Microelectronics Reliability 42 (2002) 1133–1140 www.elsevier.com/locate/microrel
Analysis and design of thin film resonator ladder filters q A.T. Kollias, J.N. Avaritsiotis
*
Department of Electrical and Computer Engineering, Division of Computer Science, National Technical University of Athens, 9 Iroon Polytechniou Str., Zographou, Athens 157 73, Greece Received 22 December 2001; received in revised form 12 February 2002
Abstract This work presents a new simulation method for the analysis and design of bandpass filters based on thin film resonators (TFRs). The method is based on linear two-port network theory and can be applied for of ladder filters in general. Analytical equations for the case of TFR ladder filters are extracted by the proper modeling of the TFR. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction The next generation of radio subsystems requires miniature bandpass filters with high performance, which can be integrated n the same chip with other radio subsystems. At the present time the majority of radio communication tranceivers take advantage of the high quality factor of SAW filters available to reach adequate frequency selection in their RF and IF filtering stages. The aforesaid electro-acoustical filters are discrete components and consequently they do not allow further miniaturization. Due to their large dimensions SAW filters cannot be integrated in the same chip with other radio subsystems. A solution to the problem of miniaturization is the fabrication of miniature bandpass filters based on thin film resonators (TFRs) that can be included on the same chip with other subsystems. These integrated electro-acoustical filters present comparable performance with SAW filters for example high quality factors. The ladder configuration is a common realization of bandpass filters based on TFRs [1]. This work establishes new procedures for the analysis of ladder filters based on TFRs. Using linear two port network
q
An earlier version of this paper was published in Proceedings of the 15th Annual European Passive Components Conference (CARTS-Europe 2001), Copenhagen, 15–19 October 2001, pp. 237–242. * Corresponding author. Tel.: +30-1-7722-547; fax: +30-17722-548. E-mail address:
[email protected] (J.N. Avaritsiotis).
theory, analytical expressions are presented concerning the filter’s insertion losses. Finally the critical parameters, which influence the filter’s performance, are presented.
2. Two-port network theory application The network shown in Fig. 1 is the building block of a ladder filter. It consists of two TFRs X1, X2, called the shunt and series resonators, with Z1 , Z2 impedances respectively. Resonance and anti-resonance frequencies of X1, X2 must be equal and that is performed by properly adjust the thickness of piezoelectric films. Cascading a number of similar building blocks results to the formation of a ladder filter. Starting from the chain matrix T of the building block " # 1 þ ZZ21 Z2 T ¼ ð1Þ 1 1 Z1 it is easy to obtain the chain matrix of a k-element ladder network "
#" # k1 k1 1 þ ZZ21 Z2 t12 t11 T ¼ k ¼ k1 k1 k 1 t21 t22 t21 t22 1 Z1 3 2 k1 k1 k1 k1 þ Z11 t12 Z2 t11 þ t12 1 þ ZZ21 t11 7 6 Tk ¼ 4 5 Z2 k1 1 k1 k1 k1 1 þ Z1 t21 þ Z1 t22 Z2 t21 þ t22 k
k t11
0026-2714/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 6 - 2 7 1 4 ( 0 2 ) 0 0 0 5 6 - 2
k t12
#
"
ð2Þ
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and Z2 ¼ q exp ð j/ab Þ Z1
ð6Þ
Then the IL are given by 2 IL ¼ n P 2 P 2 o1=2 1þ 1 þ 2 X 1
¼
k X
qi1 ðP1 þ P2 Þ
ð7Þ
i¼1
X 2
¼
k X
qi1 ðR1 þ R2 Þ
i¼1
P1 ¼ lk1 cos ðði 1Þ/ab Þ þ lkiþ1 cos ði/ab Þ i
Fig. 1. The building block of a ladder filter.
Finally it is easy to show by induction that, take the form k t11 ¼ lk1 þ
k X
lkiþ1
i¼1 k ¼ t12
k X
kki
i¼1
Z2i Z1i1
Z2i Z1i
¼ Z2
¼ lk1 þ k X i¼1
kki
Z2 Z1
X k
Z2i1 Z1i1
lkiþ1
i¼1
i1 X i1 k k X Z2 1 Z2 ¼ ¼ kki kki i Z Z Z1i1 1 1 i¼1 i¼1 i1 k X Z2 k ¼ lk1 t22 i Z1i1 i¼1
k k k k t11 ; t21 ; t12 ; t22
Z2i1 Z1i1
P2 ¼
Zload k qa ki cos ðði 1Þ/ab /b Þ þ kk qb Zload i
cos ðði 1Þ/ab þ /a Þ
R1 ¼ lk1 sin ðði 1Þ/ab Þ þ lkiþ1 sin ði/ab Þ i
ð3Þ
k t21
R2 ¼
Zload k qa ki sin ðði 1Þ/ab /b Þ þ kk qb Zload i
sin ðði 1Þ/ab þ /a Þ Now it is apparent that the performance of the filter heavily depends upon the magnitude impedance ratio q and phase difference /ab .
where lki , kki are number sequences computed from the following recurrence relations: ; lki ¼ lk1 i
4. Analysis of the impedance ratio
i¼1
k1 lki ¼ kk1 þ lk1 i1 ; i1 þ li
; lki ¼ lk1 i
þ lk1 ; kki ¼ kk1 i i ; kki ¼ lk1 i
i ¼ 2; . . . ; k
i¼kþ1
4.1. The phase difference ð4Þ
i ¼ 1; . . . ; k 1
i¼k
3. Filter transfer function As soon as the chain matrix of the k-element ladder filter has been evaluated analytically, it is straightforward to obtain the filter’s insertion losses (IL) over the frequency spectrum S21 ¼
k k þ t22 þ t11
2 k t12 Zload
k þ ðZload t12 Þ
ð5Þ
IL ¼ 20 log jS21 j Eq. (5) takes a more comprehensive form by substituting Eq. (2) and assuming that Z2 ¼ qa exp ð j/a Þ;
Z1 ¼ qb exp ð j/b Þ
Phase difference /ab depends only on resonance, antiresonance frequencies and quality factors of TFRs. This can be shown using the Butterworth–Van Dyke (BVD) model [2] after considerable algebric manipulations. The variation of /ab over the frequency spectrum defines the filter’s IL both in rejection band and passband. The /ab can vary between approximately 180° and 180° for considerable high values of Q. Moderate values of Q lower the extreme values of /ab as shown in Fig. 2. In the vicinity of central frequency /ab is approximately a straight line which crosses zero at central frequency as shown in Fig. 3 assuming that matching condition fp1 ¼ fs2
ð8Þ
is satisfied. It is reasonable to assume that /ab takes zero value near the central frequency. A frequency mismatch leads to the formation of an extended zero phase difference around central frequency as shown in Fig. 4.
A.T. Kollias, J.N. Avaritsiotis / Microelectronics Reliability 42 (2002) 1133–1140
Fig. 2. Phase difference /ab for several values of quality factor Q.
Fig. 3. Phase difference /ab in the neighborhood of central frequency.
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Fig. 4. Phase difference /ab in the case of resonance, anti-resonance frequency mismatch.
5. The magnitude ratio The relative position of the resonance, anti-resonance frequencies usually described by the effective electromechanical coupling factor p f s 2 fp 2
ð9Þ keff ¼ p fs tan 2 fp and it is characteristic of the piezoelectric material [3]. Typical values of electromechanical coupling factors are shown in Table 1, for common piezoelectric thin films [1]. The magnitude impedance ratio is shown in Fig. 5 for several values of Q. The position of the two maximum 2 values of the ratio depends on the keff value. A minimum is observed at the central frequency assuming again that matching condition (8) is satisfied. As easily observed in Fig. 5 the extremes of magnitude ratio depend on Q Table 1 Typical values of electromechanical coupling factors for common piezoelectric thin films Material
2 keff
ZnO AlN
0.06 0.075
values. If matching condition (8) is not satisfied the minimum is not observed at central frequency as shown in Fig. 6. 6. Insertion losses 6.1. Minimum insertion losses As the values of Q increase the IL of the filter are minimized. The above statement can be proved formally in the case of minimum IL as soon as matching condition is met.
2 IL ¼ 20 log P 1
where X 1
¼
k X i¼1
c
i1
Zload qa min k k1 k þ ki li þ liþ1 þ qb max Zload ð10Þ
c¼
qa min qb min
As Q value increases the value of c decreases (as shown in Fig. 6) and IL are minimized. From Eq. (10) is evident that the variation of minimum IL as a function of Q
A.T. Kollias, J.N. Avaritsiotis / Microelectronics Reliability 42 (2002) 1133–1140
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Fig. 5. Magnitude impedance ratio for several values of quality factor Q.
Fig. 6. Magnitude impedance ratio in the case of resonance, anti-resonance frequency mismatch.
(Fig. 7) is not linear but has the form shown in Fig. 8, which shows that if high performance filters is be achieved then TFRs with quality factors up to 3000 must be constructed [1].
6.2. Out of band rejection At rejection band phase difference is zero as shown in Fig. 2. Then IL are given from
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Fig. 7. Minimum insertion losses IL as a function of quality factor Q.
Fig. 8. Insertion losses IL at rejection band as a function of the number of building blocks.
IL ¼ 20 log n X 1
X 2
¼ ¼
k X i¼1 k X i¼1
2 ð1 þ
P
1Þ
2
þð
P
2Þ
2
þ lkiþ1 qi1 lk1 i qi1 kki
Zload q a qb Zload
Magnitude impedance ratio, at rejection band, is defined by motional capacitance of TFRs [4]. If the latter are chosen so
o1=2 ð11Þ
Zload q pffiffiffiffiffiffiffiffiffiffi ¼ a () qa qb ¼ Zload qb Zload then IL are given from
ð12Þ
A.T. Kollias, J.N. Avaritsiotis / Microelectronics Reliability 42 (2002) 1133–1140
Fig. 9. Narrow band frequency response of a three-element ladder filter when design condition
Fig. 10. Narrow band frequency response of a three-element ladder filter when design condition
pffiffiffiffiffiffiffiffiffiffi qa qb ¼ Zload is met.
pffiffiffiffiffiffiffiffiffiffi qa qb ¼ Zload is not met.
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IL ¼ 20 log X 1
¼
k X
2 P ð1 þ 1 Þ þ lkiþ1 qi1 lk1 i
ð13Þ
i¼1
In Fig. 8 is shown the IL at rejection band as a function of the number of building blocks. Equation is a filter design condition and defines the IL over the frequency spectrum and not strictly at rejection band.
termines the filter’s IL in passband. Furthermore there is a possibility to control the passband ripple by mismatching the resonance, anti-resonance frequencies with an additional increase in filter’s bandwidth. For a given number of building blocks, out of band rejection is determined by the choice of motional capacitances. The only limitation may be encountered refers to filter’s bandwidth. The latter determined only by effective electromechanical coupling factor which is constant for a specific piezoelectric material.
6.3. The effect of motional capacitances The narrow band frequency response of a ladder filter based on TFRs is shown in Fig. 9. The electrode areas of the TFR have been chosen so that the motional capacitances Ca , Cb meet the design condition for load impedance Zload ¼ 50 X. It is apparent that the response is symmetrical around the central frequency as shown in Fig. 9. However, for different values of impedance load where design condition are no more satisfied the response looses its symmetry as shown in Fig. 10. A major conclusion is that the motional capacitance of a TFR plays a dominant role in the design of high performance bandpass filters [3].
7. Conclusions The analysis presented in this work, highlights the critical parameters that affect filter’s characteristics. It is apparent that ladder filters offer some flexibility in overall filter design. The number of building blocks de-
Acknowledgements The authors acknowledge the financial support of Institute of Communication and Computer Systems (ICCS) Project Archimedes.
References [1] Lakin KM. Thin film resonators and filters. In: IEEE Ultrasonics Symposium, 1999. p. 895–906. [2] Naik RS et al. Electromechanical coupling constant extraction of thin-film piezoelectric materials using a bulk acoustic wave resonator. IEEE Trans Ultrason Ferroelectr Frequen Control 1998;45(1):257–63. [3] Lakin KM, Kline GR, McCarron KT. High-Q microwave acoustic resonators and filters. IEEE Trans Microwave Theory Techniques 1993;41(12):2139–46. [4] Lakin KM, Kline GR, McCarron KT. In: Proceedings of the 1992 IEEE International Ultrasonics Symposium, 1992. p. 471–6.