very sharp cutoffs near to the passband. The attenuation returns to relatively small values outside these zeros, and more resonators are necessary for increasing the attenuation there. For this reason filters with 5 or 7 resonators are chosen here as representatives for ladder filters. The large number of resonators induces larger sensitivity to the FBAR losses (for example the filter with 7 resonators has twice narrower passband if losses are taken into account). The stopband attenuation of the lattice filters depends on their design: It may increase smoothly or the filter may have transmission at prescribed position. The fewer number of resonators makes these filters less sensitive with respect to losses. Lattice filters have typically wider passband compared with ladder

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filters. The general conclusion from the comparison is that both filter classes may have advantages in different applications depending on the requirements and thorough investigations are necessary in every specific case. 5) Some remarks concerning the design of the filters. It has been already commented how to position the FBAR resonance frequencies: keeping ωpa = ωsb or placing small difference between these frequencies in the case of “1+1” filter; by calculation of their values using special iterative procedures in the cases of “2+1” and “2+2” filters. The next task in all filters is to position the frequency ω0, at which equation (11) is satisfied. It is the frequency of the extra maximum of the frequency response and defines the exact passband position and maximum ripples. The right place of this frequency depends on the coupling factor of the used resonators and can be found by few simulation of the filter frequency response, varying the product τaτb of the time constants. The value of the product τaτb is also a result from these simulations. The stopband attenuation depends basically on the ratio of the time constants c = τa/τb or their relative difference . They ⁄√ . The ratio c defines the existence and the positions of the transmission zeros. The are related by the formula √ zeros, if they exist, are two and approximately symmetrical around the passband in the cases of “1+1” and “2+2” filters and their positions can be chosen again by simulation of the frequency response. There is no choice practically for c for the “2+1” filter – it should be between 1 and 1.023. The time constants τa and τb can be calculated easily after determining their product and ratio – they are √ ⁄ , where  is the product τaτb. √ and All simulations can be done using the formulas for the filter gain and for the impedances of the resonators (including combined resonators). The normalizing frequency could be determined after finalizing the adjustment of the passband – it must fix the normalized passband center frequency to its real desired value. The normalizing impedance can be determined after design of the resonators and matching their normalized and real parallel capacitances Cp. VI. CONCLUSION A detailed consideration of lattice FBAR filters is done in the paper. It begins with presentation of the basic structure of these filters and an option for its modification containing twice lower number of resonators. The theoretical explanation of both versions of the lattice filters, presented in the paper, allows elucidation of their basic properties: transfer function and frequency response; limitation of the passband bandwidth; the opportunities for adjustment of the passband by varying the terminating resistors in the basic circuit or internal resistors Ra and Rb in the modified circuit; creating transmission zeros at a desired position. The last two properties can be realized easier in the modified circuit, since resistors Ra and Rb are separated from the terminals. The fewer number of resonators makes the modified circuit less sensitive with respect to FBAR losses as it is discussed in the last section of the paper. A significant part of the paper is dedicated to improving the frequency response by setting two goals: extending the passband bandwidth and increasing the stopband attenuation. It is shown that a small violation of the requirement for equality of the series resonance frequency of one of the resonators and the parallel resonance of the other FBAR widens the passband, however the stopband attenuation slightly decreases. Better results are achieved with the other considered approach: using parallel or series combination of two resonators in one or in both arms of the filter. This approach is more appropriate for the modified circuit, since the number of the resonators in this case is still not high. The larger number of resonators does not increase the theoretical frequency band, where the passband is placed – the area, where the resonators in the arms have reactances with opposite signs. However this method allows extending the area, where the frequency response is approximately flat with small ripples, which determines the real passband. Probably the filter, designated as “2+1”, is most interesting. Its benefits, achieved at certain conditions are: widest passband, very good stopband attenuation, and relatively low sensitivity to FBAR losses. The comparison of the frequency responses of lattice and ladder FBAR filters shows that the ladder ones need more resonators (> 5 as it can be seen from the considered examples) for achieving good attenuation in wider area outside of the passband and their major advantage is the very sharp cutoff close to the passband. In contrast, the modified lattice filter has fewer resonators, wider passband and allows to place their transmission zeros at desired position. In general, the differences in the frequency responses define different area of application of both classes of FBAR filters. ACKNOWLEDGEMENTS

This works is sponsored by the National Science Fund of the Bulgarian Ministry of Education, Youth and Science; Contract DDVU 02/6 from 17.12.2010. REFERENCES

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